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

Closed Markovian networks of queues that have the product form in their stationary probability distributions are useful in the performance evaluation and design of computer and telecommunication systems. Therefore, the efficient computation of the partition function — the key element of the solution in product form — has attracted considerable effort. We present a new and broadly applicable method for calculating the partition function. This method can be applied to very large networks, which were previously computationally intractable. Most of the paper details applications of this approach to a network class which arose in modeling an interactive processor. We show that the partition function and derivatives such as mean values (response times, CPU utilizations, etc.) may be represented by integrals and their ratios. The integrands contain a parameter N which is large for large networks. Next, the classical techniques of asymptotic analysis are applied to derive three main power series expansions in descending powers of N to correspond to normal, high, and very high usage. This work emphasizes multiple terms in the expansions for precision and error analyses.