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Passage Times in Fluid Models with Application to Risk Processes

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  • V. Ramaswami

    (AT&T Labs-Research)

Abstract

An efficient quadratically convergent algorithm has been derived earlier by Ahn and Ramaswami for computing the busy period distribution of the canonical fluid flow model. In this paper, we derive formulae for a variety of passage time distributions in the canonical fluid flow model in terms of its busy period distribution and that of its reflection about the time axis. These include several passage time distributions with taboo not only of the fluid level 0 but also of a set [a, ∞) of levels. These are fundamental to the analysis of a large set of complex applied probability models, and their use is illustrated in the context of a general insurance risk model with Markovian arrival of claims and phase type distributed claim sizes, a context in which we have also introduced some new ideas that make the analysis very transparent.

Suggested Citation

  • V. Ramaswami, 2006. "Passage Times in Fluid Models with Application to Risk Processes," Methodology and Computing in Applied Probability, Springer, vol. 8(4), pages 497-515, December.
    • Handle: RePEc:spr:metcap:v:8:y:2006:i:4:d:10.1007_s11009-006-0426-9
    DOI: 10.1007/s11009-006-0426-9
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    References listed on IDEAS

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    1. Søren Asmussen, 1994. "Busy period analysis, rare events and transient behavior in fluid flow models," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-31, January.
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    Cited by:

    1. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    2. Mehmet Akif Yazici & Nail Akar, 2017. "The finite/infinite horizon ruin problem with multi-threshold premiums: a Markov fluid queue approach," Annals of Operations Research, Springer, vol. 252(1), pages 85-99, May.
    3. Yonit Barron, 2016. "Performance analysis of a reflected fluid production/inventory model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 1-31, February.
    4. Yonit Barron, 2023. "Integrating Replenishment Policy and Maintenance Services in a Stochastic Inventory System with Bilateral Movements," Mathematics, MDPI, vol. 11(4), pages 1-35, February.
    5. Nigel Bean & Małgorzata O’Reilly, 2008. "Performance measures of a multi-layer Markovian fluid model," Annals of Operations Research, Springer, vol. 160(1), pages 99-120, April.
    6. Ahn, Soohan & Badescu, Andrei L. & Cheung, Eric C.K. & Kim, Jeong-Rae, 2018. "An IBNR–RBNS insurance risk model with marked Poisson arrivals," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 26-42.
    7. Yonit Barron, 2016. "Performance analysis of a reflected fluid production/inventory model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 1-31, February.
    8. Yonit Barron & Dror Hermel, 2017. "Shortage decision policies for a fluid production model with MAP arrivals," International Journal of Production Research, Taylor & Francis Journals, vol. 55(14), pages 3946-3969, July.
    9. Jin, Can & Li, Shuanming & Wu, Xueyuan, 2016. "On the occupation times in a delayed Sparre Andersen risk model with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 304-316.
    10. Cheung, Eric C.K. & Landriault, David, 2010. "A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 127-134, February.
    11. Hédi Nabli & Itidel Abdallah, 2023. "Stochastic Fluid Models with Upward Jumps and Phase Transitions," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    12. Yonit Barron, 2022. "A probabilistic approach to the stochastic fluid cash management balance problem," Annals of Operations Research, Springer, vol. 312(2), pages 607-645, May.
    13. Barron, Yonit, 2016. "Clearing control policies for MAP inventory process with lost sales," European Journal of Operational Research, Elsevier, vol. 251(2), pages 495-508.
    14. V. Ramaswami & Douglas Woolford & David Stanford, 2008. "The erlangization method for Markovian fluid flows," Annals of Operations Research, Springer, vol. 160(1), pages 215-225, April.
    15. A. S. Dibu & M. J. Jacob & Apostolos D. Papaioannou & Lewis Ramsden, 2021. "Delayed Capital Injections for a Risk Process with Markovian Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 1057-1076, September.
    16. Yonit Barron & David Perry & Wolfgang Stadje, 2016. "A make-to-stock production/inventory model with MAP arrivals and phase-type demands," Annals of Operations Research, Springer, vol. 241(1), pages 373-409, June.
    17. Albrecher, Hansjörg & Badescu, Andrei & Landriault, David, 2008. "On the dual risk model with tax payments," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1086-1094, June.
    18. Bruno Sericola & Marie-Ange Remiche, 2011. "Maximum Level and Hitting Probabilities in Stochastic Fluid Flows Using Matrix Differential Riccati Equations," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 307-328, June.

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