Large Deviations, Long-Range Dependence, and Queues

Abstract

The objective of this chapter is to demonstrate the development of a large deviations theory for processes with long range dependence. Our starting point is the integral relationship between the standard Brownian motion, B, and the fractional Brownian motion, B H (with H being the Hurst parameter). This integral is known to preserve self-similarity, and produce long-range dependence in B H when H ∈ (1/2, 1). To extend beyond the class of Gaussian processes, we replace the Brownian motion by other processes, in particular, a class of point processes that we call sample-path processes, denoted X. We shall use the integral as a filter that takes as input a process X that has short-range dependence and outputs a process with long-range dependence, denoted Y. This way, we show that a general theory follows for Y to satisfy a moderate deviations principle (MDP), based on on the MDP of X. Applying this approach to queueing systems, we can derive results, in terms of the MDP, regarding the asymptotic behavior of queues fed with the long-range dependent process Y.