Large Deviation Bounds for Single Class Queueing Networks and Their Calculation

Abstract

We investigate large deviations for the behavior of single class queueing networks. The starting point is a sample large deviation principle on the path-space of network primitives describing the cumulative external arrivals, service time requirements and routing decisions. The behavior of the network, capturing the cumulative total arrivals, idle times and queue lengths, is characterized by a path-space fixed point equation containing the network primitives. The mapping from the network primitives to the set of fixed points is partially upper semicontinuous. This set-valued continuity allows us to derive large deviation bounds for the network behavior in the form of variational problems. The analysis is carried out on the doubly-infinite time axis R and can directly capture stationary and non-Markovian situations. By relaxing the fixed point equation the upper bounds and minimizing paths can be approximated with piecewise linear paths. For a class of typical rate functions we specify sequences of finite dimensional minimization problems which permit the calculation of large deviation rates and minimizing paths for the tail probabilities of queue lengths. We illustrate the approach with an example.