Performance Characteristics of Discrete-Time Queue With Variant Working Vacations

Performance Characteristics of Discrete-Time Queue With Variant Working Vacations

P. Vijaya Laxmi (Department of Applied Mathematics, Andhra University, Visakhapatnam, India) and Rajesh P. (Department of Mathematics, Vignan's Institute of Information Technology, Visakhapatnam, India)
DOI: 10.4018/IJORIS.2020040101

Abstract

This article analyzes an infinite buffer discrete-time single server queueing system with variant working vacations in which customers arrive according to a geometric process. As soon as the system becomes empty, the server takes working vacations. The server will take a maximum number K of working vacations until either he finds at least on customer in the queue or the server has exhaustively taken all the vacations. The service times during regular busy period, working vacation period and vacation times are assumed to be geometrically distributed. The probability generating function of the steady-state probabilities and the closed form expressions of the system size when the server is in different states have been derived. In addition, some other performance measures, their monotonicity with respect to K and a cost model are presented to determine the optimal service rate during working vacation.
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Introduction

Unlike the classical vacation queues where the server suspends the services temporarily, in working vacation (WV) queues the server is active during the vacation period which is called working vacation (WV), Servi and Finn (2002). Wu and Takagi (2006) generalized Servi and Finn's (2002)M/M/1/WV queue to an M/G/1/WV queue. Banik et al. (2007) studied a general input GI/M/1/N/WV queue. The stochastic decomposition results of an M/M/1 queue with WV have been derived by Liu et al. (2007) and the corresponding M/G/1 queue was studied by Li et al. (2009).

The analysis of discrete time queueing models has received considerable attention in view of their application in practical problems that arise from communication and computer systems including time-division multiple access (TDMA) schemes, asynchronous transfer mode (ATM), multiplexers in the broadband integrated services digital network (B-ISDN), management in service system and electronic commerce, etc. Past work on discrete-time queues is found in Meisling (1958), Hunter (1983), Takagi (1993). Tian and Zhang (2002), Alfa (2003). Li and Tian (2007) considered a discrete-time GI/Geo/1 queue with WV and vacation interruption. Li et al. (2007) considered a discrete-time GI/Geo/1 queue with MWV under Early Arrival System (EAS) and Late Arrival System (LAS) schemes. Li and Tian (2008) and Tian et al. (2008) have analyzed a Geo/Geo/1 queue with single working vacation (SWV) and multiple working vacation (MWV), respectively. A discrete-time renewal input finite buffer batch service queue with MWVs has been studied by Vijaya Laxmi and Jyothsna (2014) using the supplementary variable techniqeue and the corresponding queue with balking and SWV has been presented by Vijaya Laxmi et al. (2015). Recently, a retrial queue with working vacation for the batch arrival IJORIS.2020040101.m01 queue has been analyzed by Upadhyaya (2015) under EAS scheme.

In variant working vacation (VWV) policy, unlike the SWV or MWVs, a fixed number of consecutive vacations, say K, are taken by the server if the system remains empty at the end of previous vacation termination epoch. This kind of vacation schedule is investigated by Zhang and Tian (2001) for the Geo/G/1 queue with multiple adaptive vacations. Banik (2009) studied the infinite-buffer single server queue with variant of multiple vacation policy and batch Markovian arrival process by using matrix analytic method. For more literature on this work, see Ke and Chang (2009), Ke et al. (2010) and Wang et al. (2011). Zhang and Hou (2011) analyzed a steady-state GI/M/1/N queue with a variant multiple working vacation (VWV) by using matrix analytic method. Yue et al. (2014) analyzed the M/M/1 queueing system with impatient customers and VV and obtained the closed-form expressions of the mean system sizes when the server is in different states using probability generating functions. A finite buffer M/M/1 queue with VWV, balking and reneging has been analyzed by Vijaya Laxmi and Jyothsna (2014) obtaining the steady state probabilities using matrix form solutions.

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