Large deviation asymptotics of stationary queues

The paper is a survey of main asymptotic results playing an important role in the quality of service estimation (QoS) of stationary systems. The asymptotics of the probability that the workload/queue-size process with heavy tail exceeds an increasing level is considered. Similar results for the systems with Levy input process and light-tailed service time are given. The proofs are based on the methods of large deviations theory and illustrated in detail by the $M/M/1$ system. The asymptotics of the overflow probability within regeneration cycle is considered, including the multiserver systems. An asymptotic analysis of system with the long-range dependent input is discussed, with focus on fractional Brownian process. The ties between the long-range dependence of a queue-size process and the moment properties of the embedded process of the regenerations are discussed.