Analysis of discrete-time batch service renewal input queue with multiple working vacations V. Goswami a,⇑ , G.B. Mund b a School of Computer Application, KIIT University, Bhubaneswar 751 024, India b School of Computer Engineering, KIIT University, Bhubaneswar 751 024, India article info Article history: Received 5 January 2011 Received in revised form 26 April 2011 Accepted 28 April 2011 Available online 5 May 2011 Keywords: Discrete-time Multiple working vacations Batch service Supplementary variable Waiting-time abstract This paper investigates a discrete-time single server batch service queue with multiple working vacations wherein arrivals occur according to a discrete-time renewal process. The server works with a different service rate rather than completely stopping during the vacation period. The service is performed in batches and the server takes a vacation when the system does not have any waiting customers at a ser- vice completion epoch or a vacation completion epoch. We present a recursive method, using the supple- mentary variable technique to obtain the steady-state queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The displacement operator method is used to solve simulta- neous non-homogeneous difference equations. Some performance measures and waiting-time distribu- tion in the system have also been discussed. Finally, numerical results showing the effect of model parameters on key performance measures are presented. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Discrete-time queueing systems with server vacations have been studied extensively due to their potential applications to a variety of slotted digital computer communication systems. These queueing models are more accurate and efficient to analyze and design digital transmitting systems than their continuous-time counterparts because of their clock-driven operations. Queueing systems with server vacations have been studied extensively in surveys of Doshi (1986), Takagi (1991) and Tian and Zhang (2006). In the study of vacation models, the server is generally assumed to stop service completely during vacation period. However, there are numerous situations where the server remains active during the vacation period and serve the customers at a different service rate. Servi and Finn (2002) introduced the concept of multiple working vacations. They studied an M/M/1 queue with multiple working vacations (M/M/1/WV). Wu and Takagi (2006) generalized Servi and Finn’s (2002) M/M/1/WV queue to an M/G/1/WV queue. Baba (2005) extended this study to a renewal input GI/M/1 queue with working vacations and derived the steady state system length distributions at an arrival and arbitrary epoches. Takagi (1993) has presented a comprehensive analysis of dis- crete-time Geo/G/1 queues with a variety of vacation policies. The Geo/Geo/1 queue with single and multiple working vacations have been discussed in Li and Tian (2008) and Tian et al. (2008) respectively. Alfa (2003) analyzed a series of models with non- exhaustive service in which both the vacation time and service time follow phase type distribution. Tian and Zhang (2002) inves- tigated a GI/Geo/1 queue with multiple vacations. Samanta, Gupta, and Sharma (2007) studied a finite buffer GI/Geo/1/N queue with multiple vacations. Chae, Lim, and Yang (2009) studied the GI/M/ 1 queue and GI/Geo/1 queue with single working vacation (SWV). The discrete time infinite buffer GI/Geo/1 queue with multiple working vacations and vacation interruption have been studied in Li, Tian, and Liu (2007) and Li and Tian (2007) respectively. The discrete-time finite buffer GI/Geo/1 queue with multiple work- ing vacations has been discussed by Goswami and Mund (2010). All the above studies on discrete-time single server queues have been carried out under the assumption that a server serves the cus- tomer one at a time. However, there are many instances where the services are carried out in batches to enhance the performance of the system. Over the last several years the discrete-time single ser- ver batch service queues without vacations have been studied in Gupta and Goswami (2002), Chaudhry and Chang (2004), Alfa and He (2008) and Yi et al. (2007). The continuous time infinite buffer single server batch service queue with multiple vacations has been analyzed by Choi and Han (1994). But, discrete-time queueing systems are better suited than their continuous-time counterparts to evaluate system performance measures in com- puter and digital telecommunication networks. Moreover, the modeling of discrete-time queues is more involved and quite dif- ferent from the analysis used for the corresponding continuous- time queueing models. The advantage of analyzing discrete-time queues is that it provides the results of the corresponding 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.04.018 ⇑ Corresponding author. Tel.: +91 9437306628; fax: +91 6742725113. E-mail addresses: veena_goswami@yahoo.com (V. Goswami), mundgb@yahoo. com (G.B. Mund). Computers & Industrial Engineering 61 (2011) 629–636 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie