Akcelik & Associates Pty Ltd PO Box 1075G, Greythorn, Vic 3104 AUSTRALIA [email protected] ® Management Systems Registered to ISO 9001 ABN 79 088 889 687 REPRINT A comparison of three delay models for sign-controlled intersections R. AKÇELIK, B. CHRISTENSEN and E. CHUNG REFERENCE: AKÇELIK, R., CHRISTENSEN, B. and CHUNG, E. (1998). A comparison of three delay models for sign-controlled intersections. In: Third International Symposium on Highway Capacity, Copenhagen, Denmark, 22-27 June 1998, Volume 1 (Edited by R. Rysgaard). Road Directorate, Ministry of Transport, Denmark, pp 35-56. NOTE: This paper is related to the intersection analysis methodology used in the SIDRA INTERSECTION software. Since the publication of this paper, many related aspects of the traffic model have been further developed in later versions of SIDRA INTERSECTION. Though some aspects of this paper may be outdated, this reprint is provided as a record of important aspects of the SIDRA INTERSECTION software, and in order to promote software assessment and further research. © Akcelik and Associates Pty Ltd / www.sidrasolutions.com PO Box 1075G, Greythorn Victoria 3104, Australia Email: [email protected] Third International Symposium on Highway Capacity, Copenhagen,Denmark, 22-27 June L998 A comparisonof three delay modelsfor sign-controlledintersections Authors: Rahmi Akqelik Chief ResearchScientist, ARRB Transport ResearchLtd Bente Christensen Traffic EngineerrOslo Vei, Norway Bdward Chung SeniorResearch Scientist, ARRB TransportResearch Ltd Contact : Rahmi Akgelik, ARRB Transport ResearchLtd, 500Burwood Highway, Vermont SouthVIC 3133,Australia Ph: (613)98811567, Fx: (613)98878104, Email: [email protected] Abstract Resultsof an evaluationof three analyticaldelay models for unsignalisedintersections are presented.The delay modelsstudied are the Highway CapacityManual Chapter 10 (HCM 94) model, the Akgelik - Troutbeck model, and the SIDRA 5 model. These models are applicableto sign-controlledintersections and roundabouts.The evaluation work reportedin this paper is for sign-controlledintersections. Each delay model was used with its associatedcapacity model. The models were comparedby means of extensive tests using the microscopic simulation program ModelC for a basic gap- acceptancecase. Delays predicted by the current form of each model were first compared against simulated delays. Modified forms of the three models were then calibratedagainst the simulationdata. Generally,the modified models improved delay predictions to a small extent. Overall, the SIDRA and Akgelik-Troutbeck models indicatedsimilar levels of predictionability whereasthe HCM 94 model displayedpoor performance. Improved prediction of capacitiesappeared to give larger levels of improvementin delay prediction. The HCM 97 delayand capacitymodels gave similar results compared with the HCM 94 models. Similar model comparison work is recommended usins real-life data collected at sisn-controlled intersections and roundabouts. 1. INTRODUCTION This paperpresents the resultsof an evaluationof threeanalytical models for predicting delay at unsignalisedintersections. In increasinglevel of complexity,the delay models studiedare: o the Highway CapacityManual Chapter10 (HCM 94) delaymodel which is basedon a simple queuingtheory method(TRB 1994); r Akgelik - Troutbeckmodel (1991)based on a delaymodel originallyproposed by Troutbeck(1986, 1989)and is derivedby an extensionof the simplequeuing theory methodusing a minimum delayparameter based on gap-acceptancemodelling; and o the delay model used in the SIDRA 5 packagewhich is basedon gap-acceptance, queuing theory and overflow queue methods (Akgelik 1994; Akgelik and Besley 1998;Akgelik andChung 1994aAkgelik, Chung and Besley 1998). Thesemodels are applicable to sign-controlled(two-way stop or give-way)intersections and roundabouts. The evaluation work reported in this paper is for sign-controlled intersections. Each delay model was evaluatedas usedwith its associatedcapacity model. The three models were comparedby meansof extensivesimulation tests using the microscopic simulationprogram ModelC for a basicmajor-minor gap-acceptance case employing the M3A arrival headwaydistribution model (Akgelik and Chung 1994b). While all three models are time-dependent,the evaluationwas carried out using their steady-stateforms in order to be consistent with the simulation methodology. Accordingly, undersaturatedentry traffic conditions(degrees of saturationup to 90 per cent) were considered. Delay predictionsby publishedforms of the models were first compared against simulated delays. Modified forms of the three models were consideredand each model was then calibratedagainst the simulation data. Model calibrationswere performedusing the statisticalanalysis package SPSS and Microsoft Excel. The prediction abilities of the original and modified forms of the three models formed the basisof model evaluation(Christensen 1991). All threemodels have common elements based on queuingtheory. The simplequeuing theory (HCM 94) model is widely used in the literature for modelling delay at unsignalisedintersections (sign-controlled intersections and roundabouts). It is the basis of delay modelsused in softwarepackages such as ARCADY (Hollis, Semmensand Denniss 1980) and PICADY (Semmens 1980). The Akgelik-Troutbeckmodel incorporatesthe minimum delay parameterbased on gap-acceptancemodelling. It is used in the Australian roundabout guide for predicting delays at roundabouts (AUSTROADS 1993). The SIDRA model differs from the othertwo in the useof the overflowdelay conceptas a basis of model structure and a signal analogy method for deriving various gap- acceptancerelationships. The same model structureis also used for signalised intersectionsand roundabouts(Akgelik and Besley 1998;Akgelik and Chung 1994a, 1995;Akgelik, Chungand Besley 199Ja,b,1998).The signalanalogy concept converts block and unblock periodsin the gap acceptanceprocess into equivalentred and green time periods. Unblock periodsoccur when headwayin the major traffic streamis equal or greater than the critical gap while block periods are continuous intervals of no acceptablegaps (Akgellk 1994;Akgelik, Chung and Besley 1998). The modelsgiven in this paperassume zero initial queueddemand. The model structure for the more general case with non-zero initial queue given in Akgelik, Chung and Besley(1997a,1998) is applicableto all modelsconsidered here. The delay consideredin this paperis the stop-linedelay which includesstopped delay, queue move-up delay and the delay associatedwith deceleratingfrom the approach negotiation speed to zero speedand acceleratingback to the exit negotiation speed. It doesnot include the geometricdelay. Referto Akgelik, Chung and Besley (1998) for a detaileddiscussion of different delaydefinitions. 2. LIST OF SYMBOLS d = average stop-line delay per vehicle consideringall vehicles queued and unqueued(s) (not including the geometricdelay for the purposesof this paper) dr, dz = first (non-overflow)and second(overflow) termsof the delay formula (s) dz, = second-termdelay predicted by a steady-statemodel (s) d',-' = minimum delay (s) (the valueof d at x = 0) k6 = second-termdelay parameter | = lost time in the SIDRA capacityand delaymodels (s) Ilm = minimum number of minor stream vehicles that can depart under heavy major streamflow conditions(veh/min) ge = arrival flow of the entry (minor stream)lane (veh/h) gm = total arrival flow of the major stream(pcu/s) (sum of flow ratesin all lanes of all higher priority conflicting streamsadjusted allowing for any heavy vehicle effects) Q" = entry lane (minor stream)capacity (veh/h) Qe = basicgap-acceptance capacity (vehftr) Q'n = minimum capacity(vehft) s = saturationflow rate of the entry lane (veh/h)(s = 3600/B) s g = averagecapacity per cycle (per unblockperiod) (veh/cycle) T1 = durationof the demandflow (analysis)period (hours) x = degreeof saturationof the entry lane (demandflow rateI capacity= Q"/ Q.) xo = degreeof saturationbelow which the second-termdelay is zero (dz = 0) y = flow ratio of the entry lane (arrivalflow/saturation flow = q" /s) cr = critical gap (s) p follow-up (saturation)headway (s) )," a parameterin the exponentialarrival headwaydistribution model A minimum arrival (intra-bunch)headway in the major traffic stream(s); a,r, for the major streamand A" for the minor (entry)stream (p proportion of free (unbunched)vehicles in the major traffic stream;q,,, for the major streamand q. for the minor (entry)stream 3. CAPACITY MODELS The assessmentof the each delay model was performedusing the associatedcapacity model from the relevantpublication. The models are given below (seethe Notations sectionfor definitionsof parametersused in theseexpressions). The SIDRA (Akgelik 1994)capacity formula is expressedby: Q" = max (Qg, Q,") ( la) 3600<p. q. ,I Qe = I + B- lr.-l'ta-Am) (lb) )" Q,n = min (q", 60 n,,,) ( lc) n Q*Q. 0.98 l' = subiectto tt-A. (ld) (l - A,n qJ I = 0.5F ( le) The Akgelik - Troutbeck delay model is used with the following capacity formula describedby Troutbeck(1986, 1989): 36oog-q- e-r(o-^'") e" = for q^> 0 (2) 1- e-r0 = 3600/9 Jbrq*=g The HCM 94 capacity formula given in Chapter 10 of the US Highway Capacity Manual for two-way stop sign control (TRB 1994)is expressedby: 36:00 (a - o'5'"Y' e" = s-9me r"'\ B) (3) 0 The HCM 94 capacityformula assumesa simple negativeexponential distribution of arrival headways(AkEelik 1994). Proportion bunched in the major stream (9.) and minor (entry) stream (<p")are calculatedfrom: