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Urban travel time models : Vancouver (BC) case study Nutakor, Christopher K. 1992

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URBAN TRAVEL TIME MODELS: VANCOUVER (BC) CASE STUDYbyCHRISTOPHER K. NUTAKORB.Sc. The University of Science and Technology, Ghana, 1989A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of Civil EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober, 1992© Christopher Nutakor, 1992In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely available forreference and study. I further agree that permission for scholarly purposes may be grantedby the head of my department or by his or her representatives. It is understood that copyingor publication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Civil EngineeringThe University of British ColumbiaVancouver, CanadaDateABSTRACTTravel time survey data obtained from the City of Vancouver’s Engineering Departmenthave revealed that, travel times in the City of Vancouver have remained fairly constantover the past three decades, although traffic volumes and the number of traffic controldevices have increased. This pattern of travel times is contrary to the predicted traveltime behaviour of traditional travel time models, The traditional travel time modelsgenerally predict travel time as increasing with increasing traffic volume. This thesisinvestigated the reasons for the observed travel time trends. It also investigated thevalidity of three traditional time models, using data collected on a few arterial streets inthe City of Vancouver. The results from the research indicated that, the observed traveltime trends are principally due to increases in vehicle speeds and increases in the capacityof the street network.The three traditional travel time models which were investigated for validity are the BPRmodel, the GVRD model and the Davidson model. None of the three models investigatedprovided a good fit for data collected on the arterial streets. Based on the data collectedon one of the arterials, revised forms of the models were developed. The revised modelswere validated against data collected on three other arterials in the City of Vancouver. Inall cases, the validation process proved satisfactory.11TABLE OF CONTENTSABSTRACT iiTABLE OF CONTENTS iiiLIST OF FIGURES viiLIST OF TABLES viiiACKNOWLEDGEMENT xi1. INTRODUCTION 11.1 Defining the problem 11.2 Uses of travel time information 21.3 Factors affecting travel time 31.4 Measurement of travel time 52. LITERATURE REVIEW 72.1 Travel time prediction 72.2 Travel time algorithms 81112.2.1 Travel time prediction using vehicle sensordata2.2.2 Travel time prediction for route guidance2.3 Travel time models2.3.1 Travel time prediction model on congested roads2.3.2 The BPR travel time model2.3.3 Davidson’s travel time model2.3.4 The GVRD travel time model2.3.5 Haase’s travel time model2.3.6 Smeed’s travel time model2.4 The usable models2.5 Travel time studies in the City of Vancouver2.5.1 Travel time survey procedure2.5.2 Comparison of travel time data for Vancouver3. RESEARCH METHODOLOGY3.1 Testing the applicable models3.1.1 Description of the data collection sites3.1.2 Data collection procedure3.1.4 Testing the effectiveness of the models and test results3.2 Modification of the models3.2.1 Determination of Capacity values891414151618192021212224282829• . 3135• •• 4344iv3.2.1.1 Measurement of Saturation flow 463.2.2 Determination of free flow travel time . 503.2.2.1 Measurement of free flow speed 513.2.3 Estimation of the model parameters 523.2.3.1 Estimation of the parameters for the BPR model . 533.2.3.2 Estimation of the parameter for the Davidson’smodel 543.2.3.3 Estimation of the parameters for the GVRDmodel 543.3 Validation of the revised models 563,3.1 The validation procedure and results . 574. DISCUSSION OF RESULTS 634.1 Capacity and Free flow travel time 634.2 The travel time patterns 644.2.1 Increase in vehicle speeds . . 654.2.2 Increase in capacities . . 664.3 The travel time models 674.3.1 The BPR model 684.3.2 The GVRD model 694.3,3 The Davidson model 69V5. FURTHER ESEARCH 716, CONCLUSION 72REFERENCES 75Appendix A TABLE OF RESULTS 78Appendix B EXPECTED TRAVEL TIMES AND VARIANCES USING THEREVISED TRAVEL TIME MODELS 88B.1 Estimates using the BPR model 90B2Cstimates using the GVRD model 91B.3 Estimates using the Davidson model 92viLIST OF FIGURESFigure 2.1 Graphical comparism of travel times over the years in the City ofVancouver using five minute isochrones 27Figure 3.1 (Testing the models) Fitting the old prediction models with observeddata, collected between 41st Avenue and 49th Avenue on Oak Street . . 38Figure 3.2 (Testing the models) Fitting the old prediction models with observeddata collected between 49th Avenue and 57th Avenue on Oak Street . . . 39Figure 3.3 (Testing the models) Fitting the old prediction models with observeddata, collected between Clark Street and Fraser Street on 12th Avenue . 40Figure 3.4 (Testing the models) Fitting the old prediction models with observeddata, collected between 16th Avenue and King Edward Avenue on ArbutusStreet 41Figure 3.5 (Validating the revised models) Fitting the revised models with datacollected on 12th Avenue, between Fraser Street and Clark Street 60Figure 3.6 (Validating the revised models) Fitting the revised models with datacollected on Arbutus Street, between King Edward Avenue and 16thAvenue 61Figure 3.7 (Validating the revised models) Fitting the revised models with datacollected on Oak Street, between 49th Avenue and 57th Avenue Street . . 62viiLIST OF TABLESTable 2.1 Flow parameters for various road types 18Table 2.2 Average travel time comparisons for some major streets in Vancouveroutside the Downtown peninsula 25Table 2.3 Average travel time comparison for some streets in Vancouver withinthe Downtown peninsula 26Table 3.1 Signal timing plans in operation at the various approachintersections 31Table 3.2 Traffic flow and speed data collected, between 41st Avenue and 49thAvenue on Oak Street, with computed travel times 33Table 3.3 Predicted travel times using the old travel time models, with trafficflow data collected between 41st Avenue and 49th Avenue on OakStreet 34Table 3.4 Values of the means, variances and sample sizes for the observed andpredicted data with the old models 42Table 3.5 Results from testing the old travel time models 43Table 3.6 Results of saturation flow studies at the intersection of 41st Avenue onOak Street, for through lanes in the northbound direction during themorning peak period 49Table 3.7 Results of saturation flow studies for shared lane(right and through) atviiithe intersection of 41st Avenue on Oak Street in the northbound directionduring the morning peak period 50Table 3.8 Results of free flow travel time studies on Oak Street in thenorthbound direction, after 1 lp.m 52Table 3.9 Values of the means, variances and sample sizes for the predicted datawith the revised models 59Table 3.10 Results from validation of the revised models 59Table A. 1 Traffic flow and speed data, collected between Fraser Street and ClarkStreet on 12th Avenue, with computed travel times 78Table A.2 Traffic flow and speed data, collected between 49th Avenue and 57thAvenue on Oak Street, with computed travel times 79Table A.3 Traffic flow and speed data, collected between 16th Avenue and KingEdward Avenue on Arbutus Street, with computed travel times 80Table A.4 Predicted travel times, using the old travel time models, with trafficflow data collected between Clark Street and Fraser Street on 12th Avenue81Table A.5 Predicted travel times, using the old travel time models, with trafficflow data collected between 16th Avenue and King Edward Avenue onArbutus Street 82Table A.6 Predicted travel times, using the old travel time models, with trafficflow data collected between 57th Avenue and 49th Avenue on OakStreet 83ixTable A.7 Predicted travel times, using the revised travel time models, withtraffic flow data collected between Fraser Street and Clark Street on 12thAvenue 84Table A.8 Predicted travel times, using the revised travel time models, withtraffic flow data collected between 49th Avenue and 57th Avenue on OakStreet 85Table A.9 Predicted travel times, using the revised travel time models, withtraffic flow data collected between 16th Avenue and King Edward Avenueon Arbutus Street 86Table A. 10 Summary of Saturation flow study results and determined Capacityvalues 87xACKNOWLEDGEMENTSThe author wishes to express his heartfelt thanks to Professor Frank Navin for hissupervision and careful guidance throughout the duration of this thesis. Thanks are alsodue to Professor Gerry Brown for his constructive criticisms and Professor W.F. Caseltonfor reading through this thesis.Finally, the author would like to express his gratitude to his fellow graduate students; PaulDeleur, Tarek Sayed, Mohsen Ghazel and Norman Stang for their encouragement andcontributions.God bless you all.xiChapter 1INTRODUCTION1.1 Defining the problemTravel time, is a central factor for planning the transportation system of any municipality.It is defined as: The length of time it takes to move from a given origin to a givendestination under prevailing roadway traffic and control conditions at some suitable time.The efficient management of any transportation system needs reliable travel timeinformation. Ideally, the information would have to be continous, however, it is notpossible to obtain travel time information on a continual basis by field measurementsbecause of the high cost. As a result, analysts have developed procedures for predictingtravel time from a few traffic parameters. These travel time estimating procedures haveevolved from simple mathematical equations to complex simulation models.Nevertheless, researchers have had difficulty calculating consistent values for travel timeunder varying road conditions, due to the fact that no single satisfactory theory of urbantraffic flow is available that relates the many known factors.Travel time surveys in the City of Vancouver over the past three decades have revealedthat, travel times on most arterial streets, as well as on some streets in the central businessdistrict (CBD) have remained fairly constant or shown slight improvements, although1Chapter 1. Introductiontraffic volumes on the streets and the number of traffic control devices have increased.These travel time trends are contrary to the predicted travel time behaviour of traditionaltravel time models. The traditional models generally predict travel time as increasing withincreasing traffic volume. These state of affairs have raised two questions:(i) Why have the travel times in the City of Vancouver remained constant?(ii) Are the old travel time models still valid or do they need to be modified?The objective of this research is to find answers to these two questions. The research wasconducted using traffic flow and travel time data collected on a few arterial streets in theCity of Vancouver during the summer of 1992, and historic data available from the Cityof Vancouver Engineering Department.1.2 Uses of travel time informationTravel time is probably the single most important variable in Transportation Engineeringand Planning. Its uses can be summarised as follows:(i)Transportation efficiency: Travel time data can be used to develop sufficiency ratings,congestion indices or other measures of route efficiency for use in programming trafficimprovements.(ii) Effects of improvements: Travel time data is useful in evaluating the effectiveness ofspecific traffic improvements on a” before and after” basis, such as the effect of parkingprohibitions, traffic signal improvements, one-way streets, or turn prohibitions.2Chapter 1. Introduction(iii) Traffic congestion and delay: Average speeds and the amount, location, duration,frequency and causes of delay in traffic stream help define problem locations wheredesign or operational improvements may increase mobility.(iv) Trends in mobility: The data can be used to evaluate the conditions of traffic mobilityand the level of service as it changes over a period of time, due to transportationimprovements and/or traffic growth.(v) Transportation models: Trip assignment to proposed new traffic or transit facilities arebased on relative travel times as well as other factors to predict mode split. It is thereforevery useful as an input into traffic assignment algorithms and also for traffic signalcontrol coordination.(vi) Economic analyses: Travel time data can be used in cost/benefit analysis, such astransit scheduling, estimating gasoline consumption and as a very useful guide for thelocation of industries.1.3 Factors affecting travel timeTravel times of vehicles in cities are influenced by many factors, and as a result traveltime from a given origin to a given destination cannot be a single fixed quantity. Thetravel time varies depending on a number of prevailing conditions and influencing factorssuch as: traffic volumes, traffic composition, traffic control devices, times of day, weatherand other factors.3Chapter 1. Introduction(i)Traffic volumes: All other factors being equal, travel times increase with increasingflow. As such travel times of vehicles during peak hours are expected to be higher thanduring non peak hours.(ii)Traffic composition: Traffic composition can significantly affect travel times ofvehicles. Average travel times over a section of a route are likely to be high when thetraffic composition has a high percentage of heavy vehicles. This is due to the fact thatheavy vehicles are generally slow and also occupy a larger road space than all othervehicle types. As a result they reduce the capacity of the streets and may increase traveltimes.(iii)Traffic Control devices: The presence of traffic control devices such as traffic signals,stop signs and yield signs delay vehicles and therefore increase the total travel time of avehicle from a given origin to a given destination.(iv)Effect of time of day: There are several reasons why travel times may depend on thetime of day. These reasons include: altered network characteristics due to bus lanes orparking restrictions which may apply for only part of the day, changes in signal linkingplans, changes in the proportion of turning movements at junctions, changes in pedestrianactivity affecting the probability of being stopped at protected crossing facilities and alsovolume change effects.(iv)Effect of weather: Factors such as heavy rain and snow have adverse effects onvisibility of drivers and general road surface condition, and therefore effect on travel time.(v)Other factors: Other factors such as road geometry, driving styles of drivers, vehicle4Chapter]. Introductioncharacteristics, age and sex of drivers also affect travel time.1.4 Measurement of travel timeSeveral techniques have been developed for measuring travel times. The choice of anyparticular measurement technique depends on the purpose of the study, the requireddegree of accuracy and the availability of equipments for the study. The techniques aredivided into four categories as summarised below.(i)Test Vehicle Method: In this method of obtaining travel rime information, a series ofruns are made through the section to obtain representative travel times. This could beachieved by adopting either one of two driving technique strategies; the ‘floating car’technique or the ‘average speed’ technique. In the floating car technique, the driverattempts to approximate the median speed by passing as many vehicles as pass him.However, there are errors associated with this method, especially on multi-lane highwaysduring periods of congested flow and on roads with very low volumes. The second‘driving strategy’ is the average speed technique, in which the driver travels at a speedthat in his own opinion is representative of the speed of the traffic at every point in time.Data is recorded by an observer in the vehicle or by a mechanical recorder. The use ofan observer with two stop-watches is the most common method. The observer starts thefirst stop watch at the beginning of the test run and allows it to run continuously,recording the cumulative lapsed time at successive control points and delay points along5Chapter 1. Introductionthe route. The second stop watch is used to determine the length of individual time delaysat each delay point. The time, location and cause of the delay is recorded on forms or byvoice recording equipment. It is possible for a driver alone to obtain the desiredinformation by using voice-recording equipment and a stop watch mounted on thedashboard of the vehicle thus eliminating the need for an observer. However, thisprocedure can be hazardous due to the driving task.(ii) The license plate method: In the license plate method, travel time information canbe obtained by stationing one or more observers at each entrance and exit of the studysection to record the time and license number of each vehicle as it passes the observationpoint. The numbers are matched later and the travel time of each vehicle is determined.The equipment used in this technique consists of synchronized stopwatches and recordingforms or voice recorders without audible time signals.(iii) Direct observation and timing method: The direct observation and timing method ofvehicles is only employed over short road sections which are such that the observer cansee both the entrance and exit points.(iv) An interview technique: This is useful when a large amount of information is neededwith little expense for field observations. For example, employees of privateestablishments or municipal agencies are asked to record their travel time to and fromwork on a particular day.6Chapter 2LiTERATURE REVIEW2.1 Travel time predictionIn general terms, travel time prediction can be defined as: An automatic computation oftravel times from classical travel time parameters. The knowledge of travel times asdiscussed previously is a critical component of information in traffic control systems, notonly for drivers but also for traffic control managers, as it provides them with informationon operational characteristics of roadways. It also serves as an important tool for watchingover the evolution of traffic quality with time. The need for travel time information andthe high cost associated with travel time measurements has led to the development ofprocedures for predicting travel time. Travel time prediction procedures have been veryuseful in Traffic Engineering and can be broadly classified into two groups: (i) PredictionModels and (ii) Prediction Algorithms. Some of these models and algorithms areapplicable to traffic guidance systems and also useful for traffic control strategy in directmeasurement and control of travel time on intersecting roads. This strategy enables trafficcontrol managers to realize intended travel time policies. Most of them have also beenused as inputs into traffic assignment algorithms used in urban transportation planning.7Chapter 2. Literature review2.2 Travel time al2orithms2.2.1 Travel time prediction using vehicle sensor dataThis travel time prediction method which utilizes vehicle sensor data was developed byT. Oda (1990). The method involves the division of the objective road section into severalsubsections to collect vehicle sensor data. The vehicle sensors are set at each subsectionof the objective road section. Data collected through the vehicle sensors are trafficvolumes, occupancy times and mean vehicle lengths. Using the vehicle sensor informationobtained in each subsection, the subsection traffic conditions are determined and thenfuture traffic conditions are predicted based on the traffic volume and occupancy timerecorded from the past to the present. The method of prediction employed is an autoregressive model with lead time. In the travel time prediction process, the mean lengthof the vehicles is computed with changing traffic conditions; in otherwords it is notassumed to be constant. This is an improvement on the conventional method whichassumes that the vehicle length is constant despite changes in traffic conditions. This newmethod was applied to a road section in Chiba Perfecture, Japan, and the results showedthat the new method is 5.6% more accurate than the conventional method. The meantravel time on each subsection is calculated as the ratio of the length of the subsection tothe mean speed over the section. The mean speed is calculated as shown by Equation 2.1.8Chapter 2. Literature reviewS= LV/C (2.1)where:S = mean vehicle speed in meters per secondL = mean vehicle length in meter per vehicleC occupancy in seconds per minuteV = traffic flow rate vehicles per minuteThe total travel time over the objective route is then obtained by totalling the travel timescalculated over the subsections.2.2.2 Travel time prediction for route guidanceThe travel time prediction algorithm developed by G. Hoffman and J. Janko (1988), is abasic part of Berlin’s LISB guidance and information system which is an individualdynamic guidance system for motorised vehicles. Although in a dynamic system routesare recommended at regular intervals, it is not sufficient to base quickest routes on traveltimes just at the moment of recommendation. This is due to the fact that the links that avehicle will pass on its journey may change their travel times between the time ofrecommendation and the time of passing. The recommendation therefore has to be basedon the travel time a vehicle has to expect at the moment it is travelling along this link,9Chapter 2. Literature reviewhence the development of the prediction algorithm. The travel time prediction processemployed in the LISB Guidance strategy is in four steps: evaluation and validation ofmeasured travel times, development of standard profiles, continuation of standard profilesand travel time prediction algorithm. Each of the steps is as described in the followingdiscussion:(i) Evaluation and validation of measured travel timesIn the prediction process, the travel time values gathered on a link, with their time ofentry into the link during the same time interval, are used to build an arithmetical mean.However, these values cannot be used without being checked, since some of these valuesmay not be reasonable. For example very low travel times which are caused by exceededspeed limits are increased to a threshold. Also high travel time values which areconsidered to be caused by the driver’s action are excluded from the analysis, while thoseconsidered not to be caused by the driver’s action are included. For example high traveltimes due to exceptionally high volumes, weather and road surface conditions areconsidered.(ii) Development of standard profilesThe second step in the prediction process is the development of long-term standardprofiles which form the basis of the whole prediction process. Each link in the roadnetwork, the time of day divided into short time intervals (usually 5mins) and theweekday are the usual characteristics of differentiation for developing the travel timestandard profiles. Information regarding weather or road surface conditions are only taken10Chapter 2. Literature reviewinto account to explain strong variations in different profiles. Therefore, an analysisshould be made to determine if variations in road surface conditions such as dry, wet oricy have significant influences on the travel time profiles. Since knowledge of travel timeprofiles will not be known at the beginning for each link and for each day of the weekan arbitrary standard profile based on the average speed on the links is established for thedifferent time periods during the weekdays and week-ends.(iil) Continuation of standard profilesStarting with a first provision of different levels of constant travel times for the differentlinks of the network, a learning system has been brought into operation. In order toreduce the influence of stochastic variations an exponential smoothing algorithm is usedas:T = aT + (1-a)t (2.2)where:T, = new value of standard profile of link 1 in time interval n.=value of standard profile of preceding day of operation of link 1 in time intervalT0 = average travel time in link 1 in time interval n of the preceding day ofoperation= a weighting parameter which is chosen such that, on one hand short rangeeffects as vacation time or road surface conditions during the winter weather11Chapter 2. Literature reviewshould influence the new travel time standard profile and on the other hand, singledisturbances that will not occur on the following day of operation should not influencethe travel time standard profile significantly. A value of o 0.25 was used in the LISBexperiment.(iv)Travel time predictionThe estimation of the expected travel time on a downstream link is based on thefollowing information:(a) The updated travel time standard profile of each link for this particular day ofoperation, and(b) The travel time data of this day for each link up to the last 5-minute-interval beforethe prediction is started.As a first step; a parameter d1 is defined as:(2.3)where:tm is average travel time value measured on link 1 in time interval n and as defmedearlier is the corresponding value of the actual travel time standard profile.As some of the prediction cycles wifi not own travel time values for each link some traveltimes will not reflect the real situation; the parameter d, defined above is then smoothedexponentially by a weighting factorJ3. A value offl = 0,20 was used in the LISB guidance12Chapter 2. Literature reviewexperiment. The smoothening relation is given as:d1 = + (1—), (2.4)If there are no more travel times for one or more time intervals n, then the value for d1is set to 1.0 so that d1 is also smoothed to 1.0. Further, to compensate for changes inthe neighbourhood of a particular link, a mean dmin of the smoothed ratios of alladjacent links is computed as:dmin = (1/a)d (2.5)From the values of and dmin an indicator D1 is calculated as:= + dm) (2.6)The prediction equation is therefore finally given as:= D x t (2.7)where T1,, is the best estimator for the expected travel time.13Chapter 2. Literature review2.3 Travel time models2.3.1 Travel time prediction model on congested roadsA method for predicting travel times on congested roads has been developed by Usamiet. al. (1983). The application of the method involves the division of the road sectionbeing considered into several subsections. The total travel time over the road section isthen computed as the sum of the predicted travel times over the subsections. The traveltime estimation procedure on each subsection involves, dividing the number of thevehicles present in the subsection, E by the traffic volume, Q(vehls), where the numberof vehicles, E, is given as the ratio of the section length, L(m), to average space headway,H(mlveh). The equation for predicting the travel time, T(sec) for the subsection of theroad is expressed asL1T.)Z_L_ (2.8)To avoid errors that may be introduced by the constant average space headway, assumedin the above formula, the equation was modified by letting the inverse of H1, or the trafficdensity, K(vehlm), be a linear function of the traffic volume Q as follows:K1= Km-aQi (2.9)14Chapter 2. Literature reviewThe prediction equation is therefore written as:LT= Km _-aL1 (2.10)where:Km and a are constants having pre-assigned valuesQ. = traffic volume in each subsectionL1 = subsection queue length2.3.2 The BPR travel time modelThe Bureau of Public Roads (BPR)(1964), developed a travel time model which is usuallyemployed as an input into capacity restraint traffic assignment algorithms. The capacityrestraint method of traffic assignment is an iterative process. The most common methodis to assign the trips on to the network and adjust the speed or travel time on the linkafter each assignment, according to some speed-volume relationship, to minimise theimbalance of volume on the link. One such speed-volume relationships commonlyemployed is the BPR model which is given as:T=T0[1 +0. 15( VfC)4] (2.11)15Chapter 2. Literature reviewwhere:T = the link travel time at the assigned volumeT0 = base travel time at zero volumeC = the practical capacity of the linkV = volume of traffic on the linkIt has been found that a reasonable balance in volume of trips on the links can beobtained after three or four assignments.2.3.3 Davidson’s travel time modelDavidson (1966), developed a travel time model which, like the BPR model is applicableas an input into traffic assignment algorithms. This model has a quasitheoretical base fromqueueing theory, under an hypothesis that a length of continuous road can be representedas a sequential queueing system. In its original form, the model was based on the ideathat if the total travel time is considered to consist of service time (To) plus a delay, thenfor a delay in a queue with random arrivals and random service, the mean arrival timethrough the queuing system is given as:T=T0[1 +q/(s—q)] (2.12)16Chapter 2. Literature reviewDavidson argued that as traffic flow on a road was not truly a single Continuousqueueing situation, the above relation Could be modified to include an adjustmentparameter j to yield:T=T0[1÷jqf(s-q)] (2.13)where:T0 = the free mean link travel times = link capacity andj = a delay parameter which may be assumed to be a function of link type andenvironmentT = the travel time on the link at the flow qDavidson’s relationship has been used in several cases. It has been used in travel timeflow studies in Toronto and Brisbane as reported in Blunden(l971). It has also been usedby Taylor (1977) to fit data collected on four lane arterial roads in Australia.Typical parameter values for the Davidson’s model as reported by Blunden (1971) are asreported in Table 2.1,17Chapter 2. Literature reviewconditions Mean free Delay parameter (j) sat.flow(Road-type) flow travel (veh/hr)timeT0(min/mile)motorways 0.8-1.0 0-0.2 2000/lanemulti-lane hwys 1.5-2.0 0.4-0.6 1800/lanefeeder and 2.0-3.0 1.0-1,5 1800/total widthcollector roadsTable 2.1 Flow parameters for various road typesSource: Blunden (1971), Table 3.92.3.4 The GVRD travel time modelThe Greater Vancouver Regional District (GVRD) has also developed a model, which isincorporated in the capacity restraint traffic assignment algorithm of the district. Themodel is basically an improvement upon the BPR model. The assignment algorithm wasdeveloped from EMME2; a transportation planning program developed in the Universityof Montreal.The GVRD travel time model for arterial streets is generally given as:T=T0[1+O.6(V/CL’°5)4] (2.14)18Chapter 2. Literature reviewwhere:T = the link travel time at the assigned volumeTo = base travel time at zero flowV = volume of traffic on the linkC practical capacity per laneL number of lanes2.3.5 Haase’s travel time modelHaase (1968), proposed a freeway travel time model which takes into account the arrivalof a vehicle on- ramp queue and its departure from the off-ramp. This model is of theform given below:T = ng[1/q1—110]+d/u+nj[1/q2 (2.15)where:T1 = total trip time for the ith carn1 = the ith car to arrive on ramp queueq0 = the average arrival rate at the on-ramp queueq1 = the average departure rate from the off-ramp queueu = effective steady-state velocity of the N cars19Chapter 2. Literature reviewd = distance travelled on the freewayHaase’s model is based on the assumption that, the total trip time of vehicles is influencedby the presence of other vehicles using the facility. In the model shown in Equation 2.15,total trip time will be low, if the average departure rate of vehicles from the off-rampqueue exceeds the average arrival rate at the on-ramp queue and vice versa.2.3.6 Smeed’s travel time modelSmeed (1968), discussed a special situation in which drivers delay starting their trips, inorder to minimise travel time. He has developed a model that includes the fraction of thecentral business area devoted to streets in predicting travel time. This model is expressedasT = t/2 + (7.409/106)A”2 (2.16)[1 — fl/331I2]lIwhere:T = average journey time measurement from the time the first vehicle enters thecentral business district (CBD)t = period over which entries to the CBD are spreadn = area of CBD in square feet; andf = fraction of CBD area devoted to streetsThe development of Smeed’s model, is based on the idea that traffic speed usually20Chapter 2. Literature reviewdecreases as flow increases. It therefore assumes that, the travelling times of each memberof a group of vehicles, using the same road system will in general be less, when theyspread the period over which they begin their journey times than when they concentratethis period as closely as possible. The model also assumes that all the vehicles enteringthe central business district street travel at the same average speed and that the area ofthe central business district has effect on the vehicles speeds.2.4 The usable modelsFor the purposes of this research, not all the models discussed above are usable. Theamount of data required for working with some of the models and algorithms is enormousand expensive and thus beyond the scope of this study. The research is therefore limitedto the study of three models namely; BPR model, Davidson’s model and the GVRDmodel.2.5 Travel time studies in the City of VancouverVancouver is Canada’s major seaport on the Pacific Ocean and an active participant inthe rapidly developing Pacific Rim Region. The metropolitan area known as the ‘LowerMainland’ consists of 13 municipalities which have an aggregate population of 1.5million. The core of this region is the central area of the City of Vancouver with a21Chapter 2. Literature reviewpopulation of 415,000. The City of Vancouver serves 1.4 million person trips per day or40 percent of 3.5 million trips within the metropolitan Vancouver area. Included withinthese trips are auto, transit and walking trips. Through trips are not included in the total.In spite of a high level of transit usage, the automobile is the predominant means oftravel.For the effective planning of the transportation system of any municipality, there is theneed for occasional travel time studies. The City of Vancouver has on the averageconducted a travel time study once in every ten years. The results from these occassionaltravel time studies has been very useful especially in the following areas:(i) For determination of general trends in mobility(ii) To serve as a guide for the evaluation and design of transportation improvements.The available travel time data for the city are for the years of 1961, 1963, 1977 and 1988.2.5.1 Travel time survey procedureFor all the years that travel time studies have been conducted in Vancouver, the surveyswere conducted in the spring (March/April) in order to relate directly to one another andalso because traffic volumes are normally close to the annual average at that time of theyear. In each year of study three main time periods were surveyed, These were themorning peak periods between 0700-0900 hours; mid-day 1300-1500 hours; and the22Chapter 2. Literature reviewevening peak periods between 1600 and 1800 hours. An additional early morning samplewas taken between 0400-0600 hours, so that peak period travel times could be comparedto the minimum or fixed delay conditions occuring in the very light traffic period atnight.The travel time measurement technique used in all the studies was the test vehiclemethod. As described earlier, the test car was driven over each route at a speed that inthe opinion of the driver was representative of the average speed of the traffic stream atthe time of each run. In addition to the driver a recorder with a stop watch rode in thetest car. The recorder started the watch at the beginning of the test run and recorded thetime at various points along the route. A minimum of three runs was obtained on eachroute in each time period except 0400-0600 hours which was sampled once.The measurement of stopped time delay; the delay due to traffic signals and congestionat traffic signals entails stationary observation at the subject intersection. The causes ofdelay were categorized in the data collection process to assist in remedying problems. Theprincipal categories of delay considered include; delay due to congestion at traffic signals,delay due to pedestrians and delay due to stop signs. The stopped delay on a link wastaken as the delay at the downstream intersection of the link.23Chapter 2. Literature review2.5.2 Comparison of travel time data for VancouverTable 2.2 shows travel time data for some streets in Vancouver outside the downtownpeninsula during the p.m. peak hour over a period of time. The information from the tableindicates that travel time over the years have generally remained fairly constant or haveshown slight improvements over time, although traffic count data have shown that trafficvolumes have been increasing over time. Similar pattern was exhibited for the a.m. peakperiod and the mid-day travel time data as well. Table 2.3 also shows the travel timedata for some streets in the Central Business District of Vancouver during the p.m. peakperiod over a period of time. The travel time trend over the years shown in this table issimilar to that of Table 2.2. The mid-day and the a.m. peak trends were also similar inthis case.Graphical comparison of the travel times over the years during the p.m. peak period isshown in Figure 2.1. This comparison is for the years of 1961, 1966,1977 and 1987-88.The comparison is made by drawing five-minute isochrones with the common origin atthe intersection of Granville Street and Georgia Street. This comparison confirms thetrends identified in Tables 2.2 and 2.3. That is travel times have been surprisinglyconsistent over the past three decades, although these were years of rapid expansion ofthe population and number of automobiles on the City’s roads.24Chapter 2. Literature reviewSTREET Distance Year/Travel time(mile)1988 1977 1968 1966 1960________mm. mm. mm. mi mm.BROADWAYBoundary - Alma 7.41 20.20 20.19 22.96 23.56 24.05Alma - Boundary 7.41 23.65 24.69 25.64 23.59 25.26MAINMarine - Prior 4.57 11.72 12.74 13.24 12.94 12.72Prior - Marine 4.57 16.82 15.78 13.38 16.04 13.22OAK19th - 70th avenue 3.18 8.22 9.37 10.84 9.58 8.257Oth-l9th avenue 3.18 7.83 7.85 8.99 8.06 7.2841STMarine- Kingsway 7.29 19.15 18.91 20.59 19.76 19.17Kingsway- Marine 7.29 17.22 18.39 19.11 18.44 18.63Table 22 Average travel time comparisons for some major streets inVancouver outside the Downtown peninsulaSource: City of Vancouver’s Engineering Department25Chapter 2. Literature reviewSTREET Distance Year/Travel time(mile)1988 1977 1967 1965 1959mm. mm. mm. mm. mm.ABBOTPender-Cordova 0.13 - 0.64 2.10 0.82 0.84Cordova-Water 0.07 0.33 0.19 0.70 0.49 -BURRARDPacific-Hastings 0.98 5.43 5.35 3.73 4.12 4.53Hastings-Pacific 0.98 4.67 4.70 4.58 4.37 4.46DAVIEThurlow-Richards 0.43 1.75 2.50 3.21 3.40 3.46Richards-Thurlow 0.43 3.04 3.77 2.22 2.78 2.82GEORGIACadero-Beatty 1.15 7.16 7.24 5.24 6.41 5.60Homer-Cadero 0.97 6.12 6.23 7.87 7.26 5.84Table 2.3 Average travel time comparison for some streets in Vancouverwithin the Downtown peninsulaSource: City of Vancouver’s Engineering Department26Chapter 2. Literature review27Chapter 3RESEARCH METHODOLOGYThe research procedure was in three main sections. The first section involved testing theusable mode’s. The second section involved the modification of the models, by fittingthem with observed data and estimating the parameters that best fitted the observed data,and the third section was the validation of the revised models.3.1 Testing the applicable modelsThe applicable models, namely the GVRD model, the BPR model and the Davidsonmodel, were all tested using traffic flow and average travel time data collected on linksof four arterial streets in the City of Vancouver. The links are 41st Avenue to 49thAvenue on Oak Street, 57th Avenue to 49th Avenue on Oak Street, Fraser Street to ClarkStreet on 12th Avenue and 16th Avenue to King Edward Avenue on Arbutus Street. Thereason for choosing two links on Oak Street was to investigate whether data collected onlinks of the same street produce similar results with the models as compared to data fromother streets.28Chapter 3. Research methodology3.1.1 Description of the data collection sitesThe link between 49th Avenue and 41st Avenue on Oak Street has three driving lanes inthe northbound direction with an exclusive left turn lane. Two of the three driving lanesserve through traffic and the third lane which is the curb lane serves right turning trafficand through traffic. For vehicles turning left however, an exclusive left turn lane isprovided. Data were collected on this link in the northbound direction, during the morningpeak period. Traffic volume counts were conducted at a section which is approximately400 metres upstream of the intersection of 41st Avenue on Oak Street.For the section between 57th Avenue and 49th Avenue on Oak Street, data was collectedduring the morning peak period in the northbound direction. The traffic volume countswere conducted at a section located at about 300 metres upstream of the intersection of49th Avenue on Oak Street. This section consists of three driving lanes, two of whichserve through traffic and the third one, which is the curb lane serves both through andright turning vehicles. An exclusive left turn lane is provided at the intersection of 49thAvenue on Oak Street in the northbound direction.The data collection between Clark Street and Fraser Street on 12th Avenue was in theEastbound direction during the evening peak period. Traffic volume observations were29Chapter 3. Research methodologymade at a distance of about 200 metres upstream of the intersection of Clark Street on12th Avenue. This section also has two driving lanes in the Eastbound direction, one ofwhich serves through traffic and the other serves through and right turning traffic. At theintersection of Clark on 12th Avenue an exclusive left turn lane is provided for vehiclesapproaching the intersection in the Eastbound direction and wishing to turn left.The section between 16th Avenue and King Edward Avenue on Arbutus Street has twodriving lanes in the northbound direction with an exclusive left turn lane. One of the twodriving lanes, similar to the case on 12th Avenue, serves through traffic with the otherserving both through and right turning traffic. The data collection was in the northbounddirection during the morning peak period with traffic flows observed at about mid blocklocations.Table 3.1 shows the signal timing plans that were in operation on the study links in thedirection of the approach intersections, during the period data were collected.30Chapter 3. Research methodologySite Direction Time of Cycle time Green time % Greenday at approach at approach time atintersection intersection approach(seconds) (seconds) intersectionOak St. Northbound Morning 75 43 574lst-49th peakperiodOak St. Northbound Morning 75 46 6149th-57th peakperiod12th Ave. Eastbound Evening 55 26 49Clark-Fraser peakperiodArbutus St. Northbound Morning 65 32 4916th-King peakEdward periodTable 3.1 Signal timing plans in operation at the various approach intersectionsSource: City of Vancouver’s Engineering Department (May, 1992)3.1.2 Data collection procedureData required for testing the models were traffic flow and travel time data. Data werecollected under clear weather conditions on all the four links described previously. Noparked vehicles were observed along any of the four links during the period data werebeing colleced. The data collection methodology was the simultaneous observation ofshort period traffic counts and measurement of the average cruising speed of the traffic31Chapter 3. Research methodologystream using the test vehicle technique. The driver of the test vehicle drove each timewith the traffic stream at a speed which in his opinion was representative of the averagespeed of the traffic stream. Speed measurements which appeared to be affected by thedownstream or the upstream traffic signals were ignored. These were speeds when the testvehicle was either accelerating or deccelerating. An observer riding in the test vehiclewith the driver recorded the observed cruising speed of the test vehicle each time it pliedthe test section. Another observer conducted traffic volume counts of continuous trafficstreams of which the test vehicle was part and recorded the time in seconds using a stopwatch. At the end of each run through the test section, the recorder in the test vehiclereported the observed cruising speed and this was recorded against the measured trafficflow before the next run. The travel times were calculated using the speed data collected.Table 3.2 shows the data collected on the link between 41st Avenue and 49th Avenue onOak Street. The predicted travel times corresponding to the measured flows are alsoshown in Table 3.3. The observed data and the corresponding predicted data for the otherthree links are included in Appendix A.The capacity values and free flow speed values used in the prediction models are thosecurrently being used by the GVRD for the links in traffic assignment. The capacity valueused for all the links is 1067 pcph per lane and the free flow speed value used is 50km/hr.32Chapter 3. Research methodologyTraffic flow speed Travel time Traffic flow Speed Travel time(veh/hr) (km\hr) (secs/km) (veh/hr) (krn/hr) (secs/km)480 72 50.0 978 67 53.7780 65 55.4 1632 58 62.1850 60 60.0 2184 49 73.51100 65 55.4 2640 42 85.71350 62 58.1 2520 34 105.01400 63 57.1 2040 50 72.01600 55 65.5 1800 55 65.51750 59 61.0 600 68 52.91850 55 65.1 1392 60 60.02100 47 76.6 2100 50 72.02200 51 70.6 1800 62 58.12250 48 75.0 640 64 56.32300 50 72.0 840 60 60.02350 49 73.5 2100 49 73.52400 47 76.6 2292 47 76.62450 44 81.8 2400 42 85.72500 42 85.7 1680 56 64.32550 41 87.8 2650 41 90.02600 40 90.0 2700 40 100.0Table 3.2 Traffic flow and speed data, collected between 41st Avenue and 49th Avenueon Oak Street, with computed travel times (May 8, 1992)33Chapter 3. Research methodologyTraffic Predicted Travel Times Traffic Predicted Travel TimesFlow (seconds/km) Flow(veh/hr)GVRD BPR Davidson (veh/hr) GVRD BPR Davidson480 72.1 72.0 78.4 2550 124.9 86.9 233.4780 72.4 72.1 83.7 2640 129.1 88.1 252.0850 72.6 72.2 85.1 2700 133.6 89.3 274,51100 73.7 72.5 91.0 978 73.0 72.3 87.61350 75.8 73.1 98.6 1632 80.0 74.2 109.41400 76.5 73.8 100.3 2184 97.2 79.1 48.21600 79.6 74.1 108.5 2640 128.3 87.8 248.01750 82.9 75.1 116.1 2520 118.7 85.1 209.51850 85.6 75.8 122.1 2040 92.1 77.6 136.42100 94.5 78.3 142.0 1800 84.2 75.4 119.02200 99.1 79.6 152.8 640 72.2 72.0 80.42250 101.7 80.3 159.1 1380 76.2 73.2 99.62350 107.3 81.9 173.9 1800 84.2 75.4 119.02400 110.4 82.8 182.8 600 72.2 72.0 80.42450 113.7 83.7 192.8 850 72.6 72.2 84.92500 117.3 84.7 204.4 2100 94.5 78.3 142.02550 121.0 85.8 217.7 2292 103.0 80.8 163.22400 110.4 82.8 182.8 1680 81.2 74.6 112.3Table 3.3 Predicted travel times, using the old travel time models, with traffic flowdata collected between 41st Avenue and 49th Avenue on Oak Street34Chapter 3. Research methodology3.1.4 Testing the effectiveness of the models and test resultsTo investigate the effectiveness of the models in predicting travel time, the observed andthe predicted data for each of the links were graphically compared as shown in Figures3.1, 3.2, 3.3 and 3.4. The graphs indicate that the predicted data have significantdeviations from the observed data for all the locations considered. None of the graphsrepresenting the predicted data are reasonably within the observed data (scatter diagrams).Instead, all the models overestimated the link travel times. An indication that, none of theprediction models fitted the data observed on any of the links.To provide stronger evidence that the models poorly fitted the observed data, a hypothesistest was performed to test the equality of the means of the observed and the predictedtravel times by the models for each link. The test procedure involved testing a nullhypotheses, that the means of the observed and the predicted travel times are equalagainst an alternative hypotheses that they are not equal. The null hypotheses is given as:H0: ‘=(3.1)The alternative hypothesis is therefore given as:H0: p1 P’2 (3.2)35Chapter 3. Research methodologyIf X and Y are the means and S and S, are the standard deviations of the sampledobserved and predicted travel times respectively, then a random variable can be definedas:= 2 2(3.3)+n and n2 are the sample sizes of the observed and the predicted travel times respectively.The Z1 value computed from the above equation is compared with a value Z2 read fromstandard tables at a defined significant level(cL). If the values of Z1 and Z2 are such thatZ1 is greater than Z2, the null hypothesis is rejected in favour of the alternativehypothesis; and if Z1 is less than Z2 the null hypothesis is accepted and the alternativehypothesis is rejected. The acceptance of the null hypothesis signifies that the modelbeing tested is a good fit for the observed data and vice versa. The tests were conductedat three different significant levels of 2%, 5% and 10%. The lower the significance levelof the test the more crude the test; in otherwords the greater the chance for a predictionmodel to be judged as a good fit for the observed data. The means and the variances ofthe observed and the predicted data are as summarised in Table 3.4, with the test resultsin Table 3.5. The results from Table 3.5 demonstrate that the null hypothesis be rejectedfor all the models even at a significance level as low as 2%. This provides a strong36Chapter 3. Research methodologyenough evidence that all the models being tested poorly fitted the observed data.The next stage of the research was calibration of the model parameters to fit the observeddata on all the links.37Chapter 3. Research methodologyC))-D0C-)cI,ci)S4-’ci)flow(veh/min)30025020015010050 AAA A AAA ±AAAA00 10 20 30 40Observed50dataAGvrd estimatesBpr estimates Davidson estimates*Figure 3.1 (Testing the models) Fitting the old prediction models with obsen’ed data,collected between 41st Avenue and 49th Avenue on Oak Street38Chaptei 3. Resedrch methodology20018016014010060Flow (veh/min)Observed data Davidson estimatesABpr estImates Gvrd estimates•••••EJFigure 32 (Testing the models) Fitting the old prediction models with observed data,collected beeen 49th Avenue and 57th Avenue on Oak StreetU,-cCC)120ciEC80AAA4010 15 20 25 30 35 40 45 5039C,)-DC0-o(I)C)EC)>ct31F-Flow (veh/min)Chapter 3. Research methodologyFigure 3.3 (Testing the models) Fitting the old prediction models with observed data,collected between Clark Street and Fraser Street on 12th Avenue25020015010050012AAA AAA14 16 18 20 22 24 26 28AObserved data Davidson estimatesBpr estimates Gvrd data—*--40Chapter 3. Research methodologyC’)-D0C)c2)C’)>F-Flow (veh/min)Figure 3.4 (Testing the models) Fitting the old prediction models with obsen’ed data,collected between 16th Avenue and King Edward Avenue on Arbutus Street.30025020015010050AAAAAAA010 12 14 16 18 20 22 24Observed dataABpr esth77atesDavidson estimatesGvrd estimates41Chapter 3. Research methodologySite ParameterObserved GVRD BPR DavidsonModel Model ModelOak Street Mean(s/km) 70.7 94.3 78.3 144.74lst-49th Variance(s/km)2 182.5 362,9 128.6 2876.2Sample size(n) 38 38 38 38Oak Street Mean(s/krn) 74.5 84.7 87.7 120.549th-57th Variance(s/km)2 180.4 96.9 149.1 601.1Sample size(n) 32 32 32 32Arbutus St. Mean(sjkm) 82.6 94.7 98.4 155.116th-King Variance(s/km)2 252.1 151.1 204.5 1536.5Edward Sample size(n) 35 35 35 3512th Ave. Mean(s/km) 69.9 87.5 77.4 135.2Clark-Fraser Variance(s/krn)2 166.4 109.2 176.9 986.8Sample size(n) 32 32 32 32Table 3.4 Values of the means, variances and sample sizes for the observed andpredicted data with the old models42Chapter 3. Research methodologyModels Sites/Z1 values Z2 at Z2 at Z2 ato=10% cL=5% cL=2%12th Ave. Oak St. Oak St. ArbutusClark-Fraser 4 lst-49th 49th-57th St.16th-KingEdwardDavidson 10.88 8.26 9.31 10.14 1.645 1.960 2.326GVRD 6.00 6.24 3.48 3.60 1.645 1.960 2.326BPR 3.29 3.22 4.14 4.36 1.645 1.960 2.326Table 3.5 Results from testing the old travel time models3.2 Modification of the modelsThe three models being investigated can be rewritten as shown below.The GVRD model as:T= T0[1 + z(V/CL9Y] (34)the BPR model as:T= T0[1 + c(V/C)] (3.5)43Chapter 3. Research methodologyand the Davidson model as:T= T0[1 + jq/(s — q)] (3.6)The modification of the models involved the estimation of the parameters n, x, y, c’, fiand j which fitted the field data. The values of capacity and free flow travel time weredirectly measured in the field.3.2.1 Determination of Capacity valuesAccording to the Highway Capacity Manual, capacity is defined as “the maximum hourlyrate at which persons or vehicles can reasonably be expected to traverse a point oruniform section of a lane or roadway during a given time period under prevailingroadway traffic and control conditions.”The capacity of arterial streets like the cases being considered are significantly influencedby factors such as: the arterial environment, the interaction between vehicles and theeffect of traffic signals. The arterial environment includes geometric characteristics ofthe facility and adjacent land uses; the number of lanes and lane width, type of median,access point density and spacing between signalised intersections, existence of parking,level of pedestrian activity and speed limit. The presence of traffic signals also44Chapter 3. Research methodologyconsiderably reduce the capacity of arterial streets. The reason being that the trafficsignals force the vehicles to stop and to remain stopped for a certain time and then releasethe vehicles in platoons. Thus the delays and speed changes result in reduction ofcapacities.The determination of the capacity of links of arterial streets are typically based on the useof an ideal maximum traffic flow rate that is adjusted to reflect site- specific conditionsthat may not be ideal. Regardless of the specific procedure used in arterial capacityanalyses; saturation flow is used as the base flow rate. Accurate estimation of arterialcapacities is therefore based on accurate estimation of saturation flows.The highway capacity manual gave an estimate of saturation flow rate under idealconditions as 1800 passenger cars per hour green per lane (pcphgpl) which can beadjusted for site specific conditions, The capacity of an arterial link is then given as:C1 = S1x(g/C) (3.7)where:C1 capacity of lane group or approach i, in vphS1 = adjusted saturation flow rate for lane group or approach i, in vphg; and(gIC)1 = green ratio for lane group or approach i45Chapter 3. Research methodologyMeasurement of saturation flow rates by various researchers and research groups havehowever demonstrated that, saturation flow rates vary widely depending on some otherlocal conditions. These conditions include weather conditions and unusual traffic mixeswhich are not considered in the Highway Capacity Manual methodology. In otherwords,the ideal value of 1800 pcphgpl might not be applicable to some areas. This is evidencedby some saturation flow studies conducted under non ideal conditions which gave valueshigher than the ideal value of 1800 pcphgl. For example, Webster et. al. (1966),reported a value of 1850 pcphgl for an average site with an effective approach width of10 feet. Miller (1969), also reported a value of 1810 pcphgl for the same approach width.All these approach widths are less than the ideal width of 12 feet. Studies by researchgroups such as the City of Edmonton in conjunction with the University of Alberta, theUniversity of Kentucky, and the Australian Road Research Board confirmed the highdegree of variability of saturation flows. Thus for this studies, it was deemed appropriateto determine saturation flow rates using local data, The study procedure is described inthe next section.3.2.1.1 Measurement of Saturation flowThe method of saturation flow measurement adopted was the one recommended by theHighway Capacity Manual. The studies were undertaken by two people; one beingassigned as the timer with a stop watch and the other observer. The crosswalk at the46Chapter 3. Research methodologyintersections was chosen as the reference point for the studies. The studies wereconducted at the approach intersections of the four links during the same time periods thatflow and speed data were collected on the links, but on different days. The timer startedthe stop watch at the beginning of the green light and the times that the fourth and thelast vehicle in the queue crossed the stop line while the traffic signal was still green wererecorded. This procedure was repeated for five different cycles. The average headwaywas then calculated from the collected data hence the saturation flow rate. The saturationflow was measured for the through lane and the curb lane which is shared by through andright turning vehicles. The left turning vehicles were not considered; since an exclusiveleft turning lane is provided at each of the intersections, as such the left turning vehiclesdid not influence the results of the studies. Tables 3.6 and 3.7 show detailed results ofthe studies at the intersection of 41st Avenue and Oak Street. The saturation flow rate forthe through lane is calculated as1360o 223 7.pcphgl (3 8)For the shared lane the saturation flow rate is given as:2.275= 158 2.pcphgl (3 . 9)47Chapter 3. Research methodologyFor the two through lanes and the one shared lane at this study location, with theproportion of green during the morning peak hours as 0.57, the capacity of the link or theapproach lanes is calculated as(2x2237+1582)xO.57=345Opcuphper 3lanes—ll5Opcphpl (3.10)The saturation flows and capacity values for the other three links are summarised inAppendix A, Table A.l048Chapter 3. Research methodologyCycles Number of Time of crossing Headway pervehicles in reference point vehiclequeue (seconds) (A-B)/C(C) (seconds)Fourth lastvehicle vehicle(B) (A)1 12 8.05 21.76 1.7132 10 7.97 16.42 1.4003 14 7.74 23.76 1.6024 10 7.5 17.82 1.7205 13 8.0 22.49 1.610Mean headway = 1.609sTable 3.6 Results of saturation flow studies at the intersection of 41st Avenueon Oak Street for through lanes in the northbound direction during the morning peakperiod (May 13, 1992)49Chapter 3. Research methodologyCycles Number of Time of crossing Headway pervehicles in reference point vehiclequeue (seconds) (A-B)/C(C) (seconds)Fourth lastvehicle vehicle(B) (A)1 9 10.02 27.22 2.442 11 8.56 26.90 2.623 12 9.64 27.48 2.234 10 10.76 23.90 2.195 12 10.33 29.21 2.36Mean headway = 2.275sTable 3.7 Results of saturation flow studies for shared lane(right and through) at theintersection of 41st Avenue on Oak Street in the northbound direction during themorning peak period (May 13, 1992)3.2.2 Determination of free flow travel timeIn testing the models the speed value used in calculating the free flow travel time was 50km/br, which is the value currently being used by the GVRD in traffic assignment for thetest section. However, this value is the posted speed limit and as such is inappropriate foruse in determing free flow travel time. The imposition of speed limit on streets is forreasons of safety and does not represent free flow speed of drivers. An accurate50Chapter 3. Research methodologydetermination of free flow speed is therefore neccesary.Free flow speed is defined as: TI the speed adopted by drivers on a segment of road wheninfluenced by local alignment but uninfluenced by other traffic.(Navin 1991). As thedefinition suggests, free flow speeds could be observed at late night or early morningbefore peak period conditions when traffic volumes are light. A representative study wastherefore conducted on Oak street to determine the average free flow speed.3.2.2.1 Measurement of free flow speedThe method adopted for the free flow speed measurement was the moving observermethod using a test vehicle. The procedure involved following a lead vehicle with the testvehicle at late night, after 11 p.m. At this time of the night, the traffic volumes aresignificantly lower than during the day and the average time headway between vehiclescould be as high as 30 minutes. The test vehicle followed the lead vehicle and simulatedthe speed of the lead vehicle by maintaining a fairly constant headway between it and thelead vehicle. The speedometer readings of the test vehicle while it was at cruising speedwere recorded by an observer in the test vehicle. Speed values affected by traffic signalsor by the presence of other vehicles were not recorded. The procedure was repeated witha number of vehicles for three nights and the results are as tabulated in Table 3.8.Speeds of some few vehicles considered to be travelling at unreasonably high speeds were51Chapter 3. Research methodologyignored. The mean of the recorded speeds calculated as 62.2km/hr was adopted as theaverage free flow speed.Lead Free flow Lead Free flowvehicle speed(km/hr) vehicle speed(km/hr)1 55 11 682 52 12 753 65 13 704 60 14 705 62 15 556 68 16 577 72 17 608 65 18 609 50 19 6510 55 20 60Average free flow speed 62.2km/hrStandard deviation = 6.63km/hrTable 3.8 Results offree flow travel time studies on Oak Street in the northbounddirection after 11 p.m. in May, 1992. Posted speed limit = 50km/hr3.2.3 Estimation of the model parametersThe parameter estimation procedure employed to determine the model parameters for the52Chapter 3. Research methodologythree models under review is the method of least squares. This method involvesestimating the values of the parameters which minimize the sum of the squares ofdifferences between the observed and predicted travel times based on the observed flows.3.2.3.1 Estimation of the parameters for the BPR modelThe BPR model (Equation 3.5) could be linearised as shown below.in ( = 1nV/C + in (3 .11)Using data collected on the link between 41st Avenue and 49th Avenue on Oak Street,the average free flow travel time (Ta,) and the capacity value (C) determined for that link,a linear regression analysis of ln(TIT0 -1) on mV/C was performed. The linear regressionanalysis which is based on the method of least square estimation produced the value offi equal to 4.03 and the value of lnc equal to -0.65; from which c was determined as0.52. The coefficient of regression, (R2) was found to be 0.86, which is large enough forthe estimated parameter values to be accepted as reasonable.With the new values of c and fi, a revised BPR model is derived as:T=T0[1+O.52(V/C)4°3] (3.12)53Chapter 3. Research methodology3.2.3.2 Estimation of the parameter for the Davidson’s modelThe Davidson’s model (Equation 3.6) could also be rewritten as(-f- - 1) = j( q (3.13)T0 s-qUsing the same field data as for the BPR model, a linear regression analysis of (TIT0 -1)on qI(s-q) produced the value ofj equal to 0.22 with an R2 value of 0.89.With the new value of j, the Davidson’s model can be written in a new form as shownin Equation (3.14).T= T0[1 + O.22(q/s—q)] (3.14)3.2.3.3 Estimation of the parameters for the GYRD modelLike the two previous models, the parameters for the GVRD model were also estimatedbased on the method of least squares, but by solving a system of equationssimultaneously. The GVRD model (Equation 3.4) like the BPR model can be written as:1n(—L-1) = ylnV1 + lnz - ylnCLx (3.15)54Chapter 3. Research methodologyAlso for a set of travel times Ti computed from observed flows and by putting(T/To-1) as W, the sum of the squares of differences (U) between observed and predictedtravel times can be written as:zV2-”[w1, — ] (3.16)(CL-”)-’For those parameters that minimize the value of U, the following partial derivatives ofU with respect to the parameters hold.1=0 (3.17)ôz (CL9y= W1VnVCL wIvIY_(CL9V12Y1nV_1nCL V’]- 0 (3.18)The value of y was obtained by performing a linear regression analysis of ln(T/T0-1) oninVi using the same data used for the earlier models. The value of y was found to be 4.03with the R2 value as 0.86. The constant term expression from Equation 3.9 was alsofound to be -16.2 and therefore the constant term can be written as:lnZ—ylnCL” = —16.2 (3.19)55Chapter 3. Research methodologyWith the value of y known, Equations 3.18 and 3.19 were solved simultaneously for theremaining parameters. The estimates of x and z were found to be 1.12 and 0.64respectively. The GVRD model can therefore be written in a new form as:T = T0[1 + 0.64(V/CL’12)°3 (3.20)3.3 Validation of the revised modelsValidation is defined as “the extent to which any measuring instrument measures what itis intended to measure’ Carmines and Zeller (1979). Validation of any measuringinstrument is therefore very important in order to ensure that it sufficiently representswhat it purports to measure.There are several procedures for validating measuring instruments and the choice of anyparticular procedure or procedures depends on the measuring instrument being considered.Some of these validation methods are face validation, hypotheses validation and externalvalidation. The method of validation considered applicable and adopted in this researchis the external validation. External validation is achieved by comparing the measuringinstrument which are the revised models in this case to other field data.The three revised models shown in Equations 3.12, 3.14 and 3.20 were derived based on56Chapter 3. Research methodologydata collected on only the link between 41st Avenue and 49th Avenue on Oak Street.Therefore to verify their applicability to other arterial links, they were validated with datacollected on the other three links which are, 57th Avenue to 49th Avenue on Oak Street,Fraser Street to Clark Street on 12th Avenue and 16th Avenue to King Edward Avenueon Arbutus Street.3.3.1 The validation procedure and resultsThe basis for testing the validity of the revised models, was by the hypothesis test of theequality of the means of the predicted and the observed data as described under section3.1.4. For the observed data from each of the sections, the corresponding predicted traveltimes (included in Appendix A) were computed using the various revised travel timemodels. The value of Z1 was calculated for each case as in equation 3.1 and comparedwith values of Z2 read from standard tables at different significant levels. The means andvariances of the predicted data sets using the revised models are shown in Table 3.9.Also an attempt was made to fit the revised models with the observed data for eachlocation. These are shown in Figures 3.5, 3.6 and 3.7.The results from the validation of the models as revealed by the hypothesis test are shownin Table 3.10. The results from this test indicate that all the models proved valid at leastat the 2% significant level of the test for all the sites investigated. For observed datafrom 12th Avenue between Fraser and Clark, the Davidson model fitted the data best at57Chapter 3. Research methodologythe 10% significant level of the test, with the GVRD model fitting the data least at the2% significant level. The BPR model however provided the best fit for the observed datafrom Oak Street at the 10% significant level of the test, with the Davidson model beingthe least fitting this time and fitted the data at the 5% significant level, For the observeddata on Arbutus Street, the BPR model was the best fitting model and fitted the observeddata at the 10% significant level of the test, while the GVRD model fitted the data leastbut also at the 10% significant level. These patterns are better illustrated in Figures 3.5,3.6 and 3.7.The revised models could be used to estimate expected values and variances of traveltimes at particular periods on arterial links in the City of Vancouver. An example of theestimation procedure is included in Appendix B.58Chapter 3. Research methodologySite ParameterGVRD BPR DavidsonModel Model ModelOak Street Mean(s/km) 68.90 70.68 75.1949th-57th Variance(s/km)2 71.57 96.77 75.52Sample size(n) 32 32 32Arbutus St. Mean(s/km) 77.50 80.11 87.4716th-King Variance(s/km)2 111.50 143.51 193.03Edward Sample size(n) 35 35 3512th Ave. Mean(s/km) 67.14 68.84 75.68Clark-Fraser Variance(s/km)2 80.6 103.98 123.97Sample size(n) 32 32 32Table 3.9 Values of the means, variances and sample sizes for the predicted data,with the revised modelsModels Sites/Z1 values Z2 at Z2 at Z2 atcL=10% c=5% oc=2%12th Ave. Oak St. ArbutusSt.Clark- 49th-57th 16th -KingFraser EdwardDavidson 0.27 1.91 1.46 1,645 1.960 2.326GVRD 2.04 1.00 1.47 1.645 1.960 2.326BPR 1.32 0.37 0.66 1.645 1.960 2.326Table 3.10 Results from validation of the revised models.59Chapter 3. Research methodologyU,C0C-)CDU,CDEI.ci)HFlow (veh/min)Figure 3.5 (Validating the revised models) Fitting the revised models with datacollected on 12th Avenue, between Fraser Street and Clark Street12011010090807060AAAA AAAAA16 18 20 22 24 26 28Observed dataADavidson estimatesBpr estimatesAGvrd estimates—*---60Chapter 3. Research methodologyC,)C0C-)0U,00F-604010 12 14 16 18 20Flow (veh/min)1601401201008022 24Observed data Davidson esthnatesABpr estfrnates Gvrd dataFigure 3.6 (Validating the revised models) Fitting the revised models with datacollected on Arbutus Street, between King Edward Avenue and 16th Avenue61Chapter 3. Research methodologyC.’)C0C)C)U)C)EC)>I—25 30 35 40Flow (veh/min)AAAAAAA110 -1009080706050-20AA45 50Obseived data Davidson esthnatesABpr estimates Gvrd estimates.Figure 3.7 (Validating the revised models) Fitting the revised models with data.collected on Oak Street, between 49th Avenue and 57th Avenue Street62Chapter 4DISCUSSION OF RESULTSThis chapter is in three main sections. The first section discusses capacities and free flowtravel time as measured in the study. The second section offers explanations for the traveltime trends, as observed in the City of Vancouver over the years, and the third sectiondiscusses the various travel time models investigated and the results produced.4.1 Capacity and Free flow travel timeThe saturation flows which were obtained from the study of through lanes for all thestreets analysed were at least 2000 pcphgpl. This value is higher than the ideal saturationflow of 1800 pcphgpl reported by the Highway Capacity Manual although the former wasunder non-ideal conditions. This could be due to the fact that on average many moresmaller cars now use the street system and also, as mentioned previously, some localfactors not considered by the Highway Capacity Manual, might have influenced theresults. As link capacities are determined based on saturation flows, the high valuesobtained for saturation flows resulted in high capacities. However, the capacity valuesused for all the Street sections studied are not significantly different from the capacityvalues currently being used by the GVRD. Records have indicated that the GVRD hasbeen updating the street capacities in their assignment algorithm over time.63Chapter 4. Discussion of resultsThe free flow travel time calculation was based on the average free flow speed measuredin the field, The average free flow speed of about 62.2 km/hr (Table 3.8) is about 20percent greater than the 50km/hr which is currently being used by the GVRD for mostarterials (similar to the ones studied) in transportation planning and traffic assignment forthe city. The average free flow speed of about 62.2 km/hr resulted in a lower free flowtravel time of about 58 seconds per kilometre as compared to the 72 seconds perkilometer produced by using the old value of 50 km/hr. The relatively high value of thefree flow speed, and for that matter, the low value of the free flow travel time, could beattributed to the improved technology which has led to the production of faster vehiclesthat are currently used on the street network.4.2 The travel time patternsThe travel time patterns which have developed over the years in the City of Vancouver,as discussed previously have remained constant or shown slight improvements (Tables 2.2and 2.3). This appears surprising, as there have been increases in traffic volumes andtraffic control devices. The results from this research offer good explanations for theobserved trends. The nature of the three travel time models being investigated showed thatthe higher the capacity values, the lower the link travel times. Also the lower the freeflow travel times, the lower the link travel times. It is therefore clear that factors thatcontribute to an increase in free flow speed hence a reduction in free flow travel time and64Chapter 4. Discussion of resultsincrease in capacities contribute to improvement in travel times. Therefore, although thetraffic volumes and traffic control devices could increase, travel times could remainconstant or even improve, so long as the speeds of vehicles and capacity of the streetnetwork increase. Some factors that contribute to increases in vehicle speeds and capacityof the street network in the City of Vancouver are discussed below.4.2.1 Increase in vehicle speedsThe free flow speed studies that were conducted for this research has revealed that theaverage driver drives at a speed approximately 20 percent higher than the speed limitunder free flow conditions. This indicates about 20 percent reduction in link travel timeper kilometer as compared to driving at the speed limit value of 50 km per hour. Policereports (City Enginering Department 1989) have also shown that drivers are becomingmore aggressive and have been driving faster. Also as mentioned earlier, the productionof faster vehicles in recent times has also contributed to the high speeds. Another factoris that, because commuters still want to maintain their travel times despite the increasein traffic volumes and traffic control devices, they drive at higher speeds at anyopportunity to make up for delays that they anticipate or they might have sufferedpreviously due to traffic control devices or congestion.65Chapter 4. Discussion of results4.2.2 Increase in capacitiesImprovements in Street capacities can significantly reduce travel times of vehicles. Therehave always been constant efforts by the City of Vancouver’s Engineering Departmentto improve the capacity of the Street network to safely accommodate the increasing trafficvolumes. Some of the measures adopted to achieve this are summarised as follows:(i) Arterial Development: The City of Vancouver’s Engineering Department has alwaysundertaken measures to improve its arterial system in order to increase capacities. Typicalexamples of these in the City of Vancouver in recent times are the Pacific Boulevard,Quebec Street, and Marpole By-pass. The development of these arterials have addedsignificant capacities to these corridors.(ii) Major left turn bay programs: Major left turn programs embarked upon by the Cityof Vancouver’s Engineering Department have contributed immensely to capacity increaseshence reducing travel times in the affected streets. A documented example of this is theimprovement in travel time on Granville Street from 71st Avenue to 6th Avenue since1977, which was partly due to the left turn bay improvements along Granville at 70thAvenue and at 41st Avenue.(iii) Curb lane regulation programs: The removal of curb parkings along some streetsections has contributed to capacity increases hence improving the travel times along theaffected streets. An example documented by the City of Vancouver’s EngineeringDepartment is the improvement in travel time since 1977 along the Knight-Clark corridor66Chapter 4. Discussion of resultswhich was partly due to the removal of curb parking along portions of this route.(iv) Street structure and maintenance programsPeriodic maintenance of the street system such as the widening of roads andreconstruction of streets are some factors that provide additional capacities to streets inthe City of Vancouver. Reports from the City’s Engineering Department have shown thatthe improvement in travel time noticed between Kingsway Avenue and Marine Drive onBoundary Road from 1977 to 1988 could be attributed in part to the widening ofBoundary Road southward from Kingsway Avenue.(v) Computerised traffic signal management systemOne of the major factors known to have contributed to travel time improvements in theCity of Vancouver, especially in the last five years, is the installation of computerisedtraffic signal management system by the City’s Engineering Department. Thiscomputerised system, which was installed in 1986 and includes all the 460 traffic signalsin Vancouver, provides an efficient signal coordination and optimisation in the city. Theeffect of this is that the capacity of the street network is increased as the average stoppeddelay of the vehicles is reduced.4.3 The travel time modelsThe three travel time models namely the BPR model, the GVRD model and the Davidsonmodel were first tested for their effectiveness in predicting travel time as discussed under67Chapter 4. Discussion of resultssection 3.1.4. The capacity value and free flow travel time used in testing the models arethe values currently being used by the Greater Vancouver Regional District for the datacollection site and as such, should be presumably correct. However, the test resultsdemonstrated that the models as tested with the parameters currently in use provedunsatisfactory (Table 3.5). Detailed discussions on each of the models is given in thefollowing sections.4.3.1 The BPR modelThe BPR model failed to satisfactorily duplicate the observed data within the levels ofsignificance of the test. The values of a and ft (Equation 3.5), which are normally usedwith this model and were used for the test of the model, are 0.15 and 4 respectively.When the model was fitted directly with observed data to estimate the parameters thatbest fit the observed data, the value offt was found to be 4.03 which is almost the sameas the commonly used value of 4. However, the value of a was found to be 0.52, whichis significantly higher than the the value of 0.15 used in testing the model. The newlyestimated value of a appeared to fit Vancouver data for arterial streets better as confirmedby the validation test results (Table 3.10). The failure of the model to reasonablyduplicate the observed data when tested might therefore be due to the inaccurate valuesof a of 0.15 and also the free flow travel time of 72 seconds per kilometer used, insteadof 58 seconds per kilometer obtained by direct measurement. The capacity value obtained68Chapter 4. Discussion of resultsby direct measurement is almost the same as the one used in testing the model and assuch could not have contributed to the poor results produced by the model when tested.4.3.2 The GVRD modelThe GVRD model (Equation 3.4) when fitted to the calibration data produced the valuesof x, z and y as 1.12, 0.64 and 4.03 respectively. The values of the estimated parametersare all sufficiently close to the old ones being used by the GVRD which are 1.05, 0.6 and4 respectively. The validation test also proved satisfactory with the slightly modified formof the GVRD model and with a free flow travel time of 58 seconds per kilometer. Thefailure of the GVRD model to satisfy the goodness of fit test conducted under section3.1.4 could be attributed solely to the inaccurate value of free flow travel time used.4.3.3 The Davidson modelThe Davidson model (Equation 3.6) produced a j value of 0.22 when fitted with theobserved data. The model also proved successful with the same j value when validatedagainst data from other sites (Table 3.10). The j value used when first testing the modelwas 0.5. This is the mean of the range of 0.4-0.6 recommended by Blunden (1971) forarterial streets. The value of the j parameter of 0.22 produced by the Vancouver data istherefore not within the recommended range for arterial streets.69Chapter 4. Discussion of resultsDavidson’s j parameter has been an object of investigation by several researchers for along time. Menon et. al. (1974) investigated the relationship between the value of j andfactors such as the number of signalised intersections per kilometer, the number of lanes,the lane width, and the environment. In particular, Menon could not find any meaningfulrelationship between the values ofj and any of the factors mentioned above.In this research, although the validation test results (Table 3.10) showed that, of all thesites considered, the model proved most successful with data from Oak Street (the samestreet from which data was collected for calibrating the j parameter), it cannot beconcluded from this research whether or not factors peculiar to Oak Street affected theresults. The failure of the model when it was first tested could therefore be attributed toan erroneous value of the j parameter used initially and also the erroneously high freeflow travel time.70Chapter 5FURTHER RESEARCHThe subject of this thesis is expandable in several directions.Inavailability of data and the high cost involved with collecting current data have madeit impossible for some of the travel time models and algorithms discussed in the literatureto be investigated. As such, with availability of funds, the research could be expanded toinclude all the models and the algorithms.In this study, the travel time models were investigated with data collected on only arterialstreets. The research could therefore be expanded to include other classes of roads suchas roads in the central business district and freeways.More accurate results may be obtained if the arterial streets are classified and average freeflow speeds measured seperately for each road class. The streets could be classified interms of some common factors such as, number of lanes, lane widths and otherenvironmental factors. This is likely to produce more accurate results. Also collection ofmore data in working with the models and comparing the results with those produced bythis research is recommended in any future study on the subject of this thesis.71Chapter 6CONCLUSIONTravel time trends over the past three decades in the City of Vancouver have beeninvestigted in this study. The travel times have remained surprisingly constant, althoughtraffic volumes and the number of traffic control devices have increased.Three traditional travel time models have also been investigated for their validity inpredicting travel time on arterial streets in the City of Vancouver, under present trafficconditions. The pertinent results of the research are summarised in the followingdiscussions.The fairly constant trend of travel times observed in the City of Vancouver over the yearsare principally due to increases in vehicle speeds and increases in the capacity of thestreet network. The increased vehicle speeds and street capacities tend to offset anydeteriorations in travel times that might have resulted from the increased traffic vioumesand the increased number of traffic control devices. The increases in vehicle speeds couldbe ascribed to the advancement in technology, which has contributed to the productionof faster vehicles, that are currently used on the street network. Another likelycontributing factor to the higher vehicle speeds, is the increased aggressiveness of driversas documented in the police reports. The increases in the capacity of the Street network,72Chapter 6. Conclusionare the results of the constant efforts made by the City of Vancouver’s EngineeringDepartment to improve the capacities of the streets. These efforts are seen in the form ofdevelopment of the arterials, major left turn bay programs, curb lane regulation programs,street and structural maintenance programs and the installation of a computerised trafficsignal management system. Also, increases in the number of smaller cars that arecurrently used on the street network have resulted in higher saturation flows, hence highercapacity values.The three traditional travel time models investigated are the BPR model, the Davidsonmodel and the GVRD model. The other travel time models and algorithms discussed inthe literature could not be investigated due to data limitations.The Davidson model (Equation 3.6), proved invalid with data collected on a few arterialstreets in the City of Vancouver. The failure of the model to provide a good fit for thedata is due to the j value of 0.5 and free flow travel time value of 72 seconds perkilometer used. Free flow travel time study conducted in the field produced a value of58 seconds per kilometer. Also a j value of 0.22 provides a good fit for the Vancouverdata. A revised form of the Davidson model developed for the arterial streets inVancouver is given in Equation 3.14.The BPR model (Equation 3.5) also failed to satisfy the goodness of fit test with the data73Chapter 6. Conclusioncollected in the City of Vancouver. Its failure can be attributed to the o value of 0.15 andalso the erroneous value of free flow travel time used. However, an c value of 0.52provides a good fit for data collected on arterial streets in the City of Vancouver. Arevised form of the BPR model derived for arterial streets in the City of Vancouver isgiven in Equation 3.12.The GVRD model (Equation 3.4) also did not provide a good fit with the Vancouver data.However, the constant model parameters, x, y, and z determined by direct calibrationagree fairly well with the ones currently in use. The failure of the model can thereforebe attributed to the erroneous value of free flow speed used when testing the model. Aslightly modified form of the GVRD model as revealed by this research is as given inEquation 3.20.It is recommended that the revised models and the new value of free flow travel timeobtained from this research be considered for use in future traffic assignments on arterialstreets in the City of Vancouver.74REFERENCES1. Ang, A.H. and Tang, H.T. (1975) “Probability concepts in Engineering Planning andDesign.” Vol. 1, Chapter 4.2. Beck, J.V. and Arnold, K.J. (1977). “Parameter Estimation in Engineering andScience.” John Wiley and sons, New York.3. Blunden, W.R. (1971). “Land-use/Transport system.” Pergamon Press, New York.4. Carmines, E.G. and Zeller, A.R. (1979). “Reliability and Validity assessment.” SagePublications, California.5. City Engineering Department (1989). “Travel time study, 1987-88.” Vancouver, B.C.6. City Engineering Department (1980). “Travel time study, 1980.” Vancouver B.C.7. Fisk, C.S. (1988) “ A transportation planning model for detailed traffic analyses.”Transportation Research Series report No. 11, U.B.C., Dept. of Civil Engineering,Vancouver, B.C.8. Fisk, C.S. (1989). “Link travel time functions for traffic assignment.” TransportationResearch Part B, Methodological, Vol. 25B, 103-113.9. Gerlough, D.L. and Huber, M.J. (1975). “Traffic flow theory, A monogragh.” SpecialReport 165, Transportation Research Board, National Research Council, Washington, D.C.10. Haase, R.H. (1968). “Increasing freeway utilization for motorists.” Paper presented at34th national meeting of Operations Research Society of America, Philadelphia.11. Henderson, D.M., Lam, J.K. and Rock Stephen (1988). “The City of Vancouver trafficmanagement system,”Institute of Transportation Engineers Journal, Vol. 58, No.6, 17-21.12. Hoffman, G. and Janko, J. (1990). “Travel times as a basic part of the LISB Guidancestrategy.” TEE Conference Publication, No. 30, 1-8.13. Hogg, R.V. and Ledolter,J. (1987). “Engineering statistics”. Macmillan, London.14. Institute of Transportation Engineers (1982). “Transportation and Traffic EngineeringHandbook.” Prentice Hall, Englewood Cliffs, New Jersey.75References15. Lucas, C. and Davidson, K.B. (1973). “A method of measuring the effectiveness ofa given assignment method and an an application to three different methods.”AustraliaRoad Research Journal, Vol. 5, No. 5, 92-99.16. Lucas, C. and Davidson, K.B. (1974). “Quantitative evaluation of traffic assignmentmethods.’ University of Queensland, Dept. of Civil Engineering. Bulletin No. 17.17. May D. Adolf (1990). “Traffic flow fundamentals.” Prentice Hall, Englewood Cliffs,New Jersey.18. Menon et. al. (1974).”Study of Davidson’s flow travel time relationship.”Proceedings[of the conference]. Australian Road Research Board. Vol. 7, part4, 5-21.19. Miller, A.J. (1969). “On the Australian Road Capacity Guide.” Highway ResearchRecord, 289, 1-13.20. Montgomery, F.O. and May, A.D. (1987). “Factors affecting travel times on urbanradial routes.” Traffic Engineering and Control, Vol. 28, No. 9, 452-458.21. National Research Council. (1985). “Highway Capacity Manual.” TransportationResearch Board, Special Report 209, Wasington D.C., Chapter 9.22. Navin, F.P.D. (1991). “Vehicle operations, traffic speed and delay models.” ForNCHRP 2-18. Research strategies for improving Highway user cost estimating strategies.23. Navin, F.P.D. (1991). “Speed : Some definitions and observations.” Proceedings ofthe Canadian Multidisciplinary Road Safety Conference VII, Vancouver, B.C.24. Oda T. (1990). “Algorithm for prediction of travel time using vehicle sensor data.”lEE Conference Publication No. 320, 40-44.25. Smeed, R.J. (1968). “Some circumstances in which vehicles will reach theirdestinations earlier by starting later.” Transportation Science Vol. 1, No. 4, 308-3 17.26. Stokes W. Roberts. (1989). “Some factors afffecting signalised intersection capacity.”Institute of Transportation Engineers Journal, Vol. 59, No. 1, 35-40.27. Taylor, M.A.P. (1977). “Parameter estimation and sensitivity of parameter values ina flow-rate/travel time relation.” Transportation Science, Vol. 11, No. 3, 275-292.28. U.S. Department of Commerce, Bureau of Public Roads (1964). “Traffic assignment76Referencesmanual.” Office of Planning, Urban Planning Division.29. Usami et. at. (1986). “Travel time prediction algorithm and signal operations atcritical intersections for controlling travel time.” lEE Conference Publication No. 260,205-208.30. Wardrop, J.G. (1952). “Some theoretical aspects of road traffic research.” Excerpt part11, Proceedings of the Institution of Civil Engineers.77Appendix ATABLE OF RESULTSTraffic speed Travel Traffic Speed Travelflow (km\hr) time flow (km/hr) time(veh/hr) (secs/km) (veh/hr) (secs/km)1638 35 102.86 1320 43 83.721590 38 94.74 1374 48 75.001566 35 102.86 1320 45 80.001500 40 90.00 1302 55 65.451500 37 97.30 1260 53 67.921548 42 85.71 1200 50 72.001494 44 81.82 1200 52 69.231380 40 90.00 1218 55 65.451362 45 80.00 1140 57 63.161338 48 75.00 1140 60 60.001116 57 63.16 1032 55 72.001104 55 65.45 1032 52 67.231080 50 72.00 960 55 65.451080 58 62.07 942 60 60.001020 55 65.45 900 58 62.071020 52 69.23 804 65 55.38Table A.] Traffic flow and speed data, collected between Fraser Street and Clark Street on 12thAvenue, with computed travel times (15/5/92)78Appendix ATraffic speed Travel Traffic Speed Travelflow (km\hr) time flow (km/hr) time(veh/hr) (secs/km) (veh/hr) (secs/km)2694 35 102.86 2208 45 75.002640 40 120.00 2112 48 80.002622 35 102.86 2280 45 65.452520 35 102.86 2082 55 67.922484 37 97.30 2250 53 72.002460 40 90.00 2100 50 69.232400 40 90.00 1950 52 65.452280 40 90.00 1980 55 63.162244 45 80.00 1800 57 60.002088 48 75.00 1776 60 60.001740 60 63.16 1560 53 69.231656 55 65.45 1500 55 65.451680 50 72.00 1524 60 60.001620 60 62.07 1440 58 62.071836 55 65.45 1200 65 55.381638 52 69.23 1020 50 72.001704 50 65.45 912 55 65.45Table A.2 Traffic flow and speed data, collected between 49th Avenue and 57th Avenue on OakStreet, with computed travel times (22/5/92)79Appendix ATraffic speed Travel Traffic Speed Travelflow (km\hr) time flow (km/lir) time(veh/hr) (secs/km) (veh/hr) (secs/km)720 50 72.00 1212 42 85.711020 40 90.00 1140 32 112.51080 40 90.00 1320 35 102.861296 35 102.86 1020 40 90.001224 40 90.00 1140 40 90.001242 30 120.00 1320 42 85.711296 30 120.00 960 50 72.001176 40 90.00 1092 40 90.001122 50 72.00 1068 50 72.001122 40 90.00 1044 45 80.001176 45 80.00 858 60 60.001140 40 90.00 1032 55 65.451140 50 72.00 1014 50 72,001098 50 72.00 960 50 72.001098 55 65.45 738 55 65.45912 40 90.00 840 65 55.38864 50 72.00 822 55 65.45774 60 60.00Table A.3 Traffic flow and speed data, collected between 16th Avenue and King Edward Avenueon Arbutus Street, with computed travel times (29/5/92)80Appendix ATraffic Predicted travel times Traffic Predicted travel timesflow (secondstkrn) flow (seconds/km)(veh/hr) (veh/hr)Davidson BPR GVRD Davidson BPR GVRDModel Model Model Model Model Model1638 218.69 86.22 112.68 1320 138.00 77.96 89.051590 199.20 84.61 108.09 1302 135.51 77.64 88.131566 190.94 83.86 105.94 1260 130.15 76.94 86.131500 172.00 81.97 100.53 1200 123.43 76.06 83.611500 172.00 81.97 100.53 1200 123.43 76.06 83.611548 185.27 83.32 104.40 1218 125.34 76.31 84.331494 170.51 81.81 100.08 1140 117.60 75.30 81.441380 147.27 79.13 92.39 1140 117.60 75.30 81.441362 144.32 78.76 91.34 1116 115.48 75.03 80.671338 140.62 78.29 90.00 1104 114.46 74.90 80.301320 138.00 77.96 89.05 1080 112.50 74.65 79.591374 146.27 79.00 92.04 1080 112.50 74.65 79.59960 104.00 73.65 76.72 1020 108.00 74.11 78.03942 102.89 73.53 76.38 1020 108.00 74.11 78.03900 100.42 73.27 75.64 1032 108.86 74.21 78.32804 95.42 72.81 74.31 1032 108.86 74.21 78.32Table A.4 Predicted travel times, using the old travel time models, with traffic flow datacollected between Clark Street and Fraser Street on 12th Avenue81Appendix ATraffic Predicted Travel Times Traffic Predicted iTavel timesflow (seconds/km) flow (seconds/km)(veh/hr)(veh/hr) Davidson BPR GVRD Davidson BPR GVRDModel Model MOdel Model Model720 101.79 76.88 76.20 1020 136.42 91.80 89.09738 103.18 77.39 76.64 1032 138.58 92.83 89.91774 106.15 78.54 77.62 1044 140.84 93.83 90.76822 110.53 80.33 79.16 1068 145.66 95.92 92.56840 112.32 81.09 79.81 1080 148.24 97.02 93.51858 114.20 81.90 80.51 1092 150.94 98.16 94,49864 114.84 82.18 80.75 1098 152.34 98.75 94.99912 120.42 84.66 82.88 1098 152.34 98.75 94.99960 126.86 87.57 85.38 1122 158.31 101.18 97.09960 126.86 87.57 85.38 1122 158.31 101.18 97.091014 135.38 91.41 88.68 1140 163.20 103.12 98.751020 136.42 91.88 89.09 1140 163.20 103.12 98.751242 200.48 115.95 109.78 1140 163.20 103.12 98.751296 230.69 124.17 116.85 1140 163.20 103.12 98.751296 230.69 124.17 116.85 1176 174.26 107.27 102.321320 248.00 128.18 120.29 1176 174.26 107.27 102.321328 248.00 128.18 120.29 1212 187.43 111.83 106.241224 192.39 113.44 107.62Table A.5 Predicted travel times, using the old travel time models, with traffic flow datacollected between 16th Avenue and King Edward Avenue on Arbutus Street82Appendix ATraffic Predicted travel times Traffic Predicted travel timesflow (seconds/km) flow (seconds/km)________(vehJhr)Davidson BPR GVRD(veh/hr)Davidson BPR GVRDModel Model Model Model Model Model2694 179.05 116.57 107.93 2250 132.00 93.57 89.392640 171.00 113.07 105.12 2100 122.40 88.33 85.172622 168.52 111.96 104.22 1950 114.55 84.12 81.772520 156.00 106.05 99.46 1980 116.00 84.88 82.392480 152.13 104.13 97.91 1800 108.00 80.78 79.082460 149.68 102.90 96.91 1776 107.05 80.31 78.702400 144.00 99.97 94.55 1740 105.68 79.65 78.172280 134.18 94.75 90.34 1656 102.67 78.27 77.062244 131.58 93.34 89.20 1680 103.50 78.65 77.362088 121.71 87.96 84.87 1620 101.45 77.74 76.632208 129.10 91.99 88.12 1836 109.47 81.50 79.662112 123.10 88.71 85.47 1638 102.06 78.00 76.842280 134.18 94.75 90.34 1704 104.35 79.04 77.672082 121.38 87.78 84.72 1560 99.53 76.93 75.971200 90.00 73.71 73.38 1500 97.71 76.21 75.391020 86.23 72.89 72.72 1524 98.43 76.49 75.62912 84.21 72.57 72.46 1440 96.00 75.57 74.88Table A.6 Predicted travel times, using the old travel time models, with traffic flow data collectedbetween 57th Avenue and 49th Avenue on Oak Street83Appendix ATraffic Predicted travel times Traffic Predicted travel timesflow (seconds/km) flow (seconds/km)(veh/hr)GVRD BPR Davidson (veh/hr) GVRD BPR DavidsonModel Model Model Model Model Model1638 92.96 97.7 109.99 1338 73.47 75.57 82.321590 89.01 93.22 103.09 1320 72.65 74.64 81.391566 87.16 91.12 100.16 1374 75.22 77.55 84.321500 82.52 85.85 93.44 1320 72.65 74.64 81.391500 82.52 85.85 93.44 1302 71.86 73.74 80.511548 85.83 89.62 98.15 1260 70.14 71.80 78.611494 82.12 85.40 92.91 1200 67.96 69.33 76.231380 75.52 77.90 84.68 1200 67.96 69.33 76.231362 74.62 76.87 83.63 1218 68.59 70.03 76.911140 66.11 67.21 74.16 1020 63.18 63.89 70.761140 66.11 67.21 74.16 1032 63.43 64.17 71.061116 65.45 66.46 73.41 1032 63.43 64.17 71.061104 65.12 66.10 73.05 960 62.06 62.16 69.341080 64.52 65.41 72.36 942 61.76 62.27 68.951080 64.52 65.41 72.36 900 61.13 61.55 68.071020 63.18 63.89 70.76 804 59.99 60.26 66.30Table A.7 Predicted travel times, using the revised travel time models, with traffic flow data,collected between Fraser Street and Clark Street on 12th Avenue84Appendix ATraffic Predicted travel times Traffic Predicted travel timesflow (seconds/km) flow (seconds/km)(veh/hr)GVRD BPR Davidson (veh/hr) GVRD BPR DavidsonModel Model Model Model Model Model2694 88.87 93.90 95.94 2088 69.06 70.86 75.622640 86.46 91.10 93.09 2208 71.85 74.10 78.242622 85.68 90.19 92.21 2112 69.58 71.46 76.112520 81.59 85.43 87.77 2280 73.76 76.33 80.042484 80.26 83.89 86.40 2082 68.93 70.71 75.502460 79.41 82.89 85.53 2250 72.94 75.37 79.272400 77.38 80.53 83.52 2100 69.31 71.16 75.862280 73.76 76.33 80.04 1950 66.39 67.76 73.082244 72.78 75.19 79.12 1980 66.93 68.38 73.601800 64.07 65.07 70.76 1704 62.87 63.67 69.471776 63.76 64.70 70.42 1560 61.42 61.97 67.761740 63.30 64.17 69.94 1500 60.92 61.39 67.111656 62.34 63.05 68.87 1524 61.10 61.61 67.371680 62.60 63.35 69.17 1440 60.47 60.88 66.511620 61.98 62.62 68.44 1200 59.19 59.38 64.381836 64.58 65.66 71.28 1020 58.62 58.72 63.641638 62.16 62.83 68.65 912 58.39 58.46 62.33Table A.8 Predicted travel times, using the revised travel time models, with traffic flow datacollected between 49th Avenue and 57th Avenue on Oak Street85Appendix ATraffic Predicted travel times Traffic Predicted travel timesflow (seconds/km) flow (seconds/km)(veh/hr)GVRD BPR Davidson (veh/hr) GVRD BPR DavidsonModel Model Model Model Model Model720 61.61 62.09 68.56 1140 80.98 84.07 90.331020 72.68 74.65 80.83 1320 99.50 105.1 120.381080 76.48 78.96 85.02 1020 72.68 74.65 80.831296 96.54 101.7 114,25 1140 80.98 84.07 90.331224 88.61 92.72 100.67 1320 99.50 105.1 120.381242 90.46 94.82 103.54 960 69.50 71.04 77.441296 96.54 101.7 114.25 1176 84.05 87.55 94.251212 87.42 91.37 98.91 1122 79.56 82.45 88.591122 79.56 82.45 88.59 1044 74.12 76.29 82.401176 84.05 87.55 94.25 858 65.31 66.29 72.961140 80.98 84.07 90.33 1032 73.39 75.45 81.601140 80.98 84.07 90.33 1014 72.34 74.26 80.461098 77.76 80.41 86.48 960 69.50 71.04 77.441098 77.76 80.41 86.48 738 61.98 62.52 69.051092 77.33 79.92 85.98 912 67.35 68.61 75.161068 75.67 78.04 84.11 864 65.52 66.53 73.19840 64.71 65.61 72.29 774 62.83 63.48 70.10822 64.15 64.98 71.66Table A.9 Predicted travel times, using the revised travel time models, with traffic flow datacollected between 16th Avenue and King Edward Avenue on Arbutus Street86Appendix ASite Saturation flows(pcphgpl)No. of Proportion Averageshared of green at capacityThrough Shared lane and approach per lanelane (Right and through intersection (pcphpl)Through traffic) lanesOak Street 2000 1680 3 0.61 115349th-57thArbutus 2040 1920 2 0.49 970Street16th-King Ed.12th Avenue 2200 2000 2 0.49 1012Clark-FraserTable A.1O Summary of Saturation flow study results and determined Capacity values87Appendix BEXPECTED VALUES OF TRAVEL TIME AND VARIANCES USING THEREVISED MODELSThe probabilistic mean travel times and variances which commuters should expect whiletravelling on arterial links in the City of Vancouver could be estimated with each of therevised models using an approximate form of the Taylor series expansion. This is asdescribed below.For a general equation of the formg(X) = X1 + X2 + X3 /X4 etc (9.1)where X1 are random variables that may be related, the expected value is given as:E[g(X)] E[g(X)] + 1/2 82g( Cov(X,X) (9.2)i1j1 I jand the variance is also given as follows:Var[g(X)J = CC1ov(X,X) (9.3)I1 j1where:C1 = ög(X)I6X88Appendix B= ög(X)IXA detailed discussion of the above formulae is found in Ang and Tang (1975) page 198.There are a number of important precautions that need to be considered when estimatingthe probabilistic mean travel times and variances using the revised models. One of theprecautions is that the time period for which the estimates are being made must be welldefined. This is due to the fact that the distribution of the random variable (V,) is veryimportant in the estimation process and it might differ markedly from one time period toanother time period (e.g. peak and off peak conditions). Also link capacities change withthe time of the day depending on which signal timing plan is in operation. More reliableestimates could be obtained if the analysis is made over short time periods and withrelatively large sample sizes.The Taylor series formula is used to estimate the probabilistic means and variances of thetravel time using the revised models and with the traffic flow data collected between 41stAvenue and 49th Avenue on Oak Street. The results however apply to only the link fromwhich the data were collected and the time period during which the data were collectedwhich in this case is the morning peak period. The mean traffic flow rate is 30.58vehicles per minute and the variance is 120.33 (vehicles! minute.)289Appendix BB.1 Estimates using the BPR modelThe revised BPR model is given as:V 4.03T0[l ÷ O.52(—) 1 (9.4)The link capacity value, C is 43 vehicles per minute and the free flow travel time T0 is58seconds per kilometer.The expected value of the travel time is given as:— 82TE(7) 58 ÷ 0.52 + l/2—Var(V) (9.5)434.03 v23.l4xl0I° (9.6)——= 9.52 x i0i°3 (9.7)6V290Appendix BBy using equations 9.7, 9.5 and the mean value of V, the expected value is obtained as71.5 seconds per kilometer.The variance is given as:Var(T) (t3T)2 x Var(V) (9.8)Using equations 9.6, 9.8 and the variance of V, the variance of the link travel time isobtained as 119.38 (seconds/kilometer)2B.2 Estimates using the GVRD modelThe revised GVRD model is given as:4.03T0[1 + O.64( ) 1 (9.9)CL112L is the number of lanes = 3C is the average practical capacity per lane 14.3 vehicles per minute per lane.The expected value of the travel time is given as:91Appendix BE(i) 58 + 0.64V°3 + lf2-—Var(V) (9.10)6.44x10 8V2= 2.32x1OV° (9.11)8v= 7.03x101° (9.12)6112Using equations 9.12, 9.10 and mean value of V, the expected mean link travel time isobtained as 77.9 seconds per kilometer. Also, by using equations 9.8, 9.11 and thevariance of V, the variance of the link travel time is found to be 176.73 (seconds!kilometer)2.B.3 Estimates using the Davidson model.The revised Davidson model is given as:V= T[1 + 0.22 ] (9.13)(C-I’)92Appendix BC is the total link capacity = 57.3 vehicles/minuteE(l) = 58 + 12.76i + 1/2-_ZVar(V) (9.13)57.3-V 8T2= 12.76 (57.3 + (9.15)W (573 - j7)2= 12.76[-- (6566.58 - 2) (9.16)6V2 (57.3 -Using equations 9.16, 9.14 and the mean value of V. the expected mean link travel timeis obtained as 66.61 seconds/kilometer. Also, by using equations 9.15, 9.8 and thevariance of V. the variance of the link travel time was found to be 296.98(seconds/kilometer)2.The Davidson model gave the least expected value of the travel time as 66.61 seconds/km.However, it produced the highest variance as 298.98 (s/km)2. The expected valuesobtained with the BPR model and the GVRD model are 71.5 s/km and 77.95 s/km withvariances as 119.38 (s/km)2 and 176.73 (s/km)2 respectively.93

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