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Performance Analysis of a Discrete-time GeoX/G/1 Queue with Single Working Vacation

Authors: Shan Gao, Zaiming Liu


This paper treats a discrete-time batch arrival queue with single working vacation. The main purpose of this paper is to present a performance analysis of this system by using the supplementary variable technique. For this purpose, we first analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. Next, we present the stationary distributions of the system length as well as some performance measures at random epochs by using the supplementary variable method. Thirdly, still based on the supplementary variable method we give the probability generating function (PGF) of the number of customers at the beginning of a busy period and give a stochastic decomposition formulae for the PGF of the stationary system length at the departure epochs. Additionally, we investigate the relation between our discretetime system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.

Keywords: discrete-time queue, supplementary variable technique, batch arrival, working vacation, stochastic decomposition

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[1] Baba Y. Analysis of a GI/M/1 queue with multiple working vacations. Oper.Res. Letters 2005; 33: 201-209.
[2] Chang SH, Choi DW. Performance analysis of a finite-buffer discretetime queue with bulk arrival, bulk service and vacations. Comput. Oper. Res 2005; 32: 2213-2234.
[3] Doshi BT. Queueing systems with vacations - a survey. Queueing Syst 1986; (1): 29-66.
[4] Kim JD, Choi DW, Chae KC. Analysis of queue-length distribution of the M/G/1 queue with working vacations. International Conference on Statistics and Related Fields, Hawaii; 2003.
[5] Li J, Tian N, Liu W. Discrete-time GI/Geo/1 queue with working vacations. Queueing Syst 2007; 56: 53-63.
[6] Li J, Tian N. The discrete-time GI/Geo/1 queue with working vacations and vacation interruption. Appl. Math. Comput 2007; 185: 1-10.
[7] Li J, Tian N, Ma Z. Performance analysis of GI/M/1 queue with working vacations and vacation interruption. Appl. Math. Model 2008; 32: 2715- 2730.
[8] Li J, Tian N, Zhang ZG, Luh H. Analysis of the M/G/1 queue with exponentially working vacations-a matrix analytic approach. Queueing Syst 2009; 61: 139-166.
[9] Li J, Liu W, Tian N. Steady-state analysis of a discrete-time batch arrival queue with working vacations. Perform. Eval 2010; 67: 897-912.
[10] Liu W, Xu X, Tian N. Stochastic decompositions in the M/M/1 queue with working vacations. Oper. Res. Lett 2007; 35: 595-600.
[11] Servi LD, Finn SG. M/M/1 queue with working vacations (M/M/1/WV). Perform. Eval 2002; 50: 41-52.
[12] Shanthikumar J.G. On stochastic decomposition in M/G/1 type queues with generalized server vacations. Operations Research 1988; 36: 566- 569.
[13] Takagi H. Queueing Analysis - A Foundation of Performance Evaluation Vacation and Priority Systems. vol. 1, North-Holland, New York; 1991.
[14] Tian N, Zhang ZG. Vacation Queueing Models: Theory and Applications, Springer-Verlag, New York; 2006.
[15] Tian N, Xu X, Ma Z. Discrete-time queueing theory. Science press, Beijing; 2008.
[16] Wu D., Takagi H. M/G/1 queue with multiple working vacations. Perform. Eval 2006; 63: 654-681.
[17] Yi X.W. et al. The Geo/G/1 queue with disasters and multiple working vacations. Stochastic Models 2007; 23: 537-549.
[18] Zhang M, Hou Z. Performance analysis of M/G/1 queue with working vacations and vacation interruption. J. Comput. Appl. Math 2010; 234(10): 2977-2985.
[19] Zhang ZG, Tian N. Discrete time Geo/G/1 queue with multiple adaptive vacations. Queueing Syst 2001; 38 (4): 419-430.