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On Properties of the Phase-type Mixed Poisson Process and its Applications to Reliability Shock Modeling

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  • Dheeraj Goyal

    (Indian Institute of Technology Jodhpur)

  • Nil Kamal Hazra

    (Indian Institute of Technology Jodhpur
    Indian Institute of Technology Jodhpur)

  • Maxim Finkelstein

    (University of the Free State
    University of Strathclyde)

Abstract

Although Poisson processes are widely used in various applications for modeling of recurrent point events, there exist obvious limitations. Several specific mixed Poisson processes (which are formally not Poisson processes any more) that were recently introduced in the literature overcome some of these limitations. In this paper, we define a general mixed Poisson process with the phase-type (PH) distribution as the mixing one. As the PH distribution is dense in the set of lifetime distributions, the new process can be used to approximate any mixed Poisson process. We study some basic stochastic properties of the new process and discuss relevant applications by considering the extreme shock model, the stochastic failure rate model and the $$\delta$$ δ -shock model.

Suggested Citation

  • Dheeraj Goyal & Nil Kamal Hazra & Maxim Finkelstein, 2022. "On Properties of the Phase-type Mixed Poisson Process and its Applications to Reliability Shock Modeling," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2933-2960, December.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:4:d:10.1007_s11009-022-09961-2
    DOI: 10.1007/s11009-022-09961-2
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    References listed on IDEAS

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    1. Jozef L. Teugels & Petra Vynckier, 1996. "The structure distribution in a mixed Poisson process," International Journal of Stochastic Analysis, Hindawi, vol. 9, pages 1-8, January.
    2. Hidetoshi Konno, 2010. "On the Exact Solution of a Generalized Polya Process," Advances in Mathematical Physics, Hindawi, vol. 2010, pages 1-12, November.
    3. Antonio Di Crescenzo & Franco Pellerey, 2019. "Some Results and Applications of Geometric Counting Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 203-233, March.
    4. Asmussen, Søren & Bladt, Mogens, 1999. "Point processes with finite-dimensional conditional probabilities," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 127-142, July.
    5. A-Hameed, M. S. & Proschan, F., 1973. "Nonstationary shock models," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 383-404, October.
    6. Gut, Allan & Hüsler, Jürg, 2005. "Realistic variation of shock models," Statistics & Probability Letters, Elsevier, vol. 74(2), pages 187-204, September.
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    Cited by:

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