Traffic model based on synchronous and asynchronous exclusion processes

Abstract

We consider an exclusive process on a closed one‐dimensional lattice. We have used a continuous time version of traffic model based on the deterministic‐stochastic approach developed by A.P. Buslaev. There are cells and particles in the closed contour. Particles can move in both directions. We consider two versions of the system. In the continuous time version, the time interval between attempts of a particle to move is distributed exponentially. The intensity of attempts depends on the particle. In the discrete time version, a particle tries to move in one of direction with a given probability at any discrete moment. The attempt is realized if the particle tries to move to a vacant cell. In the discrete time system, if two particles try to move to the same cell, then only one particle moves. We use equivalence of the continuous time system to a queueing network to find the system stationary state distribution of the system. Stationary probabilities of the discrete time system have been found under the assumptions that the number of particles is equal to two and the number of cells is even. We have proved that, under these assumptions, the system is time reversible, and in the general case, the system is not time reversible.

A Two-Contour System with Two Clusters of Different Lengths

“…In contrast to the case of clusters equal in length, which was considered previously and in which the average velocity is always the same for both clusters, for different lengths of the clusters, depending on the initial state, either the average velocity of the clusters is the same or the average velocity of one cluster is twice as large as that of the other. We note that the dependence of the average velocity of the clusters on the initial state distinguishes the two-contour system with one-directional motion from simple traffic models, which are systems with particles moving on a one-dimensional infinite or closed grid [1][2][3][4][5]9], in which the average velocity of the particles does not depend on the initial state. But the mathematical traffic models [28] which are investigated by simulation modeling and agree with the three-phase traffic flow theory developed by B. S. Kerner, exhibit, as do the contour networks, a dependence of the average velocity of motion on the initial state.…”

“…Analytical results for mathematical traffic models in which particles move according to certain rules on a one-dimensional or two-dimensional grid and which can be interpreted in terms of cellular automata [1] or random processes with prohibitions [2] were obtained, for example, in [3][4][5][6][7][8][9].…”

Invariant measure for continuous open chain of contours with discrete time

“…The article 11 studies a one‐dimensional traffic model with particles of different types. In this model, particles can move forward and backward.…”

“…In this model, particles can move forward and backward. In Reference 11, a version with discrete time and a version with continuous time are considered. The behavior of the version with discrete time is a synchronous exclusive process (in terms of Reference 10).…”

“…The behavior of the version with continuous time is an asynchronous exclusive process. In Reference 11, the discrete time model is interpreted as a discrete time queuing network as in Reference 7, and the continuous time model is interpreted as a continuous time queuing network, Figure 1. Formulas for the average velocity of particles are obtained.…”