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The Single Server Queue with Catastrophes and Geometric Reneging

Author

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  • Spiros Dimou

    (University of Athens)

  • Antonis Economou

    (University of Athens)

Abstract

We consider the single server queue with catastrophes that occur according to a Poisson process. The catastrophes force all present customers to abandon the system and render the server inoperative. Then a repair time is set on till the server becomes ready to serve again the customers. In the meanwhile the customers are accumulated according to their arrival process. Recently, several authors have investigated the reneging behavior in such systems and other related models when the customers become impatient because of the failure of the server. Two kinds of reneging have been considered: independent and binomial. In the case of independent reneging, each customer has its own patience time and abandons the system as soon as it expires. In the case of binomial reneging, the abandonment opportunities occur according to a certain point process and then all present customers decide simultaneously but independently whether they will abandon the system or not. In the present paper, we complement these studies by considering the case of geometric reneging. This case arises when the abandonment opportunities occur according to a certain point process and the customers decide sequentially whether they will leave the system or not. We derive explicit expressions and computational schemes for various performance descriptors, concerning the number of customers in system, the sojourn time of a customer, the duration and the maximum number of customers in a busy period.

Suggested Citation

  • Spiros Dimou & Antonis Economou, 2013. "The Single Server Queue with Catastrophes and Geometric Reneging," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 595-621, September.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:3:d:10.1007_s11009-011-9271-6
    DOI: 10.1007/s11009-011-9271-6
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    References listed on IDEAS

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    1. Kyriakidis, E. G., 1994. "Stationary probabilities for a simple immigration-birth-death process under the influence of total catastrophes," Statistics & Probability Letters, Elsevier, vol. 20(3), pages 239-240, June.
    2. Economou, Antonis & Fakinos, Demetrios, 2003. "A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes," European Journal of Operational Research, Elsevier, vol. 149(3), pages 625-640, September.
    3. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    4. Economou, Antonis & Kapodistria, Stella, 2010. "Synchronized abandonments in a single server unreliable queue," European Journal of Operational Research, Elsevier, vol. 203(1), pages 143-155, May.
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    Cited by:

    1. Di Crescenzo, Antonio & Giorno, Virginia & Nobile, Amelia G., 2016. "Constructing transient birth–death processes by means of suitable transformations," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 152-171.
    2. Ye Jingjing & Liu Liwei & Jiang Tao, 2016. "Analysis of a Single-Sever Queue with Disasters and Repairs Under Bernoulli Vacation Schedule," Journal of Systems Science and Information, De Gruyter, vol. 4(6), pages 547-559, December.
    3. Antonio Di Crescenzo & Virginia Giorno & Balasubramanian Krishna Kumar & Amelia G. Nobile, 2018. "A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation," Mathematics, MDPI, vol. 6(5), pages 1-23, May.
    4. F. P. Barbhuiya & Nitin Kumar & U. C. Gupta, 2019. "Batch Renewal Arrival Process Subject to Geometric Catastrophes," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 69-83, March.
    5. Gopinath Panda & Veena Goswami & Abhijit Datta Banik, 2016. "Equilibrium and Socially Optimal Balking Strategies in Markovian Queues with Vacations and Sequential Abandonment," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(05), pages 1-34, October.

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