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

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 $\lambda :[0,\infty )\to [0,{\lambda }_h]$ 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 $(S_1,Y_1),(S_2,Y_2),\dots$ 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 $N_{\lambda }^{\ast }(\cdot )$ 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^{τ(x)}g\circ W_x(t){\rm d}t\ \ \text{and}\ \ \int_0^{τ(x)}g\circ W_x(t){\rm d}N^*_λ(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.