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Analysis of the infinite server queues with semi-Markovian multivariate discounted inputs

Author

Listed:
  • Landy Rabehasaina

    (University Bourgogne Franche Comté)

  • Jae-Kyung Woo

    (University of New South Wales)

Abstract

We consider a general k-dimensional discounted infinite server queueing process (alternatively, an incurred but not reported claim process) where the multivariate inputs (claims) are given by a k-dimensional finite-state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment-generating function jointly to the state of the input at time t given the initial state of the input at time 0, asymptotic results for the first and second (matrix) moments of the process are provided. In particular, when the interarrival or service times are exponentially distributed, transient expressions for the first two moments are obtained. Also, the moment-generating function for the process with deterministic interarrival times is considered to provide more explicit expressions. Finally, we demonstrate the potential of the present model by showing how it allows us to study semi-Markovian modulated infinite server queues where the customers (claims) arrival and service (reporting delay) times depend on the state of the process immediately before and at the switching times.

Suggested Citation

  • Landy Rabehasaina & Jae-Kyung Woo, 2020. "Analysis of the infinite server queues with semi-Markovian multivariate discounted inputs," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 393-420, April.
    • Handle: RePEc:spr:queues:v:94:y:2020:i:3:d:10.1007_s11134-020-09646-y
    DOI: 10.1007/s11134-020-09646-y
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    References listed on IDEAS

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    1. Woo, Jae-Kyung, 2016. "On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 354-363.
    2. Blom, Joke & De Turck, Koen & Mandjes, Michel, 2017. "Refined large deviations asymptotics for Markov-modulated infinite-server systems," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1036-1044.
    3. Francq, Christian & Gautier, Antony, 2004. "Estimation of time-varying ARMA models with Markovian changes in regime," Statistics & Probability Letters, Elsevier, vol. 70(4), pages 243-251, December.
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