Useful stochastic bounds in time-varying queues with service and patience times having general joint distribution

TL;DR

This paper develops stochastic bounds for workload and customer arrival processes in a time-varying queue with general joint distributions of service and patience times, using a novel coupling technique.

Contribution

It introduces a new coupling method to derive stochastic upper bounds for workload functionals in non-stationary queues with general joint distributions.

Findings

Derived stochastic upper bounds for workload integrals.

Applied bounds to queues with periodic arrival rates.

Demonstrated bounds' usefulness through examples.

Abstract

Consider a first-come, first-served single server queue with an initial workload $x>0$ and customers who arrive according to an inhomogeneous Poisson process with rate function $\lambda:[0,\infty)\rightarrow[0,\lambda_h ]$ for some $\lambda_h>0$. For each $i\in\mathbb{N}$, let $S_i$ (resp., $Y_i$) be the service (resp., patience) time of the $i$'th customer and assume that $(S_1,Y_1),(S_2,Y_2),\ldots$ is an iid sequence of bivariate random vectors with non-negative coordinates. A customer joins if and only if his patience time is not less than his prospective waiting time (i.e., the left-limit of the workload process at his arrival epoch). Let $\tau(x)$ be the first time when the system becomes empty and let $N^*_\lambda(\cdot)$ be the arrival process of those who join the queue. In the present work we suggest a novel coupling technique which is applied to derive stochastic upper bounds for the functionals: \begin{equation*} \int_0^{\tau(x)}g\circ W_x(t){\rm d}t\ \ \text{and}\ \ \int_0^{\tau(x)}g\circ W_x(t){\rm d}N^*_\lambda(t)\,, \end{equation*} where $W_x(\cdot)$ is the workload process in the queue and $g(\cdot)$ is any lower semi-continuous function. We also demonstrate how to utilise these bounds via some examples under the additional assumption that $\lambda(\cdot)$ is periodic.