Lévy-driven queues

This survey addresses the class of queues with Lévy input, which covers the classical M/G/1 queue and the reflected Brownian motion as special cases. First the stationary behavior is treated, with special attention to the case of the input process having one-sided jumps (i.e., spectrally one-sided Lévy processes). Then various transient metrics are focused on (such as the transient workload distribution, the busy period, and the workload correlation function). Distinguishing between light-tailed and heavy-tailed inputs, we give an account of results on the tail of the workload distribution; in addition we present the main asymptotic results for the various transient quantities. We then extend our basic model to various more advanced queueing systems: queues with a finite buffer, queues in which the current buffer level affects the characteristics of the Lévy input (‘feedback’), and polling type of models. The last part of the survey considers networks of queues: starting with the tandem queue, we subsequently describe the stationary behavior of a general class of Lévy-driven queueing networks. At the methodological level, a variety of techniques has been used, such as transform-based techniques, martingales, rate-conservation arguments, change-of-measure, importance sampling, and large deviations.