8.
Assessing Product Reliability
8.1. Introduction 8.1.9. How can you model reliability growth?


If the Power Law applies, Repair Rates improve over time according to the formula \(\alpha t^{\beta}\) The exponent \(\beta\) lies between 0 and 1 and is called the reliability growth slope 
This repairable system model was described in
Section 8.1.7.2. The expected number of failures by time \(t\)
has the form \(M(t) = a t^b\)
and the repair rate has the form \(m(t) = a b t^{b1}\).
This will model improvement when 0 \( < b < \) 1,
with larger improvements coming when \(b\)
is smaller. As we will see in the next section on Duane Plotting, it is convenient to define
\(\beta = 1  b\) and \(\alpha = ab\) ,
and write the repair rate as
$$ m(t) = \alpha t ^{\beta} \,\, . $$
Again we have improvement when 0 \(< \beta <\) 1,
with larger improvement coming from larger values of \(\beta\).
\(\beta\)
is known as the
Duane Plot slope or the reliability
improvement Growth Slope.
In terms of the original parameters for \(M(t)\), we have $$ a = \frac{\alpha}{1\beta}\,\,\, \mbox{ and } \,\,\, b = 1\beta \,\, . $$ Use of the Power Law model for reliability growth test data generally assumes the following: 1. While the test is ongoing, system improvements are introduced that produce continual improvements in the rate of system repair. 2. Over a long enough period of time the effect of these improvements can be modeled adequately by the continuous polynomial repair rate improvement model \(\alpha t^{\beta}\). 

When an improvement test ends, the MTBF stays constant at its last achieved value 
3. When the improvement test ends at test time \(T\) and no
further improvement actions are ongoing, the repair rate has been reduced
to \(\alpha T^{\beta}\).
The repair rate remains constant from then on at this new (improved) level.
Assumption three means that when the test ends, the HPP constant repair
rate model takes over and the MTBF for the system from then on is the reciprocal
of the final repair rate or \((T^{\beta})/\alpha\).
If we estimate the expected number of failures up to time T by the actual
number observed, the estimated MTBF at the end of a reliability test (following
the Power Law) is:
$$ \mbox{ESTIMATED MTBF AT END OF TEST } = \frac{T}{r(1\beta)} \,\, , $$
with \(T\) denoting the test time,\(r\)
is the total number of test failures and \(\beta\)
is the reliability growth slope. A formula for estimating \(\beta\)
from system failure times is given in the
Analysis Section for the Power Law model.


Simulated Data Example 
Simulating NHPP Power Law Data
Step 1: User inputs the positive constants \(a\) and \(b\). Step 2: Simulate a vector of \(n\) uniform (0,1) random numbers. Call these \(U_1, \, U_2, \, U_3, \, \ldots, \, U_n\). Step 3: Calculate \(Y_1 = \left\{ \frac{1}{a} \mbox{ ln } U_1 \right\}^{1/b}\). Step \(i\): Calculate \(Y_i = \left\{ Y_{i1}^b  \frac{1}{a} \mbox{ ln } U_i \right\}^{1/b} \) for \(i\) = 2, ..., \(n\). The \(n\) numbers \(Y_1, \, Y_2, \, \ldots, \, Y_n\) are the desired repair times simulated from an NHPP Power Law process with parameters \(a, \, b\) (or \(\beta = 1b\) and \(\alpha = ab\)). We generated \(n\) = 13 random repair times using the NHPP Power Law process with \(a\) = 0.2 and \(b\) = 0.4. The resulting data and a plot of failure number versus repair times are shown below.
