Queue-Length, Waiting-Time and Service Batch Size Analysis for the Discrete-Time GI/D-MSP (a,b)
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Queue-Length, Waiting-Time and Service Batch Size Analysis for the Discrete-Time GI/D-MSP(a,b) /1/∞ Queueing System S. K. Samanta1
· R. Nandi1
Received: 13 June 2018 / Revised: 18 August 2019 / Accepted: 14 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper analyzes an infinite-buffer single-server bulk-service queueing system in which customers arrive according to a discrete-time renewal process. The customers are served under the discrete-time Markovian service process according to the general bulk-service rule. The matrix-geometric method is used to obtain the queue-length distribution at prearrival epoch. The queue-length distributions at other various time epochs are also derived based on prearrival epoch probabilities. A simple approach has been developed to compute the waiting-time distribution of an arriving customer. We also carried out closed-form analytical expression for the service batch size distribution of an arriving customer. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions and model parameters to understand the system behaviour. Keywords Queueing · General bulk-service rule · Matrix-geometric method · Service batch size distribution · Waiting time distribution Mathematics Subject Classification (2010) 60K25 · 90B22 · 68M20
1 Introduction Over the last few decades, bulk-service (frequently called batch service) queues constitute an extensive class of queueing systems. Bulk-service queues have potential applications in many areas, e.g., in blood testing procedure to detect viruses like hepatitis B (HBV), hepatitis C (HCV) and human immunodeficiency (HIV) (Abolnikov and Dukhovny 2003; Bar-Lev et al. 2007), computer components and chip production industry (Brown et al. 2010; Koo and Moon 2013), in traffic signal systems (Hall 1999), in computer networks where jobs are S. K. Samanta
[email protected] R. Nandi [email protected] 1
Department of Mathematics, National Institute of Technology Raipur, Raipur-492010, India
Methodology and Computing in Applied Probability
processed in batches (Chao and Pinedo 1995; Claeys et al. 2010), semiconductor manufacturing industry (Shanthikumar et al. 2007; Kunii et al. 2011), transportation and distribution logistics systems (Bhaskar and Lallement 2010; Schwarz and Epp 2016), and related references therein. In view of the usefulness of bulk-service queues in many application areas, several authors have used different methods and techniques to study the performance measures of such queueing systems. The study on bulk-service queueing model was originated due to Bailey (1954), who investigated M/G[s] /1 queue using the embedded Markov chain technique wherein the server can take s customers or the whole queue, whichever is less. (Neuts 1967) first introduced the general bulk-service rule having quorum ‘a’ and maximum service capacity ‘b’ to a queueing system with general service-time distribution and Poisson arrivals. An explicit expr
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