Markov network processes with string transitions

Abstract

This study introduces a Markov network process called a string-net. Its state is the vector of quantities of customers or units that move among the nodes, and a transition of the network consists of a string of instantaneous vector increments in the state. The rate of such a string transition is a product of a transition-initiation rate and a string-generation rate. The main result characterizes the stationary distribution of a string-net. Key parameters in this distribution satisfy certain "polynomial traffic equations" involving the string-generation rates. We identify sufficient conditions for the existence of a solution of the polynomial equations, and we relate these equations to a partial balance property and throughputs of the network. Other results describe the stationary behavior of a large class of string-nets in which the vectors in the strings are unit vectors and a string-generation rate is a product of Markov routing probabilities. This class includes recently studied open networks with Jackson-type transitions augmented by transitions in which a signal (or negative customer) deletes units at nodes in one or two stages. The family of string-nets contains essentially all Markov queueing network processes, aside from reversible networks, that have known formulas for their stationary distributions. We discuss old and new variations of Jackson networks with batch services, concurrent or multiple-unit movements of units, state-dependent routings and multiple types of units and routes.