Closed Queuing Systems with Exponential Servers

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.