The M^{X}/M/1 Queue with Multiple Working Vacation

Author(s)
Yutaka Baba

Affiliation(s)

Department of Mathematics Education, College of Education and Human Sciences, Yokohama National University, Yokohama, Japan.

Department of Mathematics Education, College of Education and Human Sciences, Yokohama National University, Yokohama, Japan.

ABSTRACT

We study a batch arrival M^{X}/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 M^{X}/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.

We study a batch arrival M

Cite this paper

Y. Baba, "The M^{X}/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.

Y. Baba, "The M

References

[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

[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