The bulk input M^[X] /M/1 queue with working vacations

Abstract

In this paper, we analyze a bulk input M^[X] /M/1 queue with multiple working vacations. A quasi upper triangle transition probability matrix of two-dimensional Markov chain in this model is obtained, and with the matrix analysis method, highly complicated probability generating function(PGF) of the stationary queue length is firstly derived, from which we got the stochastic decomposition result for the stationary queue length which indicates the evident relationship with that of the classical M^[X] /M/1 queue without vacation. It is important that we find the upper and the lower bounds of the stationary waiting time in the Laplace transform order using the properties of the conditional Erlang distribution. Furthermore, we gain the mean queue length and the upper and the lower bounds of the mean waiting time.

Keywords

M^[X] /M/1 queue / bulk input / multiple working vacations / stochastic decomposition / the Laplace transform order

References

[1] Baba Y.. Analysis of a GI/M/1 queue with multiple working vacations. Operations Research Letters, 2005, 33: 201-209.

[2] Banik A.D., Gupta U.C., Pathak S.S.. On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation. Applied Mathematical Modelling, 2007, 31: 1701-1710.

[3] Doshi B.T.. Queueing systems with vacations: a survey. Queueing Systems, 1986, 1(1): 29-66.

[4] Kim, J., Choi, D. & Chae, K. (2003). Analysis of queue-length distribution of the M/G/1 queue with working vacations. In: International Conference on Statistics and Related Fields, Hawaii

[5] Liu W., Xu X., Tian N.. Stochastic decompositions in the M/M/1 queue with working vacations. Operations Research Letters, 2007, 35: 595-600.

[6] Shaked M., Shanthikumar J. G.. Stochastic Orders and Their Applications, 2005, New York: Academic Press, Inc.

[7] Servi L.D., Finn S.G.. M/M/1 queue with working vacations (M/M/1WV). Performance Evaluation, 2002, 50: 41-52.

[8] Sun R., Li J.. The Foundation of Queueing Theory, 2002, Beijing: Sciences Press

[9] Takagi H.. Queueing Analysis: A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems, Part 1, 1991, Amsterdam: North-Holland Elsevier.

[10] Tian N., Zhang Z.G.. Vacation Queueing Models: Theory and Applications, 2006, New York: Springer Publisher.

[11] Wu D., Takagi H.. M/G/1 queue with multiple working vacations. Performance Evaluation, 2006, 63: 654-681.

[12] Xu X., Zhang Z.G.. Analysis of multi-server queue with a single vacation (e,d)-policy. Performance Evaluation, 2006, 63: 625-638.