–>The failure process has an independent increment, i.e. the number of failures during the time interval (t, t + s) depends on the current time t and the length of time interval s, and does not depend on the past history of the process. –> The failure rate of the process is given by P{exactly one failure in (t, t + ∆t)} = P{N(t, t + ∆t) – N(t)=1} =(t)∆t + o(∆t) where (t) is the intensity function. –> During a small interval ∆t, the probability of more than one failure is negligible, that is, P{two or more failure in (t, t+∆t)} = o(∆t) –> The initial condition is N(0) = 0.
On the basis of these assumptions, the probability of exactly n failures occurring during the time interval (0, t) for the NHPP is given by
where
Reliability Function: The reliability R(t), defined as the probability that there are no failures in the time interval (0, t), is given by
In general, the reliability R(x|t), the probability that there are no failures in the interval (t, t + x), is given by
and its density is given by
where
The variance of the NHPP can be obtained as follows:
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