M/M/∞ Transience: Tail Asymptotics of Congestion Periods

Abstract

The c-congestion period, defined as a time interval in which the number of customers is larger than c all the time, is a key quantity in the design of communication networks. Particularly in the setting of M/M/∞ systems, the analysis of the duration of the congestion period, D c, has attracted substantial attention; related quantities have been studied as well, such as the total area A c above c, and the number of arrived customers N c during a congestion period. Laplace transforms of these three random variables being known, as well as explicit formulae for their moments, this article addresses the corresponding tail asymptotics. Our work addresses the following topics. In the so-called many-flows scaling, we show that the tail asymptotics are essentially exponential in the scaling parameter. The proof techniques stem from large-deviations theory; we also identify the most likely way in which the event under consideration occurs. In the same scaling, we approximate the model by its Gaussian counterpart. Specializing to our specific model, we show that the (fairly abstract) sample-path large-deviations theorem for Gaussian processes, viz. the generalized version of Schilder's theorem, can be written in a considerably more explicit way. Relying on this result, we derive the tail asymptotics for the Gaussian counterpart. Then we use change-of-measure arguments to find upper bounds, uniform in the model parameters, on the probabilities of interest. These change-of-measures are applied to devise a number of important sampling schemes, for fast simulation of rare-event probabilities. They turn out to yield a substantial speed-up in simulation effort, compared to naïve, direct simulations.