On terminating Poisson processes in some shock models

A system subject to a point process of shocks is considered. Shocks occur in accordance with the homogeneous Poisson process. Different criteria of system failure (termination) are discussed and the corresponding probabilities of failure (accident)-free performance are derived. The described analytical approach is based on deriving integral equations for each setting and solving these equations through the Laplace transform. Some approximations are analyzed and further generalizations and applications are discussed.

[1]  Maxim Finkelstein,et al.  Laplace-transforms and fast-repair approximations for multiple availability and its generalizations , 2002, IEEE Trans. Reliab..

[2]  J. Rice.  On generalized shot noise , 1977, Advances in Applied Probability.

[3]  Terje Aven.  Condition based replacement policiesa counting process approach , 1996 .

[4]  Zehui Li,et al.  An extended extreme shock maintenance model for a deteriorating system , 2008, Reliab. Eng. Syst. Saf..

[5]  Maxim Finkelstein.  Simple bounds for terminating Poisson and renewal shock processes , 2003 .

[6]  Allan Gut.  Cumulative shock models , 1990 .

[7]  T. H. Savits,et al.  Age Dependent Minimal Repair. , 1985 .

[8]  Rafael Pérez-Ocón,et al.  Replacement policy in a system under shocks following a Markovian arrival process , 2009, Reliab. Eng. Syst. Saf..

[9]  J. Hüsler,et al.  Realistic variation of shock models , 2005 .

[10]  Mahesh D. Pandey,et al.  Discounted cost model for condition-based maintenance optimization , 2010, Reliab. Eng. Syst. Saf..

[11]  J. Cha,et al.  On a Terminating Shock Process with Independent Wear Increments , 2009, Journal of Applied Probability.

[12]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[13]  Yeh Lam,et al.  A δ-shock maintenance model for a deteriorating system , 2006, Eur. J. Oper. Res..

[14]  Toshio Nakagawa,et al.  Shock and Damage Models in Reliability Theory , 2006 .

[15]  John Yuan,et al.  Optimal maintenance policy for a Markovian system under periodic inspection , 2001, Reliab. Eng. Syst. Saf..

[16]  Frank Beichelt,et al.  Stochastic Processes and Their Applications , 2001 .

[17]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[18]  Waltraud Kahle,et al.  On a cumulative damage process and resulting first passages times: Research Articles , 2004 .

[19]  Yeh Lam,et al.  A shock model for the maintenance problem of a repairable system , 2004, Comput. Oper. Res..

[20]  Maxim Finkelstein,et al.  Failure Rate Modelling for Reliability and Risk , 2008 .

[21]  Jane M. Booker,et al.  Point Process Models With Applications to Safety and Reliability , 1990 .

[22]  Waltraud Kahle,et al.  On a cumulative damage process and resulting first passages times , 2004 .