<abstract xmlns="http://www.w3.org/1999/xhtml">We assume that a Poisson flow of vehicles arrives at isolated signalized intersection, and each vehicle, independently of others, represents a random number <italic>X</italic> of passenger car units (PCU’s). We analyze numerically the stationary distribution of the queue process {<italic>Zn</italic>}, where <italic>Zn</italic> is the number of PCU’s in a queue at the beginning of the <italic>n</italic>-th red phase, <italic>n</italic> → <italic>∞</italic>. We approximate the number <italic>Yn</italic> of PCU’s arriving during one red-green cycle by a two-parameter Negative Binomial Distribution (NBD). The well-known fact is that {<italic>Zn</italic>} 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 <italic>E</italic>[<italic>Zn</italic>] in equilibrium on the input distribution of <italic>Yn</italic> and, in particular, on the ”over dispersion” parameter <italic>γ</italic>= <italic>Var</italic>[<italic>Yn</italic>]<italic>/E</italic>[<italic>Yn</italic>]. For Poisson input, <italic>γ</italic> = 1. <italic>γ ></italic> 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 <italic>g</italic> 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.</abstract>

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 Yn 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 Yn and, in particular, on the ”over dispersion” parameter γ= Var[Yn]/E[Yn]. 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.

Mixed Vehicle Flow At Signalized Intersection: Markov Chain Analysis

Department of Mathematics, Ben Gurion University, Israel

Transport and Telecommunication Journal

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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...

Mixed Vehicle Flow At Signalized Intersection: Markov Chain Analysis

Published Online: Jun 22, 2015

Page range: 190 - 196

DOI: https://doi.org/10.1515/ttj-2015-0017

Keywords
signalized traffic light, Markov chain, average queue length, mixed vehicle input flow, negative binomial distribution, over dispersed input, virtual delay

© Transport and Telecommunication Institute

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mixed Vehicle Flow At Signalized Intersection: Markov Chain Analysis

Published Online: Jun 22, 2015

Page range: 190 - 196

DOI: https://doi.org/10.1515/ttj-2015-0017

Keywordssignalized traffic light, Markov chain, average queue length, mixed vehicle input flow, negative binomial distribution, over dispersed input, virtual delay

© Transport and Telecommunication Institute

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
signalized traffic light, Markov chain, average queue length, mixed vehicle input flow, negative binomial distribution, over dispersed input, virtual delay