• Shin, Yang Woo (Department of Statistics Changwon National University)
  • 투고 : 2015.05.08
  • 발행 : 2016.07.31


We consider the MAP/PH/c/K queue in which blocked customers retry to get service and servers may take vacations. The time interval between retrials and vacation times are of phase type (PH) distributions. Using the method of mean drift, a sufficient condition of ergodicity is provided. A condition for the system to be unstable is also given by the stochastic comparison method.


연구 과제 주관 기관 : National Research Foundation of Korea (NRF)


  1. J. R. Artalejo, Analysis of an M/G/1 queue with constant repeated attempts and server vacations, Comput. Oper. Res. 24 (1997), no. 6, 493-504.
  2. J. R. Artalejo and A. Gomez-Corral, Retrial Queueing Systems, A Computational Approach, Hidelberg, Springer-Verlag, 2008.
  3. F. Bacelli and P. Bremaud, Elements of Queueing Theory, Palm Martingale Calculus and Stochastic Recurrences, 2nd ed., Hidelberg, Springer-Verlag, 2003.
  4. L. Breuer, A. Dudin, and V. Klimenok, A Retrial BMAP/PH/N system, Queueing Syst. 40 (2002), no. 4, 433-457.
  5. G. Choudhury, Steady state analysis of an M/G/1 queue with linear retrial policy and two phase service under Bernoulli vacation schedule, Appl. Math. Model. 32 (2008), no. 12, 2480-2489.
  6. G. Choudhury and J. C. Ke, A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair, Appl. Math. Model. 36 (2012), no. 1, 255-269.
  7. J. E. Diamond and A. S. Alfa, Matrix analytic methods for a multi-server retrial queue with buffer, Top 7 (1999), no. 2, 249-266.
  8. G. I. Falin and J. G. C. Templeton, Retrial Queues, London, Chapman, Hall, 1997.
  9. A. Graham, Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Ltd., 1981.
  10. Q. M. He, H. Li, and Y. Q. Zhao, Ergodicity of the BMAP/PH/s/s + K retrial queue with PH-retrial times, Queueing Systems Theory Appl. 35 (2000), no. 1-4, 323-347.
  11. J. C. Ke, C. H. Lin, J. Y. Yang, and Z. G. Zhang, Optimal (d, c)vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs, Appl. Math. Model. 33 (2009), no. 10, 3949-3963.
  12. B. Kim, Stability of a retrial queueing network with different class of customers and restricted resource pooling, J. Ind. Manag. Optim. 7 (2011), no. 3, 753-765.
  13. J. Kim and B. Kim, A survey of retrial queueing systems, Ann. Oper. Res.; DOI 10.1007/s10479-015-2038-7.
  14. B. K. Kummar, R. Rukmani, and V. Thangaraj, An M/M/c retrial queueing system with Bernoulli vacations, J. Syst. Sci. Syst. Eng. 18 (2009), 222-242.
  15. G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modelling, Philadelphia, ASA-SIAM, 1999.
  16. D. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Comm. Statist. Stochastic Models 7 (1991), no. 1, 1-46.
  17. E. Morozov, A multiserver retrial queue: regenerative stability analysis, Queueing Syst. 56 (2007), no. 3-4, 157-168.
  18. M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach, Baltimore, Johns Hopkins University Press, 1981.
  19. Y. W. Shin, Monotonocity properties in various retrial queues and their applications, Queueing Syst. 53 (2006), 147-157.
  20. D. Stoyan, Comparison Methods for Queues and Other Stochastic Models, John Wiley & Sons, New York, 1983.
  21. H. Takagi, Queueing Analysis, Vol. 1. Vacation Systems, Elsevier Science, Amsterdam, 1991.
  22. N. Tian and Z. G. Zhang, Vacation Queuing Models: Theory and Applications, Springer, New York, 2006.
  23. R. L. Tweedie, Sufficient conditions for regularity, recurrence and ergodicity of Markov processes, Math. Proc. Cambridge Philos. Soc. 78 (1975), part 1, 125-136.
  24. X. Xu and Z. G. Zhang, Analysis of multiple-server queue with a single vacation (e, d)-policy, Performance Evaluation 63 (2006), 825-838.

피인용 문헌

  1. Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input pp.1861-9576, 2019,