Closed Queuing Systems with Exponential Servers

TLDR

The study examines the equilibrium distribution of customers in closed queuing systems with M interconnected stages and a fixed total of N customers. The authors aim to perform an asymptotic analysis of such systems when the number of stages M is large and the effective service rates are comparable. They model each stage with r_i parallel exponential servers of rate μ_i, routing customers to stage j with probability p_ij, and solve the equilibrium equations via separation of variables, extending the analysis to large‑M asymptotics. They show that the equilibrium distribution is governed by the slowest stage as N→∞, provide asymptotic and approximate marginal distributions, and illustrate the results with an example.

Abstract

The results contained herein pertain to the problem of determining the equilibrium distribution of customers in closed queuing systems composed of M interconnected stages of service. The number of customers, N, in a closed queuing system is fixed since customers pass repeatedly through the M stages with neither entrances nor exits permitted. At the ith stage there are r i parallel exponential servers all of which have the same mean service rate μ i . When service is completed at stage i, a customer proceeds directly to stage j with probability p ıj . Such closed systems are shown to be stochastically equivalent to open systems in which the number of customers cannot exceed N. The equilibrium equations for the joint probability distribution of customers are solved by a separation of variables technique. In the limit of N → ∞ it is found that the distribution of customers in the system is regulated by the stage (or stages) with the slowest effective service rate. Asymptotic expressions are given for the marginal distributions of customers in such systems. Then, an asymptotic analysis is carried out for systems with a large number of stages (i.e., M ≫ 1) all of which have comparable effective service rates. Approximate expressions are obtained for the marginal probability distributions. The details of the analysis are illustrated by an example.