Assessment of Shock Models for a Particular Class of Intershock Time Distributions

Abstract

In this paper, delta and extreme shock models and a mixed shock model which combines delta-shock and extreme shock models are studied. In particular, the interarrival times between successive shocks are assumed to belong to a class of matrix-exponential distributions which is larger than the class of phase-type distributions. The Laplace -Stieltjes transforms of the systems’ lifetimes are obtained in a matrix form. Survival functions of the systems are approximated based on the Laplace-Stieltjes transforms. The results are applied for the reliability evaluation of a certain repairable system consisting of two components.

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References

  1. Asmussen S, Bladt M (1997) Renewal theory and queueing algorithms for matrix-exponential distributions. In: Alfa A, Chakravarthy SR (eds) Matrix-analytic methods in stochastic models. Taylor and Francis, Boca Raton, pp 313–341

  2. Bladt M, Nielsen BF (2017) Matrix-exponential distributions in applied probability. Springer, New York

  3. Eryilmaz S (2012) Generalized δ-shock model via runs. Stat Probab Lett 82:326–31

  4. Eryilmaz S, Bayramoglu K (2014) Life behavior of δ-shock models for uniformly distributed interarrival times. Stat Papers 55: 841–852

  5. Jiang Y (2020) A new δ-shock model for systems subject to multiple failure types and its optimal order-replacement policy. Proceedings of the Institution of Mechanical Engineers, Part O. J Risk Reliab 234:138–150

  6. Koutras MV, Eryilmaz S (2017) Compound geometric distribution of order k. Methodol Comput Appl Probab 19:377–393

  7. Li ZH (2007) Kong XB Life behavior of δ-shock model. Stat Probab Lett 77:577–587

  8. Longman IM, Sharir M (1971) Laplace transform inversion of rational functions. Geophys. J.R. Astr. Soc. 25:299–305

  9. Parvardeh A, Balakrishnan N (2015) On mixed δ-shock models. Stat Probab Lett 102:51–60

  10. Wang GJ, Zhang YL (2005) A shock model with two-type failures and optimal replacement policy. Int J Syst Sci 36:209–214

  11. Wen Y, Cui L, Si S, Liu B (2017) A multiple warm standby delta-shock system with a repairman having multiple vacations. Communications in Stat-Simul Comput 46:3172–3186

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Acknowledgments

The authors thank the anonymous referees for their helpful comments and suggestions, which were very useful in improving the paper.

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Correspondence to Coskun Kus.

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Kus, C., Tuncel, A. & Eryilmaz, S. Assessment of Shock Models for a Particular Class of Intershock Time Distributions. Methodol Comput Appl Probab (2021). https://doi.org/10.1007/s11009-021-09847-9

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Keywords

  • Matrix-exponential distribution
  • Reliability
  • Repairable system
  • Shock model

Mathematics Subject Classification (2010)

  • Primary: 62N05 Secondary: 62E15