Downloada Comparison of Three Delay Models
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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 Though some aspects of this paper may be outdated, this reprint is provided as a record of important aspects of the SIDRA INTERSECTION software, and in order to promote software assessment and further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.
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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 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 ( For comparisonpurposes, capacity predictions by the three models for the simulation test casesare given in Table I in Section5. 4. DELAY MODELS All three delay models assessedin this paper,namely the SIDRA, Akgelik-Troutbeck and HCM 94 delaymodels use the traditionaltwo-term form (d = dr + d2),where dr and d2 have different meaningsin each model. The models are describedbelow (see the Notationssection for definitionsof parametersused in theseexpressions). The SIDRA delay model can be statedas: d = dr*d'r (5a) + 20 d," (t o.3yo ) d1 (sb) l-y forx> 1.0,sety = BQ,/3600 dz = 900Tr[x-1+ for x > x,, (5c) =Q otherwise xo = 0.14(sg)oss (sd) ka (5e) d,,, (sf) 9. - 09. r-s3600 (se) I +05 sg = },8 (5h) The AkEelik-Troutbeckdelay model is expressedby: dr =d- (6a) = " 8k.x dz 9001[x-1+ (x-1)'+' " I (6b) Q"T'- dnQ. ,K'1 = (6c) 3600 d'n from Equation (5f). The HCM 94 delaymodel is expressedby: 3600 dr = d,r' = (ta) Q" = d2 900T,[x-l+ (x-l)2+:l (7b) k,1 = 1.0 (7c) The SIDRA andAkgelik-Troutbeck models use Troutbeck (1986, 1989) minimum delay expression (Equation 5fl. The HCM 94 model implies a simple relationship for minimum delay(Equation 7a). All three models are time-dependentdelay models. This is determinedby the form of the second-termdelay (d2)expression. In fact, the second-termdelay (d2)expressions in the Akgelik-Troutbeck and the HCM 94 models can be seen as special casesof the SIDRA expression where Xo = 0 and k6 as given by Equations (6c) and (7c), respectively. The SIDRA delay model parametersxo and k6 (Equations 5d and 5e) are for sign control. For roundaboutsthe same exDressionsare used with different values of constantsas follows: xo = 0.18(rg)n'uo (8a) k6 = o.2orp" (sg)'3n y-t'ou (d-Q") (8b) The results given in this paperfor the "current SIDRA model" are basedon the use of sign-controlmodels (Equations 5d and5e). The steady-stateform of the second-termdelay expressionis relevant to an analysis (demand flow) period of indefinite (very long) duration. It yields infinite values of delay as demand flow rate approachesthe capacity value. For lower degrees of saturation,both the time-dependentand the correspondingsteady-state expressions give similar results. The following expressioncan be usedas a generalsteady-state model for the threemodels under consideration: 3600kd(x - x" ) dz. for x > x,, (e) Q" (1- x) = Q othenvise For the SIDRA model, xo and ka are given by Equations(5d) and (5e). As in the time- dependentexpressions for d2, the Akgelik-Troutbeckand HCM 94 models can be seen as special casesof the SIDRA expressionwhere Xo= 0, and k,1is given by Equations (6c) and (7c), respectively. As a simple example,Figure 1 shows delay as a function the degreeof saturationas predictedby the steady-stateand time-dependentforms of the HCM 94 delaymodel for a casewhen Q" = 600 veh/h and Ti = t h. It is seenrn Figure 1 that, as the degreeof saturationincreases, the steady-statedelay curve approachesthe vertical line at capacity (x = l) whereasthe time-dependentcurve approachesa deterministicoversaturation delayline. steady-state -time-dependent I - 6lslsyministic s90 o E' g s60 o 1.00 1.20 Fig. I - Delay as a function of the degreeof saturation as predicted by the steady- state and time-dependentforms of the HCM 94 delay model (Q. = 600 veh/h and T1= I h) 5. MODBLC SIMULATION ModelC (Chung 1993;Chung, Young and Akgelik l992a,b)is a microscopic(vehicle- by-vehicle) simulation program for the analysisof traffic performanceat unsignalised and signalised intersections. It was originally developedfor roundabouts,and subsequentlyextended to simulate simple sign-controland traffic signal control cases (Akgelikand Chung 1994a,1995). ModelC is a time-updatesimulation model, with vehiclemovements in individual lanes, basedon a car-following model. The conflicts betweenentering and opposingvehicles (minor and major streams)are resolvedby a basicgap-acceptance model as it appliesto stop-signcontrol, give-way sign control and roundabouts.The simulationtests reported in this paper were carried out for a simple gap acceptance(major-minor stream) situation. The combinationsof minor and major streamparameters for the simulationcases used in the evaluation of current models and calibration of modified models are given in Table 1. The major traffic streamheadway distribution characteristics were specifiedto representsingle-lane and multi-lane cases. In all cases,the minor (entry) traffic stream consistedof one lane, thereforethe intra-bunchheadway (A) and parameterb valuesfor the minor streamwas the sameas the singlelane major streamvalues inTable I. Three major stream arrival flow rates were simulated,namely low flow (360 veh/h), medium flow (720 veh/h)and high flow (1080veh/h). The first two casesin Table l were simulated with low and medium maior stream flow rates only due to low capacities. For each major stream flow, five entry flow rates were simulated representinga wide rangeof degreesof saturatiofl,X = 0.1, 0.3, 0.5, 0.7 and 0.9. The input flow rates to achieve these nominal degreesof saturationwere determinedby calculatingthe capacityusing the SIDRA capacityformula. The capacitiespredicted by the three capacitymodels (EquationsI, 2 and 3) are given in Table I . The simulated entry and major stream arrival flow rates, capacitiesand degreesof saturationdiffer from the input values. The duration of the warm-up and main simulation periods were set as 15 and 60 minutes, respectively. The simulationtime incrementwas 0.1 s. Each case was simulated5 times with differentrandom number streams in orderto allow for the effects of random variations.In the analyses,results from individual simulationruns were used directly rather than using an averageof the five simulation runs for each case due to problems associatedwith averagingat higher degreesof saturation. Simulation results were checkedcarefully and somedata were eliminated. Casesthat produceddegrees of saturationabove 0.9 were deletedbecause steady-state type simulationmethod does not produce appropriatedata for conditions near capacity. After all data elimination, the numberof datapoints used in modelevaluation and calibration was 392. Table I - Data for simulation test cases Minorstream Majorstream Estimatedcapacity (veh/h) Critical Followup Number lntra-bunch Param. Flow rate SIDRA Troutbeck HCM 94 gap, headway, of lanes headway, (Akeelik) c[ (s) 0 (s) A (s) p q, (pcu/h) 8.0 4.0 >2 0.5 0.8 360 481 487 494 720 245 255 271 7.0 >2 0.5 0.8 JOU 596 601 608 720 332 342 360 t).u ?q >2 0.5 0.8 360 659 bb5 672 720 407 420 440 1080 242 257 287 5.0 J.U >2 0.8 360 OJJ o.to g46* 720 561 F76 596" 1080 420. qn 1 u.o 360 813 819 946* 720 495 510 cYo 1080 250 269 420* 4.0 2.0 1 1.5 0.6 360 1295 1300 1333 720 859 873 988 1080 495 515 732 ?n 2.0 1 0.6 360 1442 1447 1474 720 1091 1108 1207 1080 751 781 988 * The HCM 94 modelis not sensitiveto the numberof maiorstream lanes. 6. PREDICTIONS BY CURRENT DELAY MODELS Figure 2a shows a comparison of the simulated degreesof saturation with those predictedusing the SIDRA (Akqelik) capacitymodel (Equation 1) using the simulated entry and major streamflow rates. Linear trendline and associatedstatistics indicate very good correspondencebetween the simulatedand predictedvalues. The degreesof saturationpredicted by the Troutbeckmodel (Equation2) arevery similar to the SIDRA predictions. The HCM capacitymodel (Equation3) generallyunderestimated degrees of saturation(overestimated capacities) as seenin Figure 2b. Figures 3, 4 and 5 present the graphs of delays predicted by the SIDRA, AkEelik- Troutbeck and HCM 94 delaymodels plotted againstsimulated delay valueswith linear trendlines and associatedstatistics. The HCM 94 delay model is seen to give poor performance.The SIDRA and AkEelik-Troutbeckmodels give similar performance. 7. MODIFIED DELAY MODELS Modified SIDRA model: Severalmodified forms of the SIDRA delaymodel were tested and the following form was found to give the bestresults. For the first delay term: d,"(l +a, (sg yr',) A. \b, (10a) l-y for x > 1.0,set y = B Q"/3600 whereat.bt, c1&ra calibration parameters, and d- is theminimum delay given by: d. = 1r-0Q"r,e[(o-4"1-]-g*r, (r0b) 3600 q-q- ), where / = 0.5 B as in Equation(l e). Equation( l0b) wILlbe referredto as the Akgelik minimum delayformula. For the seconddelay term (Equcttion5c applies): xo = o, (rg)h (10c) k6 = o, (sg)hy" (l0d) where a2,b2, as, bs, cj arethe calibrationparameters. This model differs from the currentSIDRA model in eliminatingTroutbeck's minimum delay (d,") parameterfrom the seconddelay term. As such,the model conformswith the generalSIDRA model structuremore directly. The minimum delay,d,,, used in the first delay term is calculated from Equation (f0b) rather than Troutbeck's formula (Equation5fl. The derivation of Akgelik's minimum delay formula is based on the signal analogyconcept, and as suchconforms with the generalSIDRA model structure. Y = 1 0244x na x R'z= 0.9883 c o (E .a L ,us!'f J 06 oG o 6) I oa ED o E (E o -a v.z 0.2 0.4 0.6 0.8 1 Simulateddegree of saturation(x) Fig.2a - Comparison of simulated and predicted (SIDRA) degreesof saturation 1.0 x 0.8 Y= 0.9108x .9 * =0.9412 G 0.6 G' o o c, o Et o.4 o st ta' CD = () o.2 E 0.0 0.2 0.4 0.6 0.8 1 Simulateddegree of saturation(x) Fig. 2b - Comparison of simulated and predicted (HCM 94) degreesof saturation r)U Y = 0.6343x+3.4477 R' = 0.6581 40 aa o 1O O -a" s a a o^^ aa a EJU aa TL -'^o'a a o o j' a 6 i.t-";t''' a zv .'!lr.f.t.-13{--i '. t;l .' lS.ilJl. O 10 a a 0102030405060 Simulateddelay (s) Fig. 3 - Predicted vs simulated delay for the cunent SIDRA delay model using the SIDRA / Akgelik capacity model 60 V = 0.8197x+2.0321 50 R' = 0.6864 o G| rn o- oo lt !t J(, aaa o a €"^ Jvv .. ..:. o L - !o . -.'- F :. L ^ ol t-"^t 1"'o a .rt'ilolr'f ) o 820 o .:. ao o a ctslr loo fit"t . 10 rl . 0102030405060 Simulated delay (sec) Fig. 4 - Predicted vs simulated delay for the current Akgelik-Troutbeckdelay model using the Troutbeck capacify model Y = 0.5486x+ 6.6815 ff = 0.4186 o o a o o oao O a so E30 a ti. a' o) aa lo' = o r^^ !..o" fo 30 Simulated delay (sec) Fig. 5 - Predicted vs simulated delay for the curuent HCM 94 delay model using the HCM 94 capacity model 15.0 r HCM o Akcelik Y = Q.7273x+2.6852 12.0 Linear(FCM) R2= 0.8197 I o Linear(Akcelik) t, o L On-" =ta I1T o o J 'd 6.0 = o # I o g.o 0.0 6.0 9.0 12.0 d''' (Troutbeckmodel) Fig. 6 - Comparison of the minimum delays (seconds)predicted by the Troutbeck model with those by the HCM and Akgelik models A comparisonof minimum delay estimatesfrom the Troutbeckand Akgelik models as well as the HCM minimum delay values (Equation 7a) is given in Figure 6. The Troutbeckand Akgelik minimum delaypredictions are seento be very close. Parameter9e w?s also eliminated from the second-termdelay expressiondue to the limited effectit had on regressionresults. The modified Akgelik-Troutbeckmodel form consideredfor calibrationis as follows: k6 = 51d'Q''3600- ,c (1lb) where a, b andc arethe calibrationparameters. The modified HCM 94 model form consideredfor calibrationis as follows: d1 = dm=a (r2a) Q. k,1 -b (12b) where a and b are the calibration parameters. 8. CALIBRATION OF MODIFIED DELAY MODELS The calibrationswere performedusing the non-linearregression analysis method offered by the statisticalanalysis package SPSS. SPSS results were further analysedby means of the trendline facility and visual inspectionof graphsin Excel spreadsheets.In some cases,minor manual adjustments were made to the calibration parametersin order to fine-tune the SPSS results. The calibrationmethod is describedin detailin Christensen(1997). The calibrationresults aregiven below. Modified SIDRA delay model: ul 0.5 (13a) bt= - 0.4 0.1 a2= 0.15 (l3b) bz= 0.5 0.25 (l3c) bs= 0.3 LJ - 0.5 Figure 7a presentsthe delaypredictions from the modified SIDRA delaymodel with the above parametervalues againstthe simulateddelay values. Comparisonof Figures 3 and 7a indicates that the modified model offers a small amount of improvement. Figure 7a is basedon the use of capacitiespredicted by the SIDRA / Akgelik model (Equationl). Further improvementsto the capacityand delayprediction were obtainedwhen the lost time parameterwas calculatedfrom the following formula rathertban Equation (le): I - 0.4+0.98-0.35cr (l3d) Using the lost time from Equation (I3d) in Equation (1b) for capacity and Equation (lOb) for minimum delay, R'^= 0.990 for degreeof saturation(comp^are with R' = 0.9883 in Figure 2a) and R' = 0.7471 for delay (comparewith R' = 0.7161 in Figure 7tt) were obtained. Improved delay predictionswere obtainedwhen the modified SIDRA delay model was usedwith simulatedcapacities and slightly differentcalibration parameters (a: = 0.3, bs = 0.4, c; = - 0.6). The resultsare shown in Figure 7b. In this case,comparison of predicted and simulated delays gave R' = 0.8766. This indicatesthe importance of accuratecapacity prediction for improving the accuracyof delay predictions. Although the capacitypredictions by the SIDRA model are very good as seenfrom Figure 2a, delays are sensitive to small differencesin capacities,especially at high degreesof saturation. Mo dified Akg elik-Troutb eck delay model: a = 1.0 (14) b = 0.8 L = 0.6 Figure 8 shows the averagedelays predicted by the modified Akgelik-Troutbeckmodel plotted againstthe simulateddelays. Comparisonof Figures 4 and 8 indicatesthat the modified model offers negligibleimprovement over the original model. Modified HCM 94 delay model: a =0.9 (15) b =O.J Figure 9a shows the averagedelays predicted by the modified HCM 94 model (using the HCM 94 capacity model) plotted against the simulated delays. Comparison of Figures 5 and 9a indicatesthat the modified model offers negligibleimprovement over the originalmodel. Delay predictionsby the modified HCM 94 model was improvedsubstantially when the model was used with capacitiespredicted by the SIDRA capacitymodel (Equation |b) as seenin Figure 9b. In this case,R'= 0.6568was obtained (compare with R'= 0.4388 in Figure 9a). This improvementindicates that performanceof the HCM delay model can be improvedto a good extentby using a bettercapacity model. a Y=0.8496x+2.7032 R2= 0.7161 1>O a a a o sq) aa E ': Ee^ o 6 f )n'" t3 aa .9 t. :E .,r., o tr 20 o-- = ^a s.r' ? a 30 Simulateddelay (sec) Fig. 7a - Predicted vs simulated delay for the modified SIDRA delay model using SIDRA / Akqelik capacity values y=1.0029x+1.7849 R2= 0.8766 aO. o ?a'^a.. a 6 t .tf . 1."'i'^. t' en O-lO o a 2 ^-..' .'.' 6 l tt =.g ..r:1ri " o o = df,..4 o 30 Simulateddelay (sec) Fig. 7b - Predicted vs simulated delay for the modijied SIDRA delay model using simulated capacity values ocu s Y =O.7387x+2.5119 o R2= 0.6901 9-' x4n. a' o- 0) -o . ... t.' o t30 ..1t. ta U 6) I r) o t ?: o.:"-- ^ .Y ? I 4cn ^O -"rD a t^l.i'. i.-'{ il ,a I_.----i o o =.9 a E o oo. o o l.r-;ft- =.^ LJa- v a Ptb o 10 Simulated delay (sec) Fig. 8 - Predicted vs simulated delay for the modified Akgelik-Troutbeck delay model using the Troutbeck capacify model 9. RESULTS FOR THE HCM 97 MODEL The 1997 version of the Highway CapacityManual Chapter l0 (HCM 97) retains the HCM 94 delay model for two-way stop sign control but modifies the capacity model (Kyte 1997): 3600q- e-09- Q. = (16) 1_ o- "-0 whereq* is in pcu/s. This model can be derivedfrom the Troutbeckcapacity model (Equation2) assuminga negativeexponential distribution model for major streamheadways (A,n = 0, e- = 1.0, thereforel, = Q,r,). A comparisonof the delay predictionsby the HCM 97 delaymodel (sameas HCM 94 model) with the simulateddelays is shown rn Figure 10. For major streamflows used in the simulation testsreported in this paper,the predictionsby the HCM 97 capacity model (Equation 16) are found to be very similar to those by the HCM 94 capacity model (Equation 3) as seenin Figure I l. As a result, the delay predictionsusing the HCM 94 and HCM 97 capacityestimates are similar as seenfrom the comparisonof Figures 5 and 10. Y =0.4128x+5.5779 R'z= 0.4388 o 40 s>. o sl o) =30 o I .9 E-" o = 30 Simulateddelay (sec) Fig. 9a - Predicted vs simulated delay for the modified HCM 94 delay model using tlneHCM 94 capacity model aaa o a a >\ +u aa g o E =f ct) =30 o :E !t c, t ^.'l]-.t:'il'^ : E-- ailrtO I o o . ? = i,r';-t'"- 37aF - a l7*'. . 30 Simulateddelay (sec) Fig. 9b - Predicted vs simulated delay for the modijied HCM 94 delay model using the S/DRA / Akqelik capacity model estimates Y = 0.6194x+ 6.4933 ' R =0.443 q s E306) t- (t) " ()= " .. zv ]1 l*" 30 Simulateddelay (sec) Fig. 10 - Predicted vs simulated delay for the HCM 94 / 97 delay model using the HCM 97 capacity model 1600 1400 y=10058x-11.779 R2= 0.9999 '1200 o 1000 E o (E 800 (E o t\ o) 600 = o t 400 200 0 600 800 1000 1200 1400 HCM94 capacity (veh/h) Fig. 11 - Comparison of capacity estimates from HCM 94 and HCM 97 models 10. CONCLUSION An evaluationof three existing delay modelsfor unsignalisedintersections, namely the SIDRA, AkEelik-Troutbeck.and HCM 94 models,has beenpresented. The models are applicableto sign-controlled(two-way stop or give-way)intersections and roundabouts. The evaluation work reportedin this paper is for sign-controlledintersections. Data from ModelC simulation for a basic gap-acceptancesituation at a sign controlled intersectionwere used. Each delay model was evaluatedas used with its associated capacity model. Modified forms of the three delay models were calibratedusing the samesimulation data set. Generally, the modified models offered small or negligible improvementsover the original models. Overall, the SIDRA and Akgelik-Troutbeckmodels indicatedsimilar levels of prediction ability whereasthe HCM 94 (or HCM 97) model displayedpoor performance. Improved prediction of capacitiesappeared to give larger levels of improvement in delay prediction as evident by the improvements obtained when simulatedcapacities were used to predict delays. In particular,the HCM delay model would benefit from the useof a bettercapacity model (SIDRA / Akgelik or Troutbeck). The study presentedin this paper supportsthe delay, capacity and minimum delay formulae developedby Troutbeckusing gap-acceptancemodelling. It also supportsthe correspondingformulae developedby Akgelik using a signal analogyconcept for gap- acceptance modelling and the well-known overflow concept for delay model formulation. Interestingly,the delay,capacity and minimum delay predictionsfrom the two sets of formulae are very close, with slightly better delay predictions from the modifiedSIDRA model. While the Akgelik-Troutbeckdelay model has fewer parametersthan the SIDRA delay model, an important advantageof the SIDRA model is its consistencywith models for back of queue and stop rates at both unsignalisedand signalisedintersections. Work was also undertakento study modified forms of the current SIDRA model for back of queue prediction. The findings were similar to the SIDRA delay model evaluation (Christensen1991). This paper has presentedsimulation resultsfor a basic gap-acceptanceprocess with a single opposing movement (major street through traffic) at two-way sign control assumingknown follow-up headwayand critical gap values. Applicability of resultsto roundaboutsis limited since both the capacityand delay models for roundaboutshave differences in various parametervalues, and simulation of complicated interactions amongapproach flows is neededfor roundabouts. Real-life situations where the basic gap acceptanceprocess with a single opposing movement at a sign-controlledintersection is relevantinclude the following (assuming driving on the right-handside of the road as in the USA andEurope): (i) right turning traffic from minor streetat 3-way or 4-way intersectionswhen there is no effect of the adiacentexit flow, (ii) left-turning traffic from major street when the opposing right-turn movement is controlledby give-way(yield) or stopsign, and (iii) left-turning traffic from minor streetafter moving into the median storagearea of a 3-way intersection(where two-stage crossing is possible). In termsof HCM 94, theseare Rank 2 movementswith a singleopposing movement. Evaluation of alternative delay and capacity models as reported in this paper is recommendedfor more complicatedgap-acceptance situations with multiple opposing movements at sign-controlled intersectionsand with approachflow interactions at roundabouts. 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