DOI QR코드

DOI QR Code

WAITING TIME DISTRIBUTION IN THE M/M/M RETRIAL QUEUE

  • Kim, Jeongsim ;
  • Kim, Jerim
  • Received : 2012.10.22
  • Published : 2013.09.30

Abstract

In this paper, we are concerned with the analysis of the waiting time distribution in the M/M/m retrial queue. We give expressions for the Laplace-Stieltjes transform (LST) of the waiting time distribution and then provide a numerical algorithm for calculating the LST of the waiting time distribution. Numerical inversion of the LSTs is used to calculate the waiting time distribution. Numerical results are presented to illustrate our results.

Keywords

M/M/m retrial queue;waiting time;Laplace-Stieltjes transform;first passage time

References

  1. I. Abate and W. Whitt, The Fourier-series method for inverting transforms of probability distributions, Queueing Systems Theory Appl. 10 (1992), no. 1-2, 5-87. https://doi.org/10.1007/BF01158520
  2. J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top 7 (1999), no. 2, 187-211. https://doi.org/10.1007/BF02564721
  3. J. R. Artalejo, Accessible bibliography on retrial queues, Math. Comput. Model. 30 (1999), 1-6.
  4. J. R. Artalejo, Accessible bibliography on retrial queues: progress in 2000-2009, Math. Comput. Modelling 51 (2010), no. 9-10, 1071-1081. https://doi.org/10.1016/j.mcm.2009.12.011
  5. J. R. Artalejo and A. Gomez-Corral, Waiting time in the M/M/c queue with finite retrial group, Bull. Kerala Math. Assoc. 2 (2005), no. 1, 1-17.
  6. J. R. Artalejo and A. Gomez-Corral, Retrial Queueing Systems, Springer, 2008.
  7. Q. H. Choo and B. Conolly, New results in the theory of repeated orders queueing systems, J. Appl. Probab. 16 (1979), no. 3, 631-640. https://doi.org/10.2307/3213090
  8. G. I. Falin, On the waiting time in a single-channel queueing system with secondary calls, Moscow Univ. Comput. Math. Cybernet. 4 (1977), 83-87.
  9. G. I. Falin, A survey of retrial queues, Queueing Systems Theory Appl. 7 (1990), no. 2, 127-167. https://doi.org/10.1007/BF01158472
  10. G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997.
  11. B. S. Greenberg and R. W. Wolff, An upper bound on the performance of queues with returning customers, J. Appl. Probab. 24 (1987), no. 2, 466-475. https://doi.org/10.2307/3214270
  12. T. Hanschke, A computational procedure for the variance of the waiting time in the M/M/1/1 queue with repeated attempts, In: Operations Research Proceedings, 525-532, Springer-Verlag, Berlin, 1986.
  13. T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, J. Appl. Probab. 24 (1987), no. 2, 486-494. https://doi.org/10.2307/3214272
  14. J. Keilson, J. Cozzolino, and H. Young, A service system with unfilled requests repeated, Oper. Res. 16 (1968), 1126-1137. https://doi.org/10.1287/opre.16.6.1126
  15. J. Kim and B. Kim, Waiting time distribution in an M/PH/1 retrial queue, Performance Evaluation 70 (2013), 286-299. https://doi.org/10.1016/j.peva.2012.12.003
  16. V. G. Kulkarni, Letters to the editor, J. Appl. Probab. 19 (1982), 901-905. https://doi.org/10.2307/3213849
  17. V. G. Kulkarni and H. M. Liang, Retrial queues revisited, In: Frontiers in Queueing:Models and Applications in Science and Engineering (J. H. Dshalalow, ed.), 19-34. CRC Press, Boca Raton, 1997.
  18. M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model, Queueing Systems Theory Appl. 7 (1990), 169-190. https://doi.org/10.1007/BF01158473
  19. R. I. Wilkinson, Theories for toll traffic engineering in the U.S.A., The Bell System Technical Journal 35 (1956), 421-514. https://doi.org/10.1002/j.1538-7305.1956.tb02388.x
  20. T. Yang and J. G. C. Templeton, A survey on retrial queues, Queueing Systems Theory Appl. 2 (1987), no. 3, 201-233. https://doi.org/10.1007/BF01158899

Cited by

  1. Scheduling and performance analysis under a stochastic model for electric vehicle charging stations vol.66, 2017, https://doi.org/10.1016/j.omega.2015.11.010
  2. A survey of retrial queueing systems vol.247, pp.1, 2016, https://doi.org/10.1007/s10479-015-2038-7

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)