VOLUME-BASED PROBABILISTIC APPROACHES TO DETERMINING WHEN TO TURN ON AND OFF SIGNAL COORDINATION PLANS

Transcription

1 Unversty of Nevada, Reno VOLUME-BASED PROBABILISTIC APPROACHES TO DETERMINING WHEN TO TURN ON AND OFF SIGNAL COORDINATION PLANS A dssertaton submtted n partal fulfllment of the requrements for the degree of Doctor of Phlosophy n Cvl and Envronmental Engneerng By Rasool Andalban Dr. Zong Tan/Dssertaton Advsor May, 2015

2 by Rasool Andalban 2015 All Rghts Reserved

3 THE GRADUATE SCHOOL We recommend that the dssertaton prepared under our supervson by RASOOL ANDALIBIAN enttled Volume-based Probablstc Approaches to Determnng When to Turn On and Off Sgnal Coordnaton Plans be accepted n partal fulfllment of the requrements for the degree of DOCTOR OF PHILOSOPHY Zong Tan, Ph.D., Advsor Reed Gbby, Ph.D. Commttee Member Anna Panorska, Ph.D., Commttee Member Shunfeng Song, Ph.D., Commttee Member Hao Xu, Ph.D., Commttee Member Davd Zeh, Dean, Graduate School May, 2015

4 ABSTRACT Traffc sgnal coordnaton s an engneerng tool used to enhance the qualty of traffc flow, ncrease traffc throughput, reduce travel tme and delay, and reduce number of stops by provdng good progresson along major arterals. Whle vehcles on major arterals beneft from sgnal coordnaton, sde-street vehcles may nfer addtonal delays snce coordnaton plans demand longer cycle lengths than natural cycle lengths. Therefore, traffc engneers always face a sgnfcant queston that s at what tme of day sgnal coordnaton plans should be actvated to maxmze the benefts of vehcles on major arterals and mnmze the dsutlty accrued to sde-street vehcles? Bascally, traffc volume, ntersecton spacng, and pedestran volume are key factors n determnng when coordnaton plans should be mplemented. A few attempts have been prevously made to develop models through consderaton of those factors to determne when adjacent sgnals should be coordnated. However, t s shown that n practce these models are neffcent and nstead traffc engneers should use ther experence and engneerng judgment to determne when to turn on coordnaton plans. More mportantly, prevous studes dd not consder the delay of sde-street vehcles and number of stops of major-street vehcles whch are sgnfcant measure of effectveness n assessng the performance of arterals. Ths study focused on developng gudelnes for sgnals coordnaton from two aspects: (1) the delay that sde-street vehcles experence at the stop bar wthout seeng vehcles

5 pass by on major streets durng coordnaton plans, and (2) the expected number of stops that vehcles on major streets wll make when sgnals operate n actuated modes. Afterwards, a survey was conducted to fnd out at what level of traffc volume, traffc agences tend to actvate coordnaton plans. Then, the models outputs and the results of the survey were compared wth each other and consoldated to develop volume-based gudelnes for sgnal coordnaton. Keywords: Sgnal coordnaton, Probablstc models, Traffc volume, Sde-street delay, Number of stops

6 Dedcated to My Beloved Famly

7 v AKNOWLEDGMENTS I would lke to express my deepest apprecaton to my advsor, Dr. Zong Z. Tan, for hs gudance, support, patence, and mentorshp throughout the development of ths study. Wthout hs gudance and support ths study would not have been possble. Specal thanks are expressed to other members of the doctoral commttee, Drs. Reed Gbby, Anna Panorska, Shunfeng Song, and Hao Xu. I would also lke to thank Dr. Nng Wu, who has been very supportve and helpful throughout ths study. I would lke to thank my parents and my only brother, Mohammad Reza. They have been very supportve and encouragng wth ther best wshes. I would lke to sncerely thank my wfe, Saeedeh Farvar. She always cheers me up, supports me, and stands by me.

8 v Table of Contents ABSTRACT... AKNOWLEDGMENTS... v CHAPTER 1 INTRODUCTION... 1 CHAPTER 2 LITERATURE REVIEW SIGNAL COORDINATION AND INFLUENTIAL FACTORS Traffc Volume Intersecton Spacng Pedestrans CURRENT PRACTICE AND DEVELOPED TECHNIQUES SUMMARY AND CONCLUSIONS CHAPTER 3 SIDE-STREET DELAY MODEL THEORETICAL PROBABILISTIC MODEL Pre-tmed Coordnated Sgnals Sem-Actuated Coordnated Sgnals Arrval Patterns of Vehcles Model Valdaton LEFT-TURN PHASE SEQUENCES Dual Leadng Left-turn... 28

9 v Dual Laggng Left-turn Leadng-Laggng Left-turn SENSITIVITY ANALYSIS Cycle Length g/c Rato Proporton of Non-bunched Vehcles Left-turn Interval Summary MODEL APPLICATION Sde-street Vehcles Delay Threshold Desred Probablty of Sde-street Delay Applcaton Results ITE SURVEY SUMMARY CHAPTER 4 STOP BASED APPROACH THEORETICAL PROBABILISTIC MODEL MODEL VALIDATION Sparks Blvd Tmng Plan... 56

10 v Demand Scenaros VISSIM Model Smulaton vs. Model Outputs TRAFFIC VOLUME VERSUS ACTUATED g/c RATIO STOP BASED THRESHOLDS SUMMARY CHAPTER 5 RECOMMENDED GUIDELINES TRAFFIC VOLUME REAL CASE STUDY CHAPTER 6 SUMMARY AND CONCLUSIONS REFERENCES APPENDIX A... 83

11 v Lst of Tables Table 2.1 Coordnatablty Methodologes and Ther Recommended Breakponts Table 3.1 Parameter Values for Cowan s M3 Model Table 3.2 Levels of Traffc Volume for Sgnal Coordnaton when NTH=2, NLT= Table 3.3 Levels of Traffc Volume for Sgnal Coordnaton when NTH=2, NLT= Table 3.4 Levels of Traffc Volume for Sgnal Coordnaton when NTH=3, NLT= Table 3.5 Levels of Traffc Volume for Sgnal Coordnaton when NTH=3, NLT= Table 3.6 Traffc Volume for Sgnal Coordnaton Regardng Intersecton Geometry Table 3.7 Recommended Traffc Volumes to Coordnate Sgnals Table 3.8 Traffc Volume vs. Sgnal Coordnaton: State-of-the-practce Table 4.1 Possble Combnatons for Two Stops out of Four Intersectons Table 4.2 Sgnal Tmng Parameter; Feld Data Table 4.3 Volume-to-Capacty Rato Table 4.4 Dstrbuton of Traffc Volume between Major and Mnor Streets Table 4.5 Drectonalty Dstrbuton Traffc Volume on Major and Mnor Streets Table 4.6 Basc Sgnal Tmng Informaton for Actuated Controlled Sgnals Table 4.7 Assocated g/c Rato wth Establshed Stop-Based Thresholds Table 4.8 Traffc Volume to Coordnate Sgnals for Two-lane Arterals Table 4.9 Traffc Volume to Coordnate Sgnals for Three-lane Arterals... 70

12 x Table 5.1 Tme Perods to Run Coordnaton Plans, Sparks Blvd Table 5.2 Tme Perods to Run Coordnaton Plans, Pyramd Way... 77

13 x Lst of Fgures Fgure 3.1 Arrval-departure Pattern at Sgnalzed Intersectons Fgure 3.2 Intersecton Study n VISSIM Fgure 3.3 Probablty that Delay s Greater than h; VISSIM vs. Developed Model Fgure 3.4 Rng-barrer Structure for Dual Leadng Left-turn Fgure 3.5 Rng-barrer Structure for Unequal Dual Leadng Left-turn Fgure 3.6 Rng-barrer Structure for Dual Laggng Left-turn Fgure 3.7 Rng-barrer Structure for Leadng- Laggng Left-turn Fgure 3.8 Effect of Cycle Length on OPF Fgure 3.9 Effect of g/c Rato on OPF Fgure 3.10 Effect of Proporton of Non-bunched Vehcles on OPF Fgure 3.11 Effect of Left-turn Duraton on OPF; Dual Leadng Left-turn Fgure 3.12 Effect of Left-turn Duraton on OPF; Dual Laggng Left-turn Fgure 3.13 Effect of Left-turn Duraton on OPF; leadng-laggng Left-turn Fgure 3.14 Dstrbuton of Gap Frequency and Traffc Densty Fgure 3.15 Desgnated Intersecton Inventores on Major Streets Fgure 4.1 Stop Probablty Dstrbuton for n= Fgure 4.2 Probablty of Makng More than 0.5n Stops (50 percent stops) Fgure 4.3 Expected Number of Stops... 55

14 x Fgure 4.4 Sparks Blvd, Sparks, Nevada Fgure 4.5 Sparks Blvd Modeled n VISSIM Fgure 4.6 Scatter Graph, VISSIM vs. Probablstc Model usng HCM Method Fgure 4.7 Scatter Graph, VISSIM vs. Probablstc Model usng Synchro Method Fgure 4.8 Intersecton Inventores Modeled n Synchro Fgure 4.9 Traffc Volume vs. Actuated g/c Rato: Two-lane Approaches Fgure 4.10 Traffc Volume vs. Actuated g/c Rato: Three-lane Approaches Fgure 4.11 Relatonshp between Traffc Volume and Actuated g/c Rato Fgure 5.1 Average Daly Traffc Volume on Sparks Blvd Fgure 5.2 Average Daly Traffc Volume on Pyramd Way... 77

15 1 CHAPTER 1 INTRODUCTION The current practce for sgnal tmng and sgnal coordnaton s to provde a lmted number of predetermned plans and actvated at varous tmes of the day. Typcally, coordnaton plans are commonly developed for weekday mornng, md-day, evenng, and weekend peak perods. Tmng plans are developed to accommodate the peak perod traffc volume. Durng the remanng off-peak perods, sgnals usually operate n an actuated mode or n a coordnated mode based upon engneerng judgment. Whle coordnaton generally benefts major-street traffc, mnor-street traffc often ncur sgnfcant delays ndcatng sgnal coordnaton s not benefcal to all traffc. Sgnal coordnaton s a tradeoff between motorsts who beneft and those who do not. Several studes attempted to develop models to determne when adjacent sgnals should be coordnated. In those models, the man nputs are: traffc volume, ntersecton spacng, and platoon dsperson. Nevertheless, n practce none of those models are adequate so traffc engneers are left to ther experence and judgment to determne when to coordnate sgnals. Moreover, those models do not consder two sgnfcant factors: (1) delay to sdestreet vehcles under coordnaton plans and (2) number of stops of major-street vehcles durng actuated operatons of sgnals. Ths research nvestgates sgnal coordnaton ncorporatng these two factors neglected n the lterature. The basc methodology of ths research s the development of a

16 2 probablstc model ncorporatng each of these factors and then valdatng each model usng VISSIM, mcrosmulaton software. The sde-street delay model s found to be a functon of several parameters such as arrval flow rate, arrval pattern, saturaton flow rate, green to cycle length (g/c) rato, and delay thresholds of sde-street vehcles. To use the model and n order to derve a gudelne for sgnal coordnaton, two thresholds need to be establshed; one s the delay threshold of sde-street vehcles and the other one s probablty of the establshed delay. The threshold for the delay of sde-street vehcles s determned by usng the concept of gap acceptance and rejecton at unsgnalzed ntersectons n rural areas, whch could be an adequate reflecton of vehcles delay tolerance at sgnalzed ntersectons. The threshold for the probablty of sde-street delay s nspred by the concept of 85 th percentle used n traffc speed analyss. The stop-based model s found to be a functon of two parameters; average actuated g/c rato of the arteral and number of ntersectons along the arteral. Lke the delay model, applcaton of the stop-based model needs establshng thresholds for: (1) expected number of stops, and (2) the probablty of the expected number of stops. The expected number of stops threshold s less than half of the number of ntersectons. The probablty of expected number of stops threshold s These two thresholds nfer that as long as more than 15 percent of vehcles make less than 50 percent stops the actuated sgnal controller s desred. In other words, as long as 85 percent of vehcles make less than 50 percent stops the actuated sgnal control s desred.

17 3 Ths dssertaton s organzed as follows. After the Introducton, Chapter 2 contans the lterature revew of sgnal coordnaton. Ths s followed by a chapter that develops a Sdestreet Delay Model, the proposed probablstc model related to the delay of sde-street vehcles under coordnaton plans and the assocated gudelne. Chapter 4 presents the proposed probablstc model related to the delay of sde-street vehcles under coordnaton plans and the assocated gudelne wth ths approach. Chapter 4 presents the developed probablstc model that predcts number of stops along non-coordnated arterals and the proposed sgnal tmng gudelne. The next chapter, Recommended Gudelnes deals wth the recommended gudelnes and provdes some real case studes. Fnally, Chapter 6 presents the concluson and practcal recommendatons for sgnal tmng and sgnal coordnaton.

18 4 CHAPTER 2 LITERATURE REVIEW Sgnal coordnaton s an engneerng tool to smooth traffc flow, reduce travel tme and delay, reduce traffc emsson, etc. and has been a major challenge among traffc engneers from two aspects: (1) determnaton of sgnals phasng schemes and related offsets to maxmze the bandwdth, and (2) determnaton of tme perods that sgnal coordnaton plans should be actvated. Whle the former one has receved much attenton and several methods have subsequently been developed for bandwdth optmzaton, the second one has receved less attenton because only a few models have been developed dealng wth when adjacent sgnals should be coordnated. Therefore, traffc professonals must use ther experence and best engneerng judgment due to the lack of gudelnes and relable models for determnng when sgnals should be coordnated. Ths chapter focuses on the lterature that has studed the nfluental parameters on sgnal coordnaton plans and then examnes prevous efforts on sgnal coordnaton models. 2.1 SIGNAL COORDINATION AND INFLUENTIAL FACTORS Traffc volume, ntersecton spacng, and pedestran volume are the factors that determne when sgnal coordnaton plans should actvated. Among these factors, the traffc volume outweghs the others. Traffc safety s not drectly consdered n sgnal coordnaton study. Sgnal coordnaton can mprove the safety performance of arterals dependng on ntersecton spacng, qualty of progresson, and ntersecton geometry. The qualty of

19 5 progresson s affected by several parameters ncludng cycle length, phasng sequence, offset, and corrdor speed and s out of the scope of ths dssertaton Traffc Volume It s true to state that the most nfluental factor n sgnal coordnaton s traffc volume. Bascally, coordnatng sgnals s reasonable and justfable when traffc volumes between adjacent ntersectons are very hgh e.g., durng peak hours [1]. However, the term very hgh has not been quantfed. Operaton of sgnal coordnaton plans durng low volume condtons causes sde-street vehcles to wat at the stop bar seeng no vehcles pass by on major streets n other words, they perceve large gaps between vehcles whch s a source of complants. More mportantly, when traffc volume s low, operaton of actuated sgnal control wth placng mn-recall on major streets leads to a self-coordnaton phenomenon. Self-coordnaton s a phenomenon n whch most of the tme sgnal dsplays on major streets are green due to the (very) low traffc demands on non-major streets. As a result, when vehcles travel along major streets, they ht the green at most of the ntersectons whch means operaton of sgnal coordnaton plans s not reasonable. In addton to traffc volume, drectonalty dstrbuton of traffc, percentage of left-turns, and the amount of traffc enterng, extng, or crossng from sde streets could affect the tme perods that sgnal coordnaton plans should be on. Some researchers have made an effort to develop gudelnes on sgnal coordnaton by consderng traffc volume and platoon dsperson. Hller and Rothery [2] attempted to ascertan whether or not neghborng ntersectons can be effectvely coordnated on the bass of vehcular platoons. They combned the concept of a cyclc platoon profle wth a

20 6 delay model. Robertson [3] ntegrated the delay mnmzaton concept wth a formalzed platoon dsperson model that formed the bass of TRANSYT, a traffc smulaton and sgnal tmng optmzaton program. Manar and Baas [4] studed the relatonshp between traffc volume and platoon dsperson. They showed that the platoon dsperson s low at low traffc volume condtons but t ncreases up to ts maxmum when the traffc volume gets up to the 60 to 80 percent of the lnk capacty. After that, dsperson approaches zero when the traffc volume gets near the lnk capacty. Ther fndngs proved that sgnal coordnaton durng off-peak hours needs further analyss and research snce the platoon dsperson reaches ts peak durng off-peak hours Intersecton Spacng Intersecton spacng also known as sgnal spacng/lnk length, s one of the crtcal factors n coordnatng traffc sgnals. The sgnfcance of ntersecton spacng s mostly related to the platoon dsperson, whch becomes more sgnfcant as the dstance between sgnals ncreases. Bascally, t s recommended to coordnate sgnals when they are n close proxmty of one another [1]. Manual on Unform Traffc Control Devces (MUTCD) [5] states that traffc sgnals wthn 0.5 mle of each other should be coordnated, preferably wth nterconnected controller unts. Federal Hghway Admnstraton (FHWA) recommends coordnatng sgnals wthn 0.75 mle of each other and for ntersectons at greater dstances, t states that the traffc volume and potental for platoon dsperson should be examned [6]. There are also a few studes dealng wth ntersecton spacng and deal progresson. Traffc Control Systems Handbook states that for good progresson the rato of travel tme

21 7 to cycle length should fall between 0.4 and 0.6 [7]. Change and Messer showed that deal progresson spacng s approxmately the travel tme of one-thrd to one-half the cycle length tmes the desgn speed n any generalzed arteral streets [8] Pedestrans Sgnal coordnaton mght cause pedestrans to wat longer n comparson to actuated sgnal controls due to the fact that coordnaton plans demand longer cycle lengths than natural cycle lengths. Sgnal coordnaton mght be an ssue for traffc practtoners from the pedestran safety pont of vew because some of the pedestrans mght be reluctant to wat long untl they receve ther rght-of-way to cross and subsequently they mght cross streets just as they fnd safe gaps between vehcles. Consequently, traffc practtoners mght prefer actuated sgnal controls over coordnaton plans smply because actuated sgnal controls provde shorter watng tme for pedestran once they press the pedestran buttons. Generally, traffc engneers apply two strateges for sgnal tmng regardng pedestrans: (1) tmng based on pedestran requrements, and (2) tmng based on vehcle requrements. In the second strategy, as long as the vehcle requrements meet the pedestran requrements, the operaton of the sgnal s fne. Otherwse, sgnal gets out of coordnaton (gets nto transton) temporarly to serve pedestran demands. 2.2 CURRENT PRACTICE AND DEVELOPED TECHNIQUES The current practce for sgnal tmng s to coordnate sgnals durng peak hours and to run sgnals ether actuated or fxed tmed durng off-peak hours. Ths strategy comes from the general belef that sgnals should be coordnated when traffc demand s suffcently

22 8 hgh. However, ths statement could be nterpreted dfferently. For nstance, an agency may consder 300 vehcles per hour per lane (vphpl) as hgh enough for sgnal coordnaton, whle another agency may consder 500 vphpl. In an attempt to determne at what tme of day sgnal coordnaton plans should be operated, several numerc heurstc technques have been developed. These technques determne when two adjacent sgnals should be coordnated or when a sgnal should be nterconnected wth other sgnals. Yagoda [9] developed a smple model called couplng ndex (CI) to determne when two adjacent sgnals should be coordnated. The CI s the rato of lnk volume and ntersecton spacng. CI = V D (2-1) where, CI= couplng ndex (untless) V= two-way traffc volume for the analyss tme perod (vehcles per hour (vph)) D= ntersecton spacng (ft) If CI falls below 0.3, t means sgnals should not be coordnated. For CIs that falls between 0.3 and 0.5, t s unclear whether or not sgnals should be coordnated and for CIs greater than 0.5, t s recommended to coordnate sgnals. A varant of the couplng ndex model s the mproved couplng ndex (ICI) nspred by the gravty model. The ICI accounts for the weght of the dstance squared as provded n Eq. (2-2) [10].

23 9 ICI = V D 2 (2-2) Later, the couplng ndex was further developed through consderaton of platoon nterference and lnk travel speed n addton to the lnk volume and ntersecton spacng. The new model s more complex and advanced n comparson to ts predecessors and called strength of attracton (AF) [10]. AF = I V ( S D )2 (2-3) Akn to the couplng ndex, three ranges are defned for AF. If AF falls below 0.5 sgnals should operate free. If t s greater than 2.0, t s recommended to coordnate sgnals and f t falls between 0.5 and 2, t s unclear f sgnals should be coordnated or operate free. In ths model, platoon nterference s a untless value descrbng the nterference of the platoon as t progresses down the street. For nstance, a platoon nterference factor of 2.0 can be used for roadways wthout parkng, 1.5 for roadways wth parallel parkng, and 1.0 for roadways wth angled parkng. Synchro [11] has an nternal methodology called coordnatablty factor to determne when two adjacent sgnals should be coordnated. The coordnatablty factor s a functon of travel tme, ntersecton spacng, lnk volume, vehcle platoonng, vehcle queung, and natural cycle length. The natural cycle length s defned as the cycle at whch the ntersecton operates n free mode (actuated mode). The coordnatablty factor ranges from 0 to 100 and s obtaned usng Eq. (2-4). CF = Max(CF1, CF2) + Ap + Av + Ac (2-4)

24 10 where, CF= coordnatablty factor (untless) CF1= ntal coordnatablty factor from travel tme CF2= ntal coordnatablty factor for volume per dstance Ap= platoon adjustment Av= volume adjustment Ac= cycle length adjustment Table 2.1 presents a summary of three coordnaton models ncludng couplng ndex, mproved couplng ndex, strength of attracton, and Synchro coordnatablty factor along wth ther range of outputs [12]. Table 2.1 Coordnatablty Methodologes and Ther Recommended Breakponts Methodology No Coordnaton Coordnaton Not Clear Couplng Index < 0.3 > to 0.5 Improved Couplng Index < 1.0 > to 50 Strength of Attracton < 0.5 > to 2 Synchro < 20 > to 80 Chang [13] developed a model called nterconnecton model (IM) that calculates nterconnecton desrablty ndex based upon volume varaton at the upstream ntersecton and amount of platoon dsperson occurrng between ntersectons. I = t [ X. q max q 1 + q 2 + q 3 ] (N 2) (2-5) where, I= nterconnecton desrablty ndex t= lnk travel tme (mn)

25 11 X= number of departure lanes from the upstream ntersecton q max = straght through flow from the upstream ntersecton q 1, q 2, q 3 = traffc flow arrvng at downstream ntersecton from rght-turn, left-turn, and through movements of the upstream traffc sgnal N= number of arrval lanes to the enterng lnk of downstream ntersecton The value of I vares from 0.0 to 1.0. Whle the value of 1.0 ndcates the most desrablty for nterconnecton, the value of 0.0 ndcates the least desrable condton. Generally, no nterconnecton s needed f I falls below 0.25 but for I greater than 0.50, nterconnecton s hghly recommended. For values of I n the range of 0.25 to 0.50, decson on nterconnectvty s unclear. Balke et al. [14] also developed gudelnes and procedures for settng up a closed-loop traffc sgnal system. They provded procedures of when adjacent sgnals should be coordnated through the applcaton of the nterconnecton coordnatablty ndex Some research focused on sgnal coordnaton wth respect to ntersecton spacng. Wlshre et al. recommends coordnatng sgnals when ntersecton spacng s less than 0.50 mle [15]. Gordon et al. defned coordnatng sgnals when ntersecton spacng s less than 70 tmes of arteral speed (ft/s) [15]. Chrstopher and Kddle defned a good progresson when spacng s farly unform and the rato of travel tme to cycle length falls between 0.4 and 0.6 [15]. Hook and Albers [10] studed and compared the effectveness of the mproved couplng ndex, strength of attracton, and Synchro coordnatablty factor n determnng the potental of coordnaton between two adjacent sgnals. All the ndces were tested on fve

26 12 randomly chosen lnks. In ther study, they came to the concluson that there s no absolute best factor for determnng when sgnals should be coordnated (or where progresson breaks should occur). They beleved that each method gave about the same result; the smpler methods were just as vald as the complcated ones. Fnally, they stated that engneerng judgment and experence are the best tools. In supportng Hook and Albers concluson, the Traffc Control Systems Handbook [7] states that whle coordnaton of adjacent sgnals often provdes benefts, the traffc systems engneer must decde, n each case, whether better performance wll be acheved wth coordnated or solated operatons. It s also mentoned that when a platoon of vehcles s released from a traffc sgnal, the degree to whch ths platoon has dspersed at the next sgnal (dfference from profle at releasng sgnal) n part determnes whether sgnfcant benefts can be acheved from sgnal coordnaton [4]. 2.3 SUMMARY AND CONCLUSIONS Sgnal coordnaton s a tool that traffc engneers use to ncrease the qualty of traffc flow and operatonal performance of sgnals. Despte the advantages of sgnal coordnaton, t mght accrue some dsadvantages to vehcles on non-coordnated phases and pedestrans, whch s a source of publc complants. The trade-off between advantages and dsadvantages of sgnal coordnaton determnes when sgnal coordnaton plans should be on and off. Three factors, ncludng traffc volume, ntersecton spacng, and pedestran volume, determne when mplementaton of coordnaton plans s justfable and reasonable. As for

27 13 the traffc volume, t s stated that sgnals should be coordnated when traffc volume s suffcently hgh. However, the term suffcently hgh has not been quantfed. As for the ntersecton spacng, t s stated when sgnals are n close proxmty of each other t s advantageous to coordnate them. Two values have been defned for the term close proxmty; one s 0.5 mle and the other one s 0.75 mle. As for pedestran, there s no gudelne relevant to tme of sgnal coordnaton wth respect to pedestran volume. A few models and technques have been developed to determne when two adjacent sgnals should be coordnated or when a sgnal should be nterconnected wth other sgnals. Some of these models are couplng ndex, strength of attracton, and nterconnecton model. However, the developed models are not relable and consequently, traffc engneers must use ther experence and engneerng judgment to determne at what tme of day sgnals should be coordnated.

28 14 CHAPTER 3 SIDE-STREET DELAY MODEL One of the major concerns about sgnal coordnaton plans s the delay that sde-street vehcles experence. It s a matter of concern because coordnaton plans demand longer cycle length than natural cycle length (actuated cycle length) that mght accrue addtonal delay to sde-street vehcles. Besdes, longer cycle lengths could cause sde-street vehcles, who wat at the stop bar, to observe large gaps between vehcles on major streets and when the frequency of large gaps ncreases, sde-street drvers become frustrated and start to complan. Lack of study n ths respect demands nvestgatng the effect of coordnaton plans on the delay of sde-street vehcles. Therefore, ths chapter focuses on assessng sgnal coordnaton plans from the perspectve of sde-street vehcles by developng a model predctng the probablty that sde-street vehcles awat h seconds e.g. 20 seconds, when no vehcle passes by on major streets. After that, a traffc-volume-based gudelne wll be developed statng when sgnal coordnaton plans should be mplemented n accordance wth dfferent delay thresholds. 3.1 THEORETICAL PROBABILISTIC MODEL Pre-tmed Coordnated Sgnals The probablty that a vehcle hts the red nterval at a sgnalzed ntersecton s the rato of the red nterval to the cycle length as follows: P(red) = r sd c (3-1)

29 15 where, P(red)= probablty of httng red r sd = red nterval of sde-street (sec) c= cycle length (sec) A porton of the red nterval of the sde street has an overlap wth the green nterval of the major street whch means the green nterval of the major street s less than or equal to the red nterval of the sde street dependng on the ntersecton nventory and the phasng scheme. Thus, the probablty that a sde-street vehcle arrves durng the green nterval of the major street s determned as follows: A= Event that a sde-street vehcle arrves durng red nterval B= Event that a sde-street vehcle arrves durng green nterval of major street P(B A) = P(A B) P(A) = g mj c r sd = g mj (3-2) c r sd where, g mj = green nterval of major street for drecton (sec) The green nterval of the major street conssts of two tme ntervals; (1) queue clearance tme (t dscharge n Fgure 3.1) also known as saturaton porton of green nterval and, (2) arrval-departure tme (free arrval tme). Durng the queue clearance nterval vehcles depart from the queue wth the saturaton headway and durng the arrval-departure tme nterval vehcles arrve ether n platoon or non-platoon (free) format and depart wth the

30 16 same pattern. Fgure 3.1 depcts the general arrval-departure pattern of vehcles at sgnalzed ntersectons. Fgure 3.1 Arrval-departure Pattern at Sgnalzed Intersectons The probablty that a sde-street vehcle arrves durng the queue clearance nterval of the major street s the rato of the queue clearance tme to the green nterval of the major street whch s determned usng Eq. (3-3). P(qc green) = t mj (3-3) g mj where, P(qc green)= probablty that a sde-street vehcle arrves durng the queue clearance nterval t mj = queue clearance nterval of the major street for drecton The queue clearance tme for drecton s a functon of red nterval, arrval flow rate, and saturated flow rate calculated as follows:

31 17 t mj = r mj λ s,mj λ mj (3-4) λ mj where, λ mj = major street flow rate for drecton (vehcles per second (vps)) λ s,mj = major street saturaton flow rate for drecton (vps) r mj = major street red nterval for drecton (sec) The probablty that a sde-street vehcle awat h seconds wthout seeng a vehcle passes by on a major street for the drecton of, would be the proporton of green nterval on the major street where gaps are larger than h : P (gap > h) = T h (gap > h) (3-5) g mj where, P (gap > h)= probablty that a gap on major street s larger than h for drecton T h (gap > h)= porton of the green nterval that gaps are larger than h (sec) It also can be expressed as follows: P (gap > h) = 1 T h (gap < h) (3-6) g mj where, T h (gap < h)= porton of the green nterval that gaps are smaller than h (sec) for the drecton

32 18 In the above equaton, T h (gap < h) s consstent of queue clearance tme and total tme where gaps are smaller than h durng arrval-departure tme as defned n Eq. (3-7). T h (gap < h) = t mj + φ mj (3-7) where, φ mj = a tme nterval durng arrval-departure tme n whch gaps between vehcles on major streets are smaller than h, for the drecton of Therefore, Eq. (3-6) s re-wrtten as follows: P(gap > h) = 1 t mj + φ mj (3-8) g mj The φ mj s consstent of two terms: the total gaps durng arrval-departure tme to the condton that gaps are smaller than h and the last porton of the green tme ncludng ntergreen tme durng whch a gap cannot be fully used by vehcles. The frst term s the cross product of the expected number of vehcles arrve wth the gap less than h and the expected gap sze for the gaps less than h. The second term, n average, s consdered to be half of the gap of length h at the end of the green nterval. Therefore, φ mj = N. T + h 2 (1 p ) (3-9) where, N = expected number of vehcles that arrve wth a gap smaller than h durng green nterval on drecton T = expected headway of vehcles wth the gap smaller than h on drecton (sec)

33 19 p = probablty that a gap s less than h on drecton The probablty that a gap s less than h s determned n accordance wth the arrval probablty dstrbuton functon usng Eq. (3-10). h p = f (t)dt 0 (3-10) where, f (t)= arrval probablty dstrbuton functon on drecton wth respect to λ mj The expected number of vehcles that arrve wth the gaps less than h s determned usng the bnomal dstrbuton. The bnomal dstrbuton s used because a gap s ether smaller than h or larger than h, there are a lmted number of gaps, and the order of gaps s not mportant, so: n N = x ( n x ) (p ) x (1 p ) n x = n. p (3-11) x=0 where, n= total number of vehcles that arrve durng the arrval-departure tme, and x= number of vehcles that arrval durng the green nterval wth gaps less than h. The total arrval of vehcles durng the free arrval tme s expected to be the cross product of arrval flow rate and arrval-departure tme as defned by Eq. (3-12). n = λ mj (g mj t mj ) (3-12)

34 20 The expected value of gap for those vehcles arrvng durng the arrval-departure tme wth a gap less than h seconds s determned as follows [16]: where, T = v h t. f (t) 0 v h f (t) v = hourly volume of drecton (vph) 0 = h t. f (t)dt 0 p (3-13) The numerator s the total tme of those headways that are less than h and the denomnator s the number of headways n one hour that are less than h. Therefore, the probablty that a sde-street vehcle awats more than h seconds at the stop lne seeng no vehcles pass by on major streets s determned as: p (gap > h) = 1 r mj λ s,mj λ mj λ mj + λ mj (g mj λ mj λ s,mj λ mj λ mj g mj h 0 h 0 ) tf (t)dt + h 2 (1 f (t)dt) (3-14) In Eq. (3-14) the red and green ntervals can be wrtten as a porton of cycle length as: g mj r mj = β. c (3-15) = (1 β ). c (3-16) where, β = green to cycle length rato for drecton Eq. (3-14) gves the sde-street delay probablty whch s also defned as objectve probablty functon (OPF) based on one drecton of the major street. To consder two

35 21 drectons of the major streets wth the assumpton that through traffc on both drectons start and termnate smultaneously, some adjustments should be appled as follows: The queue clearance nterval needs to be calculated based upon the maxmum queue clearance of both drectons. Other than calculatng the queue clearance nterval, the flow rate should be based on the summaton of flow rates of both drectons. t max mj = max [r mj λ s,mj λ mj j and r λ mj mj j λ s,mj j λ mj ] (3-17) j λ mj j λ mj j = λ mj j + λ mj (3-18) g mj = g mj j = g mj (3-19) where, t max mj = maxmum queue clearance tme between drecton and j on major street, (sec) λ j mj = total flow rate on major street on both drectons and j, (vps) g mj = green tme on major street, (sec) j Subsequently, λ mj s used for calculatng f j (t) tf j (t)dt and f j (t)dt. Therefore, Eq. (3-14) s re-wrtten as: p(gap > h) h = 1 t mj max + λ j mj (g mj t max mj ) tf j (t)dt + h 2 (1 fj (t)dt) (3-20) 0 0 g mj where, h

36 22 f j j (t)= arrval headway dstrbuton functon based for drectons and j, usng λ mj The probablty that sde-street vehcles awat h seconds seeng no vehcles pass by on the major street s a functon of arrval flow rate, saturaton flow rate, green to cycle length rato, cycle length, arrval pattern of vehcles on the major street, and expected delay (h) Sem-Actuated Coordnated Sgnals Actuated-coordnated controllers are used to ncrease the effcency of traffc sgnals. The great advantage of sem-actuated coordnated controllers s ther functonalty to assgn the porton of unused green tme of the non-coordnated phases to the coordnated phases when traffc volumes of the non-coordnated phases are not very hgh. Nevertheless, ths functonalty can potentally ncrease the lkelhood of observng large gaps between majorstreet vehcles by sde-street vehcles. The reason s, the effectve green tme of the coordnated phases mght get ncreased whle ther traffc volume remans constant. Under ths condton, the expected values for red and green ntervals of the major street used n the Eq. (3-14) wll be used as follow: where, g mj = E(g mj ) (3-21) r mj = E(r mj ) (3-22) E(g mj )= expected green tme of major-street through movement, (sec) E(r mj )= expected red tme of major-street through movement, (sec)

37 Arrval Patterns of Vehcles The followng, presents common arrval headway dstrbuton functons and the relevant ntegraton functons used n the Eq. (3-14) and (3-20). Cowan s M3 Cowan s M3 s one the arrval headway dstrbuton models that allows separate analyss for platoon and non-platoon arrval of vehcles. The densty functon of the model s: 0 t < Δ f(t) = { 0 t = Δ αλ e λ (t Δ) t > Δ (3-23) where, Δ= mnmum headway,(sec) α= proporton of non-bunched (free) vehcles λ = model parameter Akçelk [17] suggests usng the followng equaton for α: α = e bδλ (3-24) where, b= bunchng factor Table 3.1 presents the parameter values of the Cowan s M3 model regardng the number of lanes.

38 24 Table 3.1 Parameter Values for Cowan s M3 Model Total number of lanes Δ b α e 3.0q e 1.0q > e 0.5q λ could be nterpreted as the ntensty of the non-bunched vehcle arrval rate and s determned usng the followng equaton: λ = αλ 1 Δλ (3-25) Therefore, h f(t)dt = 1 αe λ (h Δ) 0 (3-26) h tf(t)dt = 1 λ (h + 1 (h Δ) 0 λ )αe λ (3-27) Shfted Exponental Ths dstrbuton does not account for bunchng (platoon) arrval of vehcles and t can be derved from the Cowan s M3 when α s set to be 1. 0 t < Δ f(t) = { λ λ 1 Δλ (t Δ) t 1 Δλ e (3-28) Thus, h f(t)dt = 1 αe λ(h Δ) 0 (3-29)

39 25 h tf(t)dt = 1 λ (h + 1 λ )αe λ(h Δ) 0 (3-30) Exponental Ths dstrbuton also does not account for bunchng (platoon) arrval of vehcles and t can be derved from the Cowan s M3 when α and Δ are set to be 1 and 0, respectvely. Therefore, f(t) = λe λt (3-31) h f(t)dt = 1 e λh 0 (3-32) h tf(t)dt = 1 λ (h + 1 λ )e λh 0 (3-33) Model Valdaton VISSIM s used as a tool to valdate the developed probablty model, Eq. (3-14). A two-way ntersecton s modeled n VISSIM. In the model, two detectors are placed on the major street and one detector on the mnor street. The frst detector on the major street s placed 2000 ft upstream of the ntersecton to derve the vehcle arrval pattern and the second detector s placed n the mddle of the ntersecton to measure the gap between vehcles at the ntersecton. The mnor-street detector s placed at the stop lne to account for the arrval tme of the sde-street vehcles. Fgure 3.2 shows the modeled ntersecton n VISSIM envronment.

40 26 Fgure 3.2 Intersecton Study n VISSIM The sde-street delay probablty durng the analyss perod, T, s computed by dvdng the summaton of all gaps, whch are greater than h, by the summaton of all gaps as follows: P(gap > h) = (gap gap > h) T gap T (3-34) Several demand scenaros (traffc volumes) are generated for both major and mnor streets to compare the smulaton outputs wth the developed probablty model. Fgure 3.3 compares the probabltes of delay for sde-street vehcles from VISSIM wth those from the developed model.

41 27 (a) Vol major = 100 vph (b) Vol major = 300 vph (c) Vol major = 500 vph (d) Vol major = 700 vph Fgure 3.3 Probablty that Delay s Greater than h; VISSIM vs. Developed Model It can be clearly seen that the model and VISSIM outputs are hghly matched. In addton, the most sgnfcant observaton n the valdaton process s that the sde-street traffc volume has a mnor mpact on the target probablty. 3.2 LEFT-TURN PHASE SEQUENCES There are three phase sequences regardng left-turn movements ncludng dual leadng left-turn, dual laggng left-turn, and leadng-laggng left-turn. Sequence of the left-turn phases plays a sgnfcant role n the probablty of delay experenced by sde-street

42 28 vehcles. The reason s explaned through an example: f two consecutve through vehcles proceed through an ntersecton at tmes t X and t X+1, the observed gap by a sde-street vehcle would be t X+1 t X, whle f a left-turn vehcles proceeds through an ntersecton at tme t LT to the condton that t X < t LT < t X+1, then the sde-street vehcle wll observe two gaps whch are t LT t X and t X+1 t LT. Therefore, arrval of a left-turn vehcle breaks the gap between major vehcles (t X+1 t X ), nto two smaller gaps (t LT t X, and t X+1 t LT ) whch could be smaller than h Dual Leadng Left-turn The opposng left-turn phases on the major streets start smultaneously pror to the through traffc and once one of them termnates the opposng through traffc starts. Under low volume traffc condton, the left-turn phase gaps out and the unused porton of ts green nterval s assgned to the opposng through traffc. Fgure 3.4 shows the rng-barrer structure of the dual leadng left-turn when left-turn phases termnate together. Fgure 3.5 shows the rng barrer structure of the dual leadng left-turn phases when they do not termnate together. Fgure 3.4 Rng-barrer Structure for Dual Leadng Left-turn

43 29 Fgure 3.5 Rng-barrer Structure for Unequal Dual Leadng Left-turn Generally, nequalty between the lengths of left-turn phases creates a tme lag between the onsets of through movements on major streets. The tme lag affects the observed gaps by sde-street vehcles that needs to be addressed n OPF. To account for the mentoned tme lag, a new varable (GT) s ntroduced nto the model by assumng that drecton s the heaver drecton than drecton j for both through and left-turn traffc on the major street. GT = { (g j mj,lt j j + t mj ) (g mj,lt + t mj ) f (g mj,lt + t mj ) > (g mj,lt 0 otherwse j + t mj ) (3-35) where, GT= the tme dfference between end of left-turn phases of major streets, (sec) g mj,lt = effectve green tme of left-turn movement on drecton of major street, (sec) j g mj,lt = effectve green tme of left-turn movement on drecton j of major street (sec) Thus, the probablty of delay s calculated as:

44 30 p(gap > h) = 1 j (t mj + GT) + λ j j j mj (g mj t mj h 0 j g mj h 0 GT) tf j (t)dt + h 2 (1 fj (t)dt) (3-36) For the condton that through traffc of drecton s heaver than drecton j but leftturn traffc of drecton j outnumbers drecton, Eq. (3-35) and (3-36) are modfed as follows: GT = { (g mj,lt j j + t mj ) (g mj,lt j j + t mj ) > (g mj,lt + t mj ) f (g mj,lt 0.0 otherwse + t mj ) (3-37) Consequently, the delay probablty wll be calculated from: p(gap > h) = 1 (t mj + GT) + λ j j mj (g mj t mj h 0 g mj h 0 GT) tf j (t)dt + h 2 (1 fj (t)dt) (3-38) Dual Laggng Left-turn The left-turn movements on the major streets start subsequent to the through movements and end smultaneously. Fgure 3.6 shows the rng-barrer structure for dual laggng left-turn phases. Fgure 3.6 Rng-barrer Structure for Dual Laggng Left-turn

45 31 In contrast to dual leadng left-turn phases, the calculaton of OPF for dual laggng leftturn phases s smple and done through Eq. (3-17) to (3-19). The reason s a laggng leftturn phase on major streets does not gap out Leadng-Laggng Left-turn One of the left-turn movements starts wth ts respectve through movement, and the other left-turn movement starts subsequent to the opposng through movement and ends wth t smultaneously. Generally, t s expected that ths left-turn phase sequence results n lower values for OPF n comparson wth former left-turn phasng sequences because one of the left-turn movements could truncate the green nterval of the coordnated phase. Fgure 3.7 shows the rng-barrer structure for leadng-laggng left-turn phases. Fgure 3.7 Rng-barrer Structure for Leadng- Laggng Left-turn Interference of left-turn phases wth through traffc n the leadng-laggng left-turn phasng sequence, makes the calculaton of OPF complex. To ncorporate ths complexty t s necessary to defne the departure process of vehcles durng ths phase sequence. Vehcles depart from the ntersecton as follows: 1) Through and left-turn traffc of drecton are released, 2) Through traffc for drecton j s released once left-turn of drecton termnates,

46 32 3) Once through traffc of drecton termnates, left-turn traffc of drecton j s released, 4) Through and left-turn traffc of drecton j termnate smultaneously at the barrer. OPF should be determned based upon the above four-step departure process. Defnng some new varable s deemed necessary for the calculaton of OPF: Max = max{(g mjlt Subject to g mjlt G mjlt j + t mj ) and t mj } (3-39) where, Max = tme perod that vehcles ncludng through and left-turn depart from the ntersecton at the rate of saturaton flow, (sec) g mjlt = actuated effectve green tme for the left-turn on drecton, (sec) G mjlt = desgned effectve green tme for the left-turn on drecton, (sec) The dfference between G mjlt and g mjlt s related to the theoretcal and practcal pont of vew. G mjlt j s the green tme whch s used n desgnng sgnal tmng whle g mjlt s the green tme whch s occurred n the feld. In other words, G mjlt does not consder the j fluctuaton n traffc volume whle g mjlt does. To calculate OPF, frst the total tme n whch gaps are less than h are determned for drecton :

47 33 T() = Max + λ j mj (G mj h Max) tf j (t)dt 0 (3-40) After that, the total tme n whch gaps on drecton j are smaller than h are determned for the remanng tme whch s assocated wth the left-turn of drecton j: j T(j) = g mjlt Subject to j g mjlt j G mjlt j j + λ mj (G mjlt j g mjlt h ) tf j (t)dt 0 + h h 2 (1 fj (t)dt) 0 (3-41) where, j g mjlt = actuated effectve green tme for the left-turn movement on drecton j, (sec) Therefore, OPF for the leadng-laggng left-turn phasng sequence s determned from: P(gap > h) = 1 T() + T(j) j (3-42) + G mjlt G mj where, j G mjlt = desgned effectve green tme for the left-turn on drecton j, (sec) 3.3 SENSITIVITY ANALYSIS The OPF s a functon of several varables ncludng arrval flow rate of the major streets, cycle length, effectve green to cycle length for the major street, arrval pattern of vehcles on major streets, and desred delay for sde-street vehcles. To gan some nsght nto how these varables affect OPF, a senstvty analyss needs to be conducted. To conduct the senstvty analyss some assumptons are made as follows:

48 34 Left-turn sequence s dual leadng left-turn, Sgnal s operatng n a fxed mode and subsequently left-turns do not gap out, There are two through lanes and one left-turn pocket on each approach, The dstrbuton of turnng movements on each approach for through, left-turn, and rght-turn s 0.65, 0.20, and 0.15, respectvely, and The drectonally dstrbuton of traffc on major street s 0.60 to To study the effect of above-mentoned varables on OPF, two approaches are used: (1) takng the dervatve of OPF wth respect to ts varables, and (2) applyng numercal examples. Ths secton presents the second approach whch s more tangble Cycle Length To study the effect of cycle length on OPF, three cycle lengths are consdered: 100, 120, and 140 seconds. The g/c rato for the coordnated phases s assumed to be Two values are consdered for the delay of sde-street vehcles (h): 10 and 15 seconds. The results are depcted n Fgure 3.8. Fgure 3.8 Effect of Cycle Length on OPF

49 35 The fgure clearly shows that durng low volume condtons the cycle length very slghtly affects OPF and as the traffc volume ncreases the cycle length does not affect OPF anymore. Generally, t s concluded that cycle length does not affect OPF when the g/c rato remans constant g/c Rato The effect of g/c rato on OPF s examned through consderng varous values for g/c rato ncludng 0.35, 0.40, 0.45, and The cycle length s assumed to be 120 seconds. The effect of g/c rato on OPF s depcted n Fgure 3.9. Fgure 3.9 Effect of g/c Rato on OPF Fgure 3.9 shows that as the g/c rato ncreases, the OPF ncreases as well. However, the rate of ncrease s sgnfcantly low especally for the hgher values of h. Therefore, t s concluded that OPF s not very senstve to the g/c rato Proporton of Non-bunched Vehcles The proporton of non-bunched vehcles, α, s expected to have a great mpact on OPF. To study the effect of ths varable, t s assumed that proporton of non-bunched vehcles

50 36 vares from 0.25 to 0.75 by the ncrement of 0.25 for two values of h ncludng 10 and 15seconds. The cycle length s assumed to be 120 seconds wth the g/c rato of The results are summarzed n Fgure Fgure 3.10 Effect of Proporton of Non-bunched Vehcles on OPF As shown n the fgure, proporton of non-bunched vehcles sgnfcantly affects OPF. As α ncreases the OPF decreases smply because an ncrease n α means more vehcles arrve n non-platoon format resultng n occurrence of more traffc dsperson Left-turn Interval It seems that the length of the left-turn phases on the major street affects the OPF especally when sem-actuated coordnated controllers are used. Therefore, t deems necessary to nvestgate the effect of the length of left-turn phases on OPF. Fgures 3.11 to 3.13 demonstrate the effect of the dfferent left-turn ntervals on OPF wth respect to three phasng sequences when leadng left-turn movements of major streets are operatng n actuated mode and laggng left-turn movements operate n a fxed mode. Appendx A provdes a methodology for determnng the actuated green tme.

51 37 Fgure 3.11 Effect of Left-turn Duraton on OPF; Dual Leadng Left-turn Fgure 3.12 Effect of Left-turn Duraton on OPF; Dual Laggng Left-turn Fgure 3.13 Effect of Left-turn Duraton on OPF; leadng-laggng Left-turn

52 38 From the fgures, t s realzed that the leadng-laggng phase sequence s very senstve to the length of the left-turn phases so that longer left-turn ntervals leads to hgher values of OPF Summary OPF s a functon of several parameters ncludng cycle length, g/c rato, arrval flow rate, saturaton flow rate, arrval patterns, proporton of non-bunched vehcles, and leftturn phasng sequence. Among these parameters cycle length, g/c rato, proporton of nonbunched vehcles, left-turn sequence, and length of left-turn phases are examned. It s found that whle proporton of non-bunched vehcles and left-turn phasng sequence remarkably nfluence OPF, the other varables have mnmal effect on OPF. It s also realzed that OPF s sgnfcantly senstve to the length of left-turn phases when leadnglaggng phase sequence s n place. 3.4 MODEL APPLICATION Ths secton s to defne a volume-based gudelne statng when sgnal coordnaton plans should be mplemented wth respect to the delay of sde-street vehcles. To use the OPF for the sgnal coordnaton purpose, two thresholds need to be establshed: (1) sdestreet vehcles delay (h), and (2) probablty of that delay, P(gap > h) Sde-street Vehcles Delay Threshold Sde-street vehcles delay threshold (h) s one of the crtcal varables n OPF that can produce dfferent results. More mportantly, the value of h should reflect the expected delay threshold of sde-street vehcles at sgnalzed ntersectons. There s no document to state

53 39 what value of h s assocated wth the delay threshold of vehcles at sgnalzed ntersectons. Therefore, ths study attempts to come up wth a reasonable value for h through consderaton of drvers gap acceptance and rejecton behavor at unsgnalzed ntersectons. The gap acceptance and rejecton behavor of drves at unsgnalzed ntersectons could be consdered as a reflecton of vehcles watng behavor (acceptable delay) at sgnalzed ntersecton to some extent. However, the threshold of vehcles delay at sgnalzed ntersectons s expected to be hgher than the one at unsgnalzed ntersectons because drvers are cognzant of the wat perod at a red lght. In a study [18], drvers gap acceptance and rejecton behavor was examned at rural ntersectons across the naton. In the study three categores of gaps were defned: unsafe gaps, mx of unsafe and safe gaps, and safe gaps. Whle the unsafe gaps range from 1.2 to 4.4 seconds, the mx of unsafe and safe gaps range from 4.4 to 10 seconds, and the safe gaps are the gaps larger than 10 seconds. Fgure 3.14 [18] shows the dstrbuton of gap categores wth respect to the level of traffc volume. The results of the study showed that drvers tend to reject gaps of 6.67 seconds or less.

54 40 Fgure 3.14 Dstrbuton of Gap Frequency and Traffc Densty Consderng the defned gap categores and gap rejecton, two values are establshed for h to conduct the analyss for probablty of delay: h = 10 sec; ndcatng the begnnng pont of safe gaps, and h = 15 sec; nearly twce as much as drvers gap rejecton tme, The begnnng of safe gaps, h = 10 sec, s a good threshold snce t s assumed that drvers at unsgnalzed ntersectons have a tendency to wat untl fndng a safe gap whch could be nterpreted as the mnmum threshold of sde-street vehcles to see no vehcle on major streets at sgnalzed ntersectons. Another value for h s 15 seconds, whch s nearly twce as much as the gap rejecton. Ths value s chosen based on engneerng judgment by assumng the delay threshold of sde-street vehcles at sgnalzed ntersecton s hgher than the one at unsgnalzed ntersectons.

55 Desred Probablty of Sde-street Delay After establshng a threshold for h, t deems necessary to establsh a threshold for the assocated probablty wth h, whch s P(gap > h). In ths regard, the concept of the 85 th percentle, whch s commonly used n the speed analyss study, s appled for P(gap > h) as follows: f P(gap > h) { 0.15 > 0.15 Turn on Coordnaton Plan Turn off Coordnaton Plan (3-43) The above crteron nfers sgnal coordnaton plans are justfable as long as 15 percent of the tmes or less, sde-street vehcles perceve gaps larger than h between vehcles on major streets. In other words, sgnal coordnaton plans are justfable as long as 85 percent of the tmes, sde-street vehcles perceve gaps smaller than h between vehcles on major streets. Followng up wth the defned crtera, analyses wll be conducted wth respect varous ntersecton nventores to determne the level of traffc volume at whch sgnal coordnaton plans should be on Applcaton Results To determne at what level of traffc volume sgnals should be coordnaton, t s requred to solve Eq. (3-43) wth respect to phasng sequence and ntersecton nventores. In ths study, four ntersecton nventores are defned for major streets as llustrated n Fgure An ntersecton nventory s mportant because of the saturaton flow rate and headway dstrbuton parameters.

56 42 (a) N TH = 2, N LT = 1 (b) N TH = 2, N LT = 2 (c) N TH = 3, N LT = 1 (d) N TH = 3, N LT = 2 Fgure 3.15 Desgnated Intersecton Inventores on Major Streets

57 43 To provde more comprehensve results, traffc volumes are presented for three values of h ncludng 10, 15, and 20 seconds that gves more optons to traffc engneers f they desre to pck up a threshold dfferent from the one, recommended n ths study. To conduct the analyses, the followng assumptons are made: The dstrbuton of turnng movements at each approach on major streets s 0.65, 0.20, and 0.15 for through, left-turn, and rght-turn movements, respectvely. The saturaton flow rate of the through movement s consdered to be 3440 and 4943 vph for two and three through lanes, respectvely. The saturaton flow rate for the left-turn traffc s consdered to be 1770 and 3440 vph for one and two left-turn lanes, respectvely. The drectonalty dstrbuton of traffc flow on major streets s 0.60 to The g/c rato of the through traffc s whch s the average of 0.35, 0.40, 0.45, and The length of the left-turn phase s consdered to be 20 seconds. However, the mnmum phase for the left-turn phase s set to 10 seconds. The arrval patterns of vehcles s assumed to be Cowan s M3. Sde-street phases ncludng through and left-turn movements do not gap out. The results of solvng Eq. (3-43) regardng the aforementoned assumptons and condtons are provded n Tables 3.2 to 3.5.

58 44 Table 3.2 Levels of Traffc Volume for Sgnal Coordnaton when N TH=2, N LT=1 Left-turn Control Type h= 10 sec h= 15 sec h= 20 sec Dual Leadng Dual Laggng Leadng-Laggng Table 3.3 Levels of Traffc Volume for Sgnal Coordnaton when N TH=2, N LT=2 Left-turn Control Type h= 10 sec h= 15 sec h= 20 sec Dual Leadng Dual Laggng Leadng-Laggng Table 3.4 Levels of Traffc Volume for Sgnal Coordnaton when N TH=3, N LT=1 Left-turn Control Type h= 10 sec h= 15 sec h= 20 sec Dual Leadng Dual Laggng Leadng-Laggng Table 3.5 Levels of Traffc Volume for Sgnal Coordnaton when N TH=3, N LT=2 Left-turn Control Type h= 10 sec h= 15 sec h= 20 sec Dual Leadng Dual Laggng Leadng-Laggng The tables show that leadng-laggng left-turn phase sequence results n lower level of traffc volume for sgnal coordnaton. The reason s the green tme wndow on the coordnated drectons gets truncated by left-turn movements. The leadng left-turn truncates the leadng through phase and the laggng left-turn truncates the laggng through phase.

59 45 Revewng Tables 3.2 to 3.5 shows that although left-turn phasng sequence affects the delay probablty, they are close to one another. Consequently, Tables 3.2 to 3.5 are consoldated nto one table through makng average wth respect to left-turn phase sequence as provded below: Table 3.6 Traffc Volume for Sgnal Coordnaton Regardng Intersecton Geometry Intersecton Inventory h= 10 sec h= 15 sec h= 20 sec N TH = 2, N LT = N TH = 2, N LT = N TH = 3, N LT = N TH = 3, N LT = Studyng Table 3.6 shows that the dfference between one left-turn lane and two leftturn lanes s 25 to 50 vph dependng on h. Therefore, the author smplfes Table 3.6 and makes the followng recommendaton for sgnal coordnaton: Table 3.7 Recommended Traffc Volumes to Coordnate Sgnals Arteral h= 10 sec h= 15 sec h= 20 sec Two-lane Arteral Three-lane Arteral ITE SURVEY A survey was conducted on the ITE COMMUNITY Webste n October 2013 to fnd out at what level of traffc volume traffc agences actvate coordnaton plans. The survey launched a very hot dscusson n whch many traffc expects partcpated. However, only a few responses pertnent to the survey were receved. It was found that publc agences apply varous traffc volumes rangng from 250 to 500 vehcles per hour per lane (vphpl)

60 46 as a threshold to turn on sgnal coordnaton plans. For nstance, the mnmum threshold to trgger sgnal coordnaton by some agences n Florda, San Dego, and Sacramento s 250, 300/500, and 360 vphpl, respectvely. The Cty of San Dego uses two thresholds 300 and 500 vphpl dependng on prevalng condtons [19]. Table 3.8 presents the total traffc volume for sgnal coordnaton wth respect to survey results for arterals wth two and three lanes n each drecton. Table 3.8 Traffc Volume vs. Sgnal Coordnaton: State-of-the-practce Base Traffc Volume (vphpl) AVE Two-lane Arterals (vph) Three-lane Arterals (vph) Comparng Table 3.8, the state-of-the-practce, wth Table 3.7, the fndngs of ths chapter, reveals that the approprate value for h s 10 seconds. The reason s, the traffc volumes whch are assocated wth h of 15 and 20 seconds are far below the state-of-thepractce. However, the traffc volumes assocated wth h of 10 seconds are n the range of the state-of-the-practce and near to the average. 3.6 SUMMARY Ths chapter nvestgates the effect of coordnaton plans on the delay of sde-street vehcles. A mathematcal model s developed whch estmates the probablty that sdestreet vehcles wat at the stop bar and no vehcles on major streets pass by for a certan amount of tme e.g. 10 seconds. The probablty model s a functon of several parameters ncludng cycle length, g/c rato of the coordnated phases, arrval flow rate of the coordnated phases, saturaton flow rates of the coordnated phases, the arrval pattern of

61 47 vehcles on the coordnated phases, and the sequence of left-turn movements on the major streets. The developed model s valdated through VISSIM. After valdaton of the model, the model s used to determne when sgnal coordnaton plans should be actvated. To apply the model, two thresholds are needed to establsh; one s the delay of sde-street vehcles durng whch no vehcles on major streets pass by and the other one s the probablty of the delay. As for the delay of sde-street vehcles three thresholds are examned n accordance wth the gap acceptance and rejecton behavor of drvers at rural unsgnalzed ntersectons whch are 10, 15, and 20 seconds. The probablty of delay for sgnal coordnaton s assumed to be 0.15 or less, whch s nspred by the concept of 85 th percentle used n the speed analyss studes. The results show that settng dfferent thresholds for the delay of sde-street vehcles leads to dfferent levels of traffc volume for sgnal coordnaton. For nstance, f the delay s consdered to be 10 seconds, then sgnals should go nto coordnaton when traffc volume on the peak drecton of major streets reaches 750 and 850 vph, for arterals wth two and three lanes n each drecton, respectvely. To compare the fndngs of ths chapter wth the state-of-the-practce, a survey was also conducted on the ITE webste pertnent to at what level of traffc volume traffc agences tend to trgger sgnal coordnaton plans. Among the few responses, t s realzed that traffc agences use dfferent thresholds to turn on sgnal coordnaton plans rangng from 250 to 500 vphpl. Comparng the results of ths chapter wth the state-of-the-practce shows the model outputs are acceptable from the practcal pont of vew and, more mportantly, the

62 48 best threshold for the delay of sde-street vehcles s 10 seconds that leads to traffc volume compatble wth the practce.

63 49 CHAPTER 4 STOP BASED APPROACH One of the objectves of sgnal coordnaton s to reduce the number of stops that vehcles on major streets make durng actuated operaton of sgnals. Reducton n number of stops leads to reducton n travel tme and traffc emsson. The Natonal Traffc Sgnal Report Card defnes reducton n number of stops as one of the objectves of sgnal coordnaton and also reports reductons n number of stops as an ndcaton that sgnal retmng desgns are well-done [20]. Despte the mportance of number of stops t has been neglected n the lterature. Therefore, ths chapter deals wth sgnal coordnaton from the perspectve of number of stops. 4.1 THEORETICAL PROBABILISTIC MODEL If sgnals along a corrdor are operatng free (n an actuated mode) and traffc volume s not very hgh, the arrval pattern of vehcles s assumed to be random and more mportantly, the operaton of sgnals are ndependent from each other. Thus, the probablty that a vehcle hts the green at an ntersecton s the rato of the effectve green tme to the actuated cycle length. p g = g /c (4-1) where, p g = probablty of httng green at ntersecton g = actuated effectve green tme at ntersecton

64 50 c = actuated cycle length at ntersecton Smlarly, the probablty that a vehcle hts red s the rato of red nterval to the actuated cycle length. p r = r c = 1 g c = 1 p g (4-2) where, r = actuated red tme at ntersecton Consderng the fact that the operaton of sgnals along an arteral s ndependent from each other, the probablty that a vehcle makes x stops whle travelng along an arteral wth n ntersectons (x n) can be calculated usng Eq. (4-3). where, g Pr(x) = p Z { n =1 j n. (1 p j g ) j=1 j } (4-3) Pr(x) = probablty of makng x stops n = total number of ntersectons for coordnaton analyss = ntersecton number at whch a vehcle hts green j = ntersecton number at whch a vehcles hts red Z = a set of all possble combnatons of and j for x stops out of n ntersectons All possble combnaton of and j for x stops s determned usng Eq. (4-4).

65 51 All Possble Combnatons for x Stops = ( n x ) = n! x! (n x)! (4-4) For example, the possble combnaton of makng two stops (x = 2) along an arteral wth four ntersectons (n = 4) s presented n Table (4.1). Table 4.1 Possble Combnatons for Two Stops out of Four Intersectons Event Intersecton Number GO GO STOP STOP 2 GO STOP GO STOP 3 GO STOP STOP GO 4 STOP GO GO STOP 5 STOP GO STOP GO 6 STOP STOP GO GO If all the ntersectons have a common actuated g/c rato, the probablty of httng green at all ntersectons s alke and would be consdered as p g. p 1 g = p 2 g = = p g = = p n g = p g (4-5) Then, the probablty of makng x stops s determned by usng the bnomal dstrbuton as provded n Eq. (4-6). Pr(x) = ( n x ) (1 pg ) x. (p g ) n x (4-6) The bnomal dstrbuton could also be used for estmatng the probablty of stops when the actuated g/c rato of all sgnals are almost equal (close to each other): p 1 g p 2 g p g p n g (4-7)

66 52 In ths case, the probablty of makng x stops s determned usng Eq. (4-8) wth respect to the average of p g ( = 1, 2,, n) of all ntersectons. Pr(x) = ( n x ) (1 p g ) x. (p g ) n x (4-8) Where, p g g = ( p )/n (4-9) p g = the average probablty of httng green at n ntersectons (the average of g/c ratos for n ntersectons) Fgure 4.1 shows the stop probablty dstrbuton for the case that n = 4 and three values of average g/c rato ncludng p g = 0.35, 0.50, and Fgure 4.1 Stop Probablty Dstrbuton for n=4 The graph shows that as the average actuated g/c rato ncreases the probablty of makng hgh number of stops decreases.

67 53 In analyzng an arteral or a segment of an arteral for sgnal coordnaton, t s expected that all target sgnals along the study arteral have the actuated g/c ratos close to each other. Therefore, the rest of ths chapter wll be developed wth respect to the Eq. (4-7) to (4-9). The probablty that a vehcle makes less than a partcular number of stops e.g., X stops, whle travelng along an arteral when sgnals are operatng free can be determned usng Eq. (4-10). X Pr(x X) = ( n x ) (1 p g ) x. (p g ) n x (4-10) x=0 Consequently, the probablty of makng more than X stops s defned by Eq. (4-11). X Pr(x > X) = 1 ( n x ) (1 p g ) x. (p g ) n x (4-11) x=0 Fgure 4.2 llustrates the probablty of makng less than 0.5n stops for arterals wth 4, 6, 8 and 10 ntersectons regardng dfferent g/c ratos.

68 54 Fgure 4.2 Probablty of Makng More than 0.5n Stops (50 percent stops) Expected Number of Stops Consderng the stop probablty dstrbuton, the expected number of stops would be the cross product of the number of ntersectons and the average probablty of httng red as defned n Eq. (4-12). where, n EX STOP = x. P r(x) = n. (1 p g ) (4-12) x=0 EX STOP = expected number of stops Fgure 4.3 shows the expected number of stops when n takes the values of 4, 6, 8, and 10 for varous values of g/c ratos. Ths fgure shows that vehcles on major streets wll make few stops even durng the low volume condtons (when the average g/c rato s as hgh as 0.70).

69 55 Fgure 4.3 Expected Number of Stops 4.2 MODEL VALIDATION In order to valdate the developed probablstc model, a real case study on Sparks Blvd n Sparks, Nevada, was smulated n VISSIM Sparks Blvd Sparks Blvd s one of the major arterals n the Sparks-Reno urban area n Nevada wth nearly 4.5 mles n length encompassng 10 sgnalzed ntersectons. The average corrdor volume of the peak drecton for the mornng and evenng peak hours were about 1200 and 1700 vph, respectvely and the mdday peak hour was around 700 vph. These values consst of northbound and southbound traffc volumes. The posted speed lmt at Sparks Blvd s 40 mph. Due to the speed lmt and nventory of each ntersecton, havng exclusve leftturn lane(s), each ntersecton has dual left-turn phase. Fgure 4.4 llustrates Sparks Blvd, the red crcles represents sgnalzed ntersectons.

70 56 Sparks Blvd Fgure 4.4 Sparks Blvd, Sparks, Nevada Tmng Plan All sgnals were run n fully actuated mode wth placng mn-recalls for the south and north bound through, whch are consdered the man drectons. Other sgnal tmng parameters such as phase splt, phase sequence, changng nterval, and vehcle extenson were gathered from the feld n order to produce an accurate smulaton of the feld. In VISSIM smulaton, standard detectors wth a length of ft were placed on each lane at the stop bar. The nput data on mnmum green and vehcle extenson s presented n Table 4.2. Other sgnal tmng parameters ncludng maxmum green, yellow change, and red clearance ntervals are not presented due to ther varety.

71 57 Table 4.2 Sgnal Tmng Parameter; Feld Data Movement Mn Green Vehcle Extenson (sec) (sec) Left-turn Mnor Through Major Through ~ Demand Scenaros Whle traffc volume durng a.m., mdday, and p.m. peak hours was avalable, traffc volume durng off-peak hours was unavalable. Therefore, several demand scenaros were generated based on mdday traffc volume. The mdday peak volume was chosen because the Cty of Sparks used to run sgnals free (actuated) durng mdday peak perod snce t was thought that the traffc volume was not suffcently hgh for sgnal coordnaton. However, sgnals are currently n coordnaton because traffc engneers beleve traffc volume s hgh enough for mplementng a coordnaton plan. In order to generate demand scenaros, traffc volume from mdday was ncreased and decreased as presented n Table 4.3. Ths table presents the volume to capacty (v/c) rato for through movements, whch s consdered as a dstngushng crteron between demand scenaros. The v/c ratos were determned based on the methodology presented n Hghway Capacty Manual (HCM) edton 2000, usng TRAFFIX software.

72 58 Table 4.3 Volume-to-Capacty Rato % Demand Varaton (from mdday) (v/c) NB (v/c) SB (v/c) AVE ± VISSIM Model After obtanng and preparng all necessary data ncludng ntersecton nventory, sgnal tmng parameters, traffc volumes, and speed lmt. Sparks Blvd was modeled n VISSIM as depcted n Fgure 4.5. The smulaton program was run 10 tmes wth the duraton of 1 hour for each demand scenaro. The warm-up tme was set to be 600 seconds snce the travel tme along Sparks Blvd n a congested condton s nearly 500 seconds. The number of stops for the northbound and southbound drectons was reported as a measure of effectveness and the average of 10 runs was consdered and reported as the number of stops for each demand scenaro.

73 59 Sparks/Los Altos Sparks/Dsc Dr. Sparks/Shadow Ln Sparks/Barng Sparks/O Callaghan Sparks/Prater Sparks/E Lncoln Sparks/I80 WB Sparks/I80 EB Fgure 4.5 Sparks Blvd Modeled n VISSIM