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Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers

Author

Listed:
  • Ciro D’Apice

    (Dipartimento di Scienze Aziendali—Management and Innovation Systems, University of Salerno, 132, Via Giovanni Paolo II, 84084 Fisciano, SA, Italy)

  • Maria Pia D’Arienzo

    (Dipartimento di Scienze Aziendali—Management and Innovation Systems, University of Salerno, 132, Via Giovanni Paolo II, 84084 Fisciano, SA, Italy)

  • Alexander Dudin

    (Department of Applied Mathematics and Informatics, Belarusian State University, 4, Nezavisimosti Ave., 220030 Minsk, Belarus)

  • Rosanna Manzo

    (Dipartimento di Scienze Politiche e della Comunicazione, University of Salerno, 132, Via Giovanni Paolo II, 84084 Fisciano, SA, Italy)

Abstract

A queuing system having two different servers is under study. Demands enter the system according to a Markov arrival process. Service times have phase-type distribution. Service of demands is possible only if the fixed number of energy units, probably different for two servers, is available in the system at the potential service beginning moment. Energy units arrive in the system also according to a Markov arrival process and are stored in a stock (battery) of a finite capacity. Leakage of energy units from the stock can occur. Demands waiting in the infinite buffer are impatient and can leave the buffer after an exponentially distributed waiting time. One server is the main one and permanently provides service when the buffer is not empty and the required number of energy units is available. The second server is the assistant server and is switched on or off depending on the availability of energy units and queue length according to the hysteresis strategy defined by two thresholds. The assistant server is switched on when the queue length is not less than the greater threshold and is switched off when the queue length becomes smaller than the smaller threshold. The use of the assistant server has to be paid. Thus, the problem of the optimal selection of the thresholds defining the control strategy naturally arises. To solve this problem, the study of the behavior of the system under any fixed values of the parameters of the control strategy is necessary. Such a study is given in this paper. Numerical results are presented. They illustrate the feasibility of computer realization of the developed algorithms for computation of the stationary distribution of the system states and the main key performance indicators as well as the result of solving one of the possible optimization tasks.

Suggested Citation

  • Ciro D’Apice & Maria Pia D’Arienzo & Alexander Dudin & Rosanna Manzo, 2023. "Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers," Mathematics, MDPI, vol. 11(21), pages 1-34, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4515-:d:1272600
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    References listed on IDEAS

    as
    1. Anders Rygh Swensen, 2023. "Remaining Loads in a PH/M/c Queue with Impatient Customers," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-15, March.
    2. Neuts, Marcel F. & Rao, B. M., 1992. "On the design of a finite-capacity queue with phase-type service times and hysteretic control," European Journal of Operational Research, Elsevier, vol. 62(2), pages 221-240, October.
    3. Richard F. Gebhard, 1967. "A queuing process with bilevel hysteretic service‐rate control," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(1), pages 55-67.
    4. Dimitri frosinin & L. Breuer, 2006. "Threshold policies for controlled retrial queues with heterogeneous servers," Annals of Operations Research, Springer, vol. 141(1), pages 139-162, January.
    Full references (including those not matched with items on IDEAS)

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