Finite customer-pool queues

TL;DR

This work analyzes a finite customer-pool M/G/1 queue by deriving exact transform-domain characterizations of its transient behavior. It introduces an efficient recursive scheme to compute the joint generating function $\mu_{km}(z)=\mathbb{E}_{k,m}[z^{Z(T)}]$ at a killing time $T\sim{\rm Exp}(\gamma)$, leveraging an embedded-departure-epoch decomposition and auxiliary probabilities $u_{ni}$ and $v_{ni}$. The authors extend the framework to joint transforms with workload, provide waiting-time LSTs via a Lindley recursion along with finite-mean and heavy-tailed tail results under regularly varying service times, and supply closed-form expressions in key arrival-time regimes. They further derive a generating-function approach for geometric initial counts and present explicit forms in two special cases (pure-death-like arrival schemes and Poisson arrivals stopped after $m$). The results deliver exact, tractable, transform-domain descriptions of transient queue-length and waiting-time behavior for systems with a finite potential customer pool, with connections to i.i.d. arrival-time models and reflected Markov additive processes.

Abstract

In this paper we consider an M/G/1-type queue fed by a finite customer-pool. In terms of transforms, we characterize the time-dependent distribution of the number of customers and the workload, as well as the associated waiting times.

Theorems & Definitions (9)

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