On a Discrete-Time GIX/Geo/1/N Queue with Single Working Vacation and Partial Batch Rejection

Authors: Shan Gao

Abstract:

This paper treats a discrete-time finite buffer batch arrival queue with a single working vacation and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrary and geometrically distributed. The queue is analyzed by using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at prearrival, arbitrary and outside observer-s observation epochs. We also present probability generation function (p.g.f.) of actual waiting-time distribution in the system and some performance measures.

Keywords: Discrete-time, finite buffer, single working vacation, batch arrival, partial rejection.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1083219

References:

[1] Baba, Y. Analysis of a GI/M/1 queue with multiple working vacations. Operations Research Letters 33 (2005), 201-209.
[2] Banik, AD, Gupta, UC and Pathak, SS. On the GI/M/1/N queue with multiple working vacations - analytic anaylsis and computation. Applied Mathematical Modelling 31 (2007), 1701-1710.
[3] Chae, K. C., Lim, D.E., Yang, W.S. The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation. Performance Evaluation 66(7)(2009), 356-367 .
[4] Chang, SH, Choi, DW. Performance analysis of a finite-buffer discretetime queue with bulk arrival, bulk service and vacations. Computers & Operations Research 32 (2005), 2213-2234.
[5] Doshi, BT. Queueing systems with vacations - a survey. Queueing Systems (1)(1986), 29-66.
[6] Goswami, V and Vijaya Laxmi, P. Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection. Journal of Industrial and Management Optimization 6(4)(2010), 911-927.
[7] Goswami, V and Mund, GB. Analysis of a discrete-time GI/GEO/1/N queue with multiple working vacations. Journal of Systems Science and Systems Engineering 19(3)(2010), 367-384.
[8] Kim, JD, Choi, DW and 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.
[9] Li, J, Liu, W and Tian, N. Steady-state analysis of a discrete-time batch arrival queue with working vacations. Performance Evaluation, 67 (2010), 897-912.
[10] Li, J and Tian, N. Performance analysis of a GI/M/1 queue with single working vacation. Applied Mathematics and Computation 217(2011), 4960¨C4971 .
[11] Li, J, Tian, N and Liu, W.Discrete-time GI/Geo/1 queue with working vacations. Queueing Systems 56 (2007), 53-63.
[12] Li, J, Tian, N, Zhang, ZG and Luh, H. Analysis of the M/G/1 queue with exponentially working vacations-a matrix analytic approach. Queueing Systems 61 (2009), 139-166.
[13] Liu, W, Xu, X and Tian, N. Stochastic decompositions in the M/M/1 queue with working vacations. Operations Research Letters 35 (2007), 595-600.
[14] Servi, LD and Finn, SG. M/M/1 queue with working vacations (M/M/1/WV). Performance Evaluation 50 (2002), 41-52.
[15] Shanthikumar, JG. On stochastic decomposition in M/G/1 type queues with generalized server vacations. Operations Research 36(1988), 566-569.
[16] Takagi, H. Queueing Analysis - A Foundation of Performance Evaluation Vacation and Priority Systems. vol. 1, North-Holland, New York, 1991.
[17] Tian, N., Zhang, Z.G. Vacation Queueing Models: Theory and Applications, Springer-Verlag, New York , 2006.
[18] Tian, N, Xu, X and Ma, Z. Discrete-time queueing theory. Science press, Beijing, 2008.
[19] Wu, D and Takagi, H. M/G/1 queue with multiple working vacations. Performance Evaluation 63 (2006), 654-681.
[20] Yi, XW et al. The Geo/G/1 queue with disasters and multiple working vacations. Stochastic Models 23 (2007), 537-549.
[21] Yu, MM, Tang, YH and Fu, YH. Steady state analysis and computation of the GI
[x]/Mb/1/L queue with multiple working vacations and partial batch rejection. Computers & Industrial Engineering 56 (2009), 1243-1253.
[22] Yu, MM, Tang, YH , Fu, YH and Pan, LM. GI/Geom/1/N/MWV queue with changeover time and searching for the optimum service rate in working vacation period. Journal of Computational and Applied Mathematics 235 (2011), 2170-2184.
[23] Zhang Z.G., Tian N. Discrete time Geo/G/1 queue with multiple adaptive vacations. Queueing Systems, 38(4)(2001), 419-430 .