Characterization of the value function for a differential game formulation of a queueing network optimization problem

Queueing and service networks are used as models in many important application areas, including telecommunication, manufacturing, computer networks and vehicle traffic. The control of such networks is difficult and important, and is one of the major challenges facing modern systems theory. For many problems of this sort it can be difficult to accurately estimate a system model. Service distributions may be complicated and correlated, arrival rates may vary over time, and so on. In such a case one may wish to consider a robust approach to control design. We present some analytic machinery for the characterization and construction of value functions for robust formulations of queueing network optimization problems. These results are also useful for analyzing a number of related problems, including risk-sensitive control of queues and design of ordinary (rather than robust) stabilizing controls. The formulation we use to design robust controls is an analogue of the H^∞ formulation for unconstrained linear and nonlinear systems. We consider a dynamic model whose evolution can be affected by two controls. One control attempts to keep the system in a good operating region (e.g., bounded queue lengths). Such a control may influence the system through service or routing policies. The other control, analogous to the “disturbance” in H^∞ control, determines the value of various system parameters, such as service and arrival rates. This control will attempt to degrade system performance, and the optimization problem is posed as a game