Risk-Sensitive Control for the Parallel Server Model

A Markovian queueing model is considered in which servers of various types work in parallel to process jobs from a number of classes at rates μij that depend on the class, i, and the type, j. The problem of dynamic resource allocation so as to minimize a risk-sensitive criterion is studied in a law-of-large-numbers scaling. Letting Xi(t) denote the number of class-i jobs in the system at time t, the cost is given by E exp{n[ T 0 h(X̄(t))dt + g(X̄(T ))]}, where T > 0, h and g are given functions satisfying regularity and growth conditions, and X̄ = X̄n = n−1X(n·). It is well known in an analogous context of controlled diffusion, and has been shown for some classes of stochastic networks, that the limit behavior, as n → ∞, is governed by a differential game (DG) in which the state dynamics is given by a fluid equation for the formal limit φ of X̄ , while the cost consists of ∫ T 0 h(φ(t))dt + g(φ(T )) and an additional term that originates from the underlying large-deviation rate function. We prove that a DG of this type indeed governs the asymptotic behavior, that the game has value, and that the value can be characterized by the corresponding Hamilton–Jacobi– Isaacs equation. The framework allows for both fixed and a growing number of servers N → ∞, provided N = o(n).