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A study of the relationships between road access, traffic safety and travel speed, and applications to… Li, Jian 1996

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A STUDY OF THE RELATIONSHIPS BETWEEN ROAD ACCESS, TRAFFIC SAFETY AND TRAVEL SPEED, AND APPLICATIONS TO ACCESS PLANNING AND SPEED ZONE DESIGN by JIAN LI B.Eng., B e i j i n g I n s t i t u t e of Economics, 1982 M.Eng., Asian I n s t i t u t e of Technology, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of C i v i l Engineering) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1996 © Jian L i , 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) ABSTRACT This thesis investigates the impacts of highway access on t r a f f i c safety and t r a f f i c operation e f f i c i e n c y , and develops a model f o r planning access provisions. A comprehensive data base was created f o r the analyses by taking advantage of an e x i s t i n g photographic log of selected s e c t i o n a l highway information f o r one Canadian province. The data base includes information on access types, access density, t r a f f i c volume, speed l i m i t , geometry, and accident records. A review of the e x i s t i n g t r a f f i c s i t u a t i o n and an examination of some established relationships of access and accidents indicated a need to update current knowledge of the impact of highway access. Accident models are developed with p a r t i c u l a r emphasis on access and i t s combined e f f e c t s with t r a f f i c volume and road geometric factors on highways. Furthermore, a conceptual hazard model i s developed which includes t r a f f i c c o n f l i c t s as additional information to extend the knowledge of access as a road hazard. The l i t e r a t u r e on access and t r a f f i c operation r e l a t i o n s h i p s i s sparse, p a r t i c u l a r l y on two-lane highways, perhaps because of the high cost of comprehensive data c o l l e c t i o n f o r t h i s complex problem. Consequently the present i n v e s t i g a t i o n i s r e s t r i c t e d to the use of a surrogate of average t r a v e l speed to define access and speed r e l a t i o n s h i p . In addition to the inve s t i g a t i o n of access impact, a framework for determining the optimal number of access points i s formulated to help to set up access planning c r i t e r i a . The optimization process i s the ii a p p l i c a t i o n of the defined relationships between access and t r a f f i c safety and operations i n highway planning and design. The process minimizes the t o t a l s o c i a l cost as an integer nonlinear programming problem, solved by piecewise approximation. F i n a l l y , the relevant issues of access and speed zone design are addressed, i n which p a r t i c u l a r attention i s given to deriving speed t r a n s i t i o n zone length. The derived accident models show that accesses are major contri b u t i n g factors of accidents on a l l types of highways. In some s i t u a t i o n s , the combined e f f e c t s of access and geometric factors undermine further the safety l e v e l . The analysis f o r two-lane highway indicates that non-linear relationships may exist between accesses and tr a v e l speed. The derived optimization framework for planning access provisions provides a quantitative base for functional c l a s s i f i c a t i o n of roads. In t r a f f i c control design, a procedure of speed zone design i s outlined with consideration of access provisions. iii TABLE OF CONTENTS PAGE ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ACKNOWLEDGMENTS 1 Introduction 1 1.1 The Background 1 1.2 Access and T r a f f i c Safety 5 1.2.1 Access and Safety at the Macroscopic Level 6 1.2.2 Access and Safety at the Microscopic Level 17 1.3 Access and T r a f f i c Operations 23 1.4 Optimal Access points 24 1.5 Scope of the Study 25 2 The Data 29 2.1 Highway Sections 29 2.2 Access Data 33 2.3 T r a f f i c Volume Data 34 2.4 Highway Safety Data 35 2.5 Accident Data 36 3 The E f f e c t of Access on Highway Safety 37 3.1 Examination of Established Relationship 37 3.2 Model Construction 39 3.3 Results and Discussion 55 3.4 A Conceptual Model of Road Hazards 74 3.4.1 Introduction 75 iv i i i v v i i ix x i i 3.4.2 Baysian Method with Information 78 from T r a f f i c C o n f l i c t s 4 The E f f e c t of Access on T r a f f i c Operations 87 4.1 Introduction 87 4.2 The Photolog Data 91 4.3 Data Analysis 98 4.4 Results and Discussion 110 4.5 Summary Conclusion 116 5. A Framework for Planning Optimal Number of Access Points 120 5.1 Model Construction 120 5.1.1 General Discussion 120 5.1.2 Decision Variables 122 5.1.3 Independent Variables 124 5.1.4 The Objective Function 124 5.1.5 The Constraints 126 5.2 Solving Nonlinear Integer Programming 128 5.2.1 General Discussion 128 5.2.2 Piecewise Integer Programming 129 5.3 Model Application 137 6. Access and Speed Zone Design 144 6.1 Speed T r a n s i t i o n Zone 145 6.2 Location of R-3 Sign 149 6.3 Outline of Tr a n s i t i o n Zone Design Procedure 154 7. Implications and Conclusion 157 7.1 Implication of the Research f o r C l a s s i f i c a t i o n 158 7.2 Implication of the Research for Level of Service 160 7.3 Summary Conclusions and Applications of Research 163 V 7.4 Future Research 169 REFERENCES 170 APPENDIX A 179 APPENDIX B 181 vi LIST OF TABLES Table 1.1 Accident Rates Correspondent to Dif f e r e n t Levels of 8 Access Control (multilane Divided Rural highways) Table 1.2 Accident Rates Correspondent to Dif f e r e n t Levels of 10 Access Control (Two-lane/Four-lane Rural Highways) Table 1.3 Accident Rates Correspondent to Di f f e r e n t Levels of 12 Access Density (Two-lane Rural Highways) Table 1.4 F a t a l i t i e s and Injuries at Dif f e r e n t Access c o n t r o l 14 Level (Two-rlane Rural and Urban Highways) Table 2.1 D i s t r i b u t i o n of Sample Size by AADT 33 (Two-lane Rural Highways) Table 3.1 Independent Variables Used i n Model Construction 39 Table 3.2 Accident Estimation Models 43 Table 3.3 Co r r e l a t i o n Matrix for ACDR0-5 51 (Two-lane Rural Highways) Table 3.4 Co r r e l a t i o n Matrix for ACDD 0- 5 51 (Two-lane Rural Highways) Table 3.5 Co r r e l a t i o n Matrix for SEVR 0- 5 52 (Two-lane Rural Highways) Table 3.6 Co r r e l a t i o n Matrix for SEVF 0- 5 52 (Two-lane Rural Highways) Table 3.7 Access Type Equivalency Factors f o r Two-lane Rural 74 Highways (80 km/h, average degree of Curvature = 5 degrees) Table 3.8 T r a f f i c C o n f l i c t Data 83 Table 3.9 Baysian Estimation Results (1) 84 Table 3.10 Baysian Estimation Results (2) 86 Table 4.1 Variable D e f i n i t i o n 98 Table 4.2 I n i t i a l Model for SPD 105 Table 4.3 Co r r e l a t i o n Matrix for C o e f f i c i e n t Estimates 106 Table 4.4 Modified Model for Transformed Variable SPD 2 106 vii Table 4.5 Estimated Speed Reduction Factors 115 Table 5.1 Optimal Number of Unsignalized Intersections on a 142 (Four-km Section of Two-lane Highway) Table 5.2 Actual Number of Unsignalized Intersections for Some 143 (Two-lane Highway Sections of Length about 4-km) table 6.1 Minimum Zone Length and Corresponding Travel "Time 148 Table 6.2 Minimum T r a n s i t i o n Zone Length 149 i Table 6.3 Distance Traveled between Two Speed Limits 153 Table 7.1 Level of Service C l a s s i f i c a t i o n 161 Table 7.2 The E f f e c t of Access Density on Level of Service 162 viii LIST OF FIGURES Figure 1.1 Accident Rate on 4-lane Divided Non-interstate 9 Highways by Number of At-grade Intersections per Mile and Number of Business per Mile (Two-lane Rural Highways) Figure 1.2 Accident Rate Versus T r a f f i c C o n f l i c t s 12 (Two-lane Rural Highways) Figure 1.3 Structure of the Study 28 Figure 2.1 Sample Size Representation 31 Figure 2.2 Sample of Section D e f i n i t i o n 32 Figure 3.1 Access Density vs. Accident Rate 38 Figure 3.2 Comparison of Predicted and Observed Values 46 Figure 3.3 Component Plot f o r the Dependent Variable, ACDR 49 Figure 3.4 Estimated Relationship between Accident Rate and 57 Unsignalized Intersection Density by Horizontal Curvature (Two-lane Rural Highways) Figure 3.5 Estimated Relationship between Accident Rate and Private 58 Access Density by Horizontal Curvature (Two-lane Rural Highways) Figure 3.6 Estimated Relationship between Accident Rate and Roadside 59 Pullout Density by Horizontal Curvature (Two-lane Rural Highways) Figure 3.7 Estimated Relationship between Accident Rate and business 60 Access Density by Speed Limit (Two-lane Rural Highways) Figure 3.8 Estimated Relationship between Accident Rate and 61 Signalized Intersection Density by Speed Limit (Two-larte Suburban Highways) Figure 3.9 Estimated Relationship between Accident Rate and 62 Signalized Intersection Density by Grade (Two-lane Suburban Highways) Figure 3.10 Estimated Relationship between Accident Rate and 63 Unsignalized Intersection Density by Speed Limit (Two-lane Suburban Highways) Figure 3.11 Estimated Relationship between Accident Rate and 64 Unsignalized Intersection Density by Grade (Two-lane Suburban Highways) Figure 3.12 Estimated Relationship between Accident Rate and Business Access Density by Speed Limit (Two-lane Suburban Highways) 65 Figure 3.13 Estimated Relationship between Accident Rate and 66 Business Access Density by Grade (Two^lane Suburban Highways) Figure 3.14 Estimated Relationship between Accident Rate and 67 Signalized Intersection Density (Four-lane Rural Highways) Figure 3.15 Estimated Relationship between Accident Rate and 68 Unsignalized Intersection Density (Four-lane Rural Highways) Figure 3.16 Estimated Relationship between Accident Rate and 69 Private Driveway Density by Horizontal Curvature (Four-lane Rural Highways) Figure 3.17 Estimated Relationship between Accident Rate and 70 si g n a l i z e d Intersection Density (Four-lane Suburban Highways) Figure 3.18 Estimated Relationship between Accident Rate and 71 Unsignalized Intersection Density (Four-lane Suburban Highways) Figure 3.19 Estimated Relationship between Accident Rate and 72 Business Access Density (Four-lane Suburban Highways) Figure 4.1 Travel Speed-Flow Rate Relationship 96 Figure 4.2 Speed-Volume on Road Sections with HC < 5 degree 100 Figure 4.3 Speed-Volume on Road Sections with HC >= 5 degree 101 Figure 4.4 Speed-Volume on Road Sections with Speed Limit <= 70 km/h 102 Figure 4.5 Speed-Volume on Road Sections with Speed Limit >= 80 km/h 103 Figure 4.6 Residual Analysis of the I n i t i a l Model 107 Figure 4.7 Residual Analysis of the Modified Model 108 Figure 4.8 Estimated Relationship between Unsignalized Intersection 112 and Travel Speed by T r a f f i c Volume Figure 4.9 Estimated Relationship between Business Access and 113 Travel Speed by T r a f f i c Volume Figure 4.10 Estimated Relationship between Private Driveway and 114 Travel Speed by T r a f f i c Volume X Figure 4. 11 Estimated Relationship between Speed and T r a f f i c Volume by Access Points 117 Figure 5. 1 Polygonal Approximation of a Nonlinear Function 130 Figure 5. 2 S o c i a l Cost vs. Access Points 138 Figure 6. 1 Relationship between Access Density and Speed Zone 147 Figure 6. 2 Minimum Length of T r a n s i t i o n Zone 148 Figure 6. 3 Warning Sign Location Fundamentals 151 Figure 6. 4 Deceleration-Speed Relation 152 Figure 6. 5 T r a n s i t i o n Zone Design Procedure to Mitigate for High Access Density 155 Figure 6. 6 Speed T r a n s i t i o n Zone Sketch 156 Figure 7. 1 Access and Functional C l a s s i f i c a t i o n 159 xi ACKNOWLEDGMENT I would l i k e to thank the members of my committee: Dr. F.P.D. Navin f o r h i s kind help i n various aspects, Dr. W.F. Caselton f o r his he l p f u l suggestions, and Dr. S.O. Russell and Dr. S. Pendakur f o r t h e i r comments. I would l i k e to express special appreciation to my supervisor, Dr. G. Brown for his academic contribution throughout the progress of the t h e s i s . I would also l i k e to thank the Ministry of Transportation and Highways of B r i t i s h Columbia f or providing p a r t i a l f i n a n c i a l support to th i s research. I am thankful to my parents, wife, and son for t h e i r patience and encouragement. xii Chapter One INTRODUCTION 1.1 The Background Recent concerns over pervasive problems of road safety and de t e r i o r a t i n g l e v e l s of service of highways have i n t e n s i f i e d i n t e r e s t i n roadway access management as an e f f e c t i v e non-capital intensive means to ameliorate concerns of t r a f f i c safety and roadway t r a f f i c operations. Access gives u t i l i t y to a road network, providing benefits to road users and to those who make use of the abutting lands. On the other hand, access can be a majqr problem to main road t r a f f i c i n terms of delay and accidents and the cost implications of these. A review of the l i t e r a t u r e indicates substantial research has been done i n the past on various aspects of t h i s problem as r e f l e c t e d i n e x i s t i n g references. The l i t e r a t u r e review points to a need for new studies on the relationship of access density and roadway t r a f f i c to: (1) update past knowledge of the subject; (2) examine the subject at both the s t r a t e g i c (or planning) l e v e l and the t a c t i c a l (or t r a f f i c engineering) l e v e l ; (3) cover as many road classes as p r a c t i c a l l y possible; (4) derive an optimization framework i n planning access points. Thus, the o v e r a l l purpose of the present study i s to investigate the impact of various road access types on t r a f f i c safety and operations; and from the empirical studies undertaken i n the project to develop an optimization model for planning access provisions. 1 Adequate t r a f f i c access for abutting lands and e f f i c i e n t t r a f f i c flow on main roads are two fundamental, but contradictory, functions of a highway system. S p e c i f i c a l l y , speed differences between through t r a f f i c and accessing t r a f f i c , and vehicle maneuvers (crossing, merging, and Weaving) set up p o t e n t i a l f or t r a f f i c delays, d r i v e r confusion and motor v e h i c l e accidents. Reducing access points per u n i t distance (access density) can enhance t r a f f i c service on the main road and reduce accidents. On the other hand, increasing access points reduces d r i v e r delay f o r accessing t r a f f i c . The c l a s s i c a l approach to r e s o l v i n g t h i s dichotomy i s to develop a h i e r a r c h i c a l highway system i n which each class w i l l enhance one function (through t r a f f i c vs. access t r a f f i c ) over another. Including a l l s p e c i a l i z e d functions i n one network system w i l l i d e a l l y provide the functions needed for a l l t r a f f i c . Some classes mainly serve the movement of high speed inter-urban t r a f f i c (freeway and a r t e r i a l classes) while other classes mainly serve abutting land access ( c o l l e c t o r roads and l o c a l roads). The hierarchy range i n p r a c t i c e consists of f u l l access control ( i . e . , no access) p a r t i a l access c o n t r o l and no access c o n t r o l . This functional c l a s s i f i c a t i o n of highways and streets has been widely accepted and used by highway a u t h o r i t i e s . In the functional c l a s s i f i c a t i o n f o r urban and r u r a l roads as documented i n the Manual of Geometric Design Standards for Canadian Roads (Roads and Transportation Association of Canada, 1986) the freeway c l a s s i f i c a t i o n i s r e l a t i v e l y straightforward: with "no accesses" allowed. However, for the lower classes of roads the r e l a t i v e p r i o r i t y suggested for through t r a f f i c versus access t r a f f i c i s not well defined. This i s p a r t i c u l a r l y the case for two-lane r u r a l roads (nominally 2 " a r t e r i a l " class) f or which there i s no s p e c i f i c guidelines f o r the type of access, nor the density of access associated with t h i s c l a s s . A second problem with the functional c l a s s i f i c a t i o n system as documented i s the absence of a safety c r i t e r i o n i n the d e f i n i t i o n . Conventibnal wisdom i s that freeways are the safest c l a s s , but again the lower classes are not well defined i n terms of r e l a t i v e safety. Insights gained i n the study of access and t r a f f i c may be useful to r e f i n e the c l a s s i f i c a t i o n of roadways and i s consequently one objective of the present study. At the planning l e v e l of analysis, the project objective i s to assess the balance between delay and hazard r e s u l t i n g from a large number of access points (access density) which provide the a c c e s s i b i l i t y needed for abutting lands. Since increased access density provides u t i l i t y f o r various land use a c t i v i t i e s such as neighborhood, commercial and i n d u s t r i a l economic a c t i v i t i e s i t i s important for the purposes of this present study to determine the appropriate access density which i s complementary to a desired l e v e l of t r a f f i c service. Given the need f o r access i n order to provide u t i l i t y to the road system, t r a f f i c operational imperatives can be reduced to the need to provide optimal l e v e l s of road safety and t r a f f i c flows consistent with land use u t i l i t y planning. The operation problem then becomes one of balancing the benefits of access f o r entering/existing t r a f f i c against the d i s r u p t i o n of flow and the accident p o t e n t i a l f or main road t r a f f i c . One prime objective of t h i s study i s to develop an optimizing framework which could be used to i d e n t i f y appropriate access points consistent with t r a f f i c flow l e v e l s and which would minimize t o t a l cost of delay and 3 accidents. Such a framework w i l l provide the t r a f f i c planner with some quantitative guidelines for planning access points for a road section, and c o i n c i d e n t a l l y allow r e v i s i o n i n the highway c l a s s i f i c a t i o n system. At the t r a f f i c operations and road design l e v e l of a n a l y s i s , the concerns are with questions re l a t e d to the r e l a t i v e mix of access types, the r e l a t i v e l o c a t i o n of access points, the e f f e c t of access points on accidents and t r a f f i c e f f i c i e n c y . Since there are many road, d r i v e r , and v e h i c l e factors which can impact on accident p o t e n t i a l and t r a f f i c operation e f f i c i e n c y , the role of access as a negative influence needs to be separated from other fa c t o r s . Some previous studies (e.g. Dart and Mann, 1970) have documented the r e l a t i o n s h i p between accident rates and access points. However, most previous studies were conducted decades ago. One objective of the present study i s to examine the understanding of the access/accident r e l a t i o n s h i p from past studies, and update the quantitative r e l a t i o n s h i p between access and accidents, using an e x i s t i n g data base from one Canadian province. Unlike access/acqident studies, r e l a t i v e l y few references are a v a i l a b l e which examine the e f f e c t of access on operational e f f i c i e n c y . The Highway Capacity Manual (3rd e d i t i o n , 1994) recommends a speed reduction factor f o r the crude number of access points per mile for multilane r u r a l and suburban highways, but not for two-lane highways. Therefore, a further objective of t h i s present study i s to determine the quantitative r e l a t i o n s h i p between access and t r a f f i c e f f i c i e n c y f o r two-lane highways. The t h i r d objective i s to improve speed zone design, p a r t i c u l a r l y speed t r a n s i t i o n zone design, with consideration of actual access density and land use development. 4 1.2 Access and T r a f f i c Safety Studies conducted to reveal the r e l a t i o n s h i p between access and accidents, such as Kihlberg and Tharp (1968), Box and Associates (1970), and Dart and Mann (1970), have found that access c o n t r o l has an important impact on t r a f f i c safety. However, access i s often taken to be a broad concept, with implications f o r both highway planning and t r a f f i c operations. At the planning l e v e l , access provides u t i l i t y f o r various a c t i v i t i e s , such as neighborhood, commercial, and i n d u s t r i a l a c t i v i t i e s , and i s intimately linked to land use considerations. In t r a f f i c operational terms, access can mean entry to t r a f f i c f o r i n d i v i d u a l v e h i c l e s , or to e x i t from the main t r a f f i c stream. The l a t e r case may represent an "access" to l o c a l s t r e e t s . In t h i s sense of t r a f f i c operation, we may define "access" as the "way" of vehicles moving from one transportation f a c i l i t y to another. The r e l a t i o n s h i p between access and safety can then be described for both cases: the undesirable events (accidents) of providing u t i l i t y of transportation f a c i l i t i e s , and the undesirable events r e s u l t i n g from vehicles moving from one f a c i l i t y to another. In the f i r s t case, we deal with the issue at a macroscopic l e v e l , the concern being centered mainly on the frequency and magnitude of the interference r e s u l t i n g to through t r a f f i c from vehicles j o i n i n g to or crossing the through t r a f f i c . S p e c i f i c a l l y , i t i s the r e l a t i o n s h i p between accident occurrence and the frequency and spacing of access points. In the second case, we deal with the issue at a microscopic l e v e l , as the r e l a t i o n s h i p between accident occurrence and the geometric design elements of highways. 5 1.2.1 Access and Safety at the Macroscopic Level At the macroscopic l e v e l , p r o v i s i o n of access co n t r o l i s v i t a l to a road i n terms of service and safety. As stated i n AASHTO (1990), "With co n t r o l of access, entrances and e x i t s are located at points best s u i t e d to f i t t r a f f i c and land-use needs and are designed to enable vehicles to enter and leave s a f e l y with a minimum of interference with through t r a f f i c " . " I f access points are adequately spaced, and entering and e x i t i n g volumes are l i g h t , the street or highway functions e f f i c i e n t l y . If access points are numerous, and entering and e x i t i n g volumes are heavy, the capacity and safety of the f a c i l i t y are reduced" (AASHTO, 1990). Thus trade-offs must be made to e f f e c t i v e l y serve through t r a f f i c and land-use establishments while at the same time observing a safety constraint. One recent report (Bochner, 1991) provides guidance i n planning s i t e access i n urban areas. However, Bochner's emphasis i s put on the f o r e c a s t i n g of t r a f f i c generation, d i s t r i b u t i o n , and assignment. The safety factor was mentioned but the actual r e l a t i o n s h i p between access and accidents was not investigated. AASHTO i n i t s study (1990) indicated that "the most s i g n i f i c a n t f a c t o r c o n t r i b u t i n g to safety i s the p r o v i s i o n of f u l l access c o n t r o l " . The b e n e f i c i a l e f f e c t of t h i s factor has been documented i n many reports, such as i n Interstate System Accident Research (U.S. Department of Transportation/Federal Highway Administration, 1970). One of the p r i n c i p a l findings of the FHA study i s that the absence of access con t r o l i n v a r i a b l y increased accident rates. Furthermore, other studies 6 (Stover et a l , 1970, and OECD, 1976) indicated that the number of access points and the amount of t r a f f i c entering the t r a f f i c stream at these points determines the o v e r a l l accident rates for any p a r t i c u l a r length of highway. A study i n v o l v i n g r u r a l highways i n the USA found roughly a seven percent r i s e i n accident rates f o r each a d d i t i o n a l access point per mile (Highway Users Federation f o r Safety and M o b i l i t y , 1970). Besides, access-controlled motorways of up to date design standards co n s t i t u t e the safest and most e f f i c i e n t routes with s u b s t a n t i a l l y lower accident r i s k s when compared to other road classes (OECD, 1986, and Prisk, 1957). The provision of r e s t r i c t e d access by-passes at seven s i t e s i n the UK resulted i n a net reduction of in j u r y accidents of the order of 25 percent (Newland and Newby, 1962). The effectiveness of access c o n t r o l increases as average d a i l y t r a f f i c increases (Smith, W. and Associates, 1961). Roy Jorgensen Associates, Inc. (1978) presented a systematic analysis on access control f o r the following categories of roads: multilane divided r u r a l highways, multilane undivided r u r a l highways, two-lane r u r a l highways, and urban a r t e r i a l highways. The following review summarizes the Jorgensen report. In addition, however, r e s i d e n t i a l c o l l e c t o r roads are also discussed. 1.2.1.1. Multilane Divided Rural Highways AASHTO (1990) indicated that the accident rate increases with an increase of business a c t i v i t y and at-grade i n t e r s e c t i o n density. Figure 1.1 shows the r e l a t i o n s h i p for 4-lane divided highways. The figure shows 7 that the influence of in t e r s e c t i o n s increases with the increase of i n t e r s e c t i o n density, and the e f f e c t of business a c t i v i t y increases at the same rate f o r each i n t e r s e c t i o n access density. It i s also demonstrated i n Kihlberg and Tharp's report (1968) that as the access control increases, the accident rate on four-lane divided r u r a l highways decreases. Results showing accident rates f o r d i f f e r e n t l e v e l s of access control are shown i n Table 1.1. As one would expect, with increasing access density, the increase i n single v e h i c l e accident (e.g., run-off road) i s less than the increase i n multivehicle accident, because of the increased c o n f l i c t between vehicles entering and e x i t i n g with p a r t i a l or no access control (higher density of access). Access Control Accident Rates Total M u l t i - v e h i c l e Single-vehicle F u l l 0.67 0.38 0.31 P a r t i a l 0.91 0.58 0.33 None 1.84 1.38 0.42 * Mean Annual No. of Accidents per 0.3-mile (0.4827 km) Segment. Source: Kihlberg and Tharp (1968) Table 1.1 Accident Rates Correspondent to Different Levels of Access Control(Multilane Divided Rural Highways) 8 Source: AASHTO (1990). Figure 1.1 Accident Rate on 4-lane Divided Non-Interstate Highways by Number of At-Grade Intersections per Mile and Number of Businesses per Mile 9 1.2.1.2 Multilane Undivided Rural Highways The Kihlberg and Tharp study (1968) included a l l r u r a l highways. A decrease i n accident rates from those with no access c o n t r o l , to p a r t i a l , to f u l l , f o r divided four-lane f a c i l i t i e s was observed. Table 1.2 shows smoothed rates (mean annual number of accidents per 0.3-mile segment) f o r r u r a l highways i n Ohio with a range of t r a f f i c volumes from 4,600 to 6,899 ADT. In comparing Table 1.2 and 1.1 i t i s c l e a r that accesses on two-lane highways have a dramatic impact on the accident rate compared to multilane divided highways, p a r t i c u l a r l y on multivehicle accident rates, although two-lane highways with no access control appear to be safer than multilane divided highways with no access c o n t r o l . In addition, four lane undivided highways as i l l u s t r a t e d by Table 1.2 have a very much higher accident rate than a i l multilane divided highways shown i n Table 1.1; as one would expect. Highway Type Total M u l t i -Vehicle Single-Vehicle 2-lane, no access control 1.75 1.22 0.45 2-lane, p a r t i a l access control 1.39 1.06 0.32 4-lane, no median, no access control 2.56 1.98 0.48 Source: Kihlberg and Tharp (1968) Table 1.2 Accident Rates Correspondent to Different Levels of Access Control (2-lane/4-lane Rural Highways) 10 1.2.1.3 Two-lane Rural Highways Several studies have shown that as the roadside development increases and the number of entrances increases, the accident rate increases. Dart and Mann (1970) found that the accident rate increases as the c o n f l i c t s per mile increase for r u r a l highways i n Louisiana, as shown i n Figure 1.2. T r a f f i c c o n f l i c t s were defined by Dart and Mann as the t o t a l number of t r a f f i c access points (on both sides) per mile of highway section and included only minor road i n t e r s e c t i o n s and p r i n c i p a l access driveways to abutting property along highway sections. Intersections with major roads were considered as break points between study sections. Fee et a l (1970) employed an e s p e c i a l l y large accident data base with an example of the results shown i n Table 1.3 which relates access points per km to the accident rate. The data shown are for an ADT of 8,140 v e h i c l e s . It may be noted from Table 1.3 that a t e n - f o l d increase in the number of accesses to the highway more than doubles the accident rate; a 100-fold increasing access points raises the accident rate approximately 14 times. In the study mentioned i n section 1.2.1.1 and shown i n Table 1.2, Kihlberg and Tharp (1968) demonstrated c l e a r l y that: (a) two-lane highways had lower accident rates than four-lane highways when there was no median and no access c o n t r o l ; (b) access control had the most powerful accident reducing e f f e c t , and p a r t i a l control access was also e f f e c t i v e . 11 (0 O > c o to c 8 2.4 -2.2 1.8 = 1.6 1.4 1.2 1 -0 0 . , o y , o o / ° o o / , i , i , i , i I 1 1 0 24 4 8 12 16 20 Traffic Conflicts per Mile Source: Dart and Mann (1970) Figure 1.2 Accident Rate Versus T r a f f i c C o n f l i c t s (Two-lane Rural Highways) Access points per km Businesses per km * Accident Rate ... 0.6 78 „ 6.0 167 12.0 60.0 1060 * Rate i s number of accidents per 100-million vehicle-km. Source: Fee et a l (1970) Table 1.3 Accident Rates Correspondent to Different Levels of Access Density(2-lane Rural Highways) 12 Prisk (1957) found that head-on, angle, and pedestrian c o l l i s i o n s comprise 10 percent of a l l freeway accidents, while f a c i l i t i e s with p a r t i a l or no access control experience 35 to 50 percent accidents i n those categories. The t o t a l accident rate, f a t a l i t y rate, and i n j u r y rate on f u l l access-control f a c i l i t i e s were found to be lower than on p a r t i a l access control or no control f o r both r u r a l and urban f a c i l i t i e s . The following r e s u l t s were found f o r the period 1949 to 1955 as shown i n Table 1.4. As expected, freeways with f u l l access c o n t r o l have the most e f f e c t on reducing f a t a l accidents. Schoppert (1957) demonstrated that access to the highway by driveways or intersections i s d i r e c t l y Correlated to accidents at a l l ADT l e v e l s . The number of access points i s a reasonably good pre d i c t o r of the number of p o t e n t i a l accidents within an ADT group, although the most important factor i n the p r e d i c t i o n of t r a f f i c accidents i s the vehicle volume on the highway. The number of accidents increases with the number of situ a t i o n s presenting a change i n conditions and thus requiring a decision on the part of the motor v e h i c l e operator. Accidents increase when (a) vehicle volumes increase, (b) access points increase, and (c) sight distance i s impaired and/or cross section i s reduced. While volume i s the best predictor, the number of points of access i s second i n importance. Design features such as lane width, shoulder width, and sight r e s t r i c t i o n s are t h i r d . Raff (1953) found that the more roadside establishments per mile, the higher the accident rate, although t h i s was s i g n i f i c a n t only f or one set of data (two-lane tangent sections). 13 Type F a t a l i t i e s per 100 mvm* Injuries per * mvm Total Accidents mvm* Rural: f u l l c o n t r o l 3.4 1.36 1.56 Rural: p a r t i a l control 6.3 1.48 2.09 Rural: no control 10.3 2.14 3.44 Urban: f u l l c o n t r o l 2.2 0.89 1.94 Urban: p a r t i a l Control 5.6 1.64 5.23 Urban: no cont r o l 4.1 2.61 5.01 * mvm represents m i l l i o n - v e h i c l e - m i l e s . Source: Prisk (1957) Table 1.4 F a t a l i t i e s and Injuries at Diff e r e n t Access Control Level (2-larte Rural and Urban Highways) 1.2.1.4 Urban A r t e r i a l s Generally, a l l major urban a r t e r i a l studies reviewed found control of roadside access factors to be second only to exposure ( t r a f f i c volume) i n importance for road safety. S p e c i f i c a l l y , four safety r e l a t i o n s h i p s were established by several studies: A. As accesses increase, the t o t a l accident and in j u r y rates increase. Cribbins and Aray et a l (1967) used an access point index as a measure of roadside access points and t h e i r volumes. The access point index i s the estimated t o t a l of a l l movements entering or leaving the s i t e from commercial and i n d u s t r i a l roadside development, private 14 drives, and i n t e r s e c t i n g roadways expressed on a per mile bas i s . Thus, the index i s proportional to the number of c o n f l i c t points along the f a c i l i t y and the exposure at the points. The independent variables used i n the p r e d i c t i n g equation are: access-point index, s i g n a l i z e d openings, speed l i m i t , volume, and l e v e l of service. B. As the number of commercial driveways per mile and/or commercial units per mile increases, the accident rate increases. Mulinazzi and Michael (1967) conducted research i n which 100 study sections were s t r a t i f i e d by type of a r t e r i a l . It was concluded that the number of median and heavy volume commercial driveways per mile was s i g n i f i c a n t l y r e l a t e d to the accident rate f o r low-volume (1,200 to 5,800 ADT) sections. The accidents per mile for an urban a r t e r i a l can be expressed using the following v a r i a b l e s : volume (ADT) on the section i n thousands of v e h i c l e s ; number of heavy volume in t e r s e c t i o n s per mile; number of t r a f f i c signals per mile; number of heavy and medium volume commercial d r i v e r s per mile; parking allowed; and number of 4-way inter s e c t i o n s per mile. Head (1959) developed several l i n e a r equations f o r a r t e r i a l s , grouped by ADT ranges and number of lanes (two or four ) . The following variables are employed i n the equations: ADT (the average d a i l y t r a f f i c divided by 100); the number of commercial units per mile; the number of inter s e c t i o n s per mile; the number of t r a f f i c signals per mile; the indicated speed; and the pavement width i n feet. In each equation, the number of commercial units per mile was s i g n i f i c a n t l y related to the 15 accident rate. Also, commercial units were found to be the most important predictor of accidents. C. There i s no s i g n i f i c a n t d i f f e r e n c e i n accident rates between roads having no access points and those having access points serving only noncommercial purposes. Stover et a l (1970) used Kipp's o r i g i n a l research (1952) to investigate the influence of medial and marginal accesses on a r t e r i a l operations. They found that l i t t l e - u s e d access points act s i m i l a r l y to no access points at a l l . D. As the number of int e r s e c t i o n s per mile increases, the t o t a l accident and i n j u r y accident rates increase. In Cribbins and Aray et a l ' s report (1967), a l l i n t e r s e c t i o n openings per mile were tested, only s i g n a l i z e d i n t e r s e c t i o n openings were s i g n i f i c a n t and used i n the f i n a l regression equation. In Cribbins and Horn et a l ' s report (1967), in t e r s e c t i o n s were grouped as: (a) s i g n a l i z e d with l e f t - t u r n storage; (b) s i g n a l i z e d without l e f t - t u r n storage; (c) a l l i n t e r s e c t i o n s with l e f t - t u r n storage; and (d) a l l i n t e r s e c t i o n s without l e f t - t u r n storage, each on a per mile basis. A l l were found to be s i g n i f i c a n t l y related to the accident rate. 16 1.2.1.5 Res i d e n t i a l C o l l e c t o r Roads Daff and White (1990) revealed that more than h a l f of a l l the accidents involved a c o l l e c t o r road, and a high proportion of mid-blocks (at l e a s t 46 percent) involved c o l l i s i o n s with parked cars or driveway maneuvers, and would probably not have occurred i f parking and r e s i d e n t i a l driveways were not present. 1.2.2 Access and Safety at the Microscope Level At t h i s l e v e l , geometric design of access f a c i l i t i e s i s considered. The access f a c i l i t i e s are driveways, ramps, speed-change lanes, i n t e r s e c t i o n s , and medians. 1.2.2.1 Driveways It was found that 11 percent of the c i t y ' s t o t a l accidents were accounted for by accidents at driveways. Of these accidents, approximately two thirds involved l e f t turns into or out of driveways (Box, 1969). A number of studies of driveway accidents have been made, i n an attempt to r e l a t e accidents to driveway c h a r a c t e r i s t i c s . As Stover et a l (1970) indicated, the conclusions of such studies have rather general terms. For example, roadways along which l e f t turns into or out of the driveways are permitted have had higher accident rates than those with median b a r r i e r s which r e s t r i c t access to one d i r e c t i o n of the through 17 lanes. Box (1969) reported a p o s i t i v e c o r r e l a t i o n between uncontrolled driveway width and accident rates. These r e s u l t s , however, were not adequate to e s t a b l i s h a precise, quantitative r e l a t i o n s h i p between driveway width and accident rate. Results of other studies have been affected by high degrees of v a r i a b i l i t y and other biases r e s u l t i n g from lack of information on various types of driveways. In another paper, Box (1970) indicated that t r a d i t i o n a l "abbreviated data-processing tabulation systems provide too coarse a summation to completely e s t a b l i s h driveway influences on accident". On the other hand, Stover et a l (1970) argued that i t i s questionable whether driveway accidents a c t u a l l y constitute the most appropriate c r i t e r i a for driveway design. He believed that, at least for major roadways, the e f f i c i e n t movement of t r a f f i c i s a primary consideration. This i s complemented by findings r e l a t i v e to t r a f f i c safety. It i s because: (1) " c o n f l i c t s from poor or inadequate control of access may have reduced the e f f i c i e n c y long before i t i s r e f l e c t e d i n s t a t i s t i c a l l y s i g n i f i c a n t accident rates"; (2) " p o t e n t i a l c o n f l i c t s which account for much of accident hazard are also those c o n f l i c t s to which other i n e f f i c i e n c i e s of the t r a f f i c stream can be a t t r i b u t e d , so that, i f these i n e f f i c i e n c i e s are corrected, many of the accident hazards w i l l be a l s o . " 1.2.2.2 Ramps Some studies were made to determine which geometric features play important roles i n ramp safety. The measure of safety used i n ramp studies i s accident rates per m i l l i o n ramp vehicles, in which the number 18 of vehicles using the ramp i s counted, while distance traveled along the ramp i s not considered. Lundy (1967) investigated 722 freeway ramps, and he found that the accident rates of on-ramps were cons i s t e n t l y lower than off-ramp accident rates. Diamond ramps have the lowest accident rates, and the s c i s s o r s 1 and l e f t side ramps have the highest rates. On s c i s s o r s ramps, the primary concentration of accidents i s at the s c i s s o r s or cross-over (ramp with l o c a l road) f a c i l i t y . Straight ramps (on and o f f ) have 12 percent lower o v e r a l l accident rate than curved ramps. Yates (1970) also found that urban loop ramps with higher curvature have accident rates higher than those with low curvature. But r u r a l loop ramps with low curvature have accident rate higher than those with high curvature. For on-ramps, i t i s shown that 52 percent of accidents occurred i n the merge area, and 48 percent i n the ramp area (Lundy, 1967). Concerning the effectiveness of entrance ramp controls, Blumentritt (1981) indicated that using responsive ramp controls would reduce accidents by 10 to 33 percent. A technical synthesis, NCHRP Report 35 (1976), i d e n t i f i e d the more successful design and operating practices used at freeway off-ramp terminals. The safety problem was addressed. The main points are as follows: 1 Lundy denned in his study (1967): " A scissors ramp is one that has opposing traffic crossing the ramp traffic. The ramp traffic has the right of way and a stop sign is placed to stop the crossing vehicle." 19 o Diamond ramps have the lowest accident rate, s c i s s o r s and l e f t -handed ex i t ramps have the highest rates. o 44 percent of the accidents occurred i n the diverge area, 56 percent i n the ramp area (Lundy, 1967). o The extent to which geometric and t r a f f i c c h a r a c t e r i s t i c s (19 elements) are judged to have contributed to accidents was examined. As expected, an increase i n t r a f f i c volume r e s u l t s i n an increase i n accidents, geometric alone apparently accounts f o r only a small portion of the variance i n accidents. Wrong-way entry i s another type of incident occurring at o f f -ramps. Five percent of a l l f a t a l i t i e s on i n t e r s t a t e highway are a t t r i b u t e d to accidents r e s u l t i n g from wrong-way movements. The wrong-way accidents usually occur at night, and 75 percent of the wrong-way drivers who cause accidents have been drinking excessively (NCHRP Report 35, 1976). 1.2.2.3 Speed-Change Lanes C i r i l l o (1970) conducted research to analyze speed-change lane length. Results indicated that increasing the length of acceleration lanes w i l l reduce accident rates i f the percentage of merging vehicles i s greater than six percent of mainline volume. Increased length of deceleration lanes w i l l also reduce accident safety but to a l e s s e r degree. As the percentage of t r a f f i c entering or leaving the mainline through a speed-change lane increases, the accident increases. Lundy 20 (1967) concluded those acceleration lane lengths greater than 800 feet can be expected to have below average accident rates, deceleration lane lengths with 900 feet or longer have lower accident rates. The shorter radius, large c e n t r a l angle curved off-ramps seem to have lower rates than the ramps with median range r a d i i and d e l t a s . C i r i l l o et a l (1969) found that a 100-ft increase (up to 2,600 f t ) i n the stopping sight distance decreased annual accidents per 1,000 vehicles per day (vpd) an amount of 0.0008 (accidents/year/vpd) f o r a deceleration lane. 1.2.2.4 Intersections AASHTO (1990) recognized that i n t e r s e c t i o n s are access points. Some e a r l i e r studies, e.g., Peterson and Michael (1965), indicated that i n t e r s e c t i o n accidents increased when: o Percent green time on the bypass decreased, o Bypass or cross-street ADT increased, o Percent l e f t turn from the bypass increased, o Maximum approach speed increased, o Number of i n t e r s e c t i o n approaches increased, o Total width of driveways within 200 f t of the i n t e r s e c t i o n increased. Some studies have addressed the r o l e of i n t e r s e c t i o n sight distance i n producing accidents, and revealed the r e l a t i o n s h i p between accidents and i n t e r s e c t i o n sight distance. Wu (1973) studied the r e l a t i o n s h i p between accident rate and what he c a l l e d " c lear v i s i o n 21 right-of-way" at 192 si g n a l i z e d i n t e r s e c t i o n s . He concludes that i n t e r s e c t i o n s where v i s i o n i s poor have s i g n i f i c a n t l y higher accident rates, but no s p e c i f i c numbers are given and no s t a t i s t i c a l tests are c i t e d . David and Norman (1975) studied the r e l a t i o n s h i p between accident rate and various i n t e r s e c t i o n geometric and t r a f f i c features. The study revealed s i g n i f i c a n t accident rate differences between "obstructed" and "c l e a r " i n t e r s e c t i o n s . However, the r e s u l t s are reported without regard f o r number of legs, number of lanes, type of c o n t r o l , presence of turning lanes and speed l i m i t . 1.2.2.5 Medians and Left-Turn Lanes Highway Users Federation for Safety and Mo b i l i t y (1970) indicated that main-road l e f t - t u r n storage lanes can s i g n i f i c a n t l y reduce accident rates. Some studies also found that the presence of l e f t - t u r n storage lanes at median openings reduces the number of rear-end c o l l i s i o n s on urban a r t e r i a l (Cribbins et a l , 1967; Sawhill and Neuzil, 1963). Several studies also concluded that at intersections the presence of l e f t - t u r n lanes reduces the accidents (Shaw and Michael, 1963; Thomas, 1966; Tamburri and Hammer, 1968; and Hoffman, 1974). Cribbins (1967) conducted a study i n North Carolina, and he found that as t r a f f i c volumes increase, use of median openings r a p i d l y becomes hazardous. When combined with intensive roadside development, use of median openings under high-volume conditions becomes more hazardous. 22 In short, though there are quite a few studies on access and safety, the majority of them were conducted decades ago. The r e l a t i o n s h i p needs to be re-examined, considering changed t e c h n i c a l , s o c i a l and economic circumstances. 1.3 Access and T r a f f i c Operations Although some e f f o r t s have been made to deal with the e f f e c t s of access on t r a f f i c operations, they are e i t h e r not intended to e s t a b l i s h quantitative r e l a t i o n s h i p s , or intended unsuccessfully to e s t a b l i s h the r e l a t i o n s h i p s . One exception i s that the newest r e v i s i o n of Chapter 7 of the Highway Capacity Manual (HCM), 1992, which does e s t a b l i s h t h i s r e l a t i o n s h i p on multilane highways. Chapter 7 of the Highway Capacity Manual (Revised Chapter 7 of HCM, 1992) has considered the e f f e c t of access points on multilane r u r a l and suburban highways i n i t s procedure of analyzing highway l e v e l of service. The research conducted by R e i l l y et a l (1989) f o r the Revised Chapter 7 of HCM found that the number of access points has an important influence on free-flow speed, i n that f o r every 10 access points per mile, t r a v e l speed w i l l be reduced by 2.5 mph. The access point i s a composite v a r i a b l e made up of the separate influences of d i f f e r e n t c h a r a c t e r i s t i c s of access. However, Chapter 8 of the HCM, Two-Lane Highways, has not yet been updated to include the e f f e c t of access points on t r a f f i c operations. 23 One study conducted i n the UK (Brocklebank, 1992) on two-lane r u r a l highways found that access variables are generally d i f f i c u l t to f i t i n models of t r a f f i c operations. The composite v a r i a b l e ( i n t e r s e c t i o n s , lay-bys, and accesses) was not s i g n i f i c a n t i n the UK study. Non-residential accesses and lay-bys separate access variables were e i t h e r not s i g n i f i c a n t or bccasionally s i g n i f i c a n t . The i n t e r s e c t i o n v a r i a b l e was rather unstable with a c o e f f i c i e n t value -1.8 (km/h). One possible explanation f o r the r e s u l t i s that there are so many other independent variables considered i n the model (a t o t a l of 31 independent v a r i a b l e s ) , the e f f e c t of access variables may be subsumed. In any case the r e s u l t s do not f u l l y conform to our a p r i o r i expectation, and a further study i s perceived to be needed. In addition, two-lane highways i n urban and suburban areas should also be considered because the higher t r a f f i c and more roadside developments w i l l i n t e n s i f y the influence of access points. Glennon et a l (1975) have developed techniques to counteract vehicle xielay due to access points, and although i t was recognized that the mainstream through t r a f f i c delay i s caused by turning vehicles, no quantitative r e l a t i o n s h i p has been established. 1.4 Optimal Number of Access P o i n t s Most previous studies concentrated on minimum distance between access points. They are highway design oriented. For example, OECD (1971) recommended 250 metres to be the minimum spacing between intersections on a r t e r i a l s t r e e t s . NAASRA (1972) recommended 24 i n t e r s e c t i o n s at a minimum of 350 to 550 metres. Another e a r l i e r guideline (Swedish National Board of Urban Planning, 1968) s p e c i f i e d minimum i n t e r s e c t i o n spacing from 300 to 600 metres, depending on road status. S i m i l a r minimum access spacing c r i t e r i a f o r driveways were also recommended by Glennon (1975). In general, the above studies are based on stop sight requirements. On the other hand, Stover et a l (1970) raised i n t e r s e c t i o n spacing issue which was mainly concerned with the economic and l e g a l implications of spacing. Two other studies u t i l i z e d q u a n t i t a t i v e methods to derive the spacing problem. They are Del Mistro (1980) and Del Mistro and Fieldwick (1981). Both of them related accident estimates to access spacing by s t a t i s t i c a l a n a l y s is. It was found that i n t e r s e c t i o n should be spaced not less than 250 m apart (Del Mistro and Fieldwick, 1981). Del Mistro (1980) recommended the number of i n t e r s e c t i o n s which are permissible i n terms of the population of the r e s i d e n t i a l development and through t r a f f i c . B a s i c a l l y , only safety was taken account i n those studies. The impact of access on t r a f f i c operation was not considered. Besides, most of these studies concentrated on minimum access spacing rather than optimal access spacing. This perhaps was because they were geometric design oriented. 1.5 Scope of the Study A complete highway system consists of three components: highway f a c i l i t y ; v e h i c l e ; and d r i v e r . An access point i s a part of the highway f a c i l i t y . To study the performance of a highway f a c i l i t y , i t i s 25 desirable to also take the veh i c l e and d r i v e r into account. This i s because d i f f e r e n t types of vehicles combined with various d r i v e r c h a r a c t e r i s t i c s w i l l r e s u l t i n d i f f e r e n t l e v e l s of performance for the same highway f a c i l i t y . However, complete study of a l l of these aspects would require enormous resources, which i s beyond the scope of t h i s work. Thus the present study follows the general engineering p r a c t i c e of "fre e z i n g " some factors while focusing only on the major fa c t o r of concern, and making appropriate assumptions on those "frozen" f a c t o r s . We assume passenger cars as the general veh i c l e type (passenger vehicle) and l e t an ordinary d r i v e r represent general d r i v e r c h a r a c t e r i s t i c s . Therefore, only the highway f a c i l i t y (access points) w i l l be s p e c i f i c a l l y examined here. Besides, we w i l l p r i m a r i l y concentrate on the macroscope l e v e l of access provisions, and study the e f f e c t of frequency of access points on t r a f f i c safety and operations. Geometric design of the various types of access was not considered i n t h i s study. More s p e c i f i c a l l y , with the above caveat, this present study d i f f e r s from previous studies i n the following areas: (1) a l l access types are included, from s i g n a l i z e d i n t e r s e c t i o n to a g r i c u l t u r a l access, and roadside p u l l o u t . (2) The e f f e c t s of i n d i v i d u a l access types on safety and t r a f f i c operations are investigated, instead of using an aggregated access points measure, such as general access points. (3) The j o i n t e f f e c t s of access and other road c h a r a c t e r i s t i c s are analyzed. (4) Several accident measures are used to develop s t a t i s t i c a l models to f i n d the best s t a t i s t i c a l representation of accidents i n the r e l a t i o n s h i p . (5) Local data are used for analysis to s p e c i f i c a l l y reveal the r e l a t i o n s h i p between access and accidents on one P r o v i n c i a l highway 26 network. (6) Apply the quantitative models of access-accidents and access-operation into access planning and access design procedure. Figure 1.3 shows the basic structure of the thesis. In Chapter 2, the data to be used i n the analyses i s described s p e c i f i c a l l y . Chapter 3 estimates the r e l a t i o n s h i p s between accesses and accidents f o r d i f f e r e n t highway cla s s e s . Furthermore, a conceptual hazard model i s suggested which includes t r a f f i c c o n f l i c t s information i n safety a n a l y s i s . Chapter 4 estimates the re l a t i o n s h i p s between accesses and t r a f f i c operations ( i . e . , t r a v e l speed, f or two-lane highways). A t r a v e l speed model i s constructed. In chapter 5, an optimization model i s formulated to obtain the optimal number of access points for highway sections based on inputs from previous chapters. The objective of the model i s to minimize the t o t a l s o c i a l cost. A "piecewise" l i n e a r approximation technique i s employed to solve the integer nonlinear programming problem i n the model. In chapter 6, access and speed zone design, p a r t i c u l a r l y speed t r a n s i t i o n zone design, i s discussed. -Figure 1.3 also shows that the thesis contains three l e v e l s of study: operation (chapters 3 and 4), planning (chapter 5), and design (chapter 6). F i n a l l y , Chapter 7 summarizes the res u l t s and possible future research. 27 Chapter 2 Data Base „, , - Accident Chapters j ^ f y ^ Operation: HighwayN Access l-Study y Planning: Design: Access & Accident Model Ba> /GS Met! iod Chapter 4 Chapter 5 Optimal Access Points Chapter 6 Speed Reduction Design Traffic Operation Model Legend: Actual Study B l o c k Relevant Studies B l o c k C Type o f the Study Figure 1.3 Structure of the Study Chapter Two THE DATA The data used i n the study are extracted from the e x i s t i n g photolog of selected sections of highways i n B r i t i s h Columbia. For the purpose of c o l l e c t i n g accident, t r a f f i c and access data, a three year period (1988-1990) was used. The data base consists of four major parts: access, t r a f f i c , road c h a r a c t e r i s t i c s , and accidents. Since each s i n g l e highway section w i l l be used as a data point, we s t a r t with the d e f i n i t i o n of highway section. 2.1 Highway Sections About twenty numbered highway routes i n the province of B r i t i s h Columbia were selected for study. The t o t a l tested road length was approximately 750 km, and divided into 376 sections. The following c r i t e r i a were adopted i n s e l e c t i n g highway routes and d e f i n i n g sections: (1) geographic representation of the p r o v i n c i a l highways; (2) homogeneity of roadway c h a r a c t e r i s t i c s within a section; (3) consistent road condition during the study period; (4) a wide range of t r a f f i c volume representation (average annual d a i l y t r a f f i c , AADT); (5) no construction work i n process i n the section. The section length ranged from 0.3 km to 13.8 km, with an average value of 2.23 km per section. Road sections were grouped on the basis of road c l a s s i f i c a t i o n , i . e . , four-lane r u r a l road, four-lane suburban road, two-lane r u r a l road, and two-lane suburban road. Figure 2.1 shows 29 the sample siz e representation i n each category. B a s i c a l l y , the sample size i s greater than f i v e percent of the population i n each road category. Number of sections and the average length of the sections i n each category are also given i n the fig u r e . The section boundaries were selected at locations where no or very few accidents were recorded during the study period with the aid of an accident histogram (Figure 2.2). The section boundaries were defined i n thi s manner to ensure that hazardous locations ( i . e . , black spots) were f u l l y represented i n only one section. Other considerations i n d e f i n i n g section boundaries were the consistency of t r a f f i c c o n t r o l (e.g., speed l i m i t ) , and road c h a r a c t e r i s t i c s (e.g., with or without a u x i l i a r y lane). T r a f f i c volume i s another consideration i n section d e f i n i t i o n . An e f f o r t was made to include a wide range of t r a f f i c volume i n road sections. For example, Table 2.1 shows the sample s i z e d i s t r i b u t i o n by t r a f f i c volume range and i t s r e l a t i o n to the p r o v i n c i a l road inventory for two-lane r u r a l highways. It can be seen that the sample used i n t h i s study represents s i x percent of the t o t a l two-lane r u r a l highway system i n B r i t i s h Columbia, and that the d i s t r i b u t i o n of the observations r e f l e c t s reasonably well the d i s t r i b u t i o n of t r a f f i c volume classes on the road system as defined by the AADT. 30 Total Sample Size Tested road Length = 750 km Percentage of total = = 7% No. of sections = 376 Average section length = 2.23 km General 4-lane highways Tested road length =101 km Percentage of total = 18 % No. of sections = 82 Average section length = 1.2 km General 2-lane highways Tested road length = 659 km Percentage of total = 7 % No. of sections = 295 Average section length = 2.2 km 4-lane rural highways Tested road length = 57 km Percentage of total = 5 0 % No. of sections = 36 Average section length = 1.6 km 4-lane suburban hiqhway s Tested road length = 45 km Percentage of total = 10% No. of sections = 46 Average section length = 1 km 2-lane rural highways Tested road length = 560 km Percentage of total = 6 % No. of sections = 164 Average section length = 3.4 km 2-lane suburban highways Tested road length = 99 km Percentage of total = 20 % No. of sections =131 Average section length = 0.8 km 2-lanesuburban highways (AAD< = 10.000 Tested road length = 29 km Percentage of total = -No. of sections = 34 Average section length = 0.9 km 2-lanesuburban highways (AADT> 10,000) Tested road length = 70 km Percentage of total = -No. of sections = 97 Average section length = 0.7 km Figure 2.1 Sample Size Representation Highway 9 7 , Segment 1110, 9km-17km, Speed Limit = 80 km/h, 2-lane Landmark Kilometer 9.0 Crossing Road 20 Road 19 Crossing Road 18 Road 17 Road 15 Road 14 Road 13 Road 12 Crossing Road 11 Road 10 Road 9 Road 8 Road 7 Crossing Road 6 Road 5 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 Accident Section No. ^ Frequency p p p P P P I P p p I P 11 P P II p 1111 p p p I M P I PP r— P i i p p Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Legend I = Injury P = Properly Damage Only F Public Road Business Access Private Access Figure 2.2 Sample of Section Definition 32 1 1 AADT Inventory Sample (km) % of No. of Obs. (km) Inventory 1 -5000 7625.1 520.9 6.8% 133 % of T o t a l 83.1% 93.0% 81.1% 5001 - 10000 1261.5 28.4 2.3% 22 % of T o t a l 13.8% 5.1% 13.4% 10001 and 284.1 10.7 3.8% 9 over 3.1% 1.9% 5.5% % of T o t a l Total 9167.1 560 6.1% 164 Table 2.1 D i s t r i b u t i o n of Sample Size by AADT (Two-lane r u r a l highway) 2.2 Access Data Ten types of road access were i d e n t i f i e d f o r general road c l a s s i f i c a t i o n . They are: (1) s i g n a l i z e d i n t e r s e c t i o n ; (2) four-way unsignalized i n t e r s e c t i o n ; (3) three-way unsignalized i n t e r s e c t i o n ; (4) commercial access; (5) i n d u s t r i a l access; (6), r e s i d e n t i a l access; (7) a g r i c u l t u r a l access; (8) roadside pul l o u t ; (9) on-ramp and (10) o f f -ramp. Aft e r preliminary model t e s t i n g , i t was l a t e r determined to group some types according to functional c h a r a c t e r i s t i c s of access to s i m p l i f y the complexity of the model. At the end, s i x access types were used i n the a n a l y s i s : (1) s i g n a l i z e d i n t e r s e c t i o n ; (2) Unsignalized i n t e r s e c t i o n (including four-way and three-way i n t e r s e c t i o n s ) ; (3) business access (including commercial and i n d u s t r i a l ) ; (4) private access (including r e s i d e n t i a l and a g r i c u l t u r a l ) ; (5) roadside pul l o u t ; and (6) ramp. Note that the word " i n t e r s e c t i o n " i n t h i s study i s defined as the general area where two or more public roads j o i n or cross. This d e f i n i t i o n w i l l 33 d i s t i n g u i s h the i n t e r s e c t i o n from other access types, such as p r i v a t e access, where one private driveway i s connected to one p u b l i c road. The coding of access data was done with photologging technology. The photologging involves taking a series of photographs of a roadway section at 20 metres' i n t e r v a l s frtim a s p e c i a l l y equipped v e h i c l e . Besides the camera system the vehicle i s also equipped with s p e c i a l instruments to record road c h a r a c t e r i s t i c s , such as grade and h o r i z o n t a l curvature. The type and l o c a t i o n of accesses were recorded by viewing the study route. The access l o c a t i o n reference was recorded using the vehicle's odometer reading, which was l a t e r adjusted to match the accident l o c a t i o n reference. Access points were summarized f o r each road section by type, and then the density (number of access per km) of each access type was c a l c u l a t e d for s t a t i s t i c a l modeling purposes. 2.3 T r a f f i c Volume and Speed Two main t r a f f i c variables were observed, speed l i m i t and average annual d a i l y t r a f f i c (AADT). Speed l i m i t was obtained from the photolog, and AADT was Obtained from published count s t a t i o n reports. It should be noted that the accessing t r a f f i c volumes of each access point were not a v a i l a b l e , thus only major road t r a f f i c was included. However, i t i s believed that the grouping of accesses by functional c h a r a c t e r i s t i c s as outlined above provides a proxy for access volumes. For example, access volume i s highest for group one access ( s i g n a l i z e d i n t e r s e c t i o n s ) and least f or group f i v e (roadside p u l l o u t s ) . 34 Average t r a v e l speed of t r a f f i c i s the measurement that w i l l be used i n Chapter 4 to study the re l a t i o n s h i p between access and t r a f f i c operations. However, i t i s not d i r e c t l y given i n the photolog data base. Travel speed of the test v e h i c l e i s employed as the surrogate of the average t r a v e l speed of main road t r a f f i c . The d e t a i l discussion of t h i s representation w i l l be presented i n Chapter 4. 2.4 Road C h a r a c t e r i s t i c s Some road c h a r a c t e r i s t i c s * information was also a v a i l a b l e from the photolog, such as median type, grade, traverse slope, horizontal curvature, and the d i r e c t i o n of curvature. The l a t t e r measure gives a record i f the curvature i s l e f t or r i g h t . On the basis of th i s measurement, the frequency of change i n d i r e c t i o n per kilometre within the study segment was calcu l a t e d . The intention of recording t h i s information i s to r e f l e c t the amount of driver's attention needed to traverse a given segment. Horizontal curvature i s defined as the ce n t r a l angle of a 100 metre arc. This measurement, adapted from the Imperial d e f i n i t i o n erf curve, i s used rather than radius because the former i s the measure recorded i n the photolog database of B r i t i s h Columbia used for t h i s research. The other road c h a r a c t e r i s t i c considered to be important i s the presence of a u x i l i a r y lanes i n the section. It was defined as a dummy vari a b l e i n the data base with 1 i n d i c a t i n g an a u x i l i a r y lane, 0 otherwise. Some studies have found s i g n i f i c a n t c o r r e l a t i o n between accidents and lane and shoulder widths (e.g., Craus, Livneh and Ishai, 1991). However, because the lane and shoulder widths are l a r g e l y f i x e d throughout the study area (lane width = 3.6m, and 35 paved shoulder width = 2.0ra), these variables were not included i n the analysis. 2.5 Accident Data Accident records are a v a i l a b l e for a l l selected routes. A f t e r the determination of section boundaries, section length was a v a i l a b l e to c a l c u l a t e accident measures for each section. Five measures of accident were generated to f i n d the best representation of accidents i n the analysis. These measures are accident rate (ACDR, ac c i d e n t s / m i l l i o n -vehicle-kilometre), accident frequency (ACDD, accidents/km), accident se v e r i t y r a t i o (ASR, a weighted se v e r i t y r a t i o with weight factor 100 assigned to f a t a l accident, 10 assigned to i n j u r y accident, and 1 assigned to property-damage-only accident), severe accident rate (SEVR, f a t a l plus i n j u r y accidents/million-vehicle-kilometre), and severe accident frequency (SEVF, f a t a l plus i n j u r y accidents/km). As indicated e a r l i e r , three years (1988-1990) accident records are coded into the data base. 36 Chapter Three THE EFFECT OF ACCESS ON HIGHWAY SAFETY The f i r s t section of t h i s chapter examines the established r e l a t i o n s h i p between access and accidents. Sections 2 and 3 construct accident models using multiple regression analysis for two-lane r u r a l , two-lane suburban, four-lane r u r a l , and four-lane suburban highways. In section four, we w i l l discuss the hypothesis of a hazard model. 3.1 Examination of Established Relationship To see i f the established relationships between accidents and access are s t i l l v a l i d i n the current s i t u a t i o n , a preliminary test was conducted for two-lane r u r a l highways. A s i m i l a r graph as Figure 1.2 was p l o t t e d using our i n i t i a l data (shown i n Figure 3.1). To do t h i s , s i m i l a r access types (minor in t e r s e c t i o n s , business access, and private access) were selected and aggregated as access points per mile (same d e f i n i t i o n as i n Figure 1.2: number of minor int e r s e c t i o n s and p r i n c i p a l access driveways per mile). However, i t seems that the l i n e a r c o r r e l a t i o n between accident rate and access i s not s i g n i f i c a n t i n our s i t u a t i o n , which indicates that many other contributing factors other than access points should also be considered. Thus, i t i s presumed that t h i s circumstance i s applicable to other highway classes as well. Therefore, a complete reexamination of the relationships between access and accidents i s needed. 37 6 E i : <D > o z, a C -g o a < a v • • • - A . 10 15 20 Access Density (No./mile) 25 30 35 00 Figure 3.1 Access Density vs. Accident Rate (2-lane Rural Highways) 3.2 Model Construction F i r s t of a l l , independent variables that w i l l be used i n the model construction are summarized i n Table 3.1. These variables include three types: access, t r a f f i c , and road c h a r a c t e r i s t i c s . Dependent v a r i a b l e s are those accident measurement variables described i n section 2.5. Variable Description Density of s i g n a l i z e d i n t e r s e c t i o n (No./km) x? Density of unsignalized i n t e r s e c t i o n (No./km) X-* Density of business access (No./km) x* Density of private access (No./km) Xs Density of roadside pullout (No./km) Xfi Density of on/off ramp (No./km) x 7 Median type ( s o l i d double l i n e = l , painted median=2, barrier=3, wide grass median=4) X R Speed l i m i t (SPL, km/h) X Q Average Annual Daily T r a f f i c (AADT) x i n Grade (percent) X n Traverse slope (percent) Xl2 Frequency of changing d i r e c t i o n of curvature (No. of changes/km) Horizontal curvature (degrees) x14 Dummy variable (with a u x i l i a r y lane = 1, without a u x i l i a r y lane = 0) x-is Section length (km) Table 3.1 Independent Variables Used i n Model Construction A multiple l i n e a r regression technique was employed to develop models describing the combined e f f e c t of access, t r a f f i c and road c h a r a c t e r i s t i c s on highway safety. To f i n d the best model, a stepwise 39 procedure (backward elimination) was implemented. The c r i t e r i a of 2 choosing the best model were R , standard error (SE), mean absolute error (MAE), F-test, and t-Test. The c a l i b r a t i o n procedure started with the simplest model form, i . e . , a l l variables were entered into the model i n t h e i r o r i g i n a l forms. The advantage of estimating t h i s model i s that the d i r e c t l i n e a r relationships between the dependent and independent variables can be revealed, and the major contributory factors can be detected. However, since the actual r e l a t i o n s h i p i s very complex i n most s i t u a t i o n s , a l i n e a r model can only explain a small portion of the v a r i a t i o n i n the dependent v a r i a b l e . Therefore, a polynomial model was produced to increase the explanatory power of the model and to allow f o r i n t e r a c t i o n of access and road c h a r a c t e r i s t i c s ' v a r i a b l e s . As i s the case with many polynomial ordinary least squares (OLS) models, heteroscedasticity can pose a s i g n i f i c a n t s t a t i s t i c a l problem. In this study, a combination of variable transformation and weighting was used to normalize estimation bias. It i s noted ( C a r r o l l and Ruppert, 1988) that transformations and weighting could be combined together when the variance depends on a covariate, or when i t i s unclear whether a transformation or weighting i s preferable. Both conditions were present i n the development of the models. As a r e s u l t , a weighted least squares (WLS) method incorporating with transformation (in the form of square root) was applied to estimate the models whenever necessary. The r e s u l t s of regression models f o r accident rate (ACDR), severe accident rate (SEVR), accident severity r a t i o (ASR), accident frequency 40 (ACDD), and severe accident frequency (SEVF) are summarized i n Table 3.2. The table shows the resu l t s of the stepwise regression estimation. A l l variables are s i g n i f i c a n t at the 0.05 l e v e l ( t - T e s t ) 1 . The F-test for each model i s also s a t i s f a c t o r y at 0.01 l e v e l of s i g n i f i c a n t s , i n d i c a t i n g that a l l independent v a r i a b l e s ' c o e f f i c i e n t s are c o l l e c t i v e l y s i g n i f i c a n t l y d i f f e r e n t from zero. The c o e f f i c i e n t s of v a r i a t i o n (R ) of the models appear s a t i s f a c t o r y f o r most models, that i s greater than 50 2 percent. It should be noted that the R r e s u l t i n g from a stepwise regression estimation i s always smaller than that r e s u l t i n g from a standard regression estimation, due to the exclusion of some independent variables based on the elimination c r i t e r i a ( i n th i s case, the 0.05 l e v e l of s i g n i f i c a n c e ) . Therefore, compared with the explanatory power of previous models i n th i s f i e l d , our re s u l t s seem to be a marked improvement over previous models. Preliminary review of the re s u l t s indicated that the ASR model provided poor c o r r e l a t i o n between independent variables and the dependent variable (ASR), probably due to the a r b i t r a r y weighting factors assigned to severe accidents. Some c o e f f i c i e n t s may appear to have the wrong sign i n the equations; for example, the SEVF model i n A1N4 group (general 4-lane road) has items +1.7224X {-0.3834X*. The r e s u l t indicates that i f there are f i v e or more si g n a l i z e d intersections per km (X^), there w i l l be a negative impact on the dependent v a r i a b l e . The reasons f o r th i s are: f i r s t , the regression model i s li m i t e d by the range of data, and i t i s d i f f i c u l t to extend the result beyond the data range that b a s i c a l l y 1 Individual t values are not reported upon here, a = 0.05 The statistics software ensures all variables meeting t test, 41 includes road sections with no more than f i v e s i g n a l i z e d i n t e r s e c t i o n s per km. Since our general 4-lane road includes only r u r a l and suburban road sections, i t i s unusual to have s i g n a l i z e d i n t e r s e c t i o n density greater than f i v e per kilometre. Second, i t may also be true, l o g i c a l l y , that more s i g n a l i z e d i n t e r s e c t i o n s w i l l reduce accidents and s e v e r i t y of accidents because s i g n a l i z e d i n t e r s e c t i o n s are safer than unsignalized i n t e r s e c t i o n with the same t r a f f i c volumes. Higher s i g n a l i z e d i n t e r s e c t i o n density w i l l slow down average t r a v e l speed which has the e f f e c t of reducing accident s e v e r i t y . To show the analysis procedure the models f o r two-lane r u r a l highways ( i d e n t i f i e d as A2 group i n Table 3.2) were selected for discussion and are presented below. Figure 3.2 shows comparisons of observed and predicted values by each model, except ASR model, i n A2 group. The graphs present schematic i l l u s t r a t i o n s of the amount of v a r i a t i o n i n the observed data, along with the best f i t l i n e s representing the equations for two-lane r u r a l highways. It i s noted that the accident frequency model (ACDD and SEVF) explains more v a r i a t i o n than the accident rate model (ACDR and SEVR); and the a l l - a c c i d e n t s models (ACDD and ACDR) can explain more v a r i a t i o n than the severe-accident-only models (SEVR and SEVF). The possible explanations are as follows: i n the f i r s t case, accident rates were ca l c u l a t e d based on t r a f f i c volume (AADT). However, the access volume was not a v a i l a b l e as indicated e a r l i e r . Thus AADT only represents t r a f f i c volumes on the main road. The incomplete information about t r a f f i c volumes may r e s u l t i n an imperfect accident measure i n accident 42 Group Regression Equations R^ SE M A E F* A l ACDR05 = 1.8540 + 0.2405^1 + 0.01102X2 + 0.0120X 4 X 1 3 -0.000074X 9 0.68 0.30 0.22 19.8 ACDD1S = 0.0022^ + 0.000196X 2X 1 3 +1.46818 x 10~*X4X9 0.78 0.00 0.00 43.1 ASR 0 5 = 0.1040X, + 0.0263X8 + 0.2471X1 0 + 0.0756XU - 0.0669Xn 0.98 0.24 0.17 448.2 SEVR 0 5 = 0.8216 + 0.2584X, + 0.2316X2 - 0 . 0 1 5 3 ^ -0.009469X 3 +0.000028e'3 -0.000034X 9 + 0.0545X13 0.62 0.19 0.12 9.1 SEVF = 5.1930+ 4.0964X t + 0.3541X4 + 0.000355X9 -0.602IXU 0.47 3.99 3.14 8.6 A2 ACDR"5 = 0.7040+ 0.322(Wr" + O.OO2466X 3 0 5 X 8 + O.O13O.X'405A'13 + 0 .0456T 5 0 5 X, 3 - 0 . 0 0 5 4 2 9 X 9 ° 5 + 0.1487 Xw - 0.1083A'U + 0.1565X," 5 0.54 0.07 0.05 24.55 ACDD0 5 = 0.0223 + 0.0087X\5 + 0.000129X^Xt + 0.0012X° 5Xn + 0.0015X 5 ° 5 Xa + 0.0009\X9 - 0 . 0 1 9 4 X ; , 5 0.82 0.02 0.01 124.5 ASR a 5 = 1.4172 + 0.3366e*5 -0.6717X 1° 0 5 + 0.1070X1 2 + 1.6942X14 0.20 0.98 0.63 11.4 SEVR 0 5 = 0.2714+ 0.1391X205 + O.OO17X305X8 + 0.0094X; 5Z i 3 + 0.0303X°5X1 3 + 0.0629X1D 0.45 0.05 0.03 27.9 SEVF05 = -0.4390+0.2995XZ* + 0 .002124X 3 ° 5 X 8 + 0 . 0 2 0 8 X 4 a 5 ^ 1 3 + 0 . 0 3 4 6 Z 5 a 5 Z 1 3 + 0 . 0 1 8 8 4 X ° 5 + 0.0679X 1 0 + 0 .211Uf u 0.66 0.07 0.05 46.2 Variable Description: Density of sig n a l i z e d i n t e r s e c t i o n (No./km) x 9 Average Annual Daily T r a f f i c (AADT) Density of unsignalized i n t e r s e c t i o n (No./km) xio Grade (percent) x 3 Density of business access (No./km) X n Traverse slope (percent) X 4 Density of private access (No./km) *12 Frequency of changing d i r e c t i o n of curvature (No. of changes/km) Density of roadside pullout (No./km) *13 Horizontal curvature (degrees) Density of on/off ramp (No./km) X 1 4 Dummy variable (with a u x i l i a r y lane = 1, without a u x i l i a r y lane = 0) Median type ( s o l i d double l i n e = l , Xl5 Section length (km) painted median=2, barrier=3, wide grass median=4) x 8 Speed l i m i t (SPL, km/h) Note: A l = 4-lane rural road; A2 = 2-lane rural road. 4^  Table 3.2 Accident Estimation Models (I) Group Regression Equations " r 2 SE M A E F* A 3 a ACDR05 = 0.9423 + 0.6996Jf,"5 + 0.2982X°5 + 0.0437X3 - 0.0083X8 + 0.0886X10 + 0.0164X1 2 0.61 0.48 0.35 35.4 ACDD0i = -0.0162+ 0.1005X°S - 0.0409X 2 + 0.1023X" 5 + 0.000069eXi + 0.0066X 3 + 4.6099x 10"6X, ^_ + 0.0108X 1 0 0.67 0.06 0.04 . 39.5 SEVR 0 5 = 0.6060+ 0.3941X,05 + 0.1858X205 + 0.0242X3 - 0.0026X,05 + 0.0529X,0 0.49 0.32 0.24 26.5 SEVF05 = 0.3973-1.5020X, + 2.9955 X™ -0.7707X 2 + 1.6373X,05 + 0.000907eX2 + 0.0036X2 0.48 0.80 0.49 20.9 A 3 1 a ACDR" = 0.4138 + 0.7510X,05 + 0.3142X2°5 + 0.0737X, + 0.1047X10 0.64 0.49 0.36 43.3 ACDD" = 0.0950X,05 + 0.0398X2°5 + 0.0089X, + 3.9527 x 10 6 X 9 + O.OO27X120 0.91 0.06 0.05 186.2 SEVR" = 0.2616+ 0.3669X"5 + 0.1917X205 + 0.0436X3 + 0.0117X,2 0.56 0.32 0.24 31.7 SEVF" = 1.1083 + 0.7581X,05 + 0.2739X,05 + 0.0784X3 - 0.0492X2 A 3 2 b ACDR05 - 0.9400 + 0.8622^ + 0.1345X2 - 0.000073X9 + 0.0378X1 2 0.79 0.29 0.21 31.8 ACDD05 = 0.0743+ 0.0768^ +0.0082X2 - 0.0008X8 + 4.6226 x 1 0 ^ X 9 +0.0022X 1 2 0.81 0.02 0.02 28.8 SEVR™ = 0.5496 + 0.3529Xi + 0.0809X 2 - 0.000056X9 + 0.0301X12 0.59 0.26 0.19 13.1 V a r i a b l e D e s c r i p t i o n : D e n s i t y o f s i g n a l i z e d i n t e r s e c t i o n ( N o . / k m ) x 9 A v e r a g e A n n u a l D a i l y T r a f f i c ( A A D T ) D e n s i t y o f u n s i g n a l i z e d i n t e r s e c t i o n ( N o . / k m ) X i o G r a d e ( p e r c e n t ) x 3 D e n s i t y o f b u s i n e s s a c c e s s ( N o . / k m ) X l l T r a v e r s e s l o p e ( p e r c e n t ) X 4 D e n s i t y o f p r i v a t e a c c e s s ( N o . / k m ) x 1 2 F r e q u e n c y o f c h a n g i n g d i r e c t i o n o f c u r v a t u r e ( N o . o f c h a n g e s / k m ) D e n s i t y o f r o a d s i d e p u l l o u t ( N o . / k m ) H o r i z o n t a l c u r v a t u r e ( d e g r e e s ) x 6 D e n s i t y o f o n / o f f r a m p ( N o . / k m ) Dummy v a r i a b l e ( w i t h a u x i l i a r y l a n e = 1 , - w i t h o u t a u x i l i a r y l a n e = 0) M e d i a n t y p e ( s o l i d d o u b l e l i n e = l , * 1 5 S e c t i o n l e n g t h (km) p a i n t e d m e d i a n = 2 , b a r r i e r = 3 , w i d e g r a s s m e d i a n = 4 ) *8 S p e e d l i m i t ( S P L , k m / h ) Note: A3 = 2-lane suburban road for whole A A D T range; A31 = 2-lane suburban road with A A D T > 10000; A32 = 2-lane suburban road with A A D T < 10000. a. No equation for ASR. b. No equations for A S R and SEVF. Table 3.2 Accident Estimation Models (H) Group Regression Equations SE M A E F* A4 ACDR = 2.7606 + 0.4209X, + 0.2652X° 5 + 0.005439X32 - 0.012176X,05 0.78 0.43 0.30 27.1 ACDD = 0.0182 + 0.0134^ + 0.0046X, + O.OOOII9Z3 2 0.74 0.01 0.01 27.3 ASR = 3.3342 + 1.1764eX l - 1.0696x 10 'V 3 0.27 2.76 2.00 6.3 SEVR = 1.1973 + 0.3010^ + 0.0610Z2 - 0.005533X9 0.65 0.22 0.15 18.7 SEVF = 2.2457 + 3.1693Z, + 0.7038A'2 0.62 2.13 1.54 24.3 A2N3 ACDR05 = 0.7241 + 0.3327 Xx +0.1007 X2 + 0.0632A r 3 + 0.2147 A 5 - 9.0119 x 10"° X9 + 0.1120X,,, - 0.0672A' 1 1 + 0.02\SX13 + 0.0241A'1 3 0.53 0.50 0.34 37.2 ACDD" = 0.0871X,05 + 0.0304X205 + 0.0149Z305 + 0.000526Ar9os +0.0012^ 0.93 0.05 0.03 830.3 ASR" = 1.8294 + 0.142LY," - 0 . 0 7 3 0 ^ + 0.3337A'° 5 + 0.5617A52 -0.2348A 1 0 + 0 . 0 5 0 9 ^ + 0 . 4 0 2 7 ^ 0.20 0.10 0.06 11.5 SEVR05 = 0.5775 + 0.2816X, 0 5 + 0 .187LY 2 0 5 + 0.0299A , 3 + 0.1755X; - 0 .002407X 9 ° 5 - 0 . 0 5 3 8 Z , , + 0 .0146X 1 2 0.45 0.35 0.24 35.5 SEVF05 = 0.79 30 + 0.7508X, 0 5 + 0 . 4 1 0 6 ^ 2 ° 5 + 0.1503AT3a5 + 0.3513X, + 0.004889X, 0 5 - 0.185 \XX, 0.59 0.73 0.49 71.1 A1N4 ACDR = 0.2278 +1.7871X, - 0 . 3 7 6 8 X 2 + 0.1330X2 + 6.3964 x 10"' ex' + 0 .0353X 4 + 0 .1303X 6 0.77 0.39 0.28 42.1 ACDD = -0.0105 + 0.0195Z, + 0.001Uf3 + 1.1343x10% 0.82 0.00 0.01 118.7 ASR = -7.4867 + 4.1696X, - 1.6065X2 + 0.0360eX 2 + 0.5934X3 + 0.001416A"2 + LO488Z10 0.32 3.12 2.16 7.1 SEVR = 0.5053 + 0.3516^ + 0.0130X4 - 0.000015X9 0.52 0.27 0.20 28.2 SEVF05 = -0.0431+1.7224X, -0.3834A', 2 + 0 .3265X 2 + 0.005787A'3 2 + 0 . 2 9 9 9 7 6 ^ ° 5 - 0 . 3 4 5 9 X 7 + 0 .060739^8 -0 .0482^,2 . 0.61 0.66 0.50 15.6 V a r i a b l e D e s c r i p t i o n : x 2 x 3 x 4 x 5 x 6 x 7 x 8 Note: D e n s i t y o f s i g n a l i z e d i n t e r s e c t i o n ( N o . / k m ) D e n s i t y o f u n s i g n a l i z e d i n t e r s e c t i o n ( N o . / k m ) D e n s i t y o f b u s i n e s s a c c e s s ( N o . / k m ) D e n s i t y o f p r i v a t e a c c e s s ( N o . / k m ) D e n s i t y o f r o a d s i d e p u l l o u t ( N o . / k m ) D e n s i t y o f o n / o f f r a m p ( N o . / k m ) X 9 X l O X l l X12 Xl3 Xl4 M e d i a n t y p e ( s o l i d d o u b l e l i n e = l , X15 p a i n t e d m e d i a n = 2 , b a r r i e r = 3 , w i d e g r a s s m e d i a n = 4 ) S p e e d l i m i t ( S P L , k m / h ) A4 = 4-lane suburban road; A2N3 = general 24ane road (rural + suburban), A v e r a g e A n n u a l D a i l y T r a f f i c ( A A D T ) G r a d e ( p e r c e n t ) T r a v e r s e s l o p e ( p e r c e n t ) F r e q u e n c y o f c h a n g i n g d i r e c t i o n o f c u r v a t u r e ( N o . o f c h a n g e s / k m ) H o r i z o n t a l c u r v a t u r e ( d e g r e e s ) Dummy v a r i a b l e ( w i t h a u x i l i a r y l a n e w i t h o u t a u x i l i a r y l a n e = 0) S e c t i o n l e n g t h (km) A1N4 = general 4-lane road (rural + suburban). = 1, Table 3.2 Accident Estimation Models (HI) Predicted Figure 3.2 (cont'd) Comparison of Predicted and Observed Values 47 rate models. On the other hand, i n accident frequency models, AADT was only used as an i n d i v i d u a l independent v a r i a b l e , the dependent v a r i a b l e (accident frequency) could be p r e c i s e l y measured, therefore, the r e s u l t i n g uncertainty i s much lower than that i n accident rate model. In the second case, the reason that a l l - a c c i d e n t models are superior to severe-accident-only models i s perhaps because more accident information was included i n the former models. As we know, property-damage-only accidents are a major portion of a l l accidents, and severe accidents do not occur i n every road section. Many zero values i n the observations of the dependent v a r i a b l e (accident measure) w i l l provide l i t t l e unsafety information to model construction. Therefore, the explanation a b i l i t y of such models w i l l be affected. The component e f f e c t of each independent variable can be represented by "component + r e s i d u a l " p l o t s . Figure 3.3 shows sample plots for four access related v a r i a b l e s , (unsignalized i n t e r s e c t i o n density, Figure 3.3.a), X ^ X J - J (private access density and horizontal curvature, Figure 3.3.b) i n the accident rate (ACDR) model. The s t r a i g h t (component) l i n e s i n the graph are defined as: p j ( X y - X j ) , which m u l t i p l i e s the centered value of Xj by the associated value of i t s regression c o e f f i c i e n t 3 j . The closeness of the scattered points (component + residual) around the component l i n e f or X ? / 5 (Figure 3.3.a) indicates that unsignalized i n t e r s e c t i o n density i s s i g n i f i c a n t l y c o n t r i b u t i n g i n explaining the v a r i a t i o n i n accident rate. The componenteffect of the i n t e r a c t i o n v a r i a b l e of private access density 48 and h o r i z o n t a l curvature, X ^ , provides less s i g n i f i c a n t c o n t r i b u t i o n to the explanatory power of the model (Figure 3.3.b). The c o r r e l a t i o n analyses showed no s i g n i f i c a n t m u l t i c o l l i n e a r i t y problem among the independent v a r i a b l e s . Table 3.3 through 3.6 show c o r r e l a t i o n matrixes of ACDR, ACDD, SEVR, and SEVF models for two-lane r u r a l highways. It i s l o g i c a l that some variables be c l o s e l y correlated, such as speed l i m i t Xg and horizontal curvature X 1 3 ; but when these are combined with other variables such as X ° 5 X 8 and X j 5 X 1 3 , the r e s u l t s change. The c o r r e l a t i o n analysis uses the composite variables i n the model instead of o r i g i n a l i n d i v i d u a l variables Xg and X 1 3 . The tables indicate that the e f f e c t of each variable or composite v a r i a b l e on the accident measure can be s a t i s f a c t o r i l y explained, . r e s u l t i n g i n a conclusion that any problems of m u l t i c o l l i n e a r i t y are not serious. From regression c o e f f i c i e n t s for two-lane r u r a l highways i n Table 3.2, i t can be seen that unsignalized i n t e r s e c t i o n density has the largest e f f e c t on accident occurrence. This f i n d i n g conforms to p r i o r expectation as p u b l i c roads generate high access t r a f f i c volume and r e s u l t i n more t r a f f i c c o n f l i c t s than any other access types. The composite variable of business access density (X3) and speed l i m i t (Xg) c l e a r l y shows that a change i n accidents as a r e s u l t of X3 change also depends on the speed l i m i t value. That i s , the higher the speed l i m i t , the larger the impact of the business access density on accident occurrence. 50 Const xll X"5Xt xrxn X"Xn xa9s xia xn A. 12 Const. 1.0000 .0985 .0394 -.3216 -.4751 -.6113 .0987 -.8129 -.8219 X?' . .0985 1.0000 -.2908 -.1501 -.3650 -.5609 -.1019 -.0624 .1603 X°5XS .0394 -.2908 1.0000 -.2763 .0798 .0265 .0779 .0497 -.2287 -.3216 -.1051 -.2763 1.0000 .0377 .1751 .0340 .3659 .0778 xrxn -.4751 -.3650 .0798 .0377 1.0000 .4037 -.3877 .2123 .4992 xi5 -.6113 -.5609 .0265 .1751 .4037 1.0000 .0267 .3073 .2990 x , .0987 -.1019 .0079 .0340 -.3877 .0267 1.0000 -.2701 -.2031 x, -.8129 -.0624 .0497 .3659 .2123 .3073 -.2701 1.0000 .5501 yO.S * 12. -.8219 .1603 -.2287 .0778 .4992 .2990 -.2031 .5501 1.0000 Table 3.3 Co r r e l a t i o n Matrix for ACDR0-5 (2-lane r u r a l highways) Const. V 0 . 5 V XT 0.5 y yd.i v ^ 3 8 A 4 A 13 A 5 A 13 xl X, Const. X 3 x% xrxn X, X x\> X, 1.0000 .3475 -.1424 .3457 1.0000 -.3110 -.1424 -.3110 1.0000 -.0683 -.0186 -.7328 -.9418 -.0683 .3809 .0659 3809 -.0659 1.0000 -.1949 .1160 .0508 -.1949 1.0000 .0600 .2011 .6537 .2161 0186 -.7328 -.9418 .1160 -.6537 -.2161 .0508 -.0600 .1417 .2011 -.0595 .0381 -.0967 .0381 1.0000 .5269 1417 -.0595 -.0967 .5269 1.0000 Table 3.4 Cor r e l a t i o n Matrix for ACDD0-5 (2-lane r u r a l highways) 51 Const. XIs X?X, xrxn Const. 1.0000 -.2552 -.1984 -.3346 -.0519 -.5712 -.2552 1.0000 -.2496 .0121 -.4634 -.0349 -.1984 -.2496 1.0000 -.3985 .2481 .0701 x^x^ -.3346 .0121 -.3985 1.0000 .0023 .1282 x^xn -.0519 -.4634 .2481 .0023 1.0000 -.4341 ^ 1 0 -.5712 -.0349 .0701 .1282 -.4341 1.0000 Table 3.5 Correlation Matrix for SEVR 0- 5 (2-lane r u r a l highways) ! Const Xa25 X°'5XS x:sxn X?Xa xs* ^ 1 0 *M Const. 1.0000 .4147 -.1460 -.2215 -.3082 -.8171 -.4196 .0621 XIs .4147 1.0000 -.2478 -.0844 -.5070 -.5775 -.1116 -.0922 XIs Xt 3 S -.1460 -.2478 1.0000 -.2696 .1903 -.0014 .0787 .2164 x^x^ -.2215 -.0844 -.2696 1.0000 -.0594 -.0225 .1335 .3788 X?Xa -.3082 -.5070 .1903 -.0594 1.0000 .3760 -.3546 -.2397 X™ -.8171 -.5775 -.0014 -.0225 .3760 1.0000 .1238 -.2913 -.4196 -.1116 .0787 .1335 -.3546 .1238 1.0000 .0115 k .0621 -.0922 .2164 .3788 -.2397 -.2913 .0115 1.0000 Table 3.6 Correlation Matrix for SEVF 0• 5 (2-lane r u r a l highways 52 Similar i n t e r a c t i o n e f f e c t s can be found i n the other two access v a r i a b l e s , i . e . , private access and roadside p u l l o u t , with the i n t e r a c t i o n of horizontal curvature. These r e l a t i o n s h i p s indicate that the e f f e c t of access density on accident increases with greater horizontal curvature, implying that a sight distance problem may e x i s t . The s i g n i f i c a n c e of the horizontal curvature with the p r i v a t e access and roadside pullout along with i t s i n s i g n i f i c a n c e with the unsignalized i n t e r s e c t i o n density can be used to speculate that the sight distance i s incorporated i n i n t e r s e c t i o n geometric design, but not i n pri v a t e driveway and roadside pullout design. Besides the access variables, i t i s also worth examining the e f f e c t s of other t r a f f i c and road c h a r a c t e r i s t i c v a r i a b l e s . It i s i n t e r e s t i n g that the c o e f f i c i e n t of AADT i n the accident rate (ACDR) model i s negative, whereas i t i s p o s i t i v e i n the accident frequency (ACDD) and severe accident frequency (SEVF) models. Because accident rate i s calculated by d i v i d i n g the number of accidents by AADT, a negative AADT c o e f f i c i e n t indicates the increase i n accident rate i s prop o r t i o n a l l y smaller than the increase i n AADT. Furthermore, i t i s noted that the accident frequency models (ACDD, SEVF) depict a nonlinear r e l a t i o n s h i p that the rate of accident increase decreases as t r a f f i c volume becomes larger. This f i n d i n g i s i n accordance with the findings of other researchers such as Ng and Haure (1989). Previous studies have shown that steep grades r e s u l t i n higher accident rates (e.g., Glennon, 1987, Kilhberg and Tharp, 1968). This r e l a t i o n s h i p was confirmed i n the present study as can be seen from the 53 s i g n i f i c a n t and p o s i t i v e c o e f f i c i e n t of X^q (grade). It should be pointed out that downgrades and upgrades were not distinguished i n t h i s study since a l l accident measures were ca l c u l a t e d on the basis of two-way t r a f f i c . Another geometric feature of the roadway, traverse slope, Xn, was s i g n i f i c a n t i n the a l l - a c c i d e n t models (ACDR and ACDD) , but not i n the severe-accident models (SEVR and SEVF), which indicates that the traverse slope of the road has more e f f e c t on property-damage-only accidents. The negative sign of the c o e f f i c i e n t shows a decreasing r e l a t i o n s h i p , a r e s u l t that was previously reported by Dart and Mann (1970), Carlsson and Hedman (1989). Roadways with r e l a t i v e l y f l a t traverse slopes are more accident-prone than those with some slopes. It should be noted that the value of this v a r i a b l e i n the present study ranged between 0.9 and 4.9 degree. F i n a l l y , i t i s noted that the frequency of change i n curvature d i r e c t i o n , X-12, i s p o s i t i v e l y correlated with accident rate, and the presence of a u x i l i a r y lanes X 1 4 i s p o s i t i v e l y c o r r e l a t e d with severe accident frequency. The e f f e c t of X 1 2 could not be found i n previous studies, perhaps due to lack of information. However, the p o s i t i v e e f f e c t of t h i s variable on the accident rate seems to be i n t u i t i v e l y c orrect, since larger values of t h i s v a r i a b l e r e f l e c t a more complex d r i v i n g environment. The e f f e c t of a u x i l i a r y lanes found i n t h i s study i s d i f f e r e n t from some previous studies. Kalakota et a l (1993) found that there was no s i g n i f i c a n t r e l a t i o n s h i p between a u x i l i a r y lanes and accident rates, while Hedman (1989) reported a negative r e l a t i o n s h i p 54 with the t o t a l accident rate. In our case, the p o s i t i v e c o r r e l a t i o n of the presence of a u x i l i a r y lanes with severe-accident frequency may be due to accidents r e s u l t i n g from high speed merge/diverge maneuver. Since the v a r i a b l e i s not s i g n i f i c a n t i n the a l l - a c c i d e n t models, i t implies that a u x i l i a r y lanes do not have a p o s i t i v e influence on property-damage-only accidents. 3.3 Results and Discussion As a r e s u l t of the above analyses, the r e l a t i o n s h i p s between accidents and each access type were normalized for a l l road categories. The normalization i s based on the accident rate (ACDR) model because the accident rate i s widely used i n safety analysis. This was done by varying one of the i n t e r a c t i o n variables while holding a l l other independent variables at t h e i r mean values. For example, to study the impact of business access density on accident occurrence, various speed l i m i t values (60 km/h, 70 km/h, etc.) are considered while a l l other independent variables are f i x e d at t h e i r mean values. Figure 3.4 through Figure 3.19 show the estimated r e l a t i o n s h i p s for four road categories, i . e . , four-lane r u r a l , four-lane suburban, two-lane r u r a l , and two-lane suburban. Again, two-lane r u r a l highways were selected as an example of the d e s c r i p t i o n of the e f f e c t s of access density on accidents i n the following paragraphs. The v a r i a t i o n of the accident rate with changes i n access density and horizontal curvature i s depicted i n Figure 3.4 (for unsignalized 55 i n t e r s e c t i o n ) , Figure 3.5 (for private access), and Figure 3.6 (for roadside p u l l o u t ) . A l l figures show a diminishing increase trend i n accident rate with the increase of access density. For example, Figure 3.4 shows a 68% increase i n accident rate due to a change i n unsignalized i n t e r s e c t i o n density from zero to one per kilometre. This increase i n accident rate i s only 20% when i n t e r s e c t i o n density increases from one to two in t e r s e c t i o n s per kilometre; and i t i s 8% when i n t e r s e c t i o n density increases from f i v e to s i x in t e r s e c t i o n s per kilometre. The implication of t h i s f i n d i n g i s that the safety benefit of access control i s not proportional to the percentage reduction i n access density. Another type of re l a t i o n s h i p between access density and accidents can be found i n Figure 3.7, where the use of i n t e r a c t i o n variables i s evident by the widening gaps between contour l i n e s as the value of access density increases. The increase i n accident rate due to an increase i n access density i s greater at higher speed l i m i t than that at lower speed l i m i t . In addition, the diminishing rate of increase i n accident rate i s s t i l l present i n Figures 8 as was the case i n Figures 3.4, 3.5, and 3.6. 56 1 0 1 2 3 4 5 6 7 Unsignalized Public Road Intersection Density (No./km) * See note on page 35. Figure 3.4 Estimated Relationship between Accident Rate and Unsignalized Intersection Density by Horizontal Curvature (2-lane Rural Highways) Model: ACDR All other variables are held at their mean values Horizontal Curvature 7 degrees* " ~ " 5 degrees 3 degrees ' 0 5 10 15 20 Private Access Density (No./km) * S e e n o t e o n 3 5 Figure 3.5 Estimated Relationship between Accident Rate and Private Access Density by Horizontal Curvature (2-lane Rural Highways) 1.5 0.5 1.5 Roadside Pullout Density (No./km) See note on page 35. Figure 3.6 Estimated Relationship between Accident Rate and Roadside Pullout Density by Horizontal Curvature (2-lane Rural Highways) Figure 3.7 Estimated Relationship between Accident Rate and Business Access Density by Speed Limit (2-lane Rural Highways) SPL-60 km/h .SPL=70.km/h SPL=80 km/h 0 1 2 3 4 5 6 Signalized Intersection Density (no./km) Figure 3.8 Estimated Relationship between Accident Rate and Signalized Intersection Density by Speed Limit (Two-lane Suburban Highways) 10 Grade=4% Grade-:3% Grade=2% Grade=l% 0 1 • ! • ! •  1 0 1 2 3 4 5 6 Signalized Intersection Density (No./km) Figure 3.9 Estimated Relationship between Accident Rate and Signalized Intersection Density by Grade S (Two-lane Suburban Highways) 6 1 0 1 ! : ; ; ! [ 1 0 2 4 6 8 10 12 Unsignalized Intersection Density (No./km) Figure 3.10 Estimated Relationship between Accident Rate and Unsignalized Intersection Density by Speed Limit (Two-lane Suburban Highways) 0 2 4 6 8 10 12 Unsignalized Intersection Density (No./km) Figure 3.11 Estimated Relationship between Accident Rate and Unsignalized Intersection Density by Grade (Two-lane Suburban Highways) o 1 • • ! 1 : : 1 0 5 10 15 20 25 30 35 Business Access Density (No./km) Figure 3.12 Estimated Relationship between Accident Rate and Business Access Density by Speed Limit os (Two-lane Suburban Highways) 10 Grade=4% Grade=3% Grade=2% Grade=l% 0 0 10 15 20 Business Access Density (No./km) 25 30 35 Figure 3.13 Estimated Relationship between Accident Rate and Business Access Density by Grade ON (Two-lane Suburban Highways) 0 0.5 1 1.5 2 2.5 3 Signalized Intersection Density (No./km) Figure 3.14 Estimated Relationship between Accident Rate and Signalized Intersection Density (Four-lane Rural Highways) 2.5 0 2 3 Unsignalized Intersection Density (No./km) ON 00 Figure 3.15 Estimated Relationship between Accident Rate and Unsignalized Intersection Density (Four-lane Rural Highways) 7 0 I : : i i : 0 5 10 15 20 25 30 Private Driveway Density (No./km) * See note on page 35. Figure 3.16 Estimated Relationship between Accident Rate and Private Driveway Density by Horizontal Curvature (Four-lane Rural Highways) 0 1 2 Signalized Intersection Density (No./km) Figure 3.17 Estimated Relationship between Accident Rate and Signalized Intersection Density ^ (Four-lane Suburban Highways) o 0 1 2 3 4 5 Unsignalized Intersection Density (No./km) Figure 3.18 Estimated Relationship between Accident Rate and Unsignalized Intersection Density (Four-lane Suburban Highways) 0 5 10 15 20 Business Access Density (No./km) Figure 3.19 Estimated Relationship between Accident Rate and Business Access Density (Four-lane Suburban Highways) Another way of representing the r e l a t i o n s h i p between the density of various types of access and accident rate i s using equivalency f a c t o r s . It can be seen that the curves are rather f l a t a f t e r the i n i t i a l sharp increase i n accident rates, we then approximate curves to l i n e a r functions f o r each p a i r of access and accident r e l a t i o n s h i p within a s p e c i f i e d variable range, i . e . , ignore the low l e f t portion of the curves. This i s a reasonable s i m p l i f i c a t i o n because our major in t e r e s t s are not i n very low access density s i t u a t i o n . The l i n e a r slope was then c a l c u l a t e d f o r each access type by taking an average ho r i z o n t a l curvature (degree of curve = 5) and a speed l i m i t of 80 km/h. F i n a l l y , the equivalency factors are obtained by c a l c u l a t i n g the l i n e a r slope r a t i o between each access type and a selected base access type (unsignalized i n t e r s e c t i o n ) . Note that the r e s u l t i n g equivalency factors are v a l i d only over the s p e c i f i e d access density range. For example, i n the case of private access (Figure 3.7), the l i n e a r range i s 2-16 access per kilometre. The c a l c u l a t i o n r e s u l t s f o r two-lane r u r a l highways are shown i n Table 3.7. It indicates that, i n terms of impact on accident rate, one unsignalized access i s equivalent to two business accesses, and 10 pri v a t e accesses. Similar r e s u l t s can also be found for other road categories as shown i n Figure 3.8 through Figure 3.19. However, due to the apparent nonlinear c h a r a c t e r i s t i c i n many rel a t i o n s h i p s , the equivalency factors were not ca l c u l a t e d for other road categories. 73 Access Type Applicable Change in A C D R per Equivalent Access Density unit Change in Access Factor Range Density Intersection (Base Type) 1 - 6 /km 0.34 1 Business Access 1 - 8 / k m 0.17 0.5 Private Access 2 -16 /km 0.035 0.1 Roadside Pullout 0.5 -1.5/km 0.37 1 Table 3.7 Access Type Equivalency Factors for Two-lane Rural Highways (80 km/h, average degree of curve = 5 degrees*) * See note on page 35. In conclusion, these figures could be used to determine the con t r i b u t i n g e f f e c t s Of access on accident occurrence (accident r a t e ) . Given information about access, t r a f f i c volume, speed l i m i t , and geometries of a road section, the equations presented i n Table 3.2 could be employed when the estimation of the safety (represented by d i f f e r e n t accident measures) for that road section i s required, 3.4 A Conceptual Model of Road Hazards Regression analysis was used to construct accident models on highways i n the previous sections i n which accident records alone were used as the measure of safety i n the regression models. But i t i s becoming inc r e a s i n g l y c l e a r that the use of accident s t a t i s t i c s as derived from accident reporting i s themselves a problem i n d e f i n i n g a measure of safety. On means of complementing accident data for a more comprehensive measure of t r a f f i c safety i s the use of t r a f f i c c o n f l i c t data, i n which a t r a f f i c c o n f l i c t i s a precise, observable d r i v e r response to a perceived hazard, usually i n the form of d r i v e r behavior 74 induced by a "near miss" s i t u a t i o n . It i s presumed that a combination of accident s t a t i s t i c s and "near misses" i s some combination w i l l provide a more accurate concept of road safety than accident s t a t i s t i c s alone. This section discusses a hypothesis of an Empirical Bayes hazards model by using a combination of accidents and t r a f f i c c o n f l i c t s . The discussion here i s mainly on conceptual model d e r i v a t i o n based on empirical evidence from 10 urban i n t e r s e c t i o n s . Some exercises are presented to show the p o t e n t i a l a p p l i c a t i o n of the model. 3.4.1 Introduction It i s known that an accident i s a rare event, and random var i a t i o n s are inherent i n accident s t a t i s t i c s . Therefore, h i s t o r i c a l accident data do not always r e f l e c t long-term accident c h a r a c t e r i s t i c s accurately (see, e.g., Higle and Witkowski, 1988). A s i t e with a low accident rate i n the long run may s t i l l have a high accident rate over a short period of time, and vice versa. In short, the hazard of an e n t i t y (e.g., a road section) cannot always be s a t i s f a c t o r i l y represented by the previous accident record of that e n t i t y . To overcome t h i s problem, the Empirical Bayes (EB) method for estimating hazard has been explored elsewhere (see, e.g., Abbess, J a r r e t t , and Wright, 1981; Hauer, 1992). It i s pointed out by Hauer (1992) that a hazard can be defined as the expected value of accident occurrence. As such, the randomness feature of the hazard could be represented i n the safety d e f i n i t i o n . With t h i s consideration, Bayesian methods have been used for road safety studies. In essence, Bayesian methods are distinguished from regression 75 methods by the fact that parameters are regarded as random variables having s p e c i f i c p r o b a b i l i t y d i s t r i b u t i o n s . Two kinds of clues are used to explain the hazard of an e n t i t y : f i r s t , clues contained i n t r a i t s such as t r a f f i c , geometry, and d r i v e r behavior; and second, clues of the h i s t o r y of accident occurrence. The f i r s t kind of clues might be represented by the estimation of hazard i n a reference population which the study e n t i t y belongs to. The second kind of clues could be represented by actual hazard records (e.g., number of accidents) of that e n t i t y . In Higle et a l ' s study (1988), the observed accident rate at each s i t e i s used i n combination with the gross estimate of the regional p r o b a b i l i t y d i s t r i b u t i o n to obtain the s i t e - s p e c i f i c p r o b a b i l i t y density functions by using Bayes* theorem, number of accidents at l o c a t i o n i , accident rate at l o c a t i o n i , number of vehicles passing through l o c a t i o n i , accident rate p r o b a b i l i t y density function at l o c a t i o n i , accident rate p r o b a b i l i t y density function across the region. More s p e c i f i c a l l y , the r e l a t i o n s h i p can be given by a Gamma d i s t r i b u t i o n : 76 where, = X -Vi = h -h -fAmy,)=SL^xxrle-Si'x (3-2) r ( a j where, a± and fi^ are parameters to be estimated by moments estimates (MME) or maximum l i k e l i h o o d estimates (MLE). Therefore, s i t e i i s hazardous i f the p r o b a b i l i t y i s greater than 8 that i t s expected accident rate X exceeds the observed regional accident rate X R , i . e . , P{li>xR\Ni,Vi}>8 (3.3) Although Hauer's method (1992) i s s l i g h t l y d i f f e r e n t from Higle's, the basic assumptions are the same, i . e . , (1) at any s i t e , when the accident rate i s known, the actual number of accidents follows a Poisson d i s t r i b u t i o n , and (2) the p r o b a b i l i t y d i s t r i b u t i o n of the regional accident rate i s the Gamma d i s t r i b u t i o n . He proposed an EB approach with the combination of multivariate regression method to estimate hazard of an e n t i t y . We b r i e f l y describe the approach as i n the following. A multivariate regression model i s used to estimate expected mean value, E{m}, of hazard i n the reference population as a function of the independent v a r i a b l e s . This i s based on the b e l i e f that E{m}, which depends on the independent variables i n some systematic way, can be captured by a model. Those independent variables are t r a i t s . A residual could be cal c u l a t e d f o r each observation as a res u l t of regression analysis. Next, we can estimate the variance VAR{m} i n reference 77 population using: VAR{m} = VAR{x} - E{m} (3.4) Where, VAR{x} i s the squared res i d u a l . A good estimator of the hazard m for a s p e c i f i c e n t i t y i s given by m = aE{m\+ (l-a)x with a= r E ^ (3.5) E{m} + VAR{m} where, x = recorded accidents However , both these approaches (Hauer's and Higle et al's) use only accident records from d i f f e r e n t sources to model road hazards. Another possible approach i s to modify the hazard measure of an e n t i t y , i . e . , instead of using accident records only, include also other indicators of hazard such as t r a f f i c c o n f l i c t s . 3.4.2 Baysian Method with Information from T r a f f i c C o n f l i c t s It i s believed that information from other sources could be useful to improve the estimation of hazard. I f we treat c o n f l i c t s as an intermediate stage between accident and safety, the hazard index may include t r a f f i c c o n f l i c t s as part of the measure. Here, we use t r a f f i c c o n f l i c t as a measure of accident p o t e n t i a l , rather than as "accident surrogate". For t h i s discussion, an accident surrogate may be described 78 thusly as Glauz et a l (1085) state: "accidents are so rare, s t a t i s t i c a l l y , that one must often wait for years, and for many accidents to happen, before enough data are available to enable r a t i o n a l decisions". Therefore, " i f a surrogate measure such as t r a f f i c c o n f l i c t s could be used, decisions might be made much more quickly". In t h i s study, however, i t i s assumed that accident data are generally a v a i l a b l e . The d i f f i c u l t y of using accident data i s due to random var i a t i o n s inherent i n accident occurrence. To reduce the randomness, t r a f f i c c o n f l i c t s are employed as another source of information to enrich the accident data. This i s a process of information gain implied i n the EB method. On the other hand, the more important reason of in c l u s i o n of t r a f f i c c o n f l i c t s i s that we r e a l l y want to treat i t as an indicato r of accident p o t e n t i a l . Accident p o t e n t i a l i s an unrealized hazard event (e.g., d r i v e r may take evasive action to avoid an i n c i p i e n t crash), while the accident i s a r e a l i z e d hazard event. Nevertheless, the hazardous elements l a r g e l y e x i s t i n the unrealized hazard event. It i s , therefore, proposed that to measure the f u l l degree of hazard unrealized hazard events should be considered i n conjunction with accidents. However, to use t r a f f i c c o n f l i c t information i n the analysis, i t may be necessary to convert t r a f f i c c o n f l i c t into accident expectation in order to conform with the unit of accident records. This can be r e a l i z e d by the use of a c c i d e n t / c o n f l i c t r a t i o s . The general form of Bayes' rule can then be expressed i n the representation of posterior d i s t r i b u t i o n : 79 posterior oc likelihood x prior (3.6) Where, the p r i o r i s the i n i t i a l estimate of the occurrence of hazard events before new information i s c o l l e c t e d . The l i k e l i h o o d d i s t r i b u t i o n i s the representation of information gained from another source. Here, we may specify the s i t e accident records as p r i o r , and Si t e t r a f f i c c o n f l i c t as l i k e l i h o o d . The posterior represents the modified assessment of hazard events. In f a c t , Glauz et a l (1985) examined t h i s p o s s i b i l i t y . A f t e r c a l i b r a t i n g the a c c i d e n t / c o n f l i c t r a t i o s , they estimated the expected number of accidents from t r a f f i c c o n f l i c t rate. Then they developed a so c a l l e d "minimum variance predictions" method, where Var(Am) = j j HvUJ Var{Aa) where, Am = expected accident rate with minimum variance, AQ = expected accident rate based on t r a f f i c c o n f l i c t rate, VariyA\) = variance of A \ , Aa = expected accident rate based on accident data, Var(Aa) = variance of Aa • Actually, t h i s i s a Bayesian posterior estimator (Freund 1984). Var(k\) Var Var 80 However, i t must be pointed out that t h i s expression i s based on the presupposition that the frequency densities of both t r a f f i c c o n f l i c t s and accidents are normally d i s t r i b u t e d . However, t h i s may not be true. In some cases, accidents may follow a Poisson d i s t r i b u t i o n at a s i t e . Furthermore, a more general d i s t r i b u t i o n , Gamma d i s t r i b u t i o n , may be employed to represent s i t e accidents. In Bayesian method, i f we use Gamma as the p r i o r d i s t r i b u t i o n , the conjugate l i k e l i h o o d f o r a Gamma d i s t r i b u t i o n i s a Poisson d i s t r i b u t i o n . Therefore, equation (3.6) can be expressed s p e c i f i c a l l y as [(Gamma)posterior ] oc [(Gamma) prior ] x [(Poison)likelihood ] (3.7) a -bm x ~m me -me a+x m e y ' Apparently, the parameters of posterior d i s t r i b u t i o n are a + x b,=b + l (3.8) Thus, the revised mean and variance are a, +1 a + x + 1 mean = a = —— = 6. 6 + 1 1 (3.9) a. + l a + x + 1 variance = cr = -K (P+iy To i l l u s t r a t e the application of th i s method, we take advantage of the data obtained i n a t r a f f i c c o n f l i c t study (Brown, 1994). T r a f f i c c o n f l i c t data at ten intersections i n Vancouver are av a i l a b l e . Previous 81 f i v e years' accident data are also c o l l e c t e d for the an a l y s i s . These are shown i n Table 3.8. The estimated yearly accident numbers based on t r a f f i c c o n f l i c t information are determined by using mean a c c i d e n t - c o n f l i c t r a t i o , 7t (=2.95*10~ 4), computed i n Brown's study (1994). This estimation w i l l be used as the l i k e l i h o o d i n the Baysian method. Afte r determining parameters a and b for p r i o r (Gamma) d i s t r i b u t i o n based on f i v e year's accident records, we can c a l c u l a t e new parameters f o r po s t e r i o r d i s t r i b u t i o n by using equation (3.8). Then, the revised estimation of accident number could be computed using equation (3.9). Table 3.9 shows th i s r e s u l t and the actual accident record i n the year when the t r a f f i c c o n f l i c t data were c o l l e c t e d . To make a comparison, accident sample mean, i . e . , average accident numbers based on previous f i v e years records are also included. Several observations follow. F i r s t , Baysian method provides c l o s e r estimations to actual accidents for s i x locations. Second, the mean value of these ten points given by Baysian method i s also c l o s e r to actual mean than that from the average of f i v e years' accidents. Notice that to sim p l i f y the i l l u s t r a t i o n , we employed t o t a l number of t r a f f i c c o n f l i c t s at each location. The analysis based on t r a f f i c c o n f l i c t types would give d e t a i l analysis and might provide better r e s u l t s . 82 Intersections 5-years C o n f l i c t s * Accidents (TC) Oak/SW Marine 57 119 Granville/Drake 130 121 Seymour/Drake 58 94 Dunbar/SW Marine 25 93 Nanaimo/Hastings 129 45 Semlin/2nd Ave. 12 6 Gladstone/27th Ave. 14 1 Heather/12th Ave. 84 200 Main/lOth Ave. 52 80 Blenheim/41st Ave. 52 40 * C o n f l i c t s were observed f o r a two-day period. Table 3.8 T r a f f i c C o n f l i c t Data, Vancouber, BC There are several other p o s s i b i l i t i e s i n de f i n i n g p r i o r and l i k e l i h o o d d i s t r i b u t i o n s . One i s using regional accident sample to determine parameters of the Gamma d i s t r i b u t i o n , rather than using s i t e accident records. In t h i s case, we can usually get a larger sample s i z e and probably a better estimation of parameters. Another p o s s i b i l i t y i s using estimations from t r a f f i c c o n f l i c t information across the region to determine parameters of the Gamma d i s t r i b u t i o n . In t h i s case, the t r a f f i c c o n f l i c t i s assumed to be Gamma d i s t r i b u t e d , while s i t e accidents are Poisson d i s t r i b u t e d . Although the v e r i f i c a t i o n of these d i s t r i b u t i o n s were not conducted, the exercise show some i n t e r e s t i n g 83 r e s u l t s . Intersections Sample Mean Baysian Estimation Actual Accidents Oak/SW Marine 11.4 14.8 11 Granville/Drake 26.0 23.6 36 Seymour/Drake 11.6 14.3 14 Dunbar/SW Marine 5.0 7.0 8 Nanaimo/Hastings 25.8 19.0 29 Semlin/2nd Ave. 2.4 2.2 0 Gladstone/27th Ave. 2.8 2.2 0 Heather/12th Ave. 16.8 27.0 19 Main/lOth Ave. 10.4 10.6 15 Blenheim/41st Ave. 10.4 8.9 8 Mean 12.26 12.96 14 • 1 Table 3.9 Baysian Estimation Results (1) Table 3.10 indicates that sample mean has only one best estimation. Three Bayesian estimations have two, three, and four best ones respectively. The estimation based on regional accidents as p r i o r (Gamma) d i s t r i b u t i o n produces the best mean value of the ten in t e r s e c t i o n s , while the c a l c u l a t i o n based on s i t e accidents as p r i o r (Gamma) d i s t r i b u t i o n gives the clos e t estimations to actual accidents. The sample mean method gives the poorest estimation. On the other hand, the error sum of squares (SSE) given i n Table 84 3.10 indicates that using regional TC as p r i o r (Gamma) d i s t r i b u t i o n has the smallest SSE and sample mean method provides the second smallest SSE; while using regional accidents as p r i o r (Gamma) d i s t r i b u t i o n gives the largest SSE. The r e s u l t s show a promising method to estimate hazards, though the sample is, not large enough to provide further d e t a i l e d s t a t i s t i c a l a n a l y s i s . However, since t r a f f i c c o n f l i c t s are r e l a t i v e l y frequent events at road accesses, p a r t i c u l a r l y at in t e r s e c t i o n s as defined here, i t seems reasonable to assume that further study, with s u f f i c i e n t c o n f l i c t data, may lead to a new method of measuring and de f i n i n g road safety. 85 Sample Regional Regional Site Actual Intersections Mean Accident TC Accidents Accidents as Gamma as Gamma as Gamma+ Oak/SW Marine 11.4* 19.1 12.4 14.8 11 Granville/Drake 26.0 19.4 26.4* 23.6 36 Seymour/Drake 11.6 15.6 12.6 14.3* 14 Dunbar/SW Marine 5.0 15.5 6.3 7.0* 8 Nanaimo/Hastings 25.8 8.8 26.5* 19.0 29 Seraljn/2nd Ave. 2.4 3.4 3.8 2.2 0 Gladstone/27th Ave. 2.8 2.7 . 4.2 2.2 0 Heather/12th Ave. 16.8 30.3 17.6* 27.0 19 Main/lOth Ave. 10.4 13.7* 11.5 10.6 15 Blenheim/41st Ave. 10.4 8.1* 11.5 8.9 8 Mean 12,26 13.68* 13.24 12.96 14 SSE* 173.5 956.4 164.3 364.2 * The value i s the close s t to the actual accidents i n each row. + This column i s the same as the "Baysian Estimation" column i n Table 3.9. * SSE = ^ ( j ^ - Y{) , where, Yx represents actual accidents and Y represents the estimated values. Table 3.10 Baysian Estimation Results (2) 86 Chapter 4 The E f f e c t of Access on T r a f f i c Operations In t h i s chapter a model of t r a v e l speed i s obtained on the basis of access and other geometric information for two-lane highways. 4.1 Introduction Some previous e f f o r t s have been made to determine the e f f e c t s of access on t r a f f i c operations. However, most studies e i t h e r did not intend to e s t a b l i s h quantitative r e l a t i o n s h i p s , or were unsuccessful i n e s t a b l i s h i n g the r e l a t i o n s h i p . One p a r t i a l exception i s the new Highway Capacity Manual (HCM), 1992, which does deal with the a c c e s s / t r a f f i c operations r e l a t i o n s h i p on multilane highways but not for two lane highways. This chapter examines the r e l a t i o n s h i p between access and t r a f f i c operations i n road sections on two-lane highways by means of the photolog data base. Since speed i s used as a conditionary measure of l e v e l of service for two-lane highways i n capacity c a l c u l a t i o n s , and since i t i s r e l a t i v e l y easier to obtain speed information than percent time delay along a road section, t r a v e l speed w i l l be used i n t h i s research as a univariate proxy measure for t r a f f i c operations performance. 1 In t h i s research highway speed i s e m p i r i c a l l y tested against categories of access and several other t r a f f i c and geometric road c h a r a c t e r i s t i c s . 1 The object of this study is a highway section, not a highway network. For a highway section, access points are an important influence factor. The number of access points in one section may have some effects on the rest of a network, but that is another aspect of the impact, not the focus of this study. 87 Glennon et a l (1975) developed techniques to counteract vehicle delay due to access points, and although i t was recognized that f o r the mainstream through t r a f f i c , delay i s caused by turning vehicles, no quantitative r e l a t i o n s h i p was established. Research conducted by R e i l l y et a l (1989) f o r the Revised HCM (1992) found that the number of access points on multi-lane r u r a l and suburban highways has an important influence on free-flow speed, i n that f o r every 10 access points per mile, t r a v e l speed w i l l be reduced by 2.5 mph. The access point i s defined as a Composite variable made up of the separate influences of d i f f e r e n t c h a r a c t e r i s t i c s of access. However, the HCM (1992) has not yet been updated to include the e f f e c t of access points on t r a f f i c operations f o r two lane highways. One study conducted i n the UK (Brocklebank, 1992) on two-lane r u r a l highways found that access variables are generally d i f f i c u l t to f i t to models of t r a f f i c operations. A composite variable used which synthesized i n t e r s e c t i o n s , lay-bys, and accesses was not s i g n i f i c a n t i n the UK study. Non-r e s i d e n t i a l accesses and lay-bys as separate independent access variables were found either to be not s i g n i f i c a n t or only occasionally s i g n i f i c a n t . For example, the i n t e r s e c t i o n variable was rather unstable with a c o e f f i c i e n t value -1.8 (km/)i). One possible explanation f o r the res u l t i s that there are so many other variables considered i n the model (a t o t a l of 31 independent v a r i a b l e s ) , the e f f e c t of access variables may be subsumed. In any case the results do not f u l l y conform to our a p r i o r i expectations, and further study of two lane highways i s needed. In addition, two-lane highways i n urban and suburban areas should also be investigated because the higher t r a f f i c and more roadside development on these roads w i l l undoubtedly i n t e n s i f y the influence of access points. 88 T r a d i t i o n a l l y , f i e l d data are used to quantify t r a f f i c flow r e l a t i o n s h i p s . For example, R e i l l y et a l (1989) c o l l e c t e d f i e l d data on multilane, mainly divided, highways i n several states of the USA. Video cameras were used to c o l l e c t t r a f f i c operation data (speed and flow), with the information incorporated into a comprehensive data base including geometric c h a r a c t e r i s t i c s , roadside features, and access points. In a l l , 45 s i t e s were selected, and hundreds of observation points were coded i n the data base. Regression analyses were then performed to e s t a b l i s h t r a f f i c flow r e l a t i o n s h i p s . In Brocklebank's study (1992) a license plate matching technique was employed to record t r a f f i c information. The data base i s comprehensive, i n c l u d i n g 31 flow and geometric variables with access variables among them. Because the cost of such data c o l l e c t i o n i s very high, i t impedes comprehensive studies of access and other constraint factors i n t r a f f i c operations e f f i c i e n c y ; and because of cost i s not a f e a s i b l e a l t e r n a t i v e f or t h i s present study. To reduce data c o l l e c t i o n costs, simulation models are sometimes employed to estimate t r a f f i c impacts. Venigalla et a l (1992) used the TRAF-NETSIM simulation program to compare operational e f f e c t s of non traversable medians and two-way l e f t - t u r n lanes. No r e a l data c o l l e c t i o n was conducted. A number of scenarios with varying conditions were studied, with appropriate assumptions made to s i m p l i f y simulation analysis. Driveway density was also considered i n the study, but the main focus was on comparing a l t e r n a t i v e median designs for multilane highways. Driveway density was treated as a secondary, category scaled v a r i a b l e . The Colorado Demonstration Project (1985) u t i l i z e d TRANSYT 7F 89 to evaluate the e f f e c t of access points on congestion f o r a fi v e - m i l e road section, but again, the access v a r i a b l e was measured by a category scale: c o n t r o l l e d or uncontrolled access. One disadvantage of using a simulation model f o r t h i s problem i s that there are some assumptions that may not be v e r i f i e d . For instance, i t i s not c l e a r whether delay (speed reduction) i s s o l e l y caused by turning movements ( t h i s i s generally adopted i n simulation programs) or by the turning movement and the mere existence of access points. Revision of the HCM (1992) contends that access points may reduce t r a v e l speed because d r i v e r s adjust t h e i r t r a v e l speed when they see the existence of access points (even with no turning movements involved). Apparently, t h i s psychological response cannot be simulated without more empirical data support. The simulation program r e p l i c a t e s the mechanism of speed reduction caused by turning movements by using kinematic equations. Therefore, some actual data i s needed to reveal any applicable c o r r e l a t i o n between access and speed. To do t h i s and overcome the problem of the high cost of data c o l l e c t i o n , a search was made to see i f any e x i s t i n g data sources could be used, and indicated that only small amounts of t r a v e l speed data for two-lane highways existed i n the province of B r i t i s h Columbia. Nevertheless, i t was found that i n the photolog data source much relevant a n c i l l a r y information was ava i l a b l e , and consequently i t was decided to use the photolog as a data base with further analysis of the speed v a r i a b l e to make i t useful as a proxy f o r operations performance. 90 4.2 The Photolog Data Besides geometric information (such as grade and ho r i z o n t a l curvature), the instantaneous speed of the tes t v e h i c l e and the cumulative time are also recorded i n the photolog data. While the photolog provides only the test vehicle's speed and t r a v e l time, i f the tr a v e l distance i s r e l a t i v e l y great, or, i n other words, i f successive observation points are r e l a t i v e l y d i s t a n t , the test vehicle's t r a v e l speed can be used as a surrogate f o r average t r a v e l speed i n the access and speed r e l a t i o n s h i p . Two arguments are presented here to support t h i s contention. F i r s t , we may treat the method as a type of moving v e h i c l e technique, which i s widely employed to c o l l e c t t r a v e l time data. In t h i s process, the subject moving vehi c l e i s placed i n the t r a f f i c stream to simulate an average veh i c l e i n the t r a f f i c stream. (This i s what our photolog test vehicle d r i v e r was asked to do.) Therefore, the t r a v e l time recorded on a road section by the photolog vehicle i s used as the mean tr a v e l time for a l l t r a f f i c on the section. On the basis of t h i s mean t r a v e l time, the t r a v e l speed can be computed by section length. Since section length i s well defined, i f the t r a v e l time obtained from the test v e h i c l e can be accepted as the mean t r a v e l time as many studies do (see, e.g., Nutakor, 1992), i t follows that the derived t r a v e l speed can be used as the average t r a v e l speed. In the standard moving v e h i c l e method, the test vehicle i s required to run several times i n a study section, while the data base for th i s study gives only one run for each section. To allow f o r t h i s we recognize that the sample i n th i s study does not consist of a number of runs i n a road section, but rather consists of hundreds of sections. A sing l e observation on one section may cause sampling error, one of the major sources of erro r i n most regression problems^. Sampling error arises from random v a r i a t i o n of the observations around t h e i r expected value. However, sampling err o r i s inherent to the regression model and i s normally represented as part of the random error term . Furthermore, from a t r a f f i c engineering point of view, the variance of t r a v e l speed i n a homogeneous section i s usua l l y much smaller than that by section. Which means that the range of random v a r i a t i o n of speed observations i s small and ensures the approximate representation of the magnitude of expected speed values. The second argument for the use of section t r a v e l speed as a proxy fo r average speed pertains to the fac t that the focus of t h i s study i s on the general r e l a t i v e impact of access density on the average t r a v e l speed. What we are r e a l l y interested i n i s the v a r i a t i o n of t r a v e l speed rather than the absolute value of average t r a v e l speed on a section. It i s assumed that the percentage speed change i s nearly the same for most experienced drivers when they confront a hazard or other reason f o r reducing speed. Which means that experienced driv e r s reduce speed proportional to t h e i r o r i g i n a l t r a v e l speed. Although t h i s i s not v e r i f i e d i n t h i s study, i t seems to be a reasonable approximation based on engineering judgment. Therefore, the speed change of the test v e h i c l e would approximate the general impact of access on t r a f f i c operations (speed). 3 2 Others are specification error and measurement error, see e.g., Hawkins and Weber, 1980. 3 The use of the test vehicle data instead of traffic survey data is due mainly to practical limitations of the research. The very significant cost of collecting actual travel speed data on the one hand and the increasingly pressing need to understand the problem on the other has prompted this approach to the research . 92 The study data consist of 255 sections ( s i t e s ) on 14 two-lane routes photologged i n 1990. The t o t a l lengths of selected road sections are 625 kilometres, with an average length of 2.43 kilometres. Seven types of road access were o r i g i n a l l y i d e n t i f i e d : (1) four-way unsignalized i n t e r s e c t i o n ; (2) three-way unsignalized i n t e r s e c t i o n ; (3) commercial driveway; (4) i n d u s t r i a l driveway; (5) r e s i d e n t i a l driveway; (6) a g r i c u l t u r a l driveway; and (7) roadside p u l l o u t . However, preliminary model t e s t i n g indicated that the above c l a s s i f i c a t i o n resulted i n some i n s i g n i f i c a n t access v a r i a b l e s . Three composite variables f o r access were then created as: (1) unsignalized i n t e r s e c t i o n ; (2) business access (commercial and i n d u s t r i a l driveways); (3) and pri v a t e driveway ( r e s i d e n t i a l and a g r i c u l t u r a l driveways). Roadside pul l o u t was retained i n i t s o r i g i n a l form. This c l a s s i f i c a t i o n follows a fu n c t i o n a l d e f i n i t i o n of access and presumably resembles access volume categories. (Note that the term " i n t e r s e c t i o n " i s defined here as the general area where two or more public roads j o i n or cross. This d e f i n i t i o n w i l l d i s t i n g u i s h the i n t e r s e c t i o n from other access types, such as pri v a t e access, where one private driveway i s connected to one p u b l i c road.) T r a f f i c volume i s a key variable i n any study o f t r a f f i c flow, but instantaneous t r a f f i c flow rates were not av a i l a b l e i n the photolog data. Therefore average annual d a i l y t r a f f i c (AADT), obtained from published t r a f f i c count s t a t i o n reports, was used to study the speed-flow r e l a t i o n s h i p . Since i t i s convenient to use hourly volume instead of AADT, the AADT was converted to hourly flow rate based on the date 93 and time of the photolog data. The peak hour adjustment factor was f i r s t computed. Since the actual time when the test vehicle was t r a v e l i n g was not always i n the peak period, further adjustment was made to estimate the flow rate at the time of the test v e h i c l e run over the section. Access t r a f f i c volumes, or turning t r a f f i c , of each access point were not a v a i l a b l e i n any case, thus only major road t r a f f i c was included. The cost of c o l l e c t i n g these access volumes could be much higher than that of main road t r a f f i c , considering the number of access points i n a road section and number of sections i n the study. However, i t i s believed that the grouping of accesses by functional c h a r a c t e r i s t i c s as outlined here helps a l l e v i a t e the necessity for access volumes since each access function can by and large be associated with crude v a r i a t i o n s i n volume ranges. On the other hand, the p r i o r i t y movement p r i v i l e g e of main road t r a f f i c reduces the e f f e c t of c o n f l i c t movement from access t r a f f i c . T r a f f i c controls such as stop signs are u t i l i z e d extensively on a r t e r i a l roads to ensure l a r g e l y uninterrupted movement of main t r a f f i c . It does not matter how many vehicles i n the queue at an access point are waiting for an opportunity to cross or merge into the main t r a f f i c . Only the f i r s t v e h i c l e has the p o t e n t i a l to i n t e r f e r e with main t r a f f i c flow operations. The diff e r e n c e between higher access t r a f f i c and lower access t r a f f i c volume at a minor road i s that the former case usually means more delay for the minor road access t r a f f i c . Although turning lanes might not be provided for main road t r a f f i c at a l l access points, i t i s , however, presumed that they are accommodated at access points where turning t r a f f i c from the main road constitutes a s i g n i f i c a n t portion of through t r a f f i c . This should be true i n a well-developed region. Since this study i s not a microscopic operational analysis of delay at i n d i v i d u a l access point, 1 but rather a macroscopic 9 4 analysis of the e f f e c t on t r a f f i c i n road sections, some degree of aggregation of information would seem permissible and necessary, ( i . e . , the use of access types to represent the magnitude by classes of access t r a f f i c volume). Besides, as indicated e a r l i e r , i t i s not c l e a r yet i f the speed reduction i s a c t u a l l y caused by access t r a f f i c or the mere presence of access points, and the r e s u l t of psychological response. This hypothesis, as well as the use of access types to account for access t r a f f i c volume, can reasonably be j u s t i f i e d i f the c o r r e l a t i o n between tr a v e l speed and access density which i s c l a s s i f i e d by types can be s t a t i s t i c a l l y established. Travel speed i s defined as the section length divided by over the road t r a v e l time. The instantaneous speed recorded i n the photolog data was not s u i t a b l e f or c a l c u l a t i n g the t r a v e l time, since i t measures the speed of the test vehicle at each i n d i v i d u a l point i n the section. It does not give the actual t r a v e l time i n a road section. However, the r e a l time of the survey was registered, and the section length divided by the time i n t e r v a l used for t r a v e l i n g through that section gives the t r a v e l speed of the test v e h i c l e i n that section. It i s presumed that the r e l a t i o n s h i p between the test vehicle's t r a v e l speed under d i f f e r e n t t r a f f i c conditions and the corresponding flow rate w i l l resemble the r e l a t i o n s h i p between average t r a v e l speed (space mean speed) and flow. Figure 4.1 shows the test vehicle's t r a v e l speed and flow r e l a t i o n s h i p . It appears to have the same shape and s i m i l a r magnitudes as a standard, known, speed-flow curve. This r e l a t i o n s h i p i s explored i n d e t a i l i n the next section. 95 100 0 J I I 1_ _1 I L_ 0 500 1000 1500 Traffic Volume, vph 2000 2500 Figure 4.1 Travel Speed-Flow Rate Relationship Several road c h a r a c t e r i s t i c factors were also a v a i l a b l e from the photolog. Some of them were included i n t h i s a n alysis, such as grade, traverse slope, horizontal curvature, and the d i r e c t i o n of curvature. The l a t t e r measure indicates whether the curvature i s l e f t qr r i g h t extended, and therefore the frequency of change i n d i r e c t i o n per kilometre within the study segment can be c a l c u l a t e d . The i n t e n t i o n here i s to determine the amount of a d r i v e r ' s attention needed to traverse a given segment. Grades were taken as absolute values only, and no d i r e c t i o n of grade was considered, l a r g e l y because Troutbeck (1976) indicated that car speeds depend l a r g e l y on the magnitude of the grade, rather than the d i r e c t i o n . Horizontal curvature i s defined as the c e n t r a l angle of a 100 metre arc. Average grade and h o r i z o n t a l curvature were obtained for each road section. These two variables measure average v e r t i c a l and horizontal displacement, r e s p e c t i v e l y , which are used as road alignment variables i n other studies (Duncan, 1974; Brewer et a l , 1980; McLean, 1989; and Brocklebank, 1992) to study t r a v e l speed and geometry r e l a t i o n s h i p s . Two other factors not included i n t h i s study are lane and shoulder widths simply because this information was not a v a i l a b l e i n d i g i t i z e d form i n the photolog. However, the review of the study sections indicated that the lane and shoulder widths of the selected s i t e s are l a r g e l y fixed (lane width = 3.6m, and shoulder width = 2.0m). Therefore, i t i s assumed that any influence on speed i s independent of access points, and no e f f e c t on possible influences of access. 97 A l l v a r i a b l e s employed i n the analysis are defined i n Table 4.1. Variable Definition Range Mean SPD Travel speed (km/h) 11.02-96.00 74.02 UNS Unsignalized intersection density (no./km) 0 - 8 1.01 B S N Business access density (no./km) 0 -20 1.00 P W Y Private driveway density (no./km) 0-35 3.80 RSP Roadside Pullout (no./km) 0-1.5 0.15 VOL Two-way flow rate (veh/h) 51-2,500 664 G R D Grade (percent) 0.19-6.3 1.53 TS Traverse slope (percent) 0.87 - 4.93 2.87 F C D Frequency of change in the direction of 0.83 - 25 10.38 curvature (no. of changes/km) H C Horizontal curvature (degree) 0.3 - 14.9 3.96 Table 4.1 Variable D e f i n i t i o n 4.3 Data Analysis To v e r i f y a speed-flow re l a t i o n s h i p using test vehicle speed, a 2 l i n e a r regression analysis was conducted with a r e s u l t i n g R of 0.23, and F test of 75.44, a s i g n i f i c a n t c o r r e l a t i o n . To attempt to increase 2 the R the data were disaggregated into two horizontal curvature (HC) categories: HC less than 5 degrees, and HC greater than and equal to 5 2 degrees. In the f i r s t group, the R was 0.38 and F te s t was 113.64, 2 which indicates a notable improvement. While i n the second group, R i s 0.07 and F test i s 0.024. The plots of observation points of these two 98 groups are shown i n Figures 4.2 and 4.3. C l e a r l y most road sections with higher t r a f f i c volume are designed with lower horizontal curvature as shown i n Figure 4.2 with a large number of sections showing both high speed and higher volumes with the h o r i z o n t a l curvature less than 5°. On the other hand, road sections with greater horizontal curvature usually accommodate lower t r a f f i c volume as shown i n Figure 4.3. However, there i s no apparent speed reduction f o r the few sections i n the higher horizontal curvature group. S p e c i f i c a l l y , i t seems that h o r i z o n t a l curvature greater than 5° does not greatly a f f e c t speed. This unexpected r e s u l t may be explained by the fact that at low t r a f f i c volumes, driv e r s more l i k e l y t r a v e l at t h e i r own desirable speed. Another possible reason i s that the number of observation points, 68, may be r e l a t i v e l y small. Another test was made based on speed l i m i t disaggregation. Speed l i m i t s were categorized i n two groups: speed l i m i t s less than and equal to 70 km/h, and speed l i m i t s greater than 70 km/h. The pl o t s of these 2 two groups are shown i n Figures 4.4 and 4.5. In the f i r s t group, R i s 0.34 and F test i s 0.66, so i t i s i n s i g n i f i c a n t . The second group 2 provides a better r e s u l t : R i s 0.34 and F test i s 102.81. It can be seen that i n the low speed l i m i t group t r a v e l speed varies over a wide range at low t r a f f i c volume (the l e f t portion of Figure 4.4), which ind i c a t e s , as expected, that dr i v e r s more or less ignore the speed l i m i t , and whenever the volume i s low he or she may simply drive f a s t e r . However, at the higher speed l i m i t (the second group) the actual t r a v e l speed i s more co n s i s t e n t l y higher except when the t r a f f i c volume i s very high. Again, perhaps the poor c o r r e l a t i o n i n the low speed l i m i t group can also be a t t r i b u t e d to r e l a t i v e l y few observations, 53, i n that group. 9 9 100 80 60 40 • . .... # v • • • • * • • w • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 20 500 1000 1500 2000 2500 Traffic Volume, vph Figure 4.2 Speed-Volume on Road Sections with HC < 5 degree 100 80 f . • • • 60 ' 40 20 0 0 500 1000 1500 2000 2500 Traffic Volume, vph Figure 4.3 Speed-Volume on Road Sections with HC >= 5 degree 100 80 • • • 60 40 20 0 500 1000 1500 Traffic Volume, vph 2000 2500 Figure 4.4 Speed-Volume on Road Sections with Speed Limit <= 70 km/h 100 80 60 • 0 » % 9 • m mm • mm m m m m 40 20 I 500 1000 1500 Traffic Volume, vph 2000 2500 Figure 4.5 Speed-Volume on Road Sections with Speed Limit >= 80 km/h Since the disaggregation of data does not markedly improve the resu l t s f o r a l l groups, i t was therefore decided to use the aggregated data to examine the rel a t i o n s h i p between access and speed. It i s believed that the s t a t i s t i c a l test f o r aggregated data has shown that the speed and volume rel a t i o n s h i p i s reasonably l i n e a r , and furthermore, the use of test vehicle speed i s an appropriate i n d i r e c t measure of average t r a v e l speed. Consequently, a backward stepwise multiple l i n e a r regression analysis was c a r r i e d out. The res u l t s of t h i s i n i t i a l model are shown i n Table 4.2. It can be seen that most independent variables except roadside pullout and traverse slope were s a t i s f a c t o r i l y entered into the model at s i g n i f i c a n t l e v e l 0.01. A l l variables appear with the expected signs. The F-test i s also s a t i s f a c t o r y at 0.01 l e v e l of s i g n i f i c a n c e , i n d i c a t i n g that a l l independent v a r i a b l e s ' c o e f f i c i e n t s are c o l l e c t i v e l y s i g n i f i c a n t l y d i f f e r e n t from zero. Note that the r a t i o of the c o e f f i c i e n t s between access variables ( i n the order of i n t e r s e c t i o n , business access, and private driveway) i s approximately 1:2:4. The i n s i g n i f i c a n t e f f e c t of roadside pullout on speed agrees with the e f f e c t of lay-bys i n Brocklebank's study (1992). This might be due to infrequent use of roadside pullouts i n most cases; and secondly, the poor design standards of the pullouts make i t not e a s i l y perceived by drivers so that t h e i r behaviors are not influenced by the mere existence of these pull o u t s , without c o n f l i c t i n g t r a f f i c . 104 Independent Variable Coefficient Standard Error t-Test Level of Significance Constant 108.14 2.51 43.12 0.000 UNS -1.69 0.48 -3.53 0.001 B S N -0.89 0.29 -3.07 0.002 P W Y -0.39 0.15 -2.69 0.008 V O L -0.0092 0.0013 -7.25 0.000 G R D -1.71 0.56 -3.06 0.002 F C D -1.20 0.20 -5.92 0.000 H C -2.19 0.22 -9.99 0.000 R2 0.59 SE 9.63 F-Test 52.71 (P) 0.000 Table 4.2 I n i t i a l Model f or SPD The c o r r e l a t i o n analyses showed no s i g n i f i c a n t m u l t i c o l l i n e a r i t y among the independent variables, thus producing r e l a t i v e l y unbiased estimates. Consequently, the e f f e c t Of each va r i a b l e on the accident measure provide some in s i g h t . See Table 4.3. However, an examination of residuals i n the regression model indicated r e l a t i v e l y poorer results i n the lower speed ranges, as Figure 4.6 shows. To solve this problem, a quadratic transformation was made for dependent variable SPD. Table 4.4 and Figure 4.7 show the new 105 Const. UNS B S N PWY V O L G R D F C D H C J Const. 1.0000 .0279 .0811 .1110 -.0435 -.1680 -.8053 -.6708 UNS .0279 1.0000 -.4200 .0262 -.2078 -.0163 -.0784 -.0985 B S N .0811 -.4200 1.0000 -.0310 .0436 .0545 -.1322 -.0937 P W Y .1110 .0262 -.0310 1.0000 -.2977 .0206 -.2595 -.0607 V O L -.0435 -.2078 .0436 -.2977 1.0000 -.0610 -.2829 .1723 G R D -.1680 -.0163 .0545 .0206 -.0610 1.0000 -.0613 -.3150 F C D -.8053 -.0784 -.1322 -.2595 -.2829 -.0613 1.0000 .4945 HC -.6708 -.0985 -.0937 -.0607 .1723 -.3150 .4945 1.0000 Table 4.3 Correlation Matrix f o r C o e f f i c i e n t Estimates Independent Variable Coefficient Standard Error t-Test Level of Significance Constant 10038.59 312.58 32.12 0.000 UNS -232.06 59.72 -3.89 0.000 B S N • -117.62 36.37 -3.23 0.001 P W Y -56.64 18.18 -3.12 0.002 V O L -1.28 0.16 -8.12 0.000 G R D -147.12 69.76 -2.11 0.036 F C D -145.57 25.28 -5.76 0.000 H C -293.42 27.42 -10.70 0.000 R2 0.61 SE 1200.82 F-Test 58.94 (P) 0.000 Table 4.4 Modified Model for Transformed Variable SPD 106 30 40 50 60 70 80 90 100 Predicted Values (Speed, km/h) Figure 4.6 Residual Analysis of the Initial Model o 00 -a a -2 -4 * . s \ v ; • % « r . * % » , , , • «*.* *Jt 4 6 Predicted Values, x 1,000 (speedA2) Figure 4.7 Residual Analysis of the Modified Model 8 10 regression r e s u l t s and residuals plot respectively. The quadratic transformation appears to give better (more consistent) r e s u l t s across the speed range of the study using the o r i g i n a l set of independent v a r i a b l e s . Furthermore, the r a t i o of the c o e f f i c i e n t s between access variables ( i n the order of i n t e r s e c t i o n , business access, and private driveway) i s also approximately 1:2:4. Thus the degree of inconsistency of the i n i t i a l model would appear to be r e l a t i v e l y minor, and the comparative weight of each independent va r i a b l e has not been changed gre a t l y by the transformation. Only the influence of grade i s measurably weaker i n the transformed model as shown by the ' t ' s t a t i s t i c i n Table 4.4. In p r a c t i c e , therefore, unsignalized i n t e r s e c t i o n s , business access, and private driveways w i l l reduce t r a v e l speed by approximately 1.6, 0.8, and 0.4 km/h, respectively. Compared with previous studies, the values of these results are larger reductions than those i n the multilane highway s i t u a t i o n as reported by R e i l l e y et a l (1989), which showed a speed reduction of 0.25 mph for each access point per mile ( i . e . , 0.65 km/h for each access point per km), as expected. This may be explained as follows. F i r s t , two-lane highways are not divided, and since l e f t - t u r n movements ex i s t at most access points, these have a much greater influence on t r a f f i c operations than right turn movements i n the absence of a l e f t turn lane. Second, multilane highways provide more opportunities for following vehicles to pass the vehicles slowing down to make turning movements or to avoid r i g h t turning v e h i c l e s . Third, i n terms of behavioral response to the existence of access points, vehicles on two-lane highways may be affected more than vehicles on multilane highways because the vehicles i n the center lanes may be less influenced 109 by access points than those vehicles in the curb lanes. Compared with Brocklebank's r e s u l t (1992), the i n t e r s e c t i o n e f f e c t i n our i n i t i a l model (1.6 km/h) i s marginally smaller than his (1.8 km/h). However, that e f f e c t i n our modified model w i l l be enhanced with the increases of i n t e r s e c t i o n density and flow rate. This w i l l be discussed further i n the next section. As f o r geometric factors (grade, horizontal curvature, and frequency of change i n the d i r e c t i o n of horizontal curvature), a l l indicated inverse proportional r e l a t i o n s h i p s with speed, i . e . , the increase of v a r i a b l e value (absolute value) w i l l reduce t r a v e l speed. This generally conforms .to Duncan's (1974), Brewer et a l ' s (1980), and Brocklebank's (1992) re s u l t s f or grade and curvature, although t h e i r d e f i n i t i o n s of these two variables are quite d i f f e r e n t . The e f f e c t of the frequency of change i n the d i r e c t i o n of horizontal curvature has not been found i n previous studies, perhaps due to lack of information. However, the negative e f f e c t of t h i s v ariable on t r a v e l speed seems to be i n t u i t i v e l y c orrect, as larger values of this v ariable r e f l e c t more complex d r i v i n g circumstances. 4.4 Results and Discussion The s p e c i f i c influence of each access type on t r a v e l speed was normalized by using a modified model. The normalization was done by varying access variables while holding a l l other independent variables at t h e i r mean values. A secondary independent variable (flow rate) was also employed. For example, to study the e f f e c t of i n t e r s e c t i o n density 110 on t r a v e l speed, several flow rate l e v e l s were considered while a l l other independent variables are held at t h e i r mean values. Figure 4.8 through Figure 4.10 show the variations of t r a v e l speed with changes i n i n t e r s e c t i o n density, business access density, and private driveway density from the modified model. A l l figures show a decreasing trend l i n e of t r a v e l speed with the increase of access densities as determined by the negative c o e f f i c i e n t s i n the model. Note that the estimated r e l a t i o n s h i p curves are for s p e c i f i c degrees of highway curvature. This points to the non-linear c h a r a c t e r i s t i c s i n the access-speed r e l a t i o n s h i p , i n that the curves are wider apart at higher access density than at lower access density, which indicates that the e f f e c t of access density on speed i s i n t e n s i f i e d by the increase of flow rate. Another representation of the r e l a t i o n s h i p i s shown i n Table 4.5, where speed reduction factors were calculated f or each access type. Again, a l l other variables are kept at t h e i r mean values while only access density and flows are changing. For each p a i r of access and flow combination, the speed reduction factor i s calculated as the r a t i o of the current speed and the base speed. The speed at access density equivalent to zero and flow rate equivalent to 500 vph i s chosen as the base value for each access type. The purpose of t h i s representation of the r e l a t i o n s h i p i s for the convenience of the a p p l i c a t i o n of the r e s u l t s . An example i s presented below to show how Table 4.5 could be used i n p r a c t i c e . I l l Figure 4.8 Estimated Relationship between Unsignalized Intersection and Travel Speed by Traffic Volume 80 Flow = 500 veh/h Flow =1,000 veh/h Flow =1,500 veh/h Flaw=2,Q0O.veMi Flow =2,500 veh/h 0 10 15 Business Access Density, No./km 25 Figure 4.9 Estimated Relationship between Business Access and Travel Speed by Traffic Volume 0 10 1 5 D e n s i t y . - N ^ 0 private Driveway Figure 4.1" * Private Driveway / and ^ S p e e d b y T ^ c V o ^ 1 Accesses Density (no./km) Flow Rate (veh/h) 500 1,000 1,500 2,000 2,500 Unsig. Intersection 0 1.00 0.95 0.89 0.83 0.76 2 0.96 0.91 0.85 0.78 0.71 4 0.92 0.86 0.80 0.73 0.66 6 0.88 0.82 0.75 0.68 0.60 8 0.84 0.77 0.70 0.62 0.53 Business Access 0 1.00 0.95 0.89 0.83 0.76 5 0.95 0.89 0.83 0.77 0.69 10 0.90 0.84 0.77 0.70 0.62 15 0.84 0.78 0.70 0.63 0.53 20 0.78 0.71 0.63 0.54 0.43 Private Driveway 0 1.00 0.95 0.89 0.83 0.76 7 0.97 0.91 0.85 0.79 0.72 14 0.93 0.88 0.81 0.75 0.67 21 0.90 0.84 0.77 0.70 0.62 28 0.86 0.80 0.73 0.66 0.57 35 0.82 0.76 0.69 0.60 0.51 * The speed reduction factor i s calculated as the r a t i o between each speed and the base speed. The speed at no access points and 500 vph flow rate i s chosen as the base speed. Table 4.5 Estimated Speed Reduction Factors* 115 Assume we know a two-lane road section with the conditions as follows: l e v e l t e r r a i n , no passing lanes, 1000 vph, one unsignalized i n t e r s e c t i o n per km, f i v e business accesses per km, and seven p r i v a t e driveways per km. To determine the e f f e c t of these accesses on average speed, f i r s t , f i n d the average speed f o r the base condition: no accesses, 500 vph, which can be obtained from HCM (1985) as approximately 91.7 km/h. Then, get an adjustment f a c t o r from Table 4.5: 0.93*0.89*0.91 = 0.75, which means the speed w i l l be reduced by 26 percent. The estimated average speed i s therefore equal to: 91.7*0.75 = 68.8 km/h. F i n a l l y , the estimated speed-flow r e l a t i o n s h i p by access density i s shown i n Figure 4.11, using flow as the primary v a r i a b l e and access as the secondary v a r i a b l e . Access density l e v e l s were defined by a r b i t r a r i l y assigning densities of unsignalized i n t e r s e c t i o n s , business accesses, and private driveways i n a group. The figure c l e a r l y indicates that, with higher flow rate and access density, trav e l speed drops r a p i d l y . 4.5 Summary Conclusion To estimate the e f f e c t of access on t r a v e l speed on two-lane highways, the photolog data source was employed. It i s shown that the test vehicle's t r a v e l speed can be used as a reasonable proxy f o r average t r a v e l speed i f the test vehicle's speeds are reasonably high. This was i l l u s t r a t e d by p l o t t i n g the speed-flow r e l a t i o n s h i p for the test v e h i c l e . It i s shown that the shape and the magnitude of the speed-116 0 1 1 : : — 1 0 500 1000 1500 2000 2500 3000 Traffic Volume, vph Figure 4.11 Estimated Relationship between Speed and Traffic Volume by Access Points flow curve for the test vehicle resembles the known general speed-flow r e l a t i o n s h i p well f o r two-lane highways (Figure 4.1). Access points were c l a s s i f i e d into four types: unsignalized i n t e r s e c t i o n , business access, private driveway, and roadside p u l l o u t . Except the l a s t one, a l l others are s i g n i f i c a n t i n the model. The regression analysis indicated that non-linear relationships may e x i s t between these accesses and t r a v e l speed. The higher the access density, the more speed reduction w i l l be induced. As i n t u i t i v e l y expected, the magnitude of the influence on speed i s i n the order of i n t e r s e c t i o n , business access, and private driveway, with a r a t i o of approximately 1:2:4. If a l l other independent variables remain at mean values, one unsignalized i n t e r s e c t i o n w i l l approximately reduce t r a v e l speed by 1.6 km/h, one business access w i l l reduce the speed by 0.8 km/h, and one private driveway w i l l reduce the speed by 0.4 km/h. To more p r e c i s e l y describe the e f f e c t of access on speed reduction, Table 4.5 should be used, with the following caveat: i t i s presumed that the influence of access on speed i s due to the combined e f f e c t of turning movements at access points and the mere existence of the access points. However, no attempt was made to d i f f e r e n t i a t e between these two factors i n the study. One possible use of the findings of th i s chapter i s in the estimation of road user cost v a r i a t i o n with access density. With the change of average t r a v e l speed, as a result of change i n access density, delay costs could be estimated. By including other relevant costs, such as accident costs and out-of-way trave l costs for access t r a f f i c , an 118 optimal access density might be achieved to minimize the t o t a l road user costs i n an access control program. 119 Chapter 5 A Framework for Planning Optimal Number of Access Points The study of the optimal number of access points i s necessary for highway planning and design i n terms of access density, and for highway control i n terms of access permit regulations. The aim of t h i s chapter i s to e s t a b l i s h a procedure for defining the optimal number of access points with considerations of not only t r a f f i c safety but also t r a f f i c delay. The framework developed here can not be a l l i n c l u s i v e because of data l i m i t a t i o n s ; p a r t i c u l a r l y those data related to side road delay and non-user environmental costs and benefits. The procedure i s s t r a t e g i c , di r e c t e d to questions related to highway planning, and i s concerned with a r e l a t i v e l y wide range of road sections instead of the minimum geometric distance between two adjacent access points. 5.1 Model Construction 5.1.1 General Discussion Before presenting decision variables, independent variables, the objective function and constraints, the basic form of model objective i s discussed. This discussion w i l l provide a base and a scope to elaborate on v a r i a b l e s , objective function, and constraints of the model. The model objective here i s to minimize the t o t a l s o c i a l costs, which include the t r a f f i c accident cost and delay cost induced by access points. S p e c i f i c a l l y , the delay costs consist of two parts: main t r a f f i c 120 flow delay, and minor t r a f f i c flow delay. Thus the basic form of model objective i s : Minimize Z = cjacd + c2fspd + c3JSy (5.1) where, /acd represents the access-accident function, /spd represents access-speed function for main t r a f f i c flow, / d l y represents delay function for minor t r a f f i c flow at access points and on side road, ci, C2, and C 3 represent cost c o e f f i c i e n t s . Note that / a c ( j i s obtained from Table 3.2, and / s p ( j i s obtained from Table 4.2. Only / d i y i s to be derived. In model construction, at t h i s stage, only two-lane highways are targeted for study. However, the analysis method presented here might be applied to other road types i n the future. It i s assumed that the roadside developments on both sides of the main road are homogeneous. This assumption ensures t r i p generations and a t t r a c t i o n s are evenly d i s t r i b u t e d i n the area abutting the main road. The assumption i s necessary because of data l i m i t a t i o n s , but i n any case could appear v a l i d f o r r e s i d e n t i a l areas. 121 5.1.2 Decision Variables Decision variables to solve for are access variables i l l u s t r a t e d i n previous chapters. These access variables are i n the form of number of access points f o r each access type. Five access types are discussed for two-lane highways: s i g n a l i z e d i n t e r s e c t i o n n^, unsignalized i n t e r s e c t i o n n£, business access n3» pr i v a t e access n^, and roadside pullout n5- However, s i g n a l i z i n g an i n t e r s e c t i o n i s b a s i c a l l y a t r a f f i c control issue rather than a planning issue. The need for s i g n a l i z i n g an i n t e r s e c t i o n increases with the increase of side road t r a f f i c . The i n t e r s e c t i o n must already have been planned and constructed. On the other hand, the e f f e c t of s i g n a l i z e d i n t e r s e c t i o n on t r a f f i c operation i s related to signal coordination i n a network, while t h i s study only concentrates on road sections. Therefore, the type of s i g n a l i z e d intersections i s excluded from the present study. Unsignalized i n t e r s e c t i o n i s the most basic access type on a road. Study of t h i s basic access type would provide some e s s e n t i a l i n s i g h t understanding of the problem. Other low l e v e l access types, i . e . , business access, private access, and roadside pullout, do not contribute to side road t r a f f i c delay. This i s because the side road t r a f f i c i s so low at these access points that no queue i s usually b u i l t up. Therefore, no side road t r a f f i c delay i s involved at these access points. It can be seen from Chapter 3 and Chapter 4 that the e f f e c t of these access points on / a c d and /spd i s to increase the accidents and reduce the t r a v e l speed i n the 122 whole density range. Which means that accident cost and t r a v e l time cost w i l l be increased as the numbers of these access points increase anywhere i n the p r a c t i c a l access density range. Therefore, to minimize the t o t a l cost these access de n s i t i e s should be minimized to zero. However, t h i s i s not p r a c t i c a l . It i s therefore not necessary to include these access points i n the optimization model. The provision of these accesses should be based on the spacing of unsignalized i n t e r s e c t i o n and other considerations, e.g., minimum access spacing i n terms of geometric design c r i t e r i o n . There i s another problem with business and private access provisions. E s p e c i a l l y f or business s i t e s , access to the main road i s very important. Presumably the access provision (number, type, geometric design) may a f f e c t the sales of a business and the land value of t h i s l o c a t i o n . Conceptually, i t may be argued that the more access points and the higher standard of access design, the higher the value of the adjacent land l o t . I f the p r i c e change of the land could also be regarded as a kind of s o c i a l cost, then to include business access i n the model we may consider the change of land value as a r e s u l t of access provision. The general function i s P = fiand(rh) (5.2) where, P i s average road side land p r i c e i n a section, 123 Then t h i s component could be incorporated i n our general model. However, a conversion must be made to convert t h i s p r i c e into cost. Conceptually, the lower the p r i c e , the higher the cost. However, i t i s not easy to define t h i s function and requires much more information about land p r i c e s . Due to this lack of the information at t h i s time, the function was not defined i n t h i s study. 5.1.3 Independent Variables Independent variables are those with known values. They are inputs of an optimizaiton model. Based on the discussion i n section 5.1.1, these independent variables are: average annual d a i l y t r a f f i c Aq on the main road, average annual d a i l y t r a f f i c Aq on side roads, peak hour t r a f f i c Q on the main road, peak hour t r a f f i c q on side roads, two-lane highway section length L i n km, speed l i m i t on the main road and S 2 on side roads, and geometric variables X 1 0, X 1 2 , x13 a s defined i n Chapter 3 as grade, frequency of changing d i r e c t i o n of curvature, and horizontal curvature, respectively. Besides, cost c o e f f i c i e n t s are also inputs of the model. 5.1.4 The Objective Function For the basic form of the model objective i n section 5.1.1, note that units of / a c a (access-accident function), / S p d (access-speed function for main t r a f f i c flow), and / d l y (delay function for minor t r a f f i c flow at access points and on p a r a l l e l minor road) are quite 124 d i f f e r e n t . / a c c j gives number of accidents per kilometre per day (accident r a t e ) , / s p d gives kilometre per hour ( t r a v e l speed), and / d l y gives hour per v e h i c l e (delay time). If we have cost c o e f f i c i e n t s c^ = $/accident, and C2 = C3 = $/veh-hr, we may rewrite our objective function equation (5.1) as Minimize Z = cxjac^ + ci f \ L L ci» /acd« /spd> / d l y L » A Q » A Q a r e Prescribed. / 0 i s f s p d with unsignalized i n t e r s e c t i o n points n2 = 0, km/h. The i n c l u s i o n of t h i s f actor i s to c a l c u l a t e increased t r a v e l time due to access points. Therefore, /rj represents a base speed i n the s i t u a t i o n of no access points i n a road section. As indicated e a r l i e r , / d l y i s unknown yet. In d e f i n i n g /diy» t w o components of delay w i l l be considered for minor t r a f f i c flow. The f i r s t one i s the stop delay at i n t e r s e c t i o n s , and the second i s the delay due to slower speed of t r a f f i c on p a r a l l e l minor road. In the second part, L/4n2 i s the average t r a v e l distance on minor road and i s derived i n Appendix A. The t r a v e l time i s the distance divided by speed, and the time saving i s the difference between the two. It can be expressed as t = / L J dly J a + ' (5.4) Si sj 4«2 where, Sj_, S2, L, and are prescribed / a i s average delay of minor t r a f f i c at an access point 125 Note that / a i s derived i n Appendix B and i s a nonlinear function °f n 2 . 5.1.5 The Constraints Several constraints are formulated as follows. They define the sol u t i o n space f o r the problem. The accident rate should be les s than or equal to some c r i t i c a l value, ACD, s p e c i f i e d by the au t h o r i t i e s to meet safety requirement and po l i c y . On the other hand, i t cannot take negative value. °*facd*ACD ( 5 - 5 ) The stop delay f o r minor t r a f f i c flow at access points only should be less than or equal to some c r i t i c a l value, t c , to maintain a c e r t a i n l e v e l of service (e.g., LOS D), i . e . , 0 < / <t (5.6) a c A desirable l e v e l of service on the main road should be maintained, i . e . , t r a f f i c speed on the main road should be no less than a s p e c i f i e d value, S^, which i s the average t r a v e l speed defined i n Highway Capacity Manual (1985) f or l e v e l of service of i . / J^S. (5.7) J spd r 126 T h e o r e t i c a l l y , design speed or speed l i m i t i s another constraint, i . e . , estimated average main t r a f f i c t r a v e l speed should be lower than design speed, DSP, / ,<DSP spa However, t r a v e l speed should not be constrained by access provisions; rather i t should be constrained by speed l i m i t regulation and necessary enforcement. In other words, access points should not be used as a tool to apply speed l i m i t regulation. Therefore, t h i s constraint i s excluded from the model. The spacing should meet geometric spacing c r i t e r i o n (minimum spacing). This c r i t e r i o n i s guaranteed by the s p e c i f i e d length that should be long enough for vehicles to stop before reaching the next i n t e r s e c t i o n when they pass the p r i o r i n t e r s e c t i o n at normal t r a v e l speed. Hence, we here use stop distance as the minimum spacing c r i t e r i o n , i . e . , j s2 — > 0.278(5, )f r+ 0.139-!- ( 5 . 8 ) «2 a where, n 2, L, Sj are prescribed, t r represents response time i n seconds, a represents deceleration rate that i s a function of speed, = u ( S ; l ) , km/h/sec, 127 Furthermore, the number of access points must be a nonnegative integer. It i s self-explanatory to specify the decision v a r i a b l e as an integer. « 2 >0 n.2 i s integer (5.9) The objective function and some constraints are c l e a r l y nonlinear functions of the decision v a r i a b l e s . Therefore, t h i s i s an integer nonlinear programming problem. 5.2 Solving Integer Nonlinear Programming 5.2.1 General Discussion There are r e l a t i v e l y few methods f o r obtaining integer solutions to nonlinear programming problems. Most textbooks confined t h e i r a ttention to ei t h e r integer l i n e a r programs, or nonlinear programs, see, e.g., Wismer and Chattergy (1978), Zangwill (1969). The d i f f i c u l t y i n considering nonlinear integer programming i s that, as Simmons (1975) stated, "integer programming i s too r i c h and too i n t r i c a t e a subject to be p r o f i t a b l y treated as a species of mathematical programming, i t s current s o l u t i o n methods and recent research have l i t t l e i n common with those continuous-variable problems". One method that was suggested to solve the nonlinear integer problem i s using dynamic programming, see, e.g., Cooper and Steinberg (1970), and Pfaffenberger and Walker (1976). 128 However, since there i s only one decision variable i n the current model, there i s no advantage of applying dynamic programming. The advantage of dynamic programming i s that i t treats one subproblem with one v a r i a b l e each time f o r a problem with n v a r i a b l e s . On the other hand, the d i f f i c u l t y of using dynamic programming technique i s that the c o n s t r a i n t i s s p e c i f i e d as l i n e a r function i n this technique; while i n our model i t Includes nonlinear functions i n constraints. Therefore, even i f the model i s expanded to include several decision v a r i a b l e s , e.g., number of business access points n3^ i n the future, this method i s s t i l l not applicable. 5.2.2 Piecewise Integer Programming It was indicated that non-linear problems can sometimes be treated as piecewise integer programming problems with advantage (Cooper and Steinberg, 1974; and Williams, 1985). It i s therefore pursued to see i f the method can also be used for solving an integer nonlinear problem. The procedure i s that we f i r s t t r y to solve the nonlinear problem using the method, while ignoring the integer requirement. Then we modify the procedure to include the integer requirement. The p r e r e q u i s i t e of piecewise integer programming i s that the problem can be expressed i n a separable programming form. The separable functions ensure that they can be approximated to by piecewise l i n e a r functions. An example of a piecewise l i n e a r approximation to /(x) i s the function f(x), i l l u s t r a t e d with dashed l i n e s i n the sketch (Figure 5.1). 129 We can see that a piecewise l i n e a r function i s a set of connected l i n e segments. This i s also referred to as a "polygonal l i n e " , and the approximation / ( j c ) as a "polygonal approximation". The widths of the i n t e r v a l s need not be equal. 3f 3? dg 84 Figure 5.1 A Polygonal Approximation of a Nonlinear function (From Cooper and Steinberg, 1974) We now t r y to approximate /(x) by three connected l i n e segments as shown i n Figure 5.1. At the endpoints of each subinterval f {x) - f (x) . Thus, i n the i n t e r v a l ax and a2 x - ax a2- ax 130 Then f ( x ) ^ f ( a l ) + f ( a 2 ) ~ f ( a i \ x - g l ) ax<x<a2 (5.11) a2 - ax Assume a value X i n the range of 0 and 1, for x i n the i n t e r v a l ax and a2» we have an expression as x - Xax + (1- X)a2 If we denote t h i s value of X by equation becomes x = Xlal + X2a2 where, Xl + X2 = l, \ , X2 >: 0 (5.12) jlj, and let A ^ l - X j , then the above (5.13) Subtracting ax from both sides of equation (5.13), we have x — al = Xxax + X2al - ax = (^ 1 " 1M + ^ 2 (5.14) = —X2al + X2a2 = X2(a2-ax) Then equation (5.11) becomes 131 <h-<h = f(a1)+l2f(a2)-X2f(al) = /(a 1)(l-^) + A 2 / K ) = X1f(al) + X2f(a2) S i m i l a r l y , we can get f(x) for other Oj < x < a4 : f{x) = X2f{a2) + X3f{a3), a2<x<03 I x= X2a2 + X3a3 X2 + X3 — 1, f(x)=X3f(a3)+X4f(a4), a3<x<a4 < x = X3a3 + X4a4 X3 + X4 — 1, A l l X > 0. (5.15) two subintervals, O J ^ J C ^ O J , and (5.16) (5.17) To consider x i n the whole range, we now have f(x) = X1f(a1) + X2f{a2)X+3f(a3) + X4f(a4), < x < a4 I x = Xxax + X2a2 4- Xj03 + X4a4 (5.18) Xj + X2 + X3 + X4 = 1 Again, a l l X > 0. Note that the function now becomes l i n e a r i n the new vari a b l e Xk . To ensure that x i s only i n the i n t e r v a l of adjacent a, , i . e . , in the i n t e r v a l [a,_l5 at] or [a,, alH], we s h a l l s t i p u l a t e that at most only two adjacent Xk can be po s i t i v e and a l l others must be zero. This 132 s t i p u l a t i o n also ensures equation (5.18) i s equivalent to equations (5.15) to (5.17). To incorporate t h i s a d d i t i o n a l requirement on Xk within an integer programming framework, new variables ^ , £ = 1,2,3 are introduced i n t h i s case. Each yk. i s constrained to be 0 or 1 as below K ^ yi • X2<y1+y2 (5.19) x4<y3 yi+yi + y3 = l (5.20) yk = 0 or 1, k = l, 2, 3 (5.21) Equations (5.18) through (5.21) represent the polygonal approximation f {x) of a nonlinear function / ( x ) . Up to now we excluded constraints from the discussion. If nonlinear functions also appear i n the constraints the same transformation approach w i l l be applied. That i s , they can be replaced by l i n e a r terms and a piecewise l i n e a r approximation made to the nonlinear function. Suppose we have a problem with nonlinear constraints as Minimize z = f(x) Subjectto £ , (x)<6, ; = l , . . . ,w (5.22) x>0 133 Then the general representation of t h i s model for f(x) with P connected l i n e segments ( i . e . , f o r ak, £ = 1,2, . . . , P + l ) i s : Mirurnize z = Z (ak) t-i p+i Subject to 'i = h—,m k-l P+l k-i Xk>0 ^ = 0 o r l , k =:!,...,P where (5.23) Jt=l Be aware that equation (5.24) i s the programming i s solved, the optimal to (5.24) to obtain the optimal values not used i n the programming. Once values of the Xk may be ins e r t e d of the o r i g i n a l v a r i a b l e x. Up to now the nonlinear issue has been solved. Next, we are going to include integer requirement i n the model. To do t h i s we modify the model (5.23) by s t i p u l a t i n g that only one element i n Xk be p o s i t i v e , p r e c i s e l y the value of 1, and a l l others be 0. This requirement ensures 134 that only one l i n e segment w i l l be selected. In f a c t , t h i s i s a l o g i c constraint. Xk i s a set of ordered variables within which only one va r i a b l e must be non-zero. It i s c a l l e d SOS1 set, means sp e c i a l ordered set of type 1 (Williams, 1985). Furthermore, we s t i p u l a t e that the end point of the l i n e segment should be an integer, i . e . , ak be chosen as an integer as well. We can see that the optimal so l u t i o n w i l l be ak i f Xk i s determined to be 1 by the model. This i s because f (.Gk) i s exactly equal to f(ak)=f(x) when, x = ak . In t h i s sense, we turn piecewise programming into "pointwise" programming. Ideally, ak should be assigned as 1, 2, m. I f the value range of x i s not very wide there i s no problem; otherwise, i t may r e s u l t i n enormous computation. Fortunately, the range of our decision v a r i a b l e i s r e l a t i v e l y narrow. Due to the geometric spacing constraint, the access density would not be very high. The maximum density may be just less than 10/km for a speed l i m i t of about 40 or 50 km per hour. Then, for a road section of L km, the reasonable range of number of access points i s not very wide. It i s doable. However, for other problems with a wide range of decision variable, a wider i n t e r v a l might be selected f i r s t , e.g., 5, 10, 15, ...; then, a f t e r we f i n d the optimal i n t e r v a l we can repeat the optimization procedure i n t h i s i n t e r v a l only to f i n d the optimal integer value. This kind of strategy was also suggested by Hadley (1964). In essence, the requirement on ak i s just one of the methods used for solving general integer programming problems: i . e . , enumeration 135 method. This method enumerates a l l possible Therefore, equation (5.23) can be modified as p+i Miriimize ^=^XkJ\ak) k-l P+l Subject to ^X k g t (ak)<b,, i=l,...,m p+i k-l 2 t = 0 o r l , ak = integer where p+i J^Xkak=x Another key question i s i f the so l u t i o n i s global optimal or l o c a l l y optimal. Since t h i s procedure only produces an approximate function f o r analysis, i t generally guarantees no more than a l o c a l optimum. Nevertheless, one strategy i s to solve the problem a number of times using d i f f e r e n t ak to obtain d i f f e r e n t l o c a l optima. There may be some chance to obtain a global optimum, or at lea s t one close to the global optimum (Williams, 1985). However, i t i s indicated that only way to be sure of obtaining a global optimum when a problem i s not known to be convex i s to resort to integer programming (Cooper and Steinberg, 1970; Hadley, 1964; and Williams, 1985). Otherwise, i f a problem i s known convex a global optimum would be obtained. integer s o l u t i o n s . (5.25) (5.26) 136 5.3 Model A p p l i c a t i o n The issue of separable function does not e x i s t when there i s only one decision v a r i a b l e i n the model. However, i f we could include other decision v a r i a b l e s , e.g., number of business access points, i n the model, t h i s approach i s s t i l l applicable as long as we can separate the v a r i a b l e s . This i s most probably because i t i s presumed that there are no i n t e r a c t i o n s between decision v a r i a b l e s . The v e r i f i c a t i o n of convexity of a function i s sometimes tedious and cannot be e a s i l y solved. However, graphic representation of a function when there i s only one variable (or may be two) can greatly reduce the computation. To do t h i s we assume some input values for parameters and plot our objective function of equation (5.9) i n figure 5.2. i t can be seen c l e a r l y that the function i s convex i n t h i s range. Since constraints are very complex functions, i t i s not pursued to prove that they are a convex set. Instead, the c h a r a c t e r i s t i c s of decision v a r i a b l e ( i . e . , integer and value range) l i m i t the range of f e a s i b l e values as indicated e a r l i e r . Which ensures that a f u l l enumeration approach i s applicable and w i l l guarantee a global optimal s o l u t i o n . 137 1900 1800 1700 J 1600 o 6-1500 •!-> 6 1400 g 1300 1200 1100 1000 4 5 6 7 Mitber cf Access parts 10 Figure 5.2 So c i a l Cost vs. Access Points ( i n $/day, assuming A(Q) = 8,000, A(q) = 5,000) Substitution of equations (5.4), (5.5), (5.6), (5.7), and (5.8) into equation (5.25) y i e l d s p+i Minimize z=^Xkf(ak) k-l P+l Subject to ^Xkfacd{ak)<ACD (5.27) p+i p+i ^XkgA{ak)<L, k=l p+\ Xk = 0 or 1, a. = 1,...,10 where 138 AQ + Cifdly(-)A< ( 5 . 2 8 ) (facd a n c* ^spd a r e obtained from chapter 3 and 4 respectively.) 1 ( 5 . 2 9 ) CSH~<ilTh. (Cgn i s capacity of the shared lane as defined i n Appendix B.) gA(a,) = {0.27SSitr +0.139 <72 ( 5 . 3 0 ) a n - ( 5 . 3 1 ) It makes common sense that a graphing method w i l l most often provide the most e f f i c i e n t solution for solving mathematical programming problems when decision variables are less than three. Since there i s only one variable retained i n our model at th i s stage, the easiest way of obtaining optimal solution i s graphing the model. This was attempted and the r e s u l t s obtained from piecewise programming are i d e n t i c a l to the graphing method. For example, as shown e a r l i e r , figure 5.2 indicates that the optimal value i s 3, while the same r e s u l t was achieved from a n a l y t i c a l procedure. We found that the RHS c o e f f i c i e n t (right hand side value) of a l l in e q u a l i t y constraints are inac t i v e at t h e i r upper bounds. Which means the optimal solution w i l l be v a l i d for any very large values of accident rates, delay time, and very small value of stop distance (since t h i s value i s i n the denominator). However, the lower bounds are a l l active. 139 Which means i f we specify a very low accident rate and short waiting time the optimal solution w i l l be changed. Generally, t h i s agrees with our expectation. An important aspect of optimization i s dealing with uncertainty, A b r i e f discussion only w i l l be presented here. B a s i c a l l y , three types of uncertainty could be involved i n mathematical programming. The f i r s t type makes parameters random with known p r o b a b i l i t y d i s t r i b u t i o n s . The second type makes RHS c o e f f i c i e n t s of constraints random with known p r o b a b i l i t y d i s t r i b u t i o n . The t h i r d type makes goals and constraints nondeterministic. The f i r s t and the t h i r d types of uncertainty are not s i g n i f i c a n t i n our model. In transportation planning, cost c o e f f i c i e n t s are r e l a t i v e l y stable because time value and unit accident cost are well established. Some references were found, e.g., Bruzelius (1979) f or time cost, and M i l l e r et a l (1991) for accident cost. Objective and constraint functions are deterministic rather than random for they could be sharply defined i n our model. Regarding type two uncertainty, we observed that the RHS c o e f f i c i e n t for minor t r a f f i c delay time at access points could show some degree of v a r i a t i o n . Thus, i f we know the d i s t r i b u t i o n of waiting time of minor t r a f f i c at access points, we may employ chance constraint programming to transform the o r i g i n a l constraint to conventional l i n e a r i z e d form. However, s e n s i t i v i t y analysis indicates that our model i s quite robust. For the constraint of concern, the optimal solution i s v a l i d f o r RHS c o e f f i c i e n t i n the range of about 10 seconds to i n f i n i t y . The lower bound reaches the l e v e l of service B f o r si g n a l i z e d i n t e r s e c t i o n delay. This range appears to be quite s a t i s f a c t o r y , and supports the v a l i d i t y of the model. 140 To i l l u s t r a t e the a p p l i c a t i o n of the model, the computation was made with v a r i a t i o n s of p r i n c i p a l input values, main road and side road d a i l y t r a f f i c volumes. Other parameters were assigned f i x e d mean values. The r e s u l t s are shown i n Table 5.1. Next, we may compare t h i s with the current p r a c t i c e . Table 5.2 shows some two-lane highway sections of length about four kilometres i n the database. It can be seen that the optimal number of unsignalized intersections determined by the model i s somewhat i n the p r a c t i c a l range. However, due to the lack of minor road t r a f f i c information i t cannot be determined e x p l i c i t l y whether the e x i s t i n g condition i s optimal or not. There are l i m i t a t i o n s of the model. F i r s t , other types of access points, f or example, business access, have not been included, as discussed e a r l i e r . Second, accident cost on minor roads have not been considered. Lim i t a t i o n on, the number of access points to main, road w i l l influence t r a f f i c on minor roads. This can be explained i n that the increase t r a f f i c volume and r e l a t i v e l y slow t r a v e l speed might have a p o s i t i v e impact on increasing accidents. Therefore, the other aspect of improving the model i s to take accidents occurred on minor roads into consideration. 141 A An 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 12000 400 1 1 1 1 600 1 1 1 1 1 800 2 T 1 1 1 1 1000 2 1 1 1 1 . 1 1500 2 2 2 2 1 1 1 1 1 2000 2 2 2 2 2 2 2 1 2500 2 2 2 2 2 2 2 3000 2 2 2 2 2 2 2 3500 2 2 2 2 2 2 4000 3 3 2 2 2 2 4500 3 . 3 3 3 5000 3 3 3 3 5500 3 3 6000 3 Table 5.1 Optimal Number of Intersections on a 4-km Section of Two-lane Highway 142 Section Length (km) Main Road T r a f f i c (AADT) No. of Unsignalized Intersections 4.2 751 1 4.0 1022 0 4.2 1033 1 3.9 1047 0 4.2 1055' 0 4.0 1270 0 3.8 1303 0 3.9 1305 0 4.2 1508 0 4.0 2253 0 3.9 2450 0 3.8 2586 2 3.9 3959 3 Table 5.2 Actual Number of Unsignalized Intersections f o r Some Two-lane Highway Sections of Length about 4-km (Source: Photolog database) Chapter 6 ACCESS AND SPEED ZONE DESIGN In chapter 5, we developed a model to determine the optimal number of access points on a road section. In future design, we may use the model to plan access density f o r any two-lane road sections based on t r a f f i c volume, speed, safety and delay considerations. Once an access density i s decided upon, the operational design associated with access density can be i n i t i a t e d . One important part i s design for speed. The design f o r speed ensures safe and e f f i c i e n t t r a f f i c flow on a road section, with d i f f e r e n t speeds should be s p e c i f i e d f or sections with d i f f e r e n t access d e n s i t i e s . Speed zone design i s widely practiced for optimum road performance and safety as given by ITE (1976), Parker (1985). Usually, the 85-percentile speed as determined by spot speed studies i s the p r i n c i p a l factor used by t r a f f i c engineers to determine speed l i m i t s for a speed zone. However, access i s an important factor to be considered and not well documented, although one reference published i n T r a f f i c Engineering (1961) outlined the number of access points for roadside businesses (B) per unit distance as the density index to define the speed zone: 90 km/h for three B per kilometre, 80 km/h for 4 B per km, and 70 km/h for 5 B per kilometre. Furthermore, we could also make use of some res u l t s obtained i n chapter 3 ( i . e . , figures 3.7, 3.8, 3.10, and 3.12). For example, we can check the impact of business access density on the estimated accident rate f o r each speed l i m i t and we may then select a suitable speed l i m i t on the basis of a p r e - s p e c i f i e d acceptable accident rate. 144 In addition to the need to develop design guidelines for speed zones based on access density i t i s generally the case that some aspects of speed zone design, for instance speed t r a n s i t i o n and speed sign design, have not yet been c l a r i f i e d , p a r t i c u l a r l y as a means to mitigate e x i s t i n g high l e v e l s of access density. Therefore, i n t h i s chapter, section one deals with speed t r a n s i t i o n zones, and p a r t i c u l a r l y the minimum length of speed t r a n s i t i o n zone. Section two concentrates on speed-change sign design. Section three outlines a procedure of speed t r a n s i t i o n zone design. 6.1 Speed T r a n s i t i o n Zone The review of some studies {e.g., Parker, 1985) indicated that the p r i n c i p a l guideline used i n designing t r a n s i t i o n zones (also c a l l e d "buffer zones") i s based on the Uniform Vehicle Code's (National Committee of Uniform T r a f f i c Laws and Ordinances, 1968, and 1979) recommendation that no more than six a l t e r a t i o n s per mile be used with not more than 10 mph (16 km/h) differences between zones. However, i t does not present the c r i t e r i a to determine under what condition that a t r a n s i t i o n zone i s needed and what i s the minimum length of a t r a n s i t i o n zone. Three p r i n c i p l e s are proposed here for t r a n s i t i o n zone setup: safety, e f f i c i e n c y , and convenience. Safety: a l l road users' safety, drivers and pedestrians, must be considered. E f f i c i e n c y , roadway speed p o t e n t i a l (higher speed l i m i t ) should be f u l l y used to reduce t r a v e l time. Convenience, d r i v i n g process should be performed without excessive 145 control l i m i t s . According to the above p r i n c i p l e s , two general c r i t e r i a are suggested here. F i r s t , a t r a n s i t i o n zone should be provided when the roadway section i s a t r a n s i t i o n a l s t r e t c h of highway between low access density section and high access density section, which represents the f i r s t p r i n c i p l e . This statement i s explained g r a p h i c a l l y i n Figure 6.1. The second c r i t e r i o n of s e t t i n g up t r a n s i t i o n zone i s that the difference between speed l i m i t s of adjoining road sections should be > 40 km/h. This i s based on the recommendation by National Committee of Uniform T r a f f i c Laws and Ordinances (1968 and 1979) that speed change between zones i s no more than 10 mph (16 km/h, rounded to 20 km/h). It represents above p r i n c i p l e two and three ( e f f i c i e n c y and convenience). This means that (1) i f a higher speed i s f e a s i b l e i n terms of a safety concern, keep the speed as high as possible; and (2) do not set up too many signs and l i m i t s on the road. A p a r t i c u l a r case i s when the difference i s 30 km/h, where generally, i t i s not necessary to i n s e r t a t r a n s i t i o n zone i n between adjoining sections. However, engineering judgment i s also needed to evaluate any p r a c t i c a l s i t u a t i o n . On the other hand, probably i t i s better to reset the speed l i m i t s to get the difference as 20 km/h, or 40 km/h. 146 c CO Q e/> w CD O O < Transi t ion High De nsity High Speed Low Speed \ Low \ Speed Speed Densi ty Dece l e r a t i on ^ p Dist anc e Figure 6.1 Relationship between Access Density and Speed Zone To determine the minimum length of the t r a n s i t i o n zone, f i r s t we should examine the length of t r a n s i t i o n section of access density, D^ as indicated i n Figure 6.2. Generally, the summation of the minimum length L+; and deceleration distances, D^' and D^" (the two deceleration distances may not be equal), should be equal to or greater than D D. Second, we w i l l consider the length of "speed maintaining distance", which i s the minimum length of roadway that drivers do not have to change speed, i . e . , they can t r a v e l at a constant speed for a c e r t a i n distance. This distance could be d i f f e r e n t at d i f f e r e n t speed as proposed i n Reference ( T r a f f i c Engineering, 1961): varying from 0.2 mile (0.322 km) to 0.5 mile (0.805 km) for a speed range from 20 mph (32 km/h) to 70 mph (112 km/h), see Table 6.1. It may be more convenient to adopt a " r e l i e v i n g time" defined as a time period that re l i e v e s drivers from the change of t r a f f i c c o n t r o l . They are free from making "forced" changes i n speed. As shown i n Table 6.1, the t r a v e l time for d i f f e r e n t minimum speed zone length varies between 24 seconds and 36 seconds. On the basis of this information, a period of 30 seconds i s suggested here 147 as the " r e l i e v i n g time". Figure 6.2 Minimum Length of Tr a n s i t i o n Zone Speed (mph/km/h) Min. Length of Zone (miles/km) Corresponding Travel Time (sec) 20/32.2 0.2/0.322 36 30/48.3 0.2/0.322 24 40/64.4 0.3/0.483 27 50/80.5 0.5/0.805 36 60/96.6 0.5/0.805 30 70/112.7 _ Source: T r a f f i c Engineering, 1961 Table 6.1 Minimum Zone Length and Corresponding Travel Time We can then c a l c u l a t e the minimum t r a n s i t i o n zone length for each t r a n s i t i o n speed by using the following equation 148 d = 0.278u* + 0.139a t2 (6.1) where, d = minimum t r a n s i t i o n zone l e n g t h , L t , i n metres u = t r a n s i t i o n zone speed i n km/h a = a c c e l e r a t i o n o r d e c e l e r a t i o n r a t e i n km/h/sec t = " r e l i e v i n g t i m e " i n seconds The a c c e l e r a t i o n r a t e i s z e r o i n t h i s c a s e . The r e s u l t s a r e shown i n T a b l e 6.2. T r a n s i t i o n Zone Speed (km/h) Minimum Zone L e n g t h , L t , (m) 100 830 90 750 80 670 70 580 60 500 T a b l e 6.2 Minimum T r a n s i t i o n Zone Leng t h 6.2 Location of R-3 Sign "R-3 MAXIMUM (Speed) . . . AHEAD" s i g n may be used t o g i v e advance n o t i c e o f a speed zone w i t h a lower l i m i t . The manual o f S t a n d a r d T r a f f i c S i g n s (B.C. M i n i s t r y o f T r a n s p o r t a t i o n and Highways, T r a f f i c B r a n c h , 1988) s t a t e s t h a t " t h e advance s i g n s s h a l l be e r e c t e d from 350 149 to 750 feet ahead of the t r a n s i t i o n point". However, i t does not give s u f f i c i e n t d e t a i l to select a s p e c i f i c value. The following analysis applies kinematic p r i n c i p l e s to c a l c u l a t e appropriate advance placement distance f o r the R-3 sign. The basic equation used to c a l c u l a t e distance traveled given deceleration rates and elapsed time i s d =.278 nbtr + 0.139 M b (6.2) a where, d = t r a v e l distance t r = response time in sec U.D = i n i t i a l speed i n km/h U e = f i n a l speed i n km/h a = deceleration rate Drivers' response time to the Speed Limit sign has not been found i n l i t e r a t u r e and an average value of 1.5 second i s then assumed for t r , which i s the value of brake reaction time of 90 percent of drivers (ITE, 1982) . We define the response time as the period from the time that drivers pass by the sign to the time that drivers take action to slow down. We assume that detecting, reading, and comprehending the sign w i l l occur before drivers passing the sign, and i s neglected i n t h i s analysis. Figure 6.3 i l l u s t r a t e s the parameters for lo c a t i n g the warning sing. Where, R-4 sign indicates the e f f e c t i v e speed l i m i t . In determining proper deceleration rates, various references have 150 been searched. Since very l i t t l e information i s available we assume that drivers do not respond to the R-3 sign by applying brakes when roadway i s l e v e l or u p h i l l . Coast-down operation i s expected f o r deceleration process. Some research indicates that deceleration rates without brakes are much greater at high speeds because the resistance to motion, p a r t i c u l a r l y a i r resistance, i s greater (ITE, 1982). For example, at speed of 70 mph (113 km/h), the deceleration rate i s about 3.5 km/h/s; at speed of 50 mph (80.5 km/h), the deceleration rate i s 1.7 mph/s (2.74 km/h/s). R-3 R-4 Detecting Distance Reaction Distance Real Deceleration Distance Total Deceleration Distance Figure 6.3 Warning Sign Location Fundamentals The r e l a t i o n between deceleration rates and speeds i s a polynomial function of speed and elapsed time i n the coast-down operation as proposed i n Lucas (1986): }l= a0 + a^t + c^t2 + a3t3 + a4t4 + a5t5 + a6t6 (6.3) 151 A curve f i t t i n g technique i s used to estimate the c o e f f i c i e n t s a±. Thus the expression i s r e a d i l y d i f f e r e n t i a t e d to give the deceleration — = ax + 2a2t + 3a3t2 + 4a4t3 + 5a/ + dt and may be evaluated at a number veh i c l e speed range. The resu l t s of are shown i n Figure 6.4. 6a/ (6.4) of d i s c r e t e speeds throughout the the evaluation f o r selected speeds Deceleration (kph/s) 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 y 1 30 50 70 Speed (km/h) 90 110 Figure 6.4. Deceleration-Speed Relation In addition to consider grade we assume that i f grade i s greater than or equal to -1%, no breaks are applied; i f grade i s les s than that, a minor factor, 0.2 km/h/s, w i l l be added to the deceleration rate, a, f o r each increment of one percent downgrade slope. The implied assumption i s that reasonable breaking i s expected when roadway i s downhill. Equation (6.2) can then be modified as 152 d = 0.278 nbtr + 0.139 M f c " f±e (6.5) a + gG where, g = gradient i n percent G = gravity factor Table 6.3 summarizes the distance traveled between two speed l i m i t s . The values can be used as the suggested distance between R-3 and R-4 (see Figure 6.3) . He* Mb He Gradient i o/o) 6 4 2 0 -2 -4 -6 110 90 150 160 180 210 240 260 290 80 190 220 250 290 330 370 420 100 80 140 150 170 200 230 260 290 70 180 200 230 270 320 360 410 90 70 130 140 160 190 220 240 280 60 160 180 210 250 300 340 420 80 60 110 130 150 170 200 230 270 50 150 170 190 230 280 320 380 70 50 100 110 130 160 190 220 250 Notes: * The value i s i n metres. ** Ub = I n i t i a l Speed Limit (km/h) *** u e = F i n a l Speed Limit (km/h) Table 6.3 Distance Traveled Between Two Speed Limits* 153 6.3 Outline of T r a n s i t i o n Zone Design Procedure In conclusion, on the basis of above discussions f i g u r e 6.5 presents a general procedure to set up a speed t r a n s i t i o n zone f o r sections with higher than c r i t i c a l access density, and explained below. (1) C o l l e c t roadway c h a r a c t e r i s t i c s and e x i s t i n g speed l i m i t information. (2) Review accident records with respect to frequency, severity, type, and cause. P a r t i c u l a r attention should be given to those accidents i n which unreasonable speed appears to have been a causative f a c t o r . If such a factor e x i s t s , set up a speed t r a n s i t i o n zone, and go to (7). (3) Review road-side development and t r a f f i c condition. S p e c i f i c a l l y , t h i s includes access density; parking, loading and other vehicle operations adjacent to t r a v e l lanes; turn movements and controls. I f there i s a t r a n s i t i o n a l s t r e t c h of those a c t i v i t i e s between two adjoining speed zones, set up a speed t r a n s i t i o n zone. Otherwise, reduce speed d i r e c t l y to lower l i m i t and go to step (9). (4) Check the difference between two speed l i m i t s . Generally, i f the differ e n c e i s equal to or greater than 40 km/h, set up a speed t r a n s i t i o n zone, go to (7). If the diffe r e n c e i s le s s than 40 km/h, go to (5). 154 Road Characteristics ^ and Speed Limits (8) Determine Transition (9) > \ ' Determine Placement Distance of R-3, Ld Figure 6.5 Transition Zone Design Procedure to Mitigate for High Access Density 155 (5) If the speed d i f f e r e n c e i s equal to 30 km/h, go to (6), i f not, go to (9) f o r d i r e c t i o n speed reduction (no t r a n s i t i o n zone). (6) When the di f f e r e n c e i s equal to 30 km/h, the t r a n s i t i o n zone may not be necessary, but engineering judgment i s needed to evaluate the p r a c t i c a l s i t u a t i o n . I f a t r a n s i t i o n zone i s warranted, go to (7), otherwise, go to (9). (7) Determine t r a n s i t i o n zone speed. This speed should be set at the median value of two speed l i m i t s , and rounded to nearest 10 km/h. (8) Determine t r a n s i t i o n zone length. Check minimum t r a n s i t i o n zone length suggested i n Table 6.2. (9) Determine the advance placement distance of R-3 sigh by using Table 6.3. F i n a l l y , figure 6.6 i l l u s t r a t e s speed t r a n s i t i o n zone setup by using speed and distance diagram. Transition Spaed Transition Zone Length. Lt * If Ldl > 300 m, a second R-3 sign should be erected. R-3 R-4 Dec. Length Ld2 Speed (kph) Low Speed Limit Distance (m) Figure 6.6 Speed Transition Zone Sketch 156 Chapter 7 IMPLICATIONS AND CONCLUSIONS A c l a s s i c a l concern i n both t r a f f i c engineering and t r a f f i c planning has been the safety of i n t e r s e c t i o n s , the e f f i c i e n t operation of highways at i n t e r s e c t i o n s , and the tradeoffs required i n planning road access. The contribution of the research reported on here i s the systematic study of road access, and to suggest p r a c t i c a l applications of the r e s u l t of the findings related to two main components of highway access: those of main road safety and delay. Besides s t a t i s t i c a l studies of the relationships between highway access, safety and t r a v e l speed, the research has developed an approach based on Baysian analysis for a more complete understanding of these r e l a t i o n s h i p s . The hazard model developed i n Section 3.4 holds promise not only for a more precise understanding of access safety problems but also f o r new d e f i n i t i o n s of t r a f f i c safety i n general. The Baysian updating approach developed here i s one possible mean of improving our understanding of road safety. In addition there has been to date no systematic d e f i n i t i o n of highway c l a s s i f i c a t i o n that could be t i e d to safety c r i t e r i a . This research extends highway c l a s s i f i c a t i o n c r i t e r i a to include r e l a t i v e safety l e v e l s based on the number of in t e r s e c t i o n s , or other access types per kilometre. S i m i l a r l y , highway capacity concepts have not, i n 157 the past, included the f r i c t i o n due to access as an influence on l e v e l s of service, but conversely has treated capacity as a sub-optional process for highway (or street) sections between i n t e r s e c t i o n s , or conversely for intersections i n d i v i d u a l l y (except freeway terminals). This research suggests, by implication, the need to incorporate speed reduction due to access as a delay factor i n l e v e l s of service. F i n a l l y i t i s c l e a r that much work must be accomplished to comprehensively plan highway systems to account f o r the tradeoff's between road way benefits and the respective benefits and costs to peripheral a c t i v i t i e s and to non-users who are affected by highways. This research, suggests a framework which, i t i s proposed could be used for the optimization of access points to minimize delay cost to a l l users. Further study i s needed, along with adequate data, to bring i n the benefits and costs. However, this research points to an a n a l y t i c a l procedure based on c l a s s i c a l optimization techniques which, i t i s f e l t , holds promise for future research on t h i s subject. 7.1 Implication of the Research for Highway C l a s s i f i c a t i o n Functional hierarchy of a highway system i s an important element i n highway planning. One understands that the higher c l a s s of a road the less access points should be present along the road. This understanding i s i l l u s t r a t e d by Stover and Koepke i n Figure 7.1. However, t h i s 158 understanding of the rel a t i o n s h i p between access points and functional c l a s s i f i c a t i o n i s b a s i c a l l y conceptual. The dotted l i n e represents a general trend only. It does not imply a determined quantitative r e l a t i o n s h i p . Full Access Control /N increasing movement No Access Control Movement Access Functional Classification _ Freeway Major Arterial Collector ~ Local No Access Traffic increasing access .S, No Through Traffic Figure 7.1 Access and Functional C l a s s i f i c a t i o n (Adapted from Stover and Koepke, 1988) With the framework obtained i n Chapter 5, i t may be possible to begin to quantify the access provisions for each class of road. For example, using s p e c i f i e d d a i l y t r a f f i c volume i n a functional c l a s s i f i c a t i o n system for a c e r t a i n highway cla s s (see, e.g., Highway Functional C l a s s i f i c a t i o n Study, 1992) as an input, with other given or pre s p e c i f i e d information such as speed l i m i t , one can derive appropriate access points (unsignalized intersections) for a road section i n that highway c l a s s . One such example i s given i n Table 5.1. With given t r a f f i c input and c h a r a c t e r i s t i c s of a road section, which represent 159 the features of a s p e c i f i e d c l a s s for the road section, the appropriate (optimal) access points can be determined. For example, i n planning a four kilometre a r t e r i a l road section, i f main road d a i l y t r a f f i c i s 10,000 vehicles and side road d a i l y t r a f f i c i s less than 1,500 v e h i c l e s , the appropriate number of unsignalized i n t e r s e c t i o n s i s only one i n t e r s e c t i o n (see Table 5.1). I f the d a i l y t r a f f i c on side road i s 4,000, then two unsignalized i n t e r s e c t i o n s should be planned f o r t h i s a r t e r i a l road section. The appropriateness of the access provisions i s v a l i d a t e d by the fact that i t i s derived on the basis of minimizing the t o t a l cost. Although the minimum cost i s not stated as an objective i n a highway c l a s s i f i c a t i o n system, i t i s , however, an implied objective. As stated in Highway Functional C l a s s i f i c a t i o n Study (1992), one of the objectives of the functional c l a s s i f i c a t i o n system i s to provide "safe, e f f i c i e n t and economical operation for a l l highway users". 7.2 Implication of the Research for Highway Level of Service In determining the l e v e l of service of a two-lane highway, some adjustment factors are considered i n the Highway Capacity Manual for d i r e c t i o n a l d i s t r i b u t i o n , narrow lanes and r e s t r i c t e d shoulder width, the operational e f f e c t s of grades on passenger cars, and the presence of heavy vehicles i n the upgrade t r a f f i c stream. However, there i s no consideration at a l l for access points. The impact of access on t r a f f i c operation e f f i c i e n c y f or two lane highways can be estimated using the results obtained i n Chapter 4. 160 In Chapter 4, a quantitative r e l a t i o n s h i p between access and t r a f f i c speed i s developed. This r e l a t i o n s h i p provides a basis to evaluate the l e v e l of effectiveness (service) of two-lane highways. Although percent time delay (%) i s the primary measure of l e v e l of effectiveness f o r two lane highways, average t r a v e l speed i s a secondary measure of l e v e l of effectiveness and i s c l e a r l y s p e c i f i e d i n l e v e l - o f -service c r i t e r i a (see Highway Capacity Manual). Messer (1983) summarized the s p e c i f i c a t i o n s of the l e v e l s of service i n three ways: speed-volume, percent time delayed-volume, and maximum service volumes. Levels of service are assigned i n each case. Table 7.1 shows the c l a s s i f i c a t i o n of l e v e l of service corresponding to speed and volume. These re l a t i o n s h i p s are for t r a f f i c i n a one hour period and over f l a t t e r r a i n f o r i d e a l t r a f f i c conditions. Two-Way Volume, pcph Average Speed, km/h* Level of Service Limit < 400 > 92.52 A < 700 > 88.5 B < 1,100 > 84.47 C < 1,700 > 80.45 D < 2,800 > 72.41 E * The conversion was made from the unit of miles per hour to kilometres per hour. Source: Messer, 1983 Table 7.1 Level of Service C l a s s i f i c a t i o n 161 Based on the above r e l a t i o n s h i p between speed and l e v e l of service and the r e l a t i o n s h i p between access density and speed that i s determined i n Chapter 4, the e f f e c t of access density on: l e v e l of service can be e a s i l y obtained. From Chapter 4, we have found that one unsignalized i n t e r s e c t i o n per kilometre w i l l reduce t r a v e l speed by 1.6 km/h. It i s assumed that there are no access points (unsignalized i n t e r s e c t i o n s ) f o r i d e a l t r a f f i c conditions on a road section, and t h i s can be taken as a base l e v e l f or access so that l e v e l of service A should be no access points. As access density increases, the l e v e l of service w i l l d e t e r i orate. With the speed index shown i n Table 7.1, we have the following table (Table 7.2) g i v i n g access density for each l e v e l of service below A. Access Density* (No. per km) 0 1-2 3-5 6-7 >8 Level of Service A B C D E * Access density represents unsignalized i n t e r s e c t i o n density. Note: The t r a f f i c volume range for t h i s table i s 50 - 2500 veh/h. Table 7.2 The E f f e c t of Access Density on Level of Service Note that other types of access points, such as business accesses, can be converted into unsignalized i n t e r s e c t i o n s i n terms of t h e i r e f f e c t s on t r a v e l speed. As indicated i n Chapter 4, one unsignalized i n t e r s e c t i o n i s equivalent to two business accesses, and four p r i v a t e 162 accesses r e s p e c t i v e l y . Thus l e v e l of service may be ca l c u l a t e d f o r any combination of access types and roadway capacity ( l e v e l of service 'E') be defined i n terms of access density i n combination with the well recognized factors i n f l u e n c i n g roadway capacity. In both the functional c l a s s i f i c a t i o n system and t r a f f i c e f f i c i e n c y measures, t r a f f i c safety c r i t e r i o n i s not d i r e c t l y included. For instance, i n measuring l e v e l - o f - s e r v i c e of highways only operation indexes such as speed, density, flow are considered. However, i t i s believed that the i n c l u s i o n of safety c r i t e r i o n may be necessary i n future. Safety i s surely a part of service l e v e l of a highway. Excluding safety gives an incomplete picture of r e a l "service l e v e l " of highways. Some further work i s necessary to develop a new measure of " l e v e l of serv i c e " to include not only t r a f f i c operation measures but also safety c r i t e r i o n . Here, t h i s concern i s f i r s t r aised as a re s u l t of reviewing the implications of present t r a f f i c safety and operation studies. 7.3 Summary Conclusions and Applications of Research Two main functions of a highway are to serve through t r a f f i c flow on a regional basis and to provide a c c e s s i b i l i t y to the l o c a l area. The desire f o r free flow, high speed and safety f o r through t r a f f i c usually c o n f l i c t s with the basic needs of a c c e s s i b i l i t y of l o c a l t r a f f i c . Functional c l a s s i f i c a t i o n of highways provides an appropriate way to tackle the problem. F u l l access control i s the most e f f e c t i v e method for serving through t r a f f i c , but i s lim i t e d i n usage. P a r t i a l access control i s most widely applied. Therefore, to provide an appropriate l e v e l of 163 access c o n t r o l i t i s necessary to define the r e l a t i o n s h i p between accesses and t r a f f i c safety and t r a f f i c operations and f i n d the optimal access points f o r a road section. This research has generated i n s i g h t into some refinements f o r functional road c l a s s i f i c a t i o n . In t h i s thesis we have created a data base of highway access, accident, and t r a f f i c speed f o r the province of B r i t i s h Columbia. The main use of the data base i s to estimate the impact of accesses on highway safety and operations. The i n c l u s i o n of access information i s not a v a i l a b l e i n any other data base i n the province. Therefore, i t can be used i n some access related safety and t r a f f i c operation studies. Empirical models of the r e l a t i o n s h i p between access and accidents were developed. The preliminary accident analyses ind i c a t e that a simple c o r r e l a t i o n between accident rate and access density does not r e f l e c t the actual accident experience. The change i n the structure and complexity of such r e l a t i o n s h i p s established decades ago may be due to temporal and/or s p a t i a l differences i n the c h a r a c t e r i s t i c s of the highway system, and t h i s study demonstrates that the impact of access on accidents i s s t i l l very s i g n i f i c a n t . There also seems to be a s i g n i f i c a n t i n t e r a c t i o n between the access density and some road c h a r a c t e r i s t i c s , such as roadway horizontal curvature and speed l i m i t . More s p e c i f i c a l l y , the following observations can be derived from the res u l t s of the accident analyses 1. Access points are s i g n i f i c a n t l y correlated with the occurrence of accidents f o r a l l highway classes. As expected, s i g n a l i z e d and 164 unsignalized i n t e r s e c t i o n s appear to have the most important e f f e c t s on a l l accident measures (e.g., one unsignalized i n t e r s e c t i o n i s equivalent to ten private accesses for two-lane r u r a l highway). 2. For two-lane r u r a l highways, speed l i m i t has a major impact on the r e l a t i o n s h i p between business access density and accident rate. The impact of business access on accident rate seems to i n t e n s i f y as the speed l i m i t increases. The r e l a t i v e impact of business access on accident rate was c a l c u l a t e d as one h a l f of that of unsignalized i n t e r s e c t i o n s . 3. For two-lane r u r a l highways, grade, horizontal curvature, and the frequency of change i n curve d i r e c t i o n contribute p o s i t i v e l y to accident occurrence, whereas traverse slope i s negatively co r r e l a t e d with accidents. Furthermore, the presence of an a u x i l i a r y lane has a s i g n i f i c a n t impact on the severity of accidents, perhaps because of overtaking at high speed. 4. For two-lane r u r a l highways, horizontal curvature (measured by the average degree of curve) has a major impact on the r e l a t i o n s h i p between accident occurrence and private access and roadside pullout d e n s i t i e s . The impact of private access and roadside pullout on accident rate appears to increase as the average degree of curve increases. 5. In case of incomplete information about t r a f f i c volume for a road 165 section ( e s p e c i a l l y i n the lack of the access t r a f f i c volume data), accident frequency measures, both a l l - a c c i d e n t and severe-accident-only, are generally superior to accident rate measures. This i s because we can assure that the measure of the dependent var i a b l e i s c o r r e c t l y represented. 6. In general, i t seems that the use of a l l types of accidents i n the a n a l y s i s , as opposed to using f a t a l and i n j u r y accidents only, r e s u l t i n s t a t i s t i c a l l y better p r e d i c t i o n models. This i s due to the i n c l u s i o n of more accident information i n the study road sections. Furthermore, a conceptual hazard model was proposed to improve the understanding of highway safety. The model attempted to associate t r a f f i c c o n f l i c t s with accident records by using Empirical Bayes methods. This i s an information gain process. The information on safety should be obtained from not only accident records but also other sources, e.g., t r a f f i c c o n f l i c t which contains abundant hazardous p o t e n t i a l information. The r e s u l t s of the model show i t i s a promising method to estimate hazards. Therefore, i t may be applied i n access-safety studies. In t r a f f i c operation analyses, the quantitative r e l a t i o n s h i p between access and t r a v e l speed was developed. Although no average t r a v e l speed data was available, i t was indicated that the test vehicle's t r a v e l speed could be used as a surrogate of average t r a v e l speed i f the observations of the test vehicle's speed are reasonably 166 large. It was shown that unsignalized i n t e r s e c t i o n , business access, and private driveway are s i g n i f i c a n t i n the speed model. Nonlinear r e l a t i o n s h i p s e x i s t between these accesses and t r a v e l speed. The higher the access density, the more speed reduction would be induced. As i n t u i t i v e l y expected, the magnitude of the influence on speed i s i n the order of i n t e r s e c t i o n , business access, and private driveway, with a r a t i o of approximately 1:2:4. If a l l other independent variables remain at mean values, one unsignalized i n t e r s e c t i o n w i l l approximately reduce t r a v e l speed by 1.6 km/h, one business access w i l l reduce the speed by 0.8 km/h, and one private driveway w i l l reduce the speed by 0.4 km/h. However, Table 4.4 provides a more precise description of the e f f e c t of access on speed reduction. The possible use of the findings of t h i s study i s i n the estimation of the user costs. With the change of average t r a v e l speed, as a r e s u l t of change i n access density, the delay costs could be estimated. The other important ap p l i c a t i o n i s i n the evaluation of l e v e l of service f o r two-lane highways. The e f f e c t of access density on l e v e l of service i s derived i n Table 7.2. This f i n d i n g enriches the understanding of the effectiveness of t r a f f i c operations on two-lane highways. On the basis of the relationships between accesses and t r a f f i c safety and t r a f f i c speed derived i n Chapters 3 and 4, an optimization model was developed to minimize the s o c i a l costs. From a planning point of view, the optimization model i s to provide a procedure to determine appropriate number of access points of a road section. In the model, 167 several considerations, such as permissible t r a f f i c delay, are incorporated into the constraints. The generalized model i s an integer nonlinear function. A modified piecewise integer programming technique was developed to solve the integer nonlinear programming problem. The res u l t s of the model generally agree with the expectation. Therefore, the model could provide useful suggestions or guidelines on highway access planning. For example, given d a i l y t r a f f i c volume on main road, d a i l y t r a f f i c volume of side roads, speed l i m i t s f o r main and side road, and some geometric factors as prescribed i n 5.1.3, an optimal number of unsignalized i n t e r s e c t i o n s can be derived using the model f o r a c e r t a i n length of two-lane road section. In the implementation of highway functional c l a s s i f i c a t i o n , the optimization model w i l l provide guidelines i n access provisions f o r each highway cla s s (although only two-lane highways are considered at the present). Access points i n terms of access density are also an important factor i n transportation design. It should be taken as a c r i t e r i o n i n speed zone design (see Figure 6.1). Some related issues were discussed i n Chapter 6. S p e c i f i c a l l y , the design c r i t e r i a of speed t r a n s i t i o n zone are proposed as: f i r s t , a t r a n s i t i o n zone should be provided when the roadway section i s a t r a n s i t i o n a l stretch of highway between low access density section and high access density section, and second, the difference between speed l i m i t s of adjoining road sections should be > 40 km/h. The minimum length of speed t r a n s i t i o n zone i s derived and shown i n Table 6-2. The location of reducing speed warning sign i s 168 studied. F i n a l l y , the design procedure of speed t r a n s i t i o n zone i s proposed i n Figure 6.5. 7.4 Future Research There are several aspects that the thesis may be extended. The use of the t r a v e l speed of the test v e h i c l e i n t r a f f i c operation analysis i s due to p r a c t i c a l l i m i t a t i o n of resources. Although i t seems to be acceptable as explained i n Chapter 4, the r e a l data of average speed may need to be c o l l e c t e d once resources are a v a i l a b l e to redefine the r e l a t i o n s h i p between access and average t r a v e l speed. On the other hand, th i s w i l l also v e r i f y weather the use of test vehicle speed i n t r a f f i c operation analysis i s p r a c t i c a l . I f i t i s so, the method employed i n the thesis can be taken i n other s i m i l a r s i t u a t i o n s . The optimization model for determining number of access points i s a useful tool i n highway planning and i t may be expanded i n several ways: (1) include other types of accesses, such as business access, as decision variables; (2) define land use cost as a function of number of access points, and incorporate t h i s cost into the model; (3)consider environmental cost resulted from t r a f f i c delay. The hazard model proposed i n Chapter 3 points out a new d i r e c t i o n in measuring road safety, p a r t i c u l a r l y at access points. It needs to be f i n a l i z e d when more t r a f f i c c o n f l i c t data are a v a i l a b l e . 169 REFERENCES AASHTO, (1990) "A P o l i c y on Geometric Design of Highways and Streets", American Association of State Highway and Transportation O f f i c i a l s , Washington D.C. Abbess, C , J a r r e t t , D. , and Wright, C.C. (1981) "Accidents at Blackspots: Estimating the Effectiveness of Remedial Treatment, with Special Reference to the 'Regression-to-Mean* E f f e c t " , T r a f f i c Eng. Control 22(10), 535-542. B.C. Min i s t r y of Transportation and Highways, T r a f f i c Branch, (1988) "Manual of Standard T r a f f i c Signs". Blumentritt, C.W. (1981) "Guidelines f o r Se l e c t i o n of Ramp Control Systems", NCHRP Report 232. Bochner, B.S., et a l (1991) " T r a f f i c Access and Impact Studies f o r S i t e Development", ITE, 1991. Box, P.C. (1969) "Driveway Accident Studies, Major T r a f f i c Routes", Skokie, I l l i n o i s , Public Safety Systems. Box, P.C. and Associates. (1970) " T r a f f i c Control and Roadway Elements -Their Relationship to Highway Safety", Chapter 4 Intersections, and Chapter 5 Driveways, Highway Users Federation f o r Safety and Mob i l i t y . Brewer, M., Copley, G., and Shepherd, N.R. (1980) "Speed/flow/geometry Relationships for r u r a l single carriageway roads." PTRC Summer Annual Meeting, Proceeding Seminar Q, London: PTRC, pp.239-258. Brocklebank, P.J. (1992) "Rural Single Carriageway Speed/Flow Relationships." PTRC XXth Summer Annual Meeting, Proceeding, Seminar H, London: PTRC, pp.207-220. 170 Bruzelius, N. "The Value of Travel Time", Croom Helm., London, 1979. Carlsson G. , Hedman, K-0, (1989) "A Systematic Approach to Road Safety i n Developing Countries", The World Bank. C a r r o l l , R. J . , and Ruppert, D. (1988) "Transformation and Weighting i n Regression". Chapman and H a l l , New York. C i r i l l o , J.A. (1970) "The Relationship of Accidents to Length of Speed-change Lanes and Weaving Areas on Interstate Highways", Highway Research Record No. 312. C i r i l l o , J.A., et a l (1969) "Analysis and Modelling of Relationships between Accidents and the Geometric and T r a f f i c C h a r a c t e r i s t i c s of the Interstate System", Bureau of Public Roads, Aug., 1969. "Colorado Access Control Demonstration Project, F i n a l Report." (1985) Colorado Department of Highways. Cooper, L., Steinberg, D. (1974) "Methods and Applications of Linear Programming", W.B. Saundars Company. Cooper, L., Steinberg, D. (1970) "Introduction to Methods of Optimization", W.B. Saundars. Craus, J . , Livneh, M. , and Ishai, I. (1991) " E f f e c t of Pavement and Shoulder Condition on Highway Accidents", Transportation Research Record, Transportation Research Board, Washington, D.C., No. 1318: 51-57. Cribbins, P.D., Aray, J.M., and Donaldson, J.K. (1967) "E f f e c t s of Selected Roadway and Operational C h a r a c t e r i s t i c s on Accidents on Multilane Highways", Highway Research Record No. 188. Cribbins, P.D. (1967) "Location of Median Openings on Divided Highways", T r a f f i c Eng., Vol. 37, No. 7, 1967. 171 Cribbins, P.D., et a l (1967) "Median Openings on Divided Highways: Their E f f e c t on Accident Rates and Levels of Service", Highway Research Record, No. 188. Daff, M. and White, F.J. (1990) "Road Accidents on Modern Res i d e n t i a l C o l l e c t o r Roads", T r a f f i c Eng. and Planning Proceedings, Conference of the A u s t r a l i a n Road Research Board. David, N.A., and Norman, J.R. (1975) "Motor Vehicle Accidents i n Relation to Geometric and T r a f f i c Features of Highway Intersections", FHWA Report, FHWA-RD-76-128, July, 1975. Dart, O.K., J r . , and Mann, L., J r . (1970) "Relationship of Rural Highway Geometry to Accident Rates i n Louisiana", Highway Research Record, Highway Research Board, Washington, D.C., No. 312: 1-16. Del Mistro, R.F. (1980) "The determination of the optimum number of access points to r e s i d e n t i a l areas to minimise accidents", National I n s t i t u t e f o r Transport and Road Research, CSIR, South A f r i c a . Del Mistro, R.F., and Fieldwick R. (1981) "The Contribution of T r a f f i c Volumes, Speed, Congestion, Road Section Block Length, Abutting Land use and Kerbside A c t i v i t y to Accidents on Urban A r t e r i a l Roads", International Road Federation, World Meeting (9th: 1981), Stockholm. Duncan, N.C. (1974) "Rural Speed/flow Relations." TRRL Laboratory Report 651, UK Transport and Road Research Laboratory. Federal Highway Administration, (1988) "Manual on Uniform T r a f f i c Control Devices for Streets and Highways (MUTCD)", Government P r i n t i n g O f f i c e , Washington, D.C. Fee, J.A., et a l . (1970) "Interstate System Accident Research Study-1", FHAW. Freund, J.E., 1984, Modern Elementary S t a t i s t i c s , 6th e d i t i o n , Prentice-H a l l , Inc. New Jersey. Glauz, W.D., et a l , (1985) "Expected T r a f f i c C o n f l i c t Rates and Their Use i n Pr e d i c t i n g Accidents", TRR 1026, pp 1-12. Glennon, J.C. (1987) " E f f e c t o f ' Alignment on Highway safety, Relationship Between Safety and Key Highway Features", State of the Art Report 6, Transportation Research Board, Washington, D.C. Glennon, J . C , Valenta, J . J . , Thorson.B.A. , and Azzeh J.A. (1975) "Technical Guidelines f o r the Control of Direct Access to A r t e r i a l Highways." Report No. FHWA-RD-76-86 and Report No. FHWA-RD-76-87, Federal Highway Administration, Washington, D.C. Gwynn, D.W. (1966) "Accident Rates and Control of Access", T r a f f i c Engineering, I n s t i t u t e of T r a f f i c Engineers, Washington, D.C, Vol. 37, No. 2, Nov. 1966, 18-21. Hadley, G. (1964) "Nonlinear And Dynamic Programming", Addison-Wesley Publishing Company, Inc. Hauer E. (1992) "Empirical Bayes Approach to the Estimation of 'Unsafety': The Mul t i v a r i a t e Regression Method", Accid. Anal & Prev. Vol 24, No. 5, pp. 457-477. Hawkins, C A . and Weber, J.E. " S t a t i s t i c a l Analysis" Harper & Row, Publishers, New York, 1980 Head, J.A. (1959) "Predicting T r a f f i c Accidents from Roadway Elements on Urban Extensions of State Highways", Highway Research Board B u l l e t i n 208. Headman, K-0. (1989) "Road Design and Safety", Proceedings of STRATEGIC HIGHWAY RESEARCH PROGRAM AND TRAFFIC SAFETY ON TWO CONTINENTS, Totenburg, Sweden. "Highway Capacity Manual" (1985) Special Report 209, Transportation Research Board, Washington, D.C. "Highway Functional C l a s s i f i c a t i o n Study" (1992), Mini s t r y of Transportation and Highways, B r i t i s h Columbia, V i c t o r i a . Highway Users Federation f o r Safety arid M o b i l i t y (1970) " T r a f f i c Control and Roadway Elements - Their Relationship to Highway Safety". Higle, J.L., and Witkowski, J.M. (1988) "Bayesian I d e n t i f i c a t i o n of Hazardous Locations", TRR 1185, 24-36. Hoffman, M.R. (1974) "Two-Way, Left-Turn Lanes Work!", T r a f f i c Eng., Vol . 44, Aug. 1974. Ins t i t u t e of Transportation Engineers (ITE) (1976, 1982) "Transportation and T r a f f i c Engineering Handbook", 1st and 2nd Edit i o n s , Prentice-H a l l , Inc., Englewood C l i f f s , N.J. Kalakota, K.R., Seneviratne, P.N., Islam, M.N. (1993) "Prediction of Accidents On Rural Two-lane Highways", Paper presented to Transportation Research Board 72nd Annual Meeting, Washington, D.C. Kihlberg, J.K. and Tharp, K.J. (1968) "Accident Rates as Related to Design Elements of Rural Highways", NCHRP Report 47, Transportation Research Board, Washington, D.C. Kipp, O.L. (1952) " F i n a l Report on the Minnesota Roadside Study", Highway Research Board B u l l e t i n 55. Lucas, G.G. (1986) "Road Vehicle Performance", Gordon and Breach. Lundy, R.A. (1967) "The Eff e c t of Ramp Type and Geometry on Accidents", Highway Research Record No. 163. McLean, J.R. (1989) "Two-Lane Highway T r a f f i c Operations: Theory and P r a c t i c e " , Gordon and Breach Science Publishers. Messer, C.J. (1983) "Two-Lane, Two-Way Rural Hihgway Capacity ", F i n a l Report, NCHRP Project 3-28A, Texas Transportation I n s t i t u t e , College Station, Texas. M i l l e r , T., et a l (1991) "The Costs of Highway Crashes", F i n a l report to the Federal Highway Administration, contract DTFH61-85-C-00107. Mulinazzi, T.E., and Michael, H.L. (1967) "Correlation of Design C h a r a c t e r i s t i c s and Operational Controls with Accident Rates on Urban A r t e r i a l Urban A r t e r i a l " , Eng. B u l l . of Purdue Un i v e r s i t y : P r o c , 53rd Annual Road School, Mar. 1967. NAASRA (1972) "Guide p o l i c y for geometric design of major urban roads", A u s t r a l i a . National Committee of Uniform T r a f f i c Laws and Ordinances, (1968, 1979) "Uniform Vehicle Code and Model T r a f f i c Ordinance", the Michie Company, V i r g i n i a , 1968 and Supplement III, 1979. NCHRP, 35, (1976) "Design and Control of Freeway off-ramp Terminals". Newland, V.J., and Newby, R.F. (1962) "Changes i n Accident Frequency a f t e r the Provision of By-passes", T r a f f i c Eng. and Control, Feburary, 1962. Ng, J.C.N., Hauer, E. (1989) "Accidents on Rural Two-Lane Roads: Differences Between Seven States", Transportation Research Record Transportation Research Board, Washington, D.C., No.1238: 1-9. Nutakor, C. (1992) "Urban Travel Time Models: Vancouver (BC) Case Study", M. Ap.Sc. Thesis, the University of B r i t i s h Columbia, Department of C i v i l Eng. OECD (1986) "road safety research : a synthesis", Organization f o r Economic Co-operation and Development, Paris. OECD (1971) "research into road safety at junctions i n urban areas", Organization for Economic Co-operation and Development, Paris. OECD (1976) "hazardous road locations: i d e n t i f i c a t i o n and counter measures", Organization f o r Economic Co-operation and Development, Pari s . Parker, M.R. (1985) "Synthesis of Speed Zoning Practices", prepared f o r Federal Highway Administration, July, 1985. Peterson, A.O., and Michael, H.L. (1965) "An Analysis of T r a f f i c Accidents on a High-Volume Highway", Eng. B u l l , of Pudue Univ., Vol. 49, No. 5. Pfaffenberger, R.C., and Walker, D.A. (1976) "Mathematical Programming for Economics and Business", The Iowa State University Press, Ames, Iowa. Prisk, C.W. 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(1963) "Accidents and Operational C h a r a c t e r i s t i c s on A r t e r i a l Streets with Two-Way Median Left-Turn Lanes", Highway Research Record, No. 31. Schoopert, D.W. (1957) "Predicting T r a f f i c Accidents from Roadway Elements of Rural Two-Lane Highways with Gravel Shoulders", Highway Research Board B u l l e t i n 158. Shaw, R.B. and Michael, H.L. (1968) "Evaluation of Delays and Accidents at Intersections to Warrant Construction of a Median Lane", Highway Research Record, No.257. Simmons, D.M. (1975) "Nonlinear Programming f o r Operations Research", Prentice-Hall Inc. Smith, W. and Associates (1961) "Future Highways and Urban Growth", New Haven. Stover, V.G., Adkins, W.G., and Goodknight, J.C. (1970) "Guidelines f o r Medical and Marginal Access Control on Major Roadways", NCHRP Report 93, Transportation Research Board, Washington, D.C. Swedish National Board of Urban Planning (1968) "The Scaft Guidelines 1968: P r i n c i p l e s for urban planning with respect to road safety". Tamburri, T.N., and Hammer, C.G. (1968) "Evaluation of Minor Improvements: (Part 5) Left-Turn Channelization", State of C a l i f o r n i a , D i v i s i o n of Highways, T r a f f i c Department. Thomas, R.C. (1966) "Continuous Left-Turn Channelization and Accidents", T r a f f i c Eng., Vol. 37, No. 3, D e c , 1966. - T r a f f i c Engineering , (1961) "An Informational Report on Speed Zoning", T r a f f i c Engineering, July, 1961. Troutbeck, R.J. (1976) "Analysis of Free Speeds." Proceedings 8th A u s t r a l i a n Road Research Board Conference, 8, Session 23, pp.40-5. U.S. Department of Transportation/Federal Highway Admisnitration, (1970) "Interstate System Accident Research". Venigalla, M.M. , Margiotta, R. , Chatterjee, A., Rathi, A., and Clarke, D.B. 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Englewood C l i f f s , N.J. 178 APPENDIX A Average Travel Distance on p a r a l l e l Minor Road Several assumptions were made to determine the average t r a v e l distance on p a r a l l e l minor road: (1) Drivers w i l l take advantage of higher speed l i m i t on main road whenever they can. (2) The o r i g i n s of t r a v e l l e r s are d i s t r i b u t e d evenly along the minor road. (3) The c h a r a c t e r i s t i c s of adjacent road sections are homogeneious with the one under study. (4) The access points are parted at the same distance. The f i r s t assumption ensures that drivers w i l l turn on to the main road at the f i r s t access point they pass. The second assumption leads the conclusion that the average t r a v e l distance on the minor road to the closest access point w i l l be one fourth of the length between two adjacent access points, see sketch below. access 1 access 2 <-1/4 1/4 the average t r a v e l distance to the cl o s e s t access point = 1/4 The t h i r d and fourth assumptions specify the l o c a t i o n of access points 179 i n such a way as shown i n the following sketch adjacent section section length = L 1 V I -/—>> 7 ^ adjacent section Apparently, i f the number of access points i n the section i s n then the distance between adjacent access points i s L/n. While the assumption two holds, the average t r a v e l distance i s L/4n. 180 APPENDIX B Minor T r a f f i c Delay at Access Points and on Minor Road S p e c i f i c a l l y , fa i s prescribed as follows. According to the ITE handbook (1982), the average delay can be expressed as / =_L_ (B-l) where, a = service rate (reserve capacity) b = sid e - s t r e e t a r r i v a l rate Let the t o t a l minor t r a f f i c a r r i v a l rate on the one side i n a road section be q, the a r r i v a l rate at an i n d i v i d u a l access point then i s q/n2, assuming that the t r a f f i c i s evenly d i s t r i b u t e d i n the section. This gives b = q/n 2. Let a = C<;H> the capacity of shared lane. From the Highway Capacity Manual (1985), c<,ff — , v<+v;+v' , (B-2) where, v^ = volume of flow rate of l e f t - t u r n movement in shared lane, i n pcph; v t = volume of flow rate of through movement i n shared lane, i n pcph; v = volume of flow rate of ri g h t - t u r n movement i n shared lane, i n pcph; 181 movement capacity of the l e f t - t u r n movement i n shared lane, i n pcph; movement capacity of the l e f t - t u r n movement i n shared lane, i n pcph; movement capacity of the l e f t - t u r n movement i n shared lane, i n pcph; Furthermore, we assume that the percentage of r i g h t turn t r a f f i c from side road i s ar, the percentage of through t r a f f i c i s <xt, and the percentage of l e f t turn t r a f f i c i s a,. Therefore, the r i g h t turn minor t r a f f i c , through minor t r a f f i c , and l e f t turn minor t r a f f i c at an access point on one side of the main road are v r = CX 1^/n 2, vt=CLtqlnl, and V( = a i ^ / « 2 respectively. The p o t e n t i a l movement capacity i n passenger cars per hour as i l l u s t r a t e d i n the Highway Capacity Manual (1985) i s based on the c o n f l i c t i n g t r a f f i c volume, V c, i n vehicles per hour, and the c r i t i c a l gap, T c, i n seconds. This capacity i s given i n Figure 10-3 in the Highway Capacity Manual. To enter the optimization model, i t i s required to convert the r e l a t i o n s h i p into an equation. Therefore, the curve was d i g i t i z e d taking the average c r i t i c a l gap = 6 second, = Exp[6.968-0.00135x(conflict traffic)] v (B-3) where, = movement capacity. For r i g h t turn movement of minor t r a f f i c , the c o n f l i c t t r a f f i c i s the 182 Cml = Gmt = Cmr = through t r a f f i c on one d i r e c t i o n of the main road, which i s Q/2. For through movement of minor t r a f f i c , the c o n f l i c t t r a f f i c i s the through t r a f f i c on both d i r e c t i o n s of the main road, which i s Q. For l e f t turn movement of minor t r a f f i c , the c o n f l i c t t r a f f i c i s the main road through t r a f f i c plus through and right turn movements of minor t r a f f i c from the opposite side of the main road, which i s Q + (at + OCr)qI. Furthermore, i t i s indicated i n HCM (1985) that "when t r a f f i c becomes congested i n a h i g h - p r i o r i t y movement, i t can impede lower p r i o r i t y movements from u t i l i z i n g gaps i n the t r a f f i c stream, and reduce the p o t e n t i a l capacity of the movement." Therefore, impedance factor P was proposed to modify the po t e n t i a l capacity of a movement. However, right turns are usually not affected. To simplify the problem, average values of P were chosen as 0.68 for l e f t turns from minor road and 0.8 for through t r a f f i c from minor road. The values are derived based on the assumption that average capacities used by e x i s t i n g demand are 40 percent f o r l e f t turns and 30 percent f o r through t r a f f i c respectively. The equation (B-3) can be s p e c i f i c a l l y expressed as (B-4) In summary, the average delay at an i n d i v i d u a l access point 183 f = (B-5) CSH - 4/n2 where, c = ; <ilni £ x p [ 6 . 9 6 8 - 0 . 0 0 1 3 5 ( g / 2 ) ] P,Exp[6.968- 0.00135(0] P,£xp{6.968-0.00135[<2+(<x,+ oc,)qI n-^ 1 Exp [6.968-0.00135(0/2)] P ,£xp[6 .968-O.OO135 (0] P,Exp {6.968 - 0.00135[g+ {<xr + cct)qln7] 184 

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