A generalization of Norton's theorem

A general framework for aggregation and decomposition of product form queueing networks with state dependent routing and servicing is presented. By analogy with electrical circuit theory, the stations are grouped into clusters or subnetworks such that the process decomposes into a global process and a local process. Moreover, the local process factorizes into the subnetworks. The global process and the local processes can be analyzed separately as if they were independent. The global process describes the behaviour of the queueing network in which each cluster is aggregated into a single station, whereas the local process describes the behaviour of the subnetworks as if they are not part of the queueing network. The decomposition and aggregation method formalized in this paper allows us to first analyze the global behaviour of the queueing network and subsequently analyze the local behaviour of the subnetworks of interest or to aggregate clusters into single stations without affecting the behaviour of the rest of the queueing network. Conditions are provided such that: the global equilibrium distribution for aggregated clusters has a product form; this form can be obtained by merely monitoring the global behaviour; the computation of a detailed distribution, including its normalizing constant, can be decomposed into the computation of a global and a local distribution; the marginal distribution for the number of jobs at the stations of a cluster can be obtained by merely solving local behaviour. As a special application, Norton's theorem for queueing networks is extended to queueing networks with state dependent routing such as due to capacity constraints at stations or at clusters of stations and state dependent servicing such as due to service delays for clusters of stations.