Effect of walking-distance on a queuing system of totally asymmetric simple exclusion process equipped with functions of site assignments

TL;DR

This paper models a queueing system with site assignment functions in a totally asymmetric simple exclusion process, analyzing how walking-distance influences site reservation and usage distributions through simulations and approximation models.

Contribution

It introduces a novel queueing model with site assignment functions in TASEP, incorporating walking-distance effects and providing an approximation model validated by simulations.

Findings

Site usage distributions have a unique shape.

Reserved sites increase with bias, following an S-shape curve.

The approximation model accurately describes site usage distributions.

Abstract

This paper proposes a totally asymmetric simple exclusion process on a traveling lane, which is equipped with a queueing system and functions of site assignments along the parking lane. In the proposed system, new particles arrive at the rear of the queue existing at the leftmost site of the system. A particle at the head of the queue selects one of the empty sites in the parking lane and reserves it for stopping at once during its travel. The arriving time and staying time in the parking sites follow half-normal distributions. The random selections of empty sites are controlled by the bias of the exponential function. Our simulation results show the unique shape of site usage distributions. In addition, the number of reserved sites is found to increase with an S-shape curve as the bias to the rightmost site increases. To describe this phenomena, we propose an approximation model, which is derived from the birth-death process and extreme order statistics. A queueing model that takes the effect of distance from the leftmost site of the traveling lane into consideration is further proposed. Our approximation model properly describes the distributions of site usage, and the proposed queueing model shows good agreement with the simulation results.