Integral representations and asymptotic expansions for closed Markovian queueing networks: Normal usage

In designing a computer system, it is vitally important to be able to predict the performance of the system. Often, quantities such as throughput, processor utilization, and response time can be predicted from a closed queueing network model. However, until now the computations involved were not feasible for large systems which call for models with many processing centers and many jobs distributed over many classes. We give a radically new approach for handling such large networks — an approach that begins with a representation of the quantities of interest as ratios of integrals. These integrals contain a large parameter reflecting the size of the network. Next, expansions of the integrals in inverse powers of this large parameter are derived. For cases in which the number of processing centers is greater than one, this is the only technique we know of that yields the complete asymptotic expansion. Our method for computing the terms of the expansion can be interpreted as decomposing the original network into a large number of small “pseudonetworks.” Our technique also yields easily computed error bounds when only the first few terms of the expansion are used. This method has been implemented in a software package with which we can analyze systems larger by several orders of magnitude than was previously possible.