A Reliability Growth Model

A system can be in one of two states: perfect (s
₀
) or imperfect (s
₁
). At any trial, the system can succeed or fail if it is in s
₁
, but will never fail if in s
₀
. After every failure, some corrective action is taken which, with a chance α, will lead to a transition to s0. Even if the corrective action does not succeed, i.e. the system is still in s1, the probability of failure reduces by a constant multiple at the next trial. Expressions for the probability that the system will be in s1 at trial n and the unconditional probability that the system fails in trial n, are derived. The latter can be expressed in terms of the former. These probabilities depend not only on the number of successes in n trials but on their sequence.