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

TL;DR

The paper tackles time-varying FCFS queues with impatient customers whose service and patience times have a general joint distribution, modeled by an inhomogeneous Poisson arrival process. It introduces a novel coupling technique that uses a sequence of LCFS-PR benchmark queues to derive stochastic upper bounds for workload functionals Aλ(g) and Aλ*(g), with g in the class of nonnegative lower semi-continuous functions, under the condition λ(t) ≤ λh. The main contributions are Theorems 1 and 2, which establish stochastic dominance of Aλ(g) and Aλ*(g) by their homogeneous counterparts, plus a supplementary framework (Theorem 3) for periodic queues that yields tractable cycle-based bounds; these are complemented by heavy-tail results (Theorems 4 and 4') and moment bounds (Theorem 5) that link stability, tail behavior, and finite moments of service and patience distributions. The results provide practical tools for bounding performance metrics in nonstationary queueing systems with balking, enabling sharper analysis beyond product-form and stationary assumptions, with applications to tail asymptotics, stability conditions, and regenerative analysis under periodic arrival rates.

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 $λ:[0,\infty)\rightarrow[0,λ_h ]$ for some $λ_h>0$. 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 $τ(x)$ be the first time when the system becomes empty and let $N^*_λ(\cdot)$ 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 $λ(\cdot)$ is periodic.

Figures (1)

• Figure 1: In both drawings: (1) The blue graph is an illustration of the workload process in an $\text{M}/\text{G}(\Psi)/1+\text{H}(\Psi)$ queue with an arrival rate $\lambda_h$. (2) The red graph is an illustration of the workload process in an $\text{M}_t/\text{G}(\Psi)/1+\text{H}(\Psi)$ queue in which the arrival process is a thinning of the arrival process in the $\text{M}/\text{G}(\Psi)/1+\text{H}(\Psi)$ queue described by the blue graph. (3) Each arrival to the $\text{M}/\text{G}(\Psi)/1+\text{H}(\Psi)$ (resp., $\text{M}_t/\text{G}(\Psi)/1+\text{H}(\Psi)$) queue is denoted by a blue (resp., red) dot on the horizontal axis at its occurrence epoch. The upper drawing illustrates that when all customers have infinite patience times, the blue graph will never go below the red graph. The lower drawing illustrates that when the customers have finite patience times (noted by thick black circles), the red graph may be above the blue graph and vice-versa.

Theorems & Definitions (21)

• Definition 1
• Theorem 1
• Theorem 2
• Definition 2
• Remark 1
• Proposition 1
• Theorem 3
• Theorem 4
• Remark 2
• Theorem 5