Excluded Volume Effect in Queueing Theory

TL;DR

This paper extends queueing theory by incorporating the excluded volume effect to model pedestrian queues more realistically, deriving exact probability distributions and analyzing how pedestrian flow parameters influence queue dynamics.

Contribution

It introduces the excluded volume effect into queueing theory and provides exact calculations of pedestrian number and waiting time distributions.

Findings

Mean pedestrian number increases with arrival and leaving probabilities.

Mean waiting time exhibits a non-monotonic relationship with service time.

Exact probability distributions for pedestrian number and waiting time are derived.

Abstract

We have introduced excluded volume effect, which is an important factor to model a realistic pedestrian queue, into queueing theory. The probability distributions of pedestrian number and pedestrian waiting time in a queue have been calculated exactly. Due to time needed to close up the queue, the mean number of pedestrians increases as pedestrian arrival probability ($\lambda$) and leaving probability ($\mu$) increase even if the ratio between them (i.e., $\rho=\lambda/\mu$) remains constant. Furthermore, at a given $\rho$, the mean waiting time does not increase monotonically with the service time (which is inverse to $\mu$), a minimum could be reached instead.