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Performance analysis of a non-preemptive priority queuing system subjected to a correlated Markovian interruption process

Authors
Journal
Computers & Operations Research
0305-0548
Publisher
Elsevier
Publication Date
Volume
35
Issue
12
Identifiers
DOI: 10.1016/j.cor.2007.06.001
Keywords
  • Service Interruption
  • Server Vacation
  • Priority Queuing System
  • Performance Analysis
  • Transient Analysis
  • Discrete-Time Queues
Disciplines
  • Mathematics

Abstract

Abstract In this paper, we consider a discrete-time queuing system with head-of-line non-preemptive priority scheduling and a single server subjected to server interruptions. We model the server interruptions by a correlated Markovian on/off process with geometrically distributed on and off periods. Two classes of traffic are considered, namely high-priority and low-priority traffic. In the first part of the paper, we derive an expression for the functional equation describing the transient evolution of this priority queuing system. This functional equation is then manipulated and transformed into a mathematical tractable form. This allows us to derive the joint probability generating function (pgf) of the system contents. From this pgf, closed-form expressions for various performance measures, such as mean and variance of system contents and customer delay can be derived. Finally, we illustrate our solution technique with some numerical examples, whereby we demonstrate the negative effect of correlation in the interruption process on the performance of both classes. Some numerical results illustrating the impact of second-order characteristics of the arrival process on mean delays are also presented. The proposed approach which is purely based on pgfs is entirely analytical and enables the derivation of not only steady-state but transient performance measures, as well. The paper presents new insights into the performance analysis of discrete-time queues with service interruption and it also covers some previously published results as a special case. Scope and purpose In this contribution, we consider a practical queuing model, with HOL priority scheduling, two classes of traffic, and a server which is subjected to a correlated Markovian interruption process. We first derive a non-linear functional equation relating the joint pgf of the system state vector between two consecutive slots. Then we outline a solution technique to solve for this functional equation. This allows us to derive the joint pgf of the system contents of both classes, from which various performance measures related to mean system contents and customer delays are derived. We also demonstrate how the proposed approach allows for derivation of transient performance measures, as well. It should be noted that detailed coverage of the transient analysis of the system is beyond the scope of this paper. To our best knowledge, this is the first initiative that aims to explore the performance of queuing systems with priority scheduling when the shared server is subjected to service interruption. The paper also generalizes the results of Walraevens et al. (Analysis of a single-server ATM queue with priority scheduling, Computers & Operations Research 2003; 30(12):1807–30) by incorporating service interruption into their original queuing model. By means of numerical results, the paper also demonstrates the effect of correlation in the service interruption process on the performance of both classes of customers. The impact of second-order characteristics of the arrival process on mean delays is also investigated.

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