Computational Procedure for System Length and Waiting-time Distribution in an $$M^{X}/D/c/N$$ M X / D / c / N Queue

Sitaram Barik (), Abhijit Datta Banik (), Mohan L. Chaudhry () and Saroja Kumar Singh ()
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Sitaram Barik: Indian Institute of Technology Bhubaneswar
Abhijit Datta Banik: Indian Institute of Technology Bhubaneswar
Mohan L. Chaudhry: Royal Military College of Canada
Saroja Kumar Singh: Ravenshaw University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 3, 973-988

Abstract: Abstract In this paper, we discuss a finite-buffer and multi-server queue wherein a batch of customers of random size arrive according to a Poisson process. Service time follows the deterministic distribution, and the queue capacity is N, excluding the number of customers with the servers. The probability generating function (pgf) of queue-length and system-length distribution at a random epoch has been derived using embedded Markov chain. The Laplace-Stieltjes transform (LST) of the waiting time of a random customer in a batch has been derived using the LST of the waiting time of the first customer of a batch. We also derive the probability density function (pdf) for the waiting-time distribution of a random customer in a batch. Performance measures, like mean queue length, mean waiting time, and the probability of blocking have been obtained. Such queueing systems find applications in the performance analysis of communication, manufacturing, and transportation systems.

Keywords: Finite-buffer; Queue; Multi-server; Poisson process; Batch arrival; Deterministic service; Roots; 60K25; 68M20; 90B22 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-025-00814-5

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