Externalities in queues as stochastic processes: The case of FCFS M/G/1

TL;DR

This paper analyzes the externalities process in FCFS M/G/1 queues, representing it as an integral of a compound Poisson process, and explores its statistical properties, convergence, and generalizations involving random workload levels.

Contribution

It introduces a novel stochastic process representation of queue externalities and derives its distributional properties, including convergence to Wiener process under certain conditions.

Findings

Externalities process is convex and represented by an integral of a compound Poisson process.

Explicit formulas for the Laplace transform, mean, and auto-covariance of the process.

Conditions for weak convergence to a Wiener process are established.

Abstract

Externalities are the costs that a user of a common resource imposes on others. For example, consider a FCFS M/G/1 queue and a customer with service demand of $x\geq0$ minutes who arrived into the system when the workload level was $v\geq0$ minutes. Let $E_v(x)$ be the total waiting time which could be saved if this customer gave up on his service demand. In this work, we analyse the \textit{externalities process} $E_v(\cdot)=\left\{E_v(x):x\geq0\right\}$. It is shown that this process can be represented by an integral of a (shifted in time by $v$ minutes) compound Poisson process with positive discrete jump distribution, so that $E_v(\cdot)$ is convex. Furthermore, we compute the LST of the finite-dimensional distributions of $E_v(\cdot)$ as well as its mean and auto-covariance functions. We also identify conditions under which, a sequence of normalized externalities processes admits a weak convergence on $\mathcal{D}[0,\infty)$ equipped with the uniform metric to an integral of a (shifted in time by $v$ minutes) standard Wiener process. Finally, we also consider the extended framework when $v$ is a general nonnegative random variable which is independent from the arrival process and the service demands. This leads to a generalization of an existing result from a previous work of Haviv and Ritov (1998).