Open, Closed, and Mixed Networks of Queues with Different Classes of Customers

Many of the network results of Jackson on arrival and service rate dependencies, of Posner and Bernholtz on different classes of customers, and of Chandy on different types of service centers are combined and extended in this paper.

Abstract

The joint equilibrium distribution of queue sizes in a network of queues containing N service centers and R classes of customers is derived. The equilibrium state probabilities have the general form P(S) = Cd(S) f_1(x_1)f_2(x_2)·f_N(x_N), where S is the state of the system, x, is the configuration of customers at the ith service center, d(S) is a function of the state of the model, f, is a function that depends on the type of the ith service center, and C is a normalizing constant. It is assumed that the equilibrium probabilities exist and are unique. Four types of service centers to model central processors, data channels, terminals, and routing delays are considered. The queueing disciplines associated with these service centers include first-come-first-served, processor sharing,
\nno queueing, and last-come-first-served. Each customer belongs to a single class of customers while awaiting or receiving service at a service center, but may change classes and service centers according to fixed probabilities at the completion of a service request. For open networks, state dependent arrival processes are considered. Closed networks are those with no exogenous arrivals. A network may be closed with respect to some classes of customers and open with respect to other classes of customers. At three of the four types of service centers, the service times of customers are governed by probability distributions having rational Laplace transforms, different classes of customers having different distributions. At first-come-first-served-type service centers, the service time distribution must be identical and exponential for all classes of customers. Examples show how different classes of customers can affect models of computer systems.