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Optimum turnout spacing on forest haul roads Anderson, Dennis Ivar 1980

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OPTIMUM TURNOUT SPACING ON FOREST HAUL ROADS  by  DENNIS IVAR ANDERSON B.S.F., U n i v e r s i t y o f B r i t i s h Columbia, 1975  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY  i n the FACULTY OF GRADUATE STUDIES Department o f F o r e s t r y  We accept  t h i s t h e s i s as conforming t o the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA April, ©  1980  Dennis Ivar Anderson, 1980  In  presenting  requirements f o r  this  thesis  an  advanced  in  degree  B r i t i s h Columbia, I agree that the available  for  reference  and  partial at  fulfilment the  may  be granted by the  representatives. of my  study. .  I  Head of my  further  permission. .  Department cf  Forestry  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  for  Columbia  j 2 k ^ ^ - ^ - ^ ^ _ —  agree  of  that  scholarly  Department or by  I t i s understood that copying or  t h i s t h e s i s f o r f i n a n c i a l gain written  University  the  L i b r a r y s h a l l make i t f r e e l y  permission f o r e x t e n s i v e copying of t h i s t h e s i s purposes  of  his  publication  s h a l l not be allowed  without  ABSTRACT  Mathematical spacing  of  models are developed  turnouts  and  t o determine the optimum  to p r e d i c t the time l o s t due t o the  a c c e l e r a t i o n and d e c e l e r a t i o n of the v e h i c l e and the vehicle  spends  completely  in  the. turnout.  d e f i n e d a method of d e r i v i n g or measuring the  expected  expected once  introduced.  The  expected  the  empty  vehicle  has  come  i s the  t o a complete h a l t i n the  spacing. .  expected  developed.  F - f a c t o r equation  The  results  shew  were  that  the  total  a t t r i b u t a b l e t o turnout spacing may be a travel  i s reduced turnout  F-factor  delay i n  d i s t a n c e the loaded v e h i c l e i s from the empty v e h i c l e ,  turnout, d i v i d e d by the turnout  the  delay  The concept o f  F - f a c t o r , a measurement of the expected  the t u r n o u t , i s  the  Previous a r t i c l e s have not  i n the turnout a t t r i b u t a b l e t o turnout spacing. the  time  empty time  (i.e.,  2 0 percent)  Two  forms  expected  model i s concerned  the  delay  time  significant  part  The  optimum  with minimizing the sum of  the t u r n o u t c o n s t r u c t i o n and maintenance c o s t s and the c o s t delays  of  but i t s s i g n i f i c a n c e  when compared to the round t r i p time.  spacing  of  of  a t t r i b u t a b l e to turnout s p a c i n g . . I f t h e r e s u l t s of the  optimum t u r n o u t spacing model are used i n the i n i t i a l design of the.road  network  important.  then  the  total  potential  savings  can  be  Implementation of the optimum turnout spacing model  can be achieved with the . u t i l i z a t i o n of t a b l e s . .  iii These  tables  can  be  utilized  c o n s t r u c t i o n o f f o r e s t haul  roads.  as  a guide i n the design  and  TABLE OF CONTENTS  PAGE LIST OF TABLES  vii  LIST OF FIGURES  ..........................................yiii  ACKNOWLEDGEMENT  X  1.0  INTBODOCTION ........................................ .  1  2.0  DEVELOPMENT OF THE OPTIMUM SPACING MODEL ............  5  2.1  FORMATION OF THE PROBLEM  ..  2.2  THE ASSUMPTIONS OF THE MODEL .....................  2.3  THE TURNOUT DELAY TIME  2.4  THEORETICAL DEVELOPMENT OF THE EXPECTED F-FACTOR . . 14 ONE LOADED VEHICLE MEETING ONE EMPTY VEHICLE  2.4.2  ONE EMPTY VEHICLE MEETING A FLEET  OF  19  A FLEET OF LOADED VEHICLES MEETING A FLEET OF  EMPTY VEHICLES  3.0  16  LOADED  VEHICLES  2.4.4  7 10  2.4.1  2.4.3  5  24  HEADWAY PROBABILITY DISTRIBUTIONS ........... . 30  2.4.4.1  UNIFORM ARRIVAL DISTRIBUTION ............. 32  2.4.4.2  EXPONENTIAL HEADWAY DISTRIBUTION ........ 33  2.4.4.3  ERLANG HEADWAY DISTRIBUTION ............  2.4.4.4  PEARSON  34  TYPE I I I HEADWAY DISTRIBUTION ... 35  2.5  DEVELOPMENT OF THE COST EQUATION  37  2.6  DISCOUNTING OF THE COST EQUATION .................  40  THE ARRIVAL DISTRIBUTION OF LOGGING TRUCKS  42  3.1  ARRIVAL DATA FROM TWO OPERATIONS .................  42  3.2  ANALYSIS OF THE ARRIVAL DATA  44  4.0  SIMULATION FACTOR 4.1  5.0  THE  COST  F51  MODELS  VARIABLES  52  AND  THE  MODIFICATION  OF  THE  F-FACTOR  5.1  T H E COST  5.2  T H E TURNOUT  5.3  MODIFICATION  59  OF TRUCK  TRANSPORTATION  CONSTRUCTION  F-FACTOR  59 COSTS  EQUATIONS  . . . . . .  OF T H E MODEL  6.1  T H E U S E O F T H E MODEL  6.2  SENSITIVITY  ANALYSIS OF  THE LOADED  6.2.2  VELOCITY  OF T H E EMPTY  6.2.3  TRAFFIC  6.2.4  EXPECTED  6.2.5  TURNOUT  6.2.6  ADJUSTED  6.2.7  THE  VEHICLE VEHICLE  . . . . . . . . . . . . . . .  CONSTRUCTION  O F T H E ROAD  96  COST  97  HAULING  . . . . . . . . . . . . . . . . . . .  COST  AND  . . . . . . . . . . . . . . . . . .  DECELERATION  OF  98  THE  VEHICLE  99  6.2.8  T H E DISCOUNT  6.2.9  T H E MAINTENANCE  RATE  101  COST  ,,  6.2.10  THE DERIVATIVE  6.2.11  THE LENGTH  6.2.12  T H E HEADWAY  PROBABILITY  6.2.13  THE TURNOUT  SPACING  OF  OF THE EXPECTED THE LOADED  SPACING DISCUSSION  93 94  LIFE  ACCELERATION  85 87  RATE  TRUCK  66 68  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  VELOCITY  USEFUL  62  68  6.2.1  FLOW  ..  AND MAINTENANCE  OF THE EXPECTED  THE USE AND TESTING  EMPTY  7.0  OF THE EXPECTED  ..  THE SIMULATION  THE  VERIFICATION  EQUATIONS  EXPECTED  6.0  FOB  F-FACTOR  VEHICLE  103  . . . . . 1 0 5  . . . . . . . . . . . 1 0 6  DISTRIBUTIONS  AND T H E OPTIMUM  . . . . . .  108  TURNOUT 1 10  AND C O N C L U S I O N S  7. 1  DISCUSSION  7.2  CONCLUSIONS  . . . 1 1 5 115 .119  vi LITERATURE  CITED  .122  APPENDICES  .124  1  ROAD  STANDARDS  2  ABBREVIATIONS,  3  ACCELERATION  4  ANALYSIS  5  SIMULATION  OF  THE  F-FACTOR  6  THE  OF  THE  VEHICLE  7  DERIVATIVES  8  OPTIMUM  9  GRAPHICAL  10  CONVERSION  OF  LENGTH  SURVEY  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 5  SYMBOLS,  AND  TURNOUT  UNITS  DECELERATION  HEADWAY  OF  AND  THE  FACTORS  A  DISTRIBUTIONS  OF  THE  VEHICLE  .131  . . . . . . . . . . . . . . . . . .  '.  . . '  128  .  1  138 5 1 175  EXPECTED  SPACING  RESULTS  OF  . . ; i i . . . . . . . . . . . . . .  F-FACTOR  COMPUTER  EQUATIONS  PROGRAM  SENSITIVITY  .. , , . .  ANALYSIS  .182 . . . . . 1 86 198 . . . . . 2 2 2  vii LIST OF  TABLES  TABLE  PAGE  I  Goodness Of F i t T e s t s — C o a s t a l  II  Goodness Of F i t T e s t s — I n t e r i o r S t u d y — B o t h  III  F-factor  Simulation  —  S i n g l e Loaded V e h i c l e And IV  F - f a c t o r Simulation  —  V  Simulation  Vehicle  Equations 2.18,  Meeting  2.19,  2.30,  —  An  Empty  V e h i c l e Meeting  Equations 2.18,  2.19,  2.33,  2. 34 .  F-factor Fleets 2.30,  Empty  53  56  Loaded V e h i c l e s Based On  VI  50  I n t e r a c t i o n Between A  2.31  F-factor  And  Scales ..  A S i n g l e Empty V e h i c l e ... An  Loaded V e h i c l e s Eased On And  The  Study ............... . 47  57 Simulation  Of And  Vehicles  —  Interaction  Based On  Between  Equations 2.18,  Two 2.19,  2. 3 1 ..............  VII  Truck Hauling  Costs .................................  VIII  Excavation  IX  Turnout Maintenance Costs ..........................  Costs Per Turnout  58 61 64 65  viii LIST OF FIGURES  FIGURE  1  PAGE  I n t e r a c t i o n Between  An  Vehicle 2  Empty  Vehicle  And  A  Loaded  ..  Diagram  Showing  17  The P h y s i c a l S i t u a t i o n Leading To The  C r i t i c a l Headway 3  Case 3--An  20  Empty V e h i c l e . S t o p s At Turnout S  3  Since Next  Turnout I s Occupied ................................... 4  Case 4 — A n  Empty V e h i c l e Remains In  Turnout  S^  Since  Next Turnout I s Occupied 5  Frequency  Histogram  29  Of Headways For C o a s t a l Study (5-  minute I n t e r v a l s ) ..................................... 6  Frequency Histogram Of Headways For I n t e r i o r Study  Frequency  48  Histogram Of Headways For I n t e r i o r Study (5-  minute I n t e r v a l s ) 8  46  (1-  minute I n t e r v a l s ) 7  25  .........  49  E f f e c t Of T r a f f i c Flow Rate On The T o t a l Expected Delay Time F o r Various Turnout Spacings ....................... 69  9  Effect  Of The Turnout Spacing  On  The  Total  Expected  Delay Time For Various T r a f f i c Flow Rates ............. 10 E f f e c t  Of  The  Velocity  Of  70  The Empty V e h i c l e On The  T o t a l Expected Delay Time F o r V a r i o u s V e l o c i t i e s Of The Loaded V e h i c l e 11 E f f e c t  Of The V e l o c i t y Of The  71 Loaded  Vehicle  On  The  T o t a l Expected Delay Time For V a r i o u s V e l o c i t i e s Of The  ix Empty V e h i c l e 12 E f f e c t  Of  The  Optimum Turnout  72 Velocity  Of The Loaded V e h i c l e On The  Spacing For Various V e l o c i t i e s  Of  The  Empty V e h i c l e ......... 13 E f f e c t  Of  The  78  Velocity  Of  The Empty V e h i c l e On The  Optimum Turnout Spacing For Various V e l o c i t i e s  Of  The  Loaded V e h i c l e 14 E f f e c t  79  Of The T r a f f i c Flow Bate On The Optimum  Turnout  Spacing 15 E f f e c t Turnout 16 E f f e c t  80 Of The Turnout C o n s t r u c t i o n Cost On The Spacing Of  The  Optimum Turnout 17 E f f e c t  Optimum 81  Adjusted  Truck  Hauling  Cost  On The  Spacing  Of The Expected  82 U s e f u l L i f e Of The Road  On  The  Optimum Turnout Spacing ............................... 18 E f f e c t  Of  The  Turnout  Spacing  On The Cost Function  Based On Equation 2.38 19 E f f e c t  Of P e r t u r b a t i o n s To The V e l o c i t y  V e h i c l e On The T o t a l 20 E f f e c t  Of  83  84 Of  The  Loaded  Expected Delay Time  P e r t u r b a t i o n s To The V e l o c i t y  90 Of The Loaded  V e h i c l e On The Optimum Turnout Spacing• w............... . 91 21 E f f e c t  Of P e r t u r b a t i o n s To The V e l o c i t y  Of  The  Loaded  V e h i c l e On The Cost D i f f e r e n c e 22 E f f e c t  Of  , 92  D e v i a t i o n s From The Optimum Turnout  On The Maximum Cost D i f f e r e n c e 23 E f f e c t  Of D e v i a t i o n s From The Optimum  On The Average Cost D i f f e r e n c e  Spacing ..113  Turnout  Spacing 114  X  ACKNOWLEDGEMENT  I  would  supervisor,  like  to  acknowledge  Mr. G.G. Young,  the a s s i s t a n c e  throughout  my  of  graduate  my  studies  program and i n the development of t h i s t h e s i s . I  would  like  suggestions  and  development  of  to  thank  constructive the t h e s i s .  extended to Mr..D. Smith being  on  my  of  committee.  M r . . H . . J o l l i f f e . f o r the h e l p f u l criticisms  in  the  initial  Furthermore, my g r a t i t u d e FEEIC  and  Finally  I  i s also  Dr. P . L . . C o t t e l l f o r would  Mr..L..Henkelman f o r w r i t i n g the GPSSV computer  like  to  program used t o  simulate the i n t e r a c t i o n between a f l e e t o f empty v e h i c l e s a f l e e t o f loaded  vehicles.  thank  and  1  1.0  INTRODUCTION  Logging  haul roads are an i n t e g r a l p a r t of the  harvesting  system.  Since h a u l i n g c o s t s account f o r approximately  percent  of  the  total  logging  cost,  networks should be analysed before and process.  The  company  invests  construction  trade  the design of the road during  the  harvesting  h a u l i n g c o s t components are interdependent.. more  the  capital  log  into  road  costs  a  off  between  the  i n the road and the r e s u l t i n g  hauling  cost.  there  is  trade  As a  maintenance  hauling  Consequently, investment  10 to 20  should  and  decrease. initial This  o f f i s r e f l e c t e d i n the road standards t h a t are used f o r  the v a r i o u s s e c t i o n s of the road network. F o r e s t companies and government agencies  include  f e a t u r e s and s e r v i c e c o n d i t i o n s of the road i n t h e i r of  the  components  of  road  standards.  r e f l e c t the r e q u i r e d p h y s i c a l and the  service  assembling  road  standards  characteristics  of  haul roads.. Some of the components that can be c o n s i d e r e d  f o r i n c l u s i o n i n a s p e c i f i c road standard maximum frequency  adverse  and  favourable  be  are:  gradients,  design  speed,  subgrade  width,  o f t u r n o u t s , and right-of-way width.. The  to which components should be i n c l u d e d i n can  The  various  the  road  decision  as  standards  complex and there are no d e f i n i t e c r i t e r i a f o r s o l v i n g  t h i s problem.  The p o t e n t i a l design elements must be  to  which  determine  Environmental  should be i n c l u d e d i n the road  evaluated standards.  c o n d i t i o n s and user f a c t o r s should be c o n s i d e r e d .  2  User  factors  include  development,  use  utilization  and a g r i c u l t u r e ) , and  of of  the  road  greater  than  maintaining  a  be  (e.g.,  S i n g l e - l a n e roads are l e s s  in  expensive  but  the c o s t of truck delays i n t u r n o u t s may  the  added  investment  double-lane. road.  considered.  of  Furthermore, the on  a  range  of  turnout  and  potential  double-lane  Since road c o n s t r u c t i o n and  optimum  be  constructing  road  maintenance  c o s t s i n c r e a s e as the.turnout spacing i s decreased, theoretical  mining  roads can be u t i l i z e d  i n c r e a s e i n the speed of the v e h i c l e s must  timber  government r e g u l a t i o n s .  the e x t r a c t i o n of timber.. construct  future  the road by others  S i n g l e - l a n e roads or double-lane  to  for  there i s  spacing.  may  be  d e s i r a b l e to determine t h i s optimum range p r i o r to d e c i d i n g  on  whether t o b u i l d a s i n g l e - l a n e or double-lane There  determined  the  critical  l a n e s should be switched calculate  or  derive  turnout d e l a y time. a  manual  road.  have been few p u b l i c a t i o n s concerned  of optimum turnout spacing on f o r e s t haul  tc  from  an The  predict  transportation  traffic  with the  roads.  11*^(1965)  flow a t which the number o f  one  to  two  but  he  did  e x p r e s s i o n to determine the United S t a t e s F o r e s t S e r v i c e  the  cost  of  truck  not  expected developed  and  trailer  The  very simple method to estimate the l o s t  time  mile.  a delay i n the  turnout  e q u i v a l e n t to the time r e q u i r e d f o r a loaded v e h i c l e t o  travel  one-half  method  turnout  assumption.  et a l . ,  topic  1947).  The  (Byrne  It  a  assumed  spacing  Porpaczy  previous  but  it  is  failed  in  to  minutes  justify  and Waelti(1976) produced an a r t i c l e  s i m i l a r o b j e c t i v e s as I l ' i n . the  there  system i n v o l v e d a  T h i s paper, as i s  the  per  this with  case: f o r  a r t i c l e s , does not have a w e l l d e f i n e d method of  3  d e r i v i n g or e s t i m a t i n g turnout Since  Il'in(1965)  and  concerned  delay  times.  Porpaczy  and  with determining  Waelti(1976)  have  written  papers  the number of l a n e s  required  f o r l o g t r a n s p o r t a t i o n , t h i s study w i l l c o n c e n t r a t e  on  the problem of d e c i d i n g on the spacing of turnouts on a s i n g l e lane road.  The  method  determine the optimum t u r n o u t spacing as a f u n c t i o n  of  to  major o b j e c t i v e of the study i s  to  develop  t r a f f i c flow r a t e , speeds of the empty and loaded  "expected  useful l i f e  a  vehicles,  of the road", turnout c o s t s , and t r u c k i n g  c o s t s . . A second o b j e c t i v e i s to analyse the s e n s i t i v i t y of the solution  and  test  whether  turnout  spacing has any  economic  s i g n i f i c a n c e with r e s p e c t t o the road design problem.. During the f a l l telephone  survey  of 1976,  the author conducted  of road standards  t h i s survey r e v e a l e d t h a t t h e r e was l e n g t h of t u r n o u t s (2  to  11  a  mail  and  (Appendix 1 ) . . R e s u l t s of extensive v a r i a t i o n i n  (30 to 150 feet) and  the  the spacing of t u r n o u t s  per m i l e ) . . T h i s r e i n f o r c e s the need t o i n v e s t i g a t e  the spacing of turnouts. The  t o t a l time a logging t r u c k spends i n t u r n o u t s  measured,  but i t i s d i f f i c u l t  time that i s d i r e c t l y number  of  reasons  to determine the p o r t i o n of  a t t r i b u t a b l e to turnout s p a c i n g . . a l o g g i n g truck may  turnout than i s necessary.  can  he  required..  may  wait  T h i s paper develops  logging  truck  the t o t a l delay time  has been determined,  road  l o n g e r i n a turnout than i s  a method t o c a l c u l a t e the  time s p e c i f i c a l l y a t t r i b u t a b l e to turnout Once  a  wait l o n g e r i n a given  F o r example, i f the  decide, to  this  For  operator expects there w i l l be d e l a y s f u r t h e r a l o n g i n the network  be  delay  spacing..  a t t r i b u t a b l e t o turnout  the c o s t of the delays must be  spacing  evaluated.  4  The  r e l e v a n t c o s t s are those t h a t are f u n c t i o n s o f the t u r n o u t  s p a c i n g : the t u r n o u t c o n s t r u c t i o n c o s t , the t u r n o u t c o s t , and the Optimum  delay  turnout  function..  cost  spacing  Simulation  attributable can  then  be  models are u t i l i z e d  to  maintenance  turnout  spacing.  derived from the c o s t to  verify  and  the s e n s i t i v i t y of the delay time, c o s t i n g , and optimum spacing  test  turnout  models.  Throughout to convert  the t e x t I m p e r i a l  Imperial units  into  U n i t s i s l o c a t e d i n Appendix 10.  u n i t s are u t i l i z e d .  the  International  A table  System  of  5  2-.0  DEVELOPMENT OF  2sJ.  FOEMATION OF  The  THE  THE  delays be  PROBLEM  determination  u n i t d i s t a n c e of road turnout  OPTIMUM SPACING MODEL  and  c o s t i s not  The  viewed  as  the  so  an  This occurs  turnout  includes  a driver  landing longer  or  realizes at  the  was  over-estimation since  his  some not  than  then  i f he  the  he  may  the  i n the  d e l a y s not  to turnout  be  period) in  by  the time  I f the a c t u a l calculation  delay cost  time  spent  of  would  in  to turnout  later  the  spacing.  queue  at  the  d r i v e slower  and  wait  expect  a delay  d e l a y time  experience  delay  Another approach  time  at  would  these include  i n f l u e n c e the round t r i p  approach t o determining attributable  must  total  since  delay  of the  d i d not  the  empty v e h i c l e would  i n the  the  cost could  unit  of can  but  determined  attributable  s i t u a t i o n s t h a t would n o t  the  of  vehicle  dump,  Furthermore,  of  per  be  utilized  of  the c o s t  v e h i c l e s must spend  could  delay  regardless  delay  or mathematical formulas.  delays  in turnouts  locations.  time  i n the t u r n o u t s cost  time  and  sum  o r manuals  The  the  estimated  result.  If  obtained.  the  per  maintenance c o s t s  records  the  simulation,  delay  costs  (dollars  This estimated  the  minimizing  truck hauling cost  turnouts..  spent  company  easily  by  time  with  c o n s t r u c t i o n and  multiplied  studies,  optimum number o f t u r n o u t s  maintenance  p r e d i c t e d from e x i s t i n g  delay  the  i s concerned  construction in turnouts.  of  a delay a t the  turnout.  time, landing  Consequently,  c o s t s would have t o e l i m i n a t e  this the  spacing..  i s t o view d e l a y c o s t s i n  terms  of  the  6  number  of  round  trips  a v e h i c l e can complete  initial  s t e p i s t o d e t e r m i n e t h e number o f c o m p l e t e  a g i v e n v e h i c l e can achieve  f o r a particular  The  can  initial  experience  arrangement  or from  an  be  estimate  i n a day.. The  turnout  determined  of  a  round  spacing.  from  good  previous  turnout  spacing  a r r a n g e m e n t . . A s i m u l a t i o n , b a s e d on d i s p a t c h i n g r u l e s , behaviour, to  l o a d i n g times,  d e t e r m i n e t h e number o f c o m p l e t e  this  starting  or decreased number for  of  schedule.  number  trips  spacing  cost  spacings. per. c u n i t  spacing  will  at  adjusted will cost  the  A  round  repeated  comparison  i s then  arrangement.  of  required to This  i s  an  trips  this  i s not  the  case,  p e r v e h i c l e p e r day c a n n o t  determined.  the  delays  landing  t o turnout  of  spacing  spacing.  may  first  the turnout  have  t o the a d j u s t e d delay  times  to  s p a c i n g an  The a p p r o a c h t o t h i s  c o s t be e q u a l  by t h e t u r n o u t  the  i n the system not a t t r i b u t a b l e t o  regardless  be t o l e t t h e d e l a y  attributable  the  one c a n a c c u r a t e l y f o r e c a s t t h e t r u c k  hauling cost i s u t i l i z e d .  multiplied  From  alter  c a n be  turnout spacing are eliminated.. Since a v e h i c l e queue  utilized  c a n be i n c r e a s e d  b e t t e r o f t h e two a p p r o a c h e s a p p e a r s t o be provided  traffic  p e r day.  This process  Since i n r e a l i t y  of complete  consistently  method  turnout  method p r o v i d e d  The  trips.  transportation  dispatching the  round  range of t u r n o u t  determine the best adequate  c a n be  t o determine whether t h e v a r i a t i o n complete  total  times,  round  arrangement the t u r n o u t  a reasonable  the  be  and u n l o a d i n g  trips  study hauling  that  are  7  2.2  THE ASSUMPTIONS OF THE MODEL  The nature o f the l o g h a u l i n g process of-way  priority  belongs t o the l o g g i n g t r u c k s .  only v e h i c l e s t h a t w i l l be considered the  right-  Therefore  the  process  are  i n the delay  l o g g i n g t r u c k s themselves and i t w i l l t h e r e f o r e be assumed  that the other logging  v e h i c l e s w i l l not hinder  trucks.  Furthermore,  w i l l not be considered turnout  spacing  as  the  delays  they  are  number  during  of  the  at the l a n d i n g and dump not  solely  related  procedure, the t o t a l number of  to  vehicles,  of l o a d e r s . , Depending on the s t a r t i n g up and  s h u t t i n g down mode of the company, delays  the p r o g r e s s i o n  but depend on the l o a d i n g r a t e , the unloading  r a t e , the d i s p a t c h i n g and  d i c t a t e s that  the  first  The  there  may  be  no  turnout  few hours and l a s t few hours of the  "hauling"  day.  various  dispatching  policies  can  reflected  i n the model by l i m i t i n g the number of h a u l i n g  be hours  per day t o the " c o n f l i c t " hours.. The " c o n f l i c t " hours r e f e r t o t h a t p o r t i o n of the day where there and that  empty the  v e h i c l e s over a s e c t i o n of road.. empty  constant  Variation  in  The  turnout  turnout  which  construction  construction  costs  cost  is  could  assumed.  be p a r t i a l l y  s e c t i o n t o be analysed  into  each with a p a r t i c u l a r turnout c o n s t r u c t i o n c o s t .  discounting  amortization  I t s h a l l be assumed  delay.  accounted f o r by d i v i d i n g the road subsections,  loaded  v e h i c l e w i l l always stop a t the turnout  w i l l y i e l d the l e a s t A  i s movement o f both  of  of road  costs costs  is  included  in  the  model  w i l l not be considered.,  t r a f f i c flow r a t e w i l l be assumed..  but  A fixed  8  Further  limitations  constraining The  will  some o f t h e speed  be  the  loaded  to  characteristics  f o l l o w i n g assumptions s h a l l  1. a l l o f  added  the  model  by  of the v e h i c l e s .  be i n c l u d e d :  vehicles travel  a t t h e same  constant  velocity. 2.  A l l o f t h e empty velocity  3.  except  travel  of turnouts w i l l  the  vehicle  empty  having  4. The empty constant The  the  allow  t o a c c e l e r a t e to i t s designated  speed  will  a turnout..  accelerate  or  decelerate  unit  be d e v e l o p e d o f the t o t a l  i n four parts: expected  delay  time  of the expected  delay  time  4. d e t e r m i n a t i o n  o f the s o l u t i o n s t o t h e c o s t  consist  deceleration  turnout. is  expected  will  turnouts,  of  delay  equations  time  of the delay the  empty  p l u s an e x p r e s s i o n f o r The  developed  per v e h i c l e  i n the turnout  of the cost  and  a  d i s t a n c e o f road  determination  equation  at  rate.  model w i l l  total  constant  t o always  3. d e t e r m i n a t i o n  The  same  a turnout..  be s u f f i c i e n t  to decelerate into  vehicle  1. d e t e r m i n a t i o n per  at  when d e c e l e r a t i n g i n t o  The s p a c i n g  before  2.  vehicles  equations.  per v e h i c l e per unit caused  by  the  the  following  acceleration  v e h i c l e e n t e r i n g and l e a v i n g the  expected  delay  model t o d e s c r i b e t h e d e l a y t i m e in  distance  section.  The  in  the  i n the turnout general  cost  delay  cost  equation i s : Cost  = turnout  construction  attributable  to  turnout  cost  +  spacing  +  turnout  maintenance c o s t . . The  optimum  turnout  spacing  i s determined  by t a k i n g t h e f i r s t  d e r i v a t i v e of the c o s t f u n c t i o n with spacing and which  this  utilizing  respect  a search technique  derivative  is  to  the  turnout  to f i n d the values f o r  equal to z e r o .  values must be checked to determine whether-  Furthermore, these they  maximum o r minimum value of the c o s t f u n c t i o n .  calculate  a  10  2_. 3  THE  TORNOOT DELAY TIME  In  the p r e v i o u s s e c t i o n the g e n e r a l f o r m u l a t i o n and  assumptions of the model were o u t l i n e d .  The f i r s t  phase i s  determine the delay time per v e h i c l e per u n i t d i s t a n c e of The  time  required  for  a  vehicle,  travelling  speed, to t r a v e r s e a u n i t d i s t a n c e of road i s the v e l o c i t y of the v e h i c l e . an  expected  "n"  turnout  u n i t l e n g t h of road.  The  road.  the  inverse  to  of  experience  while p r o g r e s s i n g along  t o t a l time r e q u i r e d  to  at a constant  The empty v e h i c l e w i l l  delays  basic  this  travel  this  s e c t i o n i s the u n i n t e r r u p t e d t r a v e l time plus the time spent i n the  turnouts  plus  the  time  lost  in  the  a c c e l e r a t i o n and  d e c e l e r a t i o n of the empty v e h i c l e . From the b a s i c theory particle  (Meriam,  of  1971)  it  the can  rectilinear be  easily  motion shown  of  a  t h a t the  s t o p p i n g d i s t a n c e r e q u i r e d f o r an empty v e h i c l e , t r a v e l l i n g  at  velocity V , i s : 2  D  s  and  = V|/2a  2.1  0  i t s corresponding T  5  = \ /a  A listing  2^2  D  of the symbols u t i l i z e d  Appendix 2..  Similarly,  a c c e l e r a t i o n time D  «  =  T  A  =  stopping time i s :  V  |/  2  a  (T ) fl  the  i n the formulas  acceleration  i s located i n  distance  (D^)  and  are:  A  ^  and  time  \  ' A  ^  a  The  next  in  the  step i s to d e r i v e an turnout.  expression  for  the  delay  This delay time i s not a constant  but  11 depends on the d i s t a n c e the loaded vehicle  from  the  empty  when the empty v e h i c l e comes to a complete stop i n the  turnout. ratio  vehicle i s  The  delay i n the turnout can be  of the d i s t a n c e between the two  spacing.  This r a t i o w i l l  be  referred  represented  v e h i c l e s and to  as  by  the  the t u r n o u t  the  F-factor.  Thus: F  =  D/S  where: F  = F-factor  D  = d i s t a n c e between the two v e h i c l e s  S  = d i s t a n c e between t u r n o u t s .  The  t u r n o u t spacing i s assumed to be uniform..  The F - f a c t o r i s  utilized  r a t h e r than d i s t a n c e or time s i n c e i t g i v e s  concept  of  the  effect  that  turnout  Furthermore, p r e v i o u s a r t i c l e s subject  have  utilized  (Byrne  et a l . ,  F-factor  Utilizing  the above concepts  expected  1947)  on  the  the same r a t i o to r e p r e s e n t t h i s d e l a y .  expected  the  better  s p a c i n g has on d e l a y s .  The t h e o r e t i c a l development of an equation  calculate  a  will  be  expected  discussed  for calculating  in  the  an e x p r e s s i o n can  the  next s e c t i o n . be  derived  to  delay time per turnout i n c i d e n t . .  The  delay time per turnout i n c i d e n t c o n s i s t s of  the  time  l o s t due to v e h i c l e a c c e l e r a t i o n and d e c e l e r a t i o n :  a  A  2  a  f l  "1  - l 2 V  + . 0  2 a  a  and the expected  D.  1  time the empty v e h i c l e spends in the t u r n o u t :  (SF)/V, The  expected t  delay time per t u r n o u t i n c i d e n t  = V ( a + a ) / ( 2 a a ) + (SF)/V,  where :  2  fl  D  fl  D  (t) i s t h e r e f o r e : 2^5  12 V, = v e l o c i t y of loaded F  vehicle  = expected F - f a c t o r . Since  the t o t a l  there  are n expected  expected  delays per u n i t d i s t a n c e then  delay time per u n i t d i s t a n c e per v e h i c l e (T~)  is: T The  = n[ (V,, {a +a }/{2a a }) + (SF)/V fl  D  A  0  ]  J  2^6  n delays c o n s i s t o f :  1. The c o n f l i c t s with the i n i t i a l in  number of  loaded  vehicles  the u n i t d i s t a n c e , which i s equal to t h e : u n i t d i s t a n c e  of road  times  the  traffic  v e l o c i t y of the loaded  flow  rate  divided  by  the  vehicle,  plus 2. The  number  of loaded v e h i c l e s e n t e r i n g t h e : u n i t d i s t a n c e  while the empty v e h i c l e t r a v e r s e s road,  which  is  equal  to  the  the  unit  delay  distance  time  u n i n t e r r u p t e d t r a v e l time m u l t i p l i e d by the  plus  traffic  of the flow  rate. The e x p r e s s i o n f o r n thus becomes: 1  H  n  v  + H  SF + n — +  -,  ,  2  V  A D  a  a  or H n  =  J,  H +  1  -  HSF  v.  HV  £  2 a  (a^+a ) D  -  I  2.7  A 0 a  where : H = traffic (  flow r a t e of loaded  vehicles  (i.e.,  vehicles  per hour (vph)).  Substituting  equation 2.7  into  equation  y i e l d s an expression f o r t h e . t o t a l expected  2.6  and s i m p l i f y i n g  delay time t h a t  is  13  not  d i r e c t l y a f u n c t i o n of n. T  Once  = the  H  H  \  (a„ +a ) SF" 0  — +—  2a„ a total  delay  V, J  0  and  the  2  1-  2 a  (a„+a ) - l 0  «  a  2.8  o  time per u n i t d i s t a n c e  been c a l c u l a t e d the t o t a l delay calculated  HSF HV  per v e h i c l e has  c o s t per u n i t d i s t a n c e  can  by m u l t i p l y i n g equation 2.8 by the t r a f f i c flow  be rate  hauling c o s t . . The use of equation 2.8 i n determining  the optimum turnout spacing  w i l l be discussed  i n section  2.5..  14 2..4  THEORETICAL DEVELOPMENT OF  The  calculation  analysis  of  streams on the  the  of  the  THE  expected  interactions  a single-lane  real t r a f f i c  road.  The  traffic  the  The  network  objective  movement of t r a f f i c  rather  Wohl and  are  applicable  will  description  than the  of  for  high  involve  the  the  total  c o n f l i c t s between i n d i v i d u a l  traffic  these  methods  d e n s i t i e s . . Since l o g  densities  Car-following theories  involve  intervehicle  another  approach  found.  general,  it  is  assumed that  the  each v e h i c l e depend to some extent on of each o f the  preceeding v e h i c l e s ;  constrained  by  c h a r a c t e r i s t i c s of the approach i s too Traffic  the.  detailed  engineers situations  control,  left-turn  and  Drew (1968)).  The  traffic  c o n s t r a i n s the model  f o r use have  the  speed  thus, a  and  driver's  response  vehicles  and  1967).  the This  i n t h i s study. , probability  theory  to  merging  traffic,  intersection  storage areas  (Wohl and  Martin(1967),  delay situation  problem  is  where  one  movement of another t r a f f i c developed  of  characteristics  Martin,  utilized  for  turnout delay  (Wohl  relationships.  speed c h a r a c t e r i s t i c s  surrounding  road  analyse d e l a y  The  translate  behave l i k e  traffic  merging  to  an  traffic  analogies  low  must be  is  is  Martin(1967) suggested that  h a u l i n g systems i n v o l v e  In  separate  Fluid-flow  r e s u l t i s the  vehicles.. only  two  involves  s i t u a t i o n i n t o a workable mathematical model.  p r i n c i p l e that  flow of f l u i d s .  F-factor  between  S e v e r a l approaches are a v a i l a b l e . the  EXPECTED F-FACTOR  herein  will  similar traffic  to  the  stream  stream. utilize:  probability  15 theory. .  This  approach  involves  the i n t e r a c t i o n between the  "average" loaded v e h i c l e and the "average" empty this  approach there are c e r t a i n i n h e r e n t  vehicle..  assumptions.. Namely,  a uniform v e l o c i t y and r a t e of a c c e l e r a t i o n of the v e h i c l e s assumed.  It  will  vehicles  w i l l be  be  f u r t h e r assumed that the length  neglected  as  well  In  as  the  length  is  of the of  the  t u r n o u t s . . The e f f e c t of these assumptions on the:model w i l l be discussed  later.  The  three  situations  that  will  be  discussed  meeting o f one loaded and one empty v e h i c l e , between  a s i n g l e empty and a f l e e t  conflicts vehicles. ,  between a group  of  empty  the  are the  interactions  of loaded v e h i c l e s , and the and  a  group  of  loaded  16 2. 4.1  ONE LOADED VEHICLE MEETING ONE EMPTY VEHICLE  The  situation  empty  vehicle  Figure  1..  pertaining  and  A  one  backwards  problem.  The  first  vehicles  will  meet  utilize  a  location  loaded  vehicle  i s to  (point B) Once  the  moment  the  A.  As  in  f o r solving t h i s where  the two  i f the empty v e h i c l e does not  this  has  been  accomplished  the  empty v e h i c l e must begin to d e c e l e r a t e i n  empty  vehicle  v e h i c l e , having backtracked point  illustrated  determine  order t o stop a t the c r i t i c a l t u r n o u t this  is  approach i s u t i l i z e d  phase  turnout.. where  to the i n t e r a c t i o n between one  an  illustrated  ( S ) i s determined.. c  At  i s a t point C and the loaded equal  in  interval,  diagram,  critical  the  critical  turnout  where the two v e h i c l e s would meet i f the empty v e h i c l e into  the  critical  loaded  vehicle  spacing  from the c r i t i c a l  at  this  distance  c  turnout.  turnout when the empty v e h i c l e  The  loaded  vehicle  i 2 V  / a  o  Cf  stops  travel  the  minus the F - f a c t o r times the turnout  spacing  in  = , < cr v  x  + D  s  ) / V  2  +  cf- "  X  Thus  F S  or X  the  must  the time the empty v e h i c l e d e c e l e r a t e s . v  Furthermore,  w i l l be e x a c t l y the F - f a c t o r times the t u r n o u t  turnout. A-S  from  the  ( X ) i s defined  d i d not p u l l  distance  i s at  distance  C F  as the  the  time  = [ 2 F S V a + V V|]/[2a (V 2  D  D  (  +V ) ] a  2.9  17  _Q(2  Q  Q Q  Projected Turnout l o c a t i o n s  meeting  point  S„  Distances  |4-v,  Times  1  (X *D ) / \ C F  > K  S  x  c  ^ K — D I  I  S  ^ | I  -Deceleration -No  Deceleration  |<  (X +D C P  5  )/ \  »^-(X  C F  +D  5  ) / V ^  where  h= T S  S  J  F  stopping  = stopping =  turnout  time locations  = F-factor  X  cr = c r i t i c a l  V  ,  v  distance  = velocity z  Figure vehicle  distance of the loaded  = v e l o c i t y o f the empty  1  Interaction  between  vehicle vehicle  an  empty v e h i c l e and a loaded  18 It  i s reasonable to assume that  described  by  w i l l now before  of  F „ m  which i s d e f i n e d  y  as the  point B  is We  largest F-factor  the empty v e h i c l e can t r a v e l to the next  turnout..  The  d i s t a n c e corresponding to a F - f a c t o r equal to zero i s  denoted by  .  expression  ^=  Since 1  CO  J  F  location  an uniform d i s t r i b u t i o n on the i n t e r v a l 0-S.  introduce  critical  an  the  <. V  for F +  rr>Ay.  can  m I W  minus X„„ equals the turnout  re  CF  CO  be  spacing  ^  r  3  derived.  W Z  ^10  V  The  d e n s i t y f u n c t i o n of the F - f a c t o r can  for  each c r i t i c a l d i s t a n c e there corresponds an unique F - f a c t o r  and  the  density  of  the c r i t i c a l  over the i n t e r v a l 0-S. Pr(F,< F <F- SF) = +  be  determined  since  d i s t a n c e formula i s uniform  Thus (X . C(F  +S f )  -X . CF  = SFV /(V, *\  )/S  )  a  2. 11  where : F  = a given  F-factor  <$F = a small change i n the  F-factor.  Once the d e n s i t y f u n c t i o n of the F - f a c t o r has the  expected  F-factor  can  be  been  determined by i n t e g r a t i n g the  d e n s i t y f u n c t i o n times the F - f a c t o r between zero and  ' n\Kf>  \ F/(V  (  * \ )£F  F  mA)(  . .  p  r  ^  determined  \  F2  ~  F =  2.12  2\  T h i s expected F - f a c t o r i s r e f e r r e d to as the simple F - f a c t o r d i s t i n g u i s h i t from the expected F - f a c t o r in  the next s e c t i o n .  corresponding model  headway  developed  Those equations are r e f e r r e d to by probability  functions..  developed i n Chapter 4 i s u t i l i z e d  F - f a c t o r equation. .  equations  A  to confirm  to  their  simulation the  simple  19 2.4.2  E M £ I I VEHICLE MEETING A FLEET OF  ONE  In the incident  previous  was  section  independent  of  LOADED VEHICLES  it  was  assumed  each  turnout  any  o t h e r . . In t h i s s e c t i o n  t u r n o u t events w i l l be p a r t i a l l y dependent on each other.. situation  i n v o l v e s the  v e h i c l e and  a f l e e t of loaded  I t i s not first  in  passed the  the 2. An  the empty v e h i c l e s i n c e t h i s has  vehicle,  Once a loaded  stopped  in  the  next  loaded  a n a l y s i s of the  turnout,  v e h i c l e may  as  "the  the  same lane and  i n t e r v a l of time between s u c c e s s i v e measured from head to head  road."  There e x i s t s a c r i t i c a l  each of the  prior  loaded  vehicles  travel  to  positioned  the The  v e h i c l e i s at point  as  pass  a  headway(h ) where c  next  turnout  and  headway must  p r o b a b i l i t y of  occuring.  turnout S B.  they  critical  the f i r s t loaded v e h i c l e and at  headway  v e h i c l e s moving i n  to the c a l c u l a t i o n of the  above s i t u a t i o n s  Initially, are  just  delay i n t h a t t u r n o u t .  determined  or  proceed.  headway d i s t r i b u t i o n of the  the empty v e h i c l e can  be  two  be c l o s e enough to p r o h i b i t  i s e s s e n t i a l to solve t h i s problem.. Drew(1968) d e f i n e s  experience no  been  vehicle  empty v e h i c l e from advancing t o the next turnout  on the  the  arise:  the empty v e h i c l e may  point  empty  i n t e r a c t i o n between  previous section.  empty  s i t u a t i o n s may 1. the  the  This  vehicles.  necessary to analyse the  loaded v e h i c l e and  accomplished has  i n v e s t i g a t i o n of the meeting of an  the  c  (Figure  2).  the empty The  vehicle  second loaded  20  /O  Ol  /  K  1  V  Q  JL  Q-Q_  B Turnout l o c a t i o n s  S. l - T -  Distances  K  -0  T"T"  D — | ^ D  f  l  ^ |  Times  -No  '  I  -acceleration  M  i  acceleration  K—T.  ^  I  where : O  = l o c a t i o n o f the empty  0  = l o c a t i o n of the loaded  h  c  = critical  Situation  vehicle vehicles  headway  pertaining  t c the empty v e h i c l e e x p e r i e n c i n g no delay  «  in  turnout S^ s i n c e i t comes t o a complete h a l t a t the  as the loaded Figure 2  vehicle  Diagram  the c r i t i c a l  headway  turnout  passes the t u r n o u t .  showing  the  physical s i t u a t i o n leading to  21 The time r e q u i r e d f o r the distance D  (  is  second  equivalent  loaded  vehicle  to  travel  t o the time r e q u i r e d f o r the empty  v e h i c l e t o t r a v e l one turnout s p a c i n g .  Consequently,  T, = T +T,, + (S-D -D )/V,, S  S  A  = (2Sa a * V|{a *a })/(2a a Vj,) ft  D  fl  0  A  2^3  0  and D, = V, ( 2 S a a * V | { a + a } ) / ( 2 a a V ) A  D  ft  D  A  D  2.14  z  Since v e h i c l e arrangements are g e n e r a l l y measured i n time  units  the c r i t i c a l headway can be e a s i l y c a l c u l a t e d as: h  c  = T.+S/V, = [2Sa a (V, + \ ) + \ V | ( a fl  0  fl+  a ) ]/[ 2a a V, \ ] D  ft  0  2. 15 and the p r o b a b i l i t y t h a t a headway i s l e s s  than  the  critical  headway i s : Pr(h<h ) = c  I g(h) cSh  2. 16  Jo where: g(h) = headway p r o b a b i l i t y d e n s i t y f u n c t i o n . The  expected F - f a c t o r  headway  varies  with  (F^) f o r headways l e s s than the c r i t i c a l respect  to  the  headways l e s s than the c r i t i c a l headway. \  The  expected  F-factor  for  Thus:  hg(h)£h V, /S  expected  headway  ,2. 17  (F, )  for  c r i t i c a l headway i s the weighted  headways  than the  functions  similar  t h a t the l i m i t s of the i n t e g r a l s vary.  This i s  because an empty v e h i c l e can proceed one, two, or more  turnout  to  F  z  spacings  except  before  being  t h a t the l i m i t s of the turnout  spacing  and  forced integrals  average of  greater  into a turnout. vary  with  The r e s u l t i s  respect  to  the  the number o f t u r n o u t s the empty v e h i c l e  22 passes before being f o r c e d i n t o a t u r n o u t . . The simple F - f a c t o r w i l l be u t i l i z e d t o estimate F, s i n c e the  actual  function  complex and i t s l i m i t s are d i f f i c u l t t o determine. F, = (V, +V,, )/(2V ) simulation  model,  Thus: 2. 18  a  A  is  developed  in  Chapter H, i s u t i l i z e d t o  v e r i f y the accuracy' o f t h i s method. The expected F - f a c t o r equation empty  vehicle  and  for  the  meeting  of  one  a f l e e t o f loaded v e h i c l e s c o n s i s t s of two  parts: 1. the  component  attributable  that to  represents  the delay s i t u a t i o n  l e s s than the c r i t i c a l  The  of  delay  the  situation  where  headway.  each of these s i t u a t i o n s o c c u r i n g must be  c a l c u l a t e d , as well as the expected of  F-factor  where the headway i s  headway i s g r e a t e r than the c r i t i c a l  probability  each  average  headway and,  2. the component t h a t r e p r e s e n t s the the  the  situations.  F-factor  attributable  Consequently, the expected  to  F-factor  equation i s : F = Pr(h>h )F, c  ls.ll  + Pr(h<h )F^ c  where : h  = headway  F, = the expected F - f a c t o r f o r headways critical F^ = the  The f i r s t  turnout  independent  of  than  the  headway  expected  critical  greater  F-factor  for  headways  less  than the  headway. incident the  headway  of  an  empty  vehicle's  trip  is  p r o b a b i l i t y d i s t r i b u t i o n and the  expected F - f a c t o r r e s u l t i n g from t h i s i n c i d e n t i s the simple Ff a c t o r . . The expected F - f a c t o r r e p r e s e n t i n g  the  rest  of  the  23 turnout  i n c i d e n t s the empty v e h i c l e encounters on i t s t r i p , i s  given  by equation 2.19. . The t o t a l number of t r i p s per day  the  total  number  of  turnout  incidents  determined p r i o r to c a l c u l a t i n g expected F - f a c t o r L.The  the  per  expected  day  and  must  F-factor..  be The  i s composed of two p a r t s :  portion that represents  the f i r s t  a t r i p which i s equal to the  simple  turnout i n c i d e n t on F-factor  times  the  number of t r i p s per v e h i c l e d i v i d e d by the t o t a l number o f turnout i n c i d e n t s . plus 2. The  portion  representing  on t h e t r i p which i s  the remaining turnout  equal  to  equation 2.19  remaining number of turnout i n c i d e n t s d i v i d e d  incidents times  the  by the t o t a l  number o f turnout i n c i d e n t s . Thus: F  = [Pr(h>h )F, +Pr(h<h )F c  +[ (V  c  *V )/{2V ) t  z  ][Q /(Q +I) ]  2  3  3  ][I/(Q +I) ]  2^20  3  where: Q  3  = Hd-I = number o f headways  d  = number o f " c o n f l i c t " hours per day  I  = number o f t r i p s per day. Four  respective The  headway  probability  distributions  expected F - f a c t o r s are d i s c u s s e d  expected F - f a c t o r equations derived  referred  to  distribution. effect  of  by  their  Chapters 4  respective and 6  will  in  and  s e c t i o n 2.4.4.  i n t h i s s e c t i o n w i l l be headway discuss  probability the  relative  these equations compared to the simple F - f a c t o r and  the average F - f a c t o r r e s u l t i n g from the meeting o f of  their  vehicles..  two  fleets  24 2.4.3  A - FLEET  OF  LOADED  VEHICLES  MEETING A FLEET OF EMPTY  VEHICLES  In t h i s s e c t i o n the turnout i n c i d e n t s between a loaded  and  a  fleet  fleet  of  o f empty v e h i c l e s w i l l be examined.. The  f o u r cases t h a t w i l l  cause  an  empty  vehicle  to  utilize  a  p a r t i c u l a r turnout a r e : Case  1. An  empty  v e h i c l e may stop a t a turnout because a  loaded v e h i c l e i s approaching. Case 2. An empty v e h i c l e may remain since  in a  given  turnout  a loaded v e h i c l e i s so c l o s e as to p r o h i b i t  the empty  vehicle  from  advancing  to  the  next  turnout. Case 3. An  empty  vehicle  i s required  particular  turnout  since  occupied Case.4. An  the next  vehicle  cannot  proceed  t u r n o u t but remains at i t s present  In  the  first  two  turnout i s occupied cases,  situation  while  i n the  empty v e h i c l e i s required  situation  following  A  discussion  turnout  a is  latter  has  to  the  turnout  next since  (Figure 4 ) .  loaded  vehicle  is  two cases, s i t u a t i o n B, the  to use a p a r t i c u l a r turnout s i n c e the  next turnout i s occupied and a loaded v e h i c l e Since  at  A, the empty v e h i c l e i s  f o r c e d to use a p a r t i c u l a r turnout s i n c e a approaching  halt  (Figure 3 ) .  empty  the next  to  been  is  approaching.  d i s c u s s e d i n s e c t i o n 2.4.2, the  w i l l p r i m a r i l y i n v o l v e s i t u a t i o n B.  25  _0jQ  0_  Q  C Turnout l o c a t i o n s  S„  1  I-  J L O O  O O  B s. ' (D  1  Distances  where: O  = empty  vehicle  9  = loaded  O  = p o t e n t i a l l o c a t i o n of empty  vehicle  ^  = l o c a t i o n of s e t 1 empty  Sj  = turnout  Figure 3  vehicle  vehicles  locations  Case 3 — A n . empty v e h i c l e stops at t u r n o u t  S, o  next t u r n o u t i s occupied  since  26  Figure 3  illustrates  the case i n which an empty  t r a v e l l i n g at a constant v e l o c i t y ,  is  delayed  at  a  s i n c e one empty v e h i c l e or a group of empty v e h i c l e s occupying  the next turnout.  I f s e t 1 d i d not occupy  then  would  empty  vehicle  i n s t e a d o f being f o r c e d t o this  turnout  (set 1) i s  Herein t h i s case w i l l be r e f e r r e d  to as a Case 3 s i t u a t i o n . the  vehicle,  S  c  be able to stop at turnout S  c  utilize  turnout  turnout  ..  Maintaining  s i t u a t i o n then the empty v e h i c l e would be l o c a t e d between  p o i n t s A and B when the and E. ¥  3  The expected  loaded  vehicle  is  between  points C  F - f a c t o r f o r the Case 3 s i t u a t i o n i s :  = F, + S/S + SV, /(SV )  = F, +(V, +V )/V  Z  Z  2.21  2  where: F  3  = expected  Equations  F - f a c t o r f o r a Case 3 s i t u a t i o n .  2.12 and 2.19 can be used t o determine  an expected  F-  f a c t o r r e p r e s e n t i n g the meeting of an empty v e h i c l e and a f l e e t of  loaded  expected loaded  vehicles.  Utilizing  this  expected  d i s t a n c e the empty v e h i c l e , stopped at vehicle  equation turnouts S distance  is  can  determined. .  The  expected  d i v i d e d by  loaded  the  turnout  spacinq  F-factor  Since  plus  the  v e h i c l e can t r a v e l i n the time the empty  v e h i c l e can t r a v e l one turnout spacing d i v i d e d by spacing. .  , i s from the  the sum of equation 2.18 and the d i s t a n c e between  and S the  be  F - f a c t o r the  F,  has been approximated  the  turnout  by equation 2. 18 F  3  becomes: F  3  = 3(V, + V ) / ( 2 V ) Z  A f t e r the loaded  2. 22  Z  vehicle  has  passed  both  groups  of  v e h i c l e s and each v e h i c l e has a c c e l e r a t e d to i t s o r i g i n a l the  distance  empty speed  between the empty v e h i c l e and t h e group of empty  v e h i c l e s i s the i n t e r v a l d i s t a n c e ( D ) . . The i n t e r v a l T  distance  27 is  the  turnout spacing p l u s the d i s t a n c e an empty v e h i c l e  t r a v e l , a t a constant v e l o c i t y , i n the time  a  loaded  can  vehicle  can t r a v e l one turnout s p a c i n g . Dj = S (1 + \ /V, ) As  illustrated  stems from the  adjacent  l e s s than  turnout  the  critical  occupied  headway  to the second  and  a  turnout  since  loaded v e h i c l e i s  then  loaded  the  expected  F-factor  v e h i c l e i s e q u i v a l e n t t o the  F - f a c t o r of equation 2.17..  The expected  of  is  given  I f the headway between any two loaded v e h i c l e s i s  attributable expected  i n F i g u r e 4, Case 4 i s the s i t u a t i o n t h a t  an empty v e h i c l e w a i t i n g i n a  approaching.  two  2. 23  F - f a c t o r equation r e p r e s e n t i n g the meeting of  f l e e t s of v e h i c l e s i s composed of the p r o b a b i l i t y that each the above cases w i l l occur, times t h e i r r e s p e c t i v e  F-factor.  The  meeting of two  expected  F-factor  equation  expected  r e p r e s e n t i n g the  f l e e t s of v e h i c l e s i s t h e r e f o r e :  F = Pr(h>h |A)F, +Pr(h<h |A) F +Pr(h<h |B)F c  c  • Pr(h>h |B) (F +1+V, c  (  a  c  a  /V )  2.24  z  where: A = c o n d i t i o n a l event t h a t an empty v e h i c l e i s r e q u i r e d to utilize  a  turnout  because  a  loaded  vehicle  is  approaching B = c o n d i t i o n a l event that an empty v e h i c l e must u t i l i z e turnout  because  a  an empty v e h i c l e or a group of empty  v e h i c l e s a l r e a d y occupy the  adjacent  turnout  and  a  loaded v e h i c l e i s approaching. The expected determined  F - f a c t o r f o r each of the above cases can be  easily  but the p r o b a b i l i t y of each of the events o c c u r r i n g  i s d i f f i c u l t t o determine.  It  would  not  be  neccessary  to  28  calculate  these p r o b a b i l i t i e s provided i t can be shown t h a t the  probability  of  a  conditional  event B  is  Consequently, i t i s d e s i r a b l e t o determine an the  p r o b a b i l i t y of event B.  relatively upper  empty  and  loaded  limit  of  I f the two f l e e t s o f v e h i c l e s are  independent of each o t h e r , then the p r o b a b i l i t y o f of  small.  the  number  v e h i c l e s a r r i v i n g during any given time  i n t e r v a l , t , can be determined.. Since  f o r low  traffic  flow  r a t e s t h i s p r o b a b i l i t y i s r e l a t i v e l y s m a l l then the expected Ffactor  equation  representing  the  meeting  of a s i n g l e empty  v e h i c l e and a f l e e t o f loaded v e h i c l e s w i l l adequately d e s c r i b e the expected F - f a c t o r r e s u l t i n g from the two  fleets  of  vehicles. .  This  interactions  statement  s i m u l a t i o n model developed i n Chapter 4 .  between  i s supported by a  29  _0_0_  O  00  Time Turnout l o c a t i o n s  Turnout l o c a t i o n s  S,  S.0 I-  Turnout l o c a t i o n s  °4  o  1  t=0  |  t = S/V  I  t=h  s, 1  -I-  S„  s  f  .  ,  " 1 $ — -  ——I  where : O = empty  vehicles  •  = loaded  vehicles  ^  = l o c a t i o n of s e t 1 v e h i c l e s  S; = turnout l o c a t i o n s  Figure 4  Case 4 — An empty v e h i c l e remains i n turnout S  next turnout i s occupied  3  since  30  2.4.4  HEADWAY PROBABILITY DISTRIBUTIONS  The  theoretical  technique  described  in  section  r e q u i r e s some knowledge of the p r o b a b i l i s t i c a s p e c t s of behaviour.  have  or  of l o g g i n g  conducted  d i s t r i b u t i o n of low roads..  trucks;  studies  to  flow  along  traffic  arrival  distribution These  determine  distribution,  and  using  have  d i s t r i b u t i o n , the Pearson  Type I I I  the  highways  Gerlough (1955)  tabulated  data f o r v a r i o u s r u r a l and  the  and  rural  may  of  the  tests  distribution  may  have  been  the  in  the  proved  the: United  Wohl and  the  fact  minimum  to  be  field  States.  unsuccessful  that  allowable  prove to be  Martin(1967) u t i l i z e d  examine the frequency  of  l e v e l . . A cause f o r the f a i l u r e  between the f r o n t s of s u c c e s s i v e v e h i c l e s . flows t h i s assumption may  and  to f o u r s e t s of data  assumes there can be more than one  theoretical  Wohl  a r r i v a l frequency  the  Poisson  a r r i v a l during  i n t e r v a l e q u i v a l e n t to the minimum headway.  headway i s the  Erlang  Drew(1968).  goodness of f i t t e s t s  the f i v e percent acceptance  binomial  distribution.  be found  urban roads i n  at  negative  Type I  attempt t o f i t a Poisson d i s t r i b u t i o n  by performing  probability  distribution,  Pearson  Martin (1967) , Haight (1 963) , and  to  traffic  the a r r i v a l  various  included  D e s c r i p t i o n s of these d i s t r i b u t i o n s  a time  however,  Various authors have attempted to d e s c r i b e the headway  distributions.  The  traffic  There have been no s t u d i e s undertaken to determine  the headway d i s t r i b u t i o n engineers  2.4.2  The  time  At very low  minimum interval traffic  insignificant.  a shifted exponential  distribution  curve  of v e h i c l e headways.  In  31 h i s e m p i r i c a l example (1000 vph) the s h i f t e d e x p o n e n t i a l fits  the  data  better  than  the  curve  n o n - s h i f t e d curve f o r lower  headways while the reverse held t r u e f o r l a r g e r headways..  The  s h i f t e d e x p o n e n t i a l curve s t a t e s t h a t : Pr(h>t) = £ - u < t - Y >  2.25  where: t = time  interval  % = minimum headway. Schuhl(19 55) developed  a partially  s h i f t e d e x p o n e n t i a l curve o f  the form: Pr(h>t) = e  _ <  <t-f>/<h,-f>>  +£-ct/k£  2.26  where : h, = mean headway f o r c o n s t r a i n e d flow h^ = mean headway f o r free This  equation  v e h i c l e s t o be vehicles.. theory.  flow.  a l l o w s the speed c h a r a c t e r i s t i c s o f some o f the influenced  by  the  of  other  T h i s equation to some extent i n v o l v e s c a r - f o l l o w i n g  Wohl and Martin argued  constrained  characteristics  that  since  a l l v e h i c l e s are  a t h i g h e r flows then equation 2.26 can be r e s t a t e d  as: Pr(h>t) = Q-K-L-fi/ib-t))  2.27  Wohl and Martin t e s t e d equation 2.27 on two s e t s concluded  of  and  t h a t t h i s equation was a c c e p t a b l e at t h e ten percent  l e v e l o f s i g n i f i c a n c e , f o r a data s e t with a flow r a t e vph,  but  vph.  Consequently,  adequately  data  of  500  was r e j e c t e d f o r a data s e t with a flow rate of 1000 a general  exponential  equation  does not  d e s c r i b e these s e t s of data.  Since these authors were not a b l e t o c o n s i s t e n t l y f i t data to  a  probability  distribution  s e v e r a l d i s t r i b u t i o n s w i l l be  32 analysed.. In the a n a l y s i s i t w i l l be assumed vehicle the  will  two  headway  must  probability The  The p r o b a b i l i t y  loaded be  empty  vehicles  t h a t the  i s greater  determined..  This  time  than  than  interval  the c r i t i c a l  i s equivalent  to  the  of there being no a r r i v a l s during t h i s time p e r i o d .  expected  critical  F-factor  headway  distributions  formulas  will  that  distribution, headway  the  wait i n a turnout u n t i l a headway i s l a r g e r  c r i t i c a l headway.  between  that  be  will  be  f o r headways derived.  l e s s than the  The  probability  analysed are the uniform  the e x p o n e n t i a l headway d i s t r i b u t i o n ,  distribution,  and  the  Pearson  Type  arrival  the  Erlang  I I I headway  distribution.  2. 4 . 4 . 1  The of  UNIFORM A R I I I M  uniform p r o b a b i l i t y  l/(i-j)  distribution  over i t s i n t e r v a l  f o r an uniform the  DISTRIBUTION  (i,j).  arrival distribution  F e l l e r ( 1 966)  the  headway i s g r e a t e r than t h e c r i t i c a l Pr(h>h ) = ( 1 - h ) c  m  c  has a constant  value  showed t h a t  probability  that  the  headway i s :  f o r 0<h <1 c  where : m = number of headways. It  was determined i n Section  i s given by Q-j.  2.4.2 that the number of headways  Therefore the above equation may be  rewritten  as: Pr(h>h ) = ( 1 - h ) c  The the  2.28  Q 3  c  d e n s i t y f u n c t i o n of t h i s d i s t r i b u t i o n probability  t h a t the headway  i s less  i s the d e r i v a t i v e than  the  of  critical  33  headway.. Thus: g(h) The  3[Pr(h<h ) ] c  =  expected  critical SF^  -Q (1-h) 3-i  =  Q  3  F-factor  formula  for  headways  less  than the  headway i s d e r i v e d as f o l l o w s : Q  hg(h)<Sh  V.  g{h)6h Q  1  ] 3+*-1)  ( [ l-hc  3  Q  -  S  (Q., + 1) ( [ 1 - h  2.29  c  EXPONENTIAL HEADWAY DISTRIBUTION  2.4.4.2  The  gap  density  function  of  the  exponential  headway  distribution i s : >>[Pr(h<h ) ] c  g(h) =  = Q ec>h  This  gap  3  density  the c r i t i c a l being  Q j h  3  f u n c t i o n can be i n t e g r a t e d  between zero and  headway t o determine the p r o b a b i l i t y of a  l e s s than the c r i t i c a l headway, y i e l d i n g :  Pr(h<h ) = 1 - e - 3 c . Q  2..30  h  c  Furthermore,  the  than the c r i t i c a l  expected F - f a c t o r equation headway can be d e r i v e d  f o r headways l e s s  from the  gap  function. SF  hg(h)6h _ e r  2  !l  v, F  headway  2  c  = V, [ e -  g(h)6h Q 3 V , C  Q 3 K  /Q  h-i>]^  3  -e- ^ Q  h  c  V  (-Q h -1)+1] / [S£-Q (e-^-1)3 ] 3  C  3  2.31  density  34  ERLANG HEADWAY DISTRIBUTION  2.4.4.3  The  distinction  functions  between the gamma and E r l a n g p r o b a b i l i t y  i s t h a t the Erlang f u n c t i o n i s r e s t r i c t e d  i n t e g e r values  f o r alpha. , The  gap  density  to positive  function  of the  Erlang d i s t r i b u t i o n i s : g(h)  =  h*-ie-* ^/(oC-1) !  {oLQ f  Q  3  Wohl  and Martin(1967) showed t h a t the p r o b a b i l i t y o f a headway  being  l e s s than the c r i t i c a l headway i s :  Pr(h<h ) = j [7v(7.h) -»eht  c  oL  1  K  logging  from  v e h i c l e headway  data  collected  Sh] / [ (oc-1)! ]  fettle) / i !  = 1-e-* tThe  Xh  0  by  2.32  frequency Smith  Young(1969) have been shown  to  and be  distributions Tse(1977)  left  and  skewed  produced Boyd and  (Chapter 4 ) .  Consequently, the expected F - f a c t o r equations w i l l be developed for the  functions  t h a t are l e f t  p r o b a b i l i t y of a  skewed. . I f alpha  headway  being  less  than  e q u a l s two then the  critical  headway i s : Pr(h<h ) = 1 - e - c - ( 1 + X h )  2.33  M  c  c  where: A =  g(h) and  Q-jOC-  =  2Q  3  = A2he~  A W  the expected F - f a c t o r i s d e r i v e d hg(h)£h  = Ofi  g(h)6h  rht  e  F  a  =  as f o l l o w s :  -^c(-? 2hf-2Xh v  c  -2) A  • 2A"  -  1-e-  XV>c  (1+Ah ) c  2.34  S i m i l a r l y , the expected F - f a c t o r s and p r o b a b i l i t y f u n c t i o n s can  35  be c a l c u l a t e d  2.4.4.4  f o r other values o f a l p h a .  PEARSON TYPE I I I HEADWAY DISTRIBUTION  The  gap  density  function  of  the Pearson  Type I I I  distribution i s : g(h) = ( b [ h - c f - » e a  b < K  -  c >  )  / r<a)  where : c = minimum value of h mean = a/b + c v a r i a n c e = a/b ]7(a)  2  = gamma f u n c t i o n  = (a-1) ! For  the same  distribution  0,1,2,....  for a =  reasons  as  the value  cited  of  f o r the  Erlang  headway  'a' w i l l be set to two.. The gap  d e n s i t y f u n c t i o n becomes: g(h) = b ( h - c ) e 2  _ b<  K  _  C  c<h< o o  >  The p r o b a b i l i t y t h a t the headway  i s less  than  the  critical  headway i s : Pr(h<h ) =j^ g(h)£h = e - b c h - o (-bh- 1+cb)]^ C  c  c  Pr(h<h ) = i-e-*>ch -o (bl^+1-cb) c  and  c  2.35  the expected F - f a c t o r f o r headways l e s s than the c r i t i c a l  headway i s d e r i v e d as f o l l o w s : FS V,  j chg(h)cSh  [e~  J* g(h)5h  [ -bcVv-c> (-bh-Hcb) ]  h  c  e  b t K  ~  C )  (-bhz-2h-2/b+chb+c) ] ] ^  [ c + 2/b+e-b(h -c) ( - b h | - 2 h - 2 / b + c b h + c ) ] c  t  [ i - e - b c K - o (bh +1-cb) ] c  c  c  37  2. 5  DEVELOPMENT OF THE COST EQOATION  The those  only c o s t s t h a t are needed i n the  that  are  equation  are  f u n c t i o n s of the turnout s p a c i n g : the t u r n o u t  c o n s t r u c t i o n c o s t , the turnout cost  cost  attributable  to  maintenance c o s t , and the  turnout  spacing.  The  delay  general  cost  equation i s : C  T  = C /S + C^/S + Q HMT c  2. 37  (  where : C  T  = t o t a l c o s t a t t r i b u t a b l e t o turnout  spacing  per  unit  d i s t a n c e o f road C  c  = turnout construction cost  C  M  = t u r n o u t maintenance c o s t  H  = t r a f f i c flow r a t e  M  = adjusted t r u c k h a u l i n g c o s t  Q  = expected u s e f u l l i f e  S  = d i s t a n c e between t u r n o u t s  f  = delay time per v e h i c l e per u n i t d i s t a n c e of road.  (  Since  of the road  a v e h i c l e may have t o queue a t the landinq r e g a r d l e s s of  the turnout spacing an adjusted h a u l i n g c o s t necessary is to has  c o n d i t i o n f o r determining  t h a t the f i r s t  the optimum t u r n o u t  determined  then  a  Once  cf  the  derivative  of  the  second  the c o s t f u n c t i o n i s p o s i t i v e , the d i r e c t i o n o f  c o n c a v i t y o f the f u n c t i o n i s upwards.  Provided  the  this  second  spacing  search technique can be used t o  l o c a t e t h i s zero. . Furthermore, when the value derivative  A  d e r i v a t i v e o f the c o s t f u n c t i o n with r e s p e c t  the turnout spacing i s equal t o z e r o . been  i s utilized.  derivative  i s - positive  zero  the  value  of  represents a  38 minimum v a l u e . . available  Some  of  the numerical  are b i s e c t i o n ,  regula  falsi,  analysis  techniques  secant, and Newton's  method.. F o r f u r t h e r r e f e r e n c e c o n s u l t Conte  and deBoor(1972)  or Arden and A s t i l l ( 1 S 7 0 ) . . When d i s c o u n t i n g of maintenance c o s t s i s not used the c o s t equation C  T  becomes: 2.38  = C/S + Q HMT (  where: c  = c +c M  Three  c  forms  of  the c o s t  equation s h a l l  determining t h e optimum t u r n o u t s p a c i n g . shall  be  assumed  the expected  be u t i l i z e d i n  In t h e f i r s t  t h a t the t u r n o u t spacing i s non-uniform and  F - f a c t o r i s a f u n c t i o n o f the turnout s p a c i n g . . In  the second case the expected  F - f a c t o r w i l l be a f u n c t i o n of the  turnout s p a c i n g but the turnout spacing w i l l be assumed uniform..  case i t  The  third  but w i l l u t i l i z e  case w i l l assume uniform t u r n o u t  t o be spacing  the simple F - f a c t o r . .  I f t h e turnout spacing i s non-uniform and the expected factor  i s a f u n c t i o n of turnout s p a c i n g then the cost equation  becomes: H H + Q, HM — + — J.V, V  y(s)  2  y(S)F V" +—  V,  1-  Hy(S)F  K  E\  -i  K J 2.39  where : y (S) = turnout spacing f u n c t i o n K  F-  = 2a a / A  D  (a +a ) fl  0  and the optimum turnout spacing occurs when:  39  '&F  b [ y (S) ]•  (HQ, H2(V, +V ) j y (S)  0 = [ Y (S) ] 2  T  i  .dS. - H2F2R2V C  aCy(s) ] as  d[y<s) j  a[y (S) T  I-  2  CV, V FKH[ HV -K ]  y(S) 2  a  a  C[ R2-2HV K + H 2 V | ] Z  2.40 By m a i n t a i n i n g function  of  the  the expected turnout  s p a c i n g t o be u n i f o r m "H C  T  = -+Q S  HM  then  spacing the c o s t  "SF V,f  H" + —  —  F-factor  Lv, v J Lv, 2  and t h e optimum t u r n o u t  K  allowing  equation  HSF  +—  —  and  equation  J  2  occurs  2  turnout H  C = -+Q, HM s  —  involving  2  the simple  HSF n\~ 1— — K J L V, KJ  H " "SF V^" +  —  V  Lv,  spacing  occurs  0 = S [ MQ H K F (V, +V ) - R 2 H 2 F 2 C V 2  2  (  2  + S[ 2CV HFKV, ( K - H V ) ] + C V 2 V e  Since  the  quadratic equation.. discussed  2  simple  formula  (2HV^ K - K - H V 2 )  z  2  2.42  F-factor  equation  spacing i s :  and t h e optimum t u r n o u t 2  ]  a  Z  + S[ 2CV HFKV, (K-HV ) ] + C l f l  uniform  becomes:  when:  2  and  the t u r n o u t  2.41  i  cost equation  a  -l  HV  0 = S2[MQ H K 2 ( F + s||-) (V, *V )-R2H2F2CV  The  be  1  spacing  2  to  F-factor can  be  Z  Z  6.  when:  (2HVj, K - R 2 - H 2 V|)  i s utilized used  2.43  ]  to  The u s e a n d t h e s e n s i t i v i t y i n Chapter  -1  find  in this the  2.44 equation, the  roots  of the equations  of will  the be  40 2.6  DISCOUNTING OF THE COST EOJJATION  Turnout while delay order  to  construction  costs  costs are incurred relate  discounting  present  i s required..  are  expenses o f the present  over the l i f e of the  costs  road..  In  to f u t u r e c o s t s some form of  The present worth formula  payments made i n the f u t u r e s t a t e s  for  equal  that:  PW = [ Q (1 + i)™-1 1 / [ i ( 1 + i ) ]  2.45  m  z  where: PW = present worth Q,£ = value of the constant  cost  i  = i n t e r e s t r a t e over p e r i o d m  m  = life  This  of p r o j e c t .  formula  can  be  incorporated  into  the  cost  functions  developed i n s e c t i o n 2.5 by a l l o w i n g : Z and  = [ (1 + i ) -1 ] / [im (1+i) m  2  incorporating  cost functions. turnout  m  ]  2.46  t h i s equation i n t o the delay Since  spacing,  equation 2.46  the  inclusion  is of  independent  discounting  d e r i v a t i v e o f the c o s t f u n c t i o n w i l l r e s u l t constant..  This  process  executed  component o f the  in  an  of the into  the  additional  on equations 2.43 and 2.44  results i n : c "H —+ Q HMZ 2  S  and  —  Lv,  +  /  + S[ 2CV,,  V,  J  spacing 2  2  -l  2.47  KJ  occurs when:  HZRZFZ^ (V, +V ) - K 2 H 2 F 2 C V HFKV, (K-HV ) ] +  E\  1--  —  K  a  -  HSF  V +  —  —  v J Lv,  the optimum turnout 0 = S2[MQ  SF~~  H "  CVf\  2  (2E\  ]  K-K2-H2V|)  In t h e development of t h i s t r a n s f o r m a t i o n  2^48  process  i t was  41 assumed  that  the maintenance c o s t i s not discounted and  i s a constant t r a f f i c traffic  flow volume per  year.  In  there  reality  the  flow volume per year i s not a constant but i s v a r i a b l e  from year to year. t h i s assumption.  Chapter  6 will  d i s c u s s the  sensitivity  of  42 3.0  THE  ARRIVAL DISTRIBUTION OF LOGGING TRUCKS  la.!  ARRIVAL DATA FROM TWO  There  have  distribution  no  b a s i c data that may  interarrival  times. .  c o a s t a l B r i t i s h Columbia coastal  studies  to  determine  study,  The  be s u i t a b l e i n  two  (B.C.) and  utilizing  sample  consisted  with an average vehicles  per  determining only one and f i v e  flow hour.  the  analysis  study areas were l o c a t e d i n i n north c e n t r a l B.C..  f i v e - m i n u t e time i n t e r v a l s ,  the time the loaded l o g g i n g t r u c k s a r r i v e d The  the headway  of l o g g i n g t r u c k s , but there have been two s t u d i e s  t h a t produced of  been  OPERATIONS  of 201  rate  the  measured  log  dump.  a r r i v a l s over a eight-day p e r i o d  of  These  at  The  approximately data  are  the headway d i s t r i b u t i o n  two-and-one-half  quite. r e s t r i c t i v e for  of v e h i c l e s s i n c e there i s  frequency c l a s s f o r v e h i c l e headways o f  between  zero  minutes.  The . i n t e r i o r o p e r a t i o n c o n s i s t s of v e h i c l e s t r a v e l l i n g a company haul road and u t i l i z i n g one of two scales. 6 A.M. the  The and  first  5 P.M.  hauling  scale  was  open  while the.second  operations  had  between  s c a l e was  ceased  government  Since  the  arrival  time  at  for  operation.. February  1976  hours  f o r the day. . The  the  recorded to the nearest minute these data useful  weigh of  open u n t i l a l l  v e h i c l e s would stop a t t h e . f i r s t weigh s c a l e open..  the  provided various  appear  of  loaded it  was  scales to  on  be  was  more  headway a n a l y s i s than do the data from the c o a s t a l Observations  were  recorded  with a t o t a l sample s i z e  for  of 813  the  month  of  o b s e r v a t i o n s and  43  an average flow r a t e of approximately t h r e e v e h i c l e s per For a more d e t a i l e d d e s c r i p t i o n of the two and  Young(1969)  times of the two  and  Smith  and  s t u d i e s c o n s u l t Boyd  Tse(1977).. The  s t u d i e s w i l l be analysed  hour.  interarrival  in section  3.2..  44  1.2  ANALYSIS OF THE  ARRIVAL DATA  The primary purpose determine  of  the  whether the a r r i v a l  a known p r o b a b i l i t y equations  involve  were grouped  analysis  Since  the  expected  closeness  of  f i t of  hypothesized p r o b a b i l i t y d i s t r i b u t i o n s . to  F-factor  i n t o headway frequency c l a s s e s r a t h e r than  the  to  i n s t e a d of a r r i v a l times the data  frequency c l a s s e s . . A yj- goodness of f i t t e s t was examine  was  frequency of l o g g i n g t r u c k s f i t s  function.. headways  following  the  arrival  utilized  data  to  to  various  T h i s technique i s used  compare the expected frequency, f o r a given d i s t r i b u t i o n , to  the observed frequency. X  =  2  The c h i - s q u a r e value i s determined  £[(0;-e,)2/e. ]  as:  3.1  where: O; = observed frequency of the i  cell  e; = expected frequency of the i  cell  k  = t o t a l number of c e l l s . .  The i n i t i a l step i s to determine the c e l l limit  of  the  first  cell.  can  be  determined.  The  headway  frequency  i l l u s t r a t e d i n F i g u r e 5. the  performed  headways  is  data  A computer program w r i t t e n  the programming language BASIC f o r the  9830A c a l c u l a t o r i s used i n t h i s  of  data  t y p e : o f d i s t r i b u t i o n s most l i k e l y t o f i t the observed  can be estimated from these graphs.. in  upper  Once t h i s has been accomplished a  frequency histogram of the observed The  widths and the  Hewlett  Packard(HP)  analysis.  histogram of the c o a s t a l study i s  T h i s graph shows that  l e f t skewed.  the  frequency  Goodness o f f i t t e s t s were  f o r the e x p o n e n t i a l , s h i f t e d e x p o n e n t i a l , and  Erlang  45 probability  distributions.  The  goodness of  are located  i n Appendix 4 w h i l e T a b l e I  The  indicate  results  the observed Figure  that  widths skewed  6  into  interior  minutes..  distribution  may  data..  with c l a s s  and  but  functions  tests  than  change  shifted  graphs  indicate  observed  minute,  data. tests..  not  adequately headway.  class  that  a  Table  II  As  was  observed  minutes  e x p o n e n t i a l f u n c t i o n s were n o t do  the  probability  f i t the  were i n c r e a s e d t o f i v e  the  rejected  d e s c r i b e the data f o r The  goodness  intervals  o b s e r v a t i o n s f o r the lower  of  because  headways(i.e.,  f i t there  headways  minutes).  v a l u e s of the expected  radically  with one  distributions  may  formulas.  sensitivity  be  or  more  suitable  discussed i n Chapters  F-factor  respect  then  The  groups  s t u d y , none o f t h e t e s t e d  distributions  be  f i t  distribution  while F i g u r e 7 uses  f o r t h e one-minute c l a s s  three  If the  the  frequency  of the goodness of f i t  than the c r i t i c a l  failed  were t o o few less  results.  distributions  This figure  of these  fit  widths  exponential  headways l e s s  study.  w i d t h s o f one  When t h e c l a s s  these  headway  intervals  Both  f o r the c o a s t a l  functions,  the  one-minute  of f i v e  case  none o f t h e t e s t e d  illustrates  summarizes t h e r e s u l t s the  summarizes t h e  data.  f o r the north c e n t r a l headways  f i t calculations  to of  f o r use  the  6. .  various tested  F-factor  do  not  probability ,probability  i n the expected  of the expected  4 and  formulas  F-factor  models  will  46  E-4- in &9 ui m  I  Ef.  1/1  raf:  +  U,  m  + ut  eg in  x ut  Figure 5 (5-minute  m x  rvi x  >-  in Fl  z ca E3 Ul DC  X  rvi  Frequency histogram of headways intervals)  rvi  ui  for  ca  coastal  study  47  Table  I  G o o d n e s s Of F i t T e s t s — C o a s t a l  I  |Probability I Dist.  T  T  1  ! xj  1  Yes  125(14)  I E r l a n g (oc=2) |  Yes  | 2 5 (1 2 )  |  1  1  7|  exponi  2. 5  1  76.3  |  12  5  1  2.5  1  62.7  |  10  !  : 0 . 1 min |  Yes  125(14)  |  5  |  2. 5  |  : 0 . 5 min j  Yes  125(14)  |  5  1  2.5  |  I  : 1 . 0 min  I  Yes  125(14)  |  5  i  2.5  |  R e j e c t i o n of  I J* I lot'. 0 0 5 |  |  1  1  ! I  12  71.Q  I  12  65.8  |  12  75.2  -+ 1 1 |28.3 1 125.2 1 1  1 128.3 1 128.3 1 128.3  1 1 1 | 1 | 1 1 1  | 1 | 1 |  hypothesis  * The parenthesized number i s t h e f i n a l number o f c l a s s e s while the non-parenthesized number i s the number of f r e q u e n c y c l a s s e s • Lower c l a s s  T  ITablel  5  <  •  — 1  1  E e j e c t •|Number* C l a s s | LCL • | Computed| of Width| I cf z H y p o t h I C l a s s e s (min) | (min) + -+  | Exponential!  I Shift  Study  limit  + The number o f d e g r e e s o f freedom  frequency original  48  Figure 6 Frequency (1-minute i n t e r v a l s )  histogram  of headways f o r i n t e r i o r  study  49  rvi  ra  ra ra  ra m  EjLTl  VZJ z ra msec 3  az  u cgx  ui  ra x  ra m  ra rvi  ra  CD  rvi  x  10  X  Figure 7 (5-minute  in rvi  m  ra  ra  >Zcg  ca UJ ce u.  Frequency histogram intervals)  P4  x  Ul  in m  of headways f o r  m  ra  interior  study  50  Table  II  Goodness Of F i t  Tests- Interior =  Stud^-Both  P r o b a b i l i t y | Re j e c t • | Number*! C l a s s j LCL • | C o m p u t e d | Dist. . I of |of | Width| I Hypoth | C l a s s e s I ( m i n ) | (min) V ^ + -+ + + +  !  1  X  2  1  |«r.005| -+-  1  Yes  |80 (53)  |  1  | 0. 5  1  92.0  | 51  153.7  1  E r l a n g (cC=2) I  Yes  | 80 (48)  |  1  1 0.5  1  617.9  I 46  |53.7  1  1  I  I  1  1  ex pen  : 0 . 1 min  i  i  Yes  |80(53)  |  1  I  Yes  I 80 (52)  |  1  Exponential  Yes  121(17)  |  5  E r l a n g (oC=2) I  Yes  121(13)  |  5  I  I  :0.5  Shift  I  min  expen  0. 5  I  87.5  |  51  153.7  1  | 0.5  I  72.2  I  50  153.7  |  1  0.5  I  33.7  I  15  132.8  |  1  0.5  |  183.5  |  11  126.8  |  i  !  !  I  I  :0.1  min  I  No  121(17)  |  5  1  0.5  I  29.4  I  15  132.8  |  :0.5  min  I  No  |21(16)  |  5  1  0.5  I  17.6  |  14  131.3  |  No  120(16)  |  5  1  5.5  I  20.7  I  14  131.3  I  Exponential  Rejection  cf  hypothesis  * The parenthesized value is the c l a s s e s w h i l e the non-parenthesized number of f r e q u e n c y c l a s s e s •  !  | Table|  Exponential I  Shift  •  +  Scales  Lower c l a s s  limit  + The number o f  degrees of  freedom  f i n a l number o f number is the  frequency original  51 iUO_  SIMULATION  FOR  THE VERIFICATION OF THE EXPECTED F-FACTOR  EQUATIONS  S i m u l a t i o n can be an v a l u a b l e t o o l i n the v e r i f i c a t i o n the  accuracy  of  mathematical  w i l l be used t o determine expected  equations.  the accuracy  F - f a c t o r equations.  and  results  following  section.  of  In t h i s i n s t a n c e i t  and l i m i t a t i o n s  of  the  The t e c h n i c a l d e s c r i p t i o n of the  three s i m u l a t i o n models i s d i s c u s s e d i n features  of  the  Appendix 5  while  the  models w i l l be analysed i n the  52  4. 1  THE SIMULATION MODELS  A d i s c r e t e , c r i t i c a l event s t o c h a s t i c s i m u l a t i o n model was w r i t t e n i n the g e n e r a l purpose programming language FORTRAN f o r the IBM 370 Model 168 computer t o confirm or r e j e c t the F-factor  equation.  The  simple  program s i m u l a t e s the meeting o f one  empty v e h i c l e and one loaded v e h i c l e .  Initially,  the empty and  loaded v e h i c l e s a r e randomly separated by a d i s t a n c e o f between f i v e and t e n m i l e s . meet  i f the  The p o s i t i o n where the two v e h i c l e s  vehicles  travel  unhindered  i s determined..  f i n a l phase was t o move the empty v e h i c l e t o t h e f i r s t it  can s a f e l y  utilize.  determined  by  calculating  The  the  v e h i c l e i s from the empty v e h i c l e , stopped dividing repeated mean  The  turnout  Simultaneously, the loaded v e h i c l e was  backtracked f o r an equal time i n t e r v a l . easily  would  F-factor  distance  can  the  be  loaded  i n the turnout,  and  t h i s value by the turnout s p a c i n g . . T h i s procedure i s f o r t h e d e s i r e d number of r e p e t i t i o n s and  i s determined.  The  the  sample  program i s executed e i t h e r nine o r  twenty times and the mean and standard d e v i a t i o n o f the samples are determined..  A t - t e s t i s performed  being t h a t the mean o f the average simple  F-factor. .  The  speed  with the n u l l h y p o t h e s i s  F - f a c t o r s i s equal  and turnout spacing  were a r b i t r a r i l y chosen. . The r e s u l t s o f documented  i n Table I I I .  the  combinations  simulations  are  The r e s u l t s of the s i m u l a t i o n s , with  a one percent l e v e l o f s i g n i f i c a n c e , were not r e j e c t e d three percent of the time. .  t o the  seventy-  53  Table I I I F - f a c t o r Simulation S i n g l e Loaded Vehicle And A Equation 2 . 1 2  Sample*  r  1  T  | V e h i c l e Speed|Turn Ave. r 1 1 Spac F Loaded Empty j (mph) (mph) I + + 2  I I  1 0 0 0 0 (9) 10000(9) 10000(9) 1 0 0 0 0 (9) 1 0 0 0 0 (9) 1 0 0 0 0 (9) 10000 (20) 1 0 0 0 (20) 1 0 0 0 (20) 1000 (20) 1 0 0 0 (20)  The Interaction Between A Single Empty V e h i c l e Based On  | | | | | | | | | | |  i  10 10 30 30 30 10 10 30 30 10 10  | i | | | | | | | | | L—  20 20 30 30 40 35 35 40 40 35 35 .  |0. 1 JO. 5 10.05 | 0 . 15 | 0.3 I 0 . 15 I 0.2 | 0. 2 I 0. 3 | 0 . 15 1 0. 3  —i  S. D.  |Rejection | of I Hypothesis |oC=  +  I0.7517| 5.9 |0.7524| 4.5 I 0.99951 3.0 | 1.00251 4.4 | 0 . 8 9 2 3 | 5.2 | 0 . 6 3 9 1 | 3.8 | 0 . 6 4 5 3 | 3.4 |0.8684| 14. 5 |0.9026| 14. 1 |0.6386| 7.8 |0.6382| 10;3 —1  Eq. • F  3  xlO-3  0.05  + |0.7500| (0.75001 | 1.00001 | 1.00001 10.87501 |0.6429| |0.6429| |0.8750| 10.8750| |0.6429| |0.6429|  L ______ _ J _  L  No No s NO 5 NO s Yes Yes Yes No Yes Yes* No 5  6  The sample s p e c i f i c a t i o n s : where the non-parenthesized number i s the sample s i z e and the parenthesized number i s t h e : number of samples 1  2  The d i s t a n c e between t u r n o u t s i n miles  3 Standard d e v i a t i o n * The equation value of the expected F - f a c t o r 5  6  Hypothesis was not r e j e c t e d f o r a 10% l e v e l of s i g n i f i c a n c e Hypothesis was not r e j e c t e d f o r a 1% l e v e l of s i g n i f i c a n c e  54  A  similar  s i m u l a t i o n model was used t o t e s t the v a l i d i t y  of the expected F - f a c t o r equation r e s u l t i n g from a s i n g l e vehicle interacting model  includes  with a  fleet  of  the p r o b a b i l i s t i c  loaded  vehicles..  a  predetermined  probability  distribution..  simulates the meeting of a s i n g l e empty loaded v e h i c l e s . road  until  the  empty  t u r n o u t i t can s a f e l y to  wait  the  same  were  I t i s then  utilize.  The  The experience o f  method  equations t e s t e d i n the same Simulations  and  manner  conducted  the as  for  t a b u l a t e d i n Tables IV and V.  significance  of  the  backtracked  model  the  allows  exponential  model  expected  the the  is  F-factor  previous  model.  exponential  E r l a n g (alpha=2) headway frequency d i s t r i b u t i o n s . are  model  i n the turnout u n t i l one or more  loaded v e h i c l e s have passed.. by  The  The empty v e h i c l e progresses along an e n d l e s s  vehicle  collected  conform  v e h i c l e and a f l e e t  i t meets a loaded v e h i c l e . .  to the f i r s t  This  aspects of flow behaviour by  having the headway d i s t r i b u t i o n of the loaded v e h i c l e s to  empty  The  and  results  With a one percent l e v e l o f  F-factor  equation  was  within  a c c e p t a b l e l i m i t s n i n e t y percent of the time while the estimate of  the  E r l a n g (alpha=2)  F-factor  equation  was always w i t h i n  acceptable l i m i t s . . The range of t r a f f i c of  the  expected  F-factor  meeting o f two f l e e t s Consequently, expected  a  flow f o r which the  of  equations vehicles  must  yet  be  s i m u l a t i o n model was developed  Since  form  d e s c r i b e the determined.  t o p r e d i c t the fleets  of  t h i s model i s more complex than the p r e v i o u s  models a General Purpose.Simulation used  adequately  F - f a c t o r r e s u l t i n g from the meeting of two  vehicles. „  was  probability  System  V  (GPPSV)  program  to model t h i s s i t u a t i o n . . The c h a r a c t e r i s t i c s o f the  55  model  are  simulation  outlined  in  Appendix 5 .  The  results  of  runs are t a b u l a t e d i n Table VI.. The equations  t e s t e d by the same method as the p r e v i o u s models.. The  the were  results  i n d i c a t e that the e x p o n e n t i a l F - f a c t o r equation i s s u f f i c i e n t l y accurate  for  a  traffic  therefore  equation 2.19  practical  levels  reason  of  flow is  is  that  the  of  60 vph  sufficiently  traffic  f o r the f a i l u r e of the  100 vph,  rate  number  accurate  l e s s and for a l l  flow i n v o l v e d i n l o g h a u l i n g . t-test, of  with  a  conditional  s i t u a t i o n s has i n c r e a s e d t o the point where they a f f e c t the expected F - f a c t o r .  or  flow  rate  A of  event B delay significantly  56 Table IV F-factor Simulation -- An Empty. V e h i c l e Meeting Loaded V e h i c l e s Based On Eguations~2. 18, 2* 11M. 2. 3 0 And"2^3Jl Sample s i z e = 10000 Number of samples = 9 A c c e l e r a t i o n = 19759 mph " C o n f l i c t " hours = 11 Exponential headway d i s t r i b u t i o n x  2  I  T  1  1  1  1  1  T  | V e h i c l e Speed|Distancel T r a f f i c I Ave. | S. D. H Expect | R e j e c t i o n | T — T Between | Flow F | of | I F |X10-3| | j | | Hypothesis| |Loaded|Empty |Turnouts| Bate oc = 0.05 | | (mph) | (mph) | (miles) j (vph) + 4 + r  R  |  10  I I I I I  I I I I  I I I I I 1  30  |  40  I I I I  I I I I I  I I I I I  |  30  I |  I I I  1  I I I I  I I I I  I I I I I  |  I  I  |  |  I I I  I I I  1  |  0. 1  I  |  |  I  20  30  |  0.05  I  30  40  |  | 1  0. 2  0.3 |  1  |  | I | I I  |  1 1 1 1 1  |  1 1 1 | 1  | 1 |  1 2 4 10 60 100  |0.7476| 3.06|0.7507| I0.7492J 3.1410.7514| |0.7523| 4.46|0. 7526| | 0. 7521 | 4.18|0.7543| ) 0.7023 |5.02|0. 7034| |0.6247| 3. 16|0.6269|  No No NO No No  1 2 4 10 60 100  | 0.9980 | 5.42| 1. 0020| | 1.00561 4.23| 1.00421 1 1.0131| 3.23|1.0085| | 1.02151 6. 07| 1. 0208| | 1.09221 6. 03| 1.09711 1 1.1317| 5. 3811. 1300 I  No No Yes No Yes* No  | | j | | |  1 2 4 10 60 100  I 1. 1697| 10.66|1.1684|  1 1. 1694| |1.1690| | 1. 1804| |1.1503| | 1.0690J  6.03|1.1703| 3. 24| 1. 17371 5.26|1.1817| 5.59|1.1558J 6.03|1.07181  No No Yes No Yes* No  | | | | | |  1 2 4 10 60 100  |0.8737| |0.8799| |0.8811| 10.8841| |0.8138| |0.7062|  4.76|0.8767| 5.85|0. 8785| 4.07|0.8815| 5.18|0.8873| 7. 80|0.8165| 4.86|0.7054|  No No No No No No  | | | | | j  1 1 | L_ _ _ _ _ _ _ X  L  1  L  Yes3 2  2 2  2  2  2  2  2  2  2  2 2 2 2 2 2  1  Standard  2  Hypothesis was not r e j e c t e d f o r a 10% l e v e l o f s i g n i f i c a n c e  3  deviation  Hypothesis was not r e j e c t e d f o r a 1% l e v e l o f s i g n i f i c a n c e  |  | | | | |  i  57 Table V - - - F - f a c t o r Simulation -- I S Empty Vehicle Meeting Loaded V e h i c l e s Based On E q u a t i o n s 2. 18, 2. 19, 2.33. And~2.34 Sample s i z e = 10000 Number o f s a m p l e s = 9 A c c e l e r a t i o n = 19759mph " C o n f l i c t " h o u r s = 11 E r l a n g ( a l p h a = 2 ) headway d i s t r i b u t i o n 2  | V e h i c l e Speed|Distance T r a f f i c 1 Ave. | S . D . | E x ppeecctt| R e j e c t i o n p Flew r 1 1 Between F I of 1 F | X 1 0 - 33 j| |Loaded|Empty ITurnouts Kate Hypothesis I I (mph) | (mph) | (miles) (vph) oc = 0.05 I l + + H + r 1  | I I I I I  30 I I I I I  |  | I I I I I  40 l I I I I  |  40  0.3  I I I I I  1 i  1 1 30  |  0.2  !  l I I I I  * Standard 2  |  1 1  1 1  1 2 4 10 60 100  |0.8740| |0.6756| | 0. 8784 | |0.8974| | 0.9762| |0. 8317|  1 2 4 10 60 100  | 1. 1679| I 1. 1647| | 1. 1692| |1.1829| | 1. 32431 | 1.2811 |  4.46|0. 8753| 5.44|0. 8761| 6.71|0.8793| 4.79|0. 8973| 5.67|0. 9733| 6.96|0. 8345| I 6. 13| 1. 1668| 6. 18| 1.. 1674| 5.14|1. 1695| 6.18|1. 182 51 9.39|1.3259| 7.94|1.2856|  No No No No No No No No No No No No  2 2 2 2 2 2  2 2 2 2 2 2  deviation  Hypothesis was  not r e j e c t e d  f o r a 10% l e v e l o f s i g n i f i c a n c e  58 Table VI F-factor Simulation -- I n t e r a c t i o n Between F l e e t s Of - V e h i c l e s Based On Equations 2.187 2^19 2.30 2. 3,1 Empty v e h i c l e speed = 40 mph Loaded v e h i c l e speed = 30 mph A c c e l e r a t i o n = 19759 mph E x p o n e n t i a l headway d i s t r i b u t i o n x  x  Two And  2  Ave. . N umber|Distance| Sample Of |Between Size Samples | Turnouts| |(miles)  9718  4  24150  5  37174  5  35390  5  * Standard 2  3  F  I  r  i  Ave.  I I I I I I I  |S.P. |Expect|Rejection 1  |X10-3|  F  j  of  I Hypothesis | oL= 0.05  I I  I I  I I  I I  I I  I  I  0.3  I  10  0.88421  1. 03|0. 8873|  No  2  0. 3  |  20  0.87571 1.24|0.8873|  No  2  0.3  |  60  0.84831  0. 3  |  100  I  I  I  I  I  1. 70|0. 8165| I  I  0.90981 4. 57|0.7054|  Yes* Yes  deviation  Hypothesis was not r e j e c t e d f o r a 10% l e v e l o f s i g n i f i c a n c e Hypothesis was not r e j e c t e d f o r a 1% l e v e l of s i g n i f i c a n c e  59 5.0  THE COST VARIABLES AHD THE MODIFICATION OF THE EXPECTED F-  FACTOR  5.1  THE COST OF TRUCK  The  cost  equations  estimate o f t r u c k i n g elements  as  permits, and from  the  An  past  developed  costs.  wages  insurance,  oil.  TRANSPORTATION  This  of  the  depreciation,  in  Chapter 2  cost  involves  operator, repair,  r e q u i r e an such  cost  vehicle licences, maintenance,  fuel,  estimate o f these c o s t elements can be obtained  records,  production  manuals,  appraisal  manuals,  equipment r e n t a l manuals, and t e c h n i c a l r e p o r t s . The  Journal  equipment  rental  Association, 1978). . line),  of Logging Management p e r i o d i c a l l y p u b l i s h e s rates  prepared  Vancouver,  Their  B.C..  calculations  interest  (simple  by  Truck  depreciation  percent),  tires,  lubrication,  ( I n t e r n a t i o n a l Woodworkers of  America  l i v i n g p r o v i s i o n plus 30 percent percent  of  total),  r e n t a l r a t e i s based  and on  provisions for fuel costs. truck c l a s s e s  wages  (55 of  wage  ($1.30 per percent  the  plus  a  of  operator cost  of  p a y r o l l l o a d i n g ) , overhead (10  profit a  (straight  insurance  $100.00 o f investment), r e p a i r and maintenance depreciation),  Loggers  (Journal Of Logging Management,  include  15  the  (10  ten-hour Rental  percent day  of t o t a l ) .  but  there  are  The no  r a t e s are c a l c u l a t e d f o r two  (twelve and f i f t e e n f o o t bunks) and three  coastal  zones. . The  United  manual to p r e d i c t  States the  Bureau truck  o f Land Management produces a  hauling  costs  f o r Oregon  and  60 Washington The  (United  States  Bureau  of Land Management, 1977).  c o s t p r e d i c t e d f o r a White t r u c k  trailer cost  includes  a  fixed cost  (Model 4964) and  Peerless  ($6.59 per hour), an o p e r a t i n g  ($9.32 per hour), a d r i v e r ' s wage ($11.62 per hour) and an  overhead cost  ($2.75 per hour).  Smith and Tse(1977) subdivided costs  and  travelling costs.  trucking costs into  In-use c o s t s i n c l u d e f i x e d  (depreciation, i n t e r e s t , opportunity and  the  Smith  charges,  and  wage of the d r i v e r while the t r a v e l l i n g  such items as  fuel,  tabulated  o i l , tires,  the  in-use  hauling  repairs,  costs  costs  insurance)  costs include  and  maintenance.  f o r three  truck c l a s s e s  (eight, t e n , and twelve f o o t bunks) . Table VII summarizes these  three  sources.  d i r e c t l y compared s i n c e operating and  areas  the  predicted  hauling  costs  from  The r e s u l t s of these a r t i c l e s cannot be the  costs  apply  to  three  distinct  (north c e n t r a l i n t e r i o r o f B.C., c o a s t a l B.C.,  northwestern United S t a t e s ) .  Consequently, a cost  element  from one area may be higher or lower than that c o s t element f o r the  other  two a r e a s .  These c o s t s are given not as a hard and  f a s t r u l e but as a g u i d e l i n e f o r the c o s t elements t h a t be i n c l u d e d i n the c a l c u l a t i o n of the adjusted  should  hauling r a t e . .  61 Table VII  Truck Hauling  Source  |  Costs  Zone | Bunk j Size  |  |  I  8  Tse(1977)  |  I 1  Hauling Cost ~T"  T  I S t r a i g h t | O v e r t i m e | Delay | Time | | Time  | (feet) 1 + + Smith and  Hourly  1  ($)  1  ($)  r  1  I 24. 19» |  |  10  I  27.12* |  |  12  | 32. 241 |  |  J o u r n a l c f Logging| B a s i c I  12  | 34. 553 |  |  Management (1978)  2  United S t a t e s Bureau o f Land Management (1977)  ($)  +-  |  A  2  |  12  1 37. 053 |  j  !  B  2  1  12  \ 39. 553 |  |  !  Basic  1  15  | 45. 453 |  j  !  A  1  15  | 48.643 |  |  ;  B  I  15  | 52.053 |  |  | 30.28  | 20.03  | | |  |  | 32.26  Based on average hourly c o s t between ' s m a l l ' l o g and ' l a r g e ' log loads. 'Large' sawlogs had a minimum butt diameter o f fourteen inches. 1  2  -The zones r e f e r t o i s o l a t i o n areas where: A = up c o a s t areas with poor access B = isolated parts of Dean and R i v e r s Charlottes etc.  3-These a r e the r e n t a l r a t e s without p r o f i t margin. .  PSYU's  and  i n c l u d i n g the ten  Queen percent  62  5_i.2  THE  TOBNOOT CONSTBUCTION AND  Boad records*  construction manuals, and  categorize  turnout  costs  can  be  obtained  construction  costs.  can  costs 4b).  for The  Nagy(1977)  three  road  construction  ballasting  (from  they  turnouts  are  widened  c o n s t r u c t i o n c o s t i s somewhat l e s s than t h a t  rock  road.  estimated  classes  c a l c u l a t i o n s included  subgrade  Since  do  be r e l a t i v e l y e a s i l y  a n t i c i p a t e d f o r a comparable s t r e t c h of Sauder and  from company  various p u b l i c a t i o n s but seldom  u s u a l l y l o c a t e d where the road the a c t u a l turnout  MAINTENANCE COSTS  the  road  (B.C.. F o r e s t the  cost  construction  S e r v i c e - 3,  estimation  (cable s h o v e l , b u l l d o z e r and quarries  and  gravel  of  the  drill),  pits) ,  4a,  and  the the  surfacing. The  B.C..  Forest  a p p r a i s a l manuals (B.C. S e r v i c e (1977))  to  and  Forest  estimate  v a r i e t y of c o n d i t i o n s manuals  Service  Districts  have  Service(1975),  road  B.C..  construction  (terrain, s o i l s ,  road  developed Forest  c o s t s under a  classes)..  These  can be used to c a l c u l a t e the c o s t of a s t a t i o n of  then  apply  a  portion  of  this  States  Bureau  figure  to  the  road  turnout  construction cost. The  United  determined the turnout e x c a v a t i o n turnouts  (a  fifty-foot  approaches and approaches) classes. excavation  a one  under  The  Land  costs  turnout  plus  hundred-foot turnout  Management(1977)  for  two  two  twenty-five-foot  plus  classes  two  a v a r i e t y of s o i l c o n d i t i o n s and  total  cost  of  turnout  plus  the  construction grading  and  cost  of  fifty-foot side  slope  included  the  surfacing costs.  The  63 r e s u l t s o f t h e i r c o s t i n g study are shown i n T a b l e : V I I I . . Turnout maintenance maintenance  costs.  costs  can  be  cost  per  unit  The  estimated  main  estimated schedule projected These  road  surface.  the machine  The  figures  must  less  Forest  roads.  maintenance be  altered  costs to  wear  than  Service(1977)  r a t e s , p r o d u c t i o n r a t e s , and  f o r mainline and spur turnout  B.C..  road  s u r f a c e area i s not  e q u i v a l e n t s i n c e the t u r n o u t s would experience the  from  maintenance  Table IX i l l u s t r a t e s the based on these reflect  the  figures. different  d e t e r i o r a t i o n r a t e s between the turnout s u r f a c e and the roadway surface.  64 Table V I I I  Excavation Costs Per Turnout* 14-foot Subgrade  (10-foot Usable  Common Excavation  TI  Sock Excavation  C o s t / |Avg.,Cut Avg. .Cu C o s t / % Turnout|at Center Yards/ Turnout Side | L i n e - F t . . Turnout Slope  1  T  0 10 20 30 40 50 60 70 80 90 100  I  i$  7.10| 7. 10 | 8. 15| 12.45| 13.45 | 21.85| 79.00| 171.20| 208.80| 262. 15| 316.251  | | | | | | | | | j  x.  1.3 1.3 2.0 2.7 3.5 4.7 8.0 12.0 13.2 14.8 17.0  lengths:  i Standard approaches  | | | | I  | | | | | |  28 28 32 49 53 86 311 674 822 1032 1245  20-foot Subgrade "T"  I  r Side Slope  0  Cost/ Turnout  10 |  20 30 40 50 60 70 80 90  | | | | | | | |  100 |  turnout  1.3 1.3 2.0 2. 8 3.5 4.7 8. 0 12.0 13. 8 15.0 17. 0 plus  | | I  | I  | | | | | | i_ two  Width)  25-foot  2  |  Rock Excavation  T  1  Avg. Cut Avg. Cu| C o s t / at Center Yards/ I Turnout Line-Ft. Turnoutl  19.551 19.55 | 26.40| 30.22| 5 2 . 601  54.10|  269.00| 436.40| 5 4 7 . 10 | 674.351 794.501  1.7 1.7 3.0 3. 1 4.0 5.7 10. 1 14.0 16.0 18.0 20.0  | I | | | | | | | | |  77 77 104 119 207 213 1059 1718 2154 2655 3128  |$ | | I | | | | | | |  T  IAvg. Cut |Ave..Cu. |at Center|Yard/ |Line-Ft..|Turnout  1  I  243.75| 2 4 3 . 75 J 356.85| 372.45) 466.05| 407.55I 1700.40! 2560. 35! 3120.001 3502.20| 4093.051  1.0 1.0 2. 5 3. 1 4.0 5.6 10. 1 14. 0 16. 0 18. 0 20.0  | 125 | 125 | 183 | 191 | 239 ! 209 | 872 | 1313 | 1600 ! 1796 | 2099  Standard lengths: 100-foot turnout plus two approache s * From United S t a t e s Bureau o f Land Management(1977) 2  26 26 53 101 71 106 255 509 588 702 830  1  Common Excavation  -  |$  1  IAvg..Cut |Ave. Cu. |at Center!Yard/ | L i n e - F t . |Turnout  (12-foot Usable  —  r 1 1 I | I I | | I I I  1  |$ 50.70| | 50.70| | 103. 251 | 197.001 | 138. 451 | 206.701 | 461. 701 | 970.701 | 1146.601 | 1368. 90| | 1618.501  50-foot  Width)*  50-foot  65 Table IX  Turnout Maintenance  r Machine T Rate (Grader) per Shift  Cat  Cat  Cat  12  14  16  $227  277  334  Road Class  Costs  Maintenance Schedule in days  1 2  Annual Maintenance Cost per Turnout  3  r  2 miles|3 miles 4 milesl /shift |/shift /shift  Mainline  20  $11.448|$ 7.632  Spur  40  |  Mainline  20  13.9701  9. 313  Spur  40  I  4.657  Mainline  20  16.844| 11.229  Spur  40  |  3.816 $ 2.862  5.615  3. 492  4.211  assume a road width of s i x t e e n f e e t and a t u r n o u t area o f seven hundred f i f t y square f e e t . C a l c u l a t i o n s based on f o u r turnouts per mile, two hundred o p e r a t i n g days per year and e i g h t y - f i v e percent a v a i l a b i l i t y . 1  2  Based on B.C.  F o r e s t S e r v i c e (1977)  These c o s t s can be r e a d i l y a d j u s t e d t o f o r v a r i o u s maintenance p e r i o d s . 3  determine  the  costs  66 5.3  MODIFICATION  The  OF THE EXPECTED F^FACTOE E<2DATIONS  expected  F - f a c t o r e q u a t i o n s , developed  assume there i s no minimum reality  there  w a i t i n g time i n  a  i s a minimum delay p e r i o d .  i n Chapter  turnout  but i n  In t r a f f i c a n a l y s i s  the merging of t r a f f i c from a ramp i n t o the main t r a f f i c may be c l a s s i f i e d  as an i d e a l o r f o r c e d merge., An i d e a l  does  the  not  cause  main  traffic  stream  change l a n e s t o allow the merging v e h i c l e traffic  stream  but  phenomena to occur. logging  truck  a  forced  A parallel  merge  2,  stream merge  t o reduce speed or  to  enter  the  main  r e s u l t s i n one of these  can be drawn with regards to  u t i l i z i n g a turnout where an i d e a l p u l l - i n  a  will  not cause the loaded v e h i c l e t o reduce speed to a l l o w the empty v e h i c l e t o u t i l i z e the turnout but a f o r c e d p u l l - i n w i l l the  loaded v e h i c l e t o reduce speed., In the c a l c u l a t i o n o f the  F - f a c t o r i t has been assumed the v e h i c l e s t r a v e l a t a speed  except  vehicle  may  be  able  The  a forced p u l l - i n  operator  of  to an i d e a l p u l l - i n .  of a p u l l - i n from  forced  expected  vehicles  a  F - f a c t o r should only apply t o a  model does not account  f o r the length of the  l e n g t h o f the turnout i s accounted  and  loaded  turnout. .  f o r then there i s a  s e c t i o n o f road where there a r e two l a n e s such t h a t vehicle  being  pull-in.  The the  the  empty  Provided the operator o f  the empty v e h i c l e knows the l o c a t i o n of the loaded of  the  to i n c r e a s e h i s speed over a s e c t i o n o f  road so as t o change the c l a s s i f i c a t i o n  modification  constant  when the empty v e h i c l e d e c e l e r a t e s i n t o a t u r n o u t  and a c c e l e r a t e s from a t u r n o u t .  If  cause  v e h i c l e can pass each other.  the  empty  T h i s though  67 would  not In  of  will  alter  the c l a s s i f i c a t i o n of a p u l l - i n .  c o n c l u d i n g , i f the operator  the  factor  significantly  other equation  be  vehicles. .  v e h i c l e s then may  assumed  does n o t know t h e  location  a m o d i f i c a t i o n of the expected  be r e q u i r e d .  However,  in  this  t h e o p e r a t o r knows t h e l o c a t i o n  study  of the  Fi t  other  68 6.0  THE USE AND TESTING OF THE MODEL  6.1  THE USE OF THE MODEL  The from  s o l u t i o n of the optimum t u r n o u t s p a c i n g model  the  evaluation  utilization as  of  a  complex  of a search technique.  results  s e t of equations and the  T h i s s o l u t i o n can be  a g u i d e l i n e f o r determining turnout spacing and can be used  as an a i d i n determining when the road should be switched being  single-lane  to  double-lane.. Furthermore,  the model, equation 2.8, can be u t i l i z e d expected  independent  9,  10,  and  11  on  the t o t a l  the e f f e c t  the  v a r i a b l e s , t r a f f i c flow, turnout s p a c i n g , v e l o c i t y  the t o t a l  d i s t a n c e o f road. 11855  a section of  t o determine  illustrate  the loaded v e h i c l e s , and v e l o c i t y  have  from  delay time per v e h i c l e per u n i t d i s t a n c e : o f road..  F i g u r e s 8,  of  used  mph , 2  expected  of  vehicles,  delay time per v e h i c l e per u n i t  I t i s assumed the r a t e  the r a t e  the empty  of  acceleration  is  of d e c e l e r a t i o n i s 19759 mph , and the 2  headway frequency d i s t r i b u t i o n i s e x p o n e n t i a l . . The formulas t o determine  the  rates  of  developed  i n Appendix 3.  acceleration  and  d e c e l e r a t i o n are  Unless i t i s s t a t e d otherwise, i t i s  assumed t h e v e l o c i t y of the loaded  vehicle  i s 20  mph, the  v e l o c i t y o f the empty v e h i c l e i s 25 mph, the t u r n o u t spacing i s 0.1  miles,  and  the t o t a l expected of  the t r a f f i c flow r a t e i s 4 vph.. delay time  (T) i s expressed  the t o t a l t r a v e l empty time.  as a  Furthermore, percentage  Figure 8 Effect of traffic delay time f o r v a r i o u s turnout  flow r a t e on the t o t a l spacings  exp ected  70  U.  UJ X  ten i—a.  „  Lrl  S  m  J - U - K u ,  SI  m S3  Lrl rvi  Ln  rvi si  x: ID  Z  <*;  in  rc a. Ln  2  a:  Ln sa sa  -tr  Lrl  L/1  m  m  Lrl  rvi  PJ  ih  Lrl  is  _J>ui>rxrxujfe* >»—UJ i - j £ c n z : • UJ — z u - o s x : — j - t a t — u i r -  Figure 9 E f f e c t of the turnout s p a c i n g on the t o t a l delay time f o r v a r i o u s t r a f f i c flow r a t e s  expected  71  F i g u r e 10 Effect of the v e l o c i t y of the empty v e h i c l e t o t a l expected delay time f o r v a r i o u s v e l o c i t i e s of the vehicle  on the loaded  72  F i g u r e 11 E f f e c t of the v e l o c i t y of the loaded v e h i c l e on the total expected delay time f o r v a r i o u s v e l o c i t i e s of the empty vehicle  73 F i g u r e 8 i l l u s t r a t e s the e f f e c t of the t r a f f i c (1  to  20  vph)  on  turnout s p a c i n g s delay  time  the t o t a l expected  (0.05 t o  {%)  0.35  increases  the  The  traffic  total  expected  flow r a t e and the  t u r n o u t s p a c i n g i n c r e a s e , though the marginal e f f e c t as  the  traffic  flow  rate  increases.  s i t u a t i o n s , where the t r a f f i c t o t a l expected empty time. has  cn  traffic  flow  the  rates.  and  the  spacing i n c r e a s e s . 15  percent  flow r a t e i s l e s s than 5 vph, the  i s l e s s than  total The  decreases  For most l o g h a u l i n g  20 percent  of the t r a v e l  F i g u r e 9 i l l u s t r a t e s the e f f e c t t h a t  spacing  increases  delay time  rate  delay time f o r v a r i o u s  miles).  as  flow  expected total  marginal  delay  delay  time  (%)  decreases as the turnout  The t o t a l expected  of the t r a v e l empty time  turnout  time f o r v a r i o u s  expected  effect  the  delay time  i s less  than  provided the t r a f f i c  r a t e i s l e s s than 5 vph and t h e r e i s a minimum  of  5  flow  turnouts  per mile. . F i g u r e 10  i l l u s t r a t e s the e f f e c t t h a t the v e l o c i t y o f the  empty v e h i c l e (15 t o 45 mph) has on the time mph).  for  various  velocities  The t o t a l expected  delay  total  expected  delay  of the loaded v e h i c l e (10 t o 40 time  (%)  and  the  marginal  e f f e c t i n c r e a s e as the v e l o c i t y of the empty v e h i c l e i n c r e a s e s . The  total  expected  t r a v e l empty time* velocity expected  delay time i s l e s s than 20 percent of the F i g u r e 11  depicts  the  of the loaded v e h i c l e (10 t o 35 mph)  effect  that  has on the t o t a l  delay time f o r v a r i o u s v e l o c i t i e s of the empty v e h i c l e  (20 to 40 mph).  The t o t a l expected  delay time and the a b s o l u t e  value of the f i r s t d e r i v a t i v e of the f u n c t i o n decrease velocity  the  as  the  of the loaded v e h i c l e i n c r e a s e s . , Based on these  four  f i g u r e s , the g e n e r a l trends t h a t an i n c r e a s e o r decrease  i n one  74 of  the independent v a r i a b l e s have on the t o t a l  expected  delay  time can be p r e d i c t e d . Since is  the c o s t f u n c t i o n i s complex and a search  required to  advantageous  locate  to  optimum  turnout  program  that  the optimum  develop  a  spacing.  spacing,  i tis  computer program to determine the Appendix 8  can be u t i l i z e d  s p a c i n g . . Besides determining calculates  turnout  technique  documents  a  computer  t o determine the optimum  turnout  the optimum s o l u t i o n the program  the c o s t s , based on equation  2.38, f o r the optimum  turnout s p a c i n g , 50 percent of the optimum turnout s p a c i n g , 100 percent o f the optimum t u r n o u t s p a c i n g , and 200 percent of the optimum  turnout  spacing.  s e n s i t i v i t y of the optimum equation's  A  solution  12,  on the  fluctuations  i n the  13, 14, 15, 16, and 17 i l l u s t r a t e the e f f e c t  t h a t the independent v a r i a b l e s ,  rate ,  to  discussion  v a r i a b l e s i s l o c a t e d i n s e c t i o n 6.2. .  Figures  vehicle,  further  the  velocity  t h e turnout  the  velocity  of  the  loaded  of the empty v e h i c l e , the t r a f f i c  construction  h a u l i n g c o s t , and the expected the optimum turnout s p a c i n g .  cost,  useful l i f e  the adjusted  flow truck  of the road, have on  In t h e c a l c u l a t i o n of the optimum  turnout s p a c i n g i t i s assumed the r a t e of a c c e l e r a t i o n i s 19759 mph , 2  the r a t e  of  d e c e l e r a t i o n i s 11855 mph , the number o f 2  " c o n f l i c t " hours per day i s 5, the number of o p e r a t i n g days per year  i s 200, and  exponential.  the headway  frequency  Unless i t i s s t a t e d otherwise,  distribution  i t i s assumed the  v e l o c i t y of the leaded v e h i c l e i s 25 mph, the v e l o c i t y empty  vehicle  i s 40 mph, the t r a f f i c  i s $15 p e r hour, and the expected  of the  flow r a t e i s 4 vph, the  turnout c o n s t r u c t i o n c o s t i s $100, the adjusted cost  is  truck  hauling  u s e f u l l i f e of the road  75 is  20 years. F i g u r e 12 d e p i c t s the e f f e c t  loaded  vehicle  (10  to  45  that  mph)  has  the  velocity  of  on the optimum t u r n o u t  spacing f o r v a r i o u s v e l o c i t i e s of the empty v e h i c l e (25 mph).  The  effect  decreases  increases.  the  to  40  optimum turnout s p a c i n g i n c r e a s e s but the marginal  For  as  the  velocity  of  the  loaded  any given v e l o c i t y of the loaded  vehicle  v e h i c l e , the  optimum t u r n o u t spacing i n c r e a s e s as the v e l o c i t y o f the  empty  vehicle increases. Figure  13 i l l u s t r a t e s the e f f e c t t h a t the v e l o c i t y o f the  empty v e h i c l e (14 t o 45 mph) has on the optimum t u r n o u t s p a c i n g for  v a r i o u s values of the v e l o c i t y of t h e loaded v e h i c l e (10 t o  35 mph). turnout  The same g e n e r a l r e l a t i o n s h i p s spacing  and  the  independent  between  the  optimum  v a r i a b l e s , as shown i n  F i g u r e 12, are e v i d e n t i n F i g u r e 13. The  e f f e c t t h a t the t r a f f i c  flow r a t e (1 t o 20 vph) has on  the optimum turnout spacing i s i l l u s t r a t e d i n optimum  turnout  increases.  spacing  The g e n e r a l  decreases shape  h y p e r b o l i c and asymptotic Figure  of  as the  F i g u r e 14.  the t r a f f i c flow r a t e curve  appears  to  be  t o the x - a x i s and y - a x i s . .  15 d e p i c t s the e f f e c t the turnout c o n s t r u c t i o n c o s t  ($50 t o $1000) has on the optimum t u r n o u t spacing. turnout  The  The optimum  spacing i n c r e a s e s and the marginal e f f e c t decreases as  the turnout c o n s t r u c t i o n cost i n c r e a s e s . T h e . e f f e c t t h a t the adjusted h a u l i n g cost hour)  has  on  Figure  16.  The g e n e r a l  hyperbolic.  the  The  optimum  $45  per  turnout spacing i s i l l u s t r a t e d i n  shape  optimum  ($1 t o  of  turnout  adjusted hauling cost increases.  the  curve  spacing  appears decreases  to  be  as the  76  F i g u r e 17 i l l u s t r a t e s the e f f e c t t h a t the expected life  of  spacing.  the road  useful  (1 t o 25 years) has on the optimum t u r n o u t  The optimum turnout spacing decreases as the expected  useful l i f e  of the road i n c r e a s e s . . The general  curve  be d e p i c t e d by a h y p e r b o l i c f u n c t i o n . . Based on the  can  s i x f i g u r e s , t h e general e f f e c t t h a t  changes  shape  to  one  of the  o f the  independent v a r i a b l e s has on t h e optimum turnout spacing can be predicted. F i g u r e 18 i l l u s t r a t e s the c o s t f u n c t i o n , based on equation 2.38.  In t h i s diagram i t i s assumed:  1.,a  = 19759 mph  2  A  2. a  = 19759 mph  2  D  3. 5 " c o n f l i c t " hours per day 4. 200 o p e r a t i n g days per year 5. L  = 50 f e e t  6. the d r i v e r ' s r e a c t i o n time i s 2 seconds 7. H  = 4 vph  8. V, = 20 mph 9. V  2  = 25 mph  10. C  = $100  11. M  = $15 per hour  12. ,Q  ]  = 20 years  13. e x p o n e n t i a l headway frequency The  distribution..  cost f u n c t i o n curve begins r e l a t i v e l y s t e e p l y but f l a t t e n s  as the.optimum t u r n o u t spacing i s approached.. The the  right  of  to  the optimum t u r n o u t spacing i s not has steep as  the p o r t i o n to the l e f t o f the optimum. of  portion  A  further  discussion  the e f f e c t t h a t p e r t u r b a t i o n s t c the turnout spacing has on  77  the  cost  equation  i s discussed  in section  6.2.13.,  78  F i g u r e 12 E f f e c t of the v e l o c i t y of the loaded v e h i c l e on the optimum t u r n o u t s p a c i n g f o r v a r i o u s velocities o f the empty vehicle  79  i  u. • >i  a. = -  u  i  ui  m  m  ta  r  IS  m ui  v  j  ui  tsa IN  U1  rvj w  fS  LD x  ui —  Lrl  m m  eg —  tg  rn m  Lrl  IM D I  IS eg rvi  S O — r — Z«^UJ h-EKCCUJ LUU. Q r - U l u  Figure 1 3 Effect o f the v e l o c i t y of the empty v e h i c l e on the optimum t u r n o u t spacing f o r v a r i o u s v e l o c i t i e s of the l o a aed vehicle  80  F i g u r e 14 Effect turnout s p a c i n g  of  the  traffic  flow  r a t e on the optimum  81  F i g u r e 15 E f f e c t of the optimum t u r n o u t spacing  turnout  construction  cost  on  the  82  Ul X  X  ES  m  X  v.  in a vv  in i n rvi z rc x —;  ra v> rvi = CC  Ul  I ra  in  ra rvi r-  ca n ut  ra x m  eg ra in m  ui  ra r-  ra tn ui  ra m m  ts ra rvi  xi-m xr_ — i-  — zvu  f— CC EC LU  o.na.u. Qr-Ulw  Figure 16 Effect o f the optimum t u r n o u t spacing  adjusted  t r u c k h a u l i n g c o s t on the  83  F i g u r e 17 E f f e c t of the expected the optimum t u r n o u t spacing  useful l i f e  of the  road  on  84  EJ CD  eg rg  rg tg LO  eg eg  Lrt  eg eg x  eg es m  eg eg in X  eg eg ta X  eg eg Lrt  eg ra eg n rUl  eg eg Lrt rvi Ul C£ DZ  _1  I OS  eg eg eg rvi  eg eg Lrt —•  eg eg eg —  eg eg rvj eg eg Lrt  eg  Ul I  nzaui — v»i —_»n.:e  F i g u r e 18 Effect of the turnout based on equation 2,38  spacing on t h e . c o s t f u n c t i o n  85 6. 2  SENSITIVITY ANALYSIS  The p r e v i o u s c h a p t e r s have o u t l i n e d a method t o the  optimum  turnout  s p a c i n g but t h e formulas u t i l i z e d  t h a t t h e r e are no e r r o r s i n t h e independent formulas. .  Generally,  variables  are  required..  A  respect  to  calculate  the  unknown  so  and  measure  of  variations  called  variables  "true"  estimates  of  assume of  values  these  the  of the  values  are  t h e s t a b i l i t y of t h e f u n c t i o n s w i t h  in  the  value  of  the  variables i s  desirable.. This section w i l l include a s e n s i t i v i t y analysis of the  optimum  turnout  spacing  f u n c t i o n , t o t a l expected with  respect  assumptions, The  to  the  F-factor  d e l a y time f u n c t i o n , and c o s t f u n c t i o n independent  and headway  general  f u n c t i o n , expected  variables,  some  and  the  the  distributions.  format  of  this  section  i s t o d i s c u s s the  e f f e c t t h a t t h e independent v a r i a b l e s , t h e assumptions model,  of  headway  of  the  d i s t r i b u t i o n s have on t h e f u n c t i o n s .  S i m u l a t i o n i s t h e b a s i c m o d e l l i n g t o o l used i n t h e  sensitivity  a n a l y s i s s i n c e t h e e q u a t i o n s a r e g e n e r a l l y t o o complex t o a l l o w for  a  direct  comparison  method.  Discrete,  deterministic  s i m u l a t i o n models were w r i t t e n i n BASIC f o r the HP9830A c a l c u l a t o r t o s o l v e t h e expected delay  F - f a c t o r , the  method  i s to  independent  variables.  The  l o o p t h e independent v a r i a b l e s between a  lower and an upper l i m i t i n c o n s t a n t The  expected  t i m e , t h e optimum t u r n o u t s p a c i n g , and t h e c o s t f u n c t i o n  f o r a wide range of v a l u e s of the independent basic  total  desktop  variable  or  variable  increments.  b e i n g t e s t e d i s looped i n a s i m i l a r  manner but t h e f u n c t i o n i s a l s o e v a l u a t e d f o r p e r t u r b e d  values  86  of the independent v a r i a b l e . and  The d i f f e r e n c e between the a c t u a l  the perturbed values of the f u n c t i o n  the e x p e r i e n c e of the s i m u l a t i o n average,  are c a l c u l a t e d . .  i s assembled  i n t o groups,  Once the  the standard d e v i a t i o n , and the maximum value of each  group are determined.. Graphs of some o f simulations  are l o c a t e d i n Appendix  9..  the  results  of  the  87 6.2.1  VELOCITY OF THE LOADED VEHICLE  There  i s a p o t e n t i a l e r r o r between the estimated and t r u e  value o f the average be  assumed  that  v e l o c i t y of the loaded v e h i c l e .  It  will  the estimated value w i l l be w i t h i n 10 mph o f  t h e : a c t u a l value.  Therefore the s e n s i t i v i t y a n a l y s i s  will  be  based on a measurement e r r o r of 10 mph o r l e s s . It  can  be shown that f o r the simple F - f a c t o r the r a t e o f  change i n the expected  F - f a c t o r (6F) f o r a  i n the v e l o c i t y o f the loaded v e h i c l e Sf This  (<5v )  perturbation  is:  (  = 1/(2V ) Z  formula  can  be u t i l i z e d as an e s t i m a t i o n o f the r a t e o f  change f o r the expected vehicle  meeting  a  F-factor  fleet  though, does not account  of  simulation  model  f o r the  a  single  i n the v e l o c i t y  total  empty  T h i s formula,  expected  delay  time  delay time f u n c t i o n i s complex  i s utilized  d i f f e r e n c e i n the t o t a l expected perturbations  involving  loaded v e h i c l e s .  e r r o r . . S i n c e the t o t a l expected a  given  t o determine  delay time of  the  r e s u l t s o f the s i m u l a t i o n , as i l l u s t r a t e d  the percentage  T) loaded  f o r various vehicle.  The  i n F i g u r e 19,  show  t h a t the e r r o r may be l a r g e but i t c o i n c i d e s with a l a r g e e r r o r in  the  value  assumptions,  of  the  velocity  t h e independent  of  the loaded  vehicle.  v a r i a b l e s , and the range of v a l u e s  of these v a r i a b l e s a r e : 1. The headway frequency 2. a 3. S  rt  = a  d i s t r i b u t i o n i s exponential  = 19759 mph  2  0  The  = 0.05, 0. 10, 0. 15, 0. 20, 0.25, 0. 30, o r 0.35 miles  4. V, = 10, 20, 30, or 40 mph  88  5. \  = 10, 20, 30, or 40 mph  6. H  = 2, 6, o r 10 vph..  The v e l o c i t i e s of the v e h i c l e s are f u r t h e r c o n s t r a i n e d by: V  <  \  < V, + 20  (  • 10 mph mph.  As i l l u s t r a t e d difference  in  i n Figure  the  optimum  20, there i s turnout  a  spacing  smaller  percent  (£s*) than i n the  i n i t i a l percent p e r t u r b a t i o n i n V, . . The d i f f e r e n c e between the percentage v a l u e s , <£V, -Ss*r decreases as the p e r t u r b a t i o n of V , is  decreased.  same  As was the case i n the previous  independent  utilized  variables  i n this simulation.  their  range  of  the  values are  The e x t r a v a r i a b l e s r e q u i r e d a r e :  1. Q  = 2 or 10 years  2. C  = $100, $600, or $1000  3. M  = $25 or $45 per hour  (  and  simulation  4. 200 o p e r a t i n g days per year 5. 5 " c o n f l i c t " hours per day 6. The stopping d i s t a n c e i s l e s s than one-half  o f the t u r n o u t  spacing. By  themselves  significant,  large but  the  the c o s t equations illustrated  in  The  per year per mile Velocity  in  e f f e c t t h a t these  F i g u r e 21. . as  The  appear  to  be  p e r t u r b a t i o n s have on  for  values the  of  optimum  the  independent  turnout  spacing  c o s t d i f f e r e n c e i s l e s s than $50 per v e h i c l e for  values  of  V,  perturbations  based  on  a  greater true  g e n e r a l l y r e s u l t i n a cost d i f f e r e n c e o f v e h i c l e per year  V,  i s the r e a l q u e s t i o n . . This r e l a t i o n s h i p i s  v a r i a b l e s are the same simulation.  perturbations  per mile.  than  20  mph.  v e l o c i t y of 10 mph  less  than  T h i s value transformed  $100  per  into dollars  89  per  cunit,  based  on  25  cunits  d i f f e r e n c e of l e s s than $0.01  per t r i p , r e s u l t s i n a c o s t  per c u n i t - m i l e . ,  90  i i 1 + 1  z a —» i— cc —• ta x cc a.x Z2 1— OS  I  11  11 1•+•  I I+  1  UJ x  LV. x  a —  U CL  U JEX > C CCS  X CL Ul  _J X Ul  >  a ui t_ x • lu.  a  • ui  UJ K> Z Ul EK UJ U. U.  Ul _l<~'>-  XUJXUI  l-O. J E " - 2 Q X L J - K c_ — r - UJ <_ r - w  F i g u r e 19 E f f e c t of perturbations to the v e l o c i t y loaded v e h i c l e on the t o t a l expected delay time  of the  Figure 20 Effect of perturbations loaded v e h i c l e on the optimum turnout  to the spacing  velocity  of the  Figure 21 E f f e c t of p e r t u r b a t i o n s to loaded v e h i c l e on the c o s t d i f f e r e n c e  the  velocity  of  the  93  6.2.2  VELOCITY OF THE EMPTY VEHICLE  I t w i l l be assumed t h a t the p o t e n t i a l maximum e r r o r i n the measurement  of  the  same as t h e maximum vehicle,  10 mph.  v e l o c i t y of the empty v e h i c l e w i l l be the error  f o r the  velocity  of  the  loaded  The r a t e of change i n the simple F - f a c t o r f o r  a given p e r t u r b a t i o n  i n the v e l o c i t y o f the empty v e h i c l e  (&V ) Z  is: SF = -V, /(2V|) This  rate  of  change  formula  can be u t i l i z e d t o p r e d i c t the  p o t e n t i a l r a t e of change f o r the the  expected  meeting of an empty v e h i c l e and a f l e e t of loaded  Simulations  to  predict  the  (£s*) /  error  (6f) ,  t o t a l expected delay time  and t h e c o s t eguation  the  results  involving  of  these  perturbations  and  the  optimum  turnout  spacing  T  were  the  loaded  vehicle.  s i m i l a r t o the case  t o the v e l o c i t y of the loaded  percentage  vehicles.  (<Sc ) were conducted i n a s i m i l a r  simulations  1. the percentage d i f f e r e n c e  involving  i n the c a l c u l a t i o n o f the  manner as i n t h e case c f the v e l o c i t y of The  F-factor  vehicle:  i n the t o t a l expected delay time  difference  in  the  optimum  turnout  spacing were s l i g h t l y lower than i n t h e c a s e : i n v o l v i n g the v e l o c i t y o f the loaded 2. the  cost  difference  vehicle was s l i g h t l y higher than i n the case  i n v o l v i n g the v e l o c i t y o f the loaded v e h i c l e  (Appendix 9 ) .  94  6. 2.3  FLOW RATE  TRAFFIC  In t h e s e n s i t i v i t y a n a l y s i s of the t r a f f i c flow the  perturbations  functions rates  F, T,  of  1,  that , and C  2,  4,  were  used  6,  8,  and  (H)  are 1, 2, and 4 vph.. The  are t e s t e d f o r a c t u a l  T  rate  10  vph.  traffic  The  flow  r e s t o f the  independent v a r i a b l e s and t h e i r range o f values a r e the same as f o r the s i m u l a t i o n s velocity  of  resulting  involving the s e n s i t i v i t y a n a l y s i s  the  loaded  vehicle.  The  expected  of the F-factor  from the meeting of one empty v e h i c l e and a f l e e t  of  loaded v e h i c l e s experienced a change of l e s s than 0.01, f o r any of  the  The  percent e r r o r i n the  delay  tested  time  flow r a t e s .  This  difference i s i n s i g n i f i c a n t .  calculation  of  the t o t a l  expected  (&T) can be approximated by:  oT = (100)£H/H As  can  be  created can  shown be  from  large.  this  approximation formula the e r r o r  The  results  of  the  simulation  determine the percent d i f f e r e n c e i n the optimum t u r n o u t have  small  standard  deviations.  I f the p e r t u r b a t i o n  to  spacing i s less  than 100 percent of the a c t u a l flow r a t e then the maximum value of  w i l l be approximately 7 0 percent of the  turnout  spacing.  Maintaining  t h i s as a maximum  then the cost d i f f e r e n c e f u n c t i o n value  than  that  experienced  a n a l y s i s of e i t h e r V  (  100 p e r c e n t ) . . perturbation per  vehicle  or  actual  (oC ) has  in  T  a  optimum  perturbation  larger  maximum  the case o f t h e s e n s i t i v i t y  (based on a maximum p e r t u r b a t i o n o f  The maximum value of r J C , based on a 100-percent f  o f the t r a f f i c flow r a t e , per  mile  per  year  but  i s approximately the average  $400  value i s  95  approximately  $85.  96  6,2.4  EXPECTED USEFUL LIFE OF THE ROAD  The  expected  useful l i f e  of the road  (Q )  i s analysed  (  for  5, 10, 15, 20, and 25 years with p e r t u r b a t i o n s («JQ ) of 2 . 5 r 5, and  10  years.  The  remaining  s i m u l a t i o n models u t i l i z e simulations  to  test  loaded v e h i c l e .  the  the  independent  same  range  sensitivity  The r e s u l t s of the  of  simulation  difference  standard  d e v i a t i o n s f o r the v a r i o u s groupings  As  i n c r e a s e s Js^ i n c r e a s e s .  to  within  value  for  of  50  p e r t u r b a t i o n of 25 percent. s i m u l a t i o n had a of  the  value  larger  percent  and  value  coefficient  e s t i m a t i o n of Q  (  involving  the  i s not  Sc  R  )  21  percent for a  The r e s u l t s of the c o s t d i f f e r e n c e of  variation  T  as  than  the  By m a i n t a i n i n g the assumption is  I f the p e r t u r b a t i o n i s 50 of  oQ .  maximum  percent  the maximum c o s t d i f f e r e n c e i s approximately  maximum  the  to  12  (  the  and  (  35 percent of t h e a c t u a l optimum  <£s* s i m u l a t i o n .  per v e h i c l e per m i l e .  Q  then  that the maximum e r r o r i n the e s t i m a t i o n o f Q then  the  can be p r e d i c t e d  (  T h i s maximum value decreases  perturbation  results  involving  of  Provided Q  actual  of 6 s * i s approximately  a  as the  i n the optimum turnout s p a c i n g have s m a l l  100 percent o f the  turnout s p a c i n g .  values  of the v e l o c i t y o f the  percent  c$Q,  v a r i a b l e s i n the  i s $20. critical  The as  100  percent  $35 per year percent  accuracy  i n the  then  o f the  situations  v e l o c i t y of the loaded v e h i c l e , t h e v e l o c i t y o f  the empty v e h i c l e , nor the t r a f f i c  flow r a t e , s i n c e f o r a given  percent p e r t u r b a t i o n i n the expected the maximum value o f Jc  T  i s less.  useful l i f e  of  the  road  97 6.2.5  TURNOUT CONSTRUCTION COST  The included  sensitivity costs  perturbations  of of  a n a l y s i s o f the turnout c o n s t r u c t i o n $100,  $50,  $250,  $100,  and  $500,  and  $1000  $200.  The oS*  cost with  simulation  produced s i m i l a r r e s u l t s as the model i n v o l v i n g the s e n s i t i v i t y a n a l y s i s o f the expected u s e f u l percent  perturbation  life  in C resulted  of  the  i n aS% being  34 percent o f the a c t u a l optimum t u r n o u t spacing percent  perturbation  resulted  percent.. The s i m u l a t i o n  in  100-  approximately while  a  40-  <$S* being approximately 17  to determine Sc^ produced r e s u l t s t h a t  are not as s i g n i f i c a n t as the p r e v i o u s &C{ velocity  A  road..  simulations; i . e . ,  o f t h e loaded v e h i c l e , v e l o c i t y of t h e empty v e h i c l e ,  t r a f f i c flow r a t e , and expected u s e f u l l i f e 100-percent  perturbation  of  of C r e s u l t s i n a cost  l e s s than $30 per year per v e h i c l e per mile.  the  road.  A  difference of  98  6.2.6  ADJUSTED TEOCK HAOLING COST  The  s i m u l a t i o n s t o determine the d i f f e r e n c e i n the optimum  turnout s p a c i n g and the c o s t  difference  were  conducted  with  adjusted t r u c k h a u l i n g c o s t s o f $10, $15, $25, $35, and $45 per hour  and  with  Both of these and  range  perturbations  of  $2.5, $5, and $10 per hour.  s i m u l a t i o n s u t i l i z e d the same  of  values  of  the  variables  basic as  i n v o l v i n g the a n a l y s i s of the v e l o c i t y of the There was a small standard M  CSM,  and  percent £s*  of  the  assumptions  the simulations loaded  vehicle.  d e v i a t i o n w i t h i n the v a r i o u s groups,  s i m u l a t i o n to determine <5s^. F o r a given  p e r t u r b a t i o n of the independent v a r i a b l e , the r e s u l t i n g  (%) value i s l e s s i n the case i n v o l v i n g the adjusted  truck  hauling  cost  than i n the case i n v o l v i n g any o f the p r e v i o u s l y  analysed  v a r i a b l e s . . A 100-percent p e r t u r b a t i o n i n the a d j u s t e d  h a u l i n g c o s t r e s u l t s i n 6s* being  approximately  the  spacing  actual  optimum  turnout  p e r t u r b a t i o n r e s u l t s i n 6s% The  being  while  approximately  percent a 20  of  50-percent percent.  r e s u l t s of the s i m u l a t i o n s to determine the c o s t d i f f e r e n c e  had  a  larger  determine spacing. per  hour  the  c o e f f i c i e n t of v a r i a t i o n than the s i m u l a t i o n t o percent  difference  in  the  and  a  100-percent  perturbation,  turnout  the maximum c o s t  $40 per v e h i c l e per mile per  while the average d i f f e r e n c e was approximately a  optimum  In the case i n v o l v i n g an adjusted h a u l i n g c o s t o f $10  d i f f e r e n c e was approximately  of  34  p e r t u r b a t i o n t o the adjusted  year  $14.. The e f f e c t  h a u l i n g c o s t on the c o s t i s  not as s i g n i f i c a n t as i n the case i n v o l v i n g the p e r t u r b a t i o n o f the turnout c o n s t r u c t i o n cost.  99 6.2.7  THE ACCELERATION  Perturbations vehicle  will  provided  the  percent  of  to  factor. utilized utilized  not  the  OF THE EMPTY VEHICLE  deceleration  be  actual  can  rate  less  than  T h i s summary i s the  of  the  s i g n i f i c a n t l y a f f e c t the expected be of  predicted  same  to  deceleration..  w i l l cause the maximum e r r o r i n  empty  F-factor, within  50  A 50-percent  the  expected  F-  0.01 or 20 percent of the a c t u a l Fbased  range  of  on  a  simulation  values  model  that  of the v a r i a b l e s as was  i n the a n a l y s i s of the v e l o c i t y o f the loaded v e h i c l e .  A simulation that  t o the rate of  perturbations  perturbation factor  AND DECELERATION  model was u t i l i z e d  perturbations  d e c e l e r a t i o n have assumptions,  to on  either  the  the  total  constraints,  and  independent v a r i a b l e s u t i l i z e d  to  determine  rate  of  expected range  the  a c c e l e r a t i o n or  delay of  effect  time..  values  of  The the  i n the program were:  1. V, = 10, 15, 20, 25, 30, 35, o r 40 mph 2. V  a  = 10, 15, 20, 25, 30, 35, o r 40 mph  3. V, < V 4. V  2  + 10 mph  2  < V, + 20 mph  5. S  = 0.05, 0. 10,  6. H  = 2, 4, 6, 8, or 10 vph  7. a  or a  fl  r a t e of  assumed  0.20, 0.25, 0. 30,  = 14555, 24259, or 33963 mph headway  acceleration  to  distribution. or  deceleration  being  tested  was  have an a c t u a l r a t e of 14555, 19407, 24259, 29111,  or 33963 mph  2  while the p e r t u r b a t i o n s  mph .. The development of formulas t o 2  o r 0.35 miles  2  D  8. e x p o n e n t i a l The  0. 15,  were 2426, 4852, and 9704 determine  the  rate  of  100 acceleration simulation rates  and d e c e l e r a t i o n are o u t l i n e d i n Appendix 3.  r e s u l t s were grouped a c c o r d i n g  to  their  of a c c e l e r a t i o n or d e c e l e r a t i o n and t h e i r  A 66-percent p e r t u r b a t i o n be l e s s  than  simulation expected  delay  time  simulation  on  the  loaded  the  the  will  be: l e s s  was  travel  percent  empty  change  than  time.  The  i n the t o t a l  one-half  of  the  developed t o examine the e f f e c t  t o the rate of a c c e l e r a t i o n or  optimum turnout  utilized  utilized  of  that  model  range of the values were  perturbations.  expressed i n percent.  that perturbations have  respective  w i l l cause the maximum value of £T t o  percent  indicated  perturbation, A  20  The  spacing  deceleration  and c o s t f u n c t i o n s . . The  of the v a r i a b l e s and the assumptions  that  i n the model were e s s e n t i a l l y the same as those  i n the s e n s i t i v i t y a n a l y s i s of vehicle  (Section 6.2.1).  the  velocity  of the  I t was assumed that the r a t e  of a c c e l e r a t i o n eguals the r a t e of d e c e l e r a t i o n , 14555 or 24259 mph . 2  The p e r t u r b a t i o n s  maximum p e r t u r b a t i o n of  rjs*  spacing  2  was 80 percent.  was  l e s s than 7 percent  the  effect  acceleration spacing  were 2426, 4852, and 9704 mph  or  that  Since the  deceleration  have  function i s i n s i g n i f i c a n t .  maximum  o f the a c t u a l optimum  perturbations on  to the  and the  the optimum  value turnout  rate  of  turnout  10 1 6.2,8  THE DISCOUNT RATE  A s i m u l a t i o n model was developed that  the  discount  rate  has  on  f u n c t i o n and the c o s t f u n c t i o n .  t o determine  the  effect  the optimum t u r n o u t s p a c i n g  The  simulation  results  were  grouped with r e s p e c t t o t h e i r corresponding d i s c o u n t r a t e (2.5, 5.0,  7.5, 10.0, or 12.5 percent per annum) and expected  life of  of t h e road  (2, 5, 10, or 20 y e a r s ) .  useful  The range of  the v a r i a b l e s and the assumptions u t i l i z e d  values  i n the s i m u l a t i o n  were: 1. V, = 10, 20, 30, or 40 mph 2.  = 10, 20, 30, or 40 mph  3. V  2  < V  ;  4. V, <  + 20 mph + 10 mph  5. H  =3  6. C  = $100, $600, or $1100  7. M  = $25 or $45 per hour  8. a„ = a  o r 10 vph  = 19759 mph  2  0  9. e x p o n e n t i a l headway frequency  distribution  10. 200 o p e r a t i n g days per year 11..5 " c o n f l i c t " hours per day. The life  marginal e f f e c t o f of  (Appendix percent  the  cSs*  (%) decreased  as the expected  useful  road i n c r e a s e d or as t h e d i s c o u n t r a t e i n c r e a s e d  9 ) . , The maximum value of  was  approximately  of the optimum turnout spacing which corresponded  situation  i n v o l v i n g a discount r a t e  expected  useful  life  of  the  road  of  12.5  of  20  percent years.  50 to a  and an When the  d i s c o u n t r a t e was decreased t o 7.5 percent the maximum value o f  102  cSs^ became a p p r o x i m a t e l y  34 percent.  The r e s u l t s o f t h e c o s t d i f f e r e n c e larger  coefficient  marginal  effect  decreased  as  (CNC-J-)  s i m u l a t i o n had  o f v a r i a t i o n than t h e <$S* s i m u l a t i o n . . The  of  ,  f o r maximum  and  average  t h e d i s c o u n t r a t e o r t h e expected  the road i n c r e a s e d .  a  values,  u s e f u l l i f e of  The maximum v a l u e o f &C-f was o n l y $ 3 5 p e r  v e h i c l e p e r m i l e per year which was based on a d i s c o u n t r a t e o f 12.5  percent  years. of  and  an  expected  u s e f u l l i f e o f t h e road o f 2 0  F o r lower d i s c o u n t r a t e s ( i . e . , 5 percent)  the discount  significant.  rate  on  the cost  function  the  effect  i s not  very  103 6.2.9  THE  MAINTENANCE COST  A s i m u l a t i o n model was that  maintenance  turnout spacing grouped  with  costs  or  and  25  expected  The  assumptions of  decreased  velocity  the  f o r each g r o u p ,  <$S*.  To  of t h e  $10  same over  of  The  grouped  results  cost  of  $10  approximately  then  l o a d over  a  15  the value  10-mile  annum  $0.03  $12.50  per  10,  20,  15,  values  v a r i a b l e s and  those  utilized  vehicle  life  per  the  the  in  the  ( S e c t i o n 6.2.1).  o f the r o a d c$S*  for  were  was  costs,  increased.  approximately  i n the  values  o f <5s* e g u a l t o a p p r o x i m a t e l y  mile or approximately  per  average  spacing,  over  a  years  oS*  maintenance  (Appendix  period.  c f 'bCj i s  $0.02 p e r c u n i t  value when  Based of  6c cS S*  on  a  larger  When aS*  less  than (based  a  is  is  $300 on  25  maintenance  increases  r  of  9).  had  simulation.  of 50  f o r a maintenance c o s t  haul).  cunit  a  25-year  of t h e c o s t d i f f e r e n c e ,  the  per  or  (2, 5,  independent  loaded  o r $12.50 o v e r  percent  cunits  road  and  i s achieved  l e s s than  per  of the  reguired  than  year  $10.00,  optimum t u r n o u t  of v a r i a t i o n  per  costs  useful  coefficient 50  maintenance  there.was a l a r g e spread  actual  years  corresponding  variation  maximum v a l u e  20  were  (%) , f o r v a r i o u s m a i n t e n a n c e  c o s t o f $7.50 per annum i s This  simulation  $7.50,  have a maximum v a l u e the  o f the  of t h e  expected  coefficient  of  optimum  maximum  o f oS*  one,  percent  on t h e c o s t f u n c t i o n and  model were t h e same a s  effect  as  Since the  values  the  of the  marginal  effect  useful l i f e the  t o determine  results  $5.00,  before  determined. .  The  The  the  respect to t h e i r  years)  analysis  have  function..  ($0.05, $1.00, $2.50, annum)  developed  increased  to to  104 approximately  60 percent.. In the case of a maintenance c o s t of  $5 per annum over a 20 year p e r i o d the would be approximately  $0.01 per c u n i t .  maximum  value  of  6c  T  105 6.2.10  THE DERIVATIVE OF THE EXPECTED F-FACTOR  In first  the  determination  d e r i v a t i v e of the  turnout  spacing  of the optimum t u r n o u t spacing the  cost  function  with  respect  the  expected  F-factor  i s i n c l u d e d i n the optimum t u r n o u t  A s i m u l a t i o n model i s used t o determine  effect  first  the  derivative  of  the expected  equation has on the optimum turnout spacing resulting  cost  function.  function  the  F-factor and  the  The same values o f t h e independent  v a r i a b l e s were used i n t h i s s i m u l a t i o n , s e c t i o n 6.2.1  F-  derivative  spacing f u n c t i o n . that  the  must be equal t o zero.. Since t h e expected  f a c t o r i s a v a r i a b l e i n the c o s t f u n c t i o n the f i r s t of  to  as  were  utilized  in  except t h a t :  1. H = 1, 2, 4, 6, 8, or 10 vph 2. The  s h i f t e d e x p o n e n t i a l and Pearson Type I I I (a=2)  headway  d i s t r i b u t i o n s were used. For a t r a f f i c  flow r a t e o f l e s s than 4 vph the value of J s * was  20 percent of the optimum turnout s p a c i n g . . The became  approximately  increased  t o 10 vph.  51 percent when the t r a f f i c The corresponding  was only $27 per year per mile inclusion the  of  the f i r s t  calculation  insignificant.  of  maximum  per  flow r a t e was  maximum c o s t d i f f e r e n c e  vehicle..  Generally,  d e r i v a t i v e o f the expected the  optimum  value  turnout  the  F-factor i n spacing  is  106 6.2.11  THE  LENGTH OF  Throughout  THE  the  LOADED VEHICLE  development  assumed t h a t the loaded  of  v e h i c l e has  the  no  model  length.  of  the  length  of  the  loaded  has  been  of  the  Some  equations developed i n Chapter 2 can be modified effect  it  to include  vehicles  (Appendix 6).  Furthermore, the appendix i n c l u d e s a summary of the r e s u l t s a  simulation  program  written  to  v e r i f y the accuracy  expected F - f a c t o r equations which account f o r the loaded  vehicle.  factor and  for  This  delay  simulation  of  of  the  length of  the  average  F-  the  s i t u a t i o n s i n v o l v i n g the s h i f t e d  exponential  the Pearson Type III(a=2) headway d i s t r i b u t i o n s . A  simulation  exponential  tc  model  the  the Erlang(alpha=2) equation., 1. L 2. a  The  =60 D  =  was  developed  shifted exponential tc  the  Pearson  to  compare  F - f a c t o r equation  Type  III(a=2)  F-factor  19759 mph  2  20, 30,  or 40  mph  4. V  £  =  10,  20,  or 40  mph  5. V  2  < V + 20  mph  6. V,  < \*  10  mph  7. S  <  D  8. S  = 0. 10, 0. 15,  0.20,  9. H  =1,  8, or 10  (  V|/a  30,  2, 4, 6,  0. 25,  0.30,  or 0.35  miles  vph  r e s u l t s of the s i m u l a t i o n i n d i c a t e d that the of  and  feet  10,  length  the  s i m u l a t i o n assumed:  3. V, =  The  determined  the  the  approximately:  vehicle  on  the  expected  e f f e c t of F-factor  the was  107 if  = L/S  where: L = e f f e c t i v e l e n g t h of t h e v e h i c l e . For  a  turnout  expected  spacing  of  0.1  miles  the d i f f e r e n c e i n the  F - f a c t o r i s approximately 0.11.  The  e f f e c t o f the length of t h e  turnout  spacing  simulation  proved  utilized  to  to  be  determine  vehicle  the  optimum  inconsequential..  In the  this  on  conclusion  i t  was  assumed: 1. Q  = 2 or 10 years  2. M  = $25 or $45 per hour  3. C  = $100, $600, or $1100  (  4. 200 o p e r a t i n g days per year 5. 5 " c o n f l i c t " hours per day. The  maximum  value  of S s * , f o r the case i n v o l v i n g the s h i f t e d  e x p o n e n t i a l headway d i s t r i b u t i o n , was l e s s than  2  percent  the a c t u a l optimum turnout spacing while f o r t h e case the  Pearson  Type  I I I (a=2)  headway  value o f SS* was l e s s than 5 percent. of the  vehicle  will  not  involving  d i s t r i b u t i o n the maximum Consequently,  significantly  turnout s p a c i n g nor the c o s t equation.  of  affect  the l e n g t h  the  optimum  108  6.2.12 THE HEADWAY PROBABILITY  A  simulation  F-factors  model was developed to compare the expected  developed  distributions.  The  from  various  range  of  v a r i a b l e s were e s s e n t i a l l y involving  the  DISTRIBUTIONS  effect  the  headway  probability  of  independent  values same  as  of p e r t u r b a t i o n s  feet  and  The  the  compared  the  E r l a n g (alpha=2), s h i f t e d e x p o n e n t i a l , F-factor  equations.  rate  of  10  r a t e was decrease occured  vph. to  F-factor  This  4  vph._  a  vehicle  rate  exponential  of  60  1, 2, 3, 4, 6, or 10 vph. simple,  exponential,  and Pearson Type I I I (a=2)  was 0.0527 a t  a  traffic  value dropped t o 0.0054 when the A  larger  maximum  4  vph.  The  f o r a r a t e of 10 vph  difference  This d i f f e r e n c e  was the  0.027  smallest  of  was of the order o f magnitude o f  1 0 . . By i n c o r p o r a t i n g the e f f e c t of the length -3  the  and  maximum d i f f e r e n c e between the  and E r l a n g (alpha=2) F - f a c t o r s  those t e s t e d .  into  was  between the simple F - f a c t o r and the E r l a n g (alpha-2) F-  f a c t o r , the values being 0.091 for  simulation  The maximum d i f f e r e n c e between the simple  F - f a c t o r and the e x p o n e n t i a l flow  the  (Section 6.2.1).. The  of the loaded  t r a f f i c flow r a t e was  simulation  for  t o the v e l o c i t y of the  loaded v e h i c l e s on the expected F - f a c t o r s e x c e p t i o n s were t h a t the l e n g t h  the  expected F - f a c t o r equations there  of the v e h i c l e  was e s s e n t i a l l y no  d i f f e r e n c e i n the r e s u l t s . A s i m i l a r simulation effect  that  various  model was developed t o determine  headway d i s t r i b u t i o n s have on the optimum  turnout spacing  f u n c t i o n and the c o s t f u n c t i o n . , The values  the  utilized  variables  the  i n the s i m u l a t i o n  of  were e s s e n t i a l l y the  109 same as those loaded  used i n the  F-factors,  of  exponential,  The maximum value  velocity  of the  and of  to  Erlang(alpha=2),  &S* was  less  their before  than  3  the a c t u a l optimum turnout spacing and the maximum  value o f <Sc^. was approximately Consequently,  $11 per  equation.  vehicle  per  the type of headway d i s t r i b u t i o n  w i l l not s i g n i f i c a n t l y a l t e r the optimum cost  the  The r e s u l t s were grouped with r e s p e c t  simple,  being compared.  year.  of  v e h i c l e s except t h a t t h e t r a f f i c flow r a t e was 1, 2, 4,  6, or 10 vph.  percent  analysis  turnout  mile  per  utilized  spacing  nor  110 6.2,13  THE TURNOHT SPACING AND THE OPTIMOM TURNOUT SPACING  There  i s p o t e n t i a l l y a small e r r o r i n t h e measurement o f  the average turnout s p a c i n g . . A s i m u l a t i o n model was to  determine  spacing  the e f f e c t  that  perturbations  developed  t o the t u r n o u t  (0.05, 0.10, and 0.20 miles) have on the t o t a l  expected  delay time.. The p e r t u r b a t i o n s were added t o the t r u e values o f the turnout spacing 0.35  miles).  utilized  (0.05, 0. 10, 0. 15, 0.20,  0.25,  0. 30,  The range of values of the independent v a r i a b l e s  i n the s i m u l a t i o n was e s s e n t i a l l y the section  6.2.1.  illustrate  the e f f e c t that p e r t u r b a t i o n s t o the t u r n o u t  delay  time  and  d e r i v a t i v e of the f u n c t i o n increases.  delay time the  Furthermore,  the  as  total  value the  those  constructed  (Appendix 9 ) .  absolute  decrease  were  as  in  have on t h e t o t a l expected  Graphs  same  utilized  expected  and  total  the f i r s t  turnout  expected  spacing  The  of  to  spacing  delay  time  i n c r e a s e s as t h e s i z e of the p e r t u r b a t i o n i n c r e a s e s . . I f i t i s assumed t h a t the maximum p e r t u r b a t i o n o f the t u r n o u t spacing i s 20  percent then the maximum d i f f e r e n c e i n the t o t a l delay  i s approximately  30 percent of the t r a v e l empty  A s i m u l a t i o n model was developed that  time..  t o i l l u s t r a t e the  d e v i a t i o n s from the optimum t u r n o u t spacing  have on  the  independent 6.2.1  cost, equation.  range  of  effect  (25% t o 300%)  values  v a r i a b l e s was s i m i l a r t o those u t i l i z e d  of the  i n section  except:  1. headway frequency 2. _L  The  time  = 60 f e e t  3. 25 c u n i t s per t r i p  distribution  was the s h i f t e d  exponential  111 4..M  = $5, $10, $15, or $20 per hour  5. Q  = 5, 15, or 25 years  6. H  = 1, 4, or 8 vph.  The  c o s t d i f f e r e n c e was measured  The  experience of the model, averages, standard  maximum  values,  was recorded  r a t e , the adjusted life in  The values  d o l l a r s per c u n i t - m i l e ,  life  the  t r a f f i c flow and  the  optimum  road  decreased  as  and  spacing  the  coefficient  stable.. the  spacing  marginal e f f e c t  d i f f e r e n c e increased  The  vary  have on  the  the adjusted  The  useful  significantly  average  and  to the expected  with  The c o s t  Furthermore, h a u l i n g cost  was  of  maximum useful  difference  as the p e r t u r b a t i o n s  increased.  variations  from  t r u c k h a u l i n g c o s t . . The  the  t o the  the  cost  i n c r e a s e d but  expected u s e f u l l i f e o f the road  average  maximum value  increased  as the adjusted  of  flow  rate..  r a t e was s e t equal to 1 vph.  turnout  d e v i a t i o n s , and  o f the c o s t d i f f e r e n c e f u n c t i o n ,  of the c o s t equation with r e s p e c t of  cunit-mile.  and 23 d e p i c t the e f f e c t that d e v i a t i o n s  the optimum turnout values  per  with r e s p e c t t o the t r a f f i c  d i d not  r e s p e c t t o the t r a f f i c flow 22  dollars  truck h a u l i n g c o s t , and the expected  of the road.  Figures  in  grouped  increased.  r e s u l t s are  c o e f f i c i e n t o f v a r i a t i o n was 0.68 while 0.72..  For  each  of  the  groups,  the  maximum  c o s t d i f f e r e n c e was approximately three times the s i z e  of  average  the  difference  of  difference  while  the  i s approximately four and one-half  of the standard value  cost  maximum  times the value  d e v i a t i o n o f t h e c o s t d i f f e r e n c e . . The  the cost d i f f e r e n c e , given  cost  an adjusted  truck  maximum hauling  c o s t of $20 per hour, an expected u s e f u l l i f e o f the road of years,  and a 100-percent p e r t u r b a t i o n ,  5  i s approximately $0,007  112 per c u n i t - m i l e while the average per c u n i t - m i l e .  value i s approximately  $0,00 3  113  ui in uixas ini-n Lu  Z3  Lu>-  Lrl  Ul  Lrl  Ln  U J  Ln Ln rvi rvi  U J I — U ) r£» V»»U. I E U J C L _ I E K  —a  X  Ul  in UJ i—  re LK LO  1 J 1 1 * -+- X 1  1 1 1 1  * + X 1 J i l l * -*-X 1  • l l 1 1 1i 1  z  _J EC  a: <-» r-  E 3  rvi 3"  K  K  K  K  x x x x E S C S E S E S  ta• E S• E S• tsu i ca Ln ca  rvi ** **** **  m m Ui  Ui z Ul EK Ul  Ul  tr x _  Ln t «v>  CJ  _J  — i o: z —  O  Q. V^  r-  E S  rs IS  F i g u r e 22 Effect of deviations from spacing on the maximum c o s t d i f f e r e n c e  the  optimum  turnout  1.14  _j  —«  U.-IU1  Ul  ui x ce Ulr-X n ui Lu >ui  Ul  Ul  * I * 1 * I  X X v:  x x x x s s s s rararara carsicara  ra  zr  Ul  Ul  rvi  i i i i  ui ui x as La z  ns  Ul  rvi rvi  r-UI_« V L C Ul — • • IK X Ul  KJ Z3  Ul  .  Lrt  -r X I I I 1 4- X I 1 1 1 + X I I 1 I  .  .  .  IS U l 53  rvi  in rvi  rvi  m cn  zr  rvi x  ra r-  m  b J z  ui  m rvi  in tr x  i — i— in ca — UJ ca inu. J - Z - S • —zt_ui_i— • v> d — _, o. v>»- ra UJ DC Ul  Figure 23 Effect of deviations from spacing on the average cost d i f f e r e n c e  the  optimum  turnout  115  7.0  DISC0SSION AND  Iii  DISCOSSION  A  model  CONCIOSIONS  has  been  turnout s p a c i n g which vehicles,  developed  is  a  to  function  of  t r a f f i c flow r a t e , expected  acceleration  rate,  determine  deceleration  the optimum  velocities  useful l i f e  rate,  of  of the  turnout  the road,  construction  c o s t , turnout maintenance c o s t , and adjusted h a u l i n g c o s t . . results  of  the model should  the a b s o l u t e company  spacing  policy,  of  and  be u t i l i z e d  the  as a guide and not  turnouts.  terrain  The  The  road  as  network,  w i l l tend to d i c t a t e the a c t u a l  l o c a t i o n of the t u r n o u t s . Figures  12,  13,  14,  15,  16,  and  17  illustrate  the  r e l a t i o n s h i p s between the independent v a r i a b l e s and the optimum turnout  spacing.  The  optimum turnout s p a c i n g i n c r e a s e d with  i n c r e a s i n g turnout c o n s t r u c t i o n c o s t , vehicle,  and  on  as the expected  variables  will  The  as  an  overestimation  cause as dramatic an  empty  Optimum t u r n o u t  life  absolute  value  overestimated  of  of  the  road,  provided  the  the  independent  a change i n the optimum  the  independent  there i s not as dramatic  spacing,  of  underestimation  as the value of the  Consequently, turnout  not  spacing  variables. decrease  useful  the  flow r a t e , and adjusted h a u l i n g c o s t i n c r e a s e d . . Based  these. f i g u r e s ,  turnout  of  v e l o c i t y of the loaded v e h i c l e .  spacing decreased traffic  velocity  of  the  independent  s l o p e : o f the variable  curves  increases.  an a f f e c t t o the optimum independent  r a t h e r than underestimated.  This  variable though,  is  does  116 not be  imply  t h a t the values of the independent v a r i a b l e s should  overestimated. a  sensitivity  analysis  perturbation  to  the  turnout  optimum  traffic  flow  of  that  a  given  each o f the independent v a r i a b l e s w i l l spacing  differently.  The  to  loaded  vehicle  and  velocity  the  vehicle,  sensitivity potential  optimum turnout s p a c i n g .  vehicles,  and  velocity  relationships accuracy  considered.  of  the  turnout c o n s t r u c t i o n c o s t , expected  not as s e n s i t i v e as the t r a f f i c empty  affect  Generally,  the road, and adjusted h a u l i n g c o s t caused  change  percent  the  r a t e was the most s e n s i t i v e v a r i a b l e f o l l o w e d by  v e l o c i t y of the vehicle. .  showed  of  flow of  cannot  the  be  and  expected  the  velocity  compared  flow  useful l i f e  about  same  of  loaded v e h i c l e . .  estimation  G e n e r a l l y the t r a f f i c  useful l i f e  These v a r i a b l e s were  rate,  the  empty  of  These  unless  the  rate,  the  the  variables i s  speed  of  the  of the road are e a s i e r t o  p r e d i c t than the turnout c o n s t r u c t i o n c o s t and adjusted h a u l i n g cost. of  Consequently,  the  variables  the confidence i n the must be c o n s i d e r e d .  initial  G e n e r a l l y , the optimum  turnout s p a c i n g was not s i g n i f i c a n t l y a f f e c t e d rate,  headway  frequency  a c c e l e r a t i o n , r a t e of vehicle.  The  distribution  deceleration,  inclusion  of  the  and  estimation  by the  assumptions, the  discount rate  of  length  of  the  cost  in  the  maintenance  d e t e r m i n a t i o n of the optimum t u r n o u t spacing can be s i g n i f i c a n t (i.e.,  a 50 percent change i n the optimum turnout spacing f o r a  maintenance c o s t of $7.50 per annum over a 25-year The  t o t a l expected  the t r a v e l empty significance  time  span).  delay time can be a s i g n i f i c a n t (Figures  8,  i s d r a m a t i c a l l y reduced  9,  10,  and  part o f  11)..  This  when the t o t a l c y c l e time  117 i s considered.  I f the t r a v e l empty time i s one-quarter of the  t o t a l time then f o r a given T of 20 percent  of the t r a v e l empty  time t h e corresponding e f f e c t of T on the t o t a l round t r i p is  5  percent. .  the t o t a l  time  By comparing the t o t a l expected delay time t o  c y c l e time r a t h e r than to the t r a v e l empty  time  the  and 23 d e p i c t the e f f e c t that d e v i a t i o n s  from  s i g n i f i c a n c e o f the delay time i s reduced f o u r f o l d . Figures  22  the optimum turnout cunit-mile),  spacing  Road  have  on  construction  the c o s t  and  l o g hauling  h i g h l y v a r i a b l e w i t h i n and between d i f f e r e n t and  Nagy(1977)  and h a u l i n g based  i s of the order of  magnitude  maximum percent $0.01  i s i n the order  deviation  from  (Figure 22), per  total  optimum  and  a l a r g e number may be s i g n i f i c a n t .  Based  year, savings  and  100-mile  hauling  hours  15-year hours  a  o f 100  are  about  These  percentage o f  The t o t a l p o t e n t i a l  per  per c u n i t -  working  day,  haul,  days  per  day, the t o t a l p o t e n t i a l  time p e r i o d ,  per  savings  r a t e o f 4 vph, a 10-mile  F o r a s i t u a t i o n with a t r a f f i c haul,  cost on  cost.  A small  5-year time p e r i o d , 200  "conflict"  year, 5 " c o n f l i c t " vehicle  savings  a r e $50000.. Consequently, the p o t e n t i a l savings  important. an  5  This  spacing  r a t h e r than the p o t e n t i a l s a v i n g s  on a t r a f f i c flow  25 c u n i t s per t r i p ,  cunit,  Based  turnout  c a l c u l a t i o n s , though, can be m i s l e a d i n g .  mile. .  per  $0.13 per c u n i t , or about 1 percent o f  construction  must be c o n s i d e r e d  $10  13 m i l e s .  the maximum p o t e n t i a l  cunit-mile, road  of  Sauder  construction  of $1 per c u n i t - m i l e . the  c o s t s are  regions. .  shown t h a t the c o s t of road  on a haul d i s t a n c e o f approximately  estimation  the  have  ( d o l l a r s per  become  flow r a t e o f 4 vph, 200 working days per  velocity  of  the  loaded  of 20 mph, v e l o c i t y o f the empty v e h i c l e o f 40 mph, an  118  adjusted  hauling  construction  cost  and  of  maintenance  optimum t u r n o u t s p a c i n g savings,  based  $25  on  is  a  100  per cost  510  hour, of  feet..  $100 The  and  a  turnout  per turnout total  the  potential  percent d e v i a t i o n from the optimum  turnout s p a c i n g i s $68190. The be  implementing of the optimum turnout s p a c i n g model  achieved  with the u t i l i z a t i o n  of t a b l e s . . ft t a b l e approach  n e g l e c t s the n e c e s s i t y of the repeated e v a l u a t i o n of a set o f equations. i n t o the road  Consequently,  can  complex  the t a b l e s can be i n c o r p o r a t e d  standards.  Future areas o f research that can be i n v e s t i g a t e d a r e : 1. Develop  a  method to p r e d i c t the adjusted h a u l i n g c o s t ,  t u r n o u t c o n s t r u c t i o n c o s t , and  turnout maintenance c o s t .  2. I n c o r p o r a t e the e f f e c t t h a t loaded l o g g i n g  trucks  have  of the t r a v e l time of v e h i c l e s other than logging t r u c k s . 3. Develop fleets  of  variation vehicles. .  a  model  vehicles,  that i n v o l v e s the i n t e r a c t i o n between  non-uniform  i n t r a f f i c flow r a t e s , and  turnout  spacing,  daily  v a r i a t i o n i n v e l o c i t i e s of  119 7. 2  CONCLUSIONS  &  model  was  developed  t o allow an engineer t o determine  optimum t u r n o u t spacing f o r s p e c i f i c c o n d i t i o n s effect  of  and  determine  turnout spacing on h a u l i n g c o s t . . S i m u l a t i o n models  were used t o t e s t the s e n s i t i v i t y o f optimum t u r n o u t s p a c i n g t o p e r t u r b a t i o n s of the independent v a r i a b l e s . turnout s p a c i n g was most s e n s i t i v e to followed  by  Optimum  the  turnout  construction  speeds  of  spacing  cost,  was  expected  adjusted hauling cost.  the  the  loaded  less  G e n e r a l l y , optimum traffic  life  to  of  the  turnout road, and  The r e s u l t s from a s i m u l a t i o n  revealed  t h a t a 10 0 percent i n c r e a s e of the t r a f f i c  flow r a t e w i l l  a  the  maximum  spacing.  decrease  of  70  percent  of  Much o f t h i s r e l a t i o n s h i p i s due  increasing t r a f f i c simulations  that f o r r e a l i s t i c  vph and 0 . 3 miles between turnouts)  the  total  expected  significantly deviation  delay  (approximately  i n the t r a f f i c  Based  on  delay time was  found  of uniform  percent potential  to  the  percent  the model  as  a  guide  i n the  F o r example, i f  uses the optimum t u r n o u t spacing r a t h e r than a  deviation savings  from can  be  vary  spacing of t u r n o u t s , the  o f the road network can be determined..  an engineer  deviation i n  rate..  p o t e n t i a l savings of u t i l i z i n g design  also  p r o p o r t i o n a l l y ) with  flow  the concept  was  of  c o n d i t i o n s (e.g., 5  the expected  time  effect  delay time.. The  20 percent of the t r a v e l empty time.. The percent the  cause  optimum t u r n o u t  to  flow r a t e on t o t a l expected  determined  rate  and empty v e h i c l e s .  sensitive  useful  flow  the as  optimum,  then  high  1  as  the  percent  100  resulting of  the  120 transportation this  1  cost..  percent  $10000  per  In  the  potential  s i t u a t i o n c i t e d i n S e c t i o n 7.1  savings  year over a 10-mile  represents  haul a t 4 vph.  haul d i s t a n c e s can e a s i l y approach 30 miles hauled  over  several  savings through  such  approximately  routes  Since o f f - r o a d  and  wood  simultaneously,  can  be  potential  optimum turnout spacing could be i n  the  order  of $90000 per year.. The utilized  concept  of  to estimate  the  expected  the  F - f a c t o r was developed  expected  delay  of  turnout.  T h i s i s a measure of the expected  between  the  loaded  vehicle  and  the  not  completely  delay.  Two  truck  in  vehicle  empty  as  vehicle  of  the  expected  a  has  Previous a r t i c l e s have  d e f i n e d a method of d e r i v i n g or measuring  forms  a  separation distance  empty  p r o p o r t i o n of the turnout s p a c i n g when the come t o a complete h a l t i n t h e ' t u r n o u t .  a  and  F-factor  this  equation  were  developed: 1. F = (V, +V )/(2V/j) 2  : f o r one empty v e h i c l e one loaded  2. F = Pr(h>h )F c  |  + Pr(h<h )F^ c  meeting  vehicle  : f o r one empty  vehicle  meeting a f l e e t These realistic.  equations were t e s t e d by s i m u l a t i o n and found  The s i m u l a t i o n s showed that the i n t e r a c t i o n between  a f l e e t o f empty v e h i c l e s and a f l e e t o f adequately  described  by  the  loaded  expected  r e p r e s e n t i n g the meeting of an empty v e h i c l e loaded  to be  vehicles  F-factor and  a  was  equation fleet  of  vehicles.  The  expected  F - f a c t o r equation f o r one v e h i c l e meeting a  f l e e t r e q u i r e s the use of a headway d i s t r i b u t i o n f u n c t i o n . .  Two  s e t s of i n t e r a r r i v a l time data were analysed  if  to  determine  121 the  headway  distribution  probability headways  distribution. .  was  exponential  of  left  logging  The  skewed  trucks  frequency  fits  a  histogram  known of  the  but was found to f o l l o w n e i t h e r an  nor E r l a n g d i s t r i b u t i o n . ft s i m u l a t i o n model showed  t h a t the type of headway d i s t r i b u t i o n u t i l i z e d , e x p o n e n t i a l E r l a n g (<<= 2) , d i d not s i g n i f i c a n t l y Since the data Erlang,  a l t e r the expected F - f a c t o r .  appeared to f o l l o w a d i s t r i b u t i o n s i m i l a r t o the  and t h e F - f a c t o r i s r e l a t i v e l y  i n s e n s i t i v e t o the type  of headway d i s t r i b u t i o n , i t i s concluded t h a t f a c t o r w i l l adequately model a r e a l i s t i c Previous determining  or  authors the  have  number  written of  the  hauling papers  lanes  derived  F-  situation. concerned  required  with  for  log  t r a n s p o r t a t i o n . . They d i d not c a l c u l a t e o r derive an e x p r e s s i o n to  determine  the  expected turnout  delay  now a v a i l a b l e , the t o p i c of determining required  for  method should cost, The of  turnout  log  investigated.  to  predict  other  logging than  number  the  adjusted  lanes  trucks  hauling  maintenance c o s t .  t r u c k s have on the logging  of  be r e i n v e s t i g a t e d . . ft  c o n s t r u c t i o n c o s t , and turnout  e f f e c t t h a t loaded vehicles  the  t r a n s p o r t a t i o n should  be developed  time.. Since t h i s i s  travel  should  also  time be  122 LITEBATOBE CITED  Arden, B.W. and Astill, K.N. 1970. Numerical A l g o r i t h m s : Prig ins and Applications.. Addison-Wesley Publishing Company , Beading, Mass. 308 pp. Boyd, C.W. and Young, G.G. .1969. A Study of C a n f o r ' s Logging Trucks at H a r r i s o n Mills. (Unpublished). F a c u l t y of F o r e s t r y , U n i v e r s i t y of B . C . , Vancouver, B.C..62 pp. B.C.  F o r e s t S e r v i c e . 1975. P r i n c e George District Eoad Manual. B.C. F o r e s t S e r v i c e , V i c t o r i a , B.C.  - Forest 121 pp..  B.C.  F o r e s t Service..1977. A p p r a i s a l Manual - Vancouver F o r e s t D i s t r i c t . B.C. F o r e s t S e r v i c e , V i c t o r i a , B.C. 45 pp..  Byrne, J . J . , Nelson, B.J. and Googins, P.H. 1947..Cost of Hauling Logs by Motor Truck and T r a i l e r . P a c i f i c Northwest Forest S Bange Experiment Station. Portland, Oregon. .112 pp. Conte, S.D. and deBoor, C. 1972..Elementary Numerical A n a y s i s : An A l g o r i t h m i c Approach. 2nd ed. McGraw-Hill Book Company, New York..396 pp. Drew, D.R. .1968. T r a f f i c Flow Theory and C o n t r o l . McGraw-Hill Book Company, New York. 467 pp. F e l l e r , W. 1966. An I n t r o d u c t i o n t o P r o b a b i l i t y Theory and i t s Application - Volume I^.John Wiley S Sons, Inc., New York..461pp. Gerlough, D.L.. 1955. Use o f Poisson D i s t r i b u t i o n i n Highway T r a f f i c . In P o i s s o n and Traffic. pp.. 1-58. The Eno Foundation for Highway Traffic Control, Saugatuck, Connecticut. Haight, F.A. Academic  1963..Mathematical T h e o r i e s of P r e s s , New York. .242 pp..  Traffic  Matson, T.M., Smith, S.S. and Hurd, F.W. E n g i n e e r i n g . McGraw-Hill Book Company, New T  Meriam, J.L. 1971. Dynamics. New York. 480 pp.  Control..  1955.. T r a f f i c York. 647 pp.,  2nd..ed. John Wiley & Sons,  Inc.  Porpaczy, L . J . and W a e l t i , H. 1976. How t o s e l e c t f o r e s t road standards. Canadian F o r e s t I n d u s t r i e s 96 (12): 33, 36-37. Sauder, B.J. and Nagy, M.M.. 1977. Coast Logging: Highlead versus Long-reach A l t e r n a t i v e s . F o r e s t E n g i n e e r i n g Researh Institute of Canada. Technical Beport No. TB-19. Vancouver, B.C.. 51 pp. Schuhl,  A.  1955.  The  Probability  Theory  Applied  to  123 Distribution of V e h i c l e s on Two-lane Highways. In Poisson and T r a f f i c . pp. 59-75. The Eno Foundation For Highway T r a f f i c C o n t r o l , Saugatuck, C o n n e c t i c u t . Smith, D.G. and Tse, P.P. 1977. Logging Trucks: Comparison of Productivity and Costs. Forest Engineering Research Institute of Canada. T e c h n i c a l Report No. TR-18. Vancouver, B.C..43 pp. Truck Loggers A s s o c a t i o n . . 1978. Equipment Rental J o u r n a l o f Logging Management 9(1):1306-07. ,  Rates.  United States Bureau of Land Management. . 1977. Timber Production C o s t s : Schedule 20._,Bureau o f Land Management, Oregon State O f f i c e , P o r t l a n d , Oregon. 524 pp. Wohl, H. and M a r t i n , B.V. .1967. T r a f f i c Systems A n a l y s i s For Engineers and Planners. McGraw-Hill Book Company. New York. 558 pp. .  124  APPENDICES  APPENDIX 1  BOAD STANDARDS SORVEY  126 THE UNIVERSITY OF BRITISH COLUMBIA 2075 WESBROOK MALL VANCOUVER,B.C.,CANADA V6T 1W5 MacMILLAN BUILDING FACULTY OF FORESTRY Dear S i r : I am a graduate student a t the U n i v e r s i t y of B r i t i s h Columbia and c u r r e n t l y doing a study on road s t a n d a r d s . . The o b j e c t i v e of t h i s p r o j e c t i s t o e v a l u a t e the design elements used i n the d e t e r m i n a t i o n of road standards and road c l a s s i f i c a t i o n . It would be appreciated if you could forward the road s p e c i f i c a t i o n s your company employs i n the c o n s t r u c t i o n of i t s forest roads.. Some o f the design elements under c o n s i d e r a t i o n could be: 1. Road types ( i . e .  Main, secondary, branch and spur)  2..Design speed 3. Minimum s i g h t d i s t a n c e 4. Curve  (horizontal,  vertical)  radius  5. Adverse and f a v o u r a b l e grade 6. Subgrade width 8. Surface m a t e r i a l 9. D i t c h width and depth  (rock or s o i l )  10. Maximum s u r f a c i n g depth 11. Right-of-way width 12. Turnouts (length, width and number per mile) 13. C u l v e r t s (type) 14. Compaction compaction)  (equipment  utilized  and  15. Use o f roads  (winter, summer or year-round)  Thank you f o r your c o - o p e r a t i o n . Yours t r u l y , DIA/ns  Dennis I . . Anderson Graduate S t u d i e s  degree  of  127 Road Standards  Specifications  Survey  Response  | rL  Rest of Canada (Total=30)  British Columbia (Total=16) Speed I Right-of-way | Subgrade width | Surface width | Surface depth | Road Gradient | -Favourable | -adverse | Curve r a d i u s | Sight d i s t a n c e | -Horizontal | -Vertical | Superelevation | D i t c h depth | width | Turnouts l e n g t h | width | frequency| Back s l o p e | F i l l slope | Cross s l o p e | Load c a p a c i t y | Survey response -sample s i z e -response  | | |  12 (75) 10(63) 11(69) 14 (87) 2(13) 13 (81) 13(81) 13(81) 4(25) 3(19) 1(6) 1 (6) 12 (75) 6(37) 8(50) 6(37) 9(56) 3(19) 3(19) 2(13) 5(31) 27 19(70)  | | | j  | | | | j j  | | | | | | |  J |  23(77) 30(100) 21 (70) 29(97) 21 (70) 23(77) 23 (77) 19 (63) 6(20) 16(53) 1 (3) 7(23) 18 (60) 10 (33) 3(10) 3(10) 5(17) 12(40) 9(30) 4(13) 10(33) 53 30 (57)  (United S t a t e s | I I  | I I | 1 I 1  (Total=15) 8(53) 9(60) 11(73) 14(93) 11(73)  1 1 1  13(87) 13(87) 11(73) M27) 3(20) 1 (7) 1 (7) 12(80) 10(67) 9(60) 8(53) 9(60) 6(40) 6(40) 3(20) 1(7)  | | |  66 28 (42)  I I I I  | ! 1 1 1  |  * P a r e n t h e s i z e d v a l u e s r e p r e s e n t percentages o f : 1. the a f f i r m a t i v e response, o r 2. the sample s i z e i n t h e cases i n v o l v i n g response"  the  "survey  128 APPENDIX  2  ABBREVIATIONS. SYMBOLS^ AND ONITS  I ROMAN SYMBOLS AND ABBREVIATIONS  a,a ,a ft  D  a c c e l e r a t i o n and d e c e l e r a t i o n and  c  turnout c o n s t r u c t i o n  C  M  t u r n o u t maintenance c o s t  C  T  total fl  D^  s  A  cost  cost  distance  where s u b s c r i p t A and S r e f e r t o stopping  distance  and a c c e l e r a t i o n  interval  distance,  passed  both  distance  the distance  of empty v e h i c l e s a f t e r groups  between two s e t s  the loaded  of  v e h i c l e has a c c e l e r a t e d d  subscripts  D r e f e r to a c c e l e r a t i o n and d e c e l e r a t i o n  C  D,D ,D  where  empty  vehicle  has  v e h i c l e s and each  to i t s o r i g i n a l speed  number o f " c o n f l i c t " hours per day  F,F  L  t h e F - f a c t o r , which i s  the distance  vehicle  i s from t h e empty v e h i c l e , once the empty  vehicle  has come  turnout,  divided  subscript  L  to by  refers  F  t h e expected  F,  t h e expected the c r i t i c a l  2  F-j  to  halt  i n the  spacing.  the F - f a c t o r  The  when  one  f o r headways greater  than  of t h e v e h i c l e  F-factor headway  f o r headways l e s s  than  the  headway  t h e expected situation  complete  F-factor  t h e expected F - f a c t o r critical  a  the turnout  accounts f o r the l e n g t h  f  the loaded  F-factor  f o r a Case 3 turnout  delay  129 the  maximum  interaction loaded  F-factor between  resulting  one  empty  from  vehicle  the and  one  vehicle  coefficient  of  road g r a d i e n t ,  friction i n c l i n e , or  a c c e l e r a t i o n of  gravity  General Purpose S i m u l a t i o n t r a f f i c flow r a t e , the a point  grade  System V  number of v e h i c l e s  passing  per time u n i t  headway,  which  is  the  time  successive  v e h i c l e s measured from f r o n t to f r o n t  v e h i c l e headway where the to c o n s t r a i n e d critical the  and  free  interval  subscripts  1 and  between  2 refer  flow  headway where the  subscript  i n c l u s i o n of v e h i c l e l e n g t h  CL r e f e r s  i n the  to  calculation  of the c r i t i c a l headway 2a a / A  D  (a +a ) fl  the l e n g t h  D  of the  loaded  adjusted hauling  cost  the  expected  number  of  vehicle  ( i . e . , d o l l a r s per hour) delays  vehicle t r a v e l s a unit distance  while the  empty  of road  present worth "the  expected u s e f u l l i f e  of the  the number of headways during the  minimum time gap  v e h i c l e and distance  turnout  between the  the f r o n t of the  between t u r n o u t s  the c r i t i c a l  turnout  locations  the  road" " c o n f l i c t " hours back of the  second  vehicle  first  130 T,T ,T A  5  t i m e where t h e s u b s c r i p t s  4 and  S  a c c e l e r a t i o n t i m e and t h e s t o p p i n g f  total empty  expected  delay  t i m e o r headway  V,  v e l o c i t y of loaded velocity gross  c f empty  weight  X  C F  critical  Z  2  a present  II  to  time per u n i t  distance  vehicle vehicle  of the v e h i c l e  distance worth  function  O t h e r Symbols  oi,  parameters o f the E r l a n g  'Tj  minimum  ^2  chi-sguare  ^  base o f t h e n a t u r a l  headway d i s t r i b u t i o n  headway value logrithums  the  time  vehicle  t  W  refer  per  131 APPENDIX 3  The weight  ACCELERATION AND  braking f o r c e r e q u i r e d to stop a v e h i c l e i n v o l v e s of  the  surface.  of f r i c t i o n  (f) between the t i r e s  coefficient  a braking f o r c e measured along the plane of these  (0),  and  the  In t h i s i n s t a n c e i t w i l l be assumed there i s a  constant road g r a d i e n t , a constant  equating  the  v e h i c l e (W), the road g r a d i e n t or i n c l i n e  and the c o e f f i c i e n t road  DECELERATION OF A VEHICLE  conditions  braking f o r c e equation can  of f r i c t i o n , incline..  By  along the plane of the i n c l i n e  the  be  the  and  obtained:  Braking f o r c e = Wa /g = W f c o s e - W s i n © (  where: a  = d e c e l e r a t i o n (feet per second per second)  (  g  = a c c e l e r a t i o n of g r a v i t y . .  I f angle 0 to tan© and  i s r e l a t i v e l y small then sin© i s approximately cos8 approaches one.,  The  equation can  equal  be r e w r i t t e n  as: a  = g (f-tane)  ;  or a  D  = 79036 (f-G/100)  where : a^ = d e c e l e r a t i o n (mph ). 2  To account  f o r adverse  and  favourable  gradients  the  can be adjusted t o : a  D  = 79036 (f±G/100)  A.1  I f Q i s r e l a t i v e l y l a r g e then: a  0  = 79036 ( f c o s 6 ±  sine)  a  D  = 79036 (100f ±G)/(G + 100 ) o-s 2  2  A. 2  equation  132  where :  The and  D"  = h o r i z o n t a l stopping  D*  = i n c l i n e stopping  Fp  = frictional  N  = normal f o r c e  W  = weight of the  ©  = angle of the  next A.2.  step  distance  distance  force  vehicle incline  i s to determine the e r r o r  A v a r i a b l e epsilon(G) can  ( 1 0 0 f ± G ) / ( G 2 + 1002)  o,s  =  between equations  be introduced  A.1  such t h a t :  (f ± G / 1 0 0 M 1 £ ) +  or 0 = £ 2 (G + 10()2) • 2£(G + 10C)2) 2  Solutions  2  f o r e p s i l o n are  In r e a l i t y the speed. . such that below).  Therefore, the  +G2  illustrated  i n the t a b l e  below.  c o e f f i c i e n t of f r i c t i o n i s p r o p o r t i o n a l a coefficient  stopping  distance  of f r i c t i o n  or time i s  to  should be chosen  correct  (See  table  133  If  the  braking  distance  is  defined  as being along the  h o r i z o n t a l plane then: D  = D cos ©  s  s  where : Dg = h o r i z o n t a l braking D* = i n c l i n e braking the d e c e l e r a t i o n equation a and  distance  distance becomes:  = 79036 (f±G/100)  D  e p s i l o n equals  zero.  The a c c e l e r a t i o n of a v e h i c l e i s l i m i t e d by the horsepower of the v e h i c l e and the forms o f r e s i s t a n c e to v e h i c l e movement: rolling,  air,  et a l . (19 55)  gradient,  and  Byrne  engine,  and  e t a l . (1947)) .  inertia Matson  showed that the r e s i d u a l horsepower a v a i l a b l e f o r is: HP  = HP_  T  n  A  - HP_  ROLL  - HP _ A  AIR.  - HP„„  .  GftAOt  = HP -(WV {R^±20G) + 0.0026K, AV|) f  Z  /375  where: A  = f r o n t a l area of v e h i c l e i n sguare f e e t  HP  fl  HP  AlR  = r e s i d u a l horsepower = power to overcome a i r r e s i s t a n c e  HP,  = power t o overcome g r a d i e n t  HP„  = power to overcome r o l l i n g  resistance resistance  ROLL  HP  = t o t a l horsepower a v a i l a b l e  Kj  = streamlining  R  = r o l l i n g resistance  W and  T  R  factor i n pounds/ton  = v e h i c l e weight i n tons  the p o t e n t i a l a c c e l e r a t i o n i s :  (Matson  et al.(1955) acceleration  134 a The  fl  =  (14819 HP ) / (WVg,) fl  above  seldom  formula  calculates  the p o t e n t i a l a c c e l e r a t i o n but  i s the p o t e n t i a l a c c e l e r a t i o n e q u i v a l e n t t o  acceleration. modified  the  Consequently, the a c c e l e r a t i o n formula should be  to  account  for this deviation.  I f the a c c e l e r a t i o n  time or d i s t a n c e i s known then the a c c e l e r a t i o n can determined  from  basic  dynamics,  equations  S i m i l a r i l y , the s t o p p i n g d i s t a n c e or time t a b l e s this  actual  appendix  can  be  utilized,  2.3  be  easily or  developed  2.4. in  provided the c o e f f i c i e n t of  f r i c t i o n becomes a c o e f f i c i e n t of t r a c t i o n . . In the development of the model the t a b l e approach w i l l be u t i l i z e d i n the p o t e n t i a l a c c e l e r a t i o n formula approach. tables  are  favour  of  Consequently, the  used i n a s i m i l a r manner as f o r the case i n v o l v i n g  deceleration. .  Table Of E p s i l o n VS Road Gradient [%)  Road Gradient (%)  Epsilon  1  0.000049604  2  0.000199725  3  0.000449010  4  0.000798332  5  0.001245908  6  0.001791597  7  0.002434749  8  0.003174604  9  0.004009128  10  0.004938118  11  0.005959723  12  0.007072758  13  0.008274801  14  0.009565637  15  0.010942270  16  0.012403550  17  0.013946550  18  0.015570130  19  0.017271420  20  0.019049160  21  0.020899980  22  0.022822160  23  0.024812970  24  0.026870420  25  0.028991170  136  T a b l e  V e l o c i t y  iph)  5  Of  S t o p p i n g  C o e f f i c i e n t 0.  0.05  4.6  10  2.  1  Of  3  T i m e s — S e c o n d s  F r i c t i o n 0.  1.  P l u s  Or  Minus 0.25  0.30  0.  35  0.20  5  1.  1  0.9  0.8  0.  7  3  1.8  1.5  1.  3  4.6  3.0  2.  6.8  4.6  3.4  2.7  2.3  2.  0  4.6  3.6  3.0  2.  6  5.7  4.6  3.8  3.  3  5.5  4.6  3.9  6.4  5.3  4.  9.  15  13.7  20  18.  25  22.8  11.4  7.6  30  27.3  13.7  9.  1  6.  35  31.9  15.9  10.  6  8.0  40  36.4  18.  2  12.  1  9.  1  7.  45  41.0  20.  5  13.7  10.  2  8.2  50  45.5  22.  8  15.  55  50.  25.  1  16.7  60  54.7  1  G r a d i e n t  15  10  2  Road  9.  27.3  1  6.  18.  1  2  11.4  12.  2  8  13.7  6.  1  5.2  6.8  5.  9  7.6  6.  5  10.0  8.4  7.  2  10.9  9.  7.  8  9.  5  3  6  1  1  137  T a b l e Of S t o p p i n g  Velocity  Coefficient  Distances—Feet  Of F r i c t i o n 0. 15  P l u s Or Minus Road 0 . 20  0.25  Gradient  0.30  0 . 35  0.05  0 . 10  5  17.  8.  6.  4.  3. .  3.  2.  10  67.  33.  22.  17.  13. ,  11.  10.  15  150.  75.  50. .  38.  30. .  25.  21.  20  267.  134.  89.  67.  53.  45. .  38.  25  418. .  209. .  139.  104. .  84. .  70. .  60.  30  601.  301.  200. .  150.  120.  100.  36.  35  818. .  409.  273. .  205. .  164.  136.  1 17.  iph)  40  1069.  534.  356.  267.  214.  178.  153.  45  1353.  676. .  451. .  338.  271. .  225. .  193.  50  1670.  835.  557.  418. .  334.  278.  239.  55  2021.  1010..  674. .  505.  404. .  337. .  289.  60  2405.  1202.  802.  601. .  481.  401. .  344.  .  138 APPENDIX 4  ANALYSIS OF HEADWAY DISTRIBUTIONS  The a n a l y s i s of headway d i s t r i b u t i o n s i n v o l v e d two s t u d i e s , a c o a s t a l study and  a  northern  Limited). observed  The  interior analysis  frequency,  (Canadian Forest study  involved  from  Products Limited)  (Northwood the  data  Pulp  and Timber  determination  of  the c a l c u l a t i o n of the t r a f f i c flow  the rate,  and ~y* goodness o f f i t t e s t s . . The c o a s t a l study used  frequency  c l a s s e s o f f i v e minutes  frequency  while the i n t e r i o r study used  c l a s s e s o f one and f i v e minutes.. The employed  the  use  goodness  of  f i t tests  o f two computer programs, U.B.C. FREQ and a  program w r i t t e n i n BASIC f o r the HP9830A c a l c u l a t o r . . A b r i e f documentation the  of the BASIC program f o l l o w s .  With  use of "REM" statements i n the program the.model i s e a s i l y  explained.  The b a s i c  sections  of the program a r e :  1.. Input o f the lower c e l l l i m i t , the c e l l i n t e r v a l , and  the  number of c e l l s 2.. C a l c u l a t i o n  of the c e l l  3.. Input o f the parameters 4., Determining observed  boundaries of the d i s t r i b u t i o n t o be t e s t e d  frequency  5.  Determining expected frequency  6.  Regrouping  of  f i v e expected 7.  the  c e l l s such that there i s a minimum o f  values i n any c e l l  C a l c u l a t i o n o f the c h i - s q u a r e value..  139 Proqram V a r i a b l e s  Variable  Function  A  Chi-square  value  C  frequency of the observed and expected  D  frequency o f observed  data  E  frequency of expected  values  F  cell  N  i n i t i a l number of c e l l s  N1  cell  N2  initial  N3  number o f c e l l s  V1  t o t a l number of v e h i c l e s  V2  v e h i c l e s per hour  V3  number of h a u l i n g hours per day  boundaries  width value a f t e r regrouping  data  n o  Program  5 REM  Listing  GOODNESS OF FIT TEST  10 DIM F(80) ,C(2,80) ,Q$(40) ,X(2,80) ,D(80) ,B$(1) ,E(80) 15 REM  FREQUENCY OF OBSERVED DATA  20 DATA 30 DATA 60 DISP "INPUT DATE"; 70 INPUT Q$ 80 PRINT LIN2,Q$,LIN2 90 DISP"INPUT PROBABILITY DISTRIBUTION"; 100 INPUT Q$ 110 A=0 120 MAT C=ZER 130 DISP"INPUT NUMBER OF CLASSES 6 INTERVAL"; 140 INPUT  N N1 f  150 DISP"INPUT INITIAL VALUE"; 160 INPUT N2 165 REM  CALCULATION OF CELL BOUNDARIES  170 FOR 1=0 TO N-1 180 F (1 + 1)=I*N1+N2 190 NEXT I 200 F(N) = 200 210 DISP"INPUT # VEHICLES, VEH./HR.,DAY LENGTH"; 220 INPUT 71,72,73 230 DISP"IS DATA TO BE ENTERED"; 240 INPUT B$  141 245 REM  DETERMINING  THE FREQUENCY  OF THE OBSERVED DATA  250 FOR J = 1 TO N 260 IF B$="Y" THEN 300 270 C (1,J) = D (J) 280 GO TO 310 290 DISP"INPUT OBSERVED VALUE»J; 300 INPUT C(1,J) 310 NEXT J 320 C1=C3=0 325 REM  DETERMINING THE FREQUENCY  OF THE EXPECTED VALUES  330 FOR J=1 TO N 335 REM  THE PROBABILITY FUNCTION  340 C (2,J) =V1* (1- (1 + 2*F(J) *V2/60) *EXP(-2*F (J) *V2/60) ) 350 C2=C(2,J) 360 C (2 J) =C r  {2,3)  -C1  370 C1=C2 380 C3=C3+C (2, J) 390 NEXT J 400 PRINT LIN3,"THE DISTRIBUTION IS "Q$,LIN3 410 PRINT"NUMBER OF CLASSES  =  "N  420 PRINT"CLASS INTERVAL  =  "N1  430 PRINT"INITIAL VALUE  =  "N2  440 PRINT"TOTAL NUMBER OF VEHICLES  =  "V1  450 PRINT"VEHICLES PER HOUR  =  "V2  460 PRINT"LENGTH OF DAY  =  "V3  470 PRINT LIN2 480 FOR 1=1 TO N 490 PRINT J ; C ( 1 J ) ;C(2 J) r  500 NEXT J  f  510 PRINT LIN2 515 REM  GOODNESS OF FIT TEST  560 MAT D=ZER 565 MAT E=ZER 570 N3=1 575 REM  REGROUPING OF THE CELLS SUCH THAT THERE IS  576 REM  A MINIMUM FREQUENCY OF FIVE EXPECTED VALUES  577 REM  IN EACH CELL  580 FOR 1=1 TO N 590 D (N3) =D(N3) +C(2 I) r  595 E (N3) =E (N3) +C (1,1) 600 IF D(N3)<5 THEN 620 610 N3=N3+1 620 NEXT I 630 IF D(N3)>=5 THEN 660 640 N3=N3-1 645 E (N3) =E (N3)+E (N3+1) 650 D(N3) =D(N3)+D(N3+1) 660 FOR 1=1 TO N3 670 C (2,1) =D (I) 675 C(1,1)=E (I) 680 NEXT I 690 A=0 695 REM  CALCULATION OF THE CHI-SQUARE VALUE  700 FOR 1=1 TO N3 710 A=A*(C (1,1)-C (2,1) )f 2/C (2,1) 720 NEXT I 730 PRINT LIN1 740 PRINT"CELL NUMBER  OBSERVED  EXPECTED"LIN1  143 750 FOE J=1 TO N3 760 PRINT J;C(1,J) ;C(2,J)770 NEXT J 780 PRINT LIN1 790 PRINT"THE CHI-SQUARE VALUE IS "A 800 PRINT LIN2 810 END  144 C o a s t a l Study  Observed Headway  C l a s s Mark (minutes)  Frequency-Distribution  !Class  | Frequency|  4  Mark  |Frequency  (minutes)  l -  — + -  |  65  |  5  I  70  |  1  |  I  75  I  o  31  |  I  80  I  o  I  17  |  I  85  I  2  25  I  9  I  I  90  I  o  30  I  14  |  I  95  I  o  35  |  7  |  I  100  I  o  40  I  5  |  105  I  o  45  I  6  |  ! !  110  |  1  50  I  ^  |  i  115  I  o  55  I  2  I  i  117.5 +  I  6  60  I  5  |  Total  |  1-25  I  5  1 1 6  |  10  I  60  15  I  20  2  |  I  193  145 T r a f f i c Flow Rate  flow*  : 201 vehicles/83.4  hours = 2.41  vehicles/hour  193 vehicles/83.4  hours = 2.32  vehicles/hour  or  * There was a t o t a l o f 201 v e h i c l e s  observed  period  headway o b s e r v a t i o n s .  but  there  were  only  193  over  the  a c t u a l flow r a t e l i e s between the two c a l c u l a t e d values.  8-day The  146 Goodness Of F i t T e s t s — R e s u l t s  [  - J -  -+  +-  Table  LCL • Computed  P r o b a b i l i t y Program Number* C l a s s Width Of Dist. . C l a s s e s (min)  r  (min)  -+  00 5 -+  -+  +-  H  IPoisson  | EREQ  125(11)  |  5  1  0  |  316  I  9  123.6  |  |Binomial  | FREQ  125(10)  |  5  1  0  |  326  I  8  122.0  |  | Neg bino m. . |FBEQ  125(18)  |  5  1  0  |  120  |  15  132.8  1  |25(14)  |  5  1  2.5  J  76.3  |  12  128.3  1  | E r l a n g (oC=2) | Basic 1 2 5 ( 1 2 )  J  5  I  2.5  |  62.7  |  10  125.2  |  | S h i f t expon| | : 0 . 1 min | B a s i c  |25(14)  |  5  1  2.5 1  75. 2  I  12  128.3  !  |Exponential!  BASIC  |  :0.5  min | B a s i c  125(14)  |  5  1  2.5  J  71.0  I  12  128.3  |  |  :1.0  min | B a s i c  125(14)  |  5  1  2.5  |  65.8  I  12  128.3  |  L  L  _J  i—  _j  i  i  _x  ,  * The parenthesized number i s the f i n a l number of frequency classes while the non-parenthesized number i s the number o f frequency c l a s s e s . . + The number o f degrees of freedom • Lower c l a s s  limit  147  I n t e r i o r Study - F i r s t Weigh Scale  Observed Headway Frequency  Class Hark (min)  Class Mark (min)  Freq. L._  0.25| 1 | 2 I 3 I 4 I 5 I 6 I 7 I 8 | 9 I 10 |  3 54 46 29 27 36 34 25 20 18 26 18 14 16 12 28 13 11 12 14 8 9  I | | | | | | | | |  11 12 13 14 15 16 17 18 19 20 21  1 i _  Traffic  -++ || II || II II II || II II t I II II || II || II II II II II 1i || _J-J  Class Mark (min)  Freq. +-  22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 43  Distribution  | | | | | | | | | | | | | | | | | | | | | | L_  —+  -f+ 11 11 7 9 12 7 2 6 4 10 4 5 6 8 3 6 8 3 7 3 5 2  | 1 1 | | 1 1 1 I 1 1 1 1 1 1 | | | I I I I  | 1 1 | | 1 1 1 1 1 1 1 1  1 1 | | | I I I 1  44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65  -i-i  1 1  1 1 1 1 1 I I I I I I I I I | I | I I I 1  ++ 2 3 5 6 2 0 1 3 0 3 4 0 0 2 0 0 0 0 0 2 0 2  Freq,  Class Mark (min)  Freq.  |1 |1 || || |1 |1 I I |I |I |1 || |1 |1 |1 || || || || || || || ||  +~ 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 79.5+1 | | | | | |  0 0 3 2 0 0 0 1 1 0 2 1 0 0 5  Total| 661  L1  L_  Flow Rate  flow* = 681 vehicles/183.5  hours = 3.71 v e h i c l e s / h o u r  or  hours = 3.60 v e h i c l e s / h o u r  661 vehicles/183.5  * There was a t o t a l  o f 681 v e h i c l e s  only  observations.  661  headway  between t h e c a l c u l a t e d values.  observed The  actual  but  there  were  flow r a t e l i e s  148  G o o d n e s s Of F i t T e s t s — R e s u l t s  P r o g r a m Number* C l a s s LCL • Computed Width Value Of (min) C l a s s e s (min)  Probability Dist. .  v  r  —  Tablel |oC=. 0 0 5 |  \-  f  IPoisson  | FREQ  1-80 ( 2 2 )  !  1  I  o. 5 1  + 500  |  20  140.0  | FBEQ 1 L | Neg b i n o m. . | FREQ  | 80 (20)  !  1  I  o. 5  !  + 500  |  18  | 37.2  | 80 (59)  !  1  I  0. 5  1  74  I  56  I 53.7  |Exponential|  BASIC  | 80 (47)  !  1  I  0.5  1  76.8  I  45  153.7  | E r l a n g (cC=2) | BASIC  | 80(41)  1  |  0 .5  1  414.2  I  39  153.7  1  I  0.5  1  74.3  |  45  153.7  IBinoiaial  -+  -+  -+  --+-  |  :0.1  exponl min | BASIC  | 80 (47)  I j  1  :0.5  min | BASIC  | 80 (46)  !  1  1  0.5  |  55. 1  |  44  153.7  BASIC  I 17(15)  |  5  |  0.5  1  28.6  I  13  | 29.8  | E r l a n g (o(.=2) | EASIC  I 17(11)  |  5  I  0.5  1 202. 6  !  -9  |23. 6  | S h i f t expon| j : 0 . 1 min | BASIC  1 17(15)  I  5  I  o. 5  !  24.5  I  13  129.8  m i n | BASIC I 1 7 ( 1 4 )  |  5  |  0.5  1  13. 7  I  12  | 28. 3  |Shift  |Exponential!  |  :0.5  J.  * The p a r e n t h e s i z e d v a l u e i s t h e f i n a l number of frequency classes while the non-parenthesized number i s the i n i t i a l number o f c l a s s e s + The number o f d e g r e e s o f freedom • Lower c l a s s  limit  149 I n t e r i o r Study - Both Weigh S c a l e s  Observed Headway Frequency  "TT"  Class Mark (min)  ++  0. 25 | 1 | 2 1 3 1 4 i 5 1 6 I 7 1 8 1 9 1 10 | 11 | 12 | 13 | 14 | 15 | 1 6 | 17 | 18 | 19 | 20 | 21 | i  3 60 53 30 32 42 35 28 22 20 26 21 17 16 18 28 16 13 14 14 9 11  T r a f f i c Flow  flow* or  •TT"  Class Mark (min)  Freq.  1I  || || |1 |I |1 || || I | |1 |I 1| I | |1 |1 |1 || |I || |1 1| 1 |  Distribution  Freq.  —T+-  -+  22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43  I I I I  1 1 1 1 1 1 i  1 1 1 1 1 1 1 1 1 1 1 1  — +  11 I I 44 11 II 45 9 | | 46 1 0 | | 47 1 6 | | 48 7 | | 49 4 | | 50 9 M 51 4 | | 52 1 0 | | 53 5 | | 54 7 | | 55 1 0 | | 56 1 2 | | 57 5 | | 58 7 | | 59 8 | | 60 4 | |61 8 | |62 6 | |63 8 || 64 4 | |65 .  i~L  _ __—  775  I I  | 1 1 1 I I  |  I  | | I  | I  1 1 1 1 1 I  1  2 3 8 6 4 1 1 4 1 4 4 0 2 3 1 0 0 0 0 2 0 2  •TT"  Class Freq. Mark (min) -1 1 ++-  |1 |1 |1 || |J I| II  66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 79.5+1  2 0 3 2 0 0 1 1 1 0 3 1 0 0 10  |I |I |I || || || |1 I| | || | || | || j |j | || || j | | T o t a l | 775 LJ  x _—  1 1 1 1 1 1 | | | I I  | I I  |  | i  Rate  : 814 vehicles/252.2  hours = 3.23 v e h i c l e s / h o u r  775 vehicles/252.2  hours = 3.07 v e h i c l e s / h o u r  * There was a t o t a l o f 814 v e h i c l e s only  Freq.  Class Mark (min)  headway  observations'.  between t h e c a l c u l a t e d  values.  observed The  actual  but  there  flow rate  were lies  150  Goodness Of F i t T e s t s — R e s u l t s  I  T  T  1  1  T  1  P r o b a b i l i t y ! Program !Number* C l a s s ] LCL • Computed! Width! Value Dist. . | lOf %' j (min)| (min) |Classes +  1  T  |Table|  If*  +•  I  ^  2  K=- 0 0 5 | -+  f  I  1  Poisson  | FREQ  ! 80 (23)  !  ]  0. 5  • 500  !  21  141.4  |  Binomial  | FREQ  | 80 (21)  1  !  0.5  + 500  |  19  138.6  |  Neg bino m. . | FREQ  | 80(64)  1  !  0.5  85.3  |  61  |53.7  1  BASIC | 8 0 ( 5 3 )  1  0. 5  92.0  |  51  153.7  1  E r l a n g (cd=2) | BASIC | 8 0 ( 4 8 )  1  !  0.5  617.9  1  46  153.7  I  S h i f t exponj : 0 . 1 min | BASIC | 8 0 ( 5 3 )  1  !  0. 5  87.5  J  51  153.7  1  min | BASIC | 8 0 ( 5 2 )  i  0.5  72.2  |  50  153.7  |  BASIC | 2 1 ( 1 7 )  5  Exponential!  :0.5  Exponential!  I I 1  0. 5  1  33.7  1  15  132.8  I  I  183.5  I  11  126.8  |  29.4  I  15  132.8  I  17.6  |  14  131.3  !  20.7  |  14  131.3  |  E r l a n g (ct=2) | BASIC | 2 1 ( 1 3 )  1  5  | 0.5  S h i f t exponj : 0 . 1 min | BASIC | 2 1 ( 1 7 )  I  5  | 0. 5  min | BASIC 1 2 1 ( 1 6 )  5  !  0.5  BASIC | 2 0 ( 1 6 )  5  J  5.5  :0.5  Exponential!  I  * The parenthesized value i s the f i n a l number of frequency classes while the non-parenthesized number i s the i n i t i a l number of c l a s s e s + The number o f degrees of freedom • Lower c l a s s  limit  151 APPENDIX 5  The  SIMULATION OF  purpose of the  THE  f-FACTOR  simulation  models  was  to  test  v a l i d i t y and determine the l i m i t a t i o n s of the expected equations,  equations  were developed vehicles,  2.12,  2.18,  and  2.19., The  c o n s i s t e d of the i n t e r a c t i o n  the  F-factor  programs t h a t  between two  single  the meeting of a s i n g l e empty v e h i c l e and a f l e e t of  loaded v e h i c l e s , and the  interaction  between  two  fleets  of  vehicles. . The  models  do  not r e f l e c t a l l of an o p e r a t i o n ' s t u r n o u t  delay times s i n c e many of the i n h e r e n t delays i n the system excluded.. utilize  The  the  models assume t h a t the empty v e h i c l e w i l l always turnout  Furthermore  which  will  yield  the  least  of  have no l e n g t h . . The  the  delay..  the models assume the empty v e h i c l e has a constant  r a t e of a c c e l e r a t i o n , the turnout spacing i s uniform, turnouts  are  models  will  be  and  the  s p e c i f i c c h a r a c t e r i s t i c s of each  discussed  in  their  corresponding  sections.  I n t e r a c t i o n Between A Loaded V e h i c l e And  The F =  simple F - f a c t o r equation to be t e s t e d i s : (V, +\  Initially, and  the  both  velocities  )/{2V ) t  the empty v e h i c l e i s p o s i t i o n e d o p p o s i t e turnout loaded  distribution If  An Empty V e h i c l e  vehicle  over  randomly  the i n t e r v a l  vehicles they  is  were  would  to meet  located  S  0  (uniform  (5,10)) at a d i s t a n c e D1 away.  proceed  at  their  corresponding  at a point that i s at a d i s t a n c e  152 d i s t a n c e X from turnout S .. Thus: X = (V DIJ/CV/ * \ ) 2  The f i r s t  turnout  (INTE)  that  the  empty  vehicle  could  p o t e n t i a l l y use can be r e a d i l y determined as: INTE = X/S where : INTE i s an i n t e g e r Referring  to  number..  the f i g u r e below, the model t e s t s whether or not  the empty v e h i c l e can s a f e l y  utilize  If  the  vehicle  t h i s t u r n o u t , then the model  will  cannot  safely  use  turnout  check p r i o r turnouts u n t i l i t f i n d s the f i r s t empty  vehicle  determine d.  can s a f e l y use.  Now  INTE.  turnout that  the F - f a c t o r  the  can be e a s i l y  153  I  i— I  I  I  1  Distance(miles)  0  X  5  D1  10  Turnout l o c a t i o n s  S  INTE  where : X  = p o t e n t i a l meeting  D1  = initial  distance  point vehicles  are apart  time ,  Turnout  AS—"  |  I  1  0  INTE  locations  Y ^  Distances  1r-^rl D3 X  k  Y2-J-H  Y1  I  1 -T4  >|  (D  m~  '  I  1  -T4+T3  i  Distances  l<-Y6-^|  where: #=  location  of loaded  0=  location  o f empty  The v a r i a b l e computer  vehicle vehicle  names r e f e r t o t h e  program..  variable  names  used  in  the  154 Program  Variable  Variables  Function  A1  rate of acceleration  D1  i n i t i a l distance  D3  stopping  F1  coefficient  F9  F - f a c t o r f o r a s i n g l e meeting  INTE  first  N  the sample s i z e  S  distance  S3  the average F - f a c t o r  TOTAL  the sum of the F - f a c t o r s  T3  the stopping  14  time f o r empty v e h i c l e t o t r a v e l d i s t a n c e Y1  V1  speed o f loaded  V2  speed o f empty v e h i c l e  X  p o t e n t i a l meeting  y1  distance  v e h i c l e s are apart  distance of f r i c t i o n  turnout t h a t empty v e h i c l e t r i e s t o use  between t u r n o u t s f o r the sample  time  vehicle  point  from X to beginning of d e c e l e r a t i o n  zone Y2  distance  loaded  vehicle  i s from turnout when  empty v e h i c l e begins t o d e c e l e r a t e Y6  distance  loaded v e h i c l e  i s from  turnout  empty v e h i c l e has stopped i n turnout  when  155  Program  Listing  C C  SIMULATION OF ONE EMPTY VEHICLE MEETING ONE LOADED VEHICLE  C F1=0.25 AT=79036.3636*F1 Q1=BAND(SCLCCK (0.0) ) N=10000 Z1=N C C  THE VEHICLE'S CHABACTEEISTICS  C V1=10. V2=35. S=0 . 2 C C  THE STOPPING FUNCTIONS  C D3=V2**2/(2.*A1) T3=V2/A1 WRITE (6, 101) 101  FORMAT('I')  C C  THE NUMBER OF SAMPLES  C DO 1 11=1,20  156  TOT AL=0. C C  THE SAMPLE SIZE  C DO 2 1=1,N C C  INITIAL DISTANCE THE VEHICLES AEE APART  C D1=FRAND (0. 0) *5. +5. C C  DETERMINE TURNOUT THAT EMPTY VEHICLE CAN SAFELY PULL INTO  C X=V2*D1/(V1+V2) INTE=X/S Y1= X-S*INTE+D3 T4= Y1/V2 Y2=T4*V1+Y1-D3 T5= Y2/V1 C C  CAN EMPTY VEHICLE SAFELY PULL INTO THIS TURNOUT  C 4  IF(T5.GT.T3) GO TO 3  C C  TRY TO STOP EMPTY VEHICLE AT PREVIOUS TURNOUT  C Y1=Y1 + S Y2=Y2+V1*S/V2+S T5= Y2/V1 GO TO 4  157 C C  CALCULATION OF THE F-FACTOR  C Y 6 = (T5-T3) *V1  3  F9-Y6/S  C C  SUMATION OF THE F-FACTOR  C 2  TOTAL=TOTAL+F9  C C  THE AVERAGE F-FACTOR  C S3=T0TAL/Z1 1  WRITE ( 6 , 100) S3  100  FORM AT (' «,5X,F8.4) RETURN END  158 Interaction  Between  An  Empty  Vehicle  And A - F l e e t Of Loaded  Vehicles  The F - f a c t o r equations 2.19.  to be t e s t e d a r e equations  2.18 and  The four headway d i s t r i b u t i o n s t h a t are r e q u i r e d a r e the  e x p o n e n t i a l , s h i f t e d e x p o n e n t i a l , Erlang(alpha=2), Type III(a=2)..  The equations  and  Pearson  required f o r t h e s e : d i s t r i b u t i o n s  are: 1. . E x p o n e n t i a l : Pr(h<h ) = i - e - ^ < Q  c  = v, [ e - 3 ^ (-Q h -1)+1] /  \  Q  3  c  [-so,., ( e - ^ A - i )  ]  F, = (V, +V )/(2V ) L  2  2. . S h i f t e d e x p o n e n t i a l : Pr(h<h .) = CJ  n=  3.  *i  1-e- < K-*r > Qs  ^ R ^ - - ^ < K - R > ( h + ^ + ^ ) ] / [S(1-e-^<K -R )) ]  [  +  e  c  T  c  £  T  Erlan g ( a l p h a 2) : -  Pr(h<h ) = 1-e- ^(1 + 2Q h ) 2<?  c  3  c  F = V, [e- ^»c(-2Q h2-2h -1/Q ) +1/Q ] / {S[ 1  1 + 20- h ) ]}  Z  3  4.. Pearson Type CL  b  t  a  3  c  T  T  + R h +*^) -2h -|-^*H ]+[ R, •  1-e-kC  r  <--RT  h  c  c  T  >[ b (h -R ) +1 ] c  T  This model i s s i m i l a r to the one t h a t has in  the  previous  c  W [ b (h -R ) +1 ]  - bck-R >[ b (-h*- ^  F =  3  III(a=2):  Pr(h<h ) = 1 - e e  c  section..  In  determine v e h i c l e headways a r e :  this  model  been  developed  the f u n c t i o n s t o  159 1.  Exponential: Y = [-log(//) ] / * V A  where: JU = uniform  (0,1) v a r i a b l e  mean o f the e x p o n e n t i a l  distribution  = minimum headway and 2.  E r l a n g (alpha=2) and Pearson Type I I I (a=2) : Y = [s=] (-log(//)/A  +1  The c r i t i c a l headway must be determined vehicle  can  proceed  from  to check  i t s current  v e h i c l e progresses along a continuous  i f an  turnout.  road..  empty  The empty  160 Program  Variable  Variables  Function  Al  rate o f  acceleration  D1  i n i t i a l distance  D2  a distance  D3  stopping  F1  coefficient  F9  F-factor  H  t r a f f i c flow  INTE  f i r s t turnout empty v e h i c l e t r i e s t o  LENG  the  N  sample  Q  t o t a l number of v e h i c l e s during  R  alpha v a r i a b l e of E r l a n g  S  distance  S3  average F - f a c t o r of the  TOTAL  sum  T1  c r i t i c a l headway  T2  loaded v e h i c l e headway  T3  stopping  T4  time of empty v e h i c l e to t r a v e l d i s t a n c e  T5  time f o r loaded v e h i c l e t o reach turnout  VI  speed of loaded  V2  speed of empty v e h i c l e  X  p o t e n t i a l meeting  XLAH  a parameter i n the gamma f u n c t i o n  Yl  distance  from X where d e c e l e r a t i o n  zone begins  Y2  distance  loaded  turnout  v e h i c l e s are  function  apart  s i m i l a r to  D1  distance  length  of  friction  rate  of the loaded  utilize  vehicle  size the  day  function  between t u r n o u t s sample  of the F - f a c t o r s of the  sample  time Y1  vehicle  point  vehicle  is  from  when  empty v e h i c l e begins to d e c e l e r a t e the F - f a c t o r times the turnout  spacing  162 Proqram L i s t i n g  C C C  SIMULATE AN EMPTY VEHICLE AND A FLEET OF LOADED VEHICLES  C REAL H (18) , LENG , MIN DATA H/1.,2.,4. , 10.,6 0., 100., 1.,2.,4.,10. ,6 0., 100. , 1. , 12. ,4. ,10. ,60. , 100./ C C  THE VEHICLE'S CHARACTERISTICS  C S=0. 3 V1 = 30. V2=40. D0=11. F1=0.25 A1=79036.3636*F1 LENG=60./5280. MIN=LENG/V1/D0+0.000050505051 C C  THE CRITICAL HEADWAY  C T1= (S*A1*(V1 + V2) +V1*V2**2)/(A1*V1*V2*D0) +LENG/ (V1 *D0) Q1=RAND(SCLOCK(0.0) ) N=10000 C  163 C  THE STOPPING FUNCTIONS  C D3=V2**2/(2.*A1) T3=V2/A1 WRITE (6, 101) 101  FORM AT (* 1')  C C  THE NUMBER  OF SAMPLES  C DO  1 11=1,18  TOT AL=0. Q3=D0*H (11) C C  THE SAMPLE  SIZE  C DO 2 1=1,N I F ( I . G T . 1.5) GO TO 3 C C  THE I N I T I A L  DISTANCE THE FIRST TWO VEHICLES ARE APART  C D1= FRAND (0.0) *5. +5. D2=D1 GO TO 4 C C  CALCULATION OF THE VEHICLE  C 3  D4=FRAND (0.0) T2=-1.*ALOG (D4) /Q3  C  HEADWAY  164 C  IS VEHICLE SPACE HEADWAY LESS THAN THE VEHICLE'S LENGTH  C IF(T2.LT.MIN) GO TO 3 D2=D1+T2*V1*D0 C C  IS THE VEHICLE HEADWAY LESS THAN THE CRITICAL HEADWAY  C IF(T2.GT.T1) GO TO 5 C C  CALCULATE F-FACTOR FOR HEADWAYS LESS THAN CRITICAL HEADWAY  C Y6=T2*V1*D0+LENG F9=Y6/S D1=D2 GO TO 6 5  D2=D2-(X-Y1+D3)  4  IF(I.GT. 1.5) GO TO 7  C C  DETERMINING MEETING POINT OF THE FIRST ENCOUNTER  C X=V2*D2/ (V1+V2) GO TO 8 C C  DETERMINING MEETING POINT FOR THE OTHER  ENCOUNTERS  C 7  X=(V1*D2-V1*V2**2/(2. *A1) ) /(V1 + V2)  C C C  FIND FIRST TURNOUT EMPTY VEHICLE WILL TRY TO PULL INTO  165 8  INTE-X/S Y1=X-S*INTE+D3 T4=Y1/V2 Y2=T4*V1+Y1-D3 T5=Y2/V1  C C  CAN EMPTY VEHICLE SAFELY POLL INTO THIS TURNOUT  C 10  IF(T5.GT.T3) GO TO 9  C C  TRY TO STOP EMPTY VEHICLE AT THE PREVIOUS TURNOUT  C Y1=Y1 + S Y2=Y2+V1*S/V2+S T5= Y2/V1 GO TO 10 C C  CALCULATE THE F-FACTOR  C 9  Y6= (T5-T3) *V1+LENG F9=Y6/S I F ( I . L T . 1.5) GO TO 6 D1=D2  C C  SUMATION OF THE F-FACTORS  C 6  TOTAL=TOTAL+F9  2  CONTINUE Z1=N  166 C C  CALCULATION OF THE AVERAGE F-FACTOR  C S3=TOTAL/Z1 1  WRITE(6,100) H(I1),S3  100  FORMAT (• • 5X,F6.1 5X,F8.4) r  WRITE (6, 101) RETURN END  f  167  I n t e r a c t i o n Between Two  The  purpose  of  F l e e t s Of  this  model i s t o determine the range of  t r a f f i c flow r a t e s f o r which the are v a l i d .  The c  of  F-factor  expected  F-factor  equations  b a s i c equation to be t e s t e d i s :  F = Pr(h>h )F, Most  Vehicles  + Pr(h<h )F c  a  the t e s t i n g of the model w i l l i n v o l v e the  exponential  equation.  The  simulation  formulated  such  model,  written  the  vehicle.  The  formulation  GPSSV,  has  been  t h a t i t i s r e s t r i c t e d to c e r t a i n combinations  of v e h i c l e speeds, turnout spacings, Generally,  in  and  acceleration  empty v e h i c l e must t r a v e l f a s t e r than the figures  on  of the s i m u l a t i o n  the  following  model. .  rates. loaded  pages i l l u s t r a t e  the  168 Time Parameters Of The Em_p_ty_ V e h i c l e s  Decision  Point  ,  4T  I  Turnout l o c a t i o n s  I  Dump I  Times*  K  I  I  I  S;  ,  j  ,  I  J  ,  _ „ ,  T  ,__  1  S-  S ,  S _  J+u  J+3  ;  ,Jtl  ;  &^)<-B^l—C-^-B^I I  | ^ _  i ^  D  A = ( 2aS-V|) 60/ (2aV ) 2  B = (V /a) 60 2  C = (Sa-V2) 6 0 / ( a \ ) D = (S/V ) 60 2  S. = t u r n o u t l o c a t i o n s  Decision  Point  •i  i — \ — i S  Turnout l o c a t i o n s  z  I I  Times*  , u  i S,  Destination '  1  K  I  E  a i  — F  • Distances  K  E = [ (2aS + Vf) /(2aV ) ]60 + 2  (3S/V ) 60 Z  F = [ (2aS+V2) /(2aV ) ]60 + (3S/V ) 60 2  •the times are i n minutes  Z  ^ i  4S  ^1  169 Time Parameters Of The Loaded  I  I---I  i  Turnout l o c a t i o n s  N  Times*  I  I  S- .  S .,  I  Dump I  Vehicles  SI  r  R  >|  S , ;  ;  J*i  J  J*3  R = (S/V ) 60 (  S.- = t u r n o u t l o c a t i o n s J  Decision ,  Point  ,  ,,.^ _..,__. T  Turnout l o c a t i o n s  S  '  2  I I  I I  S, I  I  r  ' '  KQ^Z-N^Y^r£  Times*  Q = [ (aS+V|)/(aV ) E  (S/V, ) ]60  X = [ (4S/V ) - (aS+Vf)/(aV ) ]60 (  2  Y = [ (aS+V|)/(aV ) a  - (S/V, ) ]60  Z = [ (2S/V, ) - (aS+V|)/(aV ) ]60 2  •times are i n minutes  , Destination I  I  170  Time parameters f o r the empty v e h i c l e s & = time t o t r a v e l from dump t o f i r s t B = time t o d e c e l e r a t e i n t o turnout  d e c i s i o n point  from d e c i s i o n p o i n t  C = time t o a c c e l e r a t e and t r a v e l a t constant  velocity to  next d e c i s i o n p o i n t D = travel  time  between  decision  points  at  constant  travel  at  constant  velocity E = t r a v e l time  to  accelerate  v e l o c i t y from f i r s t turnout F = time  to t r a v e l t o landing  decision  and  to l a n d i n g (at constant  v e l o c i t y ) from  point  Time parameters f o r loaded  vehicles  E = time t o t r a v e l from l a s t turnout t o dump Q = time from d e c i s i o n p o i n t t o turnout X = time from l a n d i n g to f i r s t at  constant  decision  velocity  y = same as Q Z = time from turnout  to d e c i s i o n point  point  travelling  171 Proqram L i g t i n q  $EUN *GPSSV SPRINT--00- PAE=SIZE=B SIMULATE  *  * TUENOUT SIMULATION *  MODEL BY LAEEY A. .HENKELMAN  *  GBADOATE STUDIES  *  FOBESTBY, UBC  *  NOVEMBEB, 1977  *  * TIME UNITS=1/1000 MINUTES * * STORAGE DEFINITIONS STORAGE  # TURNOUTS 8 MAX  # VEHICLES-2  S1-S40,2  * * VARIABLE DEFINITIONS 1  VARIABLE  PH1-1  V1 TAKES 1 FBOM PEESENT TURNOUT  * * FUNCTION DEFINITIONS EXPON FUNCTION  VEHICLE HEADWAY DISTRIBUTIONS  BN1,C24  0,0/. 1 , .104/. 2,. 222/. 3,.335/. 4,. 509/. 5,. 69/. 6,. 915 . 7 , 1. 2 / . 7 5 , 1.38/. 8, 1.6/.84, 1.8 3/. 88, 2. 12/. 9,2. 3  172 .92,2.52/. 94,2.81/. 95,2. 99/. 96 ,3.2/. 97, 3.5 .98,3.9/.99,4.6/.995,5.3/.998,6.2/.999,7/1,8 * EXPO  FUNCTION  RN2,C24  0,0/. 1,. 104/. 2,. 2 22/. 3,. 335/. 4,. 509/. 5, . 69/. 6, . 91 5 . 7, 1. 2 / . 7 5, 1.38/. 8, 1. 6/.84, 1. 83/. 88 , 2. 12/. 9,2.3 .92,2.52/.94,2.81/.95,2.99/.96,3.2/.97,3.5 . 98,3.9/. 99,4.6/.995,5.3/.998,6.2,. 999,7/1,8 * THE LETTERS IN COL 62 OF THE *  ADVANCE BLOCKS REFER TO THE  THE TIME PARAMETERS ON THE ACCOMPANYING FIGURE  *  ********* ***************************************************** * *  MODEL SEGMENT 1 - EMPTY TRUCKS  *  DDD  BBB  RMULT  31,743  RANDOM NUMBER GENERATOR SEED  GENERATE  1000,FN$EXPON  GENERATE AN EMPTY TRUCK  ASSIGN  1,40,PH  STARTING AT TURNOUT # 40  ADVANCE  389  GO TO 1ST. DECISION PT.  GATE LR  PH1,AAA  IF SHUT, USE TURNOUT, AAA  TEST NE  PH1,1,EEE  IF LAST TURNOUT GO TO EEE  GATE SNF  VI,AAA  NEXT FULL? PULL IN  ADVANCE  450  DOESN'T USE TURNOUT  ASSIGN  1-,1,PH  CONSIDER NEXT TURNOUT  TR ANSFER  ,DDD LOOP  NOW..  TRUCKS CANNOT PROCEED, MUST TURNOUT AAA  ADVANCE  121  DECELERATE INTO TURNOUT  ENTER  PH1 , 1  TRUCK ENTERS TURNOUT  173  CCC  LINK  PH1 FIFO  TRUCK QUEUES IN TURNOUT  LEAVE  PH1 , 1  EMPTY TRUCK LEAVES TURNOUT  ADVANCE  329  GOES TO NEXT DECISION PT..  ASSIGN  1-,1,PH  CONSIDER NEXT TURNOUT  TEST NE  PH1 ,0,FFF  IF LAST TURNOUT GO TO FFF  TRANSFER  , DDD  LOOP  r  * LAST TURNOUT, PROCEED TO LANDING. FFF  ADVANCE  1861  TRANSFER  ,GGG  EEE  ADVANCE  1861  GGG  TERMINATE  THE LAST TURNOUT  THE LAST TURNOUT  ******************************************** MODEL SEGMENT 2 - LOADED TRUCKS  * *  GENERATE  1000,FN$EXPO  GENERATE A LOADED VEHICLE  ASSIGN  1,3,PH  REMEMBER  LOGIC S  1  SHUT GATE 1 TO EMPTY TRUCKS  ADVANCE  1229  GO TO 1ST DECISION POINT  LOGIC S  2  SHUT GATE 2 TO EMPTY TRUCKS  ADVANCE  571  TRAVEL TO 1ST TURNOUT  LOGIC R  1  PASS TURNOUT 1, SO OPEN GATE  UNLINK  1 CCC ALL r  r  GATE 3  RELEASE ALL TRUCKS AT 1  * GENERALIZED TURNOUT HHH  ADVANCE  29 ADVANCE  TO THE NEXT DECISION PT.  TEST NE  PH1,41,III  IS THIS THE LAST TURNOUT ?  LOGIC S  PH1  CLOSE GATE REMEMBERED IN PH1  X  Y  174  * LAST III  ADVANCE  571  GO TO T H E  NEXT  LOGIC  V1  OPEN  AS P A S S I N G  UNLINK  V1,CCC,ALL  BELEASE  ASSIGN  1+r 1 rPH  THINK  TEANSFEE  ,HHH  E  GATE  LINEUP  AT  ABOUT N E X T  Q  TURNOUT  TURNOUT TURNOUT  LOOP  TURNOUT  BEFORE  ADVANCE  571  GO TO L A S T  LOGIC  40  OPEN  UNLINK  4 0 , C C C , ALL  RELEASE  ADVANCE  600  GO TO  R  TOENODT  DUMP  GATE  TURNOUT (#40) 40 AS  LINEUP  PASS AT  GATE  40  DUMP  TERMINATE  ************************************************************** *  MODEL SEGMENT  3  -  TIMING  * GENERATE TERMINATE  2000000 1  * ********* ************************** *************************** * CONTROL  CARDS  START END  1  175  APPENDIX 6  Some modified  THE LENGTH OF THE VEHICLE  of  the  equations  developed  i n t h i s t h e s i s can be  to i l l u s t r a t e the e f f e c t of the l e n g t h o f the v e h i c l e .  T h i s l e n g t h w i l l not  affect  the  time  lost  due  to  vehicle  a c c e l e r a t i o n and d e c e l e r a t i o n but i t w i l l a l t e r the expected factor  equations.  The simple  allow f o r the loaded vehicle.  F - f a c t o r equation  F-  i s adjusted t o  v e h i c l e t o t r a v e l one length of the loaded  Consequently,  the  simple  F-factor  equation  will  become:  A. 3 where : L = t h e l e n g t h o f the loaded v e h i c l e . In  the  case of a s i n g l e empty v e h i c l e meeting a f l e e t o f  loaded v e h i c l e s the c r i t i c a l headway i s empty  vehicle  l e n g t h of  the  headway w i l l h  CL  must  wait  loaded  increased,  since  the  the e x t r a delay time c r e a t e d by the  vehicle.  Consequently,  the  critical  become:  = [2Sa a A  0  (V, * \ ) +V, Vf (a +a ) +2La a \ ] / [2a a V, V ] A  0  A  D  fl  0  £  A.4  The  expected  F-factor  and  the  a c c o r d i n g l y modified.. The formulas it  probability  equations  can be f u r t h e r modified  are i f  i s assumed there i s a minimum gap between v e h i c l e s . . With  a  shifted  exponential  headway  modified gap d e n s i t y f u n c t i o n becomes:  distribution  the  176  t)  g (  -  ^[Pr(t<h)]  _ , = Q - Q < t -R --t-> 0  e  3t  3  T  3  1  where : R  T  = the minimum time gap between the  back  of  loaded v e h i c l e and the f r o n t o f the next  the  first  vehicle  R +L/V, = minimum headway between v e h i c l e s . . T  The  probability  that  a  headway  is  less  than the c r i t i c a l  headway i s : Pr(h<h ) = 1 - e ~  Q 3  CL  <V~*T-  -^J  = 1-e-^ ^c-RT>  A.5  (  and the expected F - f a c t o r f o r headways l e s s than  the  critical  headway i s : V, L/V, +R + 1 / C - e - 3 < K -R > (h +L/V, +1/Q ) q  3  T  F  2  =  £  T  t  3  A. 6 A  shifted  developed  Erlang (alpha=2)  headway  from a Pearson Type I I I  parameter  "a"  is  set  to  two  density  distribution i s function..  then the gap d e n s i t y  If  the  function  becomes: g(t) = bz ( t - c ) e -  b c t  -  c<t< 0 0  c >  where: c = R +L/V, T  b = 2 / (1/Q -R -L/V ) 3  T  (  The p r o b a b i l i t y t h a t the headway  is  less  than  the  critical  headway i s : P r ( h < h ) =j^ b ( h - c ) e CL  2  CL  =  and  the  1-e-  btlr  b(K  - >6h c  'c-^T>[b(h -R )+1] c  T  A^7  expected F - f a c t o r f o r headways l e s s than the c r i t i c a l  headway i s :  177  -b(h -R ) [  e F  c  T  b  (  .  n  |  . i ! ^  +E T h c +  VT  IrL) -21V  R  M  %  2 =  1-e'  b(ht  "  RT)  C b ( h - H ) +1 ] c  f  A.8  +  '  178 Simulation  Of The  F-factor  For  The  Shifted  Exponential  And  Pearson Type I I I Headway D i s t r i b u t i o n s  The loaded  while the loaded  simulation  exponential The  model  and  include  v e h i c l e i s passing  was  used to c o n f i r m  the empty  computer program w r i t t e n t o simulate  the  required  a f l e e t of loaded  e f f e c t o f the l e n g t h  expected F - f a c t o r  (Chapter 4).  vehicles  the  meeting of  an  was  modified  to  v e h i c l e on  the  of the loaded  This  version  of  v e h i c l e was  r e a c t i o n time was The  assumed to be  The  tables  below  tabulate  the  average  10000 repeated 9 times. . A t - t e s t was  utilized  hypothesis  that  the  F-factor for  the  r e j e c t e d at the one  the  expected  of  the shifted  percent  The  length  F-factor  sample  of  The  to t e s t the  F-factor  was  l e v e l of s i g n i f i c a n c e , 16.7  rejected  for  slightly  these  two  the  only  Pearson  more  expected  not  percent  the  the F - f a c t o r equation f o r the s h i f t e d e x p o n e n t i a l It can be concluded that  null  to  headway case was  equation  of  equation  l e v e l of s i g n i f i c a n c e and  F-factor  and  size  i s equivalent  exponential  Type III(a=2) headway case was  case.  a  F-factor  samples.  at a f i v e percent time.  vehicle  approximately 2 seconds.  with  rejected,  program  60 f e e t while the d r i v e r ' s  expected F - f a c t o r of the s i m u l a t i o n s  developed  the  a minimum headway of the l e n g t h of the loaded  the loaded  average  vehicle.  equations..  d i v i d e d by i t s v e l o c i t y p l u s a r e a c t i o n time.  than  the  or r e j e c t the s h i f t e d  Pearson Type I I I (a= 2) F - f a c t o r  empty v e h i c l e and  of  length o f  v e h i c l e t h a t w i l l cause the empty v e h i c l e t o remain i n a  turnout A  e f f e c t i v e length i s t h a t p o r t i o n of the  often headway  F-factor  179 equations  adequately  empty v e h i c l e  and  a  d e s c r i b e the i n t e r a c t i o n group  of  r e s p e c t i v e headway d i s t r i b u t i o n s . .  loaded  between a s i n g l e  vehicles,  for  their  180  T a b l e : For The E^f a c t o r S i m u l a t i o n Of The I n t e r a c t i o n Between A S i n g l e Empty V e h i c l e And A F l e e t Of Loaded Vehicles Based On Equations 2 . 18 . 2 . 1 9 , A. 5 . And"A. 6 Sample s i z e = 1 0 0 0 0 Number of samples = 9 A c c e l e r a t i o n = 1 9 7 5 9 mph " C o n f l i c t ' * hours = 11 S h i f t = (L/V, + 2 seconds) L = 6 0 feet S h i f t e d e x p o n e n t i a l headway d i s t r i b u t i o n 2  V e h i c l e Speed|Distance | T r a f f i c 1 -| Between | Flow Loaded|Empty JTurnouts | Rate (mph) |(mph) |(miles) 1 (vph)  40  |  30  |  1 I I I  1 | | |  1 30  10  |  1 40  |  |  |  1 1  1 1  | j 1 I |  | 1 Standard  0.2  20  | | | | j  0.3  0.1  |  1 1 j  Ave. | S.D. ! Expect|R e j e c t i o n of F |x 1 0 " 3 | F I | | |Hypothesis | | I I oL = 0 . 0 5 1  | I i I | |  1 2 4 10 60 100  | 1. 2 2 2 6 | | 1.22721 | 1.2330| | 1.24271 | 1. 2 4 5 3 | |1.18641  4.68| 1.22581 5 . 6 9 | 1. 22831 8 . 6 4 | 1. 2 3 3 0 | 7 . 3 9 | 1. 2 4 4 8 | 4 . 451 1. 2 4 9 8 | 8.68|1.1866!  No No No No Yes No  | | | | | |  1 2 4 10 60 100  | 0. 9 125 | |0.9186| |0.9173| |0.9302| |0.8794| I 0.77791  3.60|0.9150| 4.0710.9173! 4.66|0.9213! 5.58!0.9299| 5. 6 5 | 0 . 8810J 4. 14|0.7820|  No No Yes No No Yes  I i I I | |  1 2 4 10 60 100  |0.8624| |0.8635| JO.8673| |0.8719| |0.8378| |0.77421  2 . 721 0 . 2 . 84f 0 . 3 . 0 9 | 0. 3. 9 7 | 0 . 3. 23| 0. 3. 47! 0.  Yes No No No No No  8647| 8658| 8678| 87201 8400| 7747J  2 2  2  3  2  2  3  2  2  3  3  2  2  2  deviation  2  Hypothesis was  not r e j e c t e d  3  Hypothesis was not r e j e c t e d  f o r a 10% l e v e l o f s i g n i f i c a n c e f o r a 1% l e v e l of s i g n i f i c a n c e  181  Table For The F - f a c t o r Simulation Of The I n t e r a c t i o n Between A S i n g l e Empty V e h i c l e and A F l e e t Of Loaded V e h i c l e s Based On Equations 2 . 1 8 , 2 . 1 9 , A. 7 , And A ..8 Sample s i z e = 1 0 0 0 0 ' Number of samples = 9 A c c e l e r a t i o n = 1 9 7 5 9 mph " C o n f l i c t " hours = 11 S h i f t = (L/V, + 2 seconds) L = 60 feet Pearson Type I I I headway d i s t r i b u t i o n 2  ( V e h i c l e Speed|Distance T r a f f i c ( Ave..|S.D 1 Expect R e j e c t i o n Flow 1 T Between of 1 F |X 1 0 - 3 , I I Loaded|Empty (Turnouts Rate Hypothesis I ) (mph) j (mph) | (miles) (vph) oL = 0 . 0 5 r —H—• +— + r + H 1  r  F  I I  I  40  (  30  |  1 I I I 1 I I I 1 . 1  1  1  1  1  |  |  1 i 1 1 1  30  0. 2  1 1 1 1 1 1  40  (  0.3  1 1 1 1 1  1 2 4 10 60 100  | 1. 2 2 0 7 | J 1.2268| | 1. 2 2 5 9 | | 1.2330( | 1. 4 0 1 2 | |1.3909J  1 2 4 10 60 100  | 0.91001 4. | 0 . 9 1 2 2 | 5. I 0 . 9 151 | 3 . | 0 . 9 3 3 1 | 6. | 1.03101 5. |0.9097| 4.  6. 6. 8. 5. 6. 7.  33|1.2236 6 2 f 1. 2 2 4 2 29| 1 . 2 2 6 3 3811.2395 80|1.4057 6 2 ( 1- 3 7 3 1 I 44|0.9131 74|0.9140 84|0. 9172 09|0.9356 05| 1.0260 79|0.8727  No No No Yes No Yes 2  2  2  No No Noz Noz Yes Yes 2  1 Standard d e v i a t i o n 2  Hypothesis was not r e j e c t e d f o r a 10% l e v e l o f s i g n i f i c a n c e  182 APPENDIX-?  DERIVATIVES OF THE EXPECTED F-FACTOR EQUATIONS  There  are  several  derivatives  of the expected  equations and components of the equation that are the  optimization  listed 1.  Many  of  these  required  in  d e r i v a t i v e s are  on the f o l l o w i n g pages..  Critical  headway  h  A  = [2Sa a (V, V ) V  c  Sh  0  +  2  +  (  Vf(a +a ) ] / (2a,a V V, ) fl  D  D  i  V, +Vj,  t  ^>S 2.  routines..  F-factor  V V (  A. 9  2  Exponential F-factcr F  z  F =  = D/S  (i-e ^ c  ^V, +V  _  2  h  jD  V, +\  2 V,  (Q-, + 1)  dF  f  V, *\  ah  ~dS  L  2V  3s  2  dD  as  Q Q-^jhc D  c  2V (Q, + 1) 2  3h  D(i-e- ^c) Q  c  as  S2  Q3  (0^ + 1);  A'_10  D = v, u e - Q A ( - Q - h c - i j + i ] / [-Q  3  ]  as  -v, Q | ( | ^ ) e - ^ c e - ^ ( - Q h - i ) + i ]  v, Qf ( | ^ ) e - ^ h  as  [-Q, ( e - ^ - i ) ]  -Q-7 (e-9j c-i)  Q  3  2  c  Q  K  A. 11  c  (  183  3.  E r l a n g (alpha=2)  F-factor  V + V, \ F =  2 V,  D  j  £-*h  (1+Xh ) + - ( 1 - £ - ^ [ 1 + Ah ]) ( Q + 1 ) A  c  c  c  5  v, +v  a  [2V A= 2 Q  (Q +1)  2  3  3  dF  [i-e-  2>h' c  2V, /  D[1-e-  A l n  as  - (1 + A h ) ]  (i+Ahc) ]\  —  +  Bs  x h t  /  \  s  / D X 2 h e - M / coh^ X  c  c  \  S2  S  (Q,+1)  / \ 9S V  A. 12  [e- Xtl  D = V,  (-Ahf-2h -2/ ) + 2 A 1 / t 1-ec)h  2>D  c  A  c  Q-^a  [SS J/ \  A h  c (1 + A h ) ] c  (Ah| + 2 h + 2/ )-2/A  [1-e-  c  X  ^  A  (1+Ah ) c  v  (1-Ah )/ c  Simple F  =  F-factor including the vehicle's length  (V/  )/(2V  2  (dF /c}S ) = -L/S2 L  A. 13  ) + L/S A. 1 4  184  185 6. . Pearson Type III(a=2) F - f a c t o r Let a = 2 b = 2/(1/Q_-R -L/V, ) f  c = R +L/V f  Then  (  186 APPENDIX 8  OPTIMUM TURNOUT SPACING COMPUTER PROGRAM  Program  Variable  Variables  Function  A1  deceleration  A2  acceleration  B  b parameter of Pearson Type I I I d i s t r i b u t i o n  C  "cost  D1  number of " c o n f l i c t " hours per day  D2  number of operating  F1  coefficient  F2  simple F - f a c t o r i n c l u d i n g the v e h i c l e ' s  F3  expected F - f a c t o r  F4  prime o f the expected F - f a c t o r  F7  coefficient  H  t r a f f i c flow r a t e i n vph  K  a c c e l e r a t i o n i n mph  L  the  L1  the length  L2  minimum gap between v e h i c l e s i n seconds  L3  minimum gap between v e h i c l e s i n days  M  adjusted  P1,P2, P3  components o f t h e c o s t  OJ  expected u s e f u l l i f e  Q3  number of headways during  functions"  days per year  of f r i c t i o n length  of a c c e l e r a t i o n  2  length of t h e v e h i c l e i n f e e t of t h e v e h i c l e i n miles  hauling  c o s t i n d o l l a r s p e r hour function  of the road i n years the " c o n f l i c t " hours  187 R1-R0  components o f expected expected  F-factor  F-factor  S  turnout  spacing  V1  speed of loaded v e h i c l e i n mph  V2  speed o f empty v e h i c l e i n mph  Y  turnout  Z  prime of "cost  spacing function"  and  prime  of  188 Program L i s t i n g  50 EEM PROGRAM FOR THE CALCULATION OF OPTIMUM SPACING 100 REM PROGRAM DEVELOPED  SEPT..1978 BY D.I..ANDERSON  150 DIM Y (4) ,Z (4) ,C(4,3) ,A$ (82) H (6) ,S (4,3) r  200 A$ (1,40)="DISTRIBUTION 250 A$(41,80)="  OPT. SPACING S* + 50%  S* - 50% " S* + 200%"  300 MAT Z=ZER 350 DISP "INPUT THE VEHICLE'S LENGTH-FEET"; 400 INPUT L 450 L1=LZ5280 500 DISP "MINIMUM GAP BETWEEN TRUCKS—SEC. "; 550 INPUT L3 600 DISP "INPUT NUMBER OF CONFLICT HOURS PER 650 INPUT D1 700 L2=L3/3600/Dl 750 DISP "NUMBER OF WORKING DAYS PER YEAR"; 800 INPUT D2 850 DISP "INPUT COEFFICIENT OF FRICTION"; 900 INPUT F1 925 DISP "INPUT COEFFICIENT OF ACCELERATION"; 930 INPUT F7 950 A1=79036.363636*F1 960 A2=79036.363636*F7 1000 DISP "INPUT TRAFFIC FLOW IN VPH"; 1050 INPUT H 1100 DISP "LOADED VEHICLE'S SPEED IN  MPH";  1150 INPUT V1 1200 DISP "EMPTY VEHICLE'S SPEED IN  MPH";  DAY";  189 1250 IN POT V2 1300 DISP " ENTER COST PER TURNOUT (DOLLARS) n.i 1350 INPOT C 1400 DISP " TRUCK RENTAL RATE IN $/HR"; 1450 INPOT M 1500 DISP " EXPECTED LIFE OF THE ROAD IN YRS it. 1550 INPOT Q1 1600 PRINT LIN3"CALCULATION OF OPTIMUM SPACING AND COST FUNCTION =  1650 PRINT LIN 1"NUMBER OF HOURS PER DAY  "D1  1700 PRINT "NUMBER OF WORKING DAYS PER YEAR - "D2 "F1  1750 PRINT "COEFFICIENT OF FRICTION 1775 PRINT "COEFFICIENT OF ACCELERATION  =  «F7  1800 PRINT "THE LENGTH OF THE VEHICLE  =  "L"  1850 PRINT "VEHICLES PER HOUR  =  "H  1900 PRINT "MINIMUM GAP BETWEEN VEHICLES  =  "L3»  SECONDS"  1950 PRINT "VELOCITY OF LOADED VEHICLE  =  "V1 "  MPH"  2000 PRINT "VELOCITY OF EMPTY VEHICLE  =  it v2"  MPH"  2050 PRINT "COST PER TURNOUT  =  «C"  DOLLARS"  2100 PRINT "TRUCK RENTAL RATE  =  "M"  DOLLARS/HOUR"  2150 PRINT "EXPECTED LIFE OF THE ROAD  =  "Q1"  2200 Q3=H*D1-1 2250 FOR S1=1 TO 3 2300 S3=V1+V2 2350 K= (2*A1*A2)/(A1 + A2) 2400 Y (1) =0.00001 2450 Y (3) =10.5 2500 Y (2) = (Y (1) +Y (3) ) /2 2550 FOR 11=1 TO 3 2600 GOSOB S1 OF 7450,7450,6100  FEET"  YEARS"  190 2650 NEXT 11 2700 IF Z(1)*Z(3)<0 THEN 2900 2750 Y (1) =F6*0.7 2800 Y (3) =F6*1.3 2850 Y (2) = <Y (1) +Y (3) ) /2 2900 Y4=Y (1) 2950 Y5=Y (3) 3000 Y6=2 3050 11=2 3100 IF ABS (Y (1)-Y (3) ) <0. 0001 THEN 3700 3150 IF Z(1)*Z(2)>0 THEN 3450 3200 Y (3) =Y (2) 3250 Y (2) = (Y (3) +Y (1) ) /2 3300 Z (3) =Z (2) 3350 GOSUB S1 OF 7450,7450,6100 3400 GOTO 3100 3450 Y (1) =Y (2) 3500 Y (2) = (Y (3) +Y (1) ) /2 3550 Z (1) =Z (2) 3600 GOSOB S1 OF 7450,7450,6100 3650 GOTO 3100 3700 I F S1=3 THEN 4000 3750 IF S1=2 THEN 3900 3800 GOSOB 4950 3850 GOTO 3950 3900 GOSOB 4950 3950 NEXT S1 4000 GOSOB 4950 4050 STANDARD  4100 PRINT LIN3 4150 PRINT A$ 4200 PRINT LIN1 4250 PRINT "SIMPLE" 4300 PRINT " SPACING (FEET) "S (1, 1) ; S (2, 1) ; S (3,1) ; S (4,1) 4350 PRINT " COST ($/M/VEH) "C (1, 1) ;C (2, 1) ;C (3, 1) ; C (4, 1) 4400 PRINT " " 4450 PRINT "SHIFTED EXPONENTIAL" 4500 PRINT " SPACING (FEET) "S (1, 2) ; S (2, 2) ; S (3, 2) ; S (4 ,2) 4550 PRINT " COST ($/M/VEH) "C (1 , 2) ; C (2, 2) ;C (3, 2) ; C (4, 2) 460 0 PRINT " " 4650 PRINT "PEARSON TYPE I I I " 4700 PRINT " SPACING (FEET) "S (1 , 3) ; S (2, 3) ; S (3 3) ; S (4, 3) r  4750 PRINT " COST ($/M/VEH) "C (1 ,3) ; C (2,3) ;C (3, 3) ; C (4, 3) 4800 PRINT LIN2 4850 END 4900 REM  S*  4950 J=1 5000 S (J, S1) =Y (2) *5280 5050 C(J, S1)=C/Y(2) +Q1*H*M*D2*D1*P3/P1* (H/V1 + H/V2) 5100 J=2 5150 Y (2) =Y (2) *0.5 5200 REM  50% OF S*  5250 S ( J S1)=Y (2) *5280 r  5300 GOSOB S1 OF 7450,7450,6100 5350 C (J, S1) =C/Y (2) +Q1 *H*M*D2*D 1 *P3/P 1* (H/V1+H/V2) 5400 J=3 5450 Y (2) =Y (2) *3 5500 REM  1 5055 OF S*  192 5550 S (J, S1) =Y (2) *5280 5600 GOSOB S1 OF 7450,7450,6100 5650 C(J, S1)=C/Y(2) +Q1 *H*M*D2*D1*P3/P1* (H/V1 + H/V2) 5700 J=4 5750 Y (2) =Y (2) *4/3 5800 EEM  200% OF S*  5850 S (J, S1) =Y (2) *5280 5900 GOSOB S1 OF 74 50,7450,6100 5950 C (J, S1)=C/Y(2) +Q1*H*M*D2*D1*P3/P1* (H/V1+H/V2) 6000 EETLTBN 6050 STOP 6100 EEM SOLUTION OF 1VSX FOE THE PEARSON TYPE I I I DISTEIBUTION 6150 EEM CALCOLATION OF THE EXPECTED F-FACTOR 6200 R1=(2*Y (11) *A1*A2*S3+V1*V2t2* (A1 + A2) ) / (2*A 1*A2*V1*V2*D1) 6250 F2=S3/(2*V2)+L1/Y(I1) 6300 B=2/(1/Q3-L1-L2) 6350 R2=EXF(-B*(R1-L2) ) 6375  Q6=2*Bl-2/B-L1/V1/D1+L2  640 0 R5=R2*(B*(-RIt2-R1*L1/V1/D1+L2*R1+L1*L2/V1/D1)-Q6) 6450 R6=R2*(E* (R1-L2)+1) 6500 R3=V1*D1/Y (11) * (E5+L2 + L1/V1/D1+2/B) 6550 F3=(E3+E6*F2) *Q3/(H*D1) +F2/(H*D1) 6600 GOSOB 7000 6650 EEM  THE FIEST DEEIVATIVE OF THE COST EQUATION  6700 P 1 = 1 -H*Y (11) *F3/V1-H*V2/K 6750 P2=F3/V1 + Y (11) *F4/V1 6800 P3=V2/K+Y(11)*F3/V1 6825 Q7=H/V1+H/V2 6850 Z (11) =-C/Y (11) f 2+ (M*H*Q1*D2*D1) * (P2/P 1+H*P2*P3/P1t 2) *Q7  193 6900 RETURN 6950 STOP 7000 EEM CALCULAION OF THE PEIME OF THE F-FACTOE 7050  R4=S3/(V1*V2*D1)  7075 Q8=Q 3*V1/ (H* Y (11) t 2 7100 E7=-(E2*(B*(-Blt2+L2*E1)-2*E1-2/B+L2)+L2+L1/V1/D1+2/B)*Q8) 7150 E8=Q3*V1/ (H*Y (II) ) *R2*R4*Bt2* (Rlt2-L2*R1) 7 200 R9=Q3/(H»D1) *R4*R2*Bf2* (-E1+L2) *S3/(2*V2) 7250 R0=-L1/(H*D1*Y (11)^2) 7300 F4=R7+R8+R9+R0 7350 RETURN 740 0 STOP 7450 EEM EXPECTED F-FACTOE FOB SIMPLE AND SHIFTED EXPONENTIAL 7500 F2=S3/(2*V2)+L1/Y (II) 7550 IF S1=1 THEN 7950 760 0 B1 = (2*Y(I1) *A1*A2*S3+V1*V2f 2* (A1 + A2) ) /(2*A1*A2*V1*V2*D1) 7650 B2=EXP(-Q3*(E1-L2) ) 770 0 B3=( (L1/V1/D1+L2+ 1/Q3) -B2* (B1 + L 1/V 1/D1 + 1/Q3) ) *V 1*D 1/Y (11) 7750 F3=(B2*F2 + E3) *Q3/(H*D1)+F2/(H*D1) 7800 GOSOB 8350 7850 EEM CALCOLATION OF THE FIRST DERIVATIVE OF THE COST EQOATIO 7900 GOTO 8050 7950 F3=F2 8000 F4=-L1/Y (I1)t 2 8050 P1=1-H*Y(11)*F3/V1-H*V2/K 8100 P2=F3/V1 + Y (11) *F4/V1 8125 Q6=H/V1+H/V2 8150 P3=V2/K+Y(11)*F3/V1 820 0 Z (11) =-C/Y(I1) t2+ (M*H*Q1*D2*D1) * (P2/P1+H*P2*P3/P1t2) *Q6  8250 BETUEN 830 0 STOP 8350 EEM CALCULAION OF THE PEINE OF THE F-FACTOE 8400 B4 = S3/(V1*V2*D1) 8450 B5=B3/V1/D1*Y(I1) 8500 B6=-V1*Q3*E5/H/Y(11)f2 8550 B7=V1*Q3t2*B4*B2/H/Y ( I I ) * ( B U L 1/V 1/D 1) 8600 B8=Q3*B2/H/D1*(Q3*B4*F2+L1/Y(I1)|2) 8650 E9=L 1/(H*Dl*Y(I1)t2) 8700  F4=B6+B7-E8-B9  8750 EETOBN 8800 STOP  195  Operating I n s t r u c t i o n s  Once, the computer program has been loaded i n t o the memory of the c a l c u l a t o r the program i s s t a r t e d by p r e s s i n g "RUN"  then  "EXECUTE".  will  display examples.  Once t h i s has been various  questions..  accomplished See  the  the  screen  following  pages  for  196 RUN INPUT THE VEHICLE'S LENGTH-FEET?50 MINIMUM GAP EETWEEN TRUCKS-SEC.?2 INPUT NUMBER OF CONFLICT HOURS PER DAY?5 NUMBER OF WORKING DAYS PER YEAR7200 INPUT COEFFICIENT OF FRICTION?.25 INPUT COEFFICIENT OF ACCELERATION?.25 INPUT TRAFFIC FLOW IN VPH?1 LOADED VEHICLE'S SPEED IN MPH?15 EMPTY VEHICLE'S SPEED IN MPH?30 ENTER COST PER TURNOUT (DOLLARS)?250 TRUCK RENTAL RATE IN $/HR?15 EXPECTED LIFE OF THE ROAD IN YEARS? 10  CALCULATION OF THE OPTIMUM SPACING AND NUMBER OF HOURS PER DAY NUMBER OF WORKING DAYS PER YEAR COEFFICIENT OF FRICTION COEFFICIENT OF ACCELERATION THE LENGTH OF THE VEHICLE VEHCILE'S PER HOUR MINIMUM GAP BETWEEN VEHICLE •S VELOCITY OF LOADED VEHICLE VELOCITY OF EMPTY VEHICLE COST PER TURNOUT TRUCK RENTAL RATE EXPECTED LIFE OF THE ROAD  = = = = = =  'COST FUNCTION'  5 200 0. 25 0. 25 50 1 = 2 = 15 = 30 = 250 = 15 = 10  FEET SECONDS MPH MPH DOLLARS DOLLARS PER HOUR YEARS  S* - 50%  S* + 50%  S* + 200%  1478.2 1 11 39 . 1 6  4434.64 990.40  5912. 85 1 149.22  SHIFTED EXPONENTIAL SPACING(FEET) 2955. 58 COST (S/H/VEH) 913.20  1477.79 1 139.64  4433.37 990.77  5911.16 1 149.23  PEARSON TYPE I I I SPACING (FEET) 2951. 35 COST ($/M/VEH) 913,28  1475.67 1 140.41  4427.02 991.46  5902. 70 1151.90  DISTRIBUTION SIMPLE SPACING(FEET) COST ($/M/VEH)  OPT. SPACING 2956.42 912.73  EON INPOT THE VEHICLE'S LENGTH-FEET?50 MINIMUM GAP EETWEEN TBOCKS-SEC.?2 INPOT NOMBEE OF CONFLICT HOOES PEE DAY?5 NOMBEE OF WORKING DAYS PEE YEAB7200 INPOT COEFFICIENT OF FEICTION?.25 INPOT COEFFICIENT OF ACCELEBATION?.25 INPOT TBAFFIC FLOW IN VPH?4 LOADED VEHICLE'S SPEED IN MPH?20 EMPTY VEHICLE'S SPEED IN MPH?25 ENTEB COST PEE TOENOOT (DOLLABS)?100 TRUCK RENTAL BATE IN $/HB?15 EXPECTED LIFE OF THE EOAD IN YEABS? 20  CALCOLATION OF THE OPTIMOM SPACING AND NUMBEB OF HOUES PEB DAY NUMBER OF WOBKING DAYS PEE YEAE COEFFICIENT OF FBICTION COEFFICIENT OF ACCELEBATION THE LENGTH OF THE VEHICLE VEHCILE'S PER HOUE MINIMUM GAP BETWEEN VEHICLE •S VELOCITY OF LOADED VEHICLE VELOCITY OF EMPTY VEHICLE COST PEE TUENOUT TEUCK EENTAL BATE EXPECTED LIFE OF THE EOAD  = = = = = = = = = = = =  'COST FUNCTION'  5 200 0. 25 0. 25 50 4 2 20 25 100 15 20  FEET SECONDS MPH MPH DOLLAES DOLLAES PEB HOUB YEABS  S* - 50%  S* + 50%  S* + 200%  371.21 3582. 59  185.61 4298. 54  556.82 3824.07  742.42 4311.96  SHIFTED EXPONENTIAL SPACING(FEET) 369.94 COST ($/M/VEH) 3592.66  184.97 4311.46  554.91 3834.32  739.89 4322.90  PEABSON TYPE I I I SPACING(FEET) 370.79 COST ($/M/VEH) 3584. 39  185.39 4301. 40  556.18 3827.34  741.58 4318.89  DISTBIBUTION SIMPLE SPACING(FEET) COST ($/M/VEH)  OPT. SPACING  198 APPENDIX 9  GRAPHICAL RESULTS OF THE  S i m u l a t i o n models were developed  SENSITIVITY ANALYSIS  to determine  the  perturbations  to  the independent v a r i a b l e s , headway  distributions,  and  some on the assumptions have on the  F-factor, and  t o t a l expected  cost.  The  grouped  variable  and  assumptions.  turnout  d i f f e r e n c e between the a c t u a l and  values of the f u n c t i o n s were  delay time, optimum  according the The  were to  These  frequency expected spacing, perturbed  differences  the t r u e value of the independent  perturbed average,  calculated. .  the  effect  values  standard  value o f each group were determined.  or  according  d e v i a t i o n , and Graphs  of  to  the  the maximum  some  of  the  r e s u l t s of the s i m u l a t i o n s are l o c a t e d on the f o l l o w i n g pages..  270 + 2H3 +  THE EFFECT THAT PERTURBRTIDN5 TD THE VELOCITY DF THE EMPTY VEHICLE HAVE ON THE C05T FUNCTION PERTURBATION CMPH) AVERAGE MAXIMUM  2IE + IB9 + COST IE2 DIFFERENCE IN DOLLARS PER YEAR 135 PER MILE PER VEHI CLE 108 ai  +  SM  +  27 + 0 VELOCITY OF EMPTY VEHCILE CMPH)  TRAFFIC FLDM RATE (VPH)  TRAFFIC FLOW RATE (VPH)  EXPECTED USEFUL LIFE DF THE RDRD CYEHR5)  EXPECTED USEFUL L I F E UF THE RDP.D CYEP.F.5J  g  EFFECT THHT PERTURBP.T 10N5 TD THE TURNOUT CONSTRUCTION COST HAVE ON THE OPTIMUM TURNOUT SPACING  TURNOUT CONSTRUCTION COST C$)  TRUCK  HAUL INE  COST  CJ/HR)  TRUCK HAULING C05T U/HR)  -I o  EFFECT THAT PERTURBATIONS TO THE COEFFICIENT OF  0.1  0.15  0.2  0.25  COEFFICIENT OF FRICTION  0.3  0.  HH  THE EFFECT THAT THE DI5CDUNT RATE FOR VARIOUS TIME PERIODS HA5 ON THE COST FUNCTION  EXPECTED USEFUL LIFE DF THE ROAD C YEARS >  B0  -r  EFTECT THRT MR INTENRNCE COSTS FOR VARIOUS TIME PERIODS HAVE ON THE OPTIMUM TURNOUT SPACING  72 +  TURNOUT MAINTENANCE COST C$/YEAR) 12.S  EH +  RVERRGE MRXIMUM  -+-+-+-  EE 7.S DIFFERENCE HB + IN OPTIMUM TURNOUT H0 + 5PRCING 32 2.5  2H + IE + B 0  B.S S  10  IS  EXPECTED USEFUL LIFE OF THE RORD (YERR5)  20  B0 -r  EFFECT THAT MAINTENANCE C05T5 FDR VARIOUS TIME PERIODS HAVE ON THE OPTIMUM TURNOUT SPACING  72 + EH +  AVERAGE MAXIMUM  -+-+-+-  SE +  EXPECTED USEFUL LIFE DF THE ROAD CYEARS)  TURNOUT MAINTENANCE COST (•/YEAR)  EXPECTED USEFUL LIFE DF THE ROAD CYEHR5)  EXPECTED USEFUL LIFE OF THE RORD CYERR5)  EFFECT THAT THE PRIME DF THE F-FRCTDR FDR VRRIDU5 TRAFFIC FLDW RRTE5 HA5 ON THE OPTIMUM TURNOUT SPACING  TRAFFIC FLOW RRTE CVPH)  E F F E C T THBT T H E PRIME T R A F F I C FLOW R R T E 5  33  OF T H E F - F A C T D R HR5 ON THE C 0 5 T  FOR V R R I 0 U 5 FUNCTION  30  27  2H  +  RVERHEE MAXIMUM -+-+-+E = 5 H I F T E D E X P O N E N T I A L HEADWAY DISTRIBUTION P = PEARSON T Y P E I I I C A = 2 ) HEADWAY DISTRIBUTION  COST DIFFERENCE IN DDLLHR5 PER YEAR PER M I L E PER VEHICLE  H TRRFF1C  E FLOW RflTE  CVPH)  HERDWRY DISTRIBUTION  E F F E C T HEADWAY D I S T R I B U T I O N S FOR VARIOUS T R A F F I C FLOW R A T E S HRVE ON T H E OPTIMUM TURNOUT S P A C I N G  TRAFFIC  FLOW R A T E  CVPH)  HEADWAY DISTRIBUTION COMPARISON  TURNOUT 5PRCINE C MILE5 )  APPENDIX - 10  CONVERSION FACTORS  Imperial Units  1 foot  Units  0. 3048  metres  1  1 mile  =  1. 609344  kilometres  1 cun i t  =  2. 831685  metres  1 pound  =  0. 45359237 kilograms  1 ton  =  1. 016047  =  0. 09 290304 m e t r e s  1 foot  2  1 horsepower  1  SI (Metric)  =  745. 7  I n t e r n a t i o n a l System of U n i t s  3  tonne  watts  2  

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