Stochastic Analysis of an M/G/1 Retrial Queue with FCFS

  • Mohamed Boualem
  • Mouloud Cherfaoui
  • Natalia Djellab
  • Djamil Aïssani
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)

Abstract

The main goal of this paper is to investigate stochastic analysis of a single server retrial queue with a First-Come-First-Served (FCFS) orbit and non-exponential retrial times using the monotonicity and comparability methods. We establish various results for the comparison and monotonicity of the underlying embedded Markov chain when the parameters vary. Moreover, we prove stochastic inequalities for the stationary distribution and some simple bounds for the mean characteristics of the system. We validate stochastic comparison method by presenting some numerical results illustrating the interest of the approach.

References

  1. 1.
    Artalejo, J.R., Gómez-Corral, A.: Channel idle periods in computer and telecommunication systems with customer retrials. Telecommun. Syst. 24, 29–46 (2003)CrossRefGoogle Scholar
  2. 2.
    Artalejo, J.R., Gómez-Corral, A.: Retrial Queueing Systems: A Computational Approach. Springer, Berlin (2008)CrossRefGoogle Scholar
  3. 3.
    Artalejo, J.R., López-Herrero, M. J.: Cellular mobile networks with repeated calls operating in random environment. Comput. Oper. Res. 37, 1158–1166 (2010)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Artalejo, J.R., Pla, V.: On the impact of customer balking, impatience and retrials in telecommunication systems. Comput. Math. Appl. 57, 217–229 (2009)MATHCrossRefGoogle Scholar
  5. 5.
    Boualem, M.: Insensitive bounds for the stationary distribution of a single server retrial queue with server subject to active breakdowns. Adv. Oper. Res. 2014, Article ID 985453, 12 pp. (2014)Google Scholar
  6. 6.
    Boualem, M., Djellab, N., Aïssani, D.: Stochastic inequalities for \(M/G/1\) retrial queues with vacations and constant retrial policy. Math. Comput. Model. 50, 207–212 (2009)MATHCrossRefGoogle Scholar
  7. 7.
    Boualem, M., Djellab, N., Aïssani, D.: Stochastic approximations and monotonicity of a single server feedback retrial queue. Math. Probl. Eng. 2012, Article ID 536982, 13 pp. (2012)Google Scholar
  8. 8.
    Boualem, M., Djellab, N., Aïssani, D.: Stochastic bounds for a single server queue with general retrial times. Bull. Iranian Math. Soc. 40, 183–198 (2014)MathSciNetGoogle Scholar
  9. 9.
    Bušić, A., Fourneau, J.M.: Monotonicity and performance evaluation: applications to high speed and mobile networks. Cluster Comput. 15, 401–414 (2012)CrossRefGoogle Scholar
  10. 10.
    Economou, A., López-Herrero, M.J.: Performance analysis of a cellular mobile network with retrials and guard channels using waiting and first passage time measures. Eur. Trans. Telecommun. 20, 389–401 (2009)CrossRefGoogle Scholar
  11. 11.
    Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman and Hall, London (1997)MATHCrossRefGoogle Scholar
  12. 12.
    Gómez-Corral, A.: Stochastic analysis of a single server retrial queue with the general retrial times. Nav. Res. Logist. 46, 561–581 (1999)MATHCrossRefGoogle Scholar
  13. 13.
    Houck D.J., Lai, W.S.: Traffic modeling and analysis of hybrid fiber-coax systems. Comput. Netw. ISDN Syst. 30, 821–834 (1998)CrossRefGoogle Scholar
  14. 14.
    Janssens, G.K.: The quasi-random input queueing system with repeated attempts as a model for a collision-avoidance local area network. IEEE Trans. Commun. 45, 360–364 (1997)CrossRefGoogle Scholar
  15. 15.
    Liu, X., Fapojuwo, A.O.: Performance analysis of hierarchical cellular networks with queueing and user retrials. Int. J. Commun. Syst. 19, 699–721 (2006)CrossRefGoogle Scholar
  16. 16.
    Machihara, F., Saitoh, M.: Mobile customers model with retrials. Eur. J. Oper. Res. 189, 1073–1087 (2008)MATHCrossRefGoogle Scholar
  17. 17.
    Mokdad, L., Castel-Taleb, H.: Stochastic comparisons: a methodology for the performance evaluation of fixed and mobile networks. Comput. Commun. 31, 3894–3904 (2008)CrossRefGoogle Scholar
  18. 18.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risk. Wiley, New York (2002)Google Scholar
  19. 19.
    Pustova, S.V.: Investigation of call centers as retrial queuing systems. Cybern. Syst. Anal. 46, 494–499 (2010)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ross, S.M.: Stochastic Processes. Wiley, New York (1983)MATHGoogle Scholar
  21. 21.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders and their Applications. Academic, San Diego (1994)MATHGoogle Scholar
  22. 22.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Academic, New York (2006)Google Scholar
  23. 23.
    Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, New York (1983)MATHGoogle Scholar
  24. 24.
    Van Do, T.: A new computational algorithm for retrial queues to cellular mobile systems with guard channels. Comput. Ind. Eng. 59, 865–872 (2010)CrossRefGoogle Scholar
  25. 25.
    Wüchner, P., Sztrik, J., de Meer, H.: Modeling wireless sensor networks using finite-source retrial queues with unreliable orbit. In: Hummel, K.A. et al. (eds.) PERFORM 2010 (Haring Festschrift), LNCS 6821, pp. 73–86 (2011)Google Scholar
  26. 26.
    Xue, F., Yoo, S.J.B., Yokoyama, H., Horiuchi, Y.: Performance analysis of wavelength-routed optical networks with connection request retrials. In: IEEE International Conference on Communication ICC 2005, vol. 3, pp. 1813–1818. IEEE, New York (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mohamed Boualem
    • 1
  • Mouloud Cherfaoui
    • 2
    • 3
  • Natalia Djellab
    • 4
  • Djamil Aïssani
    • 5
  1. 1.Research Unit LaMOS (Modeling and Optimization of Systems), Faculty of TechnologyUniversity of BejaiaBejaiaAlgeria
  2. 2.Research Unit LaMOS (Modeling and Optimization of Systems)University of BejaiaBejaiaAlgeria
  3. 3.Department of MathematicsUniversity of BiskraBiskraAlgeria
  4. 4.Laboratory LaPS, Department of MathematicsUniversity of AnnabaAnnabaAlgeria
  5. 5.Research Unit LaMOS (Modeling and Optimization of Systems), Faculty of Exact SciencesUniversity of BejaiaBejaiaAlgeria

Personalised recommendations