ABSTRACT. For a single-server multi-station polling system we focus on the generating function and Laplace-Stieltjes transform of the time-stationary joint queue length and workload distributions, respectively, under no further assumptions on the service discipline. We express these quantities as expressions involving the generating functions of the joint queue length distribution at visit beginnings and visit completions at the various stations. The latter is known for a broad variety of cases. Finally, we identify a workload decomposition result.
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.
Abstract. Let W = {W n : n ∈ N} be a sequence of random vectors in R d , d ≥ 1. This paper considers the logarithmic asymptotics of the extremes of W , that is, for any vector q > 0 in R d , we findWe follow the approach of the restricted large deviation principle introduced in Duffy et al. [7]. That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, 1 vn log P (W n/an ≥ uq) has a, continuous in q, limit J W (q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W , such that the probability law of W n/an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The formula for these asymptotics agrees with the seminal papers on this topic [3,6,7,9].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.