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<p>In this paper, we discuss an infinite-buffer single-server queue in which batches of random size arrive according to a Poisson process. The service time of each customer follows a general distribution that includes heavy-tailed distributions. The probability generating function (pgf) of the queue-length distribution at post-departure and random epochs have been derived using the roots’ method. The Tauberian theorem, along with the characteristics of slowly varying functions, is used to derive the asymptotic behavior of the tail probabilities of the queue-length distribution at a post-departure epoch, especially in cases where the service-time distribution follows a power-law distribution. The Laplace-Stieltjes transform (LST) as well as probability density function (pdf) of the waiting-time distribution of a random customer have been derived. The cost structure of the proposed model has been analyzed to determine the optimal customer arrival rate so that the service provider generates maximum revenue. Finally, numerical results are presented for various heavy-tailed service-time distributions.</p>

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Analysis and optimal control in an \(M^X/G/1\) queueing model with heavy-tailed service-time distribution

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Sitaram Barik,
  • A. D. Banik,
  • Mohan Chaudhry,
  • Souvik Ghosh

摘要

In this paper, we discuss an infinite-buffer single-server queue in which batches of random size arrive according to a Poisson process. The service time of each customer follows a general distribution that includes heavy-tailed distributions. The probability generating function (pgf) of the queue-length distribution at post-departure and random epochs have been derived using the roots’ method. The Tauberian theorem, along with the characteristics of slowly varying functions, is used to derive the asymptotic behavior of the tail probabilities of the queue-length distribution at a post-departure epoch, especially in cases where the service-time distribution follows a power-law distribution. The Laplace-Stieltjes transform (LST) as well as probability density function (pdf) of the waiting-time distribution of a random customer have been derived. The cost structure of the proposed model has been analyzed to determine the optimal customer arrival rate so that the service provider generates maximum revenue. Finally, numerical results are presented for various heavy-tailed service-time distributions.