Mixed Vehicle Flow At Signalized Intersection: Markov Chain Analysis

We assume that a Poisson flow of vehicles arrives at isolated signalized intersection, and each vehicle, independently of others, represents a random number X of passenger car units (PCU’s). We analyze numerically the stationary distribution of the queue process { Zn }, where Zn is the number of PCU’s in a queue at the beginning of the n -th red phase, n → ∞ . We approximate the number Y n of PCU’s arriving during one red-green cycle by a two-parameter Negative Binomial Distribution (NBD). The well-known fact is that { Zn } follow an infinite-state Markov chain. We approximate its stationary distribution using a finite-state Markov chain. We show numerically that there is a strong dependence of the mean queue length E [ Zn ] in equilibrium on the input distribution of Y n and, in particular, on the ”over dispersion” parameter γ = Var [ Y n ] /E [ Y n ]. For Poisson input, γ = 1. γ > 1 indicates presence of heavy-tailed input. In reality it means that a relatively large ”portion” of PCU’s, considerably exceeding the average, may arrive with high probability during one red-green cycle. Empirical formulas are presented for an accurate estimation of mean queue length as a function of load and g of the input flow. Using the Markov chain technique, we analyze the mean ”virtual” delay time for a car which always arrives at the beginning of the red phase.