Asymptotic analysis of product-form distributions related to large interconnection networks

AbstractA large interconnection network is modeled as a closed queueing network. The discussion uses the terminology of a multiprocessor system with partially available buses, connecting several groups of memory modules, but the same model applies to other engineering systems, such as circuit-switching connections. A particular feature of the model is that it accommodates non-uniform addressing patterns (asymmetric switch), and that the analysis handles bottlenecks well. Exact expressions for the standard performance measures — processing power and memory bandwidth — are presented, using the system partition function, with recursion relations for their computation.For large systems, we derive simple approximations of the performance measures, in terms of asymptotic estimates of the partitions functions and the steady-state probability distribution functions. A new approach is used to estimate the key quantity in the approximation: the relative weights of the system macro-states, expressed in terms of an equivalent queueing network, where each memory group is replaced by a single-server node with variable service rate. The approach is based on an integral representation of these weights in terms of the Cauchy integrals of their generating functions, and the estimation of these integrals by the saddle-point method. Numerical examples exhibit the dependence of the quality of the estimates on the system parameters