Large deviations and overflow probabilities for the general single-server queue, with applications

Abstract

Abstract We consider queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions ( a t , v t , t ∈ R + ) and a rate function I such that if ( W t , t ∈ R + ) denotes the workload process, then on the continuity set of I . In the case that a t = v t = t it has been argued heuristically, and recently proved in a fairly general context (for discrete time models) by Glynn and Whitt[8], that the queue-length distribution (that is, the distribution of supremum of the workload process Q = sup t ≥0 W t ) decays exponentially: and the decay rate δ is directly related to the rate function I . We establish conditions for a more general result to hold, where the scaling functions are not necessarily linear in t : we find that the queue-length distribution has an exponential tail only if lim t →∞ a t / v t is finite and strictly positive; otherwise, provided our conditions are satisfied, the tail probabilities decay like