Queues with Delays in Two-State Strategies and Lévy Input

Abstract

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. D time units later the timer expires, and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires, then the Lévy exponent does not change.Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

Old and New Examples of Scale Functions for Spectrally Negative Lévy Processes

“…Moreover with the advent of these new fluctuation identities and a better understanding of the analytical properties of the function W (q) came the possibility of revisiting and solving a number of classical and modern problems from applied probability, but now with the underlying source of randomness being a general spectrally negative Lévy processes. For example, in the theory of optimal stopping Avram et al (2002Avram et al ( , 2004; Alili and Kyprianou (2005) and Kyprianou (2006), in the theory of optimal control Avram et al (2007) and Loeffen (2007), in the theory of queuing and storage models Dube et al (2004) and Bekker et al (2008), in the theory of branching processes Bingham (1976) and Lambert (2007) Kyprianou and Surya (2007) and in the theory of fragmentation Krell (2007).…”

Queues with Lévy input and hysteretic control

“…The joint transform of the first-exit time from the interval [0, a) of the reflected process and the exit position of the reflected process can be found in [4,27]. In [7], this transform is rewritten in the case that only the exit position is required. In Corollary 2.1, we restate the latter result in terms of Lévy processes with only positive jumps.…”

A Lévy input fluid queue with input and workload regulation

“…Next, Kella et al [14] considered a model with workload removal, which fits into our model by taking F n (x) = 0 and by letting the spectrally positive Lévy process X 0 = X 1 = X be a Brownian motion superposed with an independent compound Poisson component. Our work is also related to papers [3,4], who considered the adaptation of the Laplace exponent of the input process at the moments when the workload process crosses level K. Bekker et al [4] analysed also the adaptation of the process at Poisson instants but without the possibility of additional regulation at these moments and without taking into account the supremum of the workload process between exponential timers.…”

“…Additionally, we consider the case F (1) (u, j) = δu (TCP). Further examples can be found in the literature, in particular in [3,4]. Note that in some of the models considered in those papers, the workload is adapted instantaneously when a high level is reached, rather than after an exponential time.…”