Template:Failure rate and failure intensity

Failure Rate and Failure Intensity

Failure rate and failure intensity are very similar terms. The term failure intensity typically refers to a process such as a reliability growth program. The system age when a system is first put into service is time [math]\displaystyle{ 0 }[/math] . Under the non-homogeneous Poisson process (NHPP), the first failure is governed by a distribution [math]\displaystyle{ F(x) }[/math] with failure rate [math]\displaystyle{ r(x) }[/math] . Each succeeding failure is governed by the intensity function [math]\displaystyle{ u(t) }[/math] of the process. Let [math]\displaystyle{ t }[/math] be the age of the system and [math]\displaystyle{ \Delta t }[/math] is very small. The probability that a system of age [math]\displaystyle{ t }[/math] fails between [math]\displaystyle{ t }[/math] and [math]\displaystyle{ t+\Delta t }[/math] is given by the intensity function [math]\displaystyle{ u(t)\Delta t }[/math] . Notice that this probability is not conditioned on not having any system failures up to time [math]\displaystyle{ t }[/math] , as is the case for a failure rate. The failure intensity [math]\displaystyle{ u(t) }[/math] for the NHPP has the same functional form as the failure rate governing the first system failure. Therefore, [math]\displaystyle{ u(t)=r(t) }[/math] , where [math]\displaystyle{ r(t) }[/math] 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:

[math]\displaystyle{ r(x)=\lambda \beta {{x}^{\beta -1}} }[/math]

Under minimal repair, the system intensity function is:

[math]\displaystyle{ u(t)=\lambda \beta {{t}^{\beta -1}} }[/math]

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. Additional information on the power law model can also be found in Chapter 13.