Integral Representations and Asymptotic Expansions for Closed Markovian Queueing Networks: Normal Usage

Closed Markovian queueing networks, which are tractable in having 661 the product form (or separability) in their stationary distribution, continue to have a profound influence on computer communication, computer systems analysis, and traffic theory.1"4 The closed networks have been used to model multiple-resource computer systems,2,5 multiprogrammed computer systems, 2 6-8 time-sharing, 2 and window flow control in computer communication networks;9,10 networks with blocking11,12 require the analysis of a large number of closed networks. Not surprisingly, considerable effort has gone into devising efficient procedures for computing the partition function,13"17 an element of the product form solution requiring significant computation. More recently, mainly spurred by parallel technological development in computer communication, there has been a focusing of effort on large closed networks11,15,18"21 with many classes of jobs and transactions and large populations in each class. The point of departure of this effort is the realization that the earlier recursive techniques for computing the partition function are severely limited in terms of computing time, memory storage, and attainable accuracy when it comes to the large networks presently demanding analysis. In an earlier paper, 2 we introduced a new approach to calculating the partition function. We showed there that the partition function could be represented as an integral containing a large parameter which, in some sense, reflected the large size of the network.