A Discrete-Time G I X / G e o /1 Queue with Multiple Working Vacations Under Late and Early Arrival System
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A Discrete-Time GI X /Geo /1 Queue with Multiple Working Vacations Under Late and Early Arrival System F. P. Barbhuiya1 · U. C. Gupta1 Received: 28 December 2018 / Revised: 28 December 2018 / Accepted: 21 May 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract This paper studies a discrete-time batch arrival GI /Geo/1 queue where the server may take multiple vacations depending on the state of the queue/system. However, during the vacation period, the server does not remain idle and serves the customers with a rate lower than the usual service rate. The vacation time and the service time during working vacations are geometrically distributed. Keeping note of the specific nature of the arrivals and departures in a discrete-time queue, we study the model under late arrival system with delayed access and early arrival system independently. We formulate the system using supplementary variable technique and apply the theory of difference equation to obtain closed-form expressions of steady-state system content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equations. We discuss the stability conditions of the system and develop few performance measures as well. Through some numerical examples, we illustrate the feasibility of our theoretical work and highlight the asymptotic behavior of the probability distributions at pre-arrival epochs. We further discuss the impact of various parameters on the performance of the system. The model considered in this paper covers a wide class of vacation and non-vacation queueing models which have been studied in the literature. Keywords Bulk arrival · Difference equation method · Discrete-time · GI /Geo/1 queue · Multiple working vacations · Supplementary variable technique Mathematics Subject Classification (2010) 60K25 · 90B22
1 Introduction Since last few decades, discrete-time queueing models with various vacation policies have drawn the attention of researchers because of its potential application in modeling F. P. Barbhuiya
[email protected] U. C. Gupta [email protected] 1
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
Methodology and Computing in Applied Probability
computer networks and digital telecommunication systems. These systems are intended to serve real-time applications where the processing of data takes place within a defined time constraint in slots or units of equal lengths. The stochastic processes involved in these systems occurs near the slot boundaries which eventually gives rise to two variations in the modeling of discrete-time queues: late arrival system with delayed access (LAS-DA or LAS) and early arrival system (EAS). A detailed study of discrete-time queues with vacations can be found in Takagi (1993). For further reference, one may also see the survey papers by Doshi (1986) and Ke et al. (2010). The discrete-time Geo/G/1 queue and GI /Geo/1 queue with multiple vacations was respectively
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