8.
Assessing Product Reliability
8.1. Introduction 8.1.2. What are the basic terms and models used for reliability evaluation?


Repair Rate models are based on counting the cumulative number of failures over time  A different approach is used
for modeling the rate of occurrence of failure incidences for a repairable
system. In this chapter, these rates are called repair rates (not
to be confused with the length of time for a repair, which is not discussed
in this chapter). Time is measured by system poweronhours from initial
turnon at time zero, to the end of system life. Failures occur as given
system ages and the system is repaired to a state that may be the same
as new, or better, or worse. The frequency of repairs may be increasing,
decreasing, or staying at a roughly constant rate.
Let \(N(t)\) be a counting function that keeps track of the cumulative number of failures a given system has had from time zero to time \(t\). \(N(t)\) is a step function that jumps up one every time a failure occurs and stays at the new level until the next failure. Every system will have its own observed \(N(t)\) function over time. If we observed the \(N(t)\) curves for a large number of similar systems and "averaged" these curves, we would have an estimate of \(M(t)\) = the expected number (average number) of cumulative failures by time \(t\) for these systems. 

The Repair Rate (or ROCOF) is the mean rate of failures per unit time  The derivative of \(M(t)\),
denoted \(m(t)\),
is defined to be the Repair Rate or the
Rate Of Occurrence Of Failures at Time \(t\),
or ROCOF.
Models for \(N(t)\), \(M(t)\), and \(m(t)\) will be described in the section on Repair Rate Models. 