Page 119 - Reliability and Maintainability of In service Pipelines

108 Reliability and Maintainability of In-Service Pipelines

4.3 Gamma Process Concept

To deal with data scarcity and uncertainties, using stochastic models for time-
dependent reliability analysis of deteriorating buried pipes can be considered. In
order to model monotonic progression of a deterioration process, the stochastic
gamma process concept can be used for modeling the reduction of pipe wall
thickness due to corrosion. The gamma process is a stochastic process with inde-
pendent, nonnegative increments having a gamma distribution with an identical
scale parameter and a time-dependent shape parameter.
A stochastic process model, such as gamma process, incorporates the temporal
uncertainty associated with the evolution of deterioration (e.g., Bogdanoff and
Kozin, 1985; Nicolai et al., 2004; van Noortwijk and Frangopol, 2004).
The gamma process is suitable to model gradual damage monotonically accu-
mulating over time, such as wear, fatigue, corrosion, crack growth, erosion, con-
sumption, creep, swell, a degrading health index, etc. For the mathematical
aspects of gamma processes, see Dufresne et al. (1991), Singpurwalla (1997), and
van der Weide (1997).
Abdel-Hameed (1975) was the first to propose the gamma process as a model
for deterioration occurring randomly in time. In his paper he called this stochastic
process the “gamma wear process.” An advantage of modeling deterioration pro-
cesses through gamma processes is that the required mathematical calculations
are relatively straightforward.

4.3.1 PROBLEM FORMULATION

The mathematical definition of the gamma process is given in Eq. (4.10). Given
that a random quantity d has a gamma distribution with shape parameter α . 0
and scale parameter λ . 0 if its probability density function is given by:
λ

α
α21 2λd
Ga d α; λ 5 d e ð4:10Þ

Γ αðÞ

Let α tðÞ be a nondecreasing, right continuous, real-valued function for t $ 0,
with αð0Þ 0. Γ αðÞ denotes gamma function of α with mathematical definition
of Γ αðÞ 5 α 2 1Þ!. The gamma process is a continuous-time stochastic process
ð

dtðÞ; t $ 0 with the following properties:
1. d 0ðÞ 5 0 with probability one;
2. d τðÞ 2 dðtÞBGaðατðÞ 2 α tðÞ; λÞ for all τ . t $ 0;
3. dðtÞ has independent increments.