Routing and Singular Control for Queueing Networks in Heavy Traffic

Abstract

The problem of routing control in an open queueing network under conditions of heavy traffic and finite (scaled) buffers is dealt with. The operating statistics can be state dependent. The sequence of scaled controlled state processes converges to a singularly controlled reflected diffusion (with the associated costs), under broad conditions. Due to the nature of the controls, a “scaling” method is introduced to obtain the convergence, since the actual sequence of processes does not necessarily converge in the Skorokhod topology. Owing to finite buffers, an extension of the reflection mapping needs to be obtained. The optimal value functions for the physical processes converge to the optimal value function of the limit process, under broad conditions. Approximations to the optimal control for the limit process are obtained, as well as properties of the sequence of physical processes. The optimal or controlled (but not necessarily optimal) limit process can be used to approximate a large variety of functionals of the optimal or controlled (but not necessarily optimal) physical processes.

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