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On a Markov‐modulated shock and wear process

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  • Jeffrey P. Kharoufeh
  • Dustin G. Mixon

Abstract

We present transient and asymptotic reliability indices for a single‐unit system that is subject to Markov‐modulated shocks and wear. The transient results are derived from the (transform) solution of an integro‐differential equation describing the joint distribution of the cumulative degradation process and the state of the modulating process. Additionally, we prove the asymptotic normality of a properly centered and time‐scaled version of the cumulative degradation at time t. This asymptotic result leads to a simple normal approximation for a properly centered and space‐scaled version of the systes lifetime distribution. Two numerical examples illustrate the quality of the normal approximation. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009

Suggested Citation

  • Jeffrey P. Kharoufeh & Dustin G. Mixon, 2009. "On a Markov‐modulated shock and wear process," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 563-576, September.
    • Handle: RePEc:wly:navres:v:56:y:2009:i:6:p:563-576
    DOI: 10.1002/nav.20366
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    References listed on IDEAS

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    1. Gang Li & Jiaowan Luo, 2005. "Shock model in Markovian environment," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(3), pages 253-260, April.
    2. Çinlar, Erhan & Shaked, Moshe & Shanthikumar, J. George, 1989. "On lifetimes influenced by a common environment," Stochastic Processes and their Applications, Elsevier, vol. 33(2), pages 347-359, December.
    3. Glynn, Peter W. & Whitt, Ward, 2002. "Necessary conditions in limit theorems for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 199-209, April.
    4. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    5. Glynn, Peter W. & Whitt, Ward, 1993. "Limit theorems for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 47(2), pages 299-314, September.
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