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APPROXIMATE ANALYSIS OF M/M/c RETRIAL QUEUE WITH SERVER VACATIONS

  • SHIN, YANG WOO (DEPARTMENT OF STATISTICS, CHANGWON NATIONAL UNIVERSITY) ;
  • MOON, DUG HEE (SCHOOL OF INDUSTRIAL AND NAVAL ARCHITECTURE ENGINEERING, CHANGWON NATIONAL UNIVERSITY)
  • Received : 2015.04.30
  • Accepted : 2015.10.30
  • Published : 2015.12.25

Abstract

We consider the M/M/c/c queues in which the customers blocked to enter the service facility retry after a random amount of time and some of idle servers can leave the vacation. The vacation time and retrial time are assumed to be of phase type distribution. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. We provide an approximation for M/M/c/c queue with general retrial time and general vacation time by approximating the general distribution with phase type distribution. Some numerical results are presented.

Acknowledgement

Grant : 친환경해양플랜트FEED 사업단

References

  1. J. R. Artalejo and A. Gomez-Corral, Retrial Queueing Systems, A Computational Approach, Hidelberg, Springer-Verlag, 2008.
  2. G. I. Falin and J. G. C. Templeton, Retrial Queues, London, Chapman and Hall, 1997.
  3. H. Takagi, Queueing Analysis, Vol. 1. Vacation Systems, Elsevier Science, Amsterdam, 1991.
  4. N. Tian and Z. G. Zhang, Vacation Queuing Models: Theory and Applications, Springer, New York, 2006.
  5. J. R. Artalejo, Analysis of an M/G/1 queue with constant repeated attempts and server vacations, Computers & Operations Research, 24 (1997), 493-504. https://doi.org/10.1016/S0305-0548(96)00076-7
  6. M. Boualem, N. Djellab and D. Aissani, Stochastic inequalities for M/G/1 retrial queues with vacations and constant retrial policy, Mathematical and Computer Modelling, 50 (2009), 207-212. https://doi.org/10.1016/j.mcm.2009.03.009
  7. G. Choudhury and J. C. Ke, A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delyed repair, Applied Mathematical Modelling, 36 (2012), 255-269. https://doi.org/10.1016/j.apm.2011.05.047
  8. J. C. Ke and F. M. Chang, Modified vacation policy for M/G/1 retrial queue with balking and feedback, Computers & Industrial Engineering, 57 (2009), 433-443. https://doi.org/10.1016/j.cie.2009.01.002
  9. B. K. Kummar, R. Rukmani and V. Thangaraj, An M/M/c retrial queueing system with Bernoulli vacations, Journal of Systems Science and Systems Engineering, 18(2) (2009), 222-242. https://doi.org/10.1007/s11518-009-5106-1
  10. G. Choudhury, Steady state analysis of an M/G/1 queue with linear retrial policy and two phase service under Bernoulli vacation schedule, Applied Mathematical Modelling, 32 (2008), 2480-2489. https://doi.org/10.1016/j.apm.2007.09.020
  11. T. Phung-Duc and K. Kawanishi, Multi-server retrial queues with after-call-work, Numerical Algebra, Control and Optimization, 1(4) (2011), 639-656. https://doi.org/10.3934/naco.2011.1.639
  12. Y. W. Shin, Algorithmic approach to Markovian multi-server retrial queue with vacations, Applied Mathematics and Computation, 250 (2015), 287-297. https://doi.org/10.1016/j.amc.2014.10.079
  13. Y.W. Shin and D. H. Moon, Approximation of M/M/c retrial queue with PH-retrial times European Journal of Operational Research, 213 (2011), 205-209. https://doi.org/10.1016/j.ejor.2011.03.024
  14. Y. W. Shin and D. H. Moon, Approximation of PH/PH/c retrial queue with PH-retrial time, Asia-Pacific Journal of Operational Research, 31(2) (2014), 140010 (21 pages).
  15. Y. W. Shin, Ergodicity of M AP/PH/c/K retrial queue with server vacations, Submitted for publication.
  16. X. Xu and Z.G. Zhang, Analysis of multiple-server queue with a single vacation (e, d)-policy, Performance Evaluation, 63 (2006), 825-838. https://doi.org/10.1016/j.peva.2005.09.003
  17. M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach, Baltimore, Johns Hopkins University Press, 1981.
  18. R. W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice-Hall, Inc. Englewood Cliffs, 1989.
  19. H. Tijms, A First Course in Stochastic Models, Wiley, 2003.
  20. Y.W. Shin, Fundamental matrix of transientQBD generator with finite states and level dependent transitions, Asia-Pacific Journal of Operational Research, 26 (2009), 697-714. https://doi.org/10.1142/S0217595909002407
  21. S. Wolfram, Mathematica, 2nd ed. Addison-Wesley, 1991.
  22. W. D. Kelton, R. P. Sadowski and D. A. Sadowski, Simulation with ARENA, 2nd Ed., New York, McGraw-Hill, 1998.
  23. W. Whitt, Approximating a point process by a renewal process, I: two basic methods, Operations Research, 30 (1982), 125-147. https://doi.org/10.1287/opre.30.1.125
  24. A. Bobbio, A. Horvath and M. Telek, Matching three moments with minimal acyclic phase type distributions, Stochastic Models, 21 (2005), 303-326. https://doi.org/10.1081/STM-200056210