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## Material Information- Title:
- Travel Time Estimation for Signalized Arterials Using Probabilistic Modeling
- Creator:
- CUI, XIAO (
*Author, Primary*) - Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Acceleration ( jstor )
Flow velocity ( jstor ) Modeling ( jstor ) Propagation delay ( jstor ) Sensitivity analysis ( jstor ) Signals ( jstor ) Speed ( jstor ) Traffic estimation ( jstor ) Travel time ( jstor ) Vehicles ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Xiao Cui. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 12/31/2008
- Resource Identifier:
- 658230404 ( OCLC )
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PAGE 1 1 TRAVEL TIME ESTIMATION FOR SIGNALI ZED ARTERIALS USING PROBABILISTIC MODELING By XIAO CUI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 PAGE 2 2 Copyright 2006 by Xiao Cui PAGE 3 3 To all who supported and helped me through the years in my research and my life PAGE 4 4 ACKNOWLEDGMENTS I would like to thank the chai r and members of my supervisory committee, especially my advisor, Dr. Lily Elefteriadou, for her helpful advice in my research and encouragement through the years. Without that, I should not have finished my study here . I would also like thank my parents for their endless love and support. They knew my dream and they stood behind me and supported me to finish my study in the US. They are great parents and I love them. Last, I also want to thank my boyfriend Lihe Wang for his support and encouragement, which motivated me to complete my study. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 LIST OF ABBREVIATIONS........................................................................................................19 ABSTRACT....................................................................................................................... ............22 CHAPTER 1 INTRODUCTION................................................................................................................. .24 1.1 Travel Time Estimation on Signalized Arterials...........................................................24 1.2 Objectives of the Research............................................................................................26 2 LITERATURE REVIEW.......................................................................................................27 2.1 Models for Estimating Arterial Travel Time................................................................27 2.1.1. Composite Analytical Models..........................................................................27 2.1.1.1 Intersection-delay based ar terial travel time models..........................28 2.1.1.2 Other arterial travel time models........................................................36 2.1.2 Arterial Travel Time Models Ba sed on In-Field Detection Technologies........41 2.2 Simulation Methods in Arterial Analysis......................................................................46 2.2.1 Delay Models in Simulation Packages..............................................................46 2.2.2 Estimating Delay and Travel Time Using Simulation......................................47 2.3 Summary of the Literature Review...............................................................................48 3 METHODOLOGY.................................................................................................................5 1 3.1 Simple Case: One Link between Two Intersections.....................................................51 3.1.1 Definition of a Link...........................................................................................52 3.1.2 Decomposition of Travel Time.........................................................................52 3.1.3 Estimation of MT..............................................................................................53 3.1.4 Estimation for QMT..........................................................................................57 3.1.5 Estimation for QWT...........................................................................................58 3.1.6 Link Travel Time for All Three Conditions......................................................59 3.1.7 Probability that a Certain Condition Occurs.....................................................59 3.2 General Case: An Arterial with Multiple Intersections.................................................61 3.2.1 The Flow Profile...............................................................................................61 3.2.1.1 Travel time estimation under non-congested conditions....................61 3.2.1.2 Travel time estimation under congested conditions...........................67 PAGE 6 6 3.2.2 Inputs and Outputs in the Model.......................................................................71 3.2.2.1 Inputs..................................................................................................71 3.3.2.2 Outputs................................................................................................72 3.3 An Example for Applying the Model............................................................................73 3.3.1 Inputs for the Model..........................................................................................73 3.3.2 Model Development..........................................................................................73 3.3.2.1 Step 1: Calculate the enteri ng flows at two intersections...................73 3.3.2.2 Step 2: Calculate the probabil ities for Conditions 1, 2, and 3............74 3.3.2.3 Step 3: Calculate the queue lengths and determine 1 _ link iVfor three conditions............................................................................................75 3.3.2.4 Step 4: Calculate the minimum a nd maximum travel time for each possible flow profile...........................................................................76 3.3.2.5 Step 5: Calculate tr avel time componentsMT,QMT ,andQWT, the total travel time for 9 flow profile s, the expected travel time, the standard deviation of travel time, and the travel time distribution.....76 4 DATA COLLECTION...........................................................................................................88 4.1 Travel Time Data Collection.........................................................................................88 4.2 Other Traffic Data Collection.......................................................................................89 5 MODEL VALIDATION........................................................................................................92 5.1 Simulate the Real Traffic Conditi on and Compare to the Field Data...........................92 5.1.1 Replicate the Real Traffic Conditions with Semi-Actuated Signal Timing......92 5.1.1.1 Site 1: Beaver_Pugh...........................................................................92 5.1.1.2 Site 2: Beaver_Sparks.........................................................................93 5.1.1.3 Site 3: Park..........................................................................................93 5.1.1.4 Site 4: Newberry rd.............................................................................94 5.1.2 Simulate the Travel Times and Compare with Filed Travel Times..................94 5.2 Compare the Model Estimation to the Simulation........................................................95 5.2.1 Replicate with Pre-timed Signal Timing...........................................................95 5.2.1.1 Site 1: Beaver_Pugh...........................................................................96 5.2.1.2 Site 2: Beaver_Sparks.........................................................................96 5.2.1.3 Site 3: Park..........................................................................................96 5.2.1.4 Site 4: Newberry Rd...........................................................................97 5.2.2 Comparisons and Conclusions..........................................................................97 6 SENSITIVITY ANALYSIS AND APPLICATION............................................................135 6.1 G/C Ratio.................................................................................................................. ..135 6.1.1 Site 1:Beaver_Pugh_AM /Case 1....................................................................135 6.1.2 Site 2:Beaver_Sparks_AM/Case 4..................................................................136 6.1.3 Site 3:Park_Mid_EB/Case 7...........................................................................136 6.1.4 Site 4: Newberry_4_30_EB/Case 11..............................................................136 PAGE 7 7 6.2 Link Length................................................................................................................ .136 6.2.1 Site 1: Beaver_Pugh_AM /Case 1..................................................................136 6.2.2 Site 2:Beaver_Sparks_AM/Case 4..................................................................137 6.2.3 Site 3:Park_Mid_EB/Case 7...........................................................................137 6.2.4 Site 4: Newberry_4_30_EB/Case 11..............................................................138 6.3 Maximum Operating Speed........................................................................................138 6.3.1 Site 1: Beaver_Pugh_AM /Case 1..................................................................138 6.3.2 Site 2:Beaver_Sparks_AM/Case 4..................................................................139 6.3.3 Site 3:Park_Mid_EB/Case 7...........................................................................139 6.3.4 Site 4: Newberry_4_30_EB/Case 11..............................................................140 6.4 Acceleration Rate / Deceleration Rate........................................................................140 6.4.1 Site 1:Beaver_Pugh_AM /Case 1....................................................................140 6.4.2 Site 2:Beaver_Sparks_AM/Case 4..................................................................141 6.4.3 Site 3:Park_Mid_EB/Case 7...........................................................................141 6.4.4 Site 4: Newberry_4_30_EB/Case 11..............................................................141 6.5 The Entering Flow Rate at Each Intersection.............................................................141 6.5.1 Site 1:Beaver_Pugh_AM /Case 1....................................................................141 6.5.2 Site 2:Beaver_Sparks_AM/Case 4..................................................................142 6.5.3 Site 3:Park_Mid_EB/Case 7...........................................................................142 6.5.4 Site 4: Newberry_4_30_EB/Case 11..............................................................143 6.6 Conclusions................................................................................................................ .143 6.6.1 g/C Ratio.........................................................................................................144 6.6.2 Link Length.....................................................................................................144 6.6.3 Maximum Operating Speed............................................................................144 6.6.4 Acceleration/ Deceleration Rate.....................................................................145 6.6.5 Entering Flow Rate at Each Intersection.........................................................145 7 CONCLUSIONS AND FU TURE RESEARCH..................................................................160 7.1 Overview of the Analytical Model..............................................................................160 7.2 Results from the Analytical Model.............................................................................160 7.3 Suggestions and Future Research...............................................................................161 7.3.1 Delay Caused by Driveways...........................................................................161 7.3.2 Actuated or Semi-Actuated Signal Timing.....................................................162 7.3.3 Travel Time Estimations for Other O/D Pairs................................................162 7.3.4 Travel Time Estimation for Spill Back Cases.................................................162 APPENDIX. SENSITIVITY ANALYSIS..................................................................................163 LIST OF REFERENCES............................................................................................................. 192 BIOGRAPHICAL SKETCH.......................................................................................................195 PAGE 8 8 LIST OF TABLES Table page 3-1 Inputs and outputs for trav el time estimation model.........................................................86 3-2 Probabilities for three cond itions at intersection 1 and 2...................................................86 3-3 Travel times, the associated probabili ties, and the expected travel times.........................87 4-1 Number of samples collected for tr avel time data from Newberry rd...............................90 4-2 Number of samples for travel time da ta for Beaver_Pugh, Beaver_Sparks, and Park......91 5-1 Tolerances and the acc eptable travel times......................................................................121 5-2 Simulated travel time comp ared to field travel time........................................................121 5-3 Travel time comparisons for Case 1 (Beaver_Pugh_AM_EB)........................................122 5-4 Travel time comparisons for Case 3 (Beaver_Pugh_PM_EB)........................................123 5-5 Travel time comparisons for Case 4 (Beaver_Sparks_AM_EB).....................................124 5-6 Travel time comparisons for Case 5 (Beaver_Sparks_Mid_EB).....................................125 5-7 Travel time comparisons for Case 6 (Beaver_Sparks_PM_EB)......................................126 5-8 Travel time comparisons for Case 7 (Park_Mid_EB)......................................................127 5-9 Travel time comparisons for Case 8 (Park_Mid_WB)....................................................128 5-10 Travel time comparisons for Case 9 (Park_PM_EB)......................................................129 5-11 Travel time comparisons for Case 10 (Park_PM_WB)...................................................130 5-12 Travel time comparisons for Case 11 (Newberry_4_30_EB)..........................................131 5-13 Travel time comparisons for Case 12 (Newberry_4_30_WB)........................................132 5-14 Travel time comparisons for Case 13 (Newberry_5_1_EB)............................................133 5-15 Travel time comparisons for Case 14 (Newberry_5_1_WB)..........................................134 PAGE 9 9 LIST OF FIGURES Figure page 2-1 Shockwave analysis......................................................................................................... ..50 2-2 Deterministic queuing analysis..........................................................................................50 3-1 Definition of a link....................................................................................................... ......77 3-2 Decomposition of travel time.............................................................................................77 3-3 The three conditions....................................................................................................... ....78 3-4 Link length................................................................................................................ .........78 3-5 MT Equation for Condition 1.............................................................................................79 3-6 MTEquation for Condition 2..............................................................................................79 3-7 MT Equation for Condition 3.............................................................................................79 3-8 Shockwave analysis to find th e interval lengths for conditions.........................................80 3-9 Probabilities of changing from one condition to another condition..................................80 3-10 Study area................................................................................................................ ...........80 3-11 0 _link qDand 1 _link qD............................................................................................................81 3-12 The trajectories for the nine flow profiles..........................................................................82 3-13 Time and space diagram.................................................................................................... 83 3-14 Residual queue due to Condition 2 is terminated..............................................................83 3-15 Increasing residual queue due to terminated Condition 2..................................................84 3-16 Residual queue calculation wh en Condition 2 is terminated.............................................84 3-17 Average queue length under congested conditions............................................................84 3-18 The discharged queue...................................................................................................... ..85 3-19 Flow at downstream intersection.......................................................................................85 3-20 Sketch for signalized arterial............................................................................................ .85 PAGE 10 10 3-21 Flow-density-speed relationship........................................................................................85 3-22 Travel time distribut ion for the example...........................................................................86 4-1 Sketch of site at Newberry Rd...........................................................................................90 4-2 Sketch for site Beaver_ Pugh.............................................................................................90 4-3 Sketch for site Beaver_ Sparks..........................................................................................90 4-4 Sketch for site Park....................................................................................................... .....90 5-1 Semi-actuated signal timing for Case 1 (Beaver_Pugh_AM_EB)....................................99 5-2 Semi-actuated signal timing for Case 2 (Beaver_Pugh_Mid_EB)....................................99 5-3 Semi-actuated signal timing for Case 3 (Beaver_Pugh_PM_EB)...................................100 5-4 Semi-actuated signal timing fo r Case 4 (Beaver_Sparks_AM_EB)................................100 5-5 Semi-actuated signal timing fo r Case 5 (Beaver_Sparks_Mid_EB)................................101 5-6 Semi-actuated signal timing fo r Case 6 (Beaver_Sparks_PM_EB)................................101 5-7 Semi-actuated signal timing for Case 7 (Park_Mid_EB) and Case 8 (Park_Mid_WB)..102 5-8 Semi-actuated signal timing for Case 9 (Park_PM_EB) and Case 10 (Park_PM_WB)..103 5-9 Semi-actuated signal timing fo r Case 11 (Newberry_4_30_EB) Case 12 (Newberry_4_30_WB) Case 13 (Newberr y_5_1_EB) Cases 14(Newberry_5_1_WB).104 5-10 Approximate A.M. signal timing for Case 1 (Beaver_Pugh_AM_EB)...........................105 5-11 Approximate P.M. signal timing for Case 3 (Beaver_Pugh_PM_EB)............................106 5-12 Approximate A.M. signal timing for Case 4 (Beaver_Sparks_AM_EB)........................106 5-13 Approximate Mid signal timing fo r Case 5 (Beaver_Sparks_Mid_EB)..........................107 5-14 Approximate P.M. signal timing for Case 6 (Beaver_Sparks_PM_EB).........................107 5-15 Approximate signal timing for Case 7 (Park_Mid_EB) Case 8 (Park_Mid_WB)..........108 5-16 Approximate signal timing for Case 9 (Park_PM_EB) and Case 10 (Park_PM_WB)...109 5-17 Approximate signal timing for Case 11 (Newberry_4_30_EB) and Case 13 (Newberry_5_1_EB)........................................................................................................110 PAGE 11 11 5-18 Approximate signal timing for Case 12 (Newberry_4_30_WB) and Case 14 (Newberry_5_1_WB)......................................................................................................111 5-19 Travel Time Distribution for Beaver_Pugh_AM (offset = 0, 10, 20, 30, 80, 90)............112 5-20 Travel Time Distribution for Beaver_Pugh_AM (offset = 40)........................................112 5-21 Travel Time Distribution for Beaver_Pugh_AM (offset = 50, 60, 70)............................113 5-22 Travel time comparisons for Beaver_Pugh_AM_EB (case 1)........................................113 5-23 Travel time comparisons for Beaver_Pugh_PM_EB (case 3).........................................113 5-24 Travel time comparisons for Beaver_Sparks_AM_EB (case 4)......................................114 5-25 Travel time comparisons for Beaver_Sparks_Mid_EB (case 5)......................................114 5-26 Travel time comparisons for Beaver_Sparks_PM_EB (case 6)......................................114 5-27 Travel time comparisons for Park_Mid_EB (case 7)......................................................115 5-28 Travel time comparisons for Park_Mid_WB (case 8).....................................................115 5-29 Travel time comparisons for Park_PM_EB (case 9).......................................................115 5-30 Travel time comparisons for Park_PM_WB (case 10)....................................................116 5-31 Travel time comparisons for Newberry_4_30_EB (case 11)..........................................116 5-32 Travel time comparisons for Newberry_4_30_WB (case 12).........................................116 5-33 Travel time comparisons for Newberry_5_1_EB (case 13)............................................117 5-34 Travel time comparisons for Newberry_5_1_WB (case 14)...........................................117 5-35 Queue length calculation for every cycle for Case 10 (Park_PM_WB)..........................118 5-36 Link length and driveways...............................................................................................12 0 6-1 Sensitivity analysis: g/C ratio versus travel time(Beaver_Pugh_AM)............................145 6-2 Sensitivity analysis: g/C ratio vers us travel time (Beaver_Sparks_AM).........................146 6-3 Sensitivity analysis: g/C ratio ve rsus travel time (Park_Mid_EB)..................................146 6-4 Sensitivity analysis g/C ratio versus travel time (Newberry_4_30_EB).........................146 6-5 Sensitivity analysis: link length versus travel time with adjusted offset (Beaver_Pugh_AM).........................................................................................................147 PAGE 12 12 6-6 Sensitivity analysis: link length versus travel time with unadjusted offset(Beaver_Pugh_AM)................................................................................................147 6-7 Sensitivity analysis: link length versus travel time with adjusted offset (Beaver_Sparks_AM)......................................................................................................147 6-8 Sensitivity analysis: link length versus travel time with unadjusted offset (Beaver_Sparks_AM)......................................................................................................148 6-9 Sensitivity analysis: link length versus trav el time with unadjusted offset for link 0 (Park_Mid_EB)................................................................................................................14 8 6-10 Sensitivity analysis: link length versus trav el time with unadjusted offset for link 1 (Park_Mid_EB)................................................................................................................14 8 6-11 Sensitivity analysis: link length ratio vers us travel time with unadjusted offset for link 0 (Newberry_4_30_EB)............................................................................................149 6-12 Sensitivity analysis: link length versus trav el time with unadjusted offset for link 1 (Newberry_4_30_EB)......................................................................................................149 6-13 Sensitivity analysis: link length versus trav el time with unadjusted offset for link 2 (Newberry_4_30_EB)......................................................................................................149 6-14 Sensitivity analysis: maximum operating sp eed versus travel time with adjusted offset (Beaver_Pugh_AM)...............................................................................................150 6-15 Sensitivity analysis: maximum operating sp eed versus travel time with unadjusted offset (Beaver_Pugh_AM)...............................................................................................150 6-16 Sensitivity analysis: maximum operating sp eed versus travel time with adjusted offset (Beaver_Sparks_AM)............................................................................................150 6-17 Sensitivity analysis: maximum operating sp eed versus travel time with unadjusted offset (Beaver_Sparks_AM)............................................................................................151 6-18 Sensitivity analysis: maximum operating sp eed versus travel time with unadjusted offset for link 0 (Park_Mid_EB)......................................................................................151 6-19 Sensitivity analysis: maximum operating sp eed versus travel time with unadjusted offset (Park_Mid_EB)......................................................................................................152 6-20 Sensitivity analysis: maximum operating sp eed versus travel time with unadjusted offset for link 0 (Newberry_4_30_EB)............................................................................152 6-21 Sensitivity analysis: maximum operating sp eed versus travel time with unadjusted offset for link 1 (Newberry_4_30_EB)............................................................................152 PAGE 13 13 6-22 Sensitivity analysis: maximum operating sp eed versus travel time with unadjusted offset for link 2 (Newberry_4_30_EB)............................................................................153 6-23 Sensitivity analysis: acceleration rate versus travel time (Beaver_Pugh_AM)...............153 6-24 Sensitivity analysis: deceleration rate versus travel time (Beaver_Pugh_AM)...............153 6-25 Sensitivity analysis: acceleration rate versus travel time (Beaver_Sparks_AM)............153 6-26 Sensitivity analysis: deceleration rate versus travel time (Beaver_Sparks_AM)............154 6-27 Sensitivity analysis: acceleration ra te versus travel time (Park_Mid_EB)......................154 6-28 Sensitivity analysis: deceleration rate versus travel time (Park_Mid_EB)......................154 6-29 Sensitivity analysis: acceleration rate versus travel time (Newberry_4_30_EB)............155 6-30 Sensitivity analysis: deceleration rate versus travel time (Newberry_4_30_EB)............155 6-31 Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Pugh_AM).......................................................................................................155 6-32 Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Pugh_AM).......................................................................................................156 6-33 Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Sparks_AM)....................................................................................................156 6-34 Travel time changes due to flow changes........................................................................156 6-35 Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Sparks_AM)....................................................................................................157 6-36 Sensitivity analysis: flow rate versus tr avel time for intersection 1(Park_Mid_EB).......157 6-37 Sensitivity analysis: flow rate versus tr avel time for intersection 2(Park_Mid_EB).......157 6-38 Sensitivity analysis: flow rate versus tr avel time for intersection 3(Park_Mid_EB).......158 6-39 Sensitivity analysis: flow rate versus travel time for intersection 1(Newberry_4_30_EB)....................................................................................................158 6-40 Sensitivity analysis: flow rate versus travel time for intersection 3(Newberry_4_30_EB)....................................................................................................158 6-41 Sensitivity analysis: flow rate versus travel time for intersection 4(Newberry_4_30_EB)....................................................................................................159 PAGE 14 14 6-42 Sensitivity analysis: flow rate versus travel time for intersection 5(Newberry_4_30_EB)....................................................................................................159 A-1 Sensitivity analysis: g/C ratio ve rsus travel time (Beaver_Pugh_PM)............................163 A-2 Sensitivity analysis: g/C ratio vers us travel time (Beaver_Sparks_Mid)........................163 A-3 Sensitivity Analysis: g/C ratio versus travel time (Beaver_Sparks_PM)........................163 A-4 Sensitivity analysis: g/C ratio ve rsus travel time (Park_Mid_WB).................................164 A-5 Sensitivity analysis: g/C ratio ve rsus travel time (Park_PM_EB)...................................164 A-6 Sensitivity analysis: g/C ratio ve rsus travel time (Park_PM_WB)..................................164 A-7 Sensitivity analysis: g/C ratio ve rsus travel time (Newberry_4_30_WB).......................165 A-8 Sensitivity analysis: g/C ratio ve rsus travel time (Newberry_5_1_EB)..........................165 A-9 Sensitivity analysis: g/C ratio ve rsus travel time (Newberry_5_1_WB).........................165 A-10 Sensitivity analysis: link length versus travel time with adjusted offset (Beaver_Pugh_PM)..........................................................................................................166 A-11 Sensitivity analysis: link length versus travel time with unadjusted offset (Beaver_Pugh_PM)..........................................................................................................166 A-12 Sensitivity analysis: link length versus travel time with adjusted offset (Beaver_Sparks_Mid)......................................................................................................166 A-13.Sensitivity analysis: link length vers us travel time with unadjusted offset (Beaver_Sparks_Mid)......................................................................................................167 A-14 Sensitivity analysis: link length versus travel time with adjusted offset (Beaver_Sparks_PM).......................................................................................................167 A-15 Sensitivity analysis: link length versus travel time with unadjusted offset (Beaver_Sparks_PM).......................................................................................................167 A-16 Sensitivity analysis: link length versus travel time with unadjusted offset (Park_Mid_WB)..............................................................................................................168 A-17 Sensitivity analysis: link length versus travel time with unadjusted offset (Park_Mid_WB)..............................................................................................................168 A-18 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Park_PM_EB).................................................................................................................16 8 PAGE 15 15 A-19 Sensitivity analysis: link length versus travel time with unadjusted offset (Park_PM_EB).................................................................................................................16 9 A-20 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................169 A-21 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................169 A-22 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................170 A-23 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................170 A-24 Sensitivity analysis: link length versus travel time with unadjusted offset (Newberry_5_1_EB)........................................................................................................170 A-25 Sensitivity analysis: link length versus travel time with unadjusted offset (Newberry_5_1_EB)........................................................................................................171 A-26 Sensitivity analysis: link length versus travel time with unadjusted offset (Newberry_5_1_EB)........................................................................................................171 A-27 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................171 A-28 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................172 A-29 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................172 A-30 Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................172 A-31 Sensitivity analysis: MOS versus travel time with adjusted offset (Beaver_Pugh_PM).173 A-32 Sensitivity analysis: MOS versus travel time with unadjusted offset (Beaver_Pugh_PM)..........................................................................................................173 A-33 Sensitivity analysis: MOS versus travel time with adjusted offset (Beaver_Sparks_Mid)......................................................................................................173 A-34 Sensitivity analysis: MOS versus travel time with unadjusted offset (Beaver_Sparks_Mid)......................................................................................................174 A-35 Sensitivity analysis: MOS versus travel time with adjusted offset (Beaver_Sparks_PM).......................................................................................................174 PAGE 16 16 A-36 Sensitivity Analysis: MOS versus tr avel time with unadjusted offset (Beaver_Sparks_PM).......................................................................................................174 A-37 Sensitivity analysis: MOS versus travel time with unadjusted offset (Park_Mid_WB)..175 A-38 Sensitivity analysis: MOS versus trav el time with unadjusted (Park_Mid_WB)............175 A-39 Sensitivity analysis: MOS versus travel time with unadjusted offset (Park_PM_EB)....175 A-40 Sensitivity analysis: MOS versus travel time with unadjusted offset (Park_PM_EB)....176 A-41 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................176 A-42 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................176 A-43 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................177 A-44 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB)....................................................................................................177 A-45 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_EB)........................................................................................................177 A-46 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_EB)........................................................................................................178 A-47 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_EB)........................................................................................................178 A-48 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................178 A-49 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................179 A-50 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................179 A-51 Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB)......................................................................................................179 A-52 Sensitivity analysis: acceleration rate versus travel time (Beaver_Pugh_PM)................180 A-53 Sensitivity analysis: deceleration rate versus travel time (Beaver_Pugh_PM)...............180 A-54 Sensitivity analysis: acceleration rate versus travel time (Beaver_Sparks_Mid)............180 PAGE 17 17 A-55 Sensitivity analysis: deceleration rate versus travel time (Beaver_Sparks_Mid)............181 A-56 Sensitivity analysis: acceleration rate versus travel time (Beaver_Sparks_PM).............181 A-57 Sensitivity analysis: deceleration rate versus travel time (Beaver_Sparks_PM).............181 A-58 Sensitivity analysis: acceleration ra te versus travel time (Park_Mid_WB).....................182 A-59 Sensitivity analysis: deceleration rate versus travel time (Park_Mid_WB)....................182 A-60 Sensitivity Analysis: acceleration rate versus travel time (Park_PM_EB)......................182 A-61 Sensitivity Analysis: deceleration rate versus Travel time (Park_PM_EB)....................183 A-62 Sensitivity analysis: acceleration rate versus travel time (Newberry_4_30_WB)...........183 A-63 Sensitivity analysis: deceleration rate versus travel time (Newberry_4_30_WB)..........183 A-64 Sensitivity analysis: acceleration ra te versus travel time (Newberry_5_1_EB)..............184 A-65 Sensitivity analysis: deceleration rate versus travel time (Newberry_5_1_EB)..............184 A-66 Sensitivity analysis: acceleration ra te versus travel time (Newberry_5_1_WB).............184 A-67 Sensitivity analysis: deceleration rate versus travel time (Newberry_5_1_WB)............185 A-68 Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Pugh_PM)........................................................................................................185 A-69 Sensitivity Analysis: flow rate versus travel time for intersection 2(Beaver_Pugh_PM)........................................................................................................185 A-70 Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Sparks_Mid)....................................................................................................186 A-71 Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Sparks_Mid)....................................................................................................186 A-72 Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Sparks_PM).....................................................................................................186 A-73 Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Sparks_PM).....................................................................................................187 A-74 Sensitivity analysis: flow rate versus tr avel time for intersection 3 (Park_Mid_WB)....187 A-75 Sensitivity analysis: flow rate versus tr avel time for intersection 2 (Park_Mid_WB)....187 A-76 Sensitivity analysis: flow rate versus tr avel Time for intersection 1 (Park_Mid_WB)...188 PAGE 18 18 A-77 Sensitivity analysis: flow rate versus tr avel time for intersection 1 (Park_PM_EB)......188 A-78 Sensitivity analysis: flow rate versus tr avel time for intersection 2 (Park_PM_EB)......188 A-79 Sensitivity analysis: flow rate versus tr avel time for intersection 3 (Park_PM_EB)......189 A-80 Sensitivity analysis: flow rate versus travel time for intersection 3 (Park_PM_WB).....189 A-81 Sensitivity analysis: flow rate versus travel time for intersection 5 (Newberry_4_30_WB)....................................................................................................189 A-82 Sensitivity analysis: flow rate versus travel time for intersection 4 (Newberry_4_30_WB)....................................................................................................190 A-83 Sensitivity analysis: flow rate versus travel time for intersection 3 (Newberry_4_30_WB)....................................................................................................190 A-84 Sensitivity analysis: flow rate versus travel time for intersection 2 (Newberry_4_30_WB)....................................................................................................190 A-85 Sensitivity analysis: flow rate versus travel time for intersection 1 (Newberry_4_30_WB)....................................................................................................191 PAGE 19 19 LIST OF ABBREVIATIONS ATT Arterial Travel Time, total travel time on the arterial MT Time in Motion, time period when the vehicl es travel before they join the queue QT Time in Queue, time period from when th e vehicles join the queue to when they leave the queue QWT Waiting Time in the Queue, time that vehicles have to wait in the queue QMT Moving Time in the Queue, time that vehi cles move when they are in the queue ABw Shockwave speed between state A and B BCw Shockwave speed between state B and C ACw Shockwave speed between state A and C Aq Arrival rate (actual flow rate) Cq At capacity flow rate Ak Density at arrival rate, can be c onverted from flow divided by speed Bk Jam density Ck Density at capacity C Cycle length r Effective red time s Total length of the link (excl ude the length of intersection) As Accelerating distance Cs Constant speed distance Ds The distance from the end point of S2 till the end of the link qD Distance over which the queue extends PAGE 20 20 1 _link iV VehiclesÂ’ entering speed (initial speed) when they enter link 1 mV VehiclesÂ’ maximum operating speed when they travel on the link aa VehiclesÂ’ acceleration rate da VehiclesÂ’ deceleration rate 1 _ link qD Distance over which the queue extends at link 1 1 _ link dV Discharging speed at link 1 1 _.link qD avgAverage distance over the queue extends at link 1 MQ Maximum queue length 0 _ link fV Leaving speed (final speed) for link 0 0 _ link qD Distance over which the queue extends at link 0 0 _ link dV The discharging speed from link 0 RQ The residual queue MQ The maximum queue 2condDiff The difference between the comple ted Condition 2 and the terminated Condition 2 1 _link if Entering (initial) flow rate at link 1 0 _link ff Final flow rate at link 0 WRf_ 1 int Flow rate at intersection 1 fr om westbound right turning movement WLf_ 1 int Flow rate at intersection 1 fr om westbound left turning movement NRf_ 1 int Flow rate at intersection 1 fr om southbound right turning movement SLf_ 1 int Flow rate at intersection 1 fr om northbound left turning movement PAGE 21 21 2 _link if Entering (initial) flow rate at link 2 1 _ link ff Final flow rate at link 1 WRf_ 2 int Flow rate at intersection 2 fr om westbound right turning movement WLf_ 2 int Flow rate at intersection 2 fr om westbound left turning movement NRf_ 2 int Flow rate at intersection 2 fr om southbound right turning movement SLf_ 2 int Flow rate at intersection 2 fr om northbound left turning movement PAGE 22 22 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TRAVEL TIME ESTIMATION FOR SIGNALI ZED ARTERIALS USING PROBABILISTIC MODELING By Xiao Cui December 2006 Chair: Lily Elefteriadou Major: Civil Engineering Travel time is an important performance meas ure for many transportation facilities in the sense that it is essential information in esta blishing Advanced Traveler Information System (ATIS), it is easy to communicate wi th travelers, and it can be used by operators to evaluate the network performance. However, most of travel time studies focus on freeways; there are only a few studies on arterial tr avel time. The objective of this re search is to develop a model for estimating travel time for signalized arterial s using probabilistic modeling. The dissertation reviews the literatu re related to arterial travel time and presents the methodology developed for estimating arterial travel time. In the methodology, travel time is defined as the sum of time spent in motion and time spent in queue. Time sp ent in queue can be further divided into moving time in queue and waiting time in queue. Vehicles are assumed to arrive at an intersection with three different conditions according to different traffic signal status; each condition has a certain probability of occurrence. The method estimates the travel time as a function of the signal status, and then considers all possible options to estimate the expected arterial travel time. To validate this model, field travel times were collected at 4 sites from State College, PA, and Gainesville, FL, during several time periods. There are a total of 14 cases with each of them representing a certain time period in a certain site. The an alytical model was app lied and validated and PAGE 23 23 sensitivity analysis was performed for each case to test how sensitive the analytical model is to several key factors. The analytical model provides travel ti me which is based on the green time, cycle length, link length, maximum operating sp eed, offset, acceleration/ deceleration rate, and the entering flow rate at each intersection. The model does not consider delay caused by driveways, and it does not analyze actuated and semi-actuated signal timing. It also does not provide travel time estimation fo r O/D pairs other than those for the through arterial movements. PAGE 24 24 CHAPTER 1 INTRODUCTION Arterials are major roadways, typically in urban areas, that serv e through traffic and provide access to surrounding prope rties. Arterials can be one-w ay or two-way streets, and intersections along the arterial ca n be un-signalized (including f our-way stop, two-way stop) or signalized. The focus of this research is on travel times for the signalized arterial. Travel time estimation depends heavily on the signal settings. Thus, the types of signals are briefly discussed here: there are three types of signal control: pre-timed, actuated, and adaptive. In pre-timed control the green interval is fixed. There are two types of actuated control, one is fully-actuated and the other is semi-actuated. In the fully actua ted, the green interval changes from cycle to cycle in accordance with traffic flow on both majo r and side streets, a nd there are detectors on the major and the minor streets which determine the traffic demand. In semi-actuated control, normally there are detectors on the side streets only, and the signal adju sts the green interval according to the demand on the major street only. Th e signal gives green time to side streets only when traffic is detected on the side streets. Like the pre-timed signal control, semi-actuated control allows for coordination al ong the arterial. Vehicles trave ling on the arterial have to stop or travel through each intersection. This depe nds on the signal timing and the time when one particular vehicle arrives at th e intersection. Therefore, vehicl e travel time can be affected dramatically by signal timing. 1.1 Travel Time Estimation on Signalized Arterials Travel time is very important since it is c onsidered a promising performance measure in evaluating traffic operations on various transpor tation facilities. Travel time estimation is essential, especially for new facilities where travel time ca nnot be measured. While several PAGE 25 25 research projects such as the Freeway Travel Time Reliability project funded by the National Science Foundation (NSF) (Elefteria dou et al, 2004) have started or have already been completed with focus on freeway travel time, few of them focus on arterial travel time. The arterial, however, is an essential transpor tation facility and accurate travel time estimation is important. Travel time is very important in establishi ng a traveler information system; accuracy of this information can be essential in delivering re al time information to travelers and can affect travelersÂ’ route choice. With accurate estima tion and immediate information delivery by radio stations and Internet, travelers would have better information about current traffic conditions. This helps travelers avoid conge sted routes if there are any. If travelers do not enter the congested route, traffic congesti on can be relieved faster, and ove rall delay in the whole traffic network can be reduced. Thus, travel time or speed has to be estimated accurately. In literature such as the Highway Capacity Manual (HCM), travel time along arterials was estimated by adding the link travel time and th e isolated intersection delay. This estimation however is not accurate since the link travel time is calculated by assuming a constant vehicle speed and instant speed changes. The intersection delay is typically estimated by an analytical model for isolated intersection delay, which does not consider the effect of upstream and downstream intersections. More recently, a few studies begin to estimate the arterial travel time more realistically. Among these studies, some of them tried to measure the travel time on arterials by using technologies su ch as loop detectors and probe vehicles. The drawbacks of the methods are that they are site-spe cific and most of them cannot be converted and applied at other locations. Thus, there is no theoretical model that can be applied generally to arterial facilities for predicting travel time accurately. PAGE 26 261.2 Objectives of the Research The objective of this research is to develop an analytical model for estimating arterial travel time. Travel time is a variable which can be affected by many factors. Vehicles entering at different points in time can meet with different traffic demand and signal status, which in turn can affect travel time greatly. To take variability into consideration, fi rst, traffic demand should be considered as one of the factors in the mode l. Second, the signal status should be known and be used in the estimation method. The ta sks of this research are as follows: To develop an analytical model for estimati ng the expected arterial travel time, the variance of travel time, and travel time di stribution based on proba bilistic modeling. The probability of encountering a sp ecific signal status can be found and applied in the model so that a certain travel time occurs with certain probability. To compare these estimates to field data. Simulation was also used to evaluate model estimations. PAGE 27 27 CHAPTER 2 LITERATURE REVIEW In this literature review, papers and research re ports related to different arterial travel time estimation methods were reviewed according to the types of methods. Analytical models and simulation methods were reviewed. In the review of analytical m odels, two types of models were reviewed. One is the composite analytical models based on intersection delay estimation or not. The other is arterial travel tim e models based on in-field detec tion technology. In the review of simulation methods, the delay models in simulati on packages and the travel time estimation by simulation were reviewed. The studies on freeway tr avel time estimation are not reviewed in this chapter since their methodologies are not related to this research and do not contribute to the development of methodology for this research. 2.1 Models for Estimating Arterial Travel Time Several models have been developed for estima ting arterial travel time. Some of them do not need the inputs from in-field detection technologies and others do. Thus, in this section, these two types of models are discussed. 2.1.1. Composite Analytical Models In the HCM 2000 Chapter 15; travel time on arterials is estimated by the following equations: AS L T T . . (2-1 ) d T L SR A 3600 (2-2) Where . . T T: travel time, sec L: link length between two intersections, ft AS: average speed along the arterial, ft/s RT: running time, sec d: control delay for through movements, sec PAGE 28 28 The basic method for estimating the link travel ti me in the HCM is to divide the link length by the average speed which is adjusted from the free flow speed. The average speed is calculated by using two terms: one is the running time which is the ideal travel time. The other is the control delay for the through movement encountered at intersections. The HCM uses the isolated intersection delay model to repr esent this control delay. It is a rough method for estimating arterial travel time; first, speed is changing over time, it cannot be accurately estimated by the average speed. Second, the isolated intersection c ontrol delay is not the ac tual control delay that vehicles encounter at intersecti ons with consideration of upstream and downstream traffic flow impacts. However, the HCM method presents a popular approach for calculating the arterial travel time, which is to calculate the arterial trav el time as a function of intersection delay, either control delay or stop delay. 2.1.1.1 Intersection-delay based art erial travel time models Before discussing the travel time models based on intersection delay, the types of intersection delay and their definitions have to be defined first. There are two types of intersection delay that are often used: one is control delay; it is the time period when the vehicle decelerates, stops, and acceler ates at an intersection. The other is the stop delay, which is the time when the vehicle completely stops at an inte rsection. In the intersect ion-delay based arterial travel time models, arterial trav el time can be decomposed into two parts: vehicleÂ’s link travel time and the control delay caused by intersections. Some previous research considered these two parts and tried to estimate them separately. Thus , intersection delay estimation is essential in determining travel time along the arterial. Ther e are several methods for estimating signalized intersection delays: the HCM intersection delay model, shockwave analysis, and stochastic queuing analysis. These methods can provide different estimates fo r intersection delay and they are reviewed in this section. PAGE 29 29 Different delay models were developed over th e years; these models are both for isolated intersections and for non-isolated intersections. This section pr esents some delay models and studies related to intersection delay. Intersection delay estimation is a broad field and hundreds of studies have been completed since the 1950Â’s. Many delay models have been pr esented in the previous studies and are briefly reviewed here: Rouphail, Tarko, and Li (1992) performed a study on traffic flow at signalized intersections. In this publication, they reviewed some of the previously developed delay models for signalized intersections. For example, Rouphail, Tarko, and Li (1992) referred to the WebsterÂ’s delay equation. Webster developed a comprehensive delay mode l in 1958 (Equation 2-1) which assumes vehicle arrivals are random and depart ures are constant. This mode l is widely used today: 22 1/3(25) 2(1) 0.65() 2(1)2(1) x CxC dx xqxq (2-3) Where : dstop delay, sec/veh C: cycle length, sec green ratio, x : degree of saturation, q: flow rate, veh/hr The first term of the equation is uniform de lay, while the second term is random delay. These two terms are the theoretical part of the equation, and the last term is the empirical correction factor. Webster found that the correction factor contributes to 5 to 15 percent of delay. After Webster, several dela y equations were developed. Among them, according to Rouphail, Tarko, and Li (1992), MillerÂ’s and HutchinsonÂ’s delay model are often cited. Thus, delay models were developed and modi fied over time to evaluate the traffic operations at intersections. Since 1985, the HCM de lay calculation has been used as the standard PAGE 30 30 delay estimation for intersections; currently, the HCM 2000 chapter 16 presents the estimation method for isolated intersection delay. The m odels are based on WebsterÂ’s equation and have been modified in each version of the HCM as follows: In the 1985 HCM, an intersection delay model was presented (1985): 2 22(1)16 0.38173(1)(1) (1)sCx dxxx x c (2-4) Where sd : the average stop delay, sec/veh C, and x are defined as before This is the first delay model for signalized in tersections in the HCM; it estimates the stop delay at intersections for a 15-minute period. It was found that the 1985 HCM delay model can overestimate delay when the degree of satura tion is high.The HCM in 1994 presented a new delay model: c mx x x x C ds1 1 173 x,1.0 Min 1 1 38 . 02 2 (2-5) Where m: an incremental calibration term representing the effect of arrival type and degree of platoon sdC, and x are defined as before The 1994 HCM model kept the form of the de lay model from 1985; however, it changed the coefficient of arrival type and degree of platoon from a cons tant value to a variable, which needed to be calibrated for each specific case. Ag ain, stop delay is selected as the performance measure for signalized intersection. In 1997, the HCM presented a new delay mode l, which differs significantly from the previous ones. The 1997 HCM delay model (1997) is 123ddPFdd (2-6) Where d: control delay, sec/veh PAGE 31 31 2d: incremental delay, sec/veh :3dinitial delay, sec/veh X C g Min C g d 1 1 50 . 02 1 (2-7) Where g : effective green for lane group, sec C: cycle length, sec X : degree of saturation 2 28 900(1)(1) kIX dTxx cT (2-8) T : duration of analysis period, hr X : degree of saturation k: incremental delay factor I : upstream filtering/metering adjustment parameter c: lane group capacity, veh/hr The 1997 HCM delay model presents an improve ment in estimating signalized intersection delays. It defined delay as a summation of thr ee forms of delay: uniform delay, incremental delay and initial delay, which is called control delay. Besides that, the coefficient changes from 0.38 to 0.50. The 2000 HCM presented the same delay model as that of the 1997 HCM. The HCM delay models are used as the standard estimation of intersection delay. However, the model is not perfect in some aspects. For ex ample, it is isolated intersection control delay which does not consider the upstream and the dow nstream effect. Thus, some studies tried to adjust or improve the HCM delay model. Quiroga and Bullock (1999) used GPS data to measure intersection co ntrol delay and tried the measurement results as a new standard in ev aluating the accuracy of new intersection delay models. With a GPS device, control delay can be calculated by fS Length Link TravelTime D _ (2-9) Where D: control delay, sec/veh fS: free flow speed, ft/s PAGE 32 32 The paper compared the control delay measured in the field to the c ontrol delay calculated by the HCM method. The conclusion is that the deceleration and acceleration lengths were much longer than usually anticipated so that the pa rt of the control dela y that takes place in acceleration and decelerati on is not negligible. Benekohal and Kim (2005) indicated that for oversaturated conditions, the HCM 2000 model does not apply the progression adjustment factor (PF) when there is an initial queue. This causes the wrong estimation that delay with initia l queue is sometimes less than delay without initial queue. Besides this, the model yields the same uniform delay values for all arrival types when there is an initial queue. This paper pr oposed a new model for estimating uniform delay for oversaturated conditions. The uniform delay is 2 2 0 1 2 1 1) ( 2 1sg C q t C Q C Q sg d (2-10) Where 1d: uniform delay, sec/veh s: saturation flow rate, veh/hr/ln g : effective green time, sec 1Q: the number of arrivals when queue in crease rate changes for the first time 2Q: the number of arrivals at the end of cycle C: cycle length, sec 1t: platoon duration time, sec 0q: overflow rate, veh/hr/ln The authors concluded that the new model gi ves a better estimati on of delay than the HCM 2000 delay model. Ahmed and Abu-Lebdeh (2005) proposed a model for estimating delay at signalized intersections that is caused by downstream tra ffic disturbance such as queue spillback. The authors suggested adding one more delay term in the HCM 2000 delay model to represent the downstream induced delay, which is called4d; the equation for 4dis PAGE 33 33 ) 1 1 ( ) 1 ( 2 21 1 1 ) 4 ( 4v h n d n dV (2-11) 1 2 2 1 2 2 1 1 ) 4 (L V L off v L v L da (2-12) Where 4d: downstream induced delay, sec/veh n: number of vehicles between upstr eam and downstream intersections ) 1 ( 4d: portion of 4dincurred by the first vehicle at th e downstream intersection, % 1L: queue length at downstream intersection, ft 2L: remaining space on the link, ft vh: effective space headway, sec 1 : speed of mid-block stopping wave, ft/s 1v: speed of mid-block starting wave, ft/s 2v: speed of stopping wave at dow nstream intersection, ft/s aV: average link speed, ft/s The authors used CORSIM and the HCM 2000 de lay model to get the intersection control delay for queue spillback conditions, and they conc luded that there is additional delay compared to the HCM model and the new delay model can estimate delay well when compared to CORSIM. A Shockwave is the boundary that marks the time-space domain of one flow state from another (May, 1990). It represents the discontinuity of speed and density. This method can be used for signalized intersections as shown in Figure 2-1. In Figure 2-1, A, B, C, and D are different flow states. In state A, vehicles arrive at the intersection; in state B, vehicles stop at the inte rsection; in state C, vehicles discharge from the intersection; in state D, there is no flow. The lines separating different states are shockwaves. The triangle area between the shockwave ABw and BCw is considered as the total stop delay for vehicles stopped at an intersection. Based on some previous research, Pueboobpapha n, and et al (2005) presented how to estimate travel speed along signalized arterials using shockwave analysis. The authors did a PAGE 34 34 modification of a conventional shockwave diagram so that it can present the traffic conditions more realistically. For example, the conventiona l shockwave analysis assumes that vehicles change from stopping to a certain speed instantly. Under this a ssumption, the authors suggested that delay at intersections w ould be estimated more accurately by modeling the speed changes over time; they modified the shockwave diagram a nd tested their results with simulated data. The results convinced the authors that the new model is more accurate. This research improved the estimation of speed changes around intersections, but in this paper it is assumed that a vehicle moves with constant speed on the link. In queuing analysis, a queuing system can be defined by five characteristics (Gross and Harris, 1998). The definitions for the five as pects of a queuing system were extended for applications in signalized intersections. Inter-arrival time distribution. Inter-arrival time is the time interval between two consecutive arriving vehicles. Service time. Service time is the time interval for on e vehicle to pass the intersection. Number of servers. It is the number of intersections that can Â“serveÂ” the vehicles. System storage capacity. This is the maximum link length that one inte rsection can accommodate vehicles. It is the maximum number of vehicles that the upstream link of the intersection can accommodate. Queue discipline. It is assumed to be FIFO under the assumption that there is only one lane, or lane changing is limited on the arterial with multiple lanes. In general, a queuing system can be wr itten by using Kendall Notation (Gross and Harris, 1998): Inter-arrival Time/Service Time/number of servers/System Capacity/ Queue Discipline PAGE 35 35 Queuing analysis can be applied to signalized intersection for estimating delay. There are two types of queuing analysis: deterministic queui ng analysis and stochastic queuing analysis. Deterministic queuing analysis is applied in the intersection delay models, so it is not discussed in detail in this section. In deterministic queuing analysis, the vehicles arrive at a deterministic rate; and the intersection serves at a deterministic rate when the signal is green and stops to serve when the signal is red. As shown in Figure 2-2, the black bars repr esent red time and the white bars represent green time. There are two lines; one is the cumula tive arrival line, which is the solid line. The other is the cumulative departure line, which is th e dashed line. The area between the arrival line and the departure line is inters ection delay for all the vehicles . Since the intersection delay models are based on deterministic queuing analys is, it is not considered as another method in estimating intersection delay in this research. Stochastic queuing analysis is different from deterministic queuing analysis in that the arrival distribution and/or the serv ice distribution ar e probabilistic. For example, M/M/1/ /FIFO presents a queuing system w ith exponential in ter-arrival and service times, one server, infinite capacity and First-In-First-Out (FIFO) queue discipline. Some studies were conducted in the past to find the perfor mance measures for different queuing systems. Performance measures are important in that they are meaningful in evaluating how the system performs. Commonly used perfor mance measures are: av erage service time and average waiting time in the system. Some widely used queuing systems such as M/M/1, M/G/1, and M/D/1 have been already studied and their eq uations for performance measures are ready to PAGE 36 36 use. Intersection stop delay can be estimated using the average waiting time provided by the stochastic queuing analysis. A study performed by Geroliminis and Skabar donis (2005) used Markov Decision Process (MDP) to predict arrival profile and queue leng th along signalized arte rials. This is an application of stochastic queuing analysis. MD P was used to model tw o consecutive traffic signals in this research; this approach can predict the arrival profile at downstream intersections, and the method can be applied to find travel time. They indicated that arte rial travel time is greatly affected by platoon size and dispersion at intersections . Platoon size and dispersion can help to establish the relationship between two consecutive intersections, if the relationship can be found, and then arrival profile at downstream inte rsections can be predicted by the entering flow at the first upstream intersection. By using MDP, that is: based on the arrival t ype on the previous inte rsection, what is the arrival type of the next intersection: n n n ni A i A P |1 1 (2-13) A nonlinear (concave) flow-density relationshi p was used to model platoon dispersion. A model was presented as the plat oon dispersion as a function of di stance. That is how platoons disperse after vehicles discharge from intersecti on. This way, the number of vehicles that can arrive at a downstream inters ection can be known. Since signal s Â“blockÂ” some vehicles from passing, a queue is created. Then with this met hod, queue length can be estimated. Therefore, it is a way to predict travel time. 2.1.1.2 Other arterial tr avel time models Levinson (1996) performed a study about travel speed on arterials. He drew the conclusion that travel speed can be affect ed by two factors: same directio n flow and cross direction flow. The impact of same direction fl ow is represented by the interac tions between vehicles in the PAGE 37 37 same direction. The impact of cr oss direction flow is measured by intersection delay. These two are important in determining travel speed and sh ould be considered in research about travel speed. This point of view identified that traffic volume and the status of the signal have great impacts on travel time variability. Lum and Fan (1998) studied the speed-density relationship fo r arterials in Singapore and proposed a travel time-density model for inte rrupted flow. This method can be used for estimating travel time. They collected data on tr affic volume and travel time to build the revised speed-density relationship. Since an arterial ha s interrupted flow, signals can affect the speeddensity relationship. To adjust the impact, the number of inte rsections along the arterial is incorporated into the relationship. Th e journey time can be calculated as f d k u tj ) exp( 1 (2-14) Where jt : journey time, sec u: journey speed, ft/s k: density, veh/ft d: minimum delay per signalized inte rsection under free flow conditions, sec f: number of intersections : parameter related to journey speed : parameter related to density Olszewski (2000) conducted a comparison betwee n this travel time estimation model from Singapore to the 1997 HCM arterial tr avel time model. Olszewski indi cated that the first part of the model presents running time and incrementa l delay at signals which depends on traffic density, the second part of the model presents stopped delay which de pends of density of signalized intersections. The comp assion shows that the two mode ls give similar results and HCM model gives lower speeds unde r un-congested conditions. The above arterial travel time estimations do not consider the variability of the travel time. The following papers address the issue of variability in travel time estimation. PAGE 38 38 Fu and Hellinga (2000) studied the variability of delays at signalized intersections. They suggest delays that individual ve hicles experienced at signalized intersections are subject to variations caused by random arriva l. In this study, delay was defined as the sum of uniform delay and overflow delay. The variation of total delay is the sum of th e variation of uniform delay and overflow delay. As shown in several previous stud ies, arterial travel time is different from the travel time of other facilities in that an arterial has intersections along it and these cause vehicles to stop and make travel time longer. Because of this, inters ection delay is also ve ry important in finding arterial travel time. Because of the importance of intersection delay, some of the studies focus on intersection delay when studying ar terial travel time so that they may simplify travel time estimation to intersection delay estimation. Lin, Kulkarni, and Mirchandani (2004) perfor med a study on arterial travel time for the ATIS. In the study, they developed a model to esti mate arterial travel time. In the model, they reduced travel time estimation to intersection delay estimation: in their paper, travel time is first defined as the sum of link travel time and inters ection delay. They conclu ded that link travel time is not so sensitive to the traffic flow when the fl ow is medium or high. They suggested that link travel time should be estimated as a constant valu e. This method is similar to urban street travel time estimation in the HCM in the way that link trav el time is not considered as a varying factor. While in this paper, they used the isolated in tersection delay model. Th erefore, they need to develop a transition matrix from conditional probability. The goal is to find what is the probability that a vehicle is in state i based on that its previous state is j . So, states need to be defined. PAGE 39 39 There are two states defined in this study: 0 and 1: 0 represents the condition where the vehicle is not delayed at the in tersection, while 1 represents th e condition where the vehicle is delayed at the intersection . Therefore a 2 2 matrix is generated as follows: i i i ip p p p P11 11 00 001 1 (2-15) Delay at intersections is calculated by us ing the conditional probability times WebsterÂ’s Delay Equation. The general model is 0 ) ( 1 , 1 ) ( 1 ), () ( 10 10 11 , 11 ) ( i iD E p p p p i i d (2-16) Where ) ( i : States that one in tersection can have ) () ( iD E: Webster Delay Equation at the intersection, s/veh This paper is considered as an improvement by the author in finding arterial travel time since it considered intersection delay with probability. Vehicles do not encounter intersection delay all the time, delay occurs with a certain probability. The transition matrix would be more realistic if the paper considered signal timing, of fset, traffic flow (in consideration of queue), speed, and arrival type (in c onsideration of coordination) when it calculated conditional probabilities. The paper applied th e model in four scenarios simulated in a simulation package. They concluded that the result is promising but needs to be verified with field data. Besides the studies that considered the variabil ity of travel time, ther e are other travel time studies which think travel time can be derived by estimating travel speed along the arterial. More recently, some studies are performed by deriving travel time which is the link length divided by travel speed. Tarko (2006) et al. developed a method for es timating travel speed along urban arterial streets. The model does not need inputs such as signal timing and geometry information about intersections. The model is derived from the HCM intersection delay model; it captures the PAGE 40 40 arterial through movement travel speed. The inve rse of the speed, called pace, equals the unit travel time. Pace can be calculated as i i s s i in F a n F a F F F a l a l a V p5 4 2 1 3 2 1 01 1 1 exp 3600 (2-17) Where iV : travel speed in direction i ,ft/s 0V : cruise speed, ft/s l: average distance between adjace nt signalized intersections, ft 1a, 2a, 3a : parameters of the model iF : average one-way flow in the analyzed direction i, veh/hr/ln 1F, 2F: average one-way flows along the arterial street ; veh/hr/ln Fs : flow crossing the major road ,veh/hr/ln . Select the stronger one-way volume on each side street and calculate the average sn : average number of through lanes in one direction on side streets in : average number of through lanes in the c onsidered direction on the major streets The estimation results show that when compar ed to field data, the model underestimates travel time by 18%. The error is removed by addi ng one more adjustment parameter. This model is developed for planning purposes which usually does not have de tailed traffic condition data. In this model, only the cruise speed, distance betw een intersections, number of through-lanes, and one-way volumes are required as inputs. The signal timing is not considered which may result in the biased estimation. Based on their previous mode ls in estimating arterial link travel speed, Xie and Cheu (2001) proposed a new model which they thought tends to improve the previous models. The model is as follows: Travel time = cruise time + signal delay (2-24) Signal delay = ) 1 ( 2 ) 1 ( 2 ) 1 ( 9 . 02 2x q x x C (2-25) PAGE 41 41 2 2 2) ( , 0 ) ( , ) ( ) ( L q g C if L q g C if q g C L q g C (2-26) Cruise time = det 1u L (2-27) Where C: cycle length, sec green ratio x : degree of saturation g : green time, sec q : flow rate, veh/hr/ln. : signal parameter 1L: Part of the link when vehicle can travel with normal speed, ft 2L: Part of the link when vehicle decele rates, it equals to link length minus1L, ft detu: Speed from detector. ft/s The model was tested with field data and the author concluded that the model gives better estimation than the previous models. 2.1.2 Arterial Travel Time Models Based on In-Field Detection Technologies Besides the arterial travel time models re viewed above, there are other methods for estimating travel time by developing a model based on in-field data that are collected. Since arterial travel time sometimes can not be estimated very accurately by some theoretical models such as the HCM model, some st udies also try to estimate arterial travel time with the aid from advanced technologies. Usually, there are several technologies to help estimate arterial travel time: they ar e probe vehicle, obse rvation through vide o technology, and loop detector. Thus, some studies were completed with the aid of technologies as mentioned above to develop arterial travel time models, some of these studies are discussed here: The National Institute of Statistical Science (NISS) performed two consecutive studies on arterial travel time; these studies were comp leted on year 1996 and 1998. In 1996, Sen, Soot, Ligas, and Tian (1996) did the first travel ti me study, which presents a comparison of some PAGE 42 42 arterial travel time models based on data co llection methods, such as probe vehicles and detectors. The authors broadly reviewed the existing travel time estimation methods as mentioned above: Probe vehicles can measure tr avel time. With the Advanced Video Image technology, travel time can be measured directly . Detectors are used not only for arterials but also for freeways; they obtain volume and occupa ncy data and then being transferred to travel time data. Assignment-type models have a drawback that estimation is for a long time period so they are not very sensitive to external changes. The authors suggested that the estimated travel time is the sum of cruise time and stopped delay. Cr uise time is the period of time when a vehicle is moving, and stopped delay is the period of tim e when a vehicle stops at a red signal or is slowly moving in a queue. Their conclusion was that variation in travel time mostly depends on variation in stopped delay. The second study from NISS was performed by Graves, Karr, and Thakuriah (1998) as an extension of the previous one; in this study, the authors focuse d on travel time variability and they performed the study on a select ed arterial in a Chicago suburba n area. Travel time data were collected with a probe vehicle and loop detectors. The findings are that travel time variability relies on the signal status and th e traffic volume when a particular vehicle enters the link. This paper presented a new variable in estimating arterial travel time: Relative Entry Time (RET). RET is related to the status of the downstream signal. In the paper, th ey developed a general travel time model and then adjusted the mode l for through, left turning, and right turning movements. Factors considered in travel time models are red time, green time, free flow travel time, time for each vehicle to clear the queue, and traffic volume. The general model is ) ( ) ( ) , (R RET n TC FFTT n RET TT (2-18) Where R E T : relative entry time, sec n: number of vehicles in the queue PAGE 43 43 FFT T : free flow travel time, sec ) ( n TC : time required for a queue of n vehicles to clear, sec R : red time, sec Models for through, left, and right movements are different. Zhang (1999) developed a model to find the jour ney speed along an arte rial; the idea is if the speed can be estimated correctly, arterial tr avel time can be found by getting the link length divided by the speed. In his model, the loop de tector provides occupancy measurement for developing the model. The model is //(1)cvcqcUUU (2-19) Where : a weighted factor wh ich is between 0 and 1 /exp{}vcfV UU C (2-20) /0.379i qc iq U o (2-21) Where io : the occupancy measurement from the detector, % and : parameters that need to be defined for each specific case One available model to estimate the actual travel time is the Â“Illinois ModelÂ” (Zhang, 1999). The model presents a regressi on model for delay estimation, two of the regression parameters are based on data from detectors. DELAY UNDT T (2-22) Where T : the link travel time, sec UNDT is calculated as fV L UNDT (2-23) Where L: the link length, ft fV : the free flow speed, ft/s grnrat o deloc DELAY3 2 1 0 Where deloc: ratio of detector setback to link length PAGE 44 44 o: detector occupancy,% grnrat: green ratio (green time over cycle length) 0 1 2 3 : regression parameters Mark, Sadek, and Dickason (2005) applied Artif icial Neural Networks (ANN) methods in predicting arterial travel time . The study tested the possibility for using ANN method based on detector data. The result shows that ANN is quite a pplicable in predicting ar terial travel time in that it updates and recalculate th e travel time when new travel time observation is obtained. Travel time can also be predicte d by analyzing historical data . To find the trend of travel time throughout a period of time and predict future travel time based on the trend discovered before, the method is called time series. Liu and Shuldiner (2005) predic ted travel time by the time se ries method. They collected field data on Massachusetts Route 9 for about 3.7 miles arterial with the aid of Automatic Vehicle Identification (AVI), whic h ensures the accuracy of field da ta for this research. With the AVI system, two cameras were used at the begi nning and ending points of the road segment. Cameras can capture the license plate of vehicles using the optical character recognition, when the two plates match, one travel ti me observation can be generated. There are three time series models used to develop a short-time travel time prediction model, they are: Auto Regressive Integrated Moving Average (ARIMA), ARIMA with traffic volumes as additional input variables, and Autore gressive Error Model. The first model is an ARIMA model that predicts travel time by the following equation: 1 1 2 2 1 1 t t t tZ C TT a TT a TT (2-28) Where tTT : travel time of the tth interval, sec 1tTT : travel time of the (t-1)th interval, sec 2tTT : travel time of the (t-2)th interval, sec 1a,2a,1C: parameters of the model 1tZ : white noise (0, ) for the (t-1)th interval PAGE 45 45 The second model is also ARIMA, but this tim e with traffic volumes as additional input variables. This model assumes that current trav el time can be affected by traffic volume as well as previous travel time, so the model is based on the first model with some additions: 1 1 1 1 2 1 2 2 1 1... t m n t m n t n t t t tZ C X b X b X b TT a TT a TT (2-29) Where :n tXtraffic volume of the (t-n)th interval, veh :1 n tX traffic volume of the (t-n-1)th interval, veh :1 m n tX traffic volume of the (t-n-m-1)th interval, veh : nlag parameter between traffi c flow and travel time :mnumber of intervals 1a,2a,1C,1b,2bÂ…mb : parameters of the model 1tZ : white noise (0, ) for the (t-1)th interval The third model is the Autoregressive Error Model: t tN b t a TT (2-30) t t tZ N N 1 (2-31) Where tTT : travel time of the tth interval, sec : t the number of the current interval tN : noise in the tth interval tZ : white noise (0, ) for the tth interval a, b, c: parameters of the model The three models were tested and they are cons idered as reliable in predicting near future travel time, however, these models have a shortcom ing in that they can only predict travel time under non-congested, no incident conditions. When th e travel time increases greatly and rapidly in a short period of time, which means there is congestion, these models are not good predictors. Du and Aultman-Hall (2006) developed a method for estimating road network travel time. In order to estimate the network travel time, the travel time of each link within the network has to be estimated. The research deals with the probl em that some links have GPS data and some do not. The method classifies the links into groups. In each group, there are links which have GPS data and other links without GPS data. An expected travel time is estimated from the links which PAGE 46 46 have GPS data. For the links inside the group an d do not have GPS data, a random travel time based on the expected value is generated and assigned to the link. 2.2 Simulation Methods in Arterial Analysis Simulation can replicate the real traffic condi tions and it can run for multiple times to get enough results for analysis. Thus, simulation met hods are very helpful in many transportation analyses. Simulation packages are often used in arterial analysis in determining intersection delay and travel time. 2.2.1 Delay Models in Simulation Packages Several simulation packages have their own in tersection delay model. Some of the models are the same as the HCM model and others ma y have different ones. The delay models for PASSER, TRANSYT-7F, and Sync hro are presented here. PASSER (2002) has several packages to suit the need for different tran sportation facilities, among them, PASSER II focuses on arterials. PASSE R II optimizes arterial progression and it is one of simulation software packag es that has its own delay mode l. The model does not consider initial queue delay so it divide d into two parts: uniform dela y and random/saturation delay: f s v C g C du / 1 ) / 1 ( 5 . 02 (2-32) cT X X X T drs4 1 1 9002 (2-33) Where ud : uniform control delay, sec/veh rsd : random and saturation delay, sec/veh g : effective green, sec C: cycle length,sec v:movement volume, veh/hr/ln s: saturation flow rate, veh/hr/ln c: capacity, or sxg/C, veh/hr/ln X : degree of saturation f: signal type (1.0 for pretimed and 0.85 for actuated), T : time period adjustment factor. PAGE 47 47 This delay model is different from the HCM 2000 delay model. In the HCM 2000 model, there are three parts: uniform delay, incremental delay, and initial delay. In PASSER, there are only two parts, uniform delay and random/satura tion delay, it does not include initial delay. PASSERÂ’s uniform delay equation is slightly different from the HCM 2000 model, and in random delay which is called incremental de lay in the HCM 2000, the upstream filtering adjustment factor I is assumed to be 0.5. TRANSYT-7F (2002) has been developed a nd its delay model has been changed over time. Currently, the latest versi on, 10.1, uses the delay model fr om the HCM 2000 as its standard for delay estimation. Synchro provides two options in delay calcu lation; users can choose the HCM signal delay or Synchro delay: the HCM signal dela y uses the delay model in HCM 2000. 2.2.2 Estimating Delay and Trave l Time Using Simulation Simulation modeling replicates the reality and provides some useful evaluations on performance measures. Signalized intersections along the arterial are essential for determining arterial operations; inappropriate signal timing and poor signal c oordination between intersections can cause unnecessary delays on the arterial. Some research was comple ted in finding appropriate methods to optimize signals so that delay can be reduced . A selected review of such papers concerning optimizing signals is discussed here. Travel time is greatly affect ed by intersection delay, whic h can be reduced by a certain amount if signals are optimized and coordinated w ith other signals. In the literature review, it is necessary to review papers that focus on signal optimization and coordination. To find the optimal signal timing and an eff ective way to coordina te signals, several simulation software packages have been develope d and applied to this research. Widely used PAGE 48 48 simulation packages include CORSIM, TR ANSYT-7F, and PASSER. TRANSYT-7F and PASSER can provide optimal cycle length, green splits for all time periods or for different time period throughout a day, etc. Si nce these simulators donÂ’t have the same algorithms underlying their optimization routines, several papers compare to find which software is suitable for certain circumstances, and provide recommendations for improving their simulation models. Lin and Courage (1996) conducted research on how certain simulation software can determine phase times for actuated signals al ong the arterial. TRANSY T and TRANSYT-7F are tested in this paper. They st ated that existing simulation m odels are not good enough and some of them are oversimplified since they are not considering some practi cal conditions on traffic operations, such as progression, and left turn movements. This paper focuses on how to determine average phase time in modeling actuated traffic signal control. In previous research, phase time calculations were aimed at reduci ng degree of saturation or queue service time. Kamarajugadda and Park (2003) noticed that in some simulation packages intersection delay is calculated based on the HCM intersection de lay model. In the model, delay is a function of multiple parameters; however, some of the variables such as volume, green time, and saturation flow rate are stochas tic variables with their own distribution. This characteristic implies that the intersection delay would be bette r presented with a distribution rather than an average value. Therefore, they developed a de lay variance equation for isolated intersections, and then expanded it to arterial intersections. Th is way, their model can optimize the arterial system better when compared with an exis ting simulation package such as Synchro. 2.3 Summary of the Literature Review From the studies reviewed in this chapter, it is found that arterial tr avel time is estimation models fall into two Â‘classesÂ’. One is to develop an analytical model; eith er based on intersection control delay estimation or in-fie ld technology. The other is to obt ain the arterial travel time by PAGE 49 49 simulation. Some of the studies me ntioned the variability of arteri al travel time but none of them estimated it. The objective of this research is to develop an analytical model for estimating the expected arterial travel time as well as the va riance of the travel time, and the travel time distribution. This methodology is discussed in more details in the next chapter. PAGE 50 50 Figure 2-1. Shockwave analysis Figure 2-2. Deterministic queuing analysis PAGE 51 51 CHAPTER 3 METHODOLOGY This chapter presents the study methodology. S ection 3.1 studies a simple case to find the travel time for an arterial li nk between two intersections. Base d on the method developed in the simple case, a generalized method is developed in Section 3.2 to model an arterial with multiple links. In Section 3.3, an example is pres ented to demonstrate the proposed methodology. Travel time in this study is defined as the sum of Â“travel time in motionÂ” and Â“time in queueÂ”: travel time in motion is the period of time when the vehicle leaves the queue until it joins another queue. Time in the queue is the pe riod of time when a vehicl e joins the queue until it discharges from the queue. This part of travel time is essential in estimating arterial travel time accurately. In some previous papers, this amount of time is considered as delay at intersections. The simplest method from the HCM is to estimat e the travel time as the sum of isolated intersection delay and the ideal travel time (li nk length divided by speed limit). However, that equation does not consider the effects from ups tream and downstream intersections. In fact, besides the delay at the inters ection, time in queue should also include moving time in queue when the queue is discharging. Thus , in this research, arterial trav el time is defined as the sum of Â“travel time in motionÂ”, Â“waiting time in queueÂ” and Â“moving time in queueÂ”. The three parts of travel time need to be estimated separately and then combined together to estimate the arterial travel time. 3.1 Simple Case: One Link between Two Intersections In this study, to estimate travel time, a simple case is first studied and a general case is developed later based on the conc lusions of the simple case. The simple case only includes one link between two intersections. Only the impact of the downstream intersection is considered. The simple case shows how to estimate the travel time for vehicles enc ountering different signal PAGE 52 52 status, regardless of what happene d at the upstream intersection. Based on this case, the general case considers both the upstream and the downstream intersections for estimating travel time. A complete list of notation is provided in Appendix A for this research. 3.1.1 Definition of a Link To estimate the arterial travel time in the si mple case, the starting point and the ending point of the segment should be defined first. As s hown in Figure 3-1, in this research, the starting point is defined as the stop line at the upstream intersection; the ending point is the stop line at the downstream intersection. 3.1.2 Decomposition of Travel Time After defining the starting and ending points fo r the simple case, travel time can be decomposed into two parts for this basic se gment as shown in Figure 3-2. The following equation defines the respective variables: Q MT T ATT (3-1) Where ATT (Arterial Travel Time): total travel time on the arterial; MT (Time in Motion): time period when the vehicles travel before they join the queue; and QT (Time in Queue): time period from when the ve hicles join the queue to when they leave the queue Time in motion is a function of link length, queue length, initial speed, the acceleration rate, and the deceleration rate. QT can be further divided into Waiting Time in the Queue and Moving Time in the Queue: Q Q QMT WT T (3-2) Where QWT (Waiting Time in the Queue): time that vehicles have to wait in the queue; and QMT (Moving Time in the Queue): time that vehi cles move when they are in the queue Thus, the entire equation for ATT is Q Q MMT WT T ATT (3-3) PAGE 53 53 In this equation, the calculations for the th ree terms are different as the signal status changes. The MT variable is different since the trajecto ry of the vehicle changes as the signal status changes. For time in queue, there are two types of queues: moving queue and stopped queue; moving queue is the queue that is discha rging when the signal status is green; stopped queue is the queue that is not discharging and vehicles are stoppe d when the current signal status is red. QWT and QMT are different for these two types of queues. Thus, to estimate arterial travel time, the compon ents of the arterial travel time need to be estimated. MTcan be calculated if the vehicle motion is known.QWT can be obtained from an intersection delay estimation since it is part of that. QMT can be calculated by estimating the average queue length, the speed for the vehicle entering the queue and the speed leaving the queue. The equations for estimating MT and QMT will be shown in the following sections, for various traffic conditions. 3.1.3 Estimation of MT Travel time can be accurately measured if the trajectory of the vehicle is recorded. MT can be different if the trajecto ry is different. Therefore, MT is estimated differently for different traffic conditions. Thus, this se ction presents the descriptions of the various conditions, and provides equations for MT for each condition. The three basi c conditions are identified and shown in Figure 3-3. Figure 3-3 shows that the vehicle trajectories ar riving at an intersection. The dashed lines represent the vehicle trajectories; the black bar re presents the effective red time and the white bar represents the effective green time of the in tersection. According to the different vehicle trajectories, the cycle can be divided into three conditions as follows: PAGE 54 54 Condition 1: The vehicle arrives to decelerate unt il it stops, and then it has to wait. Some other vehicles may have arrived already, and th e vehicle has to join th e stopped queue. As shown in Figure 3-3, from 1tto3t . Condition 2: The vehicle arrives when the curren t queue is moving and discharging: in this case the vehicle has to decelerate to join the moving queue and keep moving with the queue. As shown in Figure 3-3, from 3t to4t. Condition 3: The vehicle arrives on green time a nd there is no queue at the intersection: in this case the vehicle can pass through the intersection with it s desired speed. As shown in Figure 3-3, from 4tto5t .According to Figure 3-3, the interval length of the three conditions can be calculated as Condition 1 = r w w rw t t t t t tAB BC AB 1 2 2 3 1 3 (3-4) Condition 2 = AC AB BC BC ABw w w w w r t t t t t t 2 3 2 4 3 4 (3-5) Condition 3 = 12 4 2 5 4 5 AC BC AB BC ABw w w w rw r C t t t t t t (3-6) Where A B A ABk k q w , C B C BCk k q w , C A C A ACk k q q w ABw: shockwave speed between state A and B, mph; BCw : shockwave speed between state B and C, mph; ACw : shockwave speed between state A and C, mph; Aq: arrival rate (actual flow rate), veh/hr/ln; Cq : at capacity flow rate, veh/hr/ln; Ak: density at arrival rate, can be convert ed from flow divided by speed, veh/mi; Bk: jam density, veh/mi; Ck : density at capacity, veh/mi; r : effective red time,sec; and C: the cycle length, sec The sum of the three conditions is PAGE 55 55 Sum = Condition 1 + C ondition 2 + Condition 3 C w w w w rw r C r w w rw w w w w rw w w rwAC BC AB BC AB AB BC AB AC BC AB BC AB AB BC AB 1 1 Figure 3-3 is different from the one that appears in the literature review chapter since the vehicles trajectories after the green signal starts are not straight lines but curves. This is based on the theory of Kinematic Waves applied by Li ghthill and Whitham(1955) to traffic flows at signalized intersections, where the increase of sp eeds for the discharged vehicles are achieved through a fan of waves of all possible velocities. These speeds are shown in Figure 3-3 as a fan when the green signal starts. The speeds change the trajectories of the vehicles from straight lines to curved lines. The equations that are used befo re (equation 3-4, 3-5, an d 3-6) to calculate the time interval for three conditions will not be changed because the difference is negligible. For the simple case, the total link lengths is the distance from the starting point to the ending point of the link. As show n in Figure 3-4, it can be furthe r divided into three parts: the acceleration distance (As), the constant speed distance (Cs ), and the deceleration distance (Ds).Ds includes not only the distance for the vehi cle to decelerate, but also includes the distance over which the queue extends (qD ) at the intersection. It is not necessary that all three distances are included in each travel time, as so me of the travel times may only have one or two of them. MT Equation for Condition 1 is d m m link q d m a link i m a link i m Ma V V D a V a V V s a V V T 1 _ 2 2 1 _ 2 1 _2 2 (3-7) Where 1 _ link iV : vehiclesÂ’ entering speed (initial speed) wh en they enter link 1, as shown in Figure 3-5. 1 _ link iV can be determined by the vehicles tr ajectory in the previous link, ft/s mV : vehiclesÂ’ maximum operating speed wh en they travel on the link, ft/s aa : vehiclesÂ’ acceleration rate, 2ft/s PAGE 56 56 da : vehiclesÂ’ deceleration rate, 2/s ft 1 _link qD : distance over which the que ue extends at link 1, ft MT Equation for Condition 2 is d link d m m link q d link d m a link i m a link i ma V V V D a V V a V V s a V V TM1 _ 1 _ 2 1 _ 2 2 1 _ 2 1 _2 2 (3-8) Where 1 _link dV : discharging speed at link 1, ft/s, as shown in Figure 3-6 1 _link iV ,mV ,aa ,da ,1 _link qD are same as defined for the MTequation for Condition 1 In this condition, the vehicle movement can be very complicated since the vehicle joins the discharging queue and may go back and forth be tween stopping and moving. Thus, the motion of the vehicle is not so easy to trace, in order to study these motions; they are simplified in this study by assuming constant speed when moving in the queue: for example, if vehicleÂ’s entering speed is higher than the discharging speed, the motion is considered to be four parts. It accelerates with a constant accel eration rate until reaches the maximum speed, and then it keeps a constant speed until it decelerates from the maximum speed to the discharging speed, and then moves with the discharg ing speed in the queue.MT Equation for Condition 3 is m a link i m a link i mV a V V s a V V TM 22 1 _ 2 1 _ (3-9) Where 1 _link iV ,mV ,aa are same as defined for the MTequation for Condition 1, 1 _link iV is shown in Figure 3-7. For Condition 1 and 2, MTequations contain the term1 _link qD , which is the distance over which the queue extends. Estimation of 1 _link qD is based on queue length. Although for each vehicle the 1 _link qD in front of it is different, in this analysis the average 1 _link qD for each condition will be used. PAGE 57 57 The average1 _link qD can be estimated by graphically. In shockwave analysis, the average 1 _link qD for the stopped queue in Condition 1 a nd the moving queue in Condition 2 are approximately the same. The average 1 _link qD for the stopped queue and the moving queue is calculated as half of the maximum queue leng th as shown in the shockwave analysis. The equation is AB BC AB BC M link qw w w w r Q D avg) ( 7200 2 .1 _ (3-10) Where 1 _.link qD avg : Average queue length at link 1, mi MQ: Maximum queue length, mi ABw,BCw , and r are same as previously defined. 3.1.4 Estimation for QMT QMT is the period of time that the vehicle m oves in a queue, and it is not a motion that vehicles always experience. For the vehicles arri ving in Condition 3, the queue is zero; therefore, moving time in the queue does not exist in th is condition. However, under Condition 1 and Condition 2, QMT does exist. For Condition 1, the vehicl es join the stopped queue and when the traffic signal turns green, they begin to discharge. The time period for the vehicles to depart from the queue until they reach the stop line is QMT for Condition 1. For Condition 2, the vehicles join the moving queue and move with other vehicles in the queue. The time period when the vehicles begin to move in the queue until they depart from the stop line is QMT for Condition 2. Equations for these two conditions are developed below. For each vehicle in Condition 1, QMT depends on the position where the vehicle is in the queue. For each vehicle, the queue length equals to the distance fo r the vehicle to accelerate from zero speed. Thus, the time for this can be calculated by PAGE 58 58 a link q Qa D MT1 _2 (3-11) Where 1 _link qD and aa are as previously defined. The average 1 _link qD for vehicles arrived in Condition 1 can be used in the above equation to estimate the average QMT (Equation 3-12). Thus: a link q Qa D avg MT1 _. 2 (3-12) Where 1 _.link qD avg andaa are same as previously defined For Condition 2, the vehicle joins the moving queue. Based on the assumption of constant speed within the queue, QMT is the time period for the vehicle to move with a constant speed, and the moving distance is the average1 _link qD , so for each vehicle, QMT is 1 _ 1 _ link d link q QV D MT (3-13) Where 1 _ link qD and 1 _ link dV are same as previously defined. The average1 _ link qD for vehicles arrived in Condition 2 can be used in the above equation to estimate the average QMT (Equation 3-14). Thus: 1 _ 1 _.link d link q QV D avg MT (3-14) Where 1 _.link qD avg and 1 _ link dV are same as previously defined. 3.1.5 Estimation for QWT The vehicles arrive to join the stopped queue and have to wait until the signal turns green, thus, QWT exists only for Condition 1. Different arrivi ng times for different vehicles result in varying individual waiting times. The average QWT can be calculated as half of the effective red time of the intersection. PAGE 59 59 3.1.6 Link Travel Time for All Three Conditions As the equations forMT,QMT , and QWT of travel time are deve loped for each condition, the link travel time for the three conditions can be developed: For Condition 1: Q a link q d m m link q d m a link i m a link i m Q Q MWT a avgD a V V avgD a V a V V s a V V MT WT T ATT 1 _ 1 _ 2 2 1 _ 2 1 _2 2 2 (3-15) For Condition 2: d link q d link d m m q d link d m a link i m a link i m Q MV avgD a V V V avgD a V V a V V s a V V MT T ATT1 _ 1 _ 1 int _ 2 1 _ 2 2 1 _ 2 1 _2 2 (3-16) For Condition 3: m a link i m a link i m MV a V V s a V V T ATT 22 1 _ 2 1 _ (3-17) 3.1.7 Probability that a Certain Condition Occurs To estimate the expected value of travel time, the following probabilities should be considered: when one vehicle arrives at one inte rsection, the three conditi ons all have their own probabilities of occurrence. The th ree conditions described earlier present the three possibilities that one vehicle can meet when it arrives to an intersection and these probabilities are not equal, but should be estimated since they will be used to determine the percentage of vehicles in each condition. The arterial travel time variation is based on the probability that each condition can occur. Shockwave analysis can determine the pr oportion for each condition within one cycle. Shockwave analysis (May 1990) de fines the discontinuity of speed or density in the traffic stream. As shown in Figure 3-8, area A, B, a nd C indicate different speeds at one signalized intersection. Area A is the opera ting speed when the vehicles tr avel along the link, while area B PAGE 60 60 has zero speed when the vehicles are stopped at the inte rsection. Area C is the discharging speed when the vehicles begin to discha rge at the beginning of the green. 1t,2t,3t ,4t, and 5t are different time points that present the cha nges in queue length over the time period. 1t is the time point when the signal turns red, 2tis the time point when the signal turns green, 3t is the time point when the queue clears, 4tis the time point when the queue has dissipated at the intersection, 5t is the time point when the signal turn s red again. Therefore the time period between 1tand 3t is the time period when Condition 1 occurs. The time period between 3t and 4t is the time period when Condition 2 occurs. The time period between 4tand 5t is the time period when Condition 3 occurs. The interval lengths of the three conditions are as define d before. Condition 1 occurs during the time period when vehicles have to join the stopped queue; it is from the beginning of red signal until the queue begins to move. Th erefore the probability for Condition 1 is C r w w rw C w w w r r ConditionAB BC AB AB BC AB 1 Pr (3-18) Probability that Condition 2 occurs: Condition 2 occurs during the time period when vehicles have to join the moving queue; it is from when the stopped queue begins to move until the queue dissipates. Therefore the probability for Condition 2 is C w w w w w r ConditionAC AB BC BC AB 2 Pr (3-19) Condition 3 occurs during the time period when there is no queue present; it is from when the moving queue dissipates until the beginning of red signal for the next cycle. Therefore the probability for Condition 3 is PAGE 61 61 C w w w w rw C r C w w w w rw r C Condition Condition ConditionAC BC AB BC AB AC BC AB BC AB 1 1 1 2 Pr 1 Pr 1 3 Pr (3-20) Travel time estimation is related to the proba bility that each condition occurs, as shown above, traffic flow rate, capacity , and density are involved in th e equations for conditions. To consider arterials with multiple lanes, these th ree values will be used as per lane level. With the probabilities for all three conditions, the expected link travel time for the simple case can be developed as E [Ttotal] = E [TM + WTQ + MTQ| cond =1] P [cond = 1] + E[TM + WTQ + MTQ| cond =2] P [cond = 2] + E[TM + WTQ + MTQ| cond =3] P [cond = 3] (3-21) 3.2 General Case: An Arterial with Multiple Intersections With the link travel time model developed, the arterial travel time model can be developed by generalizing the simple case, and consid ering the interaction between successive intersections. The travel time on an arterial with multiple intersections would be very different if the vehicle meets with different traffic signal status: for example, travel times are different for two extreme cases: one is the vehicle travels on an arterial and arrive at all intersections in Condition 1 and has to stop, the other is the vehi cle travels on an arterial and arrive at all intersections in Condition 3, thus, the vehicle do es not need to reduce its speed. The travel time for these two vehicle trajectories can be very different from each other. 3.2.1 The Flow Profile 3.2.1.1 Travel time estimation under non-congested conditions As shown in Figure 3-9, for an arterial with four intersections, each of the three conditions at each intersection has a certain probability fo r encountering another condition or encountering the same condition at the downstream intersection: PAGE 62 62 As shown in Figure 3-9, there are43possible combinations. Each of them presents a group of vehicleÂ’s travel trajectory; it is called a flow profile. To find the total arterial travel time, these possibilities need to be calculated and the travel time associated with each possibility also need to be estimated. To study the arterial with se veral intersections, it is necessary to find the 9 possibilities between two consecu tive intersections, with the methodology for calculating the 9 possibilities; the methodology can be further used to find all possibilities for more than two intersections. As shown in Figure 3-10, suppose there are tw o consecutive intersec tions, intersection 1 and intersection 2, and one link, link 1 between them. There is one additional intersection, intersection 0, and one additional link, link 0. Intersection 0 and Link 0 are used to determine the entering speed at intersection 1. The 9 possibilities between two consecutive intersections are: :11PIt is the probability that the arrival vehi cle stops at two consecutive intersections. :12P It is the probability that the arrival vehicl e stops at the first intersection and joins the discharging queue at the downstream intersection. :13P It is the probability that the arrival vehi cle stops at the first intersection and arrives without queue at the downstream intersection :21P It is the probability that the vehicle joins the discharging queue at the first intersection and arrives to stop at the downstream intersection :22P It is the probability that the arrival vehicle joins the disc harging queue at two consecutive intersections :23P It is the probability that the arrival vehi cle joins the dischargi ng queue at the first intersection and arrives without queue at the downstream intersection PAGE 63 63 :31P It is the probability that the vehicle arri ves without queue at the first intersection and arrives to stop at the downstream intersection :32P It is the probability that the vehicle arri ves without queue at the first intersection and arrives to join the discharging que ue at the downstream intersection :33P It is the probability that the vehicle arrives without queue at two consecutive intersections Link travel times for the 9 different flow prof iles can be developed based on the travel time equations for the three conditions. In the equatio ns, since the entering speed is not known, the travel time cannot be finalized. Now conditions at two consecutive intersections are known, thus, the specific entering speed can be estimated, and the link travel time equation can be developed for the 9 cases. The entering speed refers to the speed that th e vehicles enter link 1. The entering speed for the vehicles arrive at Condition 1 at the intersection 1 is calculated as below: Since the vehicle encounters C ondition 1 at the first intersec tion, it is assumed that the vehicle starts to travel with an initial speed of 0 and has to travel the distance over which the queue extends to enter the next link, thus: a link i a link f link i link qa V a V V D2 22 1 _ 2 0 _ 1 _ 2 0 _ (3-22) Where 0 _ link fV: Leaving speed (final speed) for link 0, ft/s 0 _link qD : Distance over which the queue extends at link 0, mi 1 _link iV is same as previously defined The entering speed for the link is 0 _ 1 _2link q a link iD a V (3-23) Where1 _ link iV , aa , and 0 _ link qD are same as previously defined PAGE 64 64 There are two queue lengths, one is 0 _ link qD and the other is1 _ link qD . Figure 3-11 shows the different location of 0 _link qD and 1 _link qD Vehicles have a maximum operating speed as the limitation of their speeds, thus 1 _link iV has to be compared with mV , the maximum operating speed on the arterial, if 1 _ link iV > mV , use mV instead of 1 _link iV . The entering speed for the vehicles arrive at Condition 2 at the first intersection is calculated as below: Since the vehicle encounters C ondition 2 at the first intersection, its en tering speed for the link is the discharging speed at the upstream intersection: 0 _ 1 _ link d link iV V (3-24) Where 0 _ link dV : The discharging speed from link 0 Based on the assumption that the vehicles discharge with a constant speed, 0 _ link dV can be estimated by finding the speed at the time poi nt when vehicles joining the queue at intersection 1. The entering speed for the vehicles arrive at Condition 3 at the first intersection is calculated as below: Since the vehicle encounters C ondition 3 at the first intersection, its en tering speed for the link is the maximum speed at the upstream intersection: m link iV V 1 _ (3-25) With the equations for0V , the link travel time equation can be developed as follows: Link travel time for11P: PAGE 65 65 Q link q d m m link q d m a link q d m a link q d m Q Q MWT a D a V V D a V a D a V s a D a V MT WT T tt 1 _ 1 _ 2 0 _ 2 0 _ 112 2 2 2 2 (3-26) Link travel time for12P: d link q d d m m link q d d m a link q d m a link q d m Q MV D a V V V D a V V a D a V s a D a V MT T tt1 _ 1 _ 2 2 0 _ 2 0 _ 122 2 2 2 (3-27) Link travel time for13P : m a link q d m a link q d m MV a D a V s a D a V T tt 2 2 20 _ 2 0 _ 13 (3-28) Link travel time for21P: Q link q d m m link q d m a link d m a link d m Q Q MWT a D a V V D a V a V V s a V V MT WT T tt 1 _ 1 _ 2 2 0 _ 2 0 _ 212 2 2 (3-29) Link travel time for22P: d link q d link d m m link q d link d m a link d m a link d m Q MV D a V V V D a V V a V V s a V V MT T tt1 _ 1 _ 1 _ 2 1 _ 2 2 0 _ 2 0 _ 222 2 (3-30) Link travel time for23P : m a link d m a link d m MV a V V s a V V T tt 22 0 _ 2 0 _ 23 (3-31) Link travel time for31P : Q link q d m m link q d m Q Q MWT a D a V V D a V s MT WT T tt 1 _ 1 _ 2 312 2 (3-32) Link travel time for32P : PAGE 66 66 d link q d link d m m link q d link d m Q MV D a V V V D a V V s MT T tt1 _ 1 _ 1 _ 2 1 _ 2 322 (3-33) Link travel time for33P : m MV s T tt 33 (3-34) The trajectories for the nine possibilities are shown in Figure 3-12. The next step is to develop the equations to estimate the probability th at each flow profile occurs. All the vehicles arrive in Condition 1 at the firs t intersection departs when Conditi on 2 starts, these vehicles have possibility to arrive in Condition 1, 2 and 3 at the next intersection. As the same logic, the vehicles arrive in Condition 2 and 3 at the first intersection can al so distributed into Condition 1, 2 and 3 at the next intersection. To find the probability of the 9 po ssibilities can be later used as the probability of flow profiles. As shown in Figure 3-13, the probabilities of specific condition occurr ence are determined using the offset,MT, green time, red time, and cycle length. The method to calculate the proba bilities are as follows: Step 1: for vehicles arrive in Condition 1, fi nd the travel time for the first vehicle and the last vehicle. And then locate the travel time s for these two at the time bar on Figure 3-13, the time range between the two travel times are the possible travel times for the entire group of vehicles. Step 2: if the time range overlap with the condi tions at next intersection, find the length of each interval, and then calculate the percentage for each interval, that is the probability that each flow profile occurs. The same steps can be applied for vehicles arrive in Condition 2 and 3 at the first intersection also, therefore, th e 9 probabilities for the 9 flow profiles can be estimated. One problem in calculating the tr avel time is travel time de pends on the conditions that the vehicles will meet in the next intersection; however, this cannot be determined beforehand. The solution to this problem is to calculate lite rately, the detailed procedure is as follows: PAGE 67 67 Suppose the vehicle arrive in Condition 1, defi ne the trajectory of the vehicle and then calculate the travel time. And then locate the time in the time bar for the next interaction, if the time point falls into the range for Condition 1 at the next intersection, then the calculation is correct. Otherwise, continue with assumptions that the vehicle arrive in Condition 2 or 3, until the calculation is correct. This method is applied to an example and it is shown at the e nd of the chapter to demonstrate the whole process. 3.2.1.2 Travel time estimation under congested conditions From Figure 3-3, it is found that the three conditions are all co mpleted during one cycle. If the sum of the three conditions is greater than th e cycle length, not all the three conditions can be completed within one cycle. Thus, some vehicles cannot pass the intersection and they form a residual queue. Residual queue is defined as the group of vehicles that cannot pass the intersection during the first green interval afte r they arrive. In the field, there are two possibilities: a) Condition 3 is terminated; this does not create a residual queue because Condition 3 is the time difference between the di scharge of the current queue and beginning of the next cycle. b) Condition 2 is terminated, as shown in Figure 3-14 which creates a residual queue. If the flow and density do not change fr om cycle to cycle, the residual queue keeps increasing as shown in Figure 3-15. For example, if the residual queue at the first cycle is n , the new residual queue at the next cycle will be 2 n ; the next one will be 3 n , and so on. The residual queue cannot be reduced unl ess demand is reduced. As shown above, congested conditions are defi ned to occur when at least one vehicle cannot pass the intersection during th e first green interval after th ey arrive. According to this definition the approach is congested wh en condition 2 is terminated earlier. PAGE 68 68 If Condition 2 is terminated, the residual queue can be calculated based on shockwave analysis. As shown in Figure 3-16, the difference betw een when Condition 2 is completed and when it is terminated can be calculated as: Complete Condition 2 time interval: AB BC AB AC BC AB BC ABw w rw w w w w rw 1 Terminated Condition 2 time interval: r w w rw CAB BC AB The difference: r w w rw C w w rw w w w w rwAB BC AB AB BC AB AC BC AB BC AB 1 The residual queue can be calculated as: 22Cond Q Diff QM cond R Thus, 22cond M RDiff Cond Q Q (3-35) Where RQ: The residual queue, ft MQ: The maximum queue, ft 2 condDiff : The difference between the completed Condition 2 and the terminated Condition 2, sec After the residual queue is calculated, the total queue can be estimated by adding the residual queue length to th e initial queue length. The travel times can be estimated cycle by cycle and compared to the cycleÂ–based simulated travel times. Travel time varies as a function of the queue le ngth. For the first cycle, there is no residual queue so the expected travel time can be calculate d as before. Starting from the second cycle, the residual queue keeps increasing from RQto ( n -1)RQ. The residual queue has to be added to the current queue length when calculating travel ti me. It can affect travel time in two ways: PAGE 69 69 MT: this part of travel time happens on the se gment which is not occupied by queue. If the residual queue exists and keeps increasing, the segment length is reduced and MT is affected. QMT : this part of travel time occurs when the queue is discharging. Since the queue length is increased, MTQ is also affected. The queue length used in calculating MT and QMT , it is defined as the average queue length. It has to be adjusted according to the residual queue: As shown in Figure 3-17 using the second cycl e as an example, the new average queue is the average queue length of the queue formed in Condition 1 and Condition 2. The general calculation steps are as follows: For the nth cycle length, the residual que ue at the beginning of the cycle is ( n -1)RQ, and the residual queue at the end of the cycle is nRQ. The average queue length for Condition 1 is R MQ n Q ) 1 ( 2 The average queue length for Condition 2 is R R MnQ Q Q 2 Thus, the new average queue length is R MQ n Q ) 4 3 ( 2 There are two groups of vehicles. The first one is the vehicles join ing the residual queue and waiting for one more cycle to pass the intersecti on. The second one is the vehicles passing the intersection during the current cycle. Their trav el times are different since the travel time for the queued vehicle is the travel time for the un-queued vehicle plus one cycle length. The percentage of each group has to be calculated in order to calculate an expected travel time. Use the second cycle as an example: As shown in Figure 3-18, the maximum queue length for the cycle isR MQ Q , the residual queue when the cycle is ended isRQ 2, the discharged queue length PAGE 70 70 isR MQ Q , if the value is greater thanRQ, then the percentage for the first group is %R M RQ Q Q . If the value is smaller thanRQ, then the percentage for the first group is 100%. Under this situation, there are vehicles that have to wait mo re than one cycle. To generalize these for the nth cycle, the discharged queue isR M R R MQ Q Q n nQ Q ) 1 (. Thus, no matter which cycle it is, the discharged queue remains the same. With consideration of the above problems, to calculate the expected travel time for congested conditions, the steps are shown below: Starting from the first cycle, calculate the expected travel time as before. Starting from the second cycle, calculate the residual queue for each cycle and add the queue length when calculating TM and MTQ. Calculate the expected travel time. Determine the percentage of vehicles that belongs to the queued group and the un-queued group. Calculate the expected travel time as queued non TT queued TT TT E _ % _ % ) ( It is found that ifR MQ Q is smaller thanRQ, some vehicles in the residual queue cannot pass the intersection even after waiting for one cy cle. When calculating the expected travel time, this situation has to be considered. To generalize the method, the discharged queue is alwaysR MQ Q . In this queue, the percentage of vehicles arriving during a given cycle and discha rged during current cycle would be determined. Then the expected travel time can be calculated based on the travel times and percentages associated with them. PAGE 71 71 3.2.2 Inputs and Outputs in the Model This section describes the i nputs and outputs for the travel time estimation model for a small system of two consecutive in tersections and a link between them. 3.2.2.1 Inputs There are several inputs in the model to estimate travel time. They are: flow rate at the downstream intersection, entering speed at the downstream inters ection, the link length, and the queue length. Among the variables, flow rate ne eds to be estimated and the discharging speed needs to be assumed. Flow rate at the downstream intersection based on upstream flow. The flow rate for the internal arterial appro ach can be calculated ba sed on the initial flow rate: flow rate of the incoming appro aches is an input to the methodology: SL NR WL WR link f link if f f f f f_ 1 int _ 1 int _ 1 int _ 1 int 0 _ 1 _ (3-37) Where 1 _link if : Entering (initial) flow rate at li nk 1 upstream intersection, veh/hr/ln 0 _link ff : Final flow rate at link 0 dow nstream intersection, veh/hr/ln WRf_ 1 int: Flow rate at intersection 1 from we stbound right turning movement, veh/hr/ln WLf_ 1 int: Flow rate at intersection 1 from we stbound left turning movement, veh/hr/ln NRf_ 1 int: Flow rate at intersection 1 from so uthbound right turning movement, veh/hr/ln SLf_ 1 int: Flow rate at intersection 1 from nor thbound left turning movement, veh/hr/ln Discharging speed at each intersection. For the arterial travel time, the entering speed for the next intersection is a very important value in estimating travel time for multiple intersections, as shown before; discharging speed from the upstream link is the entering speed fo r the next link. Based on the previous entering speed calculation, entering speeds for different conditions at downstream intersections are: Entering speed for Condition 1: 0 _ 1 _2link q a link iD a V or mV (3-23) PAGE 72 72 Entering speed for Condition 2: 0 _ 1 _link d link iV V (3-24) Entering speed for Condition 3: m link iV V 1 _ (3-25) The entering speed for Condition 1 and 3 ar e known. However, the entering speed for Condition 2 needs to be assumed. Vehicles are discharging so that the flow is the maximum flow, according to the flow-density-speed relationship, assuming a linear speed-density relationship, the speed is approximately 2mV. It is assumed that this will be the speed for Condition 2. List of assumptions. Besides the discharging speed a ssumption, there are several other assumptions that are to be used; these values are obtaine d from four resources to get the appropriate values: Traffic Engineering Handbook (for acceleration rate), A Policy on Geometric Design of Highways and Streets (for deceleration rate), the HCM 2000 (f or Single Vehicle Spacing), and field data collection (for Maximum Operating Speed). They are as follows: Acceleration rate: 10.76 2ft/s Deceleration rate: -11.2 2ft/s Maximum Operating Speed: as the speed limit of the arterial 3.3.2.2 Outputs The outputs of the model are the expected travel time, the travel time distribution, and the variance of the travel time for the whole arteri al. A detailed summary of inputs and outputs for the model are included in table 3-1. PAGE 73 73 3.3 An Example for Applying the Model The following example illustrates an applic ation of the model. Suppose there is a signalized arterial with two intersections (Figure 3-20). The question is what is the expected travel time on link 1(between intersection 1 and intersection 2)? 3.3.1 Inputs for the Model The link length between intersection 1 and 2 is 1320 ft, or 0.25 mile. The arterial has two-way traffic, with one lane per direction. Maximum operating speed: The maximum operati ng speed is assumed to be equal to the speed limit of the arterial, which is 30 mph. Traffic signal control: There are pre-timed signa ls at each intersecti on, the cycle length is 80 seconds, the main arterial effective green tim e is 60 seconds, and effective red time is 20 seconds. The offset between intersections is 30 seconds. Traffic volume:0 _link ff is 900 veh/h/ln. At each intersection, traffic volumes are as follows: NB and SB LT is 50 veh/hr/ln, NB and SB RT is 100 veh/hr/ln, EB LT is 5% of the0 _link ff , EB RT is 10% of the0 _link ff , EB TH is 85% of the0 _link ff . Acceleration and decel eration rates are 102ft/s. 3.3.2 Model Development The goal is to find the expected travel time between intersection 1 a nd intersection 2, as well as the variance of the travel time To estim ate travel time, the following steps are to be completed: 3.3.2.1 Step 1: Calculate the entering flows at two intersections Intersection1: veh/h 915 100 50 % 5 900 % 10 900 900_ 1 int _ 1 int _ 1 int _ 1 int 0 _ 1 _ SL NR WL WR link f link if f f f f f Intersection2: veh/h 928 100 50 % 85 915_ 2 int _ 2 int _ 2 int _ 2 int 1 _ 2 _ SL NR WL WR link f link if f f f f f PAGE 74 74 Where 2 _link if : Entering (initial) flow rate at link 2 1 _link ff : Final flow rate at link 1 WRf_ 2 int: Flow rate at intersection 2 from westbound right turning movement WLf_ 2 int: Flow rate at intersection 2 fr om westbound left turning movement NRf_ 2 int: Flow rate at intersection 2 from southbound right turning movement SLf_ 2 int: Flow rate at intersection 2 fr om northbound left turning movement 3.3.2.2 Step 2: Calculate the probab ilities for Conditions 1, 2, and 3 Using intersection 1 as an example, the proba bility for Conditions 1, 2, and 3 can be calculated as A B A ABk k q w , C B C BCk k q w , C A C A ACk k q q w In this example, based on the assumption of a linear function for the speed-density relationaship (Figure 3-21), the calculations are as follows: veh/hr 9151 int _Aq , 18001 int _Cq veh/hr, 120 2 / 30 1800 2 /1 int _ 1 int _ m C CV q k veh/mile 240 2 2 / 30 1800 2 2 /1 int _ 1 int _ m C BV q k veh/mile Based on the four known values, the mathemati cal expression of the flow-density curve can be calculated, it is X X Y3600 152 1 int _Aq is 915 veh/hr, thus, 1 int _ 2 1 int _ 1 int _3600 15A A Ak k k , 254 . 01 int _Ak veh/mi Thus, 817 . 3 254 . 0 180 915 A B A ABk k q w, 00 . 15 120 240 1800 C B C BCk k q w, 39 . 7 120 254 . 0 1800 900 C A C A ACk k q q w 335 . 0 80 817 . 3 00 . 15 00 . 15 20 1 Pr C w w w R C w w w R R ConditionAB BC BC AB BC AB 173 . 0 80 ) 39 . 7 ( ) 817 . 3 00 . 15 ( )) 00 . 15 ( 817 . 3 ( 20 2 Pr C w w w w w R ConditionAC AB BC BC AB PAGE 75 75 492 . 0 173 . 0 335 . 0 1 1 3 Pr C w w w w w R C w w w R ConditionAC AB BC BC AB AB BC BC Thus, the same method of calcu lation is performed for inte rsection 2. Th e calculation results for intersection 1 and inters ection 2 are shown in Table 3-2. 3.3.2.3 Step 3: Calculate the queue lengths and determine 1 _link iVfor three conditions 1 _link qD for condition 1 and Condition 2 1 _link qD is one factor in calculating MTfor vehicles arriving at th e intersection in Condition 1 and Condition 2. The average 1 _link qD for the stopped queue in Condition 1 and the moving queue in Condition 2 are approximately the same. ft 49 . 76 817 . 3 15 ) 817 . 3 ( 15 7200 5280 20 5280 ) ( 7200 2 .1 _ AB BC AB BC M link qw w w w r Q D avg ft 08 . 75 870 . 3 15 ) 870 . 3 ( 15 7200 5280 20 5280 ) ( 7200 _ 2 .0 _ AB BC AB BC M link qw w w w r spacing veh Q D avg Calculate 1 _link iVfor Condition 1, Condition2, and Condition3: Entering speed for Condition 1: m link q a link iV D a V mph 42 . 26 ft/s 75 . 38 08 . 75 10 2 20 _ 1 _ Thus, mph 42 . 261 _link iV Entering speed for Condition 2: mph 00 . 15 2 30 20 _ 1 _ m link d link iV V V Entering speed for Condition 3: mph 301 _ m link iV V PAGE 76 76 3.3.2.4 Step 4: Calculate the minimum and maxi mum travel time for each possible flow profile. For vehicle starts from Condition 1 at the first intersection, ca lculate the travel time for the first vehicle in the group, and th en locate the time point on the travel time bar for the next intersection: 11tt27.38 sec, it is found that the time point for this travel time on the time bar for the next intersection is 54.21 sec, which is smaller than the end time point of Condition 1 on the next intersection (56.95 sec). Thus, the minimu m travel time point is 54.21 seconds. Find the time intervals for each possible flow profile and then calculate the probability. For vehicles that arrive in Condition 1 at the first intersection, there is only one time interval, the probability is 0 . 1 21 . 54 95 . 56 21 . 54 95 . 5611 P 3.3.2.5 Step 5: Calculate travel time componentsMT,QMT ,andQWT, the total travel time for 9 flow profiles, the expected travel time, the standard deviation of travel time, and the travel time distribution MTare calculated based on the trajectory of the vehicles QWT: is approximately calculated as half of the red time acco rding to shockwave analysis.QMT: is based on the queue spacing and the acceleration rate. Travel times can be calculate d for 9 different flow pr ofiles. The travel times are shown in Table 3-2. The expected travel time can be calculated fo r the three different gr oups of vehicles that arrive in different conditions at intersection 1. The trav el times, the associat ed probabilities, and the expected travel times are shown in Table 3-3. % % % ) (13 13 12 12 11 11 1Scenario TT Scenario TT Scenario TT TT ECond % % % ) (23 23 22 22 21 21 2Scenario TT Scenario TT Scenario TT TT ECond % % % ) (33 33 32 32 31 31 3Scenario TT Scenario TT Scenario TT TT ECond PAGE 77 77 The total expected travel time is calculated as follows: % 3 % 2 % 1 ) (3 2 1Cond TT Cond TT Cond TT TT ECond Cond Cond sec 94 . 35 59 . 0 29 . 32 41 . 0 29 . 41 ) ( TT E The standard deviation of travel time is sec 74 . 4 ) ( ) (2 2 2 2 i i i ix p x p x E x E The travel time distribution is shown in Figure 3-22. Figure 3-1. Definition of a link Figure 3-2. Decompos ition of travel time PAGE 78 78 Figure 3-3. The three conditions s: Total length of the link (exclude the length of intersection), ft As: Accelerating distance, ft Cs: Constant speed distance, ft Ds: The distance from the end point of S2 till the end of the link, ft qD: Distance over which the queue extends, ft Figure 3-4. Link length PAGE 79 79 Figure 3-5. MT Equation for Condition 1 Figure 3-6. MTEquation for Condition 2 Figure 3-7. MT Equation for Condition 3 PAGE 80 80 Figure 3-8. Shockwave analysis to find the interval length s for conditions Figure 3-9. Probabili ties of changing from one c ondition to another condition Figure 3-10. Study area PAGE 81 81 Figure 3-11. 0 _ link qDand 1 _ link qD PAGE 82 82 :11PFrom Condition 1 to Condition 1 Entering speed for the do wnstream intersection: 0 _2link q aD aor mV :12PFrom Condition 1 to Condition 2 Entering speed for the do wnstream intersection: 0 _2link q aD a or mV :13PFrom Condition 1 to Condition 3 Entering speed for the do wnstream intersection: 0 _2link q aD a or mV :21PFrom Condition 2 to Condition 1 Entering speed for the downstream intersection: 0 _ link dV :22PFrom Condition 2 to Condition 2 Entering speed for the do wnstream intersection: 0 _ link dV :23PFrom Condition 2 to Condition 3 Entering speed for the do wnstream intersection: 0 _ link dV :31PFrom Condition 3 to Condition 1 Entering speed for the do wnstream intersection: mV :32PFrom Condition 3 to Condition 2 Entering speed for the do wnstream intersection: mV :33PFrom Condition 3 to Condition 3 Entering speed for the do wnstream intersection: mV Figure 3-12. The trajectories for the nine flow profiles PAGE 83 83 Figure 3-13. Time and space diagram Figure 3-14. Residual queue due to Condition 2 is terminated PAGE 84 84 Figure 3-15. Increasing residual qu eue due to terminated Condition 2 Figure 3-16. Residual queue calculation when Condition 2 is terminated Figure 3-17. Average queue length under congested conditions PAGE 85 85 Figure 3-18. The discharged queue Figure 3-19. Flow at downstream intersection Figure 3-20. Sketch for signalized arterial Figure 3-21. Flow-density-speed relationship PAGE 86 86 Travel Time Distribution for the Example 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3030.0330.5532.2932.3232.8441.2946.4646.66 TT(sec)Probability Figure 3-22. Travel time di stribution for the example Table 3-1. Inputs and outputs fo r travel time estimation model Input Output 1 _ link if, capacity, density 11P, 12P,13P,21P,22P,23P,31P,32P,33P r , length Cycle _ 11tt,12tt,13tt,21tt,22tt,23tt,31tt,32tt,33tt s Expected travel time, variance of travel time, distribution of travel time 1 _ link iV,mV aa,da 0 _ link qD,1 _ link qD QWT Table 3-2. Probabilities for three conditions at intersection 1 and 2 Pr{1} Pr{2} Pr{3} Intersection 1 0.335 0.173 0.492 Intersection 2 0.337 0.179 0.484 PAGE 87 87 Table 3-3. Travel times, the associated prob abilities, and the expected travel times Probability TT Expected TT TT11 0.41 41.29 TT12 0.00 32.32 TT13 0.00 30.03 41.29 TT21 0.00 46.66 TT22 0.00 32.84 TT23 0.00 30.55 0 TT31 0.00 46.46 TT32 0.59 32.29 TT33 0.00 30.00 32.29 PAGE 88 88 CHAPTER 4 DATA COLLECTION Chapter 4 describes the data collection for this research. The objective of data collection is to collect field travel times as well as other traffic data to validate the analytical model. 4.1 Travel Time Data Collection The data collection helps validate the analytical model. Field travel time can be obtained and compared to the estimated travel time from the analytical model. There are 4 sites where field trav el time data were collected. As shown in Figure 4-1, one existing data se t from Gainesville, FL. (Newberry Road) is used. This data was provided by Dr. Washburn et al. (2006). Data were collected on the arterial segment which includes five intersections. The total length of the segment is 1,883 ft. Four cameras were placed along the segment; they are f acing to the east. Travel time and volume data used in this research were coll ected on 4/30/2005 and 5/1/2005. Travel time data were collected based on record ed videos. The method used was to record the time when one particular vehicle enters th e first intersection and when it leaves the last intersection. Table 4-1 shows the number of travel time runs that were conducted. Another 3 sites for data collection are Beaver_Pugh (Figure 4-2), Beaver_Sparks (Figure 4-3), and Park (Figure 4-4) in State Co llege, Pennsylvania. Please note that the three sketches are not to scale. On Beaver_Pugh (Figure 4-2) and Beaver_Spark s (Figure 4-3), data were collected for a link between two signalized intersections. The link length for Beaver_Pugh is 1263 ft, and that of Beaver_Sparks is 1429 ft. The two arterial segments all have two lanes, one-way traffic. Travel time data were collected during morning peak hour (7:30 am Â– 8:30 am), midday (12:30 pm1:30 pm), and the pm peak hour (5:00pm-6:00pm) in May 2004. PAGE 89 89 On Park Ave (Figure 4-4), data were collet ed for two consecutive links between three consecutive intersections. The link length betwee n North Atherton Street and Allen Street is 1420 ft, the link length between Allen Street and Grove Alley Street is 1471 ft. The arterial has two-way traffic. There is one lane for each direct ion. Travel time data were collected during the midday (12:30 pm-1:30 pm) and the pm peak hour (5:00pm-6:00pm) in May 2004. Travel time data were collected through probe vehicles. The method used was to record the time when the probe vehicle enters the first intersection and when it leaves the last intersection. In each run, the probe vehicle traveled along the arterial with the speed limit. When speed was lower than 5 mph below the speed limit, the vehi cle was considered to be delayed. The total travel time and the delay time were measured. The travel time estimated by the model will be compared to the field data and the model will be refined as appropriate. Table 4-2 shows the number of travel time runs that were conducted. 4.2 Other Traffic Data Collection Other data were collected and will be us ed as inputs for the model. They are: Link length; Number of Lanes (full lanes, turning bays) Traffic control (signal timing plan, offset); Traffic volume; and Maximum operating speed (the speed limit). The data were collected the same time while the travel times were collected at each site. PAGE 90 90 Figure 4-1. Sketch of site at Newberry Rd. Figure 4-2. Sketch for site Beaver_ Pugh Figure 4-3. Sketch for site Beaver_ Sparks Figure 4-4. Sketch for site Park Table 4-1. Number of samples collected for travel time data from Newberry Rd. Number of samples 4/30/2005 4/31/2005 EB 36 48 WB 38 31 PAGE 91 91 Table 4-2. Number of samples for travel time data for Beaver_Pugh, Beaver_Sparks, and Park Number of samples Beaver _Pugh Beaver_Sparks Park EB_AM 8 8 N/A EB_MID 7 7 7 EB_PM 7 7 5 WB_AM N/A N/A N/A WB_MID N/A N/A 7 WB_PM N/A N/A 5 Note: N/A means the data are not available for the case PAGE 92 92 CHAPTER 5 MODEL VALIDATION The analytical model is based on pre-timed signal timing. However, the signal timings from the field are all semi-actuated and these signal timings cannot be directly used in the model. Thus, they need to be approximated to pre-ti med signal timing for the model validation. Therefore, there are two steps in the model validation process. The first is to simulate the real traffic conditions in AIMS UN and compare the simulated travel times to the field travel times. This step is to make sure that AIMS UN can replicate field conditions accurately. The second step is to adjust AIMSUN to approximat e the semi-actuated to pre-timed signal timings and compare the travel times from model estimation to simulation. 5.1 Simulate the Real Traffic Condition and Compare to the Field Data AIMSUN is used as the simulation tool in this research. There are two steps in this part: first, replicate the real traffic conditions with the semi-actuated signal timing; second, compare the simulated travel times to the field travel times to make sure AIMSUN can simulate accurately. 5.1.1 Replicate the Real Traffic Condition s with Semi-Actuated Signal Timing There are 4 sites that field trav el time data were collected. Am ong the four sites, data were collected during different time periods for EB and/ or WB traffic. Thus, each time period for each direction in each site is counted as one case. There are total fourteen cases and their signal timings are all semi-actuated. 5.1.1.1 Site 1: Beaver_Pugh Site 1 is the Beaver Ave. between Pugh St. and Garner St. in State College, PA. Travel times were collected during three time periods AM, Mid and PM. Since the segment is one-way arterial, the travel times collected are only for EB. Thus, the 3 cases for Site 1 are: PAGE 93 93 Case 1: Beaver_Pugh_AM_EB. Case 2: Beaver_Pugh_Mid_EB. Case 3: Beaver_Pugh_PM_EB. The signal timings for the 3 cas es of the site are shown in Figure 5-1, Figure 5-2, and Figure 5-3. 5.1.1.2 Site 2: Beaver_Sparks Site 2 is the Beaver Ave. between Sparks St. and Atherton St. in State College, PA. Travel times were collected during three time periods AM, Mid and PM. Since the segment is one-way arterial, the travel times collected are only for EB. Thus, the 3 cases for Site 2 are: Case 4: Beaver_Sparks_AM_EB. Case 5: Beaver_Sparks_Mid_EB. Case 6: Beaver_Sparks_PM_EB. The signal timings for the 3 cas es of the site are shown in Figure 5-4, Figure 5-5, and Figure 5-6. 5.1.1.3 Site 3: Park Site 3 is the Park Ave. between Atherton St. and Grove Alley St. in State College, PA. Travel times were collected during two time peri ods Mid and PM for both EB and WB. Thus, the 4 cases for Site 3 are: Case 7: Park_Mid_EB. Case 8: Park_Mid_WB. Case 9: Park_PM_EB. Case 10: Park_PM_WB. The signal timings for the Case 7 and 8 is ar e shown in Figure 5-7 a nd that for Case 9 and 10 is shown in Figure 5-8. PAGE 94 945.1.1.4 Site 4: Newberry rd. Site 4 is the Newberry Road betw een I-75 exit ramp and the NW 66th St. in Gainesville, FL. Travel times were collected during two time periods April 30th, 2005 and May 1st, 2005 for both EB and WB. Thus, the 4 cases for Site 4 are: Case 11: Newberry_4_30_EB. Case 12: Newberry_4_30_WB. Case 13: Newberry_5_1_EB. Case 14: Newberry_5_1_WB. The signal timings for the two cases from May 1st of the site are the same as the one for April 30th.The signal timings for the 4 cases are shown in Figure 5-9. Besides signal timings, other tr affic characteristics were also collected for each site. They are listed in Chapter 4. 5.1.2 Simulate the Travel Times and Compare with Filed Travel Times AIMSUN NG 5.0 is used to replicate the real traffic condition. The simulated travel times cannot be the exact same value as the field travel times. However, they have to be statistically the same. To validate the simulated travel times equal to the field travel times , error tolerances are calculated and the acceptable travel time inte rvals are estimated based on the tolerances: n zs n s z e 2 2 (5-1) Where e: tolerance s: the standard deviation n: the number of samples The zvalue is determined by the confidence in terval (CI). If the CI is 90%, the zvalue is 1.645, if the CI is 95%, the zvalue is 1.96. For this research, the CI is set to 95%. PAGE 95 95 Based on the samples that were collected, the tolerances and the acceptable travel times for each scenario are calculated. They are presented in Table 5-1. From Table 5-1, if the travel time from the simulator is within the acceptable travel time range, it is considered an acceptable simulated trav el time. For some of the cases, the tolerance is high compared to other cases. This is because the travel time varies significantly during the field travel time collection. Fourteen cases are simula ted in AIMSUN. Each of them is simulated for 1 hour, and travel times for the stream (between the left-most intersection and the right-most intersection) are reported for ev ery 10 minutes for un-congested conditions and every cycle for congested conditions. Thus, if the site is not cong ested, in the 1 hour simulation period, 6 travel times are reported. 10 replications are made for each of the cases and travel time is averaged based on all replications. If it is congested, travel times were collected for each cycle and for 10 replications before spillback occurs. The simulated travel times for each of the cases are listed in Table 5-2, compared to the field data. Among all simulations, only th e simulated Beaver_Pugh_Mid_EB does not statistically equal the field trav el time. Because this case canno t be accurately replicated in simulation, this case is removed from model validation. 5.2 Compare the Model Estimation to the Simulation There are two steps in this part. First, appr oximate the semi-actuated to pre-timed signal timing and replace them in simulation. Second, co mpare the model estimated travel times to the simulated travel times to make sure AIMSUN can simulate accurately. 5.2.1 Replicate with Pre-timed Signal Timing In this research, the travel time estimatio n model is based on pre-timed signal timing. However, travel times are collected on arterials with semi-actuated signal timing. Thus, the semi- PAGE 96 96 actuated signal timings have to be approximated to pre-timed signal timing in order to test the capability of the analytical model. In order to obtain appropriate signal timings, there are several criteria in transforming the signal plans: Cycle length of all intersections have to be same in order to use offset in model estimation: the method is to get the sum of the green times for all intersections, use the longest cycle as the standard cycle length, and rescale all green times from sm aller cycles so that the new cycle length after the rescale is the same as the longest cycle length. Use the minimum green interval in actuated signal timing as the green interval for phases that have minimum recall, and use the maximu m green interval as the green interval for phases that have do not have minimum recall. Dual ring signal timings have to be changed to single ring signal timing since it is beyond the ability of the analytical model. 5.2.1.1 Site 1: Beaver_Pugh The approximate signal timings for Site 1 are shown in Figure 5-10 (for Case 1) and Figure 5-11 (for Case 3). According to the criteria, the green times for one intersection in Case 1(Beaver_Pugh_AM_EB) are changed because the cycle lengths of the two intersections are different in the field. 5.2.1.2 Site 2: Beaver_Sparks The approximate signal timings for Site 2 are shown in Figure 5-12 (for Case 4), Figure 5-13 (for Case 5), and Figure 5-14 (for Case 6). According to the criteria, the green times for one intersection in Case 5 (Beaver_S parks_Mid_EB) are changed becau se the cycle lengths of the two intersections are different. 5.2.1.3 Site 3: Park The approximate signal timings for Site 3 are shown in Figure 5-15 (for Case 7 and Case 8) and Figure 5-16 (f or Case 9 and Case 10). PAGE 97 975.2.1.4 Site 4: Newberry Rd. The approximate signal timings for Site 4 are shown in Figure 5-17 (for Case 11 and Case 13) and Figure 5-18 (for Case 12 and Case 14). Case 11 and 13 are for EB, Case 12 and 14 are for WB. In Figure 5-17, for the second intersection, the EB through movement has green time all the time. Thus, intersection 2 does not ex ist for EB, it works as same as an uncontrolled intersection. The third and the fifth intersection have dual ring phasing. As shown in Figure 5-17, the two phases inside one dual ring are co mbined into one single phase. The green time for the new phase is determined by averaging the green times from the two phases. In Figure 5-18, the second intersection exists since the WB thro ugh movement does not have green time all the time. The rest of the intersections are approximated the same way as in Figure 5-17. This step is to create a set of inputs that are applicable for both the simulation and the analytical model. The heavy vehicle volume is not considered in the analytical model. Thus, the volume of heavy vehicles is replaced on equal volume of passenger cars. 5.2.2 Comparisons and Conclusions In this section, the pre-timed signal timings are used to obtain simulated travel times from AIMSUN. The simulated travel time is then comp ared to the model estimates. Travel times are obtained both from AIMSUN and the analytical mo del for different offsets. The offset starts from 0 seconds and is increased by 10 seconds until it reaches the cycle length of the case. The results are shown in Tables 5-3 to Table 5-15 fo r each case. The tables provide the average travel time from the simulation and the model, the st andard deviation as estimated in the model estimated travel time, the percent difference betw een the average travel time from the simulation and the model, and the travel time estimated by the HCM 2000 travel time model. The travel time distributions of Case 1(Beaver_Pugh_AM_EB) are shown in Figure 5-19 to Figure 5-21 as a demonstration of how travel times are distributed as the offset changes. PAGE 98 98 The comparisons between the average simulated travel time and the average model estimated travel time are shown in figures 5-22 to Figure 5-34. Among all the cases, only Case 10 (Park_PM_WB) is a congested case. Unlike the ot her cases which the travel time is calculated for each offset, its travel time is calculated cycl e by cycle before spillback occurs. The queue length changes every cycle and travel time change s with it. The detailed queue length calculation is shown in Figure 5-35. Observations based on these comparisons are summarized below: Generally the trends in travel times from the analytical model are comparable with the simulation. The percent difference between the simulated and the model-estimated travel times are shown in tables Table 5-3 to 5-15. As shown in the tables, the percent difference for Beaver_Pugh site, Beaver_Sparks site, and Park site are smaller than that of the Newberry cases. The analytical model does not give good resul ts when multiple driveways are present. The percent difference for Case 11, 12, 13, an d 14 are shown in Table 5-12, Table 5-13, Table 5-14, and Table 5-15. As shown in Figure 536, the likely reason for that is the driveway densities at these sites compared to other sites are much higher. Vehicles often have to slow down or even stop prior to the dr iveway to avoid blocking the en trance/ exit at the driveways and this affects the travel time. Travel time estima tion in the analytical m odel does not consider the deceleration and acceleration at driveways. Howe ver, the microscopic simulation records the travel time of each vehicle that goes through the corridor and the delays at the driveways are considered. Thus, the travel times are quite different. The travel times from the model do not ch ange smoothly like the ones from the simulation. PAGE 99 99 In the analytical model estimation, the travel time changes as the offset changes. Travel time is calculated as the expected travel time of a ll selected travel times. If the offset increases by10 seconds, in some of the circumstances it does not change the set of selected travel times. Thus, travel times sometimes remain the same for several consecutive offsets. Max green: 100 sec Min green: 18 sec Y: 3 sec AR: 1 sec Intersection of Beaver Ave. and S. Pugh St. Yield Point: 20 sec Cycle Length: 45 sec Max green: 18sec Min green: 7 sec Y: 3 sec AR: 2 sec Max green: 100 sec Min green: 54 sec Y: 3.5 sec AR: 1.5 sec Intersection of Beaver Ave. and Garner St. Yield Point: 56 sec Cycle Length: 90 sec Max green: 26 sec Min green: 7 sec Y: 3.4 sec AR: 1.6 sec Figure 5-1. Semi-actuated signal ti ming for Case 1 (Beaver_Pugh_AM_EB) Max green: 100 sec Min green: 37 sec Y: 3 sec AR: 1 sec Intersection of Beaver Ave. and S. Pugh St. Yield Point: 3 sec Cycle Length: 80 sec Max green: 34sec Min green: 7 sec Y: 3 sec AR: 2 sec Max green: 100 sec Min green: 47 sec Y: 3.5 sec AR: 1.5 sec Intersection of Beaver Ave. and Garner St. Yield Point: 33 sec Cycle Length: 80 sec Max green: 23 sec Min green: 7 sec Y: 3.4 sec AR: 1.6 sec Figure 5-2. Semi-actuated signal timi ng for Case 2 (Beaver_Pugh_Mid_EB) PAGE 100 100 Max green: 100 sec Min green: 54 sec Y: 3 sec AR: 1 sec Intersection of Beaver Ave. and S. Pugh St. Yield Point: 20 sec Cycle Length: 100 sec Max green: 37sec Min green: 7 sec Y: 3 sec AR: 2 sec Max green: 100 sec Min green: 58 sec Y: 3.5 sec AR: 1.5 sec Intersection of Beaver Ave. and Garner St. Yield Point: 56 sec Cycle Length: 100 sec Max green: 32 sec Min green: 7 sec Y: 3.4 sec AR: 1.6 sec Figure 5-3. Semi-actuated signal ti ming for Case 3 (Beaver_Pugh_PM_EB) Max green: 100 sec Min green: 56 sec Y: 3 sec AR: 1 sec Intersection of B eaver Ave. and S. Sparks St. Yield Point: 1 sec Cycle Length: 90 sec Max green: 26sec Min green: 3 sec Y: 3 sec AR: 1 sec Max green: 100 sec Min green: 12 sec Y: 3 sec AR: 0 sec Max green: 100 sec Min green: 31 sec Y: 3 sec AR: 1 sec Intersection of Beaver Ave. and Atherton St. Yield Point: 32 sec Cycle Length: 90 sec Max green: 36 sec Min green: 7 sec Y: 3 sec AR: 1 sec Figure 5-4. Semi-actuated signal timi ng for Case 4 (Beaver_Sparks_AM_EB) PAGE 101 101 Max green: 100 sec Min green: 32 sec Y: 3 sec AR: 1 sec Intersection of B eaver Ave. and S. Sparks St. Yield Point: 1 sec Cycle Length: 65 sec Max green: 25sec Min green: 3 sec Y: 3 sec AR: 1 sec Max green: 100 sec Min green: 17 sec Y: 3 sec AR: 0 sec Max green: 100 sec Min green: 38 sec Y: 3 sec AR: 1 sec Intersection of Beaver Ave. and Atherton St. Yield Point: 32 sec Cycle Length: 90 sec Max green: 34 sec Min green: 7 sec Y: 3 sec AR: 1 sec Figure 5-5. Semi-actuated signal timi ng for Case 5 (Beaver_Sparks_Mid_EB) Max green: 100 sec Min green: 60 sec Y: 3 sec AR: 1 sec Intersection of B eaver Ave. and S. Sparks St. Yield Point: 71 sec Cycle Length: 100 sec Max green: 32sec Min green: 3 sec Y: 3 sec AR: 1 sec Max green: 100 sec Min green: 16 sec Y: 3 sec AR: 0 sec Max green: 100 sec Min green: 36 sec Y: 3 sec AR: 1 sec Intersection of Beaver Ave. and Atherton St. Yield Point: 90 sec Cycle Length: 100 sec Max green: 37 sec Min green: 7 sec Y: 3 sec AR: 1 sec Figure 5-6. Semi-actuated signal timi ng for Case 6 (Beaver_Sparks_PM_EB) PAGE 102 102 Max green: 22 sec Min green: 2 sec Y: 3 sec AR: 0 sec Max green: 100sec Min green: 38 sec Y: 3 sec AR: 2 sec Max green: 4sec Min green: 4 sec Y: 3 sec AR: 0 sec Intersection of Park Ave. and Atherton St. Yield Point: 86 sec Cycle Length: 100 sec Max green: 7 sec Min green: 2 sec Y: 3 sec AR: 0 sec Max green: 43 sec Min green: 23 sec Y: 3 sec AR: 1 sec Intersection of Park Ave. and Allen St. Yield Point: 10 sec Cycle Length: 85 sec Max green: 24sec Min green: 2 sec Y: 3 sec AR: 1 sec Max green: 7 sec Min green: 2 sec Y: 3 sec AR: 0 sec Max green: 45 sec Min green: 16 sec Y: 3 sec AR: 2 sec Intersection of Park Ave. and Grove Alley St. Yield Point: 26 sec Cycle Length: 85 sec Max green: 20 sec Min green: 2 sec Y: 3 sec AR: 2 sec Figure 5-7. Semi-actuated signal timing for Case 7 (Park_Mid_ EB) and Case 8 (Park_Mid_WB) PAGE 103 103 Max green: 19 sec Min green: 2 sec Y: 3 sec AR: 0 sec Max green: 100sec Min green: 44 sec Y: 3 sec AR: 2 sec Max green: 4sec Min green: 4 sec Y: 3 sec AR: 0 sec Intersection of Park Ave. and Atherton St. Yield Point: 68 sec Cycle Length: 100 sec Max green: 17sec Min green: 2 sec Y: 3 sec AR: 2 sec Max green: 11 sec Min green: 2 sec Y: 3 sec AR: 0 sec Max green: 43 sec Min green: 23 sec Y: 3 sec AR: 1 sec Intersection of Park Ave. and Allen St. Yield Point: 13 sec Cycle Length: 85 sec Max green: 20 sec Min green: 2 sec Y: 3 sec AR: 1 sec Max green: 7 sec Min green: 2 sec Y: 3 sec AR: 0 sec Max green: 39 sec Min green: 16 sec Y: 3 sec AR: 2 sec Intersection of Park Ave. and Grove Alley St. Yield Point: 26 sec Cycle Length: 85 sec Max green: 26 sec Min green: 2 sec Y: 3 sec AR: 2 sec Figure 5-8. Semi-actuated signal timing for Case 9 (Park_PM_EB) and Case 10 (Park_PM_WB) PAGE 104 104 Max green: 22 sec Min green: 4 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and I-75 NB off ramp (exit) Yield Point: 122 sec Cycle Length: 150 sec Max green: 200sec Min green: 118 sec Y: 3 sec AR: 2 sec Max green: 22 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 200 sec Min green: 76 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and I-75 NB off ramp (entrance) Yield Point: 122 sec Cycle Length: 150 sec Max green: 37 sec Min green: 6 sec Y: 3 sec AR: 2 sec Max green: 10 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 28 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 150 sec Min green: 90 sec Y: 3 sec AR:2 sec Max green: 100 sec Min green: 72 sec Y: 3 sec AR:2 sec Intersection of Newberry Rd. and NW 69th Street Yield Point: 121 sec Cycle Length: 150 sec Max green: 35 sec Min green: 8 sec Y: 3 sec AR:2 sec Max green: 35 sec Min green: 8 sec Y: 3 sec AR:2 sec Max green: 16 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 150 sec Min green: 72 sec Y: 3 sec AR: 2 sec Max green: 14 sec Min green: 4 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and Oaks Mall West Yield Point: 106 sec Cycle Length: 150 sec Max green: 28 sec Min green: 4 sec Y: 3 sec AR: 2 sec Figure 5-9. Semi-actuated signal timing for Case 11 (Newberry_4_30_EB) Case 12 (Newberry_4_30_WB) Case 13 (Newbe rry_5_1_EB) Cases 14(Newberry_5_1_WB) PAGE 105 105 Max green: 16 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 29 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 150 sec Min green: 72 sec Y: 3 sec AR:2 sec Max green: 150 sec Min green: 58 sec Y: 3 sec AR:2 sec Max green: 31 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 19 sec Min green: 4 sec Y: 3 sec AR:2 sec Intersection of Newberry Rd. and NW 66th Street Yield Point: 106 sec Cycle Length: 150 sec Max green: 11 sec Min green: 4 sec Y: 3 sec AR:2 sec Max green: 23 sec Min green: 4 sec Y: 3 sec AR:2 sec Figure 5-9. Continued Green: 40 sec Yellow: 3 sec All Red: 2 sec Intersection of Beaver Ave. and S. Pugh St. Cycle Length: 90 sec Green: 40sec Yellow: 3 sec All Red: 2 sec Green: 54 sec Yellow: 4 sec All Red: 1 sec Intersection of Beaver Ave. and Garner St. Cycle Length: 90 sec Green: 26 sec Yellow: 4 sec All Red: 1 sec Figure 5-10. Approximate A.M. signal timing for Case 1 (Beaver_Pugh_AM_EB) PAGE 106 106 Green: 54 sec Yellow: 3 sec All Red: 1 sec Intersection of Beaver Ave. and S. Pugh St. Cycle Length: 100 sec Green: 37sec Yellow: 3 sec All Red: 2 sec Green: 58 sec Yellow: 4 sec All Red: 1 sec Intersection of Beaver Ave. and Garner St. Cycle Length: 100 sec Green: 32 sec Yellow: 4 sec All Red: 1 sec Figure 5-11. Approximate P.M. signal timing for Case 3 (Beaver_Pugh_PM_EB) Green: 56sec Yellow: 3 sec All Red: 2 sec Intersection of B eaver Ave. and S. Sparks St. Cycle Length: 90 sec Green: 26sec Yellow: 3 sec All Red: 2 sec Green:12 sec Yellow: 3 sec All Red: 0 sec Green:31 sec Yellow: 3 sec All Red: 1 sec Intersection of Beaver Ave. and Atherton St. Cycle Length: 90 sec Green: 36 sec Yellow: 3 sec All Red: 1 sec Figure 5-12. Approximate A.M. signal ti ming for Case 4 (Beaver_Sparks_AM_EB) PAGE 107 107 Green: 52sec Yellow: 3 sec All Red: 1 sec Intersection of B eaver Ave. and S. Sparks St. Cycle Length: 90 sec Green: 40sec Yellow: 3 sec All Red: 1 sec Green:17 sec Yellow: 3 sec All Red: 0 sec Green:38 sec Yellow: 3 sec All Red: 1 sec Intersection of Beaver Ave. and Atherton St. Cycle Length: 90 sec Green: 34 sec Yellow: 3 sec All Red: 1 sec Figure 5-13. Approximate Mid signal timin g for Case 5 (Beaver_Sparks_Mid_EB) Green: 60sec Yellow: 3 sec All Red: 1 sec Intersection of B eaver Ave. and S. Sparks St. Cycle Length: 100 sec Green: 32sec Yellow: 3 sec All Red: 1 sec Green:16 sec Yellow: 3 sec All Red: 0 sec Green:36 sec Yellow: 3 sec All Red: 1 sec Intersection of Beaver Ave. and Atherton St. Cycle Length: 100 sec Green: 37 sec Yellow: 3 sec All Red: 1 sec Figure 5-14. Approximate P.M. signal tim ing for Case 6 (Beaver_Sparks_PM_EB) PAGE 108 108 Green: 22 sec Yellow: 3 sec All Red: 0 sec Green: 38 sec Yellow: 3 sec All Red: 2 sec Intersection of Park Ave. and Atherton St. Cycle Length: 100 sec Green: 27sec Yellow: 3 sec All Red: 2 sec Green: 8 sec Yellow: 3 sec All Red: 0 sec Green: 52 sec Yellow: 3 sec All Red: 1 sec Intersection of Park Ave. and Allen St. Cycle Length: 100 sec Green: 29sec Yellow: 3 sec All Red: 1 sec Green: 9 sec Yellow: 3 sec All Red: 0 sec Green: 54 sec Yellow: 3 sec All Red: 2 sec Intersection of Park Ave. and Grove Alley St. Cycle Length: 100 sec Green: 24 sec Yellow: 3 sec All Red: 2 sec Figure 5-15. Approximate signal timing for Ca se 7 (Park_Mid_EB) Case 8 (Park_Mid_WB) PAGE 109 109 Green: 19 sec Yellow: 3 sec All Red: 0 sec Green: 44 sec Yellow: 3 sec All Red: 2 sec Intersection of Park Ave. and Atherton St. Cycle Length: 100 sec Green: 24sec Yellow: 3 sec All Red: 2 sec Green: 13 sec Yellow: 3 sec All Red: 0 sec Green: 52 sec Yellow: 3 sec All Red: 1 sec Intersection of Park Ave. and Allen St. Cycle Length: 100 sec Green: 24sec Yellow: 3 sec All Red: 1 sec Green: 9 sec Yellow: 3 sec All Red: 0 sec Green: 47 sec Yellow: 3 sec All Red: 2 sec Intersection of Park Ave. and Grove Alley St. Cycle Length: 100 sec Green: 31 sec Yellow: 3 sec All Red: 2 sec Figure 5-16. Approximate signal timing for Case 9 (Park_PM_EB) and Case 10 (Park_PM_WB) PAGE 110 110 Green: 22 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and I-75 NB off ramp (exit) Cycle Length: 150 sec Green: 118 sec Y: 3 sec AR: 2 sec Green: 105 sec Y: 3 sec AR:2 sec Intersection of Newberry Rd. and NW 69th Street Cycle Length: 150 sec Green: 35 sec Y: 3 sec AR:2 sec Green: 93 sec Y: 3 sec AR: 2 sec Green: 14 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and Oaks Mall West Cycle Length: 150 sec Max green: 28 sec Y: 3 sec AR: 2 sec Green: 93 sec Y: 3 sec AR:2 sec Green: 25 sec Y: 3 sec AR:2 sec Intersection of Newberry Rd. and NW 66th Street Cycle Length: 150 sec Green: 17 sec Y: 3 sec AR:2 sec Figure 5-17. Approximate signal timing fo r Case 11 (Newberry_4_30_EB) and Case 13 (Newberry_5_1_EB) PAGE 111 111 Green: 22 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and I-75 NB off ramp (exit) Cycle Length: 150 sec Green: 118 sec Y: 3 sec AR: 2 sec Green: 22 sec Y: 3 sec AR:2 sec Green: 76 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and I-75 NB off ramp (entrance) Cycle Length: 150 sec Green: 37 sec Y: 3 sec AR: 2 sec Green: 105 sec Y: 3 sec AR:2 sec Intersection of Newberry Rd. and NW 69th Street Cycle Length: 150 sec Green: 35 sec Y: 3 sec AR:2 sec Green: 93 sec Y: 3 sec AR: 2 sec Green: 14 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and Oaks Mall West Cycle Length: 150 sec Max green: 28 sec Y: 3 sec AR: 2 sec Intersection of Newberry Rd. and NW 66th Street Cycle Length: 150 sec Green: 93 sec Y: 3 sec AR:2 sec Figure 5-18. Approximate signal timing for Case 12 (Newberry_4_30_WB) and Case 14 (Newberry_5_1_WB) PAGE 112 112 Green: 25 sec Y: 3 sec AR:2 sec Green: 17 sec Y: 3 sec AR:2 sec Figure 5-18.Contiuned Travel Time Distribution for Beaver_Pugh_AM(offset = 0, 10,20,30,80,90) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 34.4534.7435.4235.6135.7136.5853.2654.1554.32 TT(sec)Probability Figure 5-19. Travel Time Distri bution for Beaver_Pugh_AM (offset = 0, 10, 20, 30, 80, 90) Travel Time Distribution for Beaver_Pugh_AM(offset = 40) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 34.4534.7435.4235.6135.7136.5853.2654.1554.32 TT(sec)Probability Figure 5-20. Travel Time Distributi on for Beaver_Pugh_AM (offset = 40) PAGE 113 113 Travel Time Distribution for Beaver_Pugh_AM(offset = 50,60,70) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 34.4534.7435.4235.6135.7136.5853.2654.1554.32 TT(sec)Probability Figure 5-21. Travel Time Di stribution for Beaver_Pugh_AM (offset = 50, 60, 70) Beaver_Pugh_AM_travel time comparsion based on different offsets (AIMSUN vs Model) 0 10 20 30 40 50 60 70 80 0102030405060708090 offset(sec)TT(sec) AIMSUN Model Figure 5-22. Travel time comparisons for Beaver_Pugh_AM_EB (case 1) Beaver_Pugh_AM_travel time comparison based on different offsets(AIMSUN vs Model) 0 10 20 30 40 50 60 70 80 0102030405060708090100 Offset(sec)TT(sec) AIMSUN Model Figure 5-23. Travel time comparisons for Beaver_Pugh_PM_EB (case 3) PAGE 114 114 Beaver_Sparks_AM travel time comparison based on different offsets (AIMSUN vs Model) 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 0102030405060708090 Offset(sec)TT(sec) AIMSUN Model Figure 5-24. Travel time comparisons for Beaver_Sparks_AM_EB (case 4) Beaver_Sparks_Mid travel time comparison based on different offsets(AIMSUN vs Model) 0 10 20 30 40 50 60 70 80 90 100 0102030405060708090100 offset(sec)TT(sec) AIMSUN Model Figure 5-25. Travel time comparisons for Beaver_Sparks_Mid_EB (case 5) Beaver_Sparks_PM travel time compasion based on different offset (AIMSUN vs Model) 0 10 20 30 40 50 60 70 80 90 0102030405060708090100 offset(sec)TT(sec) AIMSUN Model Figure 5-26. Travel time comparisons for Beaver_Sparks_PM_EB (case 6) PAGE 115 115 Park_Mid_EB travel time comparison based on different offsets(AIMSUN vs Model) 0 20 40 60 80 100 120 140 0102030405060708090100 offset(sec)TT(sec) AIMSUN Model Figure 5-27. Travel time comparisons for Park_Mid_EB (case 7) Park_Mid_WB travel time comparison based on different offset(AIMSUN vs Model) 0 20 40 60 80 100 120 140 0102030405060708090100 offset(sec)TT(sec) AIMSUN Model Figure 5-28. Travel time comparisons for Park_Mid_WB (case 8) Park_PM_WB travel time comparison based on different offsets (AIMSUN vs Model) 0 20 40 60 80 100 120 140 0102030405060708090100 offset(sec)TT(sec) AIMSUN Model Figure 5-29. Travel time comparisons for Park_PM_EB (case 9) PAGE 116 116 Park_PM_WB cycle by cycle travel time (AIMSUN vs Model)0 50 100 150 200 250 300 350 400 450 500 1234567 cycleTT(sec) AIMSUN Model Figure 5-30. Travel time comparisons for Park_PM_WB (case 10) Newberry_4_30_EB travel time comparison based on different offsets(AIMSUN vs Model) 0 20 40 60 80 100 120 140 160 180 0102030405060708090100110120130140150 offset(sec)TT(sec) AIMSUN Model Figure 5-31. Travel time comparisons for Newberry_4_30_EB (case 11) Newberry_4_30_WB travel time comparison based on different offset(AIMSUN vs Model) 0 20 40 60 80 100 120 140 160 180 0102030405060708090100110120130140150 offset(sec)TT(sec) AIMSUN Model Figure 5-32. Travel time comparisons for Newberry_4_30_WB (case 12) PAGE 117 117 Newberry_5_1_EB travel time comparison based on different offsets(AIMSUN vs Model) 0 20 40 60 80 100 120 140 160 1800 10 20 30 40 50 60 70 80 90 1 00 110 120 1 30 140 150offset(sec)TT(sec) AIMSUN Model Figure 5-33. Travel time comparisons for Newberry_5_1_EB (case 13) Newberry_5_1_WB travel time comparison based on different offset(AIMSUN vs Model) 0 20 40 60 80 100 120 140 160 180 0102030405060708090100110120130140150 offset(sec)TT(sec) AIMSUN Model Figure 5-34. Travel time comparisons for Newberry_5_1_WB (case 14) PAGE 118 118Beginning MQ Cycle 1 End RQ Beginning R MQ Q Cycle 2 End RQ 2 Beginning R MQ Q 2 Cycle 3 End RQ 3 Beginning R MQ Q 3 Cycle 4 End RQ 4 Figure 5-35. Queue length calculation for every cycle for Case 10 (Park_PM_WB) PAGE 119 119Beginning R MQ Q 4 Cycle 5 End RQ 5 Beginning R MQ Q 5 Cycle 6 End RQ 6 Beginning R MQ Q 6 Cycle 7 End RQ 7 Figure 5-35. Continued PAGE 120 120Beaver_Pugh Total length: 1263 ft Total driveway: 4 Driveway Density: 0.00317/mile Beaver_Sparks Total length: 1429 ft Total driveway: 4 Driveway Density: 0.00280/mile Park Total length: 2891 ft Total driveway: 4 Driveway Density: 0.00138/mile Newberry Total length: 1883 ft Total driveway: 12 Driveway Density: 0.00637/mile Figure 5-36. Link length and driveways PAGE 121 121 Table 5-1. Tolerances and the acceptable travel times Tolerance at 95% CI(sec) Acceptable travel time at 95% CI(sec) Tolerance at 90% CI(sec) Acceptable travel time at 90% CI(sec) Case 1 Beaver_Pugh_AM_EB 6.14 [30.99,43.26] 5.15 [31.97,42.28] Case 2 Beaver_Pugh_Mid_EB 3.13 [32.15,38.42] 2.63 [32.65,37.92] Case 3 Beaver_Pugh_PM_EB 4.14 [31.28,39.57] 3.48 [31.95,38.91] Case 4 Beaver_Sparks_AM_EB 17.45 [54.18,89.07] 14.64 [56.98,86.27] Case 5 Beaver_Sparks_Mid_EB 31.9 [43.53,107.33] 26.77 [48.66,102.2] Case 6 Beaver_Sparks_PM_EB 13.9 [84.96,112.76] 11.67 [87.19,110.52] Case 7 Park_Mid_EB 10.32 [68.25,88.89] 8.66 [69.91,87.23] Case 8 Park_Mid_WB 20.25 [92.04,132.54] 17 [95.29,129.28] Case 9 Park_PM_EB 14.15 [74.45,102.75] 11.88 [76.72,100.48] Case 10 Park_PM_WB 76.47 [248.73,401.67] 64.18 [261.02,389.38] Case 11 Newberry 4_30_EB 6.37 [56.27,69.01] 5.34 [57.3,67.98] Case 12 Newberry 4_30_WB 5.42 [48.32,59.16] 4.55 49.19,58.29] Case 13 Newberry 5_1_EB 7.62 [58.9,74.14] 6.39 [60.13,72.92] Case 14 Newberry 5_1_WB 6.18 [52.43,64.79] 5.19 [53.43,63.80] Table 5-2. Simulated travel time compared to field travel time Average TT from AIMSUN (sec) Filed average TT (sec) Acceptable or not Case 1 Beaver_Pugh_AM_EB 36.67 37.13 Y Case 2 Beaver_Pugh_Mid_EB 40.83 35.29 N Case 3 Beaver_Pugh_PM_EB 37.83 35.43 Y Case 4 Beaver_Sparks_AM_EB 68.00 71.63 Y Case 5 Beaver_Sparks_Mid_EB 70.83 75.43 Y Case 6 Beaver_Sparks_PM_EB 91.33 98.86 Y Case 7 Park_Mid_EB 72.00 78.57 Y Case 8 Park_Mid_WB 99.00 112.29 Y Case 9 Park_PM_EB 83.5 88.6 Y Case 10 Park_PM_WB 330.33 325.2 Y Case 11 Newbarry 4_30_EB 68.83 62.64 Y Case 12 Newbarry 4_30_WB 58.83 53.74 Y Case 13 Newbarry 5_1_EB 72.67 66.52 Y Case 14 Newbarry 5_1_WB 57.17 58.61 Y Note: the averaged simulated travel times are based on 10 replications PAGE 122 122Table 5-3. Travel time comparison s for Case 1 (Beaver_Pugh_AM_EB) Offset (sec) Average TT from AIMSUN using the offsets (sec) Expected TT from model using the offsets (sec) Standard deviation for model estimated TT (sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 43.50 35.12 0.25 -19.26% 43.45 10 42.67 35.12 0.25 -17.69% 43.45 20 40.50 35.12 0.25 -13.28% 43.45 30 38.67 35.12 0.25 -9.18% 43.45 40 39.50 54.31 0.29 +37.49% 43.45 50 46.33 53.9 0.86 +16.34% 43.45 60 53.00 53.9 0.86 +1.70% 43.45 70 58.17 53.9 0.86 -7.34% 43.45 80 48.67 35.12 0.25 -27.84% 43.45 90 43.50 35.12 0.25 -19.26% 43.45 PAGE 123 123Table 5-4. Travel time comparison s for Case 3 (Beaver_Pugh_PM_EB) Offset (sec) Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 44.83 35.09 0.51 -21.73% 49.85 10 44.83 35.09 0.51 -21.73% 49.85 20 43.33 36.72 0.51 -15.26% 49.85 30 41.33 48.77 9.89 18.00% 49.85 40 40.17 56.57 0.89 40.83% 49.85 50 45.17 56.57 0.89 25.24% 49.85 60 49.67 56.57 0.89 13.89% 49.85 70 56.50 56.57 0.89 0.12% 49.85 80 55.00 42.39 9.63 -22.93% 49.85 90 45.00 35.09 0.51 -22.02% 49.85 100 44.83 35.09 0.51 -27.76% 49.85 PAGE 124 124Table 5-5. Travel time comparisons for Case 4 (Beaver_Sparks_AM_EB) Offset (sec) Offset in AIMSUN(sec) Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -50 71 67.24 0.22 -5.30% 75.77 10 -40 64.67 40.01 0.63 -38.13% 75.77 20 -30 57 39.47 0.33 -30.75% 75.77 30 -20 56.33 42.18 0.75 -25.12% 75.77 40 -10 56.67 67.24 0.22 18.65% 75.77 50 0 59 67.24 0.22 13.97% 75.77 60 10 61.5 67.24 0.22 9.33% 75.77 70 20 61.67 67.24 0.22 9.03% 75.77 80 30 64.83 67.24 0.22 3.72% 75.77 90 40 71 67.24 0.22 -5.30% 75.77 PAGE 125 125Table 5-6. Travel time comparisons for Case 5 (Beaver_Sparks_Mid_EB) Offset (sec) Offset in AIMSUN(sec) Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -62 88.5 76.93 0.52 -13.07% 73.87 10 -52 85 57.61 15.99 -32.22% 73.87 20 -42 64 45.66 2.20 -28.66% 73.87 30 -32 58.83 46.94 0.66 -20.21% 73.87 40 -22 57.17 76.93 0.52 34.56% 73.87 50 -12 57.67 76.93 0.52 33.40% 73.87 60 -2 60 76.93 0.52 28.22% 73.87 70 8 63.5 76.93 0.52 21.15% 73.87 80 18 71.33 76.93 0.52 7.85% 73.87 90 28 79.83 76.93 0.52 -3.63% 73.87 100 38 88.5 76.93 0.52 -13.07% 73.87 PAGE 126 126Table 5-7. Travel time comparisons for Case 6 (Beaver_Sparks_PM_EB) Offset (sec) Offset in AIMSUN(sec) Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -59 80.67 77.8 0.72 -3.56% 69.97 10 -49 68.17 45.66 0.56 -33.02% 69.97 20 -39 60.33 47.11 1.98 -21.91% 69.97 30 -29 59.83 48.61 0.58 -18.75% 69.97 40 -19 60.17 77.8 0.72 29.30% 69.97 50 -9 61.5 77.8 0.72 26.50% 69.97 60 1 62.67 77.8 0.72 24.14% 69.97 70 11 64.33 77.8 0.72 20.94% 69.97 80 21 66.83 77.8 0.72 16.41% 69.97 90 31 73.67 77.8 0.72 5.61% 69.97 100 41 80.67 77.8 0.72 -3.56% 69.97 PAGE 127 127Table 5-8. Travel time comparis ons for Case 7 (Park_Mid_EB) Offset (sec) Offset in AIMSUN(sec) Int. 1 Offset in AIMSUN(sec) Int. 2 Offset in AIMSUN(sec) Int. 3 Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -68 -11 -12 98.67 92.94 0.52 -5.81% 99.20 10 -68 -1 8 99.50 92.94 0.52 -6.59% 99.20 20 -68 9 28 95.83 92.94 0.52 -3.02% 99.20 30 -68 19 48 86.67 67.18 1.78 -22.49% 99.20 40 -68 29 68 87.67 67.12 11.52 -23.44% 99.20 50 -68 39 88 95.17 95.35 26.47 0.19% 99.20 60 -68 49 108 100.83 118.11 0.48 17.14% 99.20 70 -68 59 128 107.5 118.11 0.48 9.87% 99.20 80 -68 69 148 108.83 118.11 0.48 8.53% 99.20 90 -68 79 168 102.50 118.11 0.48 15.23% 99.20 100 -68 89 188 98.67 92.94 0.52 -5.81% 99.20 PAGE 128 128Table 5-9. Travel time comparisons for Case 8 (Park_Mid_WB) Offset (sec) Offset in AIMSUN(sec) Int. 3 Offset in AIMSUN(se c) Int. 2 Offset in AIMSUN( sec) Int. 1 Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 0 0 -68 96.17 105.46 2.12 9.66% 104.40 10 0 10 -48 93.00 105.46 2.12 13.40% 104.40 20 0 20 -28 86.00 85.32 18.32 -0.79% 104.40 30 0 30 -8 80.83 76.25 16.41 -5.67% 104.40 40 0 40 12 83.67 90.01 1.57 7.58% 104.40 50 0 50 32 90.33 101.47 16.74 12.33% 104.40 60 0 60 52 102.33 125.14 0.58 22.29% 104.40 70 0 70 72 110.00 125.14 0.58 13.76% 104.40 80 0 80 92 104.83 116.59 9.39 11.22% 104.40 90 0 90 112 95.33 105.46 2.12 10.63% 104.40 100 0 100 132 96.17 105.46 2.12 9.66% 104.40 PAGE 129 129Table 5-10. Travel time comparis ons for Case 9 (Park_PM_EB) Offset (sec) Offset in AIMSUN(sec) Int. 1 Offset in AIMSUN(sec) Int. 2 Offset in AIMSUN(sec) Int. 3 Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -68 -11 -12 126.63 96.76 80.56 -23.59% 135.10 10 -68 -1 8 128.44 96.76 80.56 -24.67% 135.10 20 -68 9 28 110.87 96.76 80.56 -12.73% 135.10 30 -68 19 48 113.23 73.78 33.08 -34.84% 135.10 40 -68 29 68 105.38 97.92 60.48 -7.08% 135.10 50 -68 39 88 105.73 121.89 45.37 15.28% 135.10 60 -68 49 108 109.76 121.89 45.37 11.05% 135.10 70 -68 59 128 113.39 121.89 45.37 7.50% 135.10 80 -68 69 148 125.99 121.89 45.37 -3.25% 135.10 90 -68 79 168 133.55 121.89 45.37 -8.73% 135.10 100 -68 89 188 126.33 96.76 80.56 -23.41% 135.10 PAGE 130 130Table 5-11 Travel time comparisons for Case 10 (Park_PM_WB) The nth Cycle Average TT from AIMSUN using the offsets (sec) TT from model using the offsets (sec) (without consideration of residual queue) TT from model using the offsets (sec) (with consideration of residual queue) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 1 106.763 79.59 79.59 25.89 -25.45% 113.60 2 133.67 82.28 155.28 23.14 16.17% 113.60 3 216.415 102.87 188.84 9.31 -12.74% 113.60 4 250.092 99.97 272.97 9.35 9.15% 113.60 5 341.81 96.93 295.87 9.40 -13.44% 113.60 6 356.102 93.79 366.79 9.44 3.00% 113.60 7 431.83 90.56 402.48 9.48 -6.80% 113.60 PAGE 131 131Table 5-12 Travel time comparison s for Case 11 (Newberry_4_30_EB) Offset (sec) Offset in AIMSUN (sec) Int. 1 Offset in AIMSUN (sec) Int. 3 Offset in AIMSUN (sec) Int. 4 Offset in AIMSUN (sec) Int. 5 Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -27 0 0 0 75.17 34.46 1.23 -54.16% 83.20 10 -27 10 20 30 67.00 84.90 45.52 26.72% 83.20 20 -27 20 40 60 67.17 63.04 32.98 -6.15% 83.20 30 -27 30 60 90 73.83 60.52 2.15 -18.03% 83.20 40 -27 40 80 120 92.50 60.64 28.05 -34.44% 83.20 50 -27 50 100 150 117.17 92.63 2.45 -20.94% 83.20 60 -27 60 120 180 141.50 92.63 3.59 -34.54% 83.20 70 -27 70 140 210 151.17 82.99 16.97 -45.10% 83.20 80 -27 80 160 240 138.33 33.92 3.73 -75.48% 83.20 90 -27 90 180 270 126.76 65.38 4.89 -48.42% 83.20 100 -27 100 200 300 105.28 65.38 4.20 -37.90% 83.20 110 -27 110 220 330 99.37 65.38 4.00 -34.21% 83.20 120 -27 120 240 360 107.88 81.04 35.62 -24.88% 83.20 130 -27 130 260 390 99.55 33.25 3.15 -66.60% 83.20 140 -27 140 280 420 87.29 33.25 2.75 -61.91% 83.20 150 -27 150 300 450 75.17 34.46 1.23 -54.16% 83.20 PAGE 132 132Table 5-13. Travel time comparison s for Case 12 (Newberry_4_30_WB) Offset (sec) Offset in AIMSUN (sec) Int. 1 Offset in AIMSU N (sec) Int. 2 Offset in AIMSUN (sec) Int. 3 Offset in AIMSU N (sec) Int. 4 Offset in AIMSUN (sec) Int. 5 Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -27 -27 0 0 0 62.31 32.91 2.69 -47.18% 56.49 10 13 3 20 10 0 56.43 104.08 12.29 84.44% 56.49 20 53 33 40 20 0 70.38 82.33 20.56 16.98% 56.49 30 93 63 60 30 0 101.33 105.78 1.12 4.39% 56.49 40 133 93 80 40 0 130.45 99.86 14.59 -23.45% 56.49 50 173 123 100 50 0 139.31 80.80 18.99 -42.00% 56.49 60 213 153 120 60 0 159.37 109.23 1.13 -31.46% 56.49 70 253 183 140 70 0 169.05 109.22 1.13 -35.39% 56.49 80 293 213 160 80 0 129.59 75.90 1.06 -41.43% 56.49 90 333 243 180 90 0 90.61 99.17 0.96 9.45% 56.49 100 373 273 200 100 0 83.85 97.37 0.78 16.12% 56.49 110 413 303 220 110 0 89.91 96.78 1.08 7.64% 56.49 120 453 333 240 120 0 94.62 35.21 1.10 -62.79% 56.49 130 493 363 260 130 0 90.00 35.15 1.11 -60.94% 56.49 140 533 393 280 140 0 85.56 34.40 2.07 -59.79% 56.49 150 573 423 300 150 0 62.31 32.91 2.69 -47.18% 56.49 PAGE 133 133Table 5-14. Travel time comparison s for Case 13 (Newberry_5_1_EB) Offset (sec) Offset in AIMSUN (sec) Int. 1 Offset in AIMSUN (sec) Int. 3 Offset in AIMSUN (sec) Int. 4 Offset in AIMSUN (sec) Int. 5 Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -27 0 0 0 70.67 32.68 4.02 -53.76% 99.20 10 -27 10 20 30 65.00 97.65 8.30 50.22% 99.20 20 -27 20 40 60 67.17 58.33 2.59 -13.17% 99.20 30 -27 30 60 90 75.33 58.48 2.37 -22.37% 99.20 40 -27 40 80 120 96.17 59.49 2.75 -38.14% 99.20 50 -27 50 100 150 120.00 91.99 3.45 -23.34% 99.20 60 -27 60 120 180 142.08 91.99 4.03 -35.26% 99.20 70 -27 70 140 210 146.91 83.52 19.88 -43.15% 99.20 80 -27 80 160 240 129.67 37.22 15.34 -71.30% 99.20 90 -27 90 180 270 106.38 63.38 4.51 -40.42% 99.20 100 -27 100 200 300 88.96 63.38 4.17 -28.75% 99.20 110 -27 110 220 330 94.65 63.38 4.06 -33.03% 99.20 120 -27 120 240 360 104.51 64.63 4.39 -38.16% 99.20 130 -27 130 260 390 94.55 31.12 2.81 -67.08% 99.20 140 -27 140 280 420 81.23 31.12 2.67 -61.68% 99.20 150 -27 150 300 450 70.67 32.68 4.02 -53.76% 99.20 PAGE 134 134Table 5-15. Travel time comparison s for Case 14 (Newberry_5_1_WB) Offset (sec) Offset in AIMSUN (sec) Int. 1 Offset in AIMSUN (sec) Int. 2 Offset in AIMSUN (sec) Int. 3 Offset in AIMSU N (sec) Int. 4 Offset in AIMSUN( sec) Int. 5 Average TT from AIMSUN using the offsets (sec) TT estimation from model using the offsets (sec) Standard deviation for model estimated TT(sec) Percent difference when compared to AIMSUN results (%) TT estimated by HCM arterial travel time model 0 -27 -27 0 0 0 63.78 35.29 2.56 -44.67% 64.10 10 13 3 20 10 0 59.49 107.07 7.88 79.98% 64.10 20 53 33 40 20 0 72.66 81.61 20.37 12.32% 64.10 30 93 63 60 30 0 104.58 107.01 0.68 2.32% 64.10 40 133 93 80 40 0 135.66 91.05 17.08 -32.88% 64.10 50 173 123 100 50 0 144.02 91.47 20.27 -36.49% 64.10 60 213 153 120 60 0 157.38 109.94 0.81 -30.14% 64.10 70 253 183 140 70 0 166.96 109.94 0.81 -34.15% 64.10 80 293 213 160 80 0 150.64 86.91 13.51 -42.31% 64.10 90 333 243 180 90 0 99.04 94.61 10.34 -4.47% 64.10 100 373 273 200 100 0 93.70 100.60 0.53 7.36% 64.10 110 413 303 220 110 0 91.26 100.26 0.79 9.86% 64.10 120 453 333 240 120 0 97.34 54.76 19.57 -43.74% 64.10 130 493 363 260 130 0 92.54 37.08 0.81 -59.93% 64.10 140 533 393 280 140 0 89.95 37.06 0.81 -58.80% 64.10 150 573 423 300 150 0 63.78 35.29 2.56 -44.67% 64.10 PAGE 135 135 CHAPTER 6 SENSITIVITY ANALYSIS AND APPLICATION In this chapter, sensitivity analysis is perfor med to test how sensitive the analytical model is to selected inputs. The following variables are tested: G/C ratio Link length Maximum operating speed Acceleration/ deceleration rate The entering flow rate at each intersection These variables are tested for each of the 13 cases used in chapter 5. Changing some of the variables (g/C ratio and flow rate) can lead to congested conditions. Thus, the travel times are shown for both non-congested and congested cond itions. Travel times when spillback occurs are not calculated since they are beyond the capability of the mode l. Varying values of offset are used in the previous chapter for model validation so it will not be used in this chapter. Because of the similarity of the sensitivity analysis for e ach site, only one case is selected from each of the four sites to show in this chapter. The rest of the analysis is included in Appendix B. 6.1 G/C Ratio 6.1.1 Site 1: Beaver_Pugh_AM /Case 1 The cycle length for the two in tersections is 90 seconds. Th e green time is changed to create different g/C ratios. Figur e 6-1 shows that, as expected, tr avel time decreases as the g/C ratio increases. When g/C ratio is 0.11, the arterial is congested and spillback occurs. PAGE 136 136 6.1.2 Site 2:Beaver_Sparks_AM/Case 4 The cycle length for the two in tersections is 90 seconds. Th e green time is changed to create different g/C ratios. Figur e 6-2 shows that, as expected, tr avel time decreases as the g/C ratio increases. When g/C ratio is 0.11 and 0.22, the arterial is congeste d and spillback occurs. 6.1.3 Site 3:Park_Mid_EB/Case 7 The cycle length for the two inte rsections is 100 seconds. The green times are changed to create different g/C ratios. Figur e 6-3 shows that, as expected, tr avel time decreases as the g/C ratio increases. When g/C ratio is 0.1 and 0.2, the arterial is congested and spillback occurs. 6.1.4 Site 4: Newberry_4_30_EB/Case 11 The cycle length for the two in tersections is 100 seconds. Th e green time is changed to create different g/C ratios. . Figure 6-4 shows that , as expected, travel ti me decreases as the g/C ratio increases. When g/C ratio is 0.07, 0.13, and 0.20, the arterial is congested and spillback occurs. 6.2 Link Length 6.2.1 Site 1: Beaver_Pugh_AM /Case 1 Starting from 100 ft, Link lengt h is increased by 100 ft for every run. The maximum link length is set to 2000 ft. As shown in the previous chapter, offset is a very important variable in estimating travel time. For the one link cases, there are two possible ways to do the sensitivity analysis for link length. One is to adjust the ideal offset every time when the link length is changing; the other is to keep it unadjusted. The result from the first method is shown in Figure 6-5; the result for the second method is shown in Figure 6-6. As shown in Figure 6-5, if the offset is adjusted as the link length changes, travel time increases as the link length increases. If the offset is not changed, as shown in Figure 6-6, travel time at first increases and then drops to a low valu e at 1100 ft, and then increases at 1300 ft. This PAGE 137 137 is because the travel time is calculated by averaging all possible travel times. For this case, the travel time drops at 1100 ft, the selected travel times are t12, t22, and t33. In 1000 ft, the selected travel times are t11, t21, and t32. Selected travel time sets are different and the probabilities associated with each selected travel time are ch anging at every run. Thus, travel time does not keep increasing as the link length increases if the offset is unchanged. 6.2.2 Site 2:Beaver_Sparks_AM/Case 4 Starting from 100 ft, Link lengt h is increased by 100 ft for every run. The maximum link length is set to 2000 ft. the link length is also tested in two ways as mentioned in case 1. The result from the first method is shown in Figure 67; the result for the second method is shown in Figure 6-8. As shown in Figure 67, if the offset is adjusted as the link length changes, travel time increases as the link length in creases. If the offset is not ch anged, as shown in Figure 6-8, travel time at first increases and then drops to a low value at 1500 ft, and then increases at 1800 ft. The reason for this change is the same as the one mentioned in Case 1. 6.2.3 Site 3:Park_Mid_EB/Case 7 In this case, there are two links . To see how travel time is affected in this two-link case. One link length is kept unchanged when the other one is increased by 100 ft starting from 100 ft. The maximum link length used is 2000 ft. The offset is kept unchanged. The result from the first link is shown in Figure 6-9; the result for the se cond link is shown in Figure 6-10. As shown in Figure 6-9, if the offset is unchang ed as the link length changes, travel time at first increases and then drops to a low value at 1200 ft, and then in creases at 1500 ft. As shown in Figure 6-10, if the offset is not changed, travel time at first increases and then drops to a low value at 200 ft, then increases at 400 ft. The reason for th ese changes are as explained in Case 1. PAGE 138 138 6.2.4 Site 4: Newberry_4_30_EB/Case 11 In this case, there are three links . To see how travel time is af fected in this three-link case. Two link lengths are kept unchanged when the othe r one is increased by 100 ft starting from 100 ft. The maximum link length used is 1000 ft. The o ffset is kept unchanged. The result from the first link is shown in Figure 611; the result for the second li nk is shown in Figure 6-12; the result for the third link is shown in Figure 613; As shown in Figure 6-11, if the offset is unchanged as the link length changes, travel time at first increases and then drops to a low value at 600 ft, and then increases at 700 ft. As show n in Figure 6-12, if the offset is not changed, travel time at first increases and then drops to a low value at 900 ft, and then increases at 1000 ft. As shown in Figure 6-13, if the offset is not ch anged, travel time increases all the time. It is possible since the selected travel time set does no t change for this case, as link length increases, travel time increases. 6.3 Maximum Operating Speed 6.3.1 Site 1: Beaver_Pugh_AM /Case 1 Similar to the link length sensitivity analysis , the changes in maximum operating speed can cause changes in the ideal offset and then the tr avel time. Thus, as for the link length, there are two ways to test the sensitivity; with adjusted or unadjusted offset. For this case, the maximum operating speed is 25 mph. Thus, the lower bou nd of the maximum operating speed is set to 5mph and increase by 5mph every time. Figure 6-14 shows that travel time deceases as the maximum operating speed increases when the offset is changed every run. Figure 6-15 shows how travel time changes if the offset is not changed: travel time at first decreases as the maximum operating speed increases and when the speed reaches 30mph, the trav el time increases and then decr eases again there after. The PAGE 139 139 reason is same as the one mentioned for Figure 66. Travel time does not keep decreasing as the speed increases if the offset is unchanged. 6.3.2 Site 2:Beaver_Sparks_AM/Case 4 Similarly to Case 1, the sensitivity analysis is performed in two ways and the lower bound of the maximum operating speed is set to 15mph and increase by 5mph every time until it reaches 45 mph. Figure6-16 shows that travel time deceases as the maximum operating speed increases when the offset is changed every run. Figure 6-17 shows how travel time changes if the offset is not changed: travel time at first decreases as the maximum operating speed increases and when the speed reaches 25 mph, the travel time increas es and then decreases again there after. The reason is same as the one mentioned for Figure 66. Travel time does not keep decreasing as the speed increases if the offset is unchanged. 6.3.3 Site 3:Park_Mid_EB/Case 7 As mentioned before, there are two links in this case. To see how travel time is affected in this two-link case, the maximum operating speed for one link is kept un changed when the other one is increased by 5mph starting from 15 mph until it reaches 45mph. The offset is kept unchanged. The result from the first link is shown in Figure 6-18; the result for the second link is shown in Figure 6-19. As shown in Figure 618, if the offset is unchanged as the maximum operating speed changes, travel time at first decr eases, increases to a higher value at 40 mph, and then decreases at 45mph. As shown in Figure 619, if the offset is not changed, travel time decreases when the maximum operating speed increases. PAGE 140 140 6.3.4 Site 4: Newberry_4_30_EB/Case 11 As mentioned before, there are three links in this case. To see how travel time is affected in this three-link case, the maximum operating sp eed for two links are kept unchanged when the other one is increased by 5mph starting from 25 mph until it reaches 55mph. The offset is kept unchanged. The result from the first link is shown in Figure 6-20; the result for the second link is shown in Figure 6-21; the result for the third li nk is shown in Figure 6-22. As shown in Figure 6-20, if the offset is unchanged as the maximum operating speed changes, travel time at first decreases and then increases to a higher value at 40 mph, and then decreases at 45mph. As shown in Figure 6-21, if the offset is unchange d as the maximum operating speed changes, travel time at first decreases and then increases to a hi gher value at 35 mph, and then decreases at 40mph. As shown in Figure 6-22, if the offset is not changed, travel time decreases when the maximum operating speed increases. 6.4 Acceleration Rate / Deceleration Rate 6.4.1 Site 1: Beaver_Pugh_AM /Case 1 For this case, the acceleration rate is 15.78 ft/s2. The lower bound for the analysis is set to 5 ft/s2 and it is increased by 2.5 ft/s2 until it reaches 20 ft/s2. Figure 6-23 shows that travel time slightly decreases as the acceleration rate increases. For this case, the deceleration rate is 16.83 ft/s2. The lower bound for the analysis is set to 5 ft/s2 and it is increased by 2.5 ft/s2 until it reaches 20 ft/s2. Figure 6-24 shows that travel time slightly decreases as the deceleration rate increases. PAGE 141 141 6.4.2 Site 2:Beaver_Sparks_AM/Case 4 For both the acceleration and the deceleration rate , similarly to the previous case, the lower bound for this sensitivity analysis is set to 5 ft/s2 and it is added by 2.5 ft/s2 until it researches 20 ft/s2. Figure 6-25 shows that travel time slightly decreases as the acceleration rate increases. Figure 6-26 shows that travel time slightly decreases as the deceleration rate increases. 6.4.3 Site 3:Park_Mid_EB/Case 7 For both the acceleration and the deceleration rate , similarly to the previous case, the lower bound for this sensitivity analysis is set to 5 ft/s2 and it is added by 2.5 ft/s2 until it reaches 20 ft/s2. Figure 6-27 shows that travel time slightly decreases as the acceleration rate increases. Figure 6-28 shows that travel time slightly decreases as the deceleration rate increases. 6.4.4 Site 4: Newberry_4_30_EB/Case 11 For both the acceleration and the deceleration ra te, similarly the previous case, the lower bound for this sensitivity analysis is set to 5 ft/s2 and it is added by 2.5 ft/s2 until it researches 20 ft/s2. Figure 6-29 shows that travel time slightly decreases as the acceleration rate increases. Figure 6-30 shows that travel time slightly decreases as the deceleration rate increases. 6.5 The Entering Flow Rate at Each Intersection 6.5.1 Site 1: Beaver_Pugh_AM /Case 1 Depending on the flow levels and the respectiv e green ratios and offsets, the flow rates of the side streets have a different impact on travel time. All the entering flows can affect travel time at each intersection. The sens itivity analysis for flow rate examines how the travel time varies as the entering flows at each inters ection change. For this case, which has two PAGE 142 142 intersections, the method is to keep the entering flow of one intersection unchanged and test how the travel time varies as the entering flow of the other intersection change s. For the entering flow at intersection 1, Figure 6-31 shows that travel time increases as the flow rate increases under both non-congested and congested conditions. For th e entering flow at intersection 2, Figure 632 shows that travel times remain almost th e same as flow increas es under non-congested conditions; when congestion occurs , the travel time increases. 6.5.2 Site 2:Beaver_Sparks_AM/Case 4 For this case, there are two intersections. Th e method is the same as in Case 1. For the entering flow at intersection 1, Figure 6-33 show s that travel time decreases as the flow rate increases, which is counter intuitive. This happens because when the flow increases, it results in an increase of the interval length of Conditions 1 and 2, and a decrease of Condition 3 (Figure 6-34). Since the three conditions for the other intersection remain the same, the resulting travel time changes randomly. As shown in Figure 6-34, when the entering flow rate at intersection 1 increases, the interval lengths of Conditions 1 a nd 2 increase; the possible travel times that are obtained for calculating the expected travel time are different. In this case, the obtained travel times change from t11 and t21 to t31 and t32, which result in overall lower travel times. Since the offset remains the same, the increase in flows results in more optimal travel times. For the entering flow at intersection 2, Figu re 6-34 shows travel time increases as flow increases under both congested and un-congested condition. 6.5.3 Site 3:Park_Mid_EB/Case 7 For this case, there are three intersections. The method is the same as in Case 1. For the entering flow at intersection 1, Figure 6-35 show s that travel time decreases as the flow rate increases. Similar to Figure 6-33, Figure 6-35 show s a decreasing trend in travel time as entering flow increases. The explanation for this trend is also similar to the one for Figure 6-33: when the PAGE 143 143 flow increases, it results in an increase of th e interval length of Conditions 1 and 2 and a decrease of Condition 3. Since the three conditio ns for the other intersection remain the same, the resulting travel time changes arbitrarily. Fo r intersection 2, Figure 6-36 shows travel time remains almost the same as flow increases under congested and increases under un-congested condition. For intersection 3, Figure 6-37 shows that travel time increases as flow increases under both congested and non-congested condition. 6.5.4 Site 4: Newberry_4_30_EB/Case 11 For this case, there are four intersections. The method is the same as in Case 1. For the entering flow at intersection 1, Figure 6-38 s hows travel time decreases as the flow rate increases. Similar to Figure 6-33, Figure 6-38 show s a decreasing trend in travel time as entering flow increases. The explanation for this trend is also similar to the one for Figure 6-36: when the flow increases, it results in an increase of th e interval length of Conditions 1 and 2 and a decrease of Condition 3.Since the three conditions for the other intersection remain the same, the resulting travel time changes randomly. For th e entering flow at intersection 2, Figure 6-39 shows travel time increases as flow increases un der both congested and non-congested condition. For the entering flow at intersection 3, Figure 640 shows travel time decreases as flow increases under congested and increases non-congested conditi on. For the entering flow at intersection 4, Figure 6-41 shows travel time remains almost th e same as flow increas es under congested and increases non-congested condition. Similar to Figu re 6-33 and Figure 6-38, Figure 6-41 (for the un-congested part) shows travel time decreases as the flow rate increases. The explanations are the same as that for Figure 6-38. 6.6 Conclusions Based on the sensitivity analysis for all the cases, the following conclusions can be reached: PAGE 144 144 6.6.1 g/C Ratio When the g/C ratio increases, the travel time decreases for all the cases. The longer the green time given to the major street, the shorter th e travel time is for the major street. The travel time continues to decrease while the g/C ratio can satisfy the demand; then it decreases only slightly. When the g/C ratio is extremely low, re sidual queues begin to form, congestion occurs and travel time becomes very high. There are two types of congested travel times. In the first type the vehicles arriving in the previous cycle ca n be discharged in the current cycle. This travel time can be calculated. In the other type the residual queue keeps increasing until spill back occurs. For this situation travel time cannot be calculated using their analytical model. 6.6.2 Link Length When link length increases, the travel time does not necessarily increase. The travel time depends on the offset for each case. For one-link case s, if the offset is set to the ideal offset every time as the link length changes, the travel times keep increasing as the link length increases. If the offset is unchanged, the travel times do not keep increasing as the link length increases, they increase at first, and then at a certain point they drop to a lower value. Starting from that value, the travel times increase again. Travel time is significantly affected by si gnal coordination. If the coordination is poor, even a shorter distance can cause a longer travel time. For the cases that contain more than one links, the travel times ar e tested as the link length of one link changes and while the others remain unchanged. The offsets are kept unchanged for these cases. The travel times show similar trends as the one-link cases with unadjusted offset. 6.6.3 Maximum Operating Speed The maximum operating speed has the opposite e ffect on travel times to the link length. As expected increasing, maximum operating speed results in lower travel times. PAGE 145 145 6.6.4Acceleration/ Deceleration Rate Increasing the acceleration an d deceleration rate reduces travel time. However, compared to other factors, travel time changes are relatively small. Thus, travel time is not highly affected by the acceleration and deceleration rates. 6.6.5 Entering Flow Rate at Each Intersection Before congestion occurs, there is no trend in travel time. In the methodology, flow rate is used to calculate the interval lengths for the three conditions. Condition 1 is when vehicles have to wait in the stopped queue; Condition 2 is wh en vehicles move with the discharging queue, Condition 3 is when vehicles move without a queue present. If the flow increases, the Condition 1 and Condition 2 increase, and Condition 3 decreases. For one-link cases, the upstream intersection flow can affect the travel time this way. For the downstream intersection, the travel time is not only affected by the interv al lengths for the conditions, it is also affected by the residual queue. When the flow rate increases to a certain point, a re sidual queue occurs and it causes the travel time to increase. After the resi dual queue occurs, the segment is considered as congested. Thus, the congested travel time is hi gher than the non-congested travel time. Sensitivity Analysis (TT vs G/C Ratio) for Beaver_Pugh_AM0 10 20 30 40 50 60 70 80 0.220.330.440.560.670.780.89 G/C RatioTT(sec) Figure 6-1. Sensitivity analysis: g/C rati o versus travel time(Beaver_Pugh_AM) PAGE 146 146 Sensitivity Analysis(TT vs G/C ratio) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 100 0.330.440.560.670.780.89 G/C ratioTT(sec) Figure 6-2. Sensitivity analysis: g/C ratio versus travel time (Beaver_Sparks_AM) Sensitivity Analysis(TT vs G/C ratio) for Park_Mid_EB0 20 40 60 80 100 120 140 0.30.40.50.60.70.80.9 G/C ratioTT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-3. Sensitivity analysis: g/C rati o versus travel time (Park_Mid_EB) Sensitivity Analysis (TT vs G/C ratio) for Newberry_4_30_EB0.00 50.00 100.00 150.00 200.00 250.00 0.270.330.40.470.530.60.670.730.80.870.93 G/C ratioTT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-4. Sensitivity analysis g/C ratio versus travel time (Newberry_4_30_EB) PAGE 147 147 Sensitivity Analysis (TT vs Link Length) for Beaver_Pugh_AM 0 10 20 30 40 50 60 70 80100 2 0 0 300 400 5 0 0 600 700 8 0 0 900 1000 1100 1 2 0 0 1300 1400 1 5 0 0 1600 1700 1 8 0 0 1900 2000Link Length(ft)TT(sec) Figure 6-5. Sensitivity analysis: link length ve rsus travel time with adjusted offset (Beaver_Pugh_AM) Sensitivity Analysis (TT vs Link Length) for Beaver_Pugh_AM 0 10 20 30 40 50 60 70 80100 200 300 400 500 600 700 800 900 1 00 0 1 100 1 200 1 300 1 400 1 50 0 1 600 1 700 1 800 1 900 2 00 0Link Length(ft)TT(sec) Figure 6-6. Sensitivity analysis: link length versus travel time with unadjusted offset(Beaver_Pugh_AM) Sensitivity Analysis (TT vs Link Length) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 1001 00 200 300 40 0 50 0 60 0 70 0 80 0 9 00 1 0 00 1100 1 2 00 1 3 00 1 4 00 1 5 00 1 6 00 17 00 18 0 0 1900 2000Link Length(ft)TT(sec) Figure 6-7. Sensitivity analysis: link length ve rsus travel time with adjusted offset (Beaver_Sparks_AM) PAGE 148 148 Sensitivity Analysis (TT vs Link Length) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 1001 0 0 2 0 0 3 0 0 4 0 0 5 0 0 60 0 70 0 80 0 90 0 10 0 0 1100 1200 1300 1400 1 5 00 1 6 00 1 7 00 1 8 00 1 9 00 2 0 00Link Length(ft)TT(sec) Figure 6-8. Sensitivity analysis: link length vers us travel time with unadjusted offset (Beaver_Sparks_AM) Sensitivity Analysis (TT vs Link Length) for Park_Mid_EB_link 00 20 40 60 80 100 120 140100 200 3 00 4 00 500 600 700 8 00 900 10 00 11 00 1 2 00 13 0 0 14 0 0 15 00 1 6 00 17 0 0 18 0 0 19 00 20 00Link Length(ft)TT(sec) Figure 6-9. Sensitivity analysis: link length versus travel time with unadjusted offset for link 0 (Park_Mid_EB) Sensitivity Analysis (TT vs Link Length) for Park_Mid_EB_link 10 20 40 60 80 100 120 1401 00 200 300 4 0 0 500 6 0 0 7 00 800 9 00 100 0 1 1 0 0 1 200 1 3 0 0 1 400 1500 160 0 1700 180 0 1 9 0 0 2 000Link Length(ft)TT(sec) Figure 6-10. Sensitivity analysis: link length versus travel time with unadjusted offset for link 1 (Park_Mid_EB) PAGE 149 149 Sensitivity Analysis (TT vs Link Length) for Newberry_4_30_EB_link 00.00 50.00 100.00 150.00 200.00 250.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure 6-11.Sensitivity analysis: link length ratio ve rsus travel time with unadjusted offset for link 0 (Newberry_4_30_EB) Sensitivity Analysis (TT vs Link Length) for Newberry_4_30_EB_link 10.00 50.00 100.00 150.00 200.00 250.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure 6-12. Sensitivity analysis: link length versus travel time with unadjusted offset for link 1 (Newberry_4_30_EB) Sensitivity Analysis (TT vs Link Length) for Newberry_4_30_EB_link 20.00 50.00 100.00 150.00 200.00 250.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure 6-13. Sensitivity analysis: link length versus travel time with unadjusted offset for link 2 (Newberry_4_30_EB) PAGE 150 150 Sensitivity Analysis(TT vs Link MOS) for Beaver_Pugh_AM 0 10 20 30 40 50 60 70 80 15202530354045 Maximum Operating Speed(MOS)(ft/s)TT(sec) Figure 6-14. Sensitivity analysis: maximum operatin g speed versus travel time with adjusted offset (Beaver_Pugh_AM) Sensitivity Analysis(TT vs Link MOS) for Beaver_Pugh_AM 0 10 20 30 40 50 60 70 80 15202530354045 Maximum Operating Speed(MOS)(ft/s)TT(sec) Figure 6-15. Sensitivity analysis: maximum operatin g speed versus travel time with unadjusted offset (Beaver_Pugh_AM) Sensitivity Analysis (TT vs MOS) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 100 15202530354045 MOS (Maximum Operating Speed) (mph)TT(sec) Figure 6-16. Sensitivity analysis: maximum operatin g speed versus travel time with adjusted offset (Beaver_Sparks_AM) PAGE 151 151 Sensitivity Analysis (TT vs MOS) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 100 15202530354045 MOS (Maximum Operating Speed) (mph)TT(sec) Figure 6-17. Sensitivity analysis: maximum operatin g speed versus travel time with unadjusted offset (Beaver_Sparks_AM) Sensitivity Analysis (TT vs MOS) for Park_Mid_EB_link 00 20 40 60 80 100 120 140 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) Figure 6-18. Sensitivity analysis: maximum operatin g speed versus travel time with unadjusted offset for link 0 (Park_Mid_EB) Sensitivity Analysis (TT vs MOS) for Park_Mid_EB_link 10 20 40 60 80 100 120 140 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) PAGE 152 152 Figure 6-19. Sensitivity analysis: maximum operatin g speed versus travel time with unadjusted offset (Park_Mid_EB) Sensitivity Analysis (TT vs MOS) for Newberry_4_30_EB_link00.00 50.00 100.00 150.00 200.00 250.00 25303540455055 MOS(Maximum Operating Speed) mphTT(sec) Figure 6-20. Sensitivity analysis: maximum operatin g speed versus travel time with unadjusted offset for link 0 (Newberry_4_30_EB) Sensitivity Analysis (TT vs MOS) for Newberry_4_30_EB_link10.00 50.00 100.00 150.00 200.00 250.00 25303540455055 MOS(Maximum Operating Speed) mphTT(sec) Figure 6-21. Sensitivity analysis: maximum operatin g speed versus travel time with unadjusted offset for link 1 (Newberry_4_30_EB) Sensitivity Analysis (TT vs MOS) for Newberry_4_30_EB_link20.00 50.00 100.00 150.00 200.00 250.00 25303540455055 MOS(Maximum Operating Speed) mphTT(sec) PAGE 153 153 Figure 6-22. Sensitivity analysis: maximum operatin g speed versus travel time with unadjusted offset for link 2 (Newberry_4_30_EB) Sensitivity Analysis(TT vs Acceleration Rate) for Beaver_Pugh_AM 0 10 20 30 40 50 60 70 80 57.51012.51517.520 Acceleration Rate (ft2/s)TT(sec) Figure 6-23. Sensitivity analysis: acceleration rate versus travel time (Beaver_Pugh_AM) Sensitivity Analysis (TT vs Deceleration Rate) for Beaver_Pugh_AM0 10 20 30 40 50 60 70 80 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure 6-24. Sensitivity analysis: deceleration rate versus travel time (Beaver_Pugh_AM) Sensitivity Analysis (TT vs Acceleration Rate) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 100 57.51012.51517.520 Acceleration Rate (ft2/s)TT(sec) Figure 6-25. Sensitivity analysis: acceleration ra te versus travel time (Beaver_Sparks_AM) PAGE 154 154 Sensitivity Analysis for Beaver_Sparks_AM(TT vs Deceleration Rate)0 10 20 30 40 50 60 70 80 90 100 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure 6-26. Sensitivity analysis: deceleration rate versus travel time (Beaver_Sparks_AM) Sensitivity Analysis(TT vs Acceleration Rate) for Park_Mid_EB0 20 40 60 80 100 120 140 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure 6-27. Sensitivity analysis: acceleration rate versus travel time (Park_Mid_EB) Sensitivity Analysis(TT vs Deceleration Rate) for Park_Mid_EB0 20 40 60 80 100 120 140 57.51012.51517.520 Deceleration Rate (ft2/s)TT(sec) Figure 6-28. Sensitivity analysis: deceleration rate versus travel time (Park_Mid_EB) PAGE 155 155 Sensitivity Analysis (TT vs Acceleration Rate) for Newberry_4_30_EB0 50 100 150 200 250 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure 6-29. Sensitivity analysis: acceleration ra te versus travel time (Newberry_4_30_EB) Sensitivity Analysis (TT vs Deceleration Rate) for Newberry_4_30_EB0 50 100 150 200 250 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure 6-30. Sensitivity analysis: deceleration rate versus travel time (Newberry_4_30_EB) Sensitivity Analysis(TT vs Intersection 1 Entering Flow) for Beaver_Pugh_AM0 10 20 30 40 50 60 70 80 500100015002000 Intersection 1 Entering Flow (veh/hr/ln)TT(sec) Figure 6-31. Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Pugh_AM) PAGE 156 156 Sensitivity Analysis(TT vs Intersection 2 Entering Flow) for Beaver_Pugh_AM0 10 20 30 40 50 60 70 80 25050075010001250 Intersection 2 Entering Flow (veh/hr/ln)TT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-32. Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Pugh_AM) Sensitivity Analysis (TT vs Intersection 1 entering flow) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 100 25050075010001250 Intersection Entering Flow (veh/hr/ln)TT(sec) Figure 6-33. Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Sparks_AM) Figure 6-34. Travel time changes due to flow changes PAGE 157 157 Sensitivity Analysis (TT vs Intersection 2 entering flow) for Beaver_Sparks_AM0 10 20 30 40 50 60 70 80 90 100 250500750800900 Intersection 2 Entering Flow(veh/hr/ln)TT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-35. Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Sparks_AM) Sensitivity Analysis (TT vs Intersection 1 Entering Flow) for Park_Mid_EB0 20 40 60 80 100 120 140 2505007501000 Intersection Entering Flow(veh/hr/ln)TT(sec) Figure 6-36. Sensitivity analysis: flow rate versus travel time for intersection 1(Park_Mid_EB) Sensitivity Analysis (TT vs Intersection 2 Entering Flow) for Park_Mid_EB0 20 40 60 80 100 120 140 2505007501000105011001150 Intersection 2 Entering Flow(veh/hr)TT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-37. Sensitivity analysis: flow rate versus travel time for intersection 2(Park_Mid_EB) PAGE 158 158 Sensitivity Analysis (TT vs Intersection 3 Entering Flow) for Park_Mid_EB 0 20 40 60 80 100 120 140 25050075010001050110011501200 Intersection 3 Entering Flow(veh/hr)TT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-38. Sensitivity analysis: flow rate versus travel time for intersection 3(Park_Mid_EB) Sensitivity Analysis (TTvs Intersection 1 Entering Flow) for Newberry_4_30_EB 0.00 50.00 100.00 150.00 200.00 250.00 5001000150020002500 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure 6-39. Sensitivity analysis: flow rate versus travel time for intersection 1(Newberry_4_30_EB) Sensitivity Analsysis(TT vs Intersection 3 Entering Flow) for Newberry_4_30_EB 0 50 100 150 200 250 5001000110012001300 Intersection 4 Entering Flow(veh/hr/ln)TT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-40. Sensitivity analysis: flow rate versus travel time for intersection 3(Newberry_4_30_EB) PAGE 159 159 Sensitivity Analsysis(TT vs Intersection 4 Entering Flow) for Newberry_4_30_EB 0.00 50.00 100.00 150.00 200.00 250.00 5001000110012001300 Intersection 4 Entering Flow(veh/hr/ln)TT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-41. Sensitivity analysis: flow rate versus travel time for intersection 4(Newberry_4_30_EB) Sensitivity Analsysis(TT vs Intersection 5 Entering Flow) for Newberry_4_30_EB 0.00 50.00 100.00 150.00 200.00 250.00 5001000110012001300 Intersection 4 Entering Flow(veh/hr/ln)TT(sec) Note: the dashed line is the seperation line of uncongested(Left) and congested coditions(Right) Figure 6-42. Sensitivity analysis: flow rate versus travel time for intersection 5(Newberry_4_30_EB) PAGE 160 160 CHAPTER 7 CONCLUSIONS AND FUTURE RESEARCH In this chapter, the method developed for arterial travel time estimation is summarized and the results obtained from the model are review ed. Some conclusions are reached based on the estimation results. Some suggestions for improving the model and some possible future research directions are presented at the end of the chapter. 7.1 Overview of the Analytical Model This research proposed an analytical model for estimating the expected travel time for signalized arterials. In the method, the travel time is defined as the sum of MT(travel time in motion), TQW(Waiting Time in the Queue), and TQM(Moving Time in the Queue). Using the vehicle trajectories when trave ling between two intersections, th e travel times are calculated for 9 different flow profiles. If ther e are n intersections, then there are 3n flow profiles. The expected travel time is calculated based on the probability of occurrence of each travel time for each flow profile. In this analytical model, th e green time, cycle length, link length, maximum operating speed, offset, acceleration/deceleration rate, and the entering flow rate at each intersection are used as inputs. 7.2 Results from the Analytical Model Data were collected at State College, PA and Gainesville, FL w ith 4 different data collection sites for various time periods for a total of 14 cases. Since the field data were collected on arterials with semi-actuated signal timing, field traffic conditions were firstly replicated in the simulator. After changing the signal timings to pre-timed, the travel times from the simulation were compared to the travel times estimated by the analytical model. For the three sites from State College, PA, the results show that travel times from both the simulation and the analytical model have similar trends. The average diffe rence between the simulated and the model PAGE 161 161 estimated travel times shows th at the travel time difference for Beaver_Pugh, Beaver_Sparks, and Park cases are smaller than those of Newberry cases. In the Park Ave. site, traffic conditions are congested and for some cases spillback occu rs. Travel time cannot be estimated with the analytical model for the spillback cases. For Newbe rry Rd Site, there are big differences in travel times between the simulation and the model. The differences are caused by the driveway delay which is not considered in the analytical model. The sensitivity analysis shows that 1) when the g/C ratio increases, the travel time decreases for all the cases; 2) when link length increases, the travel time does not necessarily increase; the tr avel time depends on the offset for each case; 3) the maximum operating speed has the opposite effect on travel times to the link length; higher maximum operating speed results in lower trav el times; 4) increasin g the acceleration and deceleration rate reduces travel time; but the magnit ude of the changes is re latively small; and 5) before congestion occurs, there is no trend in tr avel time as a function of flow; the congested travel time is always higher than the non-congested travel ti me and increases with flow. 7.3 Suggestions and Future Research There are several aspects that are not considered in this analytical model. These aspects would enhance the capability of the analytical model 7.3.1 Delay Caused by Driveways The first aspect is the delay caused by drivew ays. It is known that turning movements in driveways can cause additional delay. Thus, to make the estimation more accurate, it has to take the delay as an extra term and add it into the total estimated travel time. Previous studies estimate turning delays base d on different factors selected. Some common factors are flow rate on the major street, turnin g flow rate, operating sp eed, number of lanes on the major street, and driveway density. PAGE 162 162 7.3.2 Actuated or Semi-Actuated Signal Timing The signal timing in the model is pre-timed . However, semi-actuated signals are widely used in the field and the field data were collect ed from the arterials with semi-actuated signal timing. The major difference between the pre-timed and the semi-actuated signal timing is that the green interval for the minor street changes ev ery cycle due to the fluctuating traffic arrival rate. Cycle by cycle analysis would be required in the cases for actuated or semi-actuated signal timing. Thus, simulation might be more appropriate. 7.3.3 Travel Time Estimations for Other O/D Pairs The travel time estimated in this analytical model is for the through movement, which is only one pair of OD. For an arte rial with several intersections and driveways, there are several pairs of ODs. The travel time estimations for these ODs are also very helpful for travelers and operators. These travel time estimations are rela ted to travel time associated with turning maneuver which is sometimes interrupted by other conflicting movements. Thus, the travel time estimations depend on the specific turning mo vement, the conflicting movement, and the flow rate for both. Then according to these a specific travel time is calculated. 7.3.4 Travel Time Estimation for Spill Back Cases Travel time estimation for congested conditions when spill back occurs is also a possible future research topic. Such an analytical proced ure may result in an iterative process when the queues of the two successive intersections interact. Additional research should be undertaken to investigate this issue further. PAGE 163 163 APPENDIX A SENSITIVITY ANALYSIS g/C ratio Sensitivity Analysis (TT vs G/C ratio) for Beaver_Pugh_Pm0 10 20 30 40 50 60 70 80 90 0.30.40.50.60.70.80.9 G/C ratioTT(sec) Figure A-1.Sensitivity analysis: g/C rati o versus travel time (Beaver_Pugh_PM) Sensitivity Analysis (TT vs G/C ratio) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 110 0.30.40.50.60.70.80.9 G/C ratioTT(sec) Figure A-2.Sensitivity analysis: g/C ratio versus travel time (Beaver_Sparks_Mid) Sensitivity Analysis (TT vs G/C ratio) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 110 0.30.40.50.60.70.80.9 G/C ratioTT(sec) Figure A-3. Sensitivity Analysis: g/C ratio versus travel time (Beaver_Sparks_PM) PAGE 164 164 Sensitivity Analysis (TT vs G/C ratio) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 2030405060708090 G/C ratioTT(sec) Figure A-4. Sensitivity analysis: g/C ra tio versus travel time (Park_Mid_WB) Sensitivity Analysis (TT vs G/C ratio) for Park_PM_EB0 20 40 60 80 100 120 140 160 180 0.40.50.60.70.80.9 G/C ratioTT(sec) Figure A-5. Sensitivity analysis: g/C ra tio versus travel time (Park_PM_EB) Sensitivity Analysis (TT vs G/C ratio) for Park_PM_WB0 20 40 60 80 100 120 140 160 0.40.50.60.70.80.9 G/C ratioTT(sec) Figure A-6. Sensitivity analysis: g/C ra tio versus travel time (Park_PM_WB) PAGE 165 165 Sensitivity Analysis (TT vs G/C ratio) for Newberry_4_30_WB0 50 100 150 200 250 300 0.330.40.470.530.60.670.730.80.870.93 G/C ratioTT(sec) Figure A-7. Sensitivity analysis: g/C rati o versus travel time (Newberry_4_30_WB) Sensitivity Analysis (TT vs G/C ratio) for Newberry_5_1_EB0 20 40 60 80 100 120 140 160 0.270.330.40.470.530.60.670.730.80.870.93 G/C ratioTT(sec) Figure A-8. Sensitivity analysis: g/C rati o versus travel time (Newberry_5_1_EB) Sensitivity Analysis (TT vs G/C ratio) for Newberry_5_1_WB0 50 100 150 200 250 300 0.330.40.470.530.60.670.730.80.870.93 G/C ratioTT(sec) Figure A-9. Sensitivity analysis: g/C rati o versus travel time (Newberry_5_1_WB) PAGE 166 166 Link Length Sensitivity Analysis (TT vs Link Length) for Beaver_Pugh_PM0 10 20 30 40 50 60 70 80 90100 20 0 300 400 500 600 7 0 0 800 9 0 0 10 0 0 1100 1200 1300 1400 1 5 00 1600 1 7 0 0 1800 19 0 0 2000Link Length(ft)TT(sec) Figure A-10.Sensitivity analysis: link length versus travel time with adjusted offset (Beaver_Pugh_PM) Sensitivity Analysis (TT vs Link Length) for Beaver_Pugh_PM0 10 20 30 40 50 60 70 80 901 0 0 200 3 0 0 400 5 0 0 600 7 0 0 800 90 0 1000 1100 1200 1300 1 4 00 1500 1 6 0 0 1700 1 8 0 0 1900 2 0 0 0Link Length(ft)TT(sec) Figure A-11.Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Beaver_Pugh_PM) Sensitivity Analysis (TT vs Link Length) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 1101 0 0 20 0 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1 7 00 1 8 0 0 19 0 0 2000Link Length(ft)TT(sec) Figure A-12. Sensitivity analysis: link length ve rsus travel time with adjusted offset (Beaver_Sparks_Mid) PAGE 167 167 Sensitivity Analysis (TT vs Link Length) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 1101 0 0 200 300 400 50 0 60 0 700 800 900 10 0 0 11 0 0 12 0 0 130 0 140 0 15 0 0 16 0 0 17 0 0 1800 190 0 2 00 0Link Length(ft)TT(sec) Figure A-13.Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Beaver_Sparks_Mid) Sensitivity Analysis (TT vs Link Length) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 1101 0 0 2 0 0 3 0 0 4 0 0 5 0 0 60 0 7 0 0 8 0 0 90 0 1 00 0 1100 1 200 1 300 1400 1500 1 60 0 1700 1800 1 90 0 2 00 0Link Length(ft)TT(sec) Figure A-14. Sensitivity analysis: link length ve rsus travel time with adjusted offset (Beaver_Sparks_PM) Sensitivity Analysis (TT vs Link Length) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 11010 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 10 00 1100 1 2 00 13 0 0 1400 1 5 00 1600 1700 1 8 00 1900 2000Link Length(ft)TT(sec) Figure A-15. Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Beaver_Sparks_PM) PAGE 168 168 Sensitivity Analysis (TT vs Link Length) for Park_Mid_WB_link 00 20 40 60 80 100 120 140 160 180 10020030040050060070080090010001100120013001400150016001700180019002000 Link Length(ft)TT(sec) Figure A-16. Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Park_Mid_WB) Sensitivity Analysis (TT vs Link Length) for Park_Mid_WB_link 10 20 40 60 80 100 120 140 160 180100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000Link Length(ft)TT(sec) Figure A-17. Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Park_Mid_WB) Sensitivity Analysis (TT vs Link Length) for Park_PM_EB for llink 00 20 40 60 80 100 120 140 160 180100 200 300 400 500 600 700 800 9 00 1 000 1 100 1 200 1 300 1 400 1 500 1 600 1 700 1 800 1 900 2 000Link Length (ft)TT(sec) Figure A-18. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Park_PM_EB) PAGE 169 169 Sensitivity Analysis (TT vs Link Length) for Park_PM_EB for llink 10 20 40 60 80 100 120 140 160 180100 200 300 400 500 600 7 00 800 900 1000 1 1 00 1 200 1300 1 40 0 1 500 1 6 00 1 700 1 800 1 9 00 2 000Link Length (ft)TT(sec) Figure A-19. Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Park_PM_EB) Sensitivity Analysis (TT vs Link Length) for Newberry_4_30_WB_link 00.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-20. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_4_30_WB) Sensitivity Analysis (TT vs Link Length) for Newberry_4_30_WB_link 10.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-21. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_4_30_WB) PAGE 170 170 Sensitivity Analysis (TT vs Link Length) for Newberry_4_30_WB_link 20.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-22. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_4_30_WB) Sensitivity Analysis (TT vs Link Length) for Newberry_4_30_WB_link 30.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-23. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_4_30_WB) Sensitivity Analysis (TT vs Link Length) for Newberry_5_1_EB_link 00.00 50.00 100.00 150.00 200.00 250.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-24. Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Newberry_5_1_EB) PAGE 171 171 Sensitivity Analysis (TT vs Link Length) for Newberry_5_1_EB_link 10.00 50.00 100.00 150.00 200.00 250.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-25. Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Newberry_5_1_EB) Sensitivity Analysis (TT vs Link Length) for Newberry_5_1_EB_link 20.00 50.00 100.00 150.00 200.00 250.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-26. Sensitivity analysis: link length ve rsus travel time with unadjusted offset (Newberry_5_1_EB) Sensitivity Analysis (TT vs Link Length) for Newberry_5_1_WB_link00.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-27. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_5_1_WB) PAGE 172 172 Sensitivity Analysis (TT vs Link Length) for Newberry_5_1_WB_link10.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-28. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_5_1_WB) Sensitivity Analysis (TT vs Link Length) for Newberry_5_1_WB_link20.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-29. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_5_1_WB) Sensitivity Analysis (TT vs Link Length) for Newberry_5_1_WB_link30.00 50.00 100.00 150.00 200.00 250.00 300.00 1002003004005006007008009001000 Link Length(ft)TT(sec) Figure A-30. Sensitivity analysis: link length ratio versus travel time with unadjusted offset (Newberry_5_1_WB) PAGE 173 173 Maximum Operating Speed Sensitivity Analysis (TT vs MOS) for Beaver_Pugh_PM0 10 20 30 40 50 60 70 80 90 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) Figure A-31.Sensitivity analysis: MOS versus travel time with adjusted offset (Beaver_Pugh_PM) Sensitivity Analysis (TT vs MOS) for Beaver_Pugh_PM0 10 20 30 40 50 60 70 80 90 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) Figure A-32.Sensitivity analysis: MOS vers us travel time with unadjusted offset (Beaver_Pugh_PM) Sensitivity Analysis (TT vs MOS) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 110 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) Figure A-33.Sensitivity analysis: MOS versus travel time with adjusted offset (Beaver_Sparks_Mid) PAGE 174 174 Sensitivity Analysis (TT vs MOS) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 110 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) Figure A-34. Sensitivity analysis: MOS versus travel time with unadjusted offset (Beaver_Sparks_Mid) Sensitivity Analysis (TT vs MOS) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 110 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) Figure A-35. Sensitivity analysis: MOS versus travel time with adjusted offset (Beaver_Sparks_PM) Sensitivity Analysis (TT vs MOS) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 110 15202530354045 MOS(Maximum Operating Speed)(mph)TT(sec) Figure A-36. Sensitivity Analysis: MOS versus travel time with unadjusted offset (Beaver_Sparks_PM) PAGE 175 175 Sensitivity Analsysis (TT vs MOS) for Park_Mid_WB_link00 20 40 60 80 100 120 140 160 180 15202530354045 MOS(Maximum Operating Speed) (mph)TT(sec) Figure A-37. Sensitivity analysis: MOS versus travel time with unadjusted offset (Park_Mid_WB) Sensitivity Analsysis (TT vs MOS) for Park_Mid_WB_link00 20 40 60 80 100 120 140 160 180 15202530354045 MOS(Maximum Operating Speed) (mph)TT(sec) Figure A-38. Sensitivity analysis: MOS versus travel time with unadjusted (Park_Mid_WB) Sensitivity Analysis(TT vs MOS) for Park_PM_EB_link 00 20 40 60 80 100 120 140 160 180 15202530354045 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-39. Sensitivity analysis: MOS versus travel time with unadjusted offset (Park_PM_EB) PAGE 176 176 Sensitivity Analysis(TT vs MOS) for Park_PM_EB_link 10 20 40 60 80 100 120 140 160 180 15202530354045 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-40. Sensitivity analysis: MOS versus travel time with unadjusted offset (Park_PM_EB) Sensitivity Analysis (TT vs MOS) for Newberry_4_30_WB_link 00.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-41. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB) Sensitivity Analysis (TT vs MOS) for Newberry_4_30_WB_link 10.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-42. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB) PAGE 177 177 Sensitivity Analysis (TT vs MOS) for Newberry_4_30_WB_link 20.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-43. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB) Sensitivity Analysis (TT vs MOS) for Newberry_4_30_WB_link 30.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-44. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_4_30_WB) Sensitivity Analysis (TT vs MOS) for Newberry_5_1_EB_link 00.00 50.00 100.00 150.00 200.00 250.00 105.94101.4199.6498.3697.2496.4897.05 MOS(Maximum Operating Speed) mphTT(sec) Figure A-45. Sensitivity analysis: MOS vers us travel time with unadjusted offset (Newberry_5_1_EB) PAGE 178 178 Sensitivity Analysis (TT vs MOS) for Newberry_5_1_EB_link 10.00 50.00 100.00 150.00 200.00 250.00 105.94101.4199.6498.3697.2496.4897.05 MOS(Maximum Operating Speed) mphTT(sec) Figure A-46. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_EB) Sensitivity Analysis (TT vs MOS) for Newberry_5_1_EB_link 20.00 50.00 100.00 150.00 200.00 250.00 105.94101.4199.6498.3697.2496.4897.05 MOS(Maximum Operating Speed) mphTT(sec) Figure A-47. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_EB) Sensitivity Analysis (TT vs MOS) for Newberry_5_1_WB_link00.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-48. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB) PAGE 179 179 Sensitivity Analysis (TT vs MOS) for Newberry_5_1_WB_link10.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-49. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB) Sensitivity Analysis (TT vs MOS) for Newberry_5_1_WB_link20.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-50. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB) Sensitivity Analysis (TT vs MOS) for Newberry_5_1_WB_link30.00 50.00 100.00 150.00 200.00 250.00 300.00 25303540455055 MOS (Maximum Operating Speed) (mph)TT(sec) Figure A-51. Sensitivity analysis: MOS versus travel time with unadjusted offset (Newberry_5_1_WB) PAGE 180 180 Acceleration Rate / Deceleration Rate Sensitivity Analysis (TT vs Acceleration Rate) for Beaver_Pugh_PM0 10 20 30 40 50 60 70 80 90 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure A-52.Sensitivity analysis: acceleratio n rate versus travel time (Beaver_Pugh_PM) Sensitivity Analysis (TT vs Deceleration Rate) for Beaver_Pugh_PM0 10 20 30 40 50 60 70 80 90 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure A-53.Sensitivity analysis: deceleration rate versus travel time (Beaver_Pugh_PM) Sensitivity Analysis(TT vs Acceleration Rate) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 110 57.51012.51517.520 Acceleration Rate (ft2/s)TT(sec) Figure A-54.Sensitivity analysis: acceleration rate versus travel time (Beaver_Sparks_Mid) PAGE 181 181 Sensitivity Analysis (TT vs Decelartion Rate) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 110 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure A-55. Sensitivity analysis: deceleration rate versus travel time (Beaver_Sparks_Mid) Sensitivity Analysis (TTvs Aeceleration Rate) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 110 57.51012.51517.520 Acceleration Rate (ft2/s)TT(sec) Figure A-56.Sensitivity analysis: acceleration rate versus travel time (Beaver_Sparks_PM) Sensitivity Analysis (TT vs Deceleration Rate) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 110 57.51012.51517.520 Deceleration Rate (ft2/s)TT(sec) Figure A-57.Sensitivity analysis: deceleration rate versus travel time (Beaver_Sparks_PM) PAGE 182 182 Sensitivity Analysis(TT vs Acceleration Rate) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure A-58. Sensitivity analysis: acceleratio n rate versus travel time (Park_Mid_WB) Sensitivity Analysis(TT vs Deceleration Rate) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure A-59. Sensitivity analysis: deceleration rate versus travel time (Park_Mid_WB) Sensitivity Analysis (TT vs Acceleration Rate) for Park_PM_EB0 20 40 60 80 100 120 140 160 180 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure A-60. Sensitivity Analysis: acceleratio n rate versus travel time (Park_PM_EB) PAGE 183 183 Sensitivity Analysis (TT vs Deceleration Rate) for Park_PM_EB0 20 40 60 80 100 120 140 160 180 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure A-61. Sensitivity Analysis: deceleration rate versus Travel time (Park_PM_EB) Sensitivity Analysis(TT vs Acceleration Rate) for Newberry_4_30_WB0 50 100 150 200 250 300 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure A-62. Sensitivity analysis: acceleration rate versus travel time (Newberry_4_30_WB) Sensitivity Analysis(TT vs Deceleration Rate) for Newberry_4_30_WB0 50 100 150 200 250 300 57.51012.51517.520 Deceleration Rate (ft2/s)TT(sec) Figure A-63. Sensitivity analysis: deceleration rate versus travel time (Newberry_4_30_WB) PAGE 184 184 Sensitivity Analysis (TT vs Acceleration Rate) for Newberry_5_1_EB0 20 40 60 80 100 120 140 160 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure A-64. Sensitivity analysis: acceleration rate versus travel time (Newberry_5_1_EB) Sensitivity Analysis (TT vs Deceleration Rate) for Newberry_5_1_EB0 20 40 60 80 100 120 140 160 57.51012.51517.520 Deceleration Rate(ft2/s)TT(sec) Figure A-65. Sensitivity analysis: deceleration rate versus travel time (Newberry_5_1_EB) Sensitivity Analysis(TT vs Acceleration Rate) for Newberry_5_1_WB0 50 100 150 200 250 300 57.51012.51517.520 Acceleration Rate(ft2/s)TT(sec) Figure A-66. Sensitivity analysis: acceleration rate versus travel time (Newberry_5_1_WB) PAGE 185 185 Sensitivity Analysis(TT vs Deceleration Rate) for Newberry_5_1_WB0 50 100 150 200 250 300 57.51012.51517.520 Deceleration Rate (ft2/s)TT(sec) Figure A-67. Sensitivity analysis: deceleration rate versus travel time (Newberry_5_1_WB) Entering Flow rate at Each Intersection Sensitivity Analysis (TT vs Intersection 1 entering flow) for Beaver_Pugh_PM0 10 20 30 40 50 60 70 80 90 25050075010001250 Intersection 1 Entering Flow (veh/hr/ln)TT(sec) Figure A-68.Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Pugh_PM) Sensitivity Analysis (TT vs Intersection 2 entering flow) for Beaver_Pugh_PM 0 10 20 30 40 50 60 70 80 90 25050075010001250 Intersection 2 Entering Flow (veh/hr/ln)TT(sec) Figure A-69.Sensitivity Analysis: flow rate versus travel time for intersection 2(Beaver_Pugh_PM) PAGE 186 186 Sensitivity Analysis (TT vs Intersection 1 Entering Flow) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 110 25050075010001250 Intersection 1 Entering Flow (veh/hr/ln) TT(sec) Figure A-70. Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Sparks_Mid) Sensitivity Analysis (TT vs Intersection 2 Entering Flow) for Beaver_Sparks_Mid0 10 20 30 40 50 60 70 80 90 100 110 250500750800 Intersection 2 Entering Flow(veh/hr/ln)TT(sec) Figure A-71. Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Sparks_Mid) Sensitivity Analysis (TT vs Intersection 1 Entering Flow) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 110 2505007501000 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure A-72.Sensitivity analysis: flow rate versus travel time for intersection 1(Beaver_Sparks_PM) PAGE 187 187 Sensitivity Analysis (TT vs Intersection 2 Entering Flow) for Beaver_Sparks_PM0 10 20 30 40 50 60 70 80 90 100 110 2505007501000 Intersection 2 Entering Flow(veh/hr/ln)TT(sec) Figure A-73.Sensitivity analysis: flow rate versus travel time for intersection 2(Beaver_Sparks_PM) Sensitivity Analysis (TT vs Intersection 3 Entering Flow) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 25050075010001250 Intersection 3 Entering Flow(veh/hr/ln)TT(sec) Figure A-74. Sensitivity analysis: flow rate versus travel time for intersection 3 (Park_Mid_WB) Sensitivity Analysis(TT vs Intersection 2 Entering Flow) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 2505007501000125013001350 Intersection 2 Entering Flow(veh/hr/ln)TT(sec) Figure A-75. Sensitivity analysis: flow rate versus travel time for intersection 2 (Park_Mid_WB) PAGE 188 188 Sensitivity Analysis (TT vs Intersection 1 Entering Flow) for Park_Mid_WB 0 20 40 60 80 100 120 140 160 180 250500550600650 Intersection 1 Entering Flow (veh/hr/ln)TT(sec) Figure A-76. Sensitivity analysis: flow rate versus travel Time for intersection 1 (Park_Mid_WB) Sensitivity Analysis (TT vs Intersection 1 Entering Flow) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 2505007501000 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure A-77. Sensitivity analysis: flow rate versus travel time for intersection 1 (Park_PM_EB) Sensitivity Analysis (TT vs Intersection 2 Entering FLow) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 250500750100010501100 Intersection 2 Entering FLow(veh/hr/ln)TT(sec) Figure A-78. Sensitivity analysis: flow rate versus travel time for intersection 2 (Park_PM_EB) PAGE 189 189 Sensitivity Analysis (TT vs Intersection 3 Entering Flow) for Park_Mid_WB0 20 40 60 80 100 120 140 160 180 25050075010001050 Intersection 3 Entering Flow(veh/hr/ln)TT(sec) Figure A-79. Sensitivity analysis: flow rate versus travel time for intersection 3 (Park_PM_EB) Sensitivity Analysis (TT vs Intersection 1 Entering Flow) for Park_PM_WB0 20 40 60 80 100 120 140 160 180 250500550600 Intersection 1 Entering Flow (veh/hr/ln)TT(sec) Figure A-80. Sensitivity analysis: flow rate versus travel time for intersection 3 (Park_PM_WB) Sensitivity Analysis (TT vs Intersection 5 Entering Flow) for Newberry_4_30_WB 0.00 50.00 100.00 150.00 200.00 250.00 300.00 5001000150020002500 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure A-81. Sensitivity analysis: flow rate versus travel time for intersection 5 (Newberry_4_30_WB) PAGE 190 190 Sensitivity Analysis (TT vs Intersection 4 Entering Flow) for Newberry_4_30_WB 0.00 50.00 100.00 150.00 200.00 250.00 300.00 5001000110012001300 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure A-82. Sensitivity analysis: flow rate versus travel time for intersection 4 (Newberry_4_30_WB) Sensitivity Analysis (TT vs Intersection 3 Entering Flow) for Newberry_4_30_WB 0.00 50.00 100.00 150.00 200.00 250.00 300.00 50010001100120013001400 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure A-83. Sensitivity analysis: flow rate versus travel time for intersection 3 (Newberry_4_30_WB) Sensitivity Analysis (TT vs Intersection 2 Entering Flow) for Newberry_4_30_WB 0.00 50.00 100.00 150.00 200.00 250.00 300.00 50060070080090010001100 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure A-84. Sensitivity analysis: flow rate versus travel time for intersection 2 (Newberry_4_30_WB) PAGE 191 191 Sensitivity Analysis (TT vs Intersection 1 Entering Flow) for Newberry_4_30_WB 0.00 50.00 100.00 150.00 200.00 250.00 300.00 500100011001200130014001500 Intersection 1 Entering Flow(veh/hr/ln)TT(sec) Figure A-85. Sensitivity analysis: flow rate versus travel time for intersection 1 (Newberry_4_30_WB) Entering Flow rate at Each Intersection: same as Newberry_5_1_EB Entering Flow rate at Each Intersection: same as Newberry_4_30_WB PAGE 192 192 LIST OF REFERENCES Ahmed, K., and Abu-Lebdeh, G. 2005, Modelin g of Delay Induced by Downstream Traffic Disturbances at Signalized Intersections, Pr esented at TRB 84th annual meeting, National Research Council,Washington, D. C. Benekohal, R.F., and Kim, S.O. 2005, Arrival Based Uniform Delay Model for Oversaturated Signalized Intersections with Poor Progressi on, Presented at TRB 84th annual meeting, National Research Council,Washington, D. C. Du, J. and Aultman-Hall, L, 2006, Using Spatia l Analysis to Estimate Link Travel Times on Local Roads, Presented at TRB 85th annual meeting, National Research Council, Washington, D.C Elefteriadou, L., G. Natarajan, P. Johnson, J. 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She got her Master of Science degree in the same major from the University of Iowa in 200 3. Upon graduating in May 2003, she entered the Ph.D. program in transportation engineering at Penn State University in the Department of Civil and Environmental Engineer ing. In August 2004, she transferred to the Department of Civil and Coastal Engineer ing at the University of Florida. |