Capacity Analysis of Traffic-Actuated Intersections

Transcription

1 Capact Analss of Traffc-Actuated Intersectons b Zhl Tan Eng.B. n Cvl Engneerng (988) Tsnghua Unverst, Bejng, P. R. Chna Submtted to the Department of Cvl and Envronmental Engneerng n partal fulfllment of the requrements for the degree of Master of Scence n Transportaton at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September Massachusetts Insttute of Technolog. All rghts reserved. Sgnature of Author... Department of Cvl and Envronmental Engneerng August 6, 00 Certfed b... Moshe E. Ben-Akva Edmund K. Turner Professor of Cvl and Envronmental Engneerng Thess Supervsor Certfed b... Hars N. Koutsopoulos Operatons Research Analst Volpe Natonal Transportaton Sstems Center Thess Supervsor Accepted b... Oral Buukozturk Charman, Departmental Commttee on Graduate Studes

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3 3 Capact Analss of Traffc-Actuated Intersectons b Zhl Tan Submtted to the Department of Cvl and Envronmental Engneerng on August 6, 00 n partal fulfllment of the requrements for the degree of Master of Scence n Transportaton Abstract Ths thess proposes two models that estmate the capact of an ntersecton wth actuated control. The capact of an approach to or a lane group of the ntersecton s a functon of the saturaton flow rate, the green tme allocated to ths approach or lane group, and the ccle length of the ntersecton. The Mnmum Dela Model estmates the green tmes and the ccle lengths from flow rates, mnmzng the total dela at the ntersecton. Parameters, the rato of green extenson perod to queue servce tme specfc to each approach or lane group, are ntroduced nto ths model. The parameters depend on the dstrbuton of arrvals of vehcles at the ntersecton. The Hbrd Model combnes the determnstc queung model that estmates the queue servce tme and a theoretcal model that estmates the green extenson perod from the unt extenson, the flow rate, the speed lmt of the approach, and the detector length. A method convertng the left-turn traffc volume to equvalent through volume s developed. The method s appled to estmatng the capact of ntersectons wth permtted left-turn phases. The Mnmum Dela Model and the Hbrd Model are valdated at the ntersecton level b comparng the estmatons of effectve green ratos wth those smulated b MITSIM- Lab. These two models are also valdated at the network level wth real data from Irvne, Calforna. The results show that both the Mnmum Dela Model and the Hbrd model are approprate for estmatng capact of ntersectons wth actuated control. The Mnmum Dela Model s also sutable for estmatng capact of ntersectons wth adaptve control. The Emulaton Model s applcable to off-lne mesoscopc dnamc traffc assgnment. Thess Supervsor: Moshe E. Ben-Akva Edmund K. Turner Professor of Cvl and Envronmental Engneerng Thess Supervsor: Hars N. Koutsopoulos Volpe Natonal Transportaton Sstems Center

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5 5 Acknowledgment I am ndebted to a great number of people who generousl offered advse, encouragement, nspraton, and frendshp throughout m tme at MIT. I hold m utmost respect and sncere grattude to m advsor, Prof. Moshe Ben-Akva and Dr. Hars Koutsopoulos. I thank Moshe for sharng hs knowledge, for support, for the opportuntes he has provded me, for forcng me dg deeper nto m research, hs encouragement, and hs nvaluable deas. I thank Hars for sharng hs knowledge, hs frendshp, hs gudance, hs support, hs patence, and hs selfless commtment. I thank m fellow students at ITS lab. Thanks to Kunal Kunde, Rama Balakrshna, Srnvasan Sundaram and for ther techncal ad, frendshp and ther work on DnaMIT. Thanks to everone else n the ITS lab for ther frendshp and kndness. I thank the facult and staff of CTS for ther dedcated and kndness. Specal thanks to Leanne Russell for her knd support. Fnall, m greatest thanks and apprecaton go to m faml. I thank m faml for ther permanent love and support. I realze how luck I am to have them.

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7 7 Contents ABSTRACT.3 ACKNOWLEDGEMENT Lst of Tables Lst of Fgures CHAPTER INTRODUCTION.... Scope of the Thess.... Contrbutons....3 Thess Organzaton... CHAPTER LITERATURE REVIEW Introducton Pretmed Controls Actuated Controls NEMA Controller Tmng Characterstcs Adaptve Control Traffc-Responsve Sstem (SCOOT) Optmzed Polces for Adaptve Control Methods for Estmatng the Capact of Traffc-Actuated Intersectons Allsop s Method Daganzo s Method Hghwa Capact Manual Methodolog Traffc Control Sstem Handbook CHAPTER 3 CAPACITY ESTIMATION OF APPROACHES TO ISOLATED INTERSECTIONS WITH TRAFFIC-ACTUATED CONTROL The Mnmum Dela Model Assumptons Determnaton of the Intersecton Capact b Mnmzng the Total Dela Expressons of Ccle Length and Green Tmes Reformulaton of the Optmzaton Problem n terms of and Comments on the Mnmum Dela Model The Hbrd Model Treatment of Left-turn flow Rates n Capact Estmaton CHAPTER 4 IMPLEMENTATION OF CAPACITY ESTIMATION MODELS IN DYNAMIT Introducton to DnaMIT Implementaton of the Mnmum Dela Model and the Hbrd Model Averagng the Lane Group Capactes Input Data for the Capact Estmaton Models Imposng Lower Bound on Green Tmes and Intal Lane Group Capactes CHAPTER 5 VALIDATION OF THE PROPOSED MODELS Valdaton of the Proposed Models at Intersecton Level Introducton to the Two Intersectons Comparson of the Proposed Models wth those from TCS Handbook Comparson of the Green Tmes Estmated b Proposed Models wth those smulated b MITSIM-Lab at An Intersecton wth Eght Protected Phases... 59

8 Comparson of the Green Tme Estmatons b Dfferent Models at An Intersecton wth Permtted Left Turn Phases Valdaton of the Capact Estmaton Models at Network Level Introducton Comparson of the Feld Observatons and the Smulated Flows Error Statstcs Valdaton Concluson CHAPTER 6 CONCLUSION AND FUTURE STUDY Concluson Future Stud... 7 APPENDIX A GREEN TIMES AND CYCLE LENGTH FOR N PHASES. 73. The Relatonshp between λ and for Approach. 73. The Ccle Length and Green Tmes for An Intersecton wth n Phases Correctness of the Mnmum Dela Model 75 APPENDIX B Representaton of C n Terms of,,, and Representaton of d u n Terms of,,, and Representaton of d r n Terms of,,, and APPENDIX C INPUT DATA OF CAPACITY ESTIMATION. 8. Determnng Protected Left-turn or Permtted Left-turn Phases Preparaton of Input Data for Actuated Control BIBLIOGRAPHY.84

9 9 Lst of Tables Table 5- Saturaton Flow Rates (vphg) Table 5- Saturaton Flow Rates (vphg) Table 5-3 Comparson of Tmng Plans Estmated b Four Dfferent Models Table 5-4 Flow Rates nto the Intersecton... 6 Table 5-5 Effectve Green Ratos Estmated b Dfferent Models under Dfferent Scenaros at Intersecton of Irvne Center Dr and Laguna Cn... 6 Table 5-6 Flow Rates nto the Intersecton Table 5-7 Comparson of Effectve Green Ratos Estmated b Dfferent Models under Dfferent Scenaros at the Intersecton wth Permtted Left-turn Phases... 65

10 0 Lst of Fgures Fgure. Four-phase Controller Dagram 6 Fgure. Eght-phase (dual-rng) Controller Dagram 7 Fgure.3 Phase Order for Dual-rng Controller 7 Fgure.4 Actuated Phase Intervals 8 Fgure.5 A Gap-reducton Functon 0 Fgure.6 The Actuated Traffc Sgnal Strateg under whch Green Phases Termnate as soon as ther Queues Vansh 8 Fgure.7 Queue Accumulaton Polgon Illustratng Green Tme Computaton 30 Fgure 3. Intersecton Laout 34 Fgure 3. Sgnal Phases 35 Fgure 3.3 Queue Accumulaton Polgon 37 Fgure 3.4 Traffc Actuated Control Strateg 40 Fgure 4. Structure of DnaMIT 49 Fgure 4. Smulaton Process 50 Fgure 5. Intersecton of Irvne Center Dr and Laguna Cn (Protected Phasng) 55 Fgure 5. Intersecton of Laguna Cn wth One Permtted Left Turn Phase 55 Fgure 5.3 Heav Traffc Volume 6 Fgure 5.4 Normal Traffc Volume 6 Fgure 5.5 Lght Traffc Volume 63 Fgure 5.6 Heav Traffc Volume 65 Fgure 5.7 Normal Traffc Volume 66 Fgure 5.8 Lght Traffc Volume 66 Fgure 5.9 Irvne Road Network 67 Fgure 5.0 Feld Observatons v.s. Smulated Flows at Sensor Fgure 5. Feld Observatons v.s. Smulated Flows at Sensor Fgure 6. Interactons of Sgnal Control wth DTA 7

11 Chapter Introducton Urban traffc congeston s currentl severe n most ctes n the world and ntellgent transportaton sstems are beng desgned to provde real-tme control and route gudance to motorsts to optmze traffc network performance. Actuated control polces and adaptve control strateges are becomng popular because of ther potental to reduce delas at ntersectons. The advent of extremel fast methods of communcaton and computaton n the past decade has created man new opportuntes for controllng traffc on road networks. New control sstem such as SCOOT, a traffc-responsve sstem, was developed n the U.K. for optmzng network traffc performance. New control algorthms such as Optmzed Polces for Adaptve Control (OPAC), an on-lne traffc sgnal tmng optmzaton algorthm, were developed n the U.S. Improvement of the traffc control of congested networks progresses slowl because of lack of understandng of the long-term dnamcal sstem wthn whch the traffc control sstem s embedded. In a congested network wth traffc sgnals controlled automatcall accordng to actuated or adaptve control polc, there are nteractons between traffc and sgnal controls on the varous streets. In realt, these nteractons are often extremel complcated and ther medum-term effects are hard to forecast. Dnamc traffc assgnment (DTA) models have been appled to smulatng the wthn-da dnamcs of traffc, drvers route choce, and dnamc traffc control. In recent research of dnamc traffc assgnment, researchers studed mpacts of actuated control or responsve control on travelers route choce and how drvers respond to traffc control. The current research on dnamc traffc assgnment focuses on the realstc representaton of the traffc network ncludng formulatng varous actuated or adaptve traffc control. Most of the studes approxmate the responsve traffc control b formulatng delas on streets wth actuated or adaptve traffc control. Snce the realworld traffc controls are sophstcated, dela models cannot take nto account the varous control polces. In ths research, we drectl estmate the capact of approaches to ntersectons wth varous traffc controls n dnamc traffc assgnment. The capact of an approach to an ntersecton wth traffc actuated or adaptve control s a functon of the flow rates on approaches to the ntersecton. Snce traffc demands are tme-dependent, the capactes of ntersectons wth sgnal control respondng to traffc also var. Intersecton capactes estmated from hstorcal demands do not reflect the varatons of capact wthn a da. Although capact estmaton for ntersectons wth pre-tmed control has been comprehensvel studed and estmaton models exst n the lterature, those models cannot be used n estmatng capact of ntersectons wth actuated or adaptve control. For nstance, Webster s model s commonl used for desgnng tmng plans of pretmed control. However, ths model s not senstve to the desgn parameters of actuated control (Courage, 998). Therefore, the capact estmaton models for pretmed control cannot be used to determne the capactes of actuated ntersectons n dnamc traffc assgnment (DTA).

12 The traffc controls are oversmplfed n dnamc traffc assgnment because the traffccontrolled ntersectons are usuall treated as pre-tmed. For example, DnaMIT uses the pre-determned approach capactes calbrated b the method of the Hghwa Capact Manual (HCM) n traffc assgnment. In order to capture the wthn-da dnamcs of traffc, capact estmaton models for actuated or adaptve control should be developed and mplemented n dnamc traffc assgnment. The requrement of appropratel estmatng capactes n DTA motvates ths stud.. Scope of the Thess Ths thess focuses on estmaton of the capact of approaches to ntersectons wth actuated traffc control. Snce the capact of an approach s a functon of the saturaton flow rate, the green tme allocated to ths approach, and the ccle length of the ntersecton, models for determnng green tmes and ccle lengths of the actuated ntersectons are developed n the thess.. Contrbutons Major contrbutons of ths research are as follows: Alternatve models for capact estmaton A model of capact estmaton, the Mnmum Dela Model, s developed. The model estmates the green tmes and the ccle lengths from flow rates b mnmzng the total dela of crtcal movements at an solated ntersecton wth actuated control. Another model of capact estmaton, the Hbrd Model, s developed. The model for determnng the capact of actuated ntersectons n 000 Hghwa Capact Manual (HCM) s mproved b estmatng the queue servce tmes wth a determnstc queung model. The teratve procedure for determnng the queue servce tmes s elmnated n the proposed model. A method, whch converts left-turn traffc volumes to the equvalent through volumes at ntersectons wth permtted left-turn phases, s developed. The method s appled to the estmaton of capact of ntersectons wth actuated control..3 Thess Organzaton Chapter provdes an overvew of pretmed, traffc actuated and adaptve control sstems. Ths chapter also provdes a revew of varous models of computng ccle lengths and green tmes of pretmed and actuated controls. Those models nclude the HCM model for actuated traffc control and the model presented b Daganzo (000) for actuated control.

13 Chapter 3 proposes two models for capact estmaton. The green tmes are estmated as the queue servce tme and the green extenson perod from traffc volumes of the approaches to an solated ntersecton n both models. The frst model estmates the ccle length and green tmes for actuated control wth the objectve functon that mnmzes total dela of crtcal movements n a short tme nterval. The second model mproves capact estmaton model for actuated control n HCM 000. A method of adjustng left-turn volume s proposed for estmatng the capact of an ntersecton wth permtted left-turn phases. Chapter 4 proposes an teratve averagng method of updatng capactes of approaches or lane groups n DTA. Chapter 5 provdes valdaton of the proposed models at both ntersecton level and at network level. Numercal comparson of the ccle lengths and the green tmes estmated b the two proposed models wth those b the HCM model and b the model for actuated control n Traffc Control Sstems (TCS) Handbook (996). The effectve green ratos determned b the two proposed models are compared wth those smulated n Mcroscopc Traffc Smulator (MITSIM-Lab) at two ntersectons wth actuated control n Irvne, CA. A real road network from Irvne, CA wth two nterstate hghwas and four arteral roads s used to valdate the two capact estmaton models. Smulated traffc flow rates are compared wth the feld observatons. The predcton errors of the smulated flow rates b DnaMIT wth dnamc capact estmaton usng the proposed models are compared wth those b DnaMIT wth statc capact estmaton. In Chapter 6, a concluson s made of the applcablt of the two proposed models for determnng the ntersecton capactes. In addton, ths chapter descrbes the future research for modelng capact of ntersectons wth adaptve control and combned model of DTA and adaptve traffc control. 3

14 4 Chapter Lterature Revew. Introducton Traffc sgnal controls are mplemented for reducng or elmnatng conflcts at ntersectons. Sgnals accomplsh ths b allocatng green tmes among the varous users at the ntersectons. Sgnal controls var from smple methods, whch determne the tmng settngs on a tme-of-da/da-of-week bass, to complex algorthms, whch calculate the green tme allocaton n real tme based on traffc volumes. We ntroduce several basc tmng parameters before dscussng current sgnal control sstems. A ccle s the tme requred for one complete sequence of sgnal ndcatons. A phase s the porton of a sgnal ccle allocated to an combnaton of one or more traffc movements smultaneousl recevng the rght of wa. Each phase s dvded nto a number of dscretel tmed ntervals, whch s a porton of the sgnal ccle durng whch all the sgnal ndcatons reman unchanged, such as green, ellow change, and all red clearance. The splt s the percentage of a ccle length allocated to each phase n a sgnal sequence (Kell and Fullerton, 99).. Pretmed Controls The pretmed control, whch has fxed ccle lengths and preset phase ntervals, operates accordng to a predetermned schedule. The pretmed controllers are best suted for locatons wth predctable volumes and traffc patterns such as downtown areas. Tmng plans are usuall selected on a tme-of-da-of-week bass b means of tme clocks. Although pre-tmed controllers have a degree of flexblt n varng tmng plan, the can cause excessve dela to vehcles where there exsts a hgh degree of varablt n the traffc flows because pre-tmed control does not recognze or accommodate short-term fluctuatons n traffc demand and uses tmng plans determned from hstorcal demands. Pretmed sgnals assgn the rght of wa to dfferent traffc streams n accordance wth a preset tmng plan. The Webster method s used to determne the optmum ccle lengths. Snce the actuated sgnals act as a fxed sgnal when all approaches are saturated, ths method can be used to compute the ccle lengths and the green tmes for actuated traffc sgnals when the actuated controller operates as a pretmed sgnal. Webster (958) has shown that mnmum ntersecton dela s obtaned when the ccle length s obtaned b the equaton C where:.5l 5 = n = C = optmal ccle length (second); (-)

15 5 L = total lost tme per ccle (second); = the crtcal lane group volume ( th phase, vph) / saturaton flow (vph); n = number of phases. The total lost tme s the tme not used b an phase for dschargng vehcles. Total lost tme s gven as L = where: n l R = l = lost tme for phase, whch s usuall 4 seconds; R = the total all-red tme durng the ccle. (-) The total effectve green tme, avalable per ccle, s gven b G te = C L. (-3) To obtan mnmum overall dela, the total effectve green tme should be dstrbuted among the dfferent phases n proporton to ther values to get the effectve green tme for each phase, G e = n G te. (-4) The actual green tme for each phase (not ncludng ellow tme) s obtaned b G a = G e l - τ (-5) where τ s ellow tme for phase..3 Actuated Controls An actuated sgnal operates wth varable vehcular tmng and phasng ntervals that depend on traffc volumes. The sgnals are actuated b vehcular detectors placed n the roadwas. The ccle lengths and green tmes of actuated control ma var from ccle to ccle n response to demands. Actuated controllers nclude sem-actuated, full actuated, and denst controllers. In sem-actuated operaton, the man street has a green ndcaton at all tmes untl a vehcle or vehcles have arrved on one or both of the mnor approaches. The sgnal then provdes a green phase for the sde street that s retaned untl vehcles are served, or untl a preset maxmum sde-street green s reached. Non-actuated phases ma be coordnated wth nearb sgnals on the same route, or the ma functon as an solated control. Non-actuated phases usuall operate wth fxed mnmum green tmes and ma

16 6 be extended b usng green tme that s not used b actuated phases wth low demand. That s, the green duraton wll be extended beond the mnmum green tme untl a vehcle actuates the detector on the sde street. At a sem-actuated controlled ntersecton, detectors nstalled on the sde street collect nformaton for tmng the sgnal. In full actuated operatons, all sgnal phases are controlled b detector actuatons. In general, each phase has a mnmum green duraton, but t also s shorter than the maxmum green tme. A phase n the ccle ma be skpped entrel f no demand exts for that phase. The rght of wa does not return automatcall to a specfc phase under the full actuated mode unless recalled b a specal settng n the controller. That s, the controller shows green ndcaton n the phase last served untl conflctng demand appears. In denst operatons, the controllers keep track of the number of arrvals and reduce the allowable gap accordng to several rules as vehcles show up or as tme progresses. NEMA specfcatons allow gap reducton based onl upon tme watng on the red (Mcshane, 99). Ths tpe of controller also has a varable ntal nterval, thus allows a varable mnmum green. Detectors are normall place farther back of the ntersecton stoplne, partcularl on hgh-speed approaches to the ntersecton of major streets (Kell and Fullerton, 99)..3. NEMA Controller The Natonal Electronc Manufacturers Assocaton (NEMA) developed a functonal standard n the traffc control feld. NEMA controllers have smlar functonalt, whch s wdel used n the U.S. The controller operates based on phase dagrams that defne the compatble phases and the order n whch phases are dsplaed. An example of a smple four-phase dagram s show n Fgure.. The east-west movements are served frst, wth the left turns n Phase and wth through and rght movements n Phase, followed b the left and through/rght movements for the north-south street n Phases 3 and 4, respectvel. Each phase has a logc specfed for ts green tmng, whch ma be pre-tmed or demand responsve. The phase order s specfed usng the conflct clearance condton. In ths example, Phase wll wat for Phase, Phase 3 for Phase, and Phase 4 for Phase 3, and Phase for Phase 4, thus defnng the proper phase order. 3 4 Fgure. Four-phase Controller Dagram An eght-phase controller that s more flexble n the phase progress s commonl used. It supports full actuated control. Fgure. shows the tpcal phase dagram for such a controller. The eght-phase controller operates two four-phase rngs smultaneousl,

17 allowng the rngs to advance ndependentl. For example, the controller wll start b dsplang Phases and 5 because nether of Phases or conflcts wth Phases 5 or 6, and then each phase can advance to ts followng phase when t s read. The phase pars of & 5, & 6, & 5, and & 6 are compatble and thus can be actve smultaneousl. The barrer that separates the prmar street and cross-street movements mposes a restrcton on the ndependence of the rngs. Because all movements on one sde of the barrer conflct wth all movements on the other sde, the rngs must advance smultaneousl across the barrer to prevent conflctng movements from beng actve smultaneousl. 7 Bar r er Fgure. Eght-phase (dual-rng) Controller Dagram Fgure.3 shows the dfferent phase sequences of the eght-phase controller. The phasng s smlar to the four-phase controller wth the addton of alternate transton phases where left and through movements ma operate concurrentl. Whch transton phase used, f an, wll depend on the endng tmes of Phases and 5, determned ether b preset tmngs or b vehcle actuatons. Each phase has a precedng phase defned to specf the phase order. The barrer s modeled wth the complementar group condton, whch wll hold a phase passvel n green whle another phase s stll actve. In the example above, Phases and 6 are defned as complementar. Ths condton constrans ther green ntervals to end at the same tme, assurng that the phase rngs cross the barrer smultaneousl. Phases 4 and 8 are smlarl defned, modelng the return across the barrer to Phases and 5. & 5 4 & 7 & 5 & 6 3 & 7 4 & 8 & 6 3 & 8 Fgure.3 Phase Order for Dual-rng Controller

18 8 Modern traffc-actuated controllers usuall mplement a dual-rng concurrent phasng n whch each phase controls onl one movement, but two phases are generall beng dsplaed concurrentl. The term, phase group, s used n the analss of ntersecton capact, whch s set of phases that are beng dsplaed concurrentl..3. Tmng Characterstcs In an actuated phase, there are three tmng parameters: the mnmum green nterval, the unt extenson, and the maxmum green nterval. These ntervals are a functon of the tpe and confguraton of the detectors nstalled at the ntersecton. These three ntervals are shown n Fgure.4. Ths fgure shows a case that the phase termnates before t reaches the maxmum green perod because there s no vehcular actuaton n the last unt extenson perod. Total green perod Mnmum green Extenson perod Unt extenson (Passage tme) Intal Interval Actuatons Maxmum green Tme Detector actuaton on a conflctng phase Detector actuaton on phase wth rght of wa Unexpred porton of unt extenson Fgure.4 Actuated Phase Intervals The unt extenson s tme b whch a green phase could be ncreased durng the extendable porton after an actuaton on that phase (Garber and Hoel, 997). It depends on the average speed of the approachng vehcles and the dstance between the detectors and the stop lne. The unt extenson can be determned b the followng equaton e 0 = S.47v (-6)

19 9 where: e 0 = unt extenson (seconds); v = average speed (mph); S = dstance between detectors and stop lne (ft). Intal nterval s the frst porton of the green phase that s adequate to allow vehcles watng between the stop lne and the detector durng the red phase to clear the ntersecton (Garber and Hoel, 997). Ths tme depends on the number of vehcles watng, the average headwa, and the startng dela. The ntal nterval can be obtaned as I = (hn K ) where: h = average headwa (seconds); n = number of vehcles between the detectors and the stop lne; K = startng dela (seconds). (-7) Sutable values for h and K are seconds and 3.5 seconds, respectvel. The mnmum green nterval s the shortest tme that should be provde for a green nterval durng an traffc phase. In basc desgn of actuated phase ntervals, the mnmum green nterval equals the sum of the ntal nterval and the unt extenson. In the advanced desgn of NEMA controller as shown n Fgure.4 the mnmum green nterval ma be less than the ntal nterval plus one unt extenson. The maxmum green nterval s the lmt that a phase can hold green n the presence of conflctng demand. Normal range of maxmum green s between 30 and 60 seconds dependng on traffc volumes (Kell and Fullerton, 99). Webster s model for pretmed controllers can be used to compute the maxmum green nterval. The computed green ntervals are multpled b a factor rangng between.5 and.50 to obtan the maxmum green (Kell and Fullerton, 99). NEMA specfes that the maxmum green tme not begn tmng untl there s a servceable conflctng call. Therefore, a phase ma reman green for some tme before a conflctng demand appears that starts tmng the maxmum green. The logc of a tpcal sem-actuated control has the followng features (Kell and Fullerton, 99): The green duraton for the sde street starts wth a predetermned ntal nterval, whch s followed automatcall b one unt extenson e 0. If there s a sde street vehcle actuatng the detector durng ths unt extenson, the green duraton ma be extended repeatedl n the same fashon untl the maxmum allowable green nterval, s reached. If no sngle vehcle actuates the detector durng a unt extenson, the green duraton wll be termnated at the end of the unt extenson.

20 0 Accordng to ths control logc, the average ccle lengths and ccle splts of a semactuated control are not affected b the traffc on the major street. A full actuated control does not dstngush between a major street and a sde street. Ever street s treated lke a sde street that s under a sem-actuated control. The control logc for each sgnal phase s the same as the one applcable to a sde street under a semactuated control. Consequentl, the resultng average ccle lengths and ccle splts depend not onl on the sgnal tmng plans but also on the traffc pattern n each phase. In denst mode, the unt extenson tme can also be set to var as a functon of the elapsed green tme, usuall reducng the extenson tme as the maxmum tme s neared. A varable extenson length s often used because a long extenson tme s desrable at the start of the phase to ensure that vehcles are cross the ntersecton, whle a shorter extenson s desred near the end of the phase so that the phase s not extended unnecessarl (McShane et al., 990). A tpcal gap-reducton functon s show n Fgure.5. Unt extenson Maxmum gap Mnmum gap Reducton tme green tme Fgure.5 A Gap-reducton Functon Traffc-actuated controllers automatcall determne ccle lengths and phase duratons based on detecton of traffc on the varous approaches. The ccle lengths and green tmes are random varables, whch depend on the real-tme traffc demand. Therefore, the capactes of approaches to an ntersecton are random varables. A comprehensve revew of the models estmatng the tmng plans and the approach capactes s presented n Secton.5..4 Adaptve Control Adaptve control uses tmng plans that are computed n real tme, based on forecasts of traffc condtons. The detector observatons are used as nput nto a predcton algorthm. Ths control s conceved as hghl responsve wth sgnal tmng adjusted n small and frequent ntervals. Examples of adaptve control sstems are SCOOT (splt, ccle and offset optmzaton technque) developed n the U.K., OPAC (Optmzed Polces for Adaptve Control) developed n the U.S. and SCAT (Sdne Coordnated

21 Adaptve Traffc Sstem) developed n Australa. We revew the frst two control sstems n ths secton..4. Traffc-Responsve Sstem (SCOOT) SCOOT s a traffc-responsve urban traffc control sstem for optmzng network traffc performance. The sstems montor traffc condtons n a network b some form of detecton and react to the nformaton receved b mplementng approprate sgnal settngs. The SCOOT sstem adapts tself to traffc patterns and responds to traffc demands as the occur. The more recent verson of the SCOOT sstem operates b nterpretng comprehensve detector nformaton onlne and montors traffc flows contnuousl from on-street detectors. It uses ths nformaton to recalculate ts traffc predctons ever few seconds and then makes sstematc tral alteratons to current sgnal settngs. New plans are contnuousl evolved n SCOOT sstem, whch s valuable n central areas where congeston s hgh and flow patterns are complex and varable. Inductve-loop detectors measurng vehcle presence (occupanc) are placed at the upstream end of each lnk n the SCOOT network and transmt the occupanc data to the central computer. The cclc flow profles for each lnk reveal the varaton n traffc demand durng each ccle. The detectors measure the cclc flow profles that are used to optmze sgnal coordnaton and for measurng and/or predctng queues, stops and congeston on each lnk. The are used durng offset optmzaton to ensure good sgnal coordnaton. The optmzaton model determnes the sgnal tmngs that mnmze a performance ndex (PI) for each SCOOT regon, based on a weghted sum of delas and stops. For each traffc flow pattern and lnk-node arrangement a PI s calculated as (McDonald and Hounsell, 99) where: N K PI = Wω d ks (-8) = 00 N = the number of lnks; W = the overall cost per average passenger car unt (pcu) equvalent hour of dela; K = the overall cost per 00 pcu stops; ω = the dela weghtng on lnk ; d = the dela on lnk ; k = the stop weghtng on lnk ; and S = the number of stops on lnk.

22 The weghtng factors balance the relatve mportance of queues and stops. A tendenc exsts to favor somewhat longer ccle tmes b usng a heav weght for stops (Roberston, 986). A SCOOT controlled network s dvded nto a number of regons. The SCOOT optmzer updates the traffc sgnal plan on a ccle-b-ccle bass. In dong ths, the optmzer uses the prevous ccle s sgnal settngs as a seed n the search for new tmngs and makes mnor alteratons to these seed sgnal settngs. The changes to the sgnal settngs are made based on a restrcted search for a mnmum PI n the mmedate vcnt of the seed sgnal settngs, rather than b an exhaustve search for a global mnmum PI (Rakha, 995). The SCOOT adjusts sgnal tmngs n frequent small ncrements to match the latest traffc stuaton. These models ensure that queues cannot exceed a maxmum queue value for each lnk. The ccle tme optmzer estmates the optmal ccle tme for the regon. The ntersectons n each regon operate to ther own common ccle tme to ensure good sgnal coordnaton. For each regon the model calculates the degree of saturaton for all ts nodes. It dentfes the most crtcal node for each regon and calculates the optmal ccle length whch s used b all nodes. The ccle tme s determned on the bass that the most heavl loaded ntersecton n the regon should operate at a maxmum degree of saturaton of about 90%. The splt optmzer decdes whether t wll advance, postpone, or leave alone the green tmes for each stage. It seeks to balance the degree of saturaton on all approaches and to avod blockng-back. The green splt s determned b mnmzng the maxmum degree of saturaton on the approaches to that ntersecton. The level of congeston can also be ncluded as an optmzaton crteron. In ths calculaton, the current estmates b SCOOT of the queue lengths, of an congeston measured on the approaches to the ntersecton, and of the constrants mposed b mnmum green tme s taken nto account. Tpcall the lower bound mght be about 30 or 40 seconds. The upper bound s set to gve maxmum traffc capact but wthout undul long red tmes. A maxmum ccle tme of 90 to 0 seconds s tpcall used n SCOOT. The offset optmzer determnes ever ccle whether or not to alter all scheduled stage change tmes at an ntersecton. The sum of the PIs on all adjacent streets for the scheduled offset s compared wth offsets that occur a few seconds earler or later n determnng the offset. The level of congeston can also be ncorporated nto the PI. The decsons of the offset optmzer are modfed where congeston occurs; the purpose s to prevent queues of vehcles from growng to the pont where upstream ntersectons are obstructed. Snce congeston s more lkel to occur on short sectons of road, the offset optmzer acts to mprove the coordnaton on the short streets b ncreasng queue on longer streets, whch have space to store queues. SCOOT s ccle-based and not full reactve, and cannot respond to major dscrete events appropratel n real tme. It ma also be slow n evolvng wth rapdl changng traffc demands, such as durng the mornng rush hour. It ma therefore be provdng slowl evolvng old plans under such dnamc condtons. In addton, SCOOT s not effcent

23 algorthms for true real-tme control that are compatble wth a central control sstem. Further research on groupng sgnals nto sub-areas ma lead to new algorthms that are of suffcent generalt and smplct as to be attractve for on-lne use. SCOOT has been developed for use when traffc demands are moderate to heav. When traffc flows are low, t ma not be necessar to run all the stages durng ever ccle tme. However, SCOOT cannot skp traffc stages automatcall..4. Optmzed Polces for Adaptve Control OPAC s an on-lne traffc sgnal tmng optmzaton algorthm, whch was developed as a dstrbuted sstem for traffc sgnal control wthout requrng a fxed ccle tme. Sgnal tmngs are calculated to drectl mnmze performance measures, such as vehcle delas and stops, and are constraned b mnmum and maxmum phase lengths. The frst verson of OPAC solved the traffc control problem, usng a dnamc programmng algorthm. Each tme nterval s desgnated as a stage, whch s tpcall fve seconds. A rollng horzon concept was appled to the OPAC algorthm n order to use real tme flow data. The horzon s tpcall equal to the average ccle length durng whch the OPAC calculates ts swtchng decsons. The modfed verson was a smplfcaton of the algorthm usng dnamc programmng. Ths verson was reorganzed for mplementaton n real tme n a control sstem. OPAC was mproved b ncorporatng a traffc predcton model that predcts the traffc pattern over the entre stage. The horzon conssts of a head and a tal porton. In the head porton of the horzon, the algorthm has avalable real-tme vehcle arrval nformaton. In the tal porton, the flows are estmated from prevous measurements (Gartner, 99). The detectors are placed well upstream of the ntersecton n order to obtan actual arrval nformaton over the head perod. Dela s calculated based on partcular phase change decsons. In OPAC, stops were ncluded n the objectve functon, whch s tpcall a lnear combnaton of delas and stops, wth the weght of stops relatve to dela beng one. In realt, ths weght favors dela. At each ndvdual ntersecton, phase plans are generated for future mplementaton based on current traffc condtons (.e., current queues and expected arrvals) so as to mnmze the objectve functon over a decson horzon. A phase plan s a sequental lst of future swtch ponts wth each swtch pont representng the start of a certan phase at a specfc tme n the future. Traffc condtons are contnuousl montored based upon vehcle detector and phase change nformaton. The decson horzon tpcall ranges from less than thrt seconds to greater than two mnutes n length. Phase plans are contnuall regenerated for the entre decson horzon but mplemented onl for the frst three to fve seconds. Ths rollng-horzon process allows sgnal tmngs to be constantl adapted to new traffc condton. The optmzaton process s decomposed nto N stages. The total number of stages N corresponds to the horzon length. At stage, the nput state vector s I, the arrvals vector s A, the output state vector s O, and the economc return output s r. A set of transformaton functons s (Gartner, 98): 3

24 4 O = T (I, A, r ) (-9) r = R (I, A, r ) (-0) The state of the ntersecton s characterzed b the state of the sgnal and b the queue length on each of the approaches. The nput decson varable ndcates whether the sgnal s to be swtched at ths stage or to reman n ts present state. The return output s the ntersecton s performance ndex. The optmzaton process mnmzes the total performance ndex. The dnamc programmng optmzaton s carred out n backwards order,.e., startng from the last stage and back-trackng to the frst, at whch tme an optmal swtchng polc for the entre tme horzon can be determned (Gartner, 98). The recursve optmzaton functon s gven b the followng equaton, f * ( I * ) = mn{ R ( I, A, x ) f ( I, A, x )} (-) x where the return at stage s the queung dela ncurred at ths stage: a a a a ( Q A D ) = Q R ( I, A, x ) = (-) a a where: a = approach desgnaton, b drecton, a = N, S, E, W; a A = number of arrvals durng stage ; a D = number of departures (dscharges) durng stage ; a Q = the queue length on the approach at the begnnng of stage. The departure rate s a functon of the state and decson varables (Gartner, 98): D = Q 0 A f f f S S a a S a = = 0, Q A = 0, Q A > where S a s the sgnal status for approach a, defned as follows: 0 S a = f f green. red When the optmzaton s completed at stage = (-3) N N * x = = f * ( I ) = mn R ( I, A, x ) = r ( I, A, x ) (-4)

25 whch s the mnmzed total dela over the horzon for a gven nput state I. Snce the ntal condtons at stage are specfed, the optmal polc s retraced b takng a * forward pass through the stored tables of X ( I ). The polc conssts of the optmal * sequence of swtchng decsons { X, =,, N} at all stages of the optmzaton process. The OPAC strateg carres out sophstcated optmzaton n real tme and adapts to varng traffc condtons. Feld tests have shown that OPAC can provde sgnfcant benefts over well-tmed actuated controllers. Because OPAC s not a traffc-drven controller, t forms a buldng block for a dstrbuted ntellgent traffc control sstem. Unlke conventonal actuated control logc, the OPAC sstem can communcate wth neghborng controllers so as to form a flexble coordnated traffc control sstem. OPAC uses the same performance measure as objectve functon for both off-peak traffc and peak traffc. A feld test shows that when the observed volumes were extremel low the mprovements were modest, because stops were not an OPAC measure of effectveness n the frst verson of the sstem. OPAC adds to the optmzaton objectve functon a penalt, whch s composed of the weghted sum of fnal queues on each approach. The penalt s added to mnmze the fnal queues so that onl mnmal queues are transferred to the succeedng stage. At hgh level of volumes queues on some approaches ma spread to the upstream ntersecton because OPAC sstem does not mpose constrants on queues on approaches to an ntersecton. The unconstraned queues wll reduce the capact of the neghborng ntersectons..5 Methods for Estmatng the Capact of Traffc-Actuated Intersectons The ntersecton capact s a functon of flow rates on approaches to an ntersecton for use n dnamc traffc assgnment. The ntersecton capact s represented b the capactes of approaches to the ntersecton. In general, the approach capact s a functon of the green tme allocated to ths approach, the ccle length of the ntersecton, the saturaton flow rate of the approach, and the characterstcs of the approachng flows. The capact of approach a to the ntersecton can be determned as: where: c = s ( G / C) (-5) a a a c a = the capact of approach a, n vph; s a = the saturaton flow rate for approach a, n vphg; C = the ccle length of the ntersecton, n seconds; and G a = effectve green tme for approach a, n seconds. We revew three models that estmate the green tmes and ccle lengths at an ntersecton wth actuated control n ths secton. 5

26 6.5. Allsop s Method Allsop (97) formulated the capact of a sgnalzed ntersecton as a lnear programmng problem. The model can be used to desgn sgnal-tmng plans b maxmzng the capact of approaches to an ntersecton and to determne approach capactes of the ntersecton as well. Hs results appl to both ntersectons n lnked sstems where traffc arrves manl n platoons and solated ntersectons where the traffc arrves at random because no assumpton about the arrvals of traffc s made. A part of the sgnal ccle n whch one partcular set of approaches has rght of wa s called a stage. At stage j, the proporton Λ j of the ccle that s effectvel green for approach j s gven b where: Λ = m ( ) j a j = 0 λ (-6) λ j = the proporton of ccle that s effectvel green for stage ( =,,, m); a j = f approach j has rght of wa n stage ( =,,,n), and 0 f not; and a 0 j = proporton of total lost tme that s effectvel green for approach j (j =,,, n). The arrval rates on all approaches are multpled b µ. The queues and delas on all approaches wll be acceptable f Λ µ b (j=,,, n),.e., f where: m ( a j ) µ b j 0 = 0 j j λ (j=,,, n) (approach capact constrants) (-7) b j = the smallest acceptable value of Λ j when the arrval rate s s j = the saturaton flow; p j = adjustment factor of the s j, the value s chosen b the engneer; and b j = q j /p j s j. (-8) In addton, the sgnal settngs must satsf the followng constrants, λ k λ 0 0 (=,,, m) (mnmum green constrants); (-9) and λ 0 k 0 ; (-0) m = 0 q j ; λ = (ccle tme constrants) (-)

27 where: k = g M /L; (-) k 0 = L/C; (-3) where: g M = mnmum green tme for stage ( =,,, m); C = the maxmum of specfed ccle tme; and k 0 = the proporton of the ccle taken up b the lost tme, L/C. µ j *, (µ j * = Λ j /b j ) s the largest value of µ j such that the delas and queues are acceptable. Then µ j *q j s the practcal capact of approach j when the proporton of the ccle of effectve green for ths approach s Λ j. µ* can be solved as a lnear programmng problem: Subject to: Equatons (-7), (-9), (-0), and (-). Maxmze µ* Let p j be the maxmum degree of saturaton that s acceptable on approach j. The approach capact s defned as 7 c j = p s Λ C for j =,, n. (-4) j j j Allsop s method (97) does not provde a closed form expresson of µ*. In order to get optmal µ, one needs to solve the above lnear programmng problem. Hs formulaton s approprate for solvng the sgnal desgn problem. However, Allsop s method s not applcable to estmaton of the approach capactes n DTA snce a lnear program has to be solved for each ntersecton and tme nterval..5. Daganzo s Method Daganzo (000) dscussed an actuated control strateg that operates wth ccles and phases close to the mnmum whle avodng overflows. Wth ths strateg traffc dela would be reduced b a factor of two, comparng wth the pre-tmed tmng plan from Webster s model (Daganzo, 000). The control strateg s: The actuated sstems end the green phase on each approach as soon as ts queue dsspates. Fgure.6 llustrates ths strateg b a smple ntersecton wth two phases. The fgure depcts two cumulatve (vrtual) arrval curves, V (t) and V (t), startng at an nstant (t = 0) when the queue at Approach has vanshed and the queue of Approach s Q (0) >0. The saturaton flows and the lost tmes for both approaches ( s, L ) are known. The departure curves are constructed for an par of V (t) s. On the arrval and departure curves, arrowheads ndcate the order n whch ponts along the departure curves are obtaned.

28 8 N I V(t) B H V(t) Q(0) s E 0 S A C D F G J K t L G L G L new G new L new Fgure.6 The Actuated Traffc Sgnal Strateg under whch Green Phases Termnate as soon as ther Queues Vansh Daganzo (000) derved the ccle length equaton for ths actuated control strateg. The followng s the summar of the result. The average duraton of a green phase for approach, s G. Each ccle has a total lost tme L = L L. q denotes average flow n an nterval. The Ccle length s estmated as, where C = L (-5) = q / s. The effectve green rato can be determned as, / C λ = G = ( =,). (-6) The above-mentoned actuated strateg s sutable for solated ntersectons of major streets wth under-saturated condtons. If traffc becomes over-saturated for an extended perod, the strateg does not proactvel allocate more green to the approach wth the hghest flow. Therefore, a long queue ma be bult up on the man street. In realt, the actuated sgnals do not work n the wa as suggested b Daganzo. For nstance, n the smple example llustrated n Fgure.6, the Phase termnates when the queue on Approach dsspates. In practce, Approach retans green tme for an extended perod untl there s demand from Approach or nterarrval headwa of traffc on Approach s longer than the unt extenson. Therefore, nether Equaton (-6) nor Equaton (-5) s approprate for estmatng green tmes of actuated control. The above model also underestmates the average ccle length because t was essentall the ccle of an actuated sgnal for the queue to dsspate. If there s no recall from other phases, the green tme on the last phase wll extend to the maxmum green tme. The

29 actuated sgnals usuall operate at ccle length between the mnmum length and the maxmum length. However, Daganzo s method can be used to estmate the queue servce tmes of an actuated ntersecton, whch s used to estmate the capact of approaches to actuated ntersecton..5.3 Hghwa Capact Manual Methodolog Pretmed Control Chapter 6 of the HCM 000 descrbes a model for estmatng the capact and sgnal tmng plans at a sgnalzed ntersecton as a functon of the traffc characterstcs. The HCM suggests that the average ccle length and phase tmes ma be approxmated b assumng that the controller s effectve n ts objectve of keepng the crtcal approaches nearl saturated. The ccle length s gven b the followng equaton: [ X c ( v / s ] C = LX c / ) c (-7) where: C = ccle length, n sec; L = lost tme per ccle, n sec; (v/s) c = flow ratos for crtcal lane group ; and X c = crtcal rato of volume to capact for the ntersecton. The effectve green tme for a partcular phase, G, s estmated wth the followng equaton v C v C G = = (-8) s X s X where: X = rato of volume to capact for lane group ; v = demand flow rate for lane group, vph; s = saturaton flow rate for lane group, vphg; and G = effectve green tme for lane group, sec. The average ccle length can be estmated usng a user-specfed X c. The capact of a lane group can be calculated b Equaton (-5) after the ccle length of the ntersecton and green tme for lane group are determned. Ths procedure s approprate for estmatng the lane group capactes of ntersectons wth pretmed control. Traffc-Actuated Control The HCM 000 descrbes a method for estmatng the capactes of ntersectons wth actuated control, whch s senstve to varatons n desgn parameters. The models were developed based on the concept of queue accumulaton polgon (QAP) whch plots the 9

30 30 number of vehcles queued at the stop lne over the ccle. Fgure.7 depcts the QAP for a smple protected movement where the accumulaton takes place on the left sde of the trangle and the dscharge takes place on the rght sde of the trangle. The duraton of a green nterval s determned b the length of the prevous red nterval and the arrval rates. The average green tme of a phase s estmated as the sum of the queue servce tme and the phase extenson perod as shown n Fgure.7. 8 Green tme based on phase extenson tme No of Vehcles n the queue 6 4 Green tme based on target v/ c rato.0 Target v/ c 0.9 Target v/ c Red Tme (seconds) Green extenson tme Fgure.7 Queue Accumulaton Polgon Illustratng Green Tme Computaton The queue servce tme, G q, s gven b where: G q q r, q g r s f q qrr = fq (-9) s q ) ( g = red arrval rate (veh/sec) and green arrval rate (veh/sec), respectvel; = effectve red tme (sec); = saturaton flow rate (veh/sec); =.08 0.( G / G (-30) max ) where: G = the actual green (not ncludng ellow tme) (sec); and G max = maxmum green (sec). The queue calbraton factor, f q, accounts for randomness n arrvals n determnng the average queue servce tme.

31 The green extenson perod, G e, s estmated b an analtcal model, whch was developed b Ln (98) and mproved b Akcelk (994). G e s estmated assumng that vehcle arrval accordng to the bunched exponental arrval headwa dstrbuton: θ ( e0 t0 ) e Ge = (-3) ϕq θ where: q = the total arrval flow (veh/sec) for all lanes that actuate the phase under consderaton; e 0 = the unt extenson tme settng (sec); t 0 = the duraton durng whch the detector s occuped b a passng vehcle (sec), t = L ) / S (-3) ( Ld v where: L v = vehcle length, assumed to be 8 ft; A L d = detector length (ft); S A = vehcle approach speed (m/hr); = mnmum arrval (ntra-bunch) headwa (seconds); ϕ = proporton of free (unbunched) vehcles; θ = a parameter calculated as: ϕq θ = (-33) q The proporton of free vehcles n the traffc stream, ϕ, s determned b the followng relatonshp orgnall proposed b Brlton (988) b q ϕ = e (-34) where b s a bunchng factor. HCM recommends parameter values as follows: Sngle-lane case: =.5 s and b = 0.6 Mult-lane case ( lanes): = 0.5 s and b = 0.5 Mult-lane case (3 or more lanes): = 0.5 s and b = 0.8 Green tmes are subject to the constrants of mnmum and maxmum green tmes, whch are controller parameters. The capact of lane groups of an ntersecton can be estmated b Equaton (-5) after the ccle length of the ntersecton and the green tmes are determned. 3

32 3 The method does not drectl determne an average ccle length and green tmes, snce the green tme requred for each phase depends on the green tmes requred b the other phases. Thus, a crcular dependenc exsts whch should be solved teratvel. HCM recommends an teratve procedure to compute the queue servce tme and green extenson perod as follows: The logcal startng pont for the teratve process nvolves the mnmum tmes specfed for each phase. If these tmes turn out to be adequate for all phases, the ccle length wll smpl be the sum of the mnmum phase tmes for the crtcal phases. If a partcular phase demands more than ts mnmum tme, more tme should be gven to that phase. Thus, a longer red tme must be mposed on all of the other phases. Ths, n turn, wll ncrease the green tme requred for the subject phase. Although the teratve procedure applcable to off-lne capact estmaton, t s not applcable to the dnamc traffc assgnment because of the real tme nature of man applcatons. In addton to ncreasng the run tme of DTA, the procedure usuall does not converge to the actual green tmes. Therefore, a model that does not requre teratve computaton needs to be developed. The followng numercal example, whch s used n the HCM 000, demonstrates the estmaton problem of the HCM method. Consder the ntersecton of two streets wth a sngle lane n each drecton. Each approach has dentcal characterstcs, and carres 675 vehcles per hour wth no left or rght turns. The ntersecton s controlled b a two-phase actuated sgnal wth one phase for the east-west movement and the other for north-south movement. The average headwa s.0 seconds per vehcle and the lost tme per phase s 3.0 seconds. The actuated controller settngs are: Intal nterval: Unt extenson: Maxmum green: Inter-green: 0 seconds 3 seconds 46 seconds 4 seconds Usng the HCM approach, the estmated value of green extenson for each approach s 7 seconds snce the approach flow rates are the same. The soluton of the queue servce tme, a part of effectve green tme, s 39 seconds for each phase. The estmated ccle length s (39 7) 3 4 = 0 seconds assumng that the total all-red tme s 4 seconds. Ths result s slghtl lower than the ccle length determned b Webster s model (For the demands at the ntersecton, the ccle length determned b Webster s model s seconds). In ths example, the mnmum phase tme s, ntal nterval unt extenson ntergreen, whch s = 7 seconds for both phases. If a partcular phase demands more than ts mnmum tme, then more tme must be gven to that phase. Thus, a longer red tme s mposed on all of the other phases. Ths, n turn, wll ncrease the green tme requred for the subject phase. Snce there s not an objectve functon and no