Journal of Statistical Planning and Inference 113 (2003) 541–549 www.elsevier.com/locate/jspi Simple bounds for terminating Poisson and renewal shock processes M.S. Finkelstein Department of Mathematical Statistics, University of the Orange Free State, P.O. Box 339, 9300 Bloemfontein, South Africa Received 17 November 2000; received in revised form 12 October 2001; accepted 15 November 2001 Abstract A system subject to a point process of shocks is considered. The shocks occur in accordance with a renewal process or a nonhomogeneous Poisson process. Each shock independently of the previous history leads to a system failure with probability  and is survived with a complimentary probability  . A number of problems in reliability and safety analysis can be interpreted by means of this model. The exact solution for the probability of survival  W (t; ) can be obtained only in the form of innite series (renewal process of shocks). Approximate solutions and new simple bounds for the probability of survival are obtained. The introduced method is based on the notion of a stochastic hazard rate process. A supplementary characteristic in this analysis is the mean of the hazard rate process. This method makes it possible to consider a generalization important in practical applications when the probability of a system failure under the eect of a current shock depends on the time since the previous one. c  2002 Elsevier Science B.V. All rights reserved. MSC: 62G18 Keywords: Terminating Poisson process; Terminating renewal process; Shock process; Harward rate process 1. Introduction Consider an i.i.d. sequence {X n } n¿0 of lifetime random variables with distribu- tion function (DF) F (t ). Let G be a geometric variable with parameter  (indepen- dent of {X n } n¿0 ) and W (t; ) denotes the corresponding geometric sum of lifetimes E-mail address: msf@wwg3.uovs.ac.za (M.S. Finkelstein). 0378-3758/02/$-see front matter c  2002 Elsevier Science B.V. All rights reserved. PII:S0378-3758(02)00111-8