Gamma process

In addition to Federal Government (U. S. Atomic Energy Commission, U. S. Army, Bureau of Mines, Bureau of Standards, etc.) requirements, several companies, including the American Novawood Corp., American Nuclear Corp., Gamma Process Co., General Electric Co., Lockheed Aircraft Corp., Nuclear Materials and Equipment Corp., Picker X-Ray Corp., and U. S. Nuclear Corp., and the Japanese and Canadian governments have... 

Reid D., Gamma processing technology an alternative technology for terminal sterilization parenterals, J. Parenteral Sci.Technol., 1995,49,83-89. 

The lowest temperature assigned to the glass transition (1) has been termed the gamma process. Torsion pendulum measurements 1,5, 6, 7) suggest that this relaxation has two or more components. Dilatometry (8) resolved two discrete transitions at 108 and 152 K which have been labeled the 711 and 71, respectively (5,6, 7,9). Illers data on samples of various crystallinities support the conclusion that 711 and 71 arise from crystalline and amorphous phases, respectively (5, 6, 7). Several mechanisms for this transition have been discussed which involve local motions of three to five methylene units (5, 6, 7, 9). The interpretation of this relaxation as the glass transition is doubtful (2). 

Carr P, Chang E, Madan D (1998) The Variance Gamma Process and Option Pricing. European Finance Review 2 79-105. 

The Gamma process is an example of a Levy process with infinite number of jumps. The Gamma process Xp(t) is a pure jump Levy process with the intensity measure... 

The jumps in Xp(t) with a size in the interval [z, z + dz] occur v(dz). The parameter Ko determines the rate of jumps per unit time and p is the measure of jump size. Since the jumps are positive, we analyze the Gamma process by using the Laplace transform... 

The expression [P/(s + is the Laplace transform of the density of the Gamma process [15]. Therefore, we can find an explicit expression for the mesoscopic density of particles for x > 0 ... 

Van Noortwijk, J.M., Van der Weide, J.A.M., Kallen, J.M., and Pandey, M.D. 2007. Gamma processes and peaks-over-threshold distributions for time-dependent reliability. gi jlity n ineerin S sterr j 92(12) 1651-1658. 

ABSTRACT A system is subjected to difierent cracks. A crack can be only detected when its length exceeds a fixed minimum value. The length of the crack is modeled using a gamma process. When the length of the crack reaches a certain level or when the munber of cracks detected in the system attains a fixed value, the system fails and needs to be replaced immediately. A preventive maintenance is carried out when the age of the system exceeds a value T. A corrective maintenance is performed when the system fads. Under a corrective maintenance and imder a preventive maintenance, the system is replaced by a new one. Costs are associated with the corrective maintenance and with the preventive maintenance. The problem is to determine an optimal planned replacement time T, minimizing the average cost rate of the system. 

Furthermore, let a(f) be a non-decreasing, right-continuous, real valued function for i > 0 with a(0) = 0. The gamma process with shape function a t) > 0 and scale parameter /S > 0 is a continuous-time stochastic process X(f), f > 0 with the following properties... 

We denote by ij, / = 1,2,... the length of the i-th crack t units of time after its initiation. We assume that LJ follows an homogeneous gamma process with shape and scale parameters given by at and p respectively, that is, for i < i, the density function of the increment of the length of the crack / is given by... 

In order to illustrate the impact of the preventive maintenance policy. Figure 1 shows the evolution of the average maintenance cost as a function of the preventive maintenance parameter T. The parameters for the Poisson and the gamma process are a = 4.10 , /S=1.5( are millimeters per hour),... 

Noortwijk, J. (2009). A survey of the application of gamma processes in maintenance. 

First, we describe the degradation model and then we present one of the more nsed process the gamma process (van Noortwijk 2009). 

Consider a system or a component subject to an accumulation of degradation in time which can be represented by the degradation variable Z,. The degradation is supposed to be gradual and monotone. Abdel-Hameed (1975) proposed to use the gamma process as a proper model for deterioration occurring at random time-instants. 

Z,(i), t > 0 is a gamma process with shape parameter a and scale parameter f -. 

Note E and Var respectively the mean and the variance of the gamma process. 

Suppose also that the degradation variable Z,- of each component i is measurable at each decision time Tdj. This data is retrieved via a feedback of information providing from telematic systems or from workshops at each decision time Ti. We consider that each component degradation can be modelled using a gamma process. 

The considered system gradually deteriorates in a strictly increasing way and the evolution of its deterioration depends only on the deterioration level at that time. To model the deterioration a gamma process is used as it has been widely used in the literature, refer to Abdel-Hameed (1975), van Noortwijk (2009) and Grail et al. (2002). Some characteristics of this process is that it is clearly monotone increasing which is the behaviour observed in most physical deterioration... 

It is assumed that the future deterioration of the system is independent ofthe current level of deterioration and depends only on the duration of the period over which the system is allowed to deteriorate. The gamma process is well adapted as it satisfies... 

The ongoing research is to integrate this multiobjective portion with a Gamma process which has been commonly used to address issues related to continuous degradation (see Noortwijk (2009) for more details). In this way the work developed by Castanier et al. (2003) would be extended. 

The basic procedure in performance qualification (PQ) exercises is the measurement of dose distribution in real products. In gamma processing, similar dosimetry systems can be applied like in OQ exercises. In electron processing, usually film dosimeters are suggested for use with the restriction of the application of only thin film dosimeters for dose mapping in inhomogeneous products. 

Note The dosimetry systems applicable in gamma processing can also be used in X-ray processing according to the present knowledge. 

Other groups of these dosimeters contain certain dyes mixed into the basic material (in most cases a polymer), and the optical absorption of these dyes changes upon irradiation. These systems are simple to measure and apply, but their response is affected by environmental factors such as humidity, light, and temperature. Some of these systems are preferred in gamma processing, while the main application field of others is electron-beam dosimetry. 

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