MODELING TRAFFIC FLOW ON OVERSATURATED ARTERIALS A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA

Transcription

1 MODELING TRAFFIC FLOW ON OVERSATURATED ARTERIALS A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY XINKAI WU IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY FACULTY ADVISOR: HENRY X. LIU OCTOBER 2010

2 XINKAI WU 2010 ALL RIGHTS RESERVED

3 To my dear grandmother

4 Acknowledgements I would lke to express the deepest apprecaton to my advsor, Dr. Henry X. Lu. Hs nfectous enthusasm has encouraged me to carry out successful research. Wthout hs contnuous gudance and support, ths dssertaton would not have been possble. I am also greatly thankful to Dr. Panos G. Mchalopoulos. He offered me the opportunty to jon the transportaton program at the unversty and ntroduced me to traffc flow theory. Many thanks go to Dr. Davd M. Levnson. I had wonderful experences of workng wth hm. Hs energy, ambton, and sense of novel research deas have nspred me. I also want to thank Dr. Nkolas Gerolmns. As a good frend, I admre hs ntellgence and thank hm for sharng hs successful experences wth me. I also apprecate the help from Dr. Gary A. Davs. I have benefted much from hs profound knowledge n statstcs, and I respect hs ntellgence and ntegrty. Fnally, a specal thanks goes to Dr. Shash Shekhar for hs servce on my commttee. Hs nsghtful opnons have mproved ths dssertaton. Ths research could not have been done wthout the fnancal support from the Mnnesota Department of Transportaton and the Natonal Cooperatve Hghway Research Program. Specally, I would lke to acknowledge Mr. Steve Msgen and Mr. Ron Chrstopherson from the Mnnesota Department of Transportaton. I would also lke to thank my fellow colleagues and frends, especally Mr. Heng Hu, Dr. Guzhen Yu, and Mr. Sundeep Bhmreddy for ther help n data collecton and algorthm programmng. I also much apprecate Mr. Hu Xong and Mr. Xaozheng He for ther help n my lfe. Last but not least, my wholehearted grattude goes to my famly, especally to Ms. Fengqn Lan, the lovely young lady I am wllng to spend my lfe wth and who s wllng to do the same wth me.

5 Abstract Traffc congeston s a natonal ssue n the Unted States and has gotten worse n regons of all szes. Now, more and more ntersectons are operated n oversaturated stuatons where the traffc demand exceeds the capacty of the system. Although a sgnfcant amount of lterature has been devoted to how to manage oversaturated traffc sgnal systems, our understandng of the characterstcs of oversaturaton remans lmted, partcularly wth regard to dentfcaton of oversaturaton and the transton process from under-saturated condton to oversaturaton. It has become ncreasngly obvous that successful traffc management requres effcent methods to dentfy and model oversaturated condtons. Ths research moves towards a better understandng of oversaturaton, by 1) provdng coherent methodologes to quantfy oversaturaton and 2) developng a smplfed model to descrbe oversaturaton at sgnalzed ntersectons based on hgh-resoluton traffc sgnal data collected by the SMART-SIGNAL (Systematc Montorng of Arteral Road Traffc Sgnals) system. In partcular, the research focuses on the followng four areas: 1) Quantfcaton of oversaturaton: Tradtonal defntons of oversaturaton are not applcable for exstng detecton systems. Ths research crcumvents ths ssue by quantfyng the detrmental effects of oversaturaton on sgnal operatons, both temporally and spatally. In the temporal dmenson, the detrmental effect s characterzed by a resdual queue at the end of a cycle, whch occupes a porton of green tme n the next cycle. In the spatal dmenson, the detrmental effect s characterzed by a downstream spllover, whch blocks the traffc and reduces usable green tme. From these observatons, we derve two types of an oversaturaton severty ndex (OSI): one temporally-based (T-OSI) and one spatally-based (S-OSI). Both T-OSI and S-OSI are desgned to yeld a rato between the unusable green tme due to detrmental effects and the total avalable green tme n a cycle, usng hgh resoluton traffc sgnal data. T-OSI s quantfed by estmatng the resdual queue length; and S-OSI s quantfed by measurng

6 the tme perod of spllover. Snce dfferent types of OSI (T-OSI or S-OSI) pont to dfferent underlyng causes of oversaturaton, ths research has the potental to provde gudance for the mtgaton strateges of sgnal oversaturaton. 2) Real-tme queue length estmaton for congested ntersectons: To quantfy T-OSI, ths research proposes a novel shockwave-based algorthm to estmate tme-dependent queue length even when the sgnal lnks are congested wth long queues, a stuaton that the tradtonal nput-output approach for queue length estmaton cannot handle. Usng hgh-resoluton event-based traffc sgnal data, the new algorthm frst dentfes traffc state changes; and then apples Lghthll-Whtham-Rchards (LWR) shockwave theory to estmate maxmum and mnmum (.e. resdual) queue length. Ths algorthm s also applcable for other aspects of arteral performance such as travel tme, delay, and level of servce. 3) Queue-Over-Detector (QOD): To quantfy S-OSI, we study a phenomenon we call Queue-Over-Detector (QOD). QOD occurs when a vehcle stops and rests on a detector for a perod of tme creatng a large occupancy value. Ths research demonstrates that a man cause of QOD s spllover from downstream ntersectons. Thus QOD dentfcaton can be used to quantfy oversaturaton n the spatal dmenson,.e. S-OSI. Ths research also brefly studes the relatonshp between QOD and the cycle-based arteral fundamental dagram (AFD) by mcroscopcally nvestgatng ndvdual vehcle trajectores derved from event-based data. Results show that proper treatment of QOD results n a stable form of the AFD whch clearly dentfes three dfferent regmes, under-saturaton, saturaton, and over-saturaton wth queue spllovers. Achevng a stable form of the AFD s of great mportance for traffc sgnal control because of ts ablty to dentfy traffc states on a sgnal lnk. 4). Traffc flow modelng for oversaturated arterals: The culmnaton of ths research project s a smplfed traffc flow model for congested arteral networks, whch we call the shockwave profle model (SPM). Unlke conventonal macroscopc models, n whch space s often dscretzed nto small cells for numercal soluton, SPM treats each v

7 homogeneous road segment wth constant capacty as a secton; then categorzes the traffc wthn each secton smply as free-flow, saturated, or jammed. Traffc dynamcs are analytcally descrbed by tracng the shockwave fronts whch explctly separate these three traffc states. SPM s partcularly sutable for smulatng traffc flow on congested sgnalzed arterals, especally wth queue spllover problems. In SPM, queue spllover can be treated as ether extendng a red lght or creatng new smaller cycles. Snce only the essental features of arteral traffc flow,.e., queue buld-up and dsspaton, are consdered, SPM sgnfcantly smplfes arteral network desgn and mproves numercal effcency. For these reasons, we fully expect ths model to be adopted n real-tme applcatons such as arteral performance predcton and sgnal optmzaton. v

8 Table of Contents ACKNOWLEDGEMENTS... II ABSTRACT... III TABLE OF CONTENTS... VI LIST OF TABLES... VIII LIST OF FIGURES... IX ACRONYMS... XI NOTATION... XII 1 INTRODUCTION PROBLEM STATEMENT RESEARCH CONTRIBUTIONS DISSERTATION ORGANIZATION BACKGROUND SMART-SIGNAL SYSTEM ARCHITECTURE DATA COLLECTION UNIT DATA PROCESSING PROCEDURE A QUANTIFIABLE MEASURE OF OVERSATURATION THE CONVENTIONAL DEFINITION OF OVERSATURATION A QUANTIFIABLE MEASURE OF OVERSATURATION REAL-TIME QUEUE LENGTH ESTIMATION FOR CONGESTED INTERSECTIONS SHOCKWAVE ANALYSIS & BREAK POINTS IDENTIFICATION QUEUE ESTIMATION MODELS IMPLEMENTATION DISCUSSION SUMMARY QUANTIFICATION OF OVERSATURATION ALGORITHM FOR RESIDUAL QUEUE LENGTH ESTIMATION ALGORITHM FOR IDENTIFICATION OF SPILLOVER FIELD-TESTING & RESULTS SUMMARY v

9 6 TRAFFIC FLOW MODELING FOR OVERSATURATED ARTERIALS BACKGROUND THE SHOCKWAVE PROFILE MODEL (SPM) NETWORK REPRESENTATION NUMERICAL EXAMPLE COMPARING SPM & CTM FIELD TEST RESULTS SUMMARY CONCLUSIONS & FUTURE RESEARCH CONCLUSIONS FUTURE RESEARCH REFERENCES v

10 Lst of Tables TABLE 4-1 MEAN ABSOLUTE PERCENTAGE ERRORS OF THREE MODELS TABLE 4-2 ABSOLUTE ERRORS OF TIME OF MAXIMUM QUEUE LENGTH TABLE 4-3 MEAN ABSOLUTE PERCENTAGE ERRORS OF MAXIMUM QUEUE LENGTH TABLE 5-1 OVERSATURATION SEVERITY INDICES (OSI) FOR WINNETKA AVE. INTERSECTION TABLE 5-2 OVERSATURATION SEVERITY INDICES (OSI) FOR RHODE ISLAND INTERSECTION TABLE 6-1 COMPARISON OF ERROR RATES OF TRAFFIC VOLUMES FOR TWO DETECTOR.. 92 v

11 Lst of Fgures FIGURE 2-1 SMART-SIGNAL SYSTEM ARCHITECTURE (SOURCE: LIU & MA, 2009)... 6 FIGURE 2-2 DEMONSTRATION OF THE TRAFFIC DATA COLLECTION COMPONENTS... 7 FIGURE 2-3 SAMPLE DATA (SOURCE: LIU ET AL., 2010)... 8 FIGURE 2-4 DATA FLOW OF SMART-SIGNAL SYSTEM (SOURCE: LIU ET AL., 2008)... 9 FIGURE 4-1 REPRESENTATION OF SHOCKWAVES IN THE FUNDAMENTAL DIAGRAM FIGURE 4-2 SHOCKWAVE PROPAGATION FIGURE 4-3 BREAK POINTS A, B, C, & TRAFFIC SHOCKWAVES AT AN INTERSECTION FIGURE 4-4 (A) DETECTOR OCCUPANCY TIME PROFILE IN A CYCLE; (B) TIME GAP BETWEEN TWO CONSECUTIVE VEHICLES IN A CYCLE FIGURE 4-5 TWO OTHER EXAMPLES OF OCCUPANCY TIME & GAP DATA (A) OVERSATURATION; (B) PLATOON ARRIVAL FIGURE 4-6 BASIC MODEL FOR INTERSECTION QUEUE LENGTH ESTIMATION FIGURE 4-7 EXPANSION I FOR INTERSECTION QUEUE LENGTH ESTIMATION FIGURE 4-8 EXPANSION II FOR INTERSECTION QUEUE LENGTH ESTIMATION FIGURE 4-9 FLOW CHART OF IMPLEMENTATION PROCEDURE FOR INTERSECTION QUEUE LENGTH ESTIMATION FIGURE 4-10 (A) DATA COLLECTION SITE; (B) DETECTOR LAYOUT FIGURE 4-11 TESTING RESULTS MAXIMUM QUEUE LENGTH COMPARISON FIGURE 4-12 QUEUE TRAJECTORIES ESTIMATED BY THREE MODELS FIGURE 4-13 COMPARISON OF MAXIMUM QUEUE LENGTH OBTAINED FROM THE BASIC MODEL & INDEPENDENT OBSERVERS FIGURE 4-14 IMMEDIATE PLATOON ARRIVAL AFTER QUEUE DISCHARGE FIGURE 5-1 CALCULATION OF RESIDUAL QUEUE LENGTH WHEN POINT C CANNOT BE IDENTIFIED FIGURE 5-2 DATA COLLECTION SITE ON TH55 IN MINNEAPOLIS, MINNESOTA FIGURE 5-3 CYCLE-BASED AFDS: (A) DATA COLLECTED BY STOP-BAR DETECTOR; (B) DATA COLLECTED BY ADVANCE DETECTOR FIGURE 5-4 VEHICLE TRAJECTORIES FOR TWO CYCLES: (A) DATA POINT AT AM - (600, 0.213); (B) DATA POINT AT PM - (600, 0.064) FIGURE 5-5 MAXIMUM REPLACEABLE QOD-I TIME FIGURE 5-6 AFDS AFTER REMOVING QOD-I: (A) DATA COLLECTED BY ADVANCE DETECTOR; (B) DATA COLLECTED BY STOP-BAR DETECTOR FIGURE 5-7 A STABLE AFD (BASED ON TWO-WEEK S DATA: FROM NOV. 10 TH, 2008 TO NOV. 21 ST, 2008, WEEKDAYS DATA ONLY) FIGURE 5-8 TWO TYPES OF QUEUE-OVER-DETECTOR EVENTS FIGURE 5-9 SHOCKWAVE PROFILE WITH DOWNSTREAM SPILLOVER FIGURE 5-10 DETECTOR LAYOUT IN DATA COLLECTION SITE FIGURE 5-11 ESTIMATION RESULTS OF RESIDUAL QUEUE FOR EASTBOUND APPROACH AT GLENWOOD FIGURE 5-12 ESTIMATED RESIDUAL QUEUE LENGTH FOR EASTBOUND APPROACH AT BOONE AVE x

12 FIGURE 5-13 IDENTIFICATION OF QOD-II USING DETECTOR OCCUPANCY TIME DATA FIGURE 5-14 VEHICLE TRAJECTORIES IN THE CASE OF SPILLOVER FROM WINNETKA TO RHODE ISLAND FIGURE 5-15 AFD AFTER QOD-I REMOVAL FIGURE 5-16 QUEUE LENGTH PROFILE AT THE INTERSECTION OF WINNETKA FIGURE 6-1 LAYOUT OF A SIGNALIZED ARTERIAL WITH THREE INTERSECTIONS FIGURE 6-2 SHOCKWAVE PROFILE OF SINGLE INTERSECTION: (A) WITHOUT RESIDUAL QUEUES; (B) WITH A RESIDUAL QUEUE FIGURE 6-3 TWO INTERPRETIVE CASES FOR SPILLOVER: (A) I: EXTENDING THE RED PHASE; (B) II: CREATING NEW CYCLES FIGURE 6-4 SHOCKWAVE PROFILES FOR THREE INTERSECTIONS WITH SPILLOVERS FIGURE 6-5 NODES & ARCS FIGURE 6-6 (A) LAYOUT OF AN INTERSECTION WITH TURNING BAY; (B) NETWORK REPRESENTATION FIGURE 6-7 SHOCKWAVE PROFILES FOR AN INTERSECTION WITH SPILLOVER AT TURNING BAYS FIGURE 6-8 LAYOUTS OF INTERSECTIONS WITH TURNING BAYS THROUGH MOVEMENT ARE PARTIALLY BLOCKED FIGURE 6-9 (A) LAYOUT OF THE TWO-INTERSECTION EXAMPLE; (B) SIGNAL TIMING PARAMETERS; (C) INPUT FLOW RATE; (D) THE FD FOR CTM FIGURE 6-10 CELL DESIGN FOR CTM FIGURE 6-11 SECTION DESIGN FOR SPM FIGURE 6-12 (A) CTM DENSITY CONTOUR (TIME INTERVAL: 1SEC); (B) SHOCKWAVE PROFILE GENERATED BY SPM FIGURE 6-13 OUTPUT AT THE STOP-BAR FOR THE FIRST INTERSECTION FIGURE 6-14 LAYOUT OF TH55 TEST SITE FIGURE 6-15 NETWORK REPRESENTATION OF THE THREE INTERSECTIONS ON TH FIGURE 6-16 COMPARISON OF OBSERVED & SIMULATED TRAFFIC VOLUMES FIGURE 6-17 SHOCKWAVE PROFILES FOR NODES 15, 17, 18, & FIGURE 6-18 COMPARISON OF TIME LENGTHS OCCUPIED BY SPILLOVERS x

13 Acronyms AFD: CTM: LWR: MAPE: MPE: OSI: QOD: QOD-I: QOD-II: S-OSI: Arteral Fundamental Dagram Cell Transmsson Model Lghthll-Whtham-Rchards Mean Absolute Percentage Error Mean Percentage Error Oversaturaton Severty Index Queue-Over-Detector The frst type of QOD: caused by red phases The second type of QOD: caused by a spllover OSI n the spatal dmenson SMART-SIGNAL: Systematc Montorng of Arteral Road Traffc Sgnals SPM: T-OSI: Shockwave Profle Model OSI n the temporal dmenson x

14 Notaton d jam : g n (t): h s : Space headway at jammed traffc condtons Effectve green nterval n the current cycle for the through movement at ntersecton n when tme t belongs to the current cycle Saturaton dscharge tme headway : Index of cycle k: Densty k a : Average densty of arrval flow durng the th cycle k s n : Saturaton densty (when traffc flow s maxmum) at lnk L n k jam n : Jam densty at lnk L n w j n l t : Dstance from the front of shockwave w j (j = 1, 2, 3, or 4) to the stop lne at ntersecton n at tme t l ŵ n t : Queue length, defned as the dstance from the front of shockwave w 1 or w 3 to the stop lne at ntersecton n at tme t n: Index of Intersecton n q: Flow q a : Average arrval flow rate durng the th cycle q s n : q n-1 (t) : Maxmum traffc flow rate at lnk L n Arrval flow rate at the entrance of lnk L n at tme t x

15 q () n t : Departure flow rate at ntersecton n at tme t for the major approaches q () t : Departure flow rate at ntersecton n at tme t for the mnor m n approaches q n V1 (t) : q n V2 (t) : r n (t) : V1 Arrval flow rate at the entrance of secton L n V2 Arrval flow rate at the entrance of secton L n Effectve red nterval n the current cycle for the through movement at ntersecton n when tme t belongs to the current cycle t: Tme t v: Speed v f : Free-flow speed w: The velocty of a shockwave w 1 : w 2 : w 3 : w 4 : Queung wave Dscharge wave Departure wave Compresson wave w * : The absolute value of w 2 or w 4 β n : Δt Turnng percentage for the through movement at lnk L n A small tme nterval x

16 D e : L d : L max : L mn : Effectve vehcle length Dstance between advance detector and stop-bar Maxmum queue length durng the th cycle Mnmum queue length durng the th cycle L n : Lnk L n (wth length L n ) L A n : L B n : L U n : L V1 n : L V2 n : O n (t) : Turnng bay at lnk L n Downstream through secton at lnk L n (startng from separaton pont U to lnk end) Upstream through secton at lnk L n (startng from lnk entrance to separaton pont U) U Blocked lane(s) n secton L n U Unblocked lane(s) n secton L n Start tme of the effectve red (or the end of effectve green) n the current cycle for the through movement at lnk L n T: End tme of smulaton or the whole smulaton tme perod T g : T QOD end : T max : T mn : End tme of the effectve green durng the th cycle Tme nstant when QOD ends Tme nstant when the approach has the maxmum queue length durng the th cycle Tme nstant when the approach has the mnmum queue length durng the th cycle xv

17 T r : T QOD start : T A : T B : T C : End tme of the effectve red durng the th cycle Tme nstant when QOD starts Tme nstant when break pont A happens Tme nstant when break pont B happens Tme nstant when break pont B happens xv

18 1 Introducton 1.1 Problem Statement Traffc congeston s a natonal ssue n the Unted States and has gotten worse n regons of all szes (Texas Transportaton Insttute, 2009). Now, more and more urban and suburban ntersectons are operated n congested condtons and stuatons where the traffc demand exceeds the capacty of the system. Under ths condton of oversaturaton, typcal traffc control strateges do not work as effcently as necessary. As ndcated by the results of the 2007 Traffc Sgnal Operaton Self Assessment surveys (Insttute of Transportaton Engneers, 2007), the majorty of agences nvolved n the operaton and mantenance of traffc sgnal systems are already stretched thn and challenged to provde adequate servce to drvers n ther jursdctons. Oversaturated condtons present an addtonal burden for practtoners that lack adequate tools for addressng such stuatons. Although a sgnfcant amount of lterature has been devoted to how to manage oversaturated traffc sgnal systems, our understandng of the characterstcs of oversaturaton, partcularly dentfcaton of oversaturaton and the transton process from under-saturated condton to oversaturaton, s lmted. As a result, there s a crtcal need for new methods to dentfy and model oversaturaton. Ths research moves towards a better understandng of oversaturaton by provdng coherent methodologes to quantfy and model oversaturaton at sgnalzed ntersectons. Two major themes are the focus of ths dssertaton: 1). Quantfcaton of oversaturaton. Prevous research studes usng traffc data from sgnal systems to dagnose and dentfy oversaturaton are mostly qualtatve and ncomplete. Conceptual defntons of oversaturaton, whch defne an oversaturated traffc ntersecton as one where traffc demand exceeds the capacty, cannot be appled drectly to dentfy oversaturated ntersectons because traffc demand under congested 1

19 condtons s not measurable, partcularly wth fxed-locaton sensors. Furthermore, successful mtgaton strateges requre the ablty both to detect the onset of oversaturaton and to quantfy ts severty. Thus t s mperatve to have an mplementable and quantfable metrc of oversaturaton as well as a coherent methodology to dentfy such stuaton. 2). Modelng of oversaturaton. Compared to the contnuous and ntensve research efforts on modelng of freeway traffc, there have been far fewer reported efforts to study arteral traffc flow. Exstng traffc flow models are desgned to model contnuum freeway traffc, and thus have defcences when appled to congested arterals where traffc s nterrupted by traffc lghts. Ths s partcularly true for large arteral networks durng oversaturated condtons. Snce traffc flow modelng serves as the foundaton for all sgnal optmzaton schemes, there s a need for a quck and approxmate approach, wth suffcent descrptve power, to model oversaturated traffc flow dynamcs n a sgnalzed network for real-tme applcatons. The research to address these ssues s bult upon the hgh-resoluton traffc sgnal data collected and archved by the SMART-SIGNAL (Systematc Montorng of Arteral Road Traffc Sgnals) system developed by the Unversty of Mnnesota (Lu & Ma, 2009; Lu et al., 2010; Wu et al., 2010a; Wu et al., 2010b; Wu et al., 2010c). Wthout the real-tme data collected by ths system, our project would not have been possble. 1.2 Research Contrbutons Ths thess makes contrbutons n the areas of arteral performance measurement, arteral traffc flow theory, and arteral sgnal control. Specfcally: 1) It proposes a systematc approach to quantfy oversaturaton. Tradtonal defntons of oversaturaton are not applcable for exstng detecton systems. Ths research crcumvents ths ssue by quantfyng the detrmental effects of oversaturaton 2

20 on sgnal operatons, both temporally and spatally. The detrmental effect s characterzed ether temporally by a resdual queue at the end of a cycle, whch occupes a porton of green tme n the next cycle; or spatally by the spllover from downstream traffc whch reduces usable green tme due to the downstream blockage. An oversaturaton severty ndex (OSI), based ether on the temporal dmenson (T-OSI) or spatal dmenson (S-OSI), can then be calculated as the rato between the unusable green tme due to detrmental effects and the total avalable green tme n a cycle. T-OSI s quantfed by estmatng the resdual queue length; and S-OSI s quantfed by measurng the tme perod of spllover. Snce dfferent types of OSI (T-OSI or S-OSI) pont to dfferent underlyng causes of oversaturaton, ths research s expected to provde gudance for strateges to mtgate sgnal oversaturaton. 2) It develops a shockwave-based ntersecton queue length estmaton algorthm. The tradtonal nput-output approach for queue length estmaton can only handle queues that are shorter than the dstance between the vehcle detector and ntersecton stop lne, because cumulatve vehcle counts for arrval traffc are not avalable once the detector s occuped by the queue. To estmate queue length under congested condtons, a novel shockwave-based queue estmaton algorthm s developed n ths research. Usng hghresoluton event-based traffc sgnal data, the new algorthm frst dentfes traffc state changes that dstngush queue dscharge flow from upstream arrval traffc; t then apples Lghthll-Whtham-Rchards (LWR) shockwave theory to estmate maxmum and mnmum (.e. resdual) queue length. Snce ths approach can estmate tme-dependent queue length even when sgnal lnks are congested wth long queues, we use t to quantfy T-OSI. Ths algorthm can also be used to measure other aspects of arteral performance such as travel tme, delay, and level of servce. 3) It dentfes the Queue-Over-Detector (QOD) phenomenon and ts sgnfcance. Ths work s the frst to study the QOD phenomenon. We demonstrate the relatonshp between QOD and spllover, and then propose an algorthm to quantfy S-OSI by dentfyng QOD. Ths research also studes the mpact of QOD on the cycle-based arteral fundamental dagram (AFD) by mcroscopcally nvestgatng ndvdual vehcle 3

21 trajectores derved from event-based data; analyss reveals that proper treatment of QOD yelds a stable form of AFD whch clearly dentfes three dstnct regmes: undersaturaton, saturaton, and over-saturaton wth queue spllovers. Havng a stable form of AFD s of great mportance for traffc sgnal control because of ts ablty to dentfy traffc states on a sgnal lnk. 4) It bulds a shockwave profle traffc flow model for congested arteral networks. The culmnaton of ths work s a new traffc flow model for congested arteral networks, the shockwave profle model (SPM). Unlke conventonal macroscopc models, n whch space s often dscretzed nto small cells for numercal soluton, SPM treats each homogeneous road segment wth constant capacty as a secton, and then categorzes traffc n a secton nto three dstnct states: free-flow, saturated, and jammed. Traffc dynamcs are analytcally descrbed by tracng the shockwave fronts whch explctly separate the three traffc states. SPM s partcularly sutable for smulatng traffc flow on congested sgnalzed arterals, especally those wth queue spllover problems. In SPM, queue spllover s treated as ether extendng a red lght or creatng new smaller cycles. Snce t only consders the essental features of arteral traffc flow,.e., queue buld-up and dsspaton, SPM greatly smplfes the process of network desgn and mproves numercal effcency. For these reasons, adopton of ths model can reasonably be expected n a number of real-tme applcatons such as arteral performance predcton and sgnal optmzaton. 1.3 Dssertaton Organzaton The rest of the dssertaton s organzed as follows: Chapter 2 brefly revews the SMART-SIGNAL System. Chapter 3 ntroduces a quantfable measurement of oversaturaton - oversaturaton severty ndex (OSI). In Chapter 4, a novel algorthm s proposed for queue length estmaton at congested ntersectons. The algorthms to quantfy OSI n both temporal and spatal dmensons are descrbed n Chapter 5. Chapter 6 proposes our shockwave profle traffc flow model for oversaturated arterals. Chapter 7 concludes ths dssertaton and lays out the drecton of future research. 4

22 2 Background SMART-SIGNAL The SMART-SIGNAL (Systematc Montorng of Arteral Road Traffc and SIGNAL) system, developed by the Unversty of Mnnesota, offers an easly mplementable approach to collect and archve contnuous and hgh-resoluton traffc data on sgnalzed arterals (Lu et al., 2008). In ths system, a complete hstory of traffc sgnal control, ncludng all vehcle actuaton events and sgnal phase change events, s archved and stored. Based on the event-based data, arteral performance measures, such as arteral travel tme, ntersecton queue length, and level of servce, are produced. The SMART- SIGNAL system has been nstalled on 11 ntersectons along France Avenue n Hennepn County, Mnnesota, and 6 ntersectons along Trunk Hghway 55, Mnnesota, snce February Event-based traffc data are beng collected n a 24/7 mode and then archved n a database system, thus yeldng a tremendous amount of feld data avalable for research. Ths chapter wll brefly ntroduce the system archtecture, data collecton hardware, and data processng procedure of the SMART-SIGNAL. 2.1 System Archtecture As llustrated n Fgure 2-1, the SMART-SIGNAL system has three major components, data collecton, performance measurements, and performance presentaton through user nterfaces. The data collecton component collects hgh resoluton raw data drectly from the feld on an event-by-event bass. Sgnal phase change events and vehcle-detector actuaton events are acqured separately from data collecton unts located n traffc sgnal cabnets. The data are recorded n daly log fles and sent to a data server at the master cabnet by seral port communcaton. The daly log fles are fnally transmtted to a database located at the Mnnesota Transportaton Observatory (MTO) lab at the Unversty of Mnnesota through the Dgtal Subscrber Lne (DSL) or wreless communcaton. The second component of the SMART-SIGNAL system s performance measure calculaton usng the feld-collected data. Analyss of the stored data log fles 5

23 yelds a set of performance measures, one covers ntersecton level measures (e.g. queue length) and a second related to arteral level measures (e.g. travel tme). Once all performance measures have been derved from the raw data, the results are made accessble to a varety of users. UMN TRAFFIC LAB (MTO) (Web Interface) Fgure 2-1 SMART-SIGNAL System Archtecture (Source: Lu & Ma, 2009) 2.2 Data Collecton Unt The key element of the SMART-SIGNAL s the data collecton unt, whch conssts of an ndustral PC and a data acquston card. At each ntersecton, an ndustral PC wth a data acquston card s nstalled, and event data collected at each ntersecton s transmtted to the data server n the master controller cabnet through the exstng communcaton lne (n ths case, spare twsted par) between sgnalzed ntersectons. The data acquston cards (PCI-6511 from Natonal Instruments (2006)) used n the 6

24 SMART-SIGNAL system, as shown n Fgure 2-2a, have 64 nput channels. If the total number of detector nputs and sgnal phases for one ntersecton exceed 64, an addtonal data acquston card needs to be nstalled. A termnal box s used n order to lmt the nput drect current (DC) to a safe range and establsh the connecton between the data acquston card and back panel of the traffc cabnet, as shown n Fgure 2-2b. The termnal box allows dgtal voltage changes on the back panel, whch ndcate dfferent traffc events n the feld, to be captured by the data acquston card nstalled n the ndustral computer. A traffc event recorder software program, developed usng the Mcrosoft Vsual C# program, runs on the ndustral computer n the feld to record the events (for example, a phase 1 green change from ON to OFF ) nto a log fle. Fgure 2-2 Demonstraton of the Traffc Data Collecton Components Data communcaton between two controller cabnets s done usng the exstng twsted par communcaton lnes. A protocol of RS-485 s used to transmt data and synchronze tme between cabnets (B&B Electroncs, 2007). After the data n the local cabnets s transferred to the master cabnet, DSL or a wreless unt nstalled n the master cabnet s used to send the data back to the database located n the MTO. A sample of data s shown n Fgure 2-3. Each logged event starts wth a tme stamp that ncludes the date, hour, mnute, second and mllsecond based on the computer system tme, followed by dfferent types of event data ncludng phase changes, detector actuaton and pedestran calls. A complete hstory of traffc sgnal events s thus recorded. 7

25 08:09:15.012, D8 on, 7.902s 08:09:15.481, D8 off, 0.468s 08:09:16.761, G3 off, s 08:09:16.761, Y3 on, s 08:09:17.620, D9 on, 2.686s 08:09:18.151, D10 on, 2.593s 08:09:18.307, D9 off, 0.687s 08:09:18.823, D10 off, 0.671s 08:09:20.244, Y3 off, 3.482s 08:09:21.649, D22 on, s 08:09:22.008, D22 off, 0.359s 08:09:23.242, G1 on, s Detector #8 on at 08:09:15.012; Vacant tme s 7.902s Green Phase #3 off at 08:09:16.761; Green duraton tme s s Detector #9 off at 08:09:18.307; Occupy tme s 0.687s Yellow Phase #3 off at 08:09:20.244; Yellow duraton tme s 3.482s Green Phase #1 on at 08:09:23.242; Red duraton tme s s Fgure 2-3 Sample Data (Source: Lu et al., 2010) 2.3 Data Processng Procedure The raw data collected from the feld needs to be preprocessed and converted to an easyread format before the performance measures can be derved. Based on event data, the sgnal phase duraton can be calculated from the tme dfference between the start and end of a sgnal event. The tme nterval between the start and end of a vehcle actuaton event s the detector occupancy tme, and the tme nterval between the end of a vehcle actuaton event and the start of a followng vehcle actuaton event (from the same detector) s the tme gap between two consecutve vehcles crossng the detector. Further processng can be done to determne second-by-second volume, occupancy, and cycleby-cycle sgnal tmng plan. The processed data can now be used to generate performance measures, such as queue length and average vehcle delay for ntersectons and travel tme for arteral lnks. Fgure 2-4 demonstrates the data flow of the SMART-SIGNAL system. 8

26 Raw Data Volume, Occupancy Preprocessed Data... Sgnal Tmng Intersecton Queue Intersecton Performance Measures... Delay Travel Tme Arteral Performance Measures Number of stops... Fgure 2-4 Data Flow of SMART-SIGNAL System (Source: Lu et al., 2008) Although many exstng sgnal control systems are capable of generatng data to support performance assessment, most do not make t easy for the managng agences to prortze mprovements and plan for future needs. The SMART-SIGNAL System flls ths gap. The hgh-resoluton event data collected by the system s extremely valuable. As wll be explaned n the followng chapters, the data s used to estmate real-tme ntersecton queue length, dentfy queue spllover, and quantfy the severty of oversaturaton. 9

27 3 A QUANTIFIABLE MEASURE OF OVERSATURATION 3.1 The Conventonal Defnton of Oversaturaton Conventonally, an oversaturated ntersecton s defned as one where the demand exceeds the capacty (Gazs, 1964). The degree of saturaton,.e. the volume/capacty rato, s defned as: X j VOL j (3.1) CAP j where X j s the degree of saturaton for lane group j and VOL j and CAP j are the demand flow rate and capacty for lane group j, respectvely. A lane group s oversaturated when X j >1. For a sngle ntersecton wth two competng streams, Gazs (1964) expanded ths concept by proposng the followng nequalty: q1 q2 LT 1 ( ) (3.2) S1 S2 C where q 1 and q 2 are the arrval rates for two conflctng drectons, S 1 and S 2 are the saturaton flow rates for the two drectons, LT s the total lost tme, and C s the cycle length. Drect applcaton of these two defntons to detect the onset of oversaturaton and quantfy ts duraton and extent s dffcult, however. Ths s partly due to the uncertanty of the capacty and saturaton flow, and partly due to the dffculty of measurng the arrval flow usng current data collecton systems durng oversaturated stuatons (the very condton that we are tryng to dentfy). Most exstng detecton systems, 10

28 partcularly those wth nductance loop detectors, provde observatons of traffc flows at a fxed pont on a lnk when they are not fully occuped. Traffc demand s smply not measurable when a fxed-locaton detector s occuped wth a vehcular queue. Alternatvely, some researchers have characterzed oversaturaton as a stopped queue that cannot be completely dsspated durng a green cycle (Gazs, 1964), or as traffc queues that persst from cycle to cycle ether due to nsuffcent green splts or because of blockage (Abu-Lebdeh & Benekohal, 2003). However, these authors provded no methodology for measurng queue length or dentfyng the exstence of such stuatons. Another measure that can potentally be used to detect oversaturaton s green phase utlzaton. If a sgnal phase s oversaturated wth a long queue, once the green lght starts, vehcles wll contnue to dscharge at the saturaton flow rate untl the end of the effectve green. The rato between the used green tme and the green phase duraton can be treated as an ndcator of saturaton for a partcular approach. Ths concept has been ncorporated nto several adaptve traffc control systems, for example, SCATS (Sydney Coordnated Adaptve Traffc System) (Sms & Dobnson, 1980) and ACS-Lte (Adaptve Control Software Lte) (Luyanda, et al., 2003; Gettman, et al., 2007). Ths method, however, cannot estmate the degree to whch a certan traffc phase s oversaturated. The ndcator smply dentfes that green tme s nsuffcent to serve the traffc demand, but the severty of oversaturaton s unknown. In addton, t s necessary to pont out that hgh green utlzaton of a gven phase s not necessarly an ndcator of oversaturaton. In some cases, well-coordnated sgnalzed ntersectons can also generate hgh values of green utlzaton. To the best of our knowledge, prevous research studes usng traffc data from sgnal systems to dagnose and dentfy oversaturaton are mostly qualtatve and ncomplete. Conceptual defntons dscussed above are ether not applcable n the real world or have other defcences. Snce detecton of the onset of oversaturaton as well as quantfyng the severty of oversaturaton s a crtcal step before approprate mtgaton strateges can be appled, t s mperatve to have an mplementable and quantfable measure of 11

29 oversaturaton as well as a coherent methodology to dentfy such stuatons. We beleve our research begns to fll ths gap. 3.2 A Quantfable Measure of Oversaturaton Snce the general defnton of oversaturaton,.e. traffc demand exceedng the capacty of a faclty, cannot be appled drectly to detect the occurrence of oversaturaton, we propose a measure of oversaturaton by quantfyng ts detrmental effects. The detrmental effects of oversaturaton are any effects whch lead to the reducton of usable green tme n a cycle for a sgnalzed approach. These effects can be descrbed from ether a temporal or a spatal perspectve. A detrmental effect of oversaturaton n the temporal dmenson s characterzed by a resdual queue at the end of a cycle. The resdual vehcles, whch are part of a dschargng platoon and supposed to pass through the ntersecton n the current cycle, cannot be dscharged due to nsuffcent green splts, thereby creatng detrmental effects on the followng cycle by occupyng a porton of green tme. The porton of green tme utlzed by the resdual queue then becomes unusable for the traffc arrvals durng that cycle. A detrmental effect of oversaturaton n the spatal dmenson can be characterzed by a spllover from downstream traffc. When spllover happens, a downstream lnk of the ntersecton s blocked and vehcles cannot be dscharged from the ntersecton even n a green phase. Here agan, a porton of the green tme becomes unusable wth detrmental effects. Whle spllover s common when vehcular queues fully occupy a downstream lnk, oversaturated condtons occur only f spllover from a downstream ntersecton leads to unusable green tme. Therefore, n our approach, the condton of oversaturaton s characterzed 1) by a resdual queue at the end of a cycle that creates detrmental effects on the followng cycle, and/or 2) by a downstream spllover wthn a cycle that creates detrmental effects on upstream traffc facltes. To quantfy the detrmental effects n ether temporal or 12

30 spatal dmensons, we ntroduce the oversaturaton severty ndex (OSI) by usng the rato between unusable green tme and total avalable green tme n a cycle. OSI wll be a non-negatve percentage value between 0 and 100, wth 0 ndcatng no detrmental effect for sgnal operaton, and 100 denotng the worst case where all avalable green tme becomes unusable. We further dfferentate OSI nto T-OSI and S-OSI. T-OSI descrbes the detrmental effects created by resdual queue,.e. the detrmental effect n the temporal dmenson; and S-OSI descrbes the detrmental effects caused by spllover,.e. the detrmental effect n the spatal dmenson. Although both T-OSI and S-OSI can be calculated usng the rato between unusable green tme and total avalable green tme, the meanngs of unusable are dfferent. For T-OSI, the unusable green tme s the equvalent green tme to dscharge the resdual queue n the followng cycle, n whch case vehcles are dscharged at saturaton flow rate durng that tme perod. By contrast, for S-OSI, the unusable green tme s the tme perod durng whch a downstream lnk s blocked, n whch case the dscharge rate s zero. Snce T-OSI quantfes the detrmental effect of oversaturaton on the followng cycle, the persstence, duraton, and frequency of T-OSI greater than zero becomes an mportant ndcator of traffc congeston at the ntersecton level. On the other hand, S-OSI descrbes the detrmental effect of oversaturaton caused by downstream queue spllover, ndcatng the spatal extent that traffc congeston has spread network-wde. The dfferentaton of T-OSI and S-OSI may also help to dentfy the causal relatonshp of arteral traffc congeston. For example, for an ntersecton, a postve T-OSI ndcates that the avalable green tme s nsuffcent for queue dscharge and a resdual queue s generated at the end of a cycle. Subsequently, n the followng cycles, the queue may grow and spll over to an upstream ntersecton, resultng n an S-OSI at the upstream ntersecton that s postve. Clearly n ths case a postve S-OSI at the upstream ntersecton s caused by the downstream bottleneck. Note that the downstream bottleneck may lead to a stuaton n whch both the S-OSI and T-OSI are greater than 13

31 zero, smply because a porton of the green tme s wasted due to downstream blockage (.e. S-OSI> 0), and a resdual queue may be generated (.e. T-OSI> 0) due to the reducton of green tme. Here postve T-OSI and S-OSI ndcate that traffc congeston may start to spread even further upstream. Wth the proposed oversaturaton severty ndex, the focus of the dentfcaton algorthms shfts from measurng travel demand to quantfyng detrmental effects n both temporal and spatal dmensons. It s necessary to pont out that the ablty to classfy and quantfy the detrmental effects created by oversaturaton s very mportant for traffc management, because dfferent oversaturated stuatons may call for dfferent strateges to mtgate congeston. For example, for an solated ntersecton wth postve T-OSI values, extenson of green tme to dscharge resdual queue may be suffcent (Qunn, 1992). By contrast for an arteral corrdor wth multple overloaded ntersectons, smultaneous or even negatve offsets (see Pgnataro, et al. 1978) may be needed to prevent further deteroraton of the oversaturated condton. Snce the focus of ths study s to dentfy and quantfy oversaturated condtons, how to map OSI wth dfferent mtgaton strateges s left for future research. In the followng two chapters, we propose two algorthms for the dentfcaton and quantfcaton of oversaturated condtons, one that estmates resdual queue length, and a second that detects spllover condtons. Our focus here s not only to dentfy oversaturaton qualtatvely, but also to quantfy ts severty. 14

32 4 Real-Tme Queue Length Estmaton for Congested Intersectons It has long been recognzed that vehcular queue length s crucal both for measurng sgnal performance n terms of vehcle delay and stops (Webster & Cobbe, 1966; Cronje, 1983a and 1983b; Balke et al., 2005), and for sgnal optmzaton (Webster, 1958; Gazs, 1964; Newell, 1965; Green, 1968; Mchalopoulos & Stephanopolos, 1977a, b; Chang & Ln, 2000; Mrchandan & Zou, 2007). Over the years, many researchers have dedcated themselves to ths topc, and two types of queue estmaton models have been developed. The frst one, whch s based on the analyss of cumulatve traffc nput-output to a sgnal lnk, was proposed by Webster (1958) and later mproved by a number of researchers (Newell, 1965; Robertson, 1969; Gazs, 1974; May, 1965; Catlng, 1977; Akcelk, 1999; Strong, et al., 2006; Sharma, et al., 2007; Vgos et al., 2008). Ths type of model s commonly used to descrbe traffc queung processes, but, as noted by Mchalopoulos & Stephanopolos (1981), t s nsuffcent for obtanng the spatal dstrbuton of queue lengths n tme. In addton, cumulatve nput-output technques can only be used for the estmaton of queue length when the rear of the queue does not extend beyond the vehcle detector. Snce the technques can only handle relatvely short queues, applcatons of the approach are lmted. The second model s based on traffc flow theory, whch was frst demonstrated by Lghthll and Whtham (1955) and Rchards (1956) for unnterrupted flow; and later expanded by Stephanopolos & Mchalopoulos (1979 & 1981) to sgnalzed ntersectons. Essentally, ths model estmates queue lengths by tracng the trajectory of shockwaves based on contnuum traffc flow theory. Although shockwave theory-based models can successfully descrbe the complex queung process n both temporal and spatal dmensons, these elegant theoretcal models have had lmted practcal applcatons as 15

33 they requre perfect nput nformaton. Specfcally, these models assume known vehcle arrvals. Ths requrement s often mpossble to satsfy, however, because vehcle arrvals cannot be measured once a detector s fully occuped, whch s usually the case wth congested arterals. Wthout arrval nformaton, exstng shockwave models cannot be used to estmate ntersecton queue lengths. To estmate queue lengths longer than the dstance from an ntersecton stop bar to an advance detector, Muck (2002) used a lnear regresson model that descrbes the relatonshp between queue sze and fll-up tme,.e., the tme that passes from the begnnng of the red tme of a sgnal untl contnuous occupancy of a detector. The evaluaton results showed that ths method can estmate queues up to 5-10 tmes further upstream from the actual detector locaton. However, Muck s heurstc approach s based on the premse that the arrval rate of traffc flow s constant wthn a cycle. Ths means that the measured fll-up tme can be proportonally nversed to the queue buld-up. Such an approach could be problematc when the arrval rate of traffc flow fluctuates greatly, whch s not unusual wth the gatng effect of an upstream traffc sgnal. More recently, Skabardons and Gerolmns (2008) proposed a method to estmate ntersecton queue length usng aggregated loop detector data n 30-second ntervals. Ther method for determnng queue rear ends entals examnng the data for flow and occupancy changes. The problem s that 30-second aggregaton smoothes out varatons for between-vehcle gaps, makng t dffcult to dentfy a queue rear unless the arrval traffc flow s sgnfcantly dfferent wth the queue dscharge flow. In addton, snce queue length estmaton s only part of ther travel tme estmaton model, no evaluaton results are reported n ther paper to verfy the accuracy of ther queue length estmaton method. The research presented here provdes a method for estmatng long queues (meanng queues longer than the dstance between an ntersecton stop lne and the vehcle detectors ). We are partcularly nterested n the estmaton of maxmum queue length,.e. the dstance from the ntersecton stop lne to the poston of the last vehcle that has 16

34 to stop n a cycle. The proposed methodology reles on hgh resoluton traffc sgnal data, whch has become ncreasngly avalable n recent years. For example, second-by-second detector data has been used by ACS-Lte (Adaptve Control Software Lte) (Luyanda, et al., 2003). Even more sgnfcant for our work s the avalablty of event-based data (ncludng both vehcle-detector actuaton events and sgnal phase change events) can now be automatcally collected and archved by the SMART-SIGNAL system. Hgh-resoluton traffc data s valuable because t potentally allows us to recover the event hstory of a traffc sgnal; t thus can help provde a bass to analyze the relatonshp between sgnal phase changes and traffc flow durng queue formaton and dscharge. In partcular, hgh resoluton data reveals some break ponts that dentfy traffc flow pattern changes. These break ponts represent concrete postons of some major shockwave and contrbute sgnfcantly to our approach of the long queue estmaton. Indeed, we wll demonstrate wth such data that smple shockwave theory, orgnally developed by Lghthll and Whtham (1955) and Rchards (1956), can be successfully appled to estmate ntersecton queue length. It should be noted that our queue length estmaton methods are desgned to work wth typcal detector confguratons for sgnal operaton,.e., wth ether stop-bar detectors for vehcle presence detecton or advance detectors (detectors placed a few hundred feet upstream from a stop-lne) for green extenson, or both. We make no provsons for lnk nput detectors,.e., detectors placed suffcently upstream so that nput traffc flow to the traffc sgnal can be measured. Wth lnk nput detectors, nput-output approaches can be used for queue estmaton. In addton, such detectors are usually not avalable n the realworld. Ths chapter begns wth a revew of shockwave theory, followed by data descrpton and break ponts dentfcaton. The analytcal model s presented n Secton 4.2, along wth two expanson models that allow varatons of hgh-resoluton traffc sgnal data to be 17

35 used for specfc real world scenaros. The testng results are gven n Secton 4.3 followed by dscusson and conclusons. 4.1 Shockwave Analyss & Break Ponts Identfcaton Shockwave Analyss Traffc shockwave theory s derved from the Lghthll-Whtham-Rchards (LWR) traffc flow model. LWR hypotheszes that flow s a functon of densty at any pont of the road. A shockwave s defned as the moton (or propagaton) of an abrupt change (dscontnuty) n concentraton (Stephanopolos & Mchalopoulos, 1979). Traffc shockwave theory s derved from LWR model by applyng the method of characterstcs to analytcally solve the partal dfferental equaton (PDE) n the LWR model. Bascally, when characterstc curves (along whch, the densty s constant) nteract, a shockwave s formed and whose velocty can be determned by the equaton: q q q k v k v w k k k k k (4.1) where q 1 (q 2 ), k 1 (k 2 ), v 1 (v 2 ) represent, respectvely, the flow, densty and velocty of the upstream (downstream) regon. A traffc shockwave can also be llustrated wth a fundamental dagram (flow-densty curve). The tangent of the chord drawn between any two ponts on the curve defnes the shockwave speed, as shown n Fgure 4-1 (We wll explan w 1, w 2, and w 3 n the followng). Flow q s Fundamental Dagram q, k s s w 3 w 2 q a, k a k s w 1 0, k jam k jam Densty Fgure 4-1 Representaton of Shockwaves n the Fundamental Dagram 18

36 T m e (a) Dstance T g Red Green Tme (b) Dstance Red T r Green Tme ( q, k ) a a ( q, k ) a a w 1 w 1 (0, k jam ) w 1 w 2 (0, k jam ) w 1 w 2 T g Tme T g T r (c) ( q, k ) a a Dstance Red T max Green Tme Dstance (d) ( q a, ka ) 1 1 Red Green 1 T g Red Tme w 3 w 3 w 3 ( q, k ) s s w 3 w1 w2 ( q, k ) s s w1 w2 w 4 w 4 T g T r T max Tme T g T r T max 1 T g (e) Dstance Red Green 1 T mn Red Tme (f) ( q a, ka ) 1 1 Dstance Red Green Red Tme 1 T r ( q a, ka ) 1 1 w 5 w 3 w 3 w 5 w 5 w 5 (0, k jam ) w1 w2 w 4 (0, k jam ) w1 w2 w 4 T g T r T max 1 T g T mn Tme T g T r T max 1 T g T mn 1 T r Tme Fgure 4-2 Shockwave Propagaton Sgnalzed ntersectons generate multple shockwaves due to the stop-and-go traffc caused by sgnal changes. For a more complete explanaton, assume that a queue has 19

37 been fully dscharged durng the last green phase. In the followng red nterval, vehcles are forced to stop, whch creates dfferent flow and densty condtons between the arrval and stopped traffc. Such nterrupton of traffc flow, as ndcated n Fgure 4-2a, forms a queung shockwave (w 1 n Fgure 4-1) movng upstream wth velocty 0 q a w1 k jam k a (4.2) where 0 and k jam represent the jammed flow and densty; and q a and k a are the average arrval flow rate and densty durng the th cycle (subscrpt a means arrval traffc ). In Fgure 4-2a, T g and T r ndcate the end tme and the start tme, respectvely, of the effectve green durng the th cycle. Queung shockwave w 1 contnues to propagate to upstream. At the begnnng of the effectve green (T r n Fgure 4-2b), and assumng there s no congeston downstream, vehcles begn to dscharge at the saturaton flow rate, thereby, formng a second shock wave, or dscharge shockwave (w 2 n Fgure 4-1), at the stop lne that travels upstream wth speed w 2 qs 0 k k s jam where q s and k s are the saturaton flow (capacty) and densty. (4.3) Dscharge shockwave w 2 usually has hgher speed than w 1, so the two waves wll meet at tme T max (n Fgure 4-2c), whch s the tme at whch ths approach has the maxmum queue length. As soon as the two shockwaves meet, a thrd wave (known as a departure shockwave, w 3 n Fgure 4-1) s generated propagatng toward the stop lne wth speed qs w3 k s q k a a (4.4) At the end of ths cycle,.e., the start of the red phase of the next cycle, f the queue cannot be fully dscharged, a resdual queue s formed, whch bulds a fourth wave w 4 (defned as a compresson wave, Fgure 4-2d) at the stop lne movng upstream wth speed 20

38 w 4 k 0 qs jam k s (4.5) Shockwave w 4 llustrates the queue compresson process. Waves w 3 and w 4 have nverse drectons therefore they meet at tme T mn (Fgure 4-2e), whch s the tme the approach has mnmum queue length,.e., the resdual queue. As soon as the two waves meet a ffth wave w 5 forms a new queung wave of the (+1) th cycle, and moves upstream wth smlar speed as shock wave w 1. w 0 q 1 a 5 1 kjam ka (4.6) A smlar process s repeated n the followng cycles as ndcated from Fgure 4-2a to f. It s clear that the tal end of the queue follows the trajectory of shockwaves w 1 and w 3. However, the queue dynamcs can be analytcally calculated only f accurate vehcle arrvals are known. As we mentoned before, n the real world, once a vehcular queue splls back to detectors, traffc arrval nformaton at the detector locaton n the current cycle s unavalable untl the queue starts to dscharge. The shockwave moton descrbed above demonstrates the repettve queung and dschargng process at a sgnalzed ntersecton. The queues begn to accumulate at the start of a red nterval and dscharge at the begnnng of a green phase. Maxmum queue length s acheved shortly after the green start, whle mnmum queues (resdual queues) form after a red start. The repeated nature of ths process suggests that f some crucal break ponts can be dentfed utlzng detector data, the queung and dschargng process can be recovered for past cycles. As ndcated n Fgure 4-3, most waves propagate through the detector lne (f there exst waves that do not cross the detector lne, then ths lnk ether has short queues or s oversaturated, whch wll be dscussed later). Ponts A, B, and C n Fgure 4-3 represent the tmes that traffc flow changes at a detector locaton. If recognzed, these break ponts can be used to recover the queung process n ths cycle. In partcular, we are nterested to estmate the maxmum queue 21

39 length n a cycle,.e., pont H n Fgure 4-3. As wll be explaned n Secton 4.1.2, f hgh-resoluton event-based traffc sgnal data can be stored and archved, such data can be used to dentfy these ponts and estmate queue length n the mmedately precedng cycle. Dstance H Lmax v 3 Loop Detector A B C w 2 D A' w 5 L d Lmn w 1 w 4 T g TA Tr TB 1 Tmax T T C g Tmn 1 T r Tme Fgure 4-3 Break Ponts A, B, C, & Traffc Shockwaves at an Intersecton Identfcaton of Break Ponts The hgh-resoluton data utlzed n ths research was collected by the SMART-SIGNAL system. As ntroduced n Chapter 2, SMART-SIGNAL can contnuously collect and archve hgh resoluton event-based data ncludng vehcle and sgnal events (Fgure 2-2). Event data provdes the start and end tmes of every vehcle-detector actuaton event and every sgnal phase change event. Therefore, the tme dfference between the start and end of a sgnal event represents a phase nterval. The tme nterval between the start and end of a vehcle actuaton event, whch s the tme t takes for a vehcle to pass the detector, s the detector occupancy tme. By assumng an effectve vehcle length, the occupancy tme can be used to calculate travel speed. Fnally, a vehcle gap s the tme nterval between the end of a vehcle actuaton event and the start of a followng event (at 22

40 the same detector). In other words, t s the tme nterval between two consecutve vehcles crossng the detector. As noted n Secton 4.1.1, break ponts A, B, and C represent the tme nstants that a traffc condton changes wthn a cycle. (Here we defne a cycle start as the effectve red start and a cycle end as the effectve green end). In detal, the tme at whch pont A appears (T A ) s the moment that the queung shock wave w 1 propagates backward to the loop detector lne. Between T g (the end of green n the th cycle) and T A, the vehcles pass the loop detector wth the arrval traffc state (q a, k a ). By contrast, between T A and T max (the tme that maxmum queue s acheved), no vehcle can pass the loop detector because of the jam traffc condton (0, k jam ). Pont A can be used to judge whether there s a long queue or not, because f pont A does not exst, whch means that the queung shockwave does not propagate to the detector, then the queue length must be less than the dstance between the stop lne and the advance detector. Pont A s not dffcult to dentfy snce after T A, the detector s occuped for a relatvely long tme, makng the value of the detector occupancy tme s relatvely large as well. For practcal applcaton, a threshold value for pont A dentfcaton s necessary. In ths study, we observed that 3-second s long enough to check whether pont A exsts. If the detector occupancy tme s longer than 3 seconds after T g, then the ntersecton has a long queue; and vce versa. Fgure 4-4a demonstrates event-based detector occupancy tme n a sample sgnal cycle. As ndcated n the fgure, after the queue splls back to the advance detector, the occupancy tme s sgnfcantly greater than 3 seconds (about 45 seconds n ths partcular cycle). We should pont out that second-by-second occupancy data can also be used to dentfy pont A,.e., the occupancy remans 100% for more than 3 seconds. 23

41 Tme (sec) 50 Case 1 - Long Queue (a) Detector Occupancy Tme Break Pont A Break Pont B 0 7:26:07 7:26:50 7:27:33 7:28:16 7:29:00 Tme Tme (sec) Case 1 - Long Queue (b) Tme Gap Between Consecutve Vehcles Pattern I: Capacty condton ( q, k ) Pattern II: Free flow arrval ( qa, ka) Break Pont C s s :26:07 7:26:50 7:27:33 7:28:16 7:29:00 Tme Fgure 4-4 (a) Detector Occupancy Tme Profle n a Cycle; (b) Tme Gap between Two Consecutve Vehcles n a Cycle. Pont B ndcates the tme (T B ) that the dscharge shockwave passes the detector. Between effectve green start tme (T r ) and T B, the traffc state over the detector s (0, k jam ); after T B, vehcles are dscharged at saturaton flow rate and the traffc state changes to (q s, k s ). It s also not dffcult to dentfy pont B usng hgh-resoluton data. After the green starts and before T B, traffc volume s zero, and detector occupancy tme s hgh (larger than 3 seconds) or second-by-second occupancy contnues to be 100% for more than 3 seconds. After T B, queued vehcles begn to dscharge over the detector, detector occupancy tme sgnfcantly drops (as ndcated n Fgure 4-4a), makng break pont B easy to spot. Most mportantly, t s to dentfy break pont C. As wll be explaned n the next secton, we can use the tme of pont C (T C ), combned wth the dscharge shock wave w 2, to estmate the maxmum queue length and re-construct the queue formng and dschargng process. Pont C ndcates the tme (T C ) when the rear end of the queue passes the 24

42 detector. The tme duraton between T B and T C s closely related to the estmaton of maxmum queue length. As noted earler, wave w 3 s the nterface between the saturaton traffc state (q s, k s ) and the arrval traffc state (q a, k a ). Therefore, before pont C appears, vehcles dscharge at the saturaton flow rate at the locaton of the loop detector,.e., the traffc state s (q s, k s ). After the wave propagates to the detector locaton, the traffc condton becomes (q a, k a ),.e., the arrval traffc states. A threshold should be selected to dentfy the two dfferent traffc states (q s, k s ) and (q a, k a ). Based on our observaton, the tme gap between two consecutve vehcles s senstve to traffc state change. As ndcated n Fgure 4-4b, traffc separates two states by break pont C. Before T C, the tme gaps between vehcles are small (less than 2.5 seconds) and the varance s small. Ths means that most of vehcles are dscharged at the saturaton flow rate. But after T C, the vehcle gaps become much bgger and the varance s sgnfcantly ncreased. More mportantly, there usually exsts a tme lag between the saturated queue dscharge flow and newly arrved traffc, as shown n Fgure 4-4b. Ideally, statstcal analyss of vehcle gap data (for example, comparng the means and varances of tme gaps of two vehcle groups) would be used to recognze a traffc state change. However, for smplcty and practcalty, we determned a threshold value for the tme lag between saturated queue dscharge flow and arrval traffc based on our observatons. In ths study, f the value of a tme gap s greater than 2.5 seconds, we consder that the end of the queue has propagated forward and reached the detector lne, therefore T C s dentfed. Consderng the varaton of tme gaps, usng a sngle value to separate traffc states may result n large errors. In our mplementaton, f the tme gap s between 2.5 seconds and 3 seconds, the system wll contnue searchng the second and thrd ponts wth tme gaps over 2.5 seconds to make sure that the traffc state has really changed. It should also be noted that the method descrbed above s stll feasble f only second-by-second data s avalable. Ths s true because we know that a gap larger than 2.5 seconds means 0% occupancy for at least two consecutve seconds. 25

43 (a) Tme (Sec) 50 Case 2 - Oversaturaton Detector Occupaton Tme Break Pont A Break Pont B 0 7:29:06 7:29:50 7:30:33 7:31:16 7:31:59 Tme Tme (Sec) Case 2 - Oversaturaton Tme Gap Between Consecutve Vehcles No pattern changes, always dscharge at saturaton flow rate :29:06 7:29:50 7:30:33 7:31:16 7:31:59 (b) Tme (Sec) 20 Case 3 - Platoon Arrval Detector Occupancy Tme Tme Break Pont A Break Pont B 0 7:32:07 7:32:50 7:33:33 7:34:17 7:35:00 Tme Tme (Sec) Case 3 - Platoon Arrval Tme Gap Between Consecutve Vehcles Pattern I: Capacty condton ( q, k ) Pattern II: Free flow arrval ( q, k ) s a s a 5 Break Pont C :32:07 7:32:50 7:33:33 7:34:17 7:35:00 Tme Fgure 4-5 Two Other Examples of Occupancy tme & Gap Data (a) Oversaturaton; (b) Platoon Arrval 26

44 Two other examples of occupancy tme and gap data whch ndcate two nterestng stuatons are demonstrated n Fgure 4-5, one where the break pont C cannot be found; and the other where vehcles arrve as a platoon. In Fgure 4-5a, durng the green phase, the traffc pattern does not change and vehcles keep dschargng at the saturaton flow rate. Ths could occur for two reasons: 1) the vehcle queue cannot be dscharged completely and ths approach at ths cycle s possbly oversaturated; 2) a vehcle platoon arrves wthn a small tme lag so that a large vehcle gap cannot be found. The second stuaton, as ndcated n Fgure 4-5b, s that although the break pont s easy to fnd, the traffc pattern before and after the break pont s nevertheless smlar. The explanaton s that a platoon dscharged from an upstream ntersecton arrves after the queue has been dscharged. Ths stuaton, sometmes, wll lead to errors to our queue estmaton model, an ssue we dscuss n later sectons. 4.2 Queue Estmaton Models The queue estmaton models proposed here are based on the dentfcaton of breakponts A, B, and C. As ndcated n Fgure 4-3, the crucal task s fgurng out how to estmate the coordnate of pont H, the maxmum queue, n both spatal (L max ) and temporal (T max ) dmensons. Pont H s the ntersecton pont of three waves and ts coordnate can be determned by any two of them. As mentoned above, wave w 1 s a queung wave, whch hghly depends on traffc arrval. Although wave w 3 s also related to arrval flow (see Eq.(4.3)), t represents the arrval traffc after queue dscharge. Snce dschargng process s more stable compared wth queung process and the traffc state durng dschargng process can be estmated usng detector data, ths research uses waves w 2 and w 3, nstead of w 1, to dentfy the coordnate of pont H. Note that wave w 2, accordng to Eq.(4.4), has a constant velocty f we assume that saturaton flow rate (q s ), saturaton densty (k s ), and jam densty (k jam ) are known a pror. In ths secton, we present our basc model plus two expansons, especally desgned for some practcal applcatons. 27

45 4.2.1 Basc Model To estmate shockwave speeds w 2 and w 3, we need to mne more nformaton from the hgh resoluton event-based data from one sngle loop detector. w 2 can be estmated usng the dstance between advance detector and stop-bar (L d ) and the tme dfference between the green start (T r ) and the tme when dscharge wave propagates back to advance detector (T B ),.e. w 2 =L d /(T B T r ). w 2 can also be estmated usng Eq.(4.3) wth assumed saturaton flow rate (q s ), saturaton densty (k s ), and jam densty (k jam ). To estmate shockwave speed w 3, we wll need to estmate two traffc states (q s, k s ) and (q a, k a ) and apply Eq.(4.4). We know we can estmate traffc states because event-based data contan both the occupancy tme and gap between consecutve vehcles passng the loop detector. Occupancy tme s used to estmate the speed of ndvdual vehcles by assumng an effectve vehcle length; and the sum of occupancy tme and vehcle gap,.e., headway, s used to estmate the average flow. Densty can then be estmated by average flow and space mean speed. The equaton for estmatng the space mean speed (v s ), flow (q) and densty (k) s (4.7) D v e j to, j v 1 s N 1 1 N j 1 v j (4.7) q 1 1 N N 1 1 hj ( to, j tg, j ) N j1 N j1 k q vs where, t o,j and t g,j are the detector occupancy tme and tme gap of vehcle j, respectvely; v j, h j are the speed and tme headway of vehcle j, respectvely; D e s the effectve vehcle length,.e., the sum of average vehcle length and detector length, whch may requre calbraton; and N s the number of vehcles dentfed as havng the same traffc state. 28

46 As we dscussed before, the traffc state s (q s, k s ) between T B and T C, and (q a, k a ) after T C (and before T g +1 ). We estmate values of q s, k s, q a, and k a by applyng Eq.(4.7). Note that we use observed data to estmate traffc state (q s, k s ) nstead of assumng the constant values; ths s because the capacty or saturaton flow rate wll decrease f the traffc s affected by downstream ntersectons. The velocty of wave w 3 s calculated based on the two estmated traffc condtons (Eq.(4.4)). Usng estmated w 3 and w 2, the maxmum queue length L max and tme T max durng th cycle can be calculated as: ( TC T ) Lmax L B d 1 1 w2 w3 ( Lmax Ld ) Tmax TB w2 where Ld s the dstance from the stop lne to the loop detector. (4.8) The mnmum queue length L mn and tme T mn (f a resdual queue exsts) durng th cycle can be also calculated: L max 1 Tmax Tg w3 Lmn 1 1 w3 w4 1 L T mn mn Tg w4 (4.9) where w 4 s the velocty of shock wave w 4, whch has same value as velocty of the shock wave w 2 (Eq.(4.3)). The dynamc queue dschargng process after T max (before T mn ) s easy to formulate because we know the wave speed w 3 ; and the queung process before T A has been recorded by loop detectors (as we can see n Fgure 4-6, the arrval may not be constant). The problem now s to estmate the queue length between ponts A and H. Wthout any other nformaton, we can assume a constant velocty for wave w 1, whch can be estmated by 29

47 ( Lmax Ld ) w1 ( T T ) max A (4.10) Wth w 1, w 2, w 3, and w 4, the entre queue accumulaton and dscharge processes can be fully descrbed. Dstance H w 3 Loop Detector Lmax w 1 A B C w 2 D A' w 5 L d Lmn w 1 v 4 T g TA Tr TB 1 Tmax T T C g Tmn 1 T r Tme Fgure 4-6 Basc Model for Intersecton Queue Length Estmaton Expanson I Usng Second-by-second Detector Data In the basc model, event-based data s requred to dentfy the shockwave speeds; such requrement may not be satsfed n the real world. Wth second-by-second detector data and sgnal phase data avalable (such as the ACS-Lte system), we can use Model Expanson I, whch provdes a smple alternatve to calculate maxmum and mnmum queue length, wthout the need to estmate traffc states usng Eq.(4.7), as explaned below. We know that most of the vehcles passng loop detectors before T C, belong to maxmum queue L max, whle a small porton of vehcles joned the queue after the tal end of the 30

48 orgnal maxmum queue began to move. Ths small porton of vehcles theoretcally makes no contrbuton to the maxmum queue length, but these cars are affected by the queue. Therefore, an approxmaton can be made to treat all vehcles passng the detector between the green start and T C as queued vehcles. Because the number of vehcles can be easly obtaned usng second-by-second detector data, by assumng a constant jam densty, we can drectly estmate maxmum queue length (L max ) and tme (T max ) as well as wave speed w 3 under such an approxmaton. L max N L k d jam L T max max Tr w w3 L max Ld TC T max 2 (4.11) where N s the number of vehcles passng a detector between T g and T C. By usng a constant shockwave speed w 4, Eq.(4.9) can then be appled to estmate mnmum queue length (L mn ) and tme (T mn ) as well as wave speed w 1. Essentally, Expanson I s a smplfed queue estmaton model. As demonstrated n Fgure 4-7, pont H represents the true maxmum queue, whle H s an approxmaton. Theoretcally, the expanded model overestmates the maxmum queue length snce t ncludes some porton of vehcles whch do not belong to the stopped vehcle queue. 31

49 Dstance w 1 H H' w 3 Loop Detector L max w1 w 2 w 3 A B C A' D w 5 w 5 L d L mn D' w 4 T g TA Tr TB T T T 1 T max C g mn 1 T r Tme Fgure 4-7 Expanson I for Intersecton Queue Length Estmaton Expanson II Dealng wth Wred-together Detectors The expanded model II s specfcally desgned to deal wth cases where detectors n dfferent lanes (but the same approach) are wred together, or where a sngle detector covers multple lanes. In ths case, vehcles travelng n dfferent lanes are counted only once f they pass the detector at the same tme. Ths detector confguraton s common for many sgnalzed ntersectons n Mnnesota. Under such a desgn, break ponts can stll be successfully dentfed usng second-by-second or event-based detector occupancy, but the vehcle counts are no longer accurate; renderng the basc Model and Expanson I unworkable. An approxmaton s made here to deal wth ths problem. Ths approxmaton s based on the assumpton that after the tal end of a maxmum queue begns to move, no addtonal vehcles can jon the queue. In ths case, the trajectory of the last stopped vehcle s actually shockwave w 3. We can analytcally derve ths trajectory by assumng a pror known constant acceleraton speed (3.5 ft/sec, for example). 32

50 Dependng on whether the last vehcle n the queue reaches maxmum speed at pont C, the tme nterval between T max and T C (t x n Fgure 4-8) can be estmated usng the followng equaton: 2 1 v f v f v f a v f tx w2 ( TC TB tx ) f tx 2 a a a 1 v 2 f atx w2 ( TC tx ) f tx 2 a where v f and a are free flow speed and acceleraton speed, respectvely. (4.12) Then the maxmum queue length (L max ) and the tme (T max ) can be estmated by: L max Ld w2 ( TC TB tx) (4.13) T max TC tx Smlar formulatons are used to estmate the mnmum queue length (L mn ) and the tme (T mn ) as well as wave speed w 1. Clearly, the assumpton made n ths model may not be true n realty. As wth Expanson I, Expanson II wll sometmes overestmates the maxmum queue. As ndcated n Fgure 4-8, pont H s the real maxmum queue length, and estmated maxmum queue length (H ) s actually an overestmaton. 33

51 Dstance H'' w 1 H w 3 w 3 Loop Detector L max w1 A B C A' t x D w 5 w 5 L d w 2 L mn tx D'' w 4 T g TA Tr TB T 1 T max T T C g mn 1 T r Tme Fgure 4-8 Expanson II for Intersecton Queue Length Estmaton 4.3 Implementaton Implementaton Procedure Fgure 4-9 llustrates the mplementaton procedure for the queue length estmaton algorthms. The frst step s to check whether break pont A exsts. If pont A cannot be found, t s a short queue case and a smple nput-output method can be appled to estmate queue sze. Ths smple method counts number of vehcles passng loop detector, assumes saturaton flow rate dschargng when sgnal turns green, and calculates the resdual number of vehcles wthn the area between stop lne and the locaton of the advance detectors. Queue length can then be estmated by smply assumng an effectve vehcle length n the jam traffc state. If pont A exsts, then ponts B and C must be dentfed. Actually, t may not be necessary to dentfy pont B because the wave speed v 2 for most stuatons s a constant value. If pont C can be dentfed, one of the models s chosen to estmate the long queue, dependng on the resoluton of the detector 34

52 data. If pont C cannot be dentfed, as we dscussed n the last secton, the approach s consdered to be under oversaturated. The queue length under oversaturated condtons s dffcult to estmate, and requres a modfed model that wll be ntroduced n the next chapter. Start Check Break Pont A Exst? YES Identfy Pont B Check Break Pont C NO Short Queue Input-Ouput Method Exst? NO Oversaturaton YES Event-based data: Basc model Second-by-second data: Expanson I Wred detector data: Expanson II Fgure 4-9 Flow Chart of Implementaton Procedure for Intersecton Queue Length Estmaton Feld Evaluaton & Results by the Research Team We tested our models to see how well they estmated ntersecton queue lengths. The ntersecton of Trunk Hghway 55 and Rhode Ave. n Mnnesota was selected as the testng ste because of the frequent occurrence of extended queues n the west bound drecton (from Glenwood Ave. to Rhode Ave.) durng mornng peak hours. Fgure 4-10a s a map of the ntersecton (ncludng sx coordnated ntersectons on Trunk Hghway 35

53 55) and Fgure 4-10b shows the detector layout of the ntersecton. Data was collected from detectors #9 and #10 durng two mornng peak hours wth fxed cycle-length (180sec) on June 11 th, 2008 (Wednesday). The data was used to estmate the ntersecton queue length. The estmated values were then compared wth the ground truth data, whch was recorded by a camera nstalled at the ntersecton. The maxmum queue length for each cycle was manually extracted from the vdeo. Snce the data collected by the detectors was event-based, all three models (basc, expanson I, and expanson II) were appled. The results of maxmum queue length are presented n Fgure Table 4-1shows the Mean Absolute Percentage Error (MAPE) whch s calculated by 1 Observaton Estmaton MAPE 100% m (4.14) Observaton m where m s the total sample sze. As showed n Fgure 4-11, the basc model successfully estmates the maxmum queue length for both left and rght lane; and the average MAPE s around 7% (Table 4-1). However, as predcted, Expanson I and Expanson II overestmate the maxmum queue; and the MAPEs are about 14% and 20%. Regardng ther ablty to capture queue dynamcs, we can clearly see n Fgure 4-12 that the proposed models successfully descrbe queue formaton and dscharge. In addton, Table 4-2 presents the tme dfferences when the maxmum queue length s reached, usng the three proposed models and real-world observatons. The absolute errors for the three models are around 5 sec, whch s qute respectable. 36

54 Glenwood Island Ave. Rhode Island Ave. a feet feet b 2 Objectve detectors Camera 400 feet 9 10 Movng Drecton TH 55 Advanced detector Stopbar detector Major Street TH 55 Fgure 4-10 (a) Data Collecton Ste; (b) Detector Layout Dstance (ft) Queue Comparson - Lane Maxmum Queue Length Comparon - Left Lane (by Detector #10) Basc Model Expanson I Expanson II Observaton Tme 6:57:36 7:12:00 7:26:24 7:40:48 7:55:12 8:09:36 8:24:00 8:38:24 8:52:48 9:07:12 Dstance (ft) Comparson - Rght Lane Maxmum Queue Length Comparon - Rght Lane (by Detector #9) Basc Model Expanson I Expanson II Observaton Tme 6:57:36 7:12:00 7:26:24 7:40:48 7:55:12 8:09:36 8:24:00 8:38:24 8:52:48 9:07:12 Fgure 4-11 Testng Results Maxmum Queue Length Comparson 37