The state-dependent M / G / 1 queue with orbit

Article

Abstract

We consider a state-dependent single-server queue with orbit. This is a versatile model for the study of service systems, where the server needs a non-negligible time to retrieve waiting customers every time he completes a service. This situation arises typically when the customers are not physically present at a system, but they have a remote access to it, as in a call center station, a communication node, etc. We introduce a probabilistic approach for the performance evaluation of this queueing system, that we refer to as the queueing and Markov chain decomposition approach. Moreover, we discuss the applicability of this approach for the performance evaluation of other non-Markovian service systems with state dependencies.

Keywords

State-dependent queueing system Orbit Retrial queue Non-negligible retrieval time Steady-state distribution Conditional sojourn time distributions Variable arrival rate Variable service speed Queueing and Markov chain decomposition 

Mathematics Subject Classification

60K25 90B22 

Notes

Acknowledgements

We thank the associate editor, and two anonymous reviewers for their valuable comments that greatly improved the paper. We thank Professor Mor Harchol-Balter of Carnegie Mellon University for pointing us to the application of our model for the analysis of load sharing algorithms. Professor Baron’s work on this research was supported by a grant from the Natural Science and Engineering Research Council of Canada. Athanasia Manou was supported by AXA Research Fund.

References

  1. 1.
    Abouee-Mehrizi, H., Baron, O.: State-dependent \(M/G/1\) queueing systems. Queueing Syst. 82, 121–148 (2016)CrossRefGoogle Scholar
  2. 2.
    Abouee-Mehrizi, H., Baron, O., Berman, O.: Exact analysis of capacitated two-echelon inventory systems with priorities. Manuf. Serv. Oper. Manag. 16, 561–577 (2014)CrossRefGoogle Scholar
  3. 3.
    Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems. A Computational Approach. Springer, Berlin (2008)CrossRefGoogle Scholar
  4. 4.
    Baron, O., Economou, A., Manou, A.: Equilibrium customer strategies for joining the \(M/G/1\) queue with orbit. Working paper (2017)Google Scholar
  5. 5.
    Bitran, G.R., Tirupati, D.: Multiproduct queueing networks with deterministic routing: decomposition approach and the notion of interference. Manage. Sci. 34, 75–100 (1988)CrossRefGoogle Scholar
  6. 6.
    Caldentey, R.: Approximations for multi-class departure processes. Queueing Syst. 38, 205–212 (2001)CrossRefGoogle Scholar
  7. 7.
    Choi, B.D., Park, K.K., Pearce, C.E.M.: An \(M/M/1\) retrial queue with control policy and general retrial times. Queueing Syst. 14, 275–292 (1993)CrossRefGoogle Scholar
  8. 8.
    Cox, D.R.: A use of complex probabilities in the theory of stochastic processes. Proc. Camb. Philos. Soc. 51, 313–319 (1955)CrossRefGoogle Scholar
  9. 9.
    Doroudi, S., Fralix, B., Harchol-Balter, M.: Clearing analysis of phases: exact limiting probabilities for skip-free, unidirectional, quasi-birth–death processes. Stoch. Syst. 6, 420–458 (2016)CrossRefGoogle Scholar
  10. 10.
    Eager, D.L., Lazowska, E.D., Zahorjan, J.: Adaptive load sharing in homogeneous distributed systems. IEEE Trans. Softw. Eng. 12, 662–675 (1986)CrossRefGoogle Scholar
  11. 11.
    Economou, A., Manou, A. (2015) A probabilistic approach for the analysis of the \(M_n/G/1\) queue. Ann. Oper. Res.  https://doi.org/10.1007/s10479-015-1943-0
  12. 12.
    Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman and Hall, London (1997)CrossRefGoogle Scholar
  13. 13.
    Fayolle, G.: A simple telephone exchange with delayed feedbacks. In: Boxma, O.J., Cohen, J.W., Tijms, H.C. (eds.) Teletraffic Analysis and Computer Performance Evaluation, pp. 245–253. Elsevier, Amsterdam (1986)Google Scholar
  14. 14.
    Gomez-Corral, A.: Stochastic analysis of a single server retrial queue with general retrial times. Naval Res. Logist. 46, 561–581 (1999)CrossRefGoogle Scholar
  15. 15.
    Kerner, Y.: The conditional distribution of the residual service time in the \(M_n/G/1\) queue. Stoch. Models 24, 364–375 (2008)CrossRefGoogle Scholar
  16. 16.
    Lazowska, E.D., Zahorjan, J., Cheriton, D.R., Zwaencpocl, W.: File access performance of diskless workstations. Department of Computer Science, University of Washington, Seattle, Technical Report 84-06-01 (1984)Google Scholar
  17. 17.
    Manou, A., Economou, A., Karaesmen, F.: Strategic customers in a transportation station: when is it optimal to wait? Oper. Res. 62, 910–925 (2014)CrossRefGoogle Scholar
  18. 18.
    Martin, M., Artalejo, J.R.: Analysis of an \(M/G/1\) queue with two types of impatient units. Adv. Appl. Prob. 27, 840–861 (1995)CrossRefGoogle Scholar
  19. 19.
    Oz, B., Adan, I., Haviv, M.: A rate balance principle and its application to queueing models. Queueing Syst. 87, 95–111 (2017)CrossRefGoogle Scholar
  20. 20.
    Oz, B., Adan, I., Haviv, M.: The conditional distribution of the remaining service or vacation time in the \(M_n/G_n/1\) queue with vacations. Preprint (2015)Google Scholar
  21. 21.
    Oz, B.: Strategic behavior in queues. Ph.D. Thesis, The Hebrew University of Jerusalem (2016)Google Scholar
  22. 22.
    Phung-Duc, T., Rogiest, W., Wittevrongel, S.: Single server retrial queues with speed scaling: analysis and performance evaluation. J. Ind. Manag. Optim. (2017).  https://doi.org/10.3934/jimo.2017025 Google Scholar
  23. 23.
    Phung-Duc, T.: Server farms with batch arrival and staggered setup. Proceedings of the 15th Symposium on Information and Communication Technology (SoICT), Hanoi, December 4–5, 2014, pp. 240–247 (2014)Google Scholar
  24. 24.
    Phung-Duc, T.: Exact solutions for \(M/M/c/\text{ Setup }\) queues. Telecommun. Syst. 64, 309–324 (2017a)CrossRefGoogle Scholar
  25. 25.
    Phung-Duc, T.: Single server retrial queues with setup time. J. Ind. Manag. Optim. 13, 1329–1345 (2017b)CrossRefGoogle Scholar
  26. 26.
    Reiser, M., Kobayashi, H.: Accuracy of diffusion approximation for some queueing systems. IBM J. Res. Dev. 18, 110–124 (1974)CrossRefGoogle Scholar
  27. 27.
    van Doorn, E.A., Regterschot, D.J.K.: Conditional PASTA. Oper. Res. Lett. 7, 229–232 (1988)CrossRefGoogle Scholar
  28. 28.
    Wang, J., Baron, O., Scheller-Wolf, A.: \(M/M/c\) queue with two priority classes. Oper. Res. 63, 733–749 (2015)CrossRefGoogle Scholar
  29. 29.
    Yajima, M., Phung-Duc, T.: Batch arrival single-server queue with variable service speed and setup time. Queueing Syst. 86, 241–260 (2017)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Rotman School of ManagementUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  3. 3.Department of Industrial EngineeringKoç UniversityIstanbulTurkey

Personalised recommendations