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<p>In this study, we investigate a G/G/1 queue with an auto-correlated arrival process, assuming that the arrival process is a stationary Markov process with a known joint distribution for adjacent inter-arrival times. This framework encompasses several commonly used correlated models, such as the Markov-modulated model, TES model, and copula-based model, as special instances. We first develop a two-stage MacLaurin series expansion method to compute the light traffic derivatives of the moments of the waiting time. We then derive the heavy traffic limits for these moments. By using the light traffic derivatives together with the heavy traffic limits, the moments of the waiting time can be calculated based on either the MacLaurin series expansion approximation or the single-/multi-point Padé approximation. Furthermore, based on the moments of the waiting time, we calculate the moments and covariances of the departure process. Extensive numerical experiments are conducted to validate the effectiveness of our proposed method.</p>

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Queues with auto-correlated inter-arrival times

  • 期刊论文

  • 发表日期：

    2026年2月9日

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  • Qixin Wang,
  • Jian-Qiang Hu,
  • Weimin Dai

摘要

In this study, we investigate a G/G/1 queue with an auto-correlated arrival process, assuming that the arrival process is a stationary Markov process with a known joint distribution for adjacent inter-arrival times. This framework encompasses several commonly used correlated models, such as the Markov-modulated model, TES model, and copula-based model, as special instances. We first develop a two-stage MacLaurin series expansion method to compute the light traffic derivatives of the moments of the waiting time. We then derive the heavy traffic limits for these moments. By using the light traffic derivatives together with the heavy traffic limits, the moments of the waiting time can be calculated based on either the MacLaurin series expansion approximation or the single-/multi-point Padé approximation. Furthermore, based on the moments of the waiting time, we calculate the moments and covariances of the departure process. Extensive numerical experiments are conducted to validate the effectiveness of our proposed method.