Modelling and analysis of a bulk service queueing model with multiple working vacations and server breakdown

Abstract

In this paper, a single server queue with variable batch size service, Poisson bulk arrival with multiple working vacations and server breakdown is considered. In working vacation, the server works with different rates rather than completely stoping the service during the vacation period. In this model, during the working vacation the server starts the service if it finds at least one customer in the queue with a maximum of 'b' customers, otherwise the server serves with variable batch size. Service time in working vacation and in regular period follows general distribution. The probability generating function of a queue size at an arbitrary time epoch as well as other completion epochs is derived. Expected queue length in a steady state is obtained. Also a numerical illustration is presented.

Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance

“…Two major classes of single-server batch service models studied in recent papers are the discrete-time (Claeys et al 2010a;Claeys et al 2010b;Claeys et al 2013;Banerjee et al 2014;Yu and Alfa 2015;Baetens et al 2016;Baetens et al 2017;Baetens et al 2018;Panda and Goswami 2020) and continuous-time models (Saxena et al 2018;D'Arienzo et al 2019;Banerjee and Gupta 2012;Banerjee et al 2015;Yu and Tang 2018;Pradhan and Gupta 2017;Pradhan et al 2016;Pradhan and Gupta 2019;Gupta et al 2020;Gupta and Banerjee 2019;Maity and Gupta 2015;Banik 2015;Vadivu and Arumuganathan 2015;Chaudhry et al 2016;Jeyakumar and Senthilnathan 2017;Zeng and Xia 2017;Niranjan et al 2018;Gupta and Banerjee 2018;Panda et al 2018;Ayyappan and Karpagam 2018;Ayyappan and Nirmala 2018;Bank and Samanta 2020;Xie et al 2020). The variety of techniques used for the analysis includes Kolmogorov equations, Supplementary variable techniques, Roots method, Matrix-Analytic Method, Embedded Markov chain analysis, Spectral methods, Asymptotic Quasi-Toeplitz Markov chain technique and Game theory, to name a few.…”

“…A notable exception is the paper (D'Arienzo et al 2019) where a retrial system is studied. Arrival process More analytically tractable are the memoryless arrival processes (Poisson in continuous time, and Geometric in discrete time case), but several works focus on general renewal processes, batch Poisson arrivals (Pradhan and Gupta 2017;Jeyakumar and Senthilnathan 2017;Ayyappan and Karpagam 2018;Ayyappan and Nirmala 2018), and, among the most general cases, the batch Markovian arrival process (BMAP) (Banik 2015;Bank and Samanta 2020), which though received less attention. Service time distribution In the majority of cases, the service time distribution is assumed to be general (e.g.…”

“…A specific version of the GBS rule (when a = b) is the Fixed Bulk Service (FBS) policy (Claeys et al 2010a;Yu and Tang 2018;Gupta and Banerjee 2019;Xie et al 2020). A significant attention is also received by the systems with variable capacity or the so-called versatile Bulk Service rule, where the batch size taken for service is random (Pradhan et al 2016;Maity and Gupta 2015;Jeyakumar and Senthilnathan 2017;Bank and Samanta 2020). Another interesting setup is considered in a series of papers (Baetens et al 2016(Baetens et al 2017(Baetens et al , 2018(Baetens et al , 2019, where the so-called two-class service policy was introduced.…”

Continuous-Service M/M/1 Queuing Systems

“…Most recently, a host of researchers actively investigate into a variation, M [X] /G(a, b)/c systems, of the standard M/G/1 system. Batch arrivals and machine repairs are often imposed on these systems; hence, results obtained from their studies are most applicable to production systems; see, for example, [12][13][14][15][16][17].…”

An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service

“…Krishna Reddy et al [4] have discussed an M [X] /G(a, b)/1 model with an N-policy, multiple vacations and setup times. Jeyakumar and Senthilnathan [5] analyzed the bulk service queueing system with multiple working vacations and server breakdown.…”