A Viscosity Solution Approach to the Asymptotic Analysis of Queueing Systems

Abstract

We consider a system of several interconnected queues (with a single class of customers) and model the state $(X_t)$ of the system as a jump Markov process. The problem of interest is to estimate the large deviations behavior of the rescaled system $X^\varepsilon_t = \varepsilon X_{t/\varepsilon}$, corresponding to large time and large excursions of the original (unscaled) system. The techniques employed are those of the theory of viscosity solutions to Hamilton-Jacobi equations. From the point of view of large deviation theory, the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty, since an abrupt change in the jump measure occurs. From the point of view of viscosity solutions, the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations. Much of the paper is devoted to proving uniqueness of viscosity solutions for these equations, and these sections are of independent interest. While our use of test functions in proving the uniqueness is an adaptation of the usual technique, the construction of the test functions themselves via the Legendre transform is new. We obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem, which suggests the correct integrand in the large deviation "rate" functional. Since it is the treatment of the effects due to the "boundaries" that is novel, we devote the majority of the paper to the detailed development of a simple two-dimensional system that exhibits all the essential new features. However, the arguments may be applied to queueing systems that are considerably more general, and we attempt to indicate this generality as well.