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29 Chapler Three CANDIDATE MODELS The previous chapter provided a summary of the main theoretical models available for estimating critical gap, capacity, and delay at TWSC intersections. The purpose of As chapter is to summarize the methods and models that were selected as candidates for evaluation against field data. CRITICAL GAP ESTIMATION PROCEDURES Some ofthe well known estimation procedures for critical gaps have been checked for consistency in MiDer (1972) and Troutbeck (1975~. The effectiveness of the maximum likelihood memos has been evaluated in further studies by Briton and Troutbeck. In a recent report(Brilon, 1995), Briton concluded that the estimation process for critical gap distribution should be consistent. There was a strong suspicion by Me research team that a great deal (if not Me majority) of all inconsistencies found in the literature regarding the relationship of critical gaps to other parameters (such as traffic volumes, delays, delays at the stop line as service times of the imbedded queuing system, and geometric characteristics of intersections that are normally studied at different sites under different traffic volumes) might not actually exist but may instead result Dom inconsistencies in the estimation procedures. The estimation of critical gaps is not an end In itself. Critical gaps are used in models to estimate the capacity of unsignatized intersections. Estimating cntical gaps using different procedures could influence the model's capacity output. It should be guaranteed to the user community that the estimated cntical gap, In conjunction with the follow- up brne, and their respective estimation procedures give a reliable and realistic estimate of intersection capacity, i.e. an estimation procedure that is independent of the external parameters mentioned above, especially the major street volume. Further, the estimation procedure for cntical gaps, the capacity model, and the delay mode! must form one integrated system using consistent definitions and assumptions. Therefore, Briton (1995) conducted a simulation study to investigate the independence to traffic parameters of different critical gap estimation methodologies. The simulation study for testing different procedures used two traffic streams: one on the major street (volume vp) and one on the minor street (volume vn; see Figure 3~. S~mulation runs with constant traffic volume vp and vn were conducted over 10 hours of simulation. This was repeated for 46 different combinations of vp and vn (where vn < capacity3. The arrival headways were generated according to a hyper-eriang-distr~bution. The critical gaps tc and the follow-up times if were generated according to a shifted Eriang-distribution. These distnbutions reflect stochastic yet realistic traffic situations. The critical gaps were estimated from the simulated flow patterns using the following methods: . . . . . maximum likelihood procedure (e.g., Troutbeck, 1992) Meshwork (1970) Raff (1950) Harders (1976) Hewitt (1992) logy mode] Siegioch (1973) In addition, for the maximum likelihood method, an estimate for the standard deviation of the critical gap was obtained from the procedure. The SiegIoch estimation procedure was also tested with constant queues in the minor street approach. For each procedure, two different combinations of the critical gap tc and the follow-up time If were used: for 50 km/in: to = 5.S see, If = 2.6 see for 70 tenth: tc = 7.2 see, If = 3.6 see (The annotation of the cases corresponds to Harders' (1976) findings regarding the speed dependency of tc and if in Germany, a finding which has not been confirmed by this study). The stochastic distributions of critical gap and follow-up time assumed ~at: witch = 0.308 x tc city = 0.387 x if mints = Max(0.3Ixtc, 2.0) max to = Min ~ 2.16 x to, 20.0) In all simulation runs, the drivers behave consistently:
30 although tc and tf for each Diver are sampled from an assented distribution, each individual driver uses the same tC and tf value in any situation (e.g. regardless of time spent in queue). In subsequent studies it could also be shown Mat the significant conclusions, drawn from this study, are also valid if inconsistent drivers - i.e. drivers with varying critical gaps - are represented In We simulation programs. Figure 17 illustrates the simulation results using the maximum likelihood method. Figure IS shows the comparison between the regression functions of different mesons for Me case oftc = S.S see and tf= 2.6 sec. Note that the scales of the y-ax~s on Figure 17 and IS are different An unbiased estimation method should yield the same average critical gap estimate over multiple simulations, independent of the major street volume (or over traffic parameters). Based on the simulation Study, Me following conclusions were reached: . The Siegioch method' gives values close to Me hue values; however, Me variance is large, and Me results depend on Me major street flow vp. Overall, Me SiegIoch method is reasonable. However, it is not the least biased compared to over melons. ~ t-7 te~ ~ I · ~ ~ · lne maximum likelihood method had an extremely 0th correlation between the true tc value and the estimated values. For the 10-hour simulation period, the maximum deviation between estimation and true value is below 0.15 seconds. The regression function (Figure 17) of all Me values exactly coincides with the Rue value and the result does not depend on the priority street volume. The maximum likelihood estimation procedure is also not influenced by traffic volumes In the nompnority stream. · The probit estimation procedure described by Hewitt has an extremely good correlation between the hue value and He estimated values. The max~mumdev~ations are below O.2seconds,and He regression line Is horizontal and coincides with He Sue value (see Figure IS). Therefore Hewitt's estimation procedure performs extremely well. Hew~tt's procedure is also independent of He minor sheet volume. There is, however, some indication that He results show sightly more variance Man He maximum likelihood method with very small traffic volumes. . In conclusion: . . . A logit model using bow lags and gaps does not include He hue value and has a strong dependency on the major street volume. Therefore, a logit model that uses lags and gaps see~nsto produce biased results.Ashwor~1968) indicated that this bias can be predicted under some circumstances. The estimation method for critical gaps must be selected with great care. Among the tested methods, only the maximum likelihood method and the Hewitt method are consistent with respect to major street and minor street flows. The recommended method used for this project is the maximum likelihood method. The theory for implementation of this method was presented in chapter two. Inputs to the procedure are the observed accepted gap and maximum rejected gap for each driver in a sample. The distribution of critical gaps is assumed to be log-normal. An iterative procedure provides an estimated critical gap, as well as the estimated variance of He critical gap. 6.0 5.9 - x x ,,,: x ~ 5.9 x x x x In x X x 58¢ x ~ y x x 5.7 x x xx 57 ~1 ~1 ~ 1 ~ 1 ~ 1 ~ 1 ~1 ~1 ~ 0 100 200 300 400 500 600 700 800 900 Major Street Volume, veh/hr ~ = S.8 sac x Estimated Values Figure 17. Estimating the Critical Gap by the MDum Likelihood Method
31 7.0 6.5 ~ 6.0 g 5.5 :~ 5.0 4.5 Figure 18. Estimating the Cntical Gap by Various Methods CAPACITY MODELS l ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ | Hewitt _~ 0 100 200 300 400 500 600 700 800 Lo' x Major Street Volume, vehlhr Is. Two general categories of models for capacity analysis of TWSC intersections have been addressed In chapter two: gap acceptance models and empirical models. The candidate capacity models considered for testing as part of NCHRP 3-46 are summanzed in this section. Mode! I.1 The 1985 HCM memos Is based on gap acceptance theory developed by Harders. It requires an understanding of the availability of major stream gaps, He usefulness of each gap, and the relative hierarchy of the traffic streams at the intersection. It assumes random arrivals on the major street and minor stream driver behavior that is both consistent and homogeneous In accepting or rejecting major stream gaps. v e ~(~P ~) ~ ic - try c = ~ P (8S) ewpirl3600_ 1 where en is the capacity of He non-pnority stream in veer, vp is the priority flow rate In Their, tc is the critical gap In seconds, and If is the follow-up time In seconds. Mode] 1.2 Another gap acceptance capacity model, developed by Siegloch and now included In He 1994 HCM Update, is based on a somewhat different formulation of the critical gap. 3600 -v ~ / moo en = e p o (86) If where to is the zero gap and the other parameters are defined as before. Both the Harders model and the Siegioch model yield nearly identical estimates of capacity despite somewhat different formulations. Troutbeck has extended both of these models to account for non-random or platooned major stream flow patterns and for multiple lanes on the major sheet. He modified both Harders and SiegIoch models using a dichotomized headway distribution. Mode! 1.3 The modified Harders mode! is given in Equation 87. a vp e -A ~ i~ - tr - Jay eat! _] (87) a v 3600 ~ tm vp (88) where ~ is He proportion of free-vehicles, tm is the minimum gap in the major traffic stream, and the over parameters are defined as before. Model 1.4 The modified SiegIoch model is given in Equation 89. av e-A(io~im) en - (89) A If a v A = P (90) Model 1.5 An alternative formulation of capacity of TWSC intersections, developed by Kimber, is based on emp~ncally derived relationships between capacity, flow rates, and intersection geometry.
32 en = A - EBf Vp f - If ~ Vp i Vp ~ (9 l) where v is the volume on the various opposing and conflicting approaches, A, B. and C are regression coefficients, and the other parameters are defined as before. Mode! 1.6 Kyte also developed an emp~ncal mode} based on a somewhat different method of measuring capacities. Both the Kimber mode} and He Kyte mode! relate the minor stream capacity directly to the flow rates of the conflicting traffic streams. C = Deaf Vi (92) where Ai's are regression coefficients, and vi is traffic volume of movement i. DELAY MODELS There are four general categories of models for estimating average vehicle delay at TWSC intersections: gap acceptance models, queueing theory models, empirical models, and hybrid models based on t~me-dependent flows. Models for estimating average vehicle delay based on gap acceptance theory were developed by Harders and SiegIoch. Both models require simplifications ~ havoc flow assumptions to allow for practical application. Mode! 2.l Harders developed the delay mode} based on gap acceptance theory, which is also used In the 1985 HCM. The mode! is given In Equation 93. d = 3600 ( 1 - e ( P c ., ) (93' C - V where d is He average delay of He subject approach and the other parameters are defined as before. Mode] 2.2 Siegioch also proposed a delay mode} using the gap acceptance theory, and the model is given in Equation 94. 3600 ~ e-VS J./ 3600 d= ~_ v5 ~1 - X 1) t94' where x is the degree of saturation and He other parameters are defined as before. Mode! 2.3 Troutbeck used the PolIarczek-Klintchine formula Tom queueing theory for delay estimation. d = ~ [1 + XC 1 on ~1 ~ xJ where cnis the capacibrofthe subject approach and C is as defined following Equation 50; other parameters are defined as before. Mode! 2.4 Kimber and Hollis developed He following delay formulation based on time-dependent flows. A simplification of this mode! assumes steady state conditions for undersaturated flows and a coordinate transformation to account for oversaturated flows. d=~i +E+ 3600 too dim = ~ [~(F2+G)-F] t97' _ 2vo E= V (98) F = V ~ tc ~ fin) y + 2 by --~ ~ + E t99' G = 2Ty [_ _ TIC - V ~ E ] (100) h = c Co + Vo (101)
33 Y vn Mode] 2.5 Troutbeck and Akcelic developed a simplified version of He Kimber-Hollis model Hat depends on He length of the peak period, T. d = 3600 + 900 T|X - 1 + ,|(x _1)2 +(C/36oo~xl(lo This is He equation used In the 1994 HEM Update. Mode! 2.6 Kyte proposed an empirical approach directly relating delay to reserve capacity (cn - vn). d=`5le-P2(c~ ~ where ,B's are regression coefficients. MODEL EVALUATION AND SELECTION (104) A primary objective of NCHRP 3-46 is to develop new methodologies for computing capacity and level of service for stop-controlled intersections based on data that are representative of U.S. conditions. At the heart of these methodologies must be models that produce reasonably accurate forecasts of capacitor and delay (assuming that delay is the basis for determining level of serviced using input data normally available to practicing tragic engineers. What procedure is used to determine if a model is able to meet this objective? Consider the following statements: · Specification. The model must be specified using standard traffic engineering parameters. ti02' ~ . . . . Theory. The model specification must represent a sound underlying theory of traffic flow. Calibration. The model parameter must be unmated using the data Hat have been collect This process of determining the numerical values of the model parameters is called mode! calibration. Range. The model must be able to account for a wide range of traffic flow and geometric conditions likely to be encountered by the practicing traffic engineers. Validation. The accuracy of the mode! forecasts must be verified over a wide range of operating conditions. When forecasts are verified using data that was not used to calibrate the model, this process is called model validation. Qualily. The model must produce better forecasts than other competing models. If the standards inherent In these statements can be met, a model can be put forward as the core of a new methodology to forecast capacity or delay. These six statements have been translated into five evaluation criteria that are used to provide an initial assessment of the candidate models. The criteria used to assess each model is listed below. Ideally, a model should be . . . . theoretically sound easily validated with field data practical and easily applied by the practitioner produce sufficient and appropriate measures of effectiveness as output relevant in terms of common situations encountered by He practitioner. Each of the candidate models described above were evaluated using these criteria. The results of this evaluation are given in Table 3. Based on this evaluation, a few models were recommended for testing against field data. If none of the recommended models were subsequently found to adequately represent field data, then another candidate model could be tested.
34 Table 3. Model Evalaudon 1 ................................................................................................ ................................................................................................................ . TWSC CapacIb Models Model 1.1 Model 1.2 Model 1.3 Model 1.4 Model 1.5 Model 1.6 TVVSC Delay Models Model 2.1 Model 2.2 Model 2.3 Model 2.4 Model 2.5 Model 2.6 ,.......... ........... ........... :-:::::::::: :,::: .y:::: ::::~:::: P P Y Y Y Y . _ . P . Y . Y . Y l Y l .. . . . . . . . . . . . . . . . . . . . .. . . . .. .. ... ..... ... , , ~ .......................... .. .............. . ... ... . . ... . . . . . . . . . . .. . . . . . . . ::::::::::: :-:::: ::::::::::::::::: :::::::::::: :-:: :.: ::::-:-:::-:::-:-:-:-:-::: . :::::::- :~:::::::-:: :::::::::~::::: ::::-::::~::: :-: ::::::-:-:~-:-: :-:-: ........... ........ ............. ................... Y N P Y Y N P Y Y N P Y Y N P Y P Y Y Y P . Y Y Y . Y Y Y Y Y Y Y Y P Y Y Y Y Y P Y Y Y Y Y P Y Y Y Notes: Y = Yes, meets criterion; N = No, does not meet criterion; P = Partially meets criterion Recommen`ded TWSC Intersection Capacity Mo`dels Mode! 1.4 is a theoretically sound gap acceptance mode} that considers ~e effects of vehicle platoon~ng on the major street. The Harders and SiegIoch models by themselves do not o~er this advantage. Mode! ].5 is a generali~d empirical approach to capacity estimation, and is similar to the me~o~proposed by Kyte. The theoredcal basis of the models based on gap acceptance theory lead to the selection of the SiegIoch and Harders models, wid1 ~e corrections proposed by Troutbeck to account for vehicle platoon~g for test~ng agamst field data. The emp~ncal models would be calibrated only if the gap acceptance models were subsequently found to inadequately replicate field data. Recommended TWSC Intersection Delay Mo`dels All six models listed in Table 3 are capable of estimating delay for steady state conditions. However, o~ly Mode] 2.4 andMoa?eZ 2.5 account for ~e effe~ct oftime-vanation in traffic flows. This effect is becoming more unportant as analysts are more Dequently required to address more highly congested tra~c facilities. The complexity of Mode] 2.4 limits its practical application; thus Mode! 2.5 was selected for final test~ng. As discussed ~n chapter two, the accuracy of the delay estimate using any delay models is I~m~ted by the accuracy of ~e estimated capacit parameter which is a prima~y input to all delay models.