Approximate Analysis of General Queuing Networks

An approximate iterative technique for the analysis of complex queuing networks with general service times is presented. The technique is based on an application of Norton’s theorem from electrical circuit theory to queuing networks which obey local balance. The technique determines approximations of the queue length and waiting time distributions for each queue in the network. Comparison of results obtained by the approximate method with simulated and exact results shows that the approximate method has reasonable accuracy. introduction Queuing network models are being widely used in the analysis of computer systems and teleprocessing networks [ 1 , 21. Some efficient computational techniques for analyzing complex networks, limited to problems which satisfy local balance, have been described [3] . Efficient methods for the analysis of specijic networks which do not satisfy local balance exist [4]. Approximating a general network by one which satisfies local balance results, usually, in unacceptable error for computer systems analysis. Green and Tang [5] suggest that systems analysis techniques, which require modest computation time and provide results within accuracy limits of 10 to 20 percent, are adequate for the configuration phase of teleprocessing network design; subsequent phases may require more detailed, more accurate and more expensive techniques. The present work is concerned with the configuration phase. This paper presents an approximate, iterative method for determining performance values of closed queuing networks with first come, first served discipline and general service times. The method is a direct application of Norton’s theorem [ 61, and it gives exact solutions for networks which satisfy local balance. The method may be extended to networks with other disciplines and also to several classes of customers. Norton’s theorem for queuing networks Consider a closed queuing network R with M queues indexed 1 , 2 , . . ., M and with N customers (Fig. 1 ). The network is assumed to have only one class of customers. (Extension to several classes of customers is also presented in [ 61 .) The service time for any queue may depend on the state of that queue (for instance, the number of customers in that queue) but is assumed to be independent of the rest of the network. A customer branches to queue j after service in queue i with probability pij independent of the state of the system, i = 1 , 2, . . .. R1 and j = 1, 2 , . . ., M . The network is assumed to satisfy local balance [ 3, 61. For example, a queue may have an exponential service time and a first come, first served discipline or a general service time with a rational Laplace transform and a last come, first served preemptive-resume or a processor-sharing discipline, etc. Essential to Norton’s theorem [ 61 is the exact equivalence of the complement of a queue. For any queue i in the given network, i = 1 , . . ., M , there exists a reduced two-queue network consisting of queue i and its “complementary queue” Bi (Fig. 2 ) such that the equilibrium queue length and wait time distributions of queue i in the reduced network are identical with those of queue i in the given network. We have shown in [6] that the complementary queue Bi can have a first come, first served discipline, an independent exponential service time and a service rate r i (n) that is a function of the number of customers n in queue B,; the rate r i (n) is the conditional rate at which customers arrive at queue i of the given network R , given that there are N n customers in queue i (and n customers in the complementary queue Bi) . The rate ri( n ) is equal to the number of customers served per unit time in queue i of the given network when 1 ) the service time of queue i is identically zero and 2) there are n customers in the network, n = 1 , 2 , . . ., N . The complement of a set of queues can be defined in the same manner. We have shown how Norton’s theorem can be applied in the parametric analysis of queuing networks. It is often required to study the behavior of a large network as 43 JANUARY 1975 APPROXIMATE ANALYSIS OF QUEUING