A tree convolution algorithm for the solution of queueing networks

TLDR: A new algorithm called the tree convolution algorithm, for the computation of normalization constants and performance measures of product-form queueing networks, is presented and is very efficient in the solution of networks with many service centers and many sparse routing chains.

Abstract: A new algorithm called the tree convolution algorithm, for the computation of normalization constants and performance measures of product-form queueing networks, is presented. Compared to existing algorithms, the algorithm is very efficient in the solution of networks with many service centers and many sparse routing chains. (A network is said to have sparse routing chains if the chains visit, on the average, only a small fraction of all centers in the network.) In such a network, substantial time and space savings can be achieved by exploiting the network's routing information. The time and space reductions are made possible by two features of the algorithm: (1) the sequence of array convolutions to compute a normalization constant is determined according to the traversal of a tree; (2) the convolutions are performed between arrays that are smaller than arrays used by existing algorithms. The routing information of a given network is used to configure the tree to reduce the algorithm's time and space requirements; some effective heuristics for optimization are described. An exact solution of a communication network model with 64 queues and 32 routing chains is illustrated.