16.1 THE WEIBULL PROCESS AND ITS CHARACTERISTICS

The Weibull process is a non-homogeneous Poisson process with a failure rate function

Consider a repairable equipment that operates from age zero, fails at age T ₁, gets repaired immediately, operates again, fails later at age T ₂, gets repaired again, operates again and the process gets repeated. If the repair time is negligible, or the repair time is excluded from the calendar time ellapsed and only the actual time of operation is considered, a sequence of cumulative operating times-to-failure data are thus obtained; i.e.,

Consider a fixed time interval [0, t], during which the number of failures, denoted by N(0, t) or N( t) for short, is a random variable. Then N( t) is a Weibull process, if it satisfies the following two properties:

Let N( t, t + ? t) denote the number of failures in the time interval ( t, t + ? t), then

Condition 1 is

and

where o( ? t) denotes a function tending to zero more rapidly than ? t, or the higher order terms in ? t, such as the probability of two failures in ? t given by [ ?( t) ? t] ², and ?( t) is the failure rate function defined by Eq. (16.1).

Condition 2 is

N( t, t + ?