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

O.J. Boxma (Corresponding author), M. Saxena, A. J.E.M. Janssen

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

In this paper, we study a single-server polling model with two queues. Customers arrive at the queues according to two independent Poisson processes. The server spends random amounts of time in each queue, regardless of the amounts of work present at the queues. The service speed is not constant; it is assumed that the server works at speed (Formula presented.) at queue i when its current workload equals xi, i = 1, 2. We first focus on the case that all visit times are constant. In the two-queue case, we then compute the Laplace–Stieltjes transform (LST) of the steady-state joint workload distribution. Using a different method in which we exploit the independence of the workloads at visit endings, we compute the joint LST of workloads in the case of an arbitrary number of queues with constant visit times. Next, we consider a two-queue polling model with constant visit times at the first queue and general visit times at the second, and we derive the marginal workload distribution at the first queue. We also investigate the case of a two-queue polling model with exponentially distributed visit times. We determine the steady-state marginal workload distributions, and we formulate a two-dimensional Volterra integral equation for the LST of the steady-state joint workload distribution. We finally show that this equation can be solved by a fixed-point iteration.

Original languageEnglish
JournalStochastic Models
VolumeXX
Issue numberXX
DOIs
Publication statusAccepted/In press - 2020

Keywords

  • Polling model
  • time-limited
  • workload
  • workload-dependent service speed

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