The MX/M/1 Queue with Multiple Working Vacation

DOI: 10.4236/ajor.2012.22025   PDF   HTML   XML   5,587 Downloads   10,205 Views   Citations


We study a batch arrival MX/M/1 queue with multiple working vacation. The server serves customers at a lower rate rather than completely stopping service during the service period. Using a quasi upper triangular transition probability matrix of two-dimensional Markov chain and matrix analytic method, the probability generating function (PGF) of the stationary system length distribution is obtained, from which we obtain the stochastic decomposition structure of system length which indicates the relationship with that of the MX/M/1 queue without vacation. Some performance indices are derived by using the PGF of the stationary system length distribution. It is important that we obtain the Laplace Stieltjes transform (LST) of the stationary waiting time distribution. Further, we obtain the mean system length and the mean waiting time. Finally, numerical results for some special cases are presented to show the effects of system parameters.

Share and Cite:

Y. Baba, "The MX/M/1 Queue with Multiple Working Vacation," American Journal of Operations Research, Vol. 2 No. 2, 2012, pp. 217-224. doi: 10.4236/ajor.2012.22025.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. Takagi, “Queueing Analysis: A Foundation of Performance Evaluation, Vol. 1: Vacation and Priority Systems, Part 1,” Elsevier Science Publishers, Amsterdam, 1991.
[2] N. Tianand and G. Zhang, “Vacation Queueing Models- Theory and Applications,” Springer-Verlag, New York, 2006.
[3] B. Doshi, “Queueing Systems with Vacations—A Survey,” Queueing Systems, Vol. 1, No. 1, 1986, pp. 29-66. doi:10.1007/BF01149327
[4] L. Servi and S. Finn, “M/M/1 Queue with with Working Vacations (M/M/1/WV),” Performance Evaluation, Vol. 50, No. 1, 2002, pp. 41-52. doi:10.1016/S0166-5316(02)00057-3
[5] D. Wuand and H. Takagi, “M/G/1 Queue with Multiple Working Vacations,” Performance Evaluation, Vol. 64, 2006, pp. 654-681.
[6] Y. Baba, “Analysis of a GI/M/1 Queue with Multiple Working Vacations,” Operations Research Letters, Vol. 33, No. 2, 2005, pp. 201-209. doi:10.1016/j.orl.2004.05.006
[7] A. Banik, U. Gupta and S. Pathak, “On the GI/M/1/N Queue with Multiple Working Vacations-Analytic Analysis and Computation,” Applied Mathematical Modelling, Vol. 31, No. 9, 2007, pp. 1701-1710. doi:10.1016/j.apm.2006.05.010
[8] W. Liu, X. Xu and N. Tian, “Some Results on the M/M/1 Queue with Working Vacations,” Operations Research Letters, Vol. 35, No. 5, 2007, pp. 595-600. doi:10.1016/j.orl.2006.12.007
[9] J. Li, N. Tian, Z. G. Zhang and H. P. Lu, “Analysis of the M/G/1 Queue with Exponential Working Vacations-A Matrix Analytic Approach,” Queueing Systems, Vol. 61, No. 2-3, 2009, pp. 139-166. doi:10.1007/s11134-008-9103-8
[10] X. Xu, Z. Zhang and N. Tian, “Analysis for the M[X]/M/1 Working Vacation Queue,” International Journal of Information and Management Sciences, Vol. 20, 2009, pp. 379-394.
[11] M. F. Neuts, “Structured Stochastic Matrices of M/G/1 Type and Their Applications,” MarcelDekker Inc., New York, 1989.
[12] P. J. Burke, “Delay in Single-Server Queues with Batch Arrivals,” Operations Research, Vol. 23, No. 4, 1975, pp. 830-833. doi:10.1287/opre.23.4.830

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.