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Hitting probabilities and hitting times for stochastic fluid flows

Author

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  • Bean, Nigel G.
  • O'Reilly, Malgorzata M.
  • Taylor, Peter G.

Abstract

Recently there has been considerable interest in Markovian stochastic fluid flow models. A number of authors have used different methods to calculate quantities of interest. In this paper, we consider a fluid flow model, formulated so that time is preserved, and derive expressions for return probabilities to the initial level, the Laplace-Stieltjes transforms (for arguments with nonnegative real part only) and moments of the time taken to return to the initial level, excursion probabilities to high/low levels, and the Laplace-Stieltjes transforms of sojourn times in specified sets. An important feature of our results is their physical interpretation within the stochastic fluid flow environment, which is given. This allows for further implementation of our expressions in the calculation of other quantities of interest. Novel aspects of our treatment include the calculation of probability densities with respect to level and an argument under which we condition on the infimum of the levels at which a "down-up period" occurs. Significantly, these results are achieved with techniques applied directly within the fluid flow model, without the need for discretization or transformation to other equivalent models.

Suggested Citation

  • Bean, Nigel G. & O'Reilly, Malgorzata M. & Taylor, Peter G., 2005. "Hitting probabilities and hitting times for stochastic fluid flows," Stochastic Processes and their Applications, Elsevier, vol. 115(9), pages 1530-1556, September.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:9:p:1530-1556
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    References listed on IDEAS

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    1. Latouche, Guy & Taylor, Peter, 2002. "Truncation and augmentation of level-independent QBD processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 53-80, May.
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    Cited by:

    1. Samuelson, Aviva & Haigh, Andrew & O'Reilly, Małgorzata M. & Bean, Nigel G., 2017. "Stochastic model for maintenance in continuously deteriorating systems," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1169-1179.
    2. Barbara Margolius & Małgorzata M. O’Reilly, 2016. "The analysis of cyclic stochastic fluid flows with time-varying transition rates," Queueing Systems: Theory and Applications, Springer, vol. 82(1), pages 43-73, February.
    3. Bean, Nigel G. & Nguyen, Giang T. & Nielsen, Bo F. & Peralta, Oscar, 2022. "RAP-modulated fluid processes: First passages and the stationary distribution," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 308-340.
    4. 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.
    5. O’Reilly, Małgorzata M., 2014. "Multi-stage stochastic fluid models for congestion control," European Journal of Operational Research, Elsevier, vol. 238(2), pages 514-526.
    6. Nikki Sonenberg & Peter G. Taylor, 2019. "Networks of interacting stochastic fluid models with infinite and finite buffers," Queueing Systems: Theory and Applications, Springer, vol. 92(3), pages 293-322, August.
    7. Bean, Nigel G. & O’Reilly, Małgorzata M., 2014. "The stochastic fluid–fluid model: A stochastic fluid model driven by an uncountable-state process, which is a stochastic fluid model itself," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1741-1772.

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