``Failure'' problem for a stochastic point process (Q1094754)

Consider a system in which certain physical events take place at random instants \(0<T_ 1<T_ 2<..\). and denote this point process by \(\{\) N(s)\(| s\geq 0\}\). At the instant T of realization of one of these events there occurs a ``failure'' in the evolution of the system such that the time interval to the next fixed event and all the subsequent intervals between these events have new distributions. The aim of this paper is to obtain an explicit equation for the conditional probability P(T\(\leq t| N(S),s\leq t)\).