Template:Process ex rsa

Process Example
Now suppose that a system consists of many components with each component in a socket. A failure in any socket constitutes a failure of the system. Each component in a socket is a renewal process governed by its respective distribution function. When the system fails due to a failure in a socket, the component is replaced and the socket is again as good as new. The system has been repaired. Because there are many other components still operating with various ages, the system is not typically put back into a like new condition after the replacement of a single component. For example, a car is not as good as new after the replacement of a failed water pump. Therefore, distribution theory does not apply to the failures of a complex system, such as a car. In general, the intervals between failures for a complex repairable system do not follow the same distribution. Distributions apply to the components that are replaced in the sockets but not at the system level. At the system level, a distribution applies to the very first failure. There is one failure associated with a distribution. For example, the very first system failure may follow a Weibull distribution.

For many systems in a real world environment, a repair is only enough to get the system operational again. If the water pump fails on the car, the repair consists only of installing a new water pump. If a seal leaks, the seal is replaced but no additional maintenance is done, etc. This is the concept of minimal repair. For a system with many failure modes, the repair of a single failure mode does not greatly improve the system reliability from what it was just before the failure. Under minimal repair for a complex system with many failure modes, the system reliability after a repair is the same as it was just before the failure. In this case, the sequence of failure at the system level follows a non-homogeneous Poisson process (NHPP). The system age when the system is first put into service is time $$0$$. Under the NHPP, the first failure is governed by a distribution $$F(x)$$  with failure rate  $$r(x)$$. Each succeeding failure is governed by the intensity function $$u(t)$$  of the process. Let $$t$$  be the age of the system and  $$\Delta t$$  is very small. The probability that a system of age $$t$$  fails between  $$t$$  and  $$t+\Delta t$$  is given by the intensity function  $$u(t)\Delta t$$. Notice that this probability is not conditioned on not having any system failures up to time $$t$$, as is the case for a failure rate. The failure intensity $$u(t)$$  for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, $$u(t)=r(t)$$, where  $$r(t)$$  is the failure rate for the distribution function of the first system failure. If the first system failure follows the Weibull distribution, the failure rate is:


 * $$r(x)=\lambda \beta {{x}^{\beta -1}}$$

Under minimal repair, the system intensity function is:


 * $$u(t)=\lambda \beta {{t}^{\beta -1}}$$

This is the Power Law model. It can be viewed as an extension of the Weibull distribution. The Weibull distribution governs the first system failure and the Power Law model governs each succeeding system failure. If the system has a constant failure intensity $$u(t)$$  =  $$\lambda $$, then the intervals between system failures follow an exponential distribution with failure rate  $$\lambda $$. If the system operates for time $$T$$, then the random number of failures  $$N(T)$$  over  $$0$$  to  $$T$$  is given by the Power Law mean value function.


 * $$E[N(T)]=\lambda {{T}^{\beta }}$$

Therefore, the probability $$N(T)=n$$  is given by the Poisson probability.


 * $$\frac{n!};\text{ }n=0,1,2\ldots $$

This is referred to as a homogeneous Poisson process because there is no change in the intensity function. This is a special case of the Power Law model for $$\beta =1$$. The Power Law model is a generalization of the homogeneous Poisson process and allows for change in the intensity function as the repairable system ages. For the Power Law model, the failure intensity is increasing for $$\beta >1$$  (wearout), decreasing for  $$\beta <1$$  (infant morality) and constant for  $$\beta =1$$  (useful life).