A Single-Server Queue with Feedback

TLDR

Customers arrive at a counter following a Poisson process with rate λ, are served by a single server, and after service either rejoin the queue with probability p or depart with probability q, with service times drawn from distribution H(x). The paper aims to determine the stationary queue‑size distribution and related transforms for a single‑server queue with feedback. It derives explicit formulas for the queue‑size distribution, its Laplace‑Stieltjes transform, and the first two moments of the total time spent in the system.

Abstract

Let us suppose that customers arrive at a counter in accordance with a Poisson process of density λ. The customers are served by a single server in order of arrival. The service times are identically distributed, mutually independent, positive random variables with distribution function H(x). Suppose that after being served each customer either immediately joins the queue again with probability p or departs permanently with probability q (p + q =1). In this paper we shell determine for a stationary process the distribution of the queue size as well as the Laplace-Stieltjes transform and the first two moments of the distribution function of the total time spent in the system by a customer.