8.
Assessing Product Reliability
8.1. Introduction 8.1.7. What are some basic repair rate models used for repairable systems?


The repair rate for a NHPP following the Power law  A flexible model (that has been very successful in many applications) for the expected number of failures in the first \(t\) hours, \(M(t)\), is given by the polynomial $$ M(t) = at^b, \,\, \mbox{ for } \,\, a, \, b > 0 \,\, . $$ The repair rate (or ROCOF) for this model is $$ m(t) = abt^{b1} = \alpha t^{\beta}, \,\, \mbox{ for } \,\, \alpha > 0, \,\, \beta < 1 \,\, . $$  
The Power law model is very flexible and contains the HPP (exponential) model as a special case 
The HPP model has a the constant repair rate \( m(t) = \lambda\).
If we substitute an arbitrary function \(\lambda(t)\) for \(\lambda\)
we have a NonHomogeneous Poisson Process (NHPP) with Intensity Function \(\lambda\).
If
$$ \lambda(t) = m(t) = \alpha t^{\beta} \,\, , $$
then we have an NHPP with a Power Law intensity function (the "intensity
function" is another name for the repair rate \(m(t)\)).
Because of the polynomial nature of the ROCOF, this model is very flexible and can model both increasing (\(b>1\) or \(\beta < 0\)) and decreasing (\(0 < b < 1\) or \(0 < \beta < 1\)) failure rates. When \(b\) = 1 or \(\beta\) = 0, the model reduces to the HPP constant repair rate model. 

Probabilities of failure for all NHPP processes can easily be calculated based on the Poisson formula  Probabilities of a given number of failures for the NHPP model are calculated by a straightforward generalization of the formulas for the HPP. Thus, for any NHPP $$ P(N(T) = k) = \frac{M(T)^k}{k!} e^{M(T)} \,\, , $$ and for the Power Law model: $$ P(N(T)=k) = \frac{ \left[ aT^b \right]^k \mbox{exp}(aT^b)}{k!} = \frac{a^k T^{bk} \mbox{exp}(aT^b)}{k!} \,\, . $$  
The Power Law model is also called the Duane Model and the AMSAA model  Other names for the Power Law model are: the Duane Model and the AMSAA model. AMSAA stands for the United States Army Materials System Analysis Activity, where much theoretical work describing the Power Law model was performed in the 1970's.  
It is also called a Weibull Process  but this name is misleading and should be avoided 
The time to the first fail for a Power Law process has a
Weibull distribution with shape parameter \(b\)
and characteristic life \(a\).
For this reason, the Power Law model
is sometimes called a Weibull Process. This name is confusing
and should be avoided, however, since it mixes a life distribution model
applicable to the lifetimes of a nonrepairable population with a model
for the interarrival times of failures of a repairable population.
For any NHPP process with intensity function \(m(t)\), the distribution function (CDF) for the interarrival time \(\tau\) to the next failure, given a failure just occurred at time \(T\), is given by 

Once a failure occurs, the waiting time to the next failure for an NHPP has a simple CDF formula  $$ F_T(t) = 1  \mbox{exp}\left( \int_0^t m(T+\tau)d\tau \right) \,\, . $$ In particular, for the Power Law the waiting time to the next failure, given a failure at time \(T\), has distribution function $$ F_T(t) = 1  \mbox{exp}\left( a \left[(T+t)^b  T^b \right] \right) \,\, . $$ This inter arrival time CDF can be used to derive a simple algorithm for simulating NHPP Power Law Data. 