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<p>In this paper, we discuss a finite-buffer single-server queue wherein arrivals occur according to a Markovian arrival process. The server serves customers in batches according to (<i>a</i>,&#xa0;<i>b</i>) bulk-service rule, that is, the server serves in batches of maximum size “<i>b</i>” with a minimum threshold size “<i>a</i>”. The service time of each batch follows the class of distributions whose Laplace-Stieltjes transforms are rational functions (<i>R</i>-type distributions). The queue capacity is <i>N</i> excluding the batch with the server. An alternative approach for finding the queue-length probability vector at a post-departure epoch has been proposed using the roots of a characteristic equation. Queue-length probability vectors at random and pre-arrival epochs have also been obtained using the embedded Markov chain, Markov renewal theory, and the semi-Markov processes. Approximation to the phase-wise virtual waiting-time distribution (in the queue as well as in the system) of a random customer is derived using the functional relation between the vector generating function (vgf) for the queue-length probability vectors at a random epoch and the vector containing the phase-wise Laplace-Stieltjes transform (LST) of the queueing-time distribution of a random customer. Using the LST, we discuss the derivation of an approximate phase-wise probability density function of the virtual waiting-time of a random customer as well as its numerical implementations.</p>

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On the Heuristic Computational Procedures of the Virtual Waiting-Time Distribution in a Non-renewal Input Finite-Buffer Bulk-Service Queues: \(MAP/R^{(a,b)}/1/N\)

  • 期刊论文

  • 发表日期：

    2025年4月15日

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  • Abhijit Datta Banik,
  • Mohan Chaudhry,
  • Sitaram Barik,
  • Gagandeep Singh

摘要

In this paper, we discuss a finite-buffer single-server queue wherein arrivals occur according to a Markovian arrival process. The server serves customers in batches according to (a, b) bulk-service rule, that is, the server serves in batches of maximum size “b” with a minimum threshold size “a”. The service time of each batch follows the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The queue capacity is N excluding the batch with the server. An alternative approach for finding the queue-length probability vector at a post-departure epoch has been proposed using the roots of a characteristic equation. Queue-length probability vectors at random and pre-arrival epochs have also been obtained using the embedded Markov chain, Markov renewal theory, and the semi-Markov processes. Approximation to the phase-wise virtual waiting-time distribution (in the queue as well as in the system) of a random customer is derived using the functional relation between the vector generating function (vgf) for the queue-length probability vectors at a random epoch and the vector containing the phase-wise Laplace-Stieltjes transform (LST) of the queueing-time distribution of a random customer. Using the LST, we discuss the derivation of an approximate phase-wise probability density function of the virtual waiting-time of a random customer as well as its numerical implementations.