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Some Transient Results on the M/SM/1 Special Semi-Markov Model in Risk and Queueing Theories

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  • Janssen, Jacques

Abstract

We consider a usual situation in risk theory for which the arrival process is a Poisson process and the claim process a positive (J — X) process inducing a semi-Markov process. The equivalent in queueing theory is the M/SM/1 model introduced for the first time by Neuts (1966).For both models, we give an explicit expression of the probability of non-ruin on [o, t] starting with u as initial reserve and of the waiting time distribution of the last customer arrived before t. “Explicit expression†means in terms of the matrix of the aggregate claims distributions.

Suggested Citation

  • Janssen, Jacques, 1980. "Some Transient Results on the M/SM/1 Special Semi-Markov Model in Risk and Queueing Theories," ASTIN Bulletin, Cambridge University Press, vol. 11(1), pages 41-51, June.
    • Handle: RePEc:cup:astinb:v:11:y:1980:i:01:p:41-51_00
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    Cited by:

    1. Shao, Jia & Papaioannou, Apostolos D. & Pantelous, Athanasios A., 2017. "Pricing and simulating catastrophe risk bonds in a Markov-dependent environment," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 68-84.
    2. Guglielmo D’Amico & Fulvio Gismondi & Jacques Janssen & Raimondo Manca, 2015. "Discrete Time Homogeneous Markov Processes for the Study of the Basic Risk Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(4), pages 983-998, December.
    3. Schmidli, Hanspeter, 2001. "Distribution of the first ladder height of a stationary risk process perturbed by [alpha]-stable Lévy motion," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 13-20, February.

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