Gamma Process (GP) Deterioration Model
The gamma process is a continuous-time Markov process with independent and gamma distributed increments (Abdel-Hameed 1975). The gamma process is a limit of a compound Poisson process with gamma distributed increments. The limit is reached when the Poisson rate approaches infinity in any finite time interval as the size of the increment tends proportionally to zero (Dufresne et al. 1991). In other words, the gamma process model implies that deterioration progresses with frequent occurrence of very small increment. A physical example such process is the flow accelerated corrosion in nuclear piping system.
In the stationary gamma process (GP) model, the cumulative deterioration X(t) follows a gamma distribution ga(x; a t, в) with the shape parameter at and the scale parameter Ц. The mean, variance and COV of X(t) are given as
In this model, both mean and variance of the deterioration are linear in time. In RV model, the variance is quadratic in time as shown in Eq.(3). From Eq. (1), the cumulative lifetime distribution can be expressed as
FT (t) — P[X (t) >p] — 1 — GA[p;at, e] (11)
The probability density function, fff) = dFT(t)/dt, has no closed form expression, but it can be computed numerically. Similarly, the moments of the lifetime have to be evaluated by numerical integration.