Robust Feedback Control of a Single Server Queueing System

Abstract

This paper extends previous work of Ball et al. [BDKY] to control of a model of a simple queueing server. There are n queues of customers to be served by a single server who can service only one queue at a time. Each queue is subject to an unknown arrival rate, called a “disturbance” in accord with standard usage from H ∞ theory. An H ∞-type performance criterion is formulated. The resulting control problem has several novel features distinguishing it from the standard smooth case already studied in the control literature: the presence of constraining dynamics on the boundary of the state space to ensure the physical property that queue lengths remain nonnegative, and jump discontinuities in any nonconstant state-feedback law caused by the finiteness of the admissible control set (choice of queue to be served). We arrive at the solution to the appropriate Hamilton–Jacobi equation via an analogue of the stable invariant manifold for the associated Hamiltonian flow (as was done by van der Schaft for the smooth case) and relate this solution to the (lower) value of a restricted differential game, similar to that formulated by Soravia for problems without constraining dynamics. An additional example is included which shows that the projection dynamics used to maintain nonnegativity of the state variables must be handled carefully in more general models involving interactions among the different queues. Primary motivation comes from the application to traffic signal control. Other application areas, such as manufacturing systems and computer networks, are mentioned.