Advertisement

Analysis of a Batch Service Polling System in a Multi-phase Random Environment

  • Tao Jiang
  • Liwei Liu
  • Yuanyuan Zhu
Article
  • 57 Downloads

Abstract

In this paper, we consider a single-server multi-queue polling system with unlimited-size batch service (so called ‘Israeli queue’) operating in a multi-phase random environment. The polling system consists of a service region and a waiting region, and the external environment evolves through time, i.e., when the external environment is in state i, after a period time, it stays in this state or makes a transition from this state to its adjacent ones. By using matrix analytic method and spectral expansion method, stationary probabilities are derived for computations of performance measures and the conditional waiting times of customers in waiting region. In addition, some numerical examples are presented to show the impact of parameters on performance measures.

Keywords

Israeli queue Multi-phase random environment Matrix analytic method Spectral expansion method 

Mathematics Subject Classification (2010)

60K25 90B22 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors would like to thank the editor and the referees for their valuable comments and helpful suggestions to improve the quality of this paper.

References

  1. Baykal-Gursoy M, Xiao W (2004) Stochastic decomposition in M/M/\(\infty \) queues with Markov modulated service rates. Queueing Syst 48:75–88MathSciNetCrossRefzbMATHGoogle Scholar
  2. Boxma OJ, Van der Wal Y, Yechiali U (2007) Polling with gated batch service. In: Proceedings of the sixth international conference on “analysis of manufacturing systems”. Lunteren, Netherlands, pp 155–159Google Scholar
  3. Boxma OJ, Van der Wal Y, Yechiali U (2008) Polling with batch service. Stoch Model 24(4):604–625MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cordeiro JD, Kharoufeh JP (2012) The unreliable M/M/1 retrial queue in a random environment. Stoch Models 28(1):29–48MathSciNetCrossRefzbMATHGoogle Scholar
  5. D’Auria B (2008) M/M/\(\infty \) queues in semi-Markovian random environment. Queueing Syst 58:221–237MathSciNetCrossRefzbMATHGoogle Scholar
  6. Do TV (2010) An efficient computation algorithm for a multiserver feedback retrial queue with a large queueing capacity. Appl Math Model 34(8):2272–2278CrossRefzbMATHGoogle Scholar
  7. Do TV (2011) Solution for a retrial queueing problem in cellular networks with the Fractional Guard Channel policy. Math Comput Model 53:2059–2066CrossRefzbMATHGoogle Scholar
  8. Falin G (2008) The M/M/1 queue in random environment. Queueing Syst 58:65–76MathSciNetCrossRefzbMATHGoogle Scholar
  9. Grassmann WK (2003) The use of eigenvalues for finding equilibrium probabilities of certain Markovian two-dimensional queueing problems. INFORMS J Comput 15 (4):412–421MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gun L (1989) Experimental results on matrix-analytical solution techniques-extensions and comparisons. Stoch Models 5(4):669–682MathSciNetCrossRefzbMATHGoogle Scholar
  11. Jiang T, Liu L, Li J (2015) Analysis of the M/G/1 queue in multi-phase random environment with disasters. J Math Anal Appl 430:857–873MathSciNetCrossRefzbMATHGoogle Scholar
  12. Krishnamoorthy A, Jaya S, Lakshmy B (2015) Queues with interruption in random environment. Ann Oper Res 233:201–219MathSciNetCrossRefzbMATHGoogle Scholar
  13. Latouche G, Ramaswami V (1999) Introduction to matrix analytic methods in stochastic modeling. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  14. Li QL (2010) Constructive computation in stochastic models with applications: the RG-factorizations. Springer, Berlin and Tsinghua University Press, BeijingGoogle Scholar
  15. Liu Z, Yu S (2016) The M/M/C queueing system in a random environment. J Math Anal Appl 436:556–567MathSciNetCrossRefzbMATHGoogle Scholar
  16. Mitrani I, Chakka R (1995) Spectral expansion solution for a class of Markov models: application and comparison with the matrix-geometric method. Perform Eval 23:241–260CrossRefzbMATHGoogle Scholar
  17. Neuts MF (1981) Matrix-geometric solutions in stochastic models: algorithmic approach. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  18. Perel N, Yechiali U (2013) The Israeli queue with priorities. Stoch Model 29 (3):353–379MathSciNetCrossRefzbMATHGoogle Scholar
  19. Perel N, Yechiali U (2014a) The Israeli queue with infinite number of groups. Probab Eng Inf Sci 28:1–19MathSciNetCrossRefzbMATHGoogle Scholar
  20. Perel N, Yechiali U (2014b) The Israeli Queue with retrials. Queueing Syst 78:31–56MathSciNetCrossRefzbMATHGoogle Scholar
  21. Perel N, Yechiali U (2015) The Israeli Queue with a general group-joining policy. Ann Oper Res. doi: 10.1007/s10479-015-1942-1
  22. Van der Wal Y, Yechiali U (2003) Dynamic visit-order rules for batch-service polling. Probab Eng Inf Sci 17(3):351–367MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.College of Economics and ManagementShandong University of Science and TechnologyQingdaoChina
  2. 2.School of ScienceNanjing University of Science and TechnologyNanjingChina

Personalised recommendations