Lévy-driven Polling Systems and Continuous-State Branching Processes

Abstract

In this paper we consider a ring of N ≥ 1 queues served by a single server in a cyclic order. After having served a queue (according to a service discipline that may vary from queue to queue), there is a switch-over period and then the server serves the next queue and so forth. This model is known in the literature as a polling model.Each of the queues is fed by a non-decreasing Lévy process, which can be different during each of the consecutive periods within the server's cycle. The N -dimensional Lévy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams. For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch.Our analysis heavily relies on establishing a link between fluid (Lévy input) polling systems and multi-type Jiřina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the branching property for a discipline, which can be traced back to Fuhrmann and Resing. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated.1. Introduction. Consider a queueing model consisting of multiple queues attended by a single server, visiting the queues one at a time in a cyclic order. Moving from one queue to another, the server incurs a nonnegligible switch-over time. Such single-server multiple-queue models are commonly referred to as polling models. Stimulated by a wide variety of applications, polling models have been extensively studied in the literature, see [28,30,31] for a series of comprehensive surveys and [20,29] for extensive overviews of the applicability of polling models.

Two queues with time-limited polling and workload-dependent service speeds

“…However, if the service discipline in at least one queue is not of branching type, then the joint queue length distribution is almost never known; see Section 5 of Borst and Boxma [7] for a discussion of a few rare exceptions to this rule. When it comes to joint workloads, a very similar sharp division holds; now multi-type Jirina processes (continuousstate discrete-time branching processes) give rise to explicit expressions for (the transform of) the steady-state joint workload distribution [9] . We shall show that polling systems with workload-proportional service speed are quite tractable, in particular if the visit times are constant.…”

Polling Systems and Their Application to Telecommunication Networks

“…Boxma et al [109] consider a cyclic polling system with a mixed service discipline in which queue load (the amount of work that needs to be processed by the server) is controlled by a positively incremented N-dimensional Levy process. Such a process means the dependence of incoming flows in the queue.…”

Workload analysis of a two-queue fluid polling model

“…In contrast with the extensive literature on fluid queues and polling, there are very few studies focusing on polling systems with fluid input. Some exceptions are Czerniak and Yechiali [22], Boxma et al [13], Remerova et al [38], and Adan et al [3]; see also [12, Section 6]. A recent heavy-traffic analysis of a fluid model with two queues in series is Koops et al [36].…”