Queues with Lévy input and hysteretic control

Abstract

We consider a (doubly) reflected Lévy process where the Lévy exponent is controlled by a hysteretic policy consisting of two stages. In each stage there is typically a different service speed, drift parameter, or arrival rate. We determine the steady-state performance, both for systems with finite and infinite capacity. Thereby, we unify and extend many existing results in the literature, focusing on the special cases of M/G/1 queues and Brownian motion.

Queues with Dropping Functions and General Arrival Processes

“…The queueing models of type (a) and their solutions can be found, for instance, in [1–4], while the models of type (b), in [5–7]. …”

Analysis of an M|G|1|R queue with batch arrivals and two hysteretic overload control policies

“…One can find a comprehensive review of results on the hysteresis control in the work of Dshalalow (1997) and Bekker (2009). The closest to our research model and methods are those developed by Roughan and Pearce (2000), who give numerous references on the problem of the analysis of queueing systems with hysteresis control of the incoming flow intensity (hereinafter-with hysteretic load control).…”

A Lévy input model with additional state-dependent services

“…We now consider an example where it seems more natural to solve the equations developed in Section 3 directly. Consider the case when F(y) = (B − y) + with B being exponentially distributed with intensity β (note that this model reminds the hysteretic control developed in [3,4]). Moreover, we assume that X (t) = ∑ N (t) i=1 σ i − pt is a compound Poisson process with exponentially distributed service times σ i with intensity µ (see also the setup of Example 2).…”

“…For papers that deal with queuing systems driven by Lévy processes, see e.g. [2][3][4]9,10,15,[24][25][26] and references therein.…”