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Sojourn Time Analysis of a Single-Server Queue with Single- and Batch-Service Customers

Author

Listed:
  • Yusei Koyama

    (Graduate School of Science and Technology, Univercity of Tsukuba, Tsukuba 305-8573, Japan
    These authors contributed equally to this work.)

  • Ayane Nakamura

    (Graduate School of Science and Technology, Univercity of Tsukuba, Tsukuba 305-8573, Japan
    These authors contributed equally to this work.)

  • Tuan Phung-Duc

    (Institute of Systems and Information Engineering of Tsukuba, Tsukuba 305-8573, Japan
    These authors contributed equally to this work.)

Abstract

There are various types of sharing economy services, such as ride-sharing and shared-taxi rides. Motivated by these services, we consider a single-server queue in which customers probabilistically select the type of service, that is, the single service or batch service, or other services (e.g., train). In the proposed model, which is denoted by the M+M( K )/M/1 queue, we assume that the arrival process of all the customers follows a Poisson distribution, the batch size is constant, and the common service time (for the single- and batch-service customers) follows an exponential distribution. In this model, the derivation of the sojourn time distribution is challenging because the sojourn time of a batch-service customer is not determined upon arrival but depends on single customers who arrive later. This results in a two-dimensional recursion, which is not generally solvable, but we made it possible by utilizing a special structure of our model. We present an analysis using a quasi-birth-and-death process, deriving the exact and approximated sojourn time distributions (for the single-service customers, batch-service customers, and all the customers). Through numerical experiments, we demonstrate that the approximated sojourn time distribution is sufficiently accurate compared to the exact sojourn time distributions. We also present a reasonable approximation for the distribution of the total number of customers in the system, which would be challenging with a direct-conventional method. Furthermore, we presented an accurate approximation method for a more general model where the service time of single-service customers and that of batch-service customers follow two distinct distributions, based on our original model.

Suggested Citation

  • Yusei Koyama & Ayane Nakamura & Tuan Phung-Duc, 2024. "Sojourn Time Analysis of a Single-Server Queue with Single- and Batch-Service Customers," Mathematics, MDPI, vol. 12(18), pages 1-27, September.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2820-:d:1476176
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    References listed on IDEAS

    as
    1. Ayane Nakamura & Tuan Phung-Duc & Hiroyasu Ando, 2022. "Queueing analysis of a Car/Ride-Share system," Annals of Operations Research, Springer, vol. 310(2), pages 661-682, March.
    2. Ayane Nakamura & Tuan Phung-Duc, 2023. "Equilibrium Analysis for Batch Service Queueing Systems with Strategic Choice of Batch Size," Mathematics, MDPI, vol. 11(18), pages 1-22, September.
    3. A. Wang & I. Ziedins, 2018. "Probabilistic selfish routing in parallel batch and single-server queues," Queueing Systems: Theory and Applications, Springer, vol. 88(3), pages 389-407, April.
    4. Tom Van Woensel & Nico Vandaele, 2007. "Modeling Traffic Flows With Queueing Models: A Review," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 24(04), pages 435-461.
    5. J. Medhi, 1975. "Waiting Time Distribution in a Poisson Queue with a General Bulk Service Rule," Management Science, INFORMS, vol. 21(7), pages 777-782, March.
    6. Moeko Yajima & Tuan Phung-Duc, 2017. "Batch arrival single-server queue with variable service speed and setup time," Queueing Systems: Theory and Applications, Springer, vol. 86(3), pages 241-260, August.
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