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Annals of Operations Research

, Volume 274, Issue 1–2, pp 267–290 | Cite as

On a 2-class polling model with reneging and \(k_i\)-limited service

  • Kevin Granville
  • Steve DrekicEmail author
Original Research
  • 59 Downloads

Abstract

This paper analyzes a 2-class, single-server polling model operating under a \(k_i\)-limited service discipline with class-dependent switchover times. Arrivals to each class are assumed to follow a Poisson process with phase-type distributed service times. Within each queue, customers are impatient and renege (i.e., abandon the queue) if the time before entry into service exceeds an exponentially distributed patience time. We model the queueing system as a level-dependent quasi-birth-and-death process, and the steady-state joint queue length distribution as well as the per-class waiting time distributions are computed via the use of matrix analytic techniques. The impacts of reneging and choice of service time distribution are investigated through a series of numerical experiments, with a particular focus on the determination of \((k_1,k_2)\) which minimizes a cost function involving the expected time a customer spends waiting in the queue and an additional penalty cost should reneging take place.

Keywords

Polling model \(k_i\)-limited service discipline Reneging Quasi-birth-and-death process Switchover times Phase-type distribution 

Notes

Acknowledgements

Steve Drekic and Kevin Granville thank the anonymous referee and the editor-in-chief for their supportive comments and helpful suggestions. Steve Drekic and Kevin Granville also acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada through its Discovery Grants program (#RGPIN-2016-03685) and Postgraduate Scholarship-Doctoral program, respectively.

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

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