Capacity and Level of Service at Unsignalized Intersections Final Report Volume 1 - Two-Way-Stop-Controlled Intersections (1996) / Chapter Skim
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Chapter Two. Available Methodologies
Chapter Two. Available Methodologies
Pages 5-28
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From page 5... ...
This chapter documents the theoretical background of critical gap and follow-up time estimation, and various capacity and delay models Cat have been developed for analyzing TWSC intersections. CRITICAL GAP AND FOLLOW-UP TIME Definitions Gap acceptance theory relies upon three basic elements, including the size and distribution (availability)
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Such an estimation procedure should be consistent. If the minor street Livers within a specified composition of traffic streams have a given distribution of cntical gaps, We procedure should be able to reproduce this distribution closely.
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By assuming exponentially distributed priority stream gaps and a normal distributions for ta and to, Ashworth (196S,1970) found that the average critical gap to can be estimated Tom {a (mean of the accepted gaps ta in
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The distribution is reasonably general and is acceptable for most studies. The foHow~ng notations are used for subsequent equations: Yi is the logarithm of the gap accepted by the ith driver Yi is ~ if no gap was accepted xi is the logarithm of We largest gap rejected by the ith Liver xi is zero if no gap was rejected ~ is the mean ofthe distribution of the logarithms of the individual drivels critical gaps 02 iS Me variance of the distribution of the logan~ns of Me individual Diverts cntical gaps fit ~ is the probability density function for the normal distribution Fit ~ is the cumulative distribution function for the normal distribution The maximum likelihood of a sample of n drivers having accepted and largest rejected gaps of (ye x)
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The mean cntical gap to and the variance s2 can then be computed by: ~ = en ~ °.so2 S2 = [C2 (e°2 -I) It is this mean cntical gap that has been used In venous gap acceptance capacity and delay models.
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Illustration of the Basic Queueing Theory The mathematical derivation of the cap acid en for the minor stream is as follows. Let gets be the n~nberofminor stream vehicles that can enter into a major stream gap of duration I
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is the probability that n minor stream vehicles enter a gap in the major stream of duration t.
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Relationship Between Capacitor (C=) and Priority Street Volume (vp)
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Therefore, Me critical gap and follow-up time may no longer be constant values, but flow-dependent values. Also, a minor stream vehicle may enter a small gap and force Me major stream vehicle to be delayed.
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from which the basic gap acceptance capacity formulae are derived, a hierarchy of traffic streams exists at all TWSC intersections. These different levels of priority are established by traffic ndes as follows: .
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For each traffic stream, the maxi n potential capacitor en should be calculated using the method shown In chapter To using We sum of all conflicting traffic volumes with higher rank Man the rank of the traffic stream in question. To aid in correct calculations, Table 1 can tee used This table basically corresponds to the German guidelines from bow 1972 and 1991 as well as to the 1985 and 1994 HEM.
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has shown that, of the possibilities considered, Me memos described is realistic enough to represent the range of traffic volumes that occur in practical applications. ~ other words, the impedance effect of hither ranked traffic streams accounts for the unusable time to the subject movement caused by the queuing of these higher ranked streams.
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This capacity should depend on the priority traffic volume vp during the same period. To derive this relationship, observations ot~traffic operations of the intersection have to be made during periods of oversaturation on Me minor steam approach.
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In addition to the influence of priority stream traffic volumes on the minor street capacity, Me influence of the geometric layout of the intersection should also be included. To do this, the constant values could be related to road widths or visibility or even over characteristic values of the intersection layout by using another set of I~near regression analysis.
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From page 19... ...
However, the same ability to reorder gaps is not permissible for delay estimates. Using these usual gap acceptance assumptions, and assumptions about the amval patterns of the minor stream drivers and the order of the arrival of the major steam headways, the delays can be estimated.
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, en is He capacity of the minor stream to enter the intersection, given a major stream flow of vp, and DO is Adams' delay (Adams, 1936~. Arlamq' dela~r is the average delay to minor steam vehicles when minor stream flow is very low.
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Queuing theory can also be used to approximate He gap acceptance process by assuming the simple two-stream system (Figure 3) can be represented by a M/G/l queue.
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The total average delay of minor street vehicles is then d=Dq+ W In general, Me average service time for a s~ngle-channe} queuing system is the reciprocal of the capacity.
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For this situation, the average delay during ~e peak penod can be estimated as: d = D~ + E + ~ (61)
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(65) h = en ~ CnO + Vn° (6o h Y = - en (6 where en is the capacity of the intersection entry during the peak period of duration T and cnO is We capacity of the intersection entry before and after the peak period, vn is the minor street volume during the peak period of duration T
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~ ~ (78) The average delay predicted by Equation 7g is dependent on the initial queue length, the time of operation, He degree of saturation and He steady state equation coefficients.
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has developed equations for average delay using the reserve capacity concept for use when Me intersection is oversaturated. Using a revised deterministic equation to account for We effect of the initial queue before the peak and assuming the equilibrium queue length after the peak is the same as the equilibrium queue leng~before We peek period: d = -B +;B2 +b (81)
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From page 27... ...
Example of We Effect of Changing T in Me Delay Model Although more generalized capacitor models have been developed by considenng non-random major steam headway and using vaned cntical gap and follow-up time values, capacity models proposed by Harders and Siegioch can generally give reasonable results, when random major stream headway and constant critical gap and follow-up time are used. Various delay models were developed based on Me degree of saturation (either volume/capacity ratio or reserve capacity.
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