A Novel Computational Procedure for the Waiting-Time Distribution (In the Queue) for Bulk-Service Finite-Buffer Queues with Poisson Input

Mohan Chaudhry, Abhijit Datta Banik (), Sitaram Barik and Veena Goswami
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Mohan Chaudhry: Department of Mathematics and Computer Science, Royal Military College of Canada, STN Forces, P.O. Box 17000, Kingston, ON K7K 7B4, Canada
Abhijit Datta Banik: School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Permanent Campus Argul, Jatni, Khurda 752050, India
Sitaram Barik: School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Permanent Campus Argul, Jatni, Khurda 752050, India
Veena Goswami: School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar 751024, India

Mathematics, 2023, vol. 11, issue 5, 1-26

Abstract: In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under ( a , b ) -bulk service rule. The queueing system has a finite-buffer capacity ‘ N ’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distribution for a random customer. Using LSTs, we discuss the derivation of the probability density function of a random customer’s waiting time and its numerical implementations.

Keywords: Poisson input; batch service (a,b)-rule; finite-buffer queue; roots (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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