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


Repair rate (ROCOF) models and formulas  The simplest useful model for \(M(t)\) is \(M(t) = \lambda t\) and the repair rate (or ROCOF) is the constant \(m(t) = \lambda\). This model comes about when the interarrival times between failures are independent and identically distributed according to the exponential distribution, with parameter \(\lambda\). This basic model is also known as a Homogeneous Poisson Process (HPP). The following formulas apply. $$ \begin{array}{rcl} F(t) = 1  e^{\lambda t} & = & \mbox{CDF of the waiting time to the next failure} \\ & & \mbox{(or interarrival time between failures)} \\ & & \\ N(T) & = & \mbox{cumulative number of failures from time } 0 \mbox{ to time } T \\ & & \\ P\{N(t)=k\} & = & \frac{(\lambda T)^k e^{\lambda T}}{k!} \\ & & \\ M(t) = \lambda T & = & \mbox{expected number of failures by time } T \\ & & \\ M'(t) = m(t) & = & \lambda = \mbox{repair rate or ROCOF} \\ & & \\ \frac{1}{\lambda} & = & \mbox{Mean Time Between Failures (MTBF)} \end{array} $$  
HPP model fits flat portion of "bathtub" curve  Despite the simplicity of this model, it is widely used
for repairable equipment and systems throughout industry. Justification
for this comes, in part, from the shape of the empirical
Bathtub Curve. Most systems (or complex tools or equipment)
spend most of their "lifetimes" operating in the long flat constant repair rate portion of
the Bathtub Curve. The HPP is the only model that applies to that portion
of the curve, so it is the most popular model for system reliability evaluation
and reliability test planning.
Planning reliability assessment tests (under the HPP assumption) is covered in a later section, as is estimating the MTBF from system failure data and calculating upper and lower confidence limits. 

Poisson relationship  In the HPP model, the probability of having exactly \(k\) failures by time \(T\) is given by the Poisson distribution with mean \(\lambda T\) (see formula for \(P\{N(t) = k\}\) above). 