The joint equihbrmm distribution of queue sizes in a network of queues containing N service centers and R classes of customers m derived The equilibrium state probabillUes have the general form P(S) = Cd(S) fl(xl)f2(x2) . fN(x~), where S is the state of the system, x, is the con- figuration of customers at the ~th service center, d(S) is a function of the state of the model, f, is a function that depends on the type of the zth service center, and C is a normalizing constant It is assumed that the eqmhbrlum probabfl~tles exmt and are unique Four types of service centers to model central processors, data channels, terminals, and routing delays are considered The queuemg dlSclphnes associated with these service centers include first-come-first -served, processor sharing, no queueing, and last-come-first- served Each customer belongs to a single class of customers while awaiting or receiving serwce at a service center, but may change classes and service centers according to fixed probabditms 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 serwce centers, the service times of customers are governed by probablhty dmtrlbutions hawng ratmnM Laplace transforms, different classes of customers hawng different distributions At first-come-first -served-type service centers, the service time distribution must be identical and exponentml for all classes of customers. Examples show how different classes of customers can affect models of computer systems. KEY woads AND PHRASES" networks of queues, theory of queues, queuemg theory, multiprogram- ming, time-sharing, processor sharing, Markov processes