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A finite capacity GI/PH/1 queue with group services

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  • S. Chakravarthy

Abstract

We consider a finite‐capacity single‐server queue in which arrivals occur one at a time, according to a renewal process. The successive service times are mutually independent and have a common phase‐type distribution. The customers are served in groups of size at least L, a preassigned threshold value. Explicit analytic expressions for the steady‐state queue‐length densities at arrivals and at arbitrary time points, and the throughput of the system are obtained. The Laplace‐Stieltjes transform of the stationary waiting‐time distribution of an admitted customer at points of arrivals is computed. It is shown to be of phase type when the arrival process is also of phase type. Efficient algorithmic procedures for the steady‐state analysis of the model are presented. These procedures are used in arriving at an optimal value for L that minimizes the mean waiting time of an admitted customer. A conjecture on the nature of the mean waiting time is proposed.

Suggested Citation

  • S. Chakravarthy, 1992. "A finite capacity GI/PH/1 queue with group services," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(3), pages 345-357, April.
    • Handle: RePEc:wly:navres:v:39:y:1992:i:3:p:345-357
    DOI: 10.1002/1520-6750(199204)39:33.0.CO;2-V
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    Cited by:

    1. A. D. Banik & M. L. Chaudhry & U. C. Gupta, 2008. "On the Finite Buffer Queue with Renewal Input and Batch Markovian Service Process: GI/BMSP/1/N," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 559-575, December.
    2. 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.
    3. Sergei Dudin & Olga Dudina, 2023. "Analysis of a Multi-Server Queue with Group Service and Service Time Dependent on the Size of a Group as a Model of a Delivery System," Mathematics, MDPI, vol. 11(22), pages 1-20, November.

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