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Arrival and departure state distributions in the general bulk‐service queue

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  • Gérard Hébuterne
  • Catherine Rosenberg

Abstract

In this paper, we give an explicit relation between steady‐state probability distributions of the buffer occupancy at customer entrance and departure epochs, for the classical single‐server system G/G[N]/1 with batch services and for the finite capacity case. The method relies on level‐crossing arguments. For the particular case of Poisson input, we also express the loss probability in terms of state probabilities at departure epochs, yielding probabilities observed by arriving customers. This work provides the “bulk queue” version of a result established by Burke, who stated the equality between probabilities at arrival and departure epochs for systems with “unit jumps.” © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 107–118, 1999

Suggested Citation

  • Gérard Hébuterne & Catherine Rosenberg, 1999. "Arrival and departure state distributions in the general bulk‐service queue," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(1), pages 107-118, February.
    • Handle: RePEc:wly:navres:v:46:y:1999:i:1:p:107-118
    DOI: 10.1002/(SICI)1520-6750(199902)46:13.0.CO;2-Y
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    References listed on IDEAS

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    1. Dimitris Bertsimas & Daisuke Nakazato, 1995. "The Distributional Little's Law and Its Applications," Operations Research, INFORMS, vol. 43(2), pages 298-310, April.
    2. P. H. Brill & M. J. M. Posner, 1977. "Level Crossings in Point Processes Applied to Queues: Single-Server Case," Operations Research, INFORMS, vol. 25(4), pages 662-674, August.
    3. Fakinos, D., 1991. "The relation between limiting queue size distributions at arrival and departure epochs in a bulk queue," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 327-329, April.
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    1. Srinivas R. Chakravarthy & Shruti & Alexander Rumyantsev, 2021. "Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1551-1579, December.

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