Queue length estimation from probe vehicle ... - ResearchGate

ARTICLE IN PRESS6 G. Comert, M. Cetin / European Journal of Operational Research xxx (2008) xxx–xxxTable 1Two-sigma deviations at various p levelsp 0.0001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.9999# of vehicle (2r) 8.5 6.6 5.1 4.1 3.3 2.6 2.1 1.5 1.1 0.7 0.0Dev. % (2r) 75 58 45 36 29 23 18 13 10 6 05. Analysis with the overflow queueIn the previous section, the overflow queue (cycle to cycle overflow)or the residual queue at the end of green period is ignored. Inthis section, the total queue length estimation that also includesthe overflow queue due to randomness is presented. It is still assumedthat average demand is less than the capacity – undersaturatedconditions. Let N o be the overflow queue length and N r be thequeue length that occurs from new arrivals during the red period.The total queue length to be estimated, N, is the summation of N oand N r . At any given cycle, probe vehicles may or may not be presentin one or both of N o and N r , leading to four possible scenarios.Irrespective of where a probe vehicle is in the queue, the generalformulation and equations presented before are applicable as longas the marginal probability function, P(N = n), is known. Therefore,the analysis can be carried out without dealing with these scenariosindividually.The main input needed is the marginal probability of the totalqueue length, P(N = n). For a traffic light with fixed timing, someresearchers developed models that allow calculation of P(N = n)numerically. These models involve finding the complex roots ofthe pgf of overflow queue in equilibrium (Abate and Whitt,1992). The application of these models also requires numericalevaluation and approximations for pgf inversion (Van Leeuwaarden,2006). The second alternative to find P(N = n) is through simulation.In this paper, a simulation model is developed todetermine P(N = n).In this example, arrivals are assumed to be Poisson with a rateof k, and the vehicles are assumed to depart from the stopbar at aconstant rate during the green, which is assumed to be 2 secondsper vehicle. Therefore, the service rate l is equal to the green durationdivided by 2. The total queue length is defined as N = N o + N r inequilibrium conditions. The distribution of total queue length,P(N = n), is determined by running the simulation multiple times.The traffic light parameters are chosen as follows: cycle length,C = 90 seconds; red period, r = 45 seconds; and green period,g = 45 seconds. The arrival rate, k, is set to 20 vehicles per cycle.These selected parameters correspond to a v/c ratio of 0.89. Simulationis run for 180,000 seconds and replicated 1000 times. Thefirst 200-cycles in each replication is considered as the warm-upperiod and not included in calculating the equilibrium distributionof P(N = n).Once the P(N = n) is obtained, an analysis similar to the one presentedin the previous section can be performed to determine theTable 2Results for various q valuesp = 0.5q = k/l k Mean = E(N) r 2 = E(VAR(Njl p )) 2r Dev. (%)0.50 11.25 5.630 1.295 2.3 400.60 13.50 6.757 1.395 2.4 350.70 15.75 7.936 1.472 2.4 310.80 18.00 9.338 1.553 2.5 270.88 20.00 11.394 1.654 2.6 230.90 20.25 11.816 1.672 2.6 220.95 21.38 14.938 1.766 2.7 180.99 22.00 23.580 1.876 2.7 12100* σ/E(N)70605040302010λ/μ=0.6λ/μ=0.7λ/μ=0.8λ/μ=0.9λ/μ=0.9900 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Proportion of Probes, pFig. 10. Normalized errors versus proportion of probes at different q values.expected total queue length and variance. For brevity, only the resultsthat show how error changes by the proportion of probe vehiclesare presented here. Fig. 9 shows the expected total queuelength and one- and two-sigma deviations from the mean versusthe proportion of probes, p. Compared to Fig. 6, the mean is shiftedupwards due to the effect of the overflow queue, and the deviationsshow similar behavior.The results presented in Fig. 9 are summarized in Table 1 to easilysee the improvement in errors when p gets larger. The 2 rdeviations are presented in two formats: as absolute values (row2) and as percentage of the mean (row 3). The values in row 3can also be interpreted as the coefficient of variance (cv) multipliedby two. It is useful to look at both absolute deviations in terms ofthe number of vehicles and the percent deviations since the targeterror criterion can be expressed in both formats. For example, for80% probe vehicles, the expected error is about 10% and the absoluteerror is about ±1 vehicle. If the error is assumed to follow anormal distribution then, these error rates roughly correspond toa 95% confidence interval.Thus far, the results presented are for a constant arrival ratewith a v/c ratio or q equal to 0.89. To understand the impact of qon accuracy, the arrival rate is varied to generate the scenariosshown in Table 2 where the percent of probe vehicles p is kept constantat 50%. For each one of these scenarios, the simulation programis run separately to obtain P(N = n) and the equationsdescribed above are utilized to estimate the means and variances.The last two columns of Table 2 convey the same type of informationpresented in Table 1. As it can be observed, absolute deviations(2r) are not increasing significantly as v/c or q increases. On theother hand, percent deviations from the mean are decreasing substantiallywith increasing q.In order to depict a more comprehensive picture, Fig. 10 is createdto analyze the errors both as a function of probe proportions pand q = k/l values. The normalized percent error (or percent coefficientof variation) decreases with increasing proportion of probesas expected. However, the errors at different q values show a complexbehavior as a function of p. For very small p values (less than5%) the relative error gets very large when q approaches 1. This canbe attributed to the increase in the size and variance of the initialqueue. For larger p values, the relative error is smaller for larger qvalues as also indicated in the results of Table 2.Please cite this article in press as: Comert, G., Cetin, M., Queue length estimation from probe vehicle location and the impacts of samplesize, European Journal of Operational Research (2008), doi:10.1016/j.ejor.2008.06.024