Work Rates in Closed Queuing Networks with General Independent Servers

Abstract

Closed queuing networks are composed of interconnected service stages serving a fixed number of customers. Each stage consists of a queue and identical parallel servers. The path of a customer’s progress throughout the network is described by a finite Markov chain over the stage names. Service times are assumed to be mutually independent with arbitrary distribution functions. The work rate for a stage, defined as the long-run time-average amount of service time rendered by the stage, is shown to exist with probability one. Explicit expressions relating the work rates of different stages and explicit expressions for the asymptotic work rates as the number of customers becomes large are derived. Work rates are shown to depend continuously on the service-time distributions.