A Class of Closed Markovian Queuing Networks: Integral Representations, Asymptotic Expansions, and Generalizations

T h e theoretical results on the product form of the stationary distributions of large classes of Markovian queuing networks continue to have a profound influence on computer communications, computer systems analysis, and traffic theory. 1 " 4 These results make at least feasible the analysis and synthesis of the large systems of ever increasing complexity being considered in these areas. The subclass of closed * Presented at ORSA-TIMS meeting, Jan. 5-7, 1981, at Boca Raton, Fla. 599 networks of queues is more difficult to analyze than the open networks because there is no stationary independence of the network nodes. However, the incentive for investigating the closed networks does exist since they have been used to model multiple-resource computer systems,2,5 multiprogrammed computer systems,6"8 time-sharing, 2 and window flow control in computer communication networks;9,10 networks with external inputs subject to blocking require the analysis of a large number of closed networks.11,12 The closed network model that we shall use for illustrative purposes arose in the modeling of a central processor in a node of a computer network. This network is subject to a variety of processing demands. In recognition of the utility of closed networks, considerable research and commercial interest has been directed towards developing efficient procedures for computing the partition function (the normalizing constant), the only element of the product form solution requiring significant computation.13"18 However, as these existing recursive techniques are applied to the problems of particular interest in the Bell System, wherein the constituents of the closed chains are many and the number of chains are many, their shortcomings are observed to be severe in the amount of computing time and memory required and the accuracy attained.