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Two parallel queues with simultaneous service interruptions and/or disasters

Author

Listed:
  • Herwig Bruneel

    (Ghent University, SMACS Research Group, Department of Telecommunications and Information Processing)

  • Sabine Wittevrongel

    (Ghent University, SMACS Research Group, Department of Telecommunications and Information Processing)

Abstract

This paper investigates a discrete-time queueing system, which accommodates two types of customers, named type 1 and type 2. Both customer types have their own dedicated queue and their own dedicated server. The service times of all customers are equal to one time slot. Customers arrive in the system independently from slot to slot, but the numbers of arrivals of both types in any slot are not necessarily independent; their joint probability generating function (pgf) is $$A(z_1,z_2)$$ . The system operates in an unreliable environment, whereby two types of random distortions may occur, independently from slot to slot, and with given probabilities: minor breakdowns or major breakdowns. Minor breakdowns cause the temporary unavailability of both servers of the system and are referred to as service interruptions in the paper. Major breakdowns result in the simultaneous removal of all customers from the system at random instants in time and are called disasters. Whereas isolated queues with either service interruptions or disasters have been well-studied in the queueing literature, we believe the joint study of both effects, especially in the context of two parallel queues, is novel. We derive a kernel-type functional equation for the steady-state joint pgf $$U(z_1,z_2)$$ of the numbers of type-1 and type-2 customers in the system. Although solving this equation for arbitrary arrival pgfs $$A(z_1,z_2)$$ seems infeasible, we succeed in finding exact closed-form solutions for various specific classes of arrival pgfs $$A(z_1,z_2)$$ . In doing so, we notice that the stability and the behavior of the system are profoundly different, depending on whether disasters do or do not occur. We illustrate our findings abundantly with specific examples and also retrieve some existing results as special cases of our general results.

Suggested Citation

  • Herwig Bruneel & Sabine Wittevrongel, 2025. "Two parallel queues with simultaneous service interruptions and/or disasters," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 102(2), pages 245-289, December.
    • Handle: RePEc:spr:mathme:v:102:y:2025:i:2:d:10.1007_s00186-025-00907-1
    DOI: 10.1007/s00186-025-00907-1
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    References listed on IDEAS

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    1. R. Sudhesh & R. Sebasthi Priya, 2019. "An analysis of discrete-time Geo/Geo/1 queue with feedback, repair and disaster," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 35(1), pages 37-53.
    2. A. Krishnamoorthy & P. Pramod & S. Chakravarthy, 2014. "Queues with interruptions: a survey," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 290-320, April.
    3. Vijay Rajan Lumb & Indra Rani, 2022. "Analytically simple solution to discrete-time queue with catastrophes, balking and state-dependent service," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 13(2), pages 783-817, April.
    4. Herwig Bruneel, 2022. "Some thoughts on the analysis of coupled queues," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 185-187, April.
    5. Mustafa Demircioglu & Herwig Bruneel & Sabine Wittevrongel, 2021. "Analysis of a Discrete-Time Queueing Model with Disasters," Mathematics, MDPI, vol. 9(24), pages 1-22, December.
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