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5.1 Results of NHPP Modeling Using MLE
Goel-Okumoto model, which is a non-homogeneous Poisson process model, was
implemented on the failure count dataset available from telecom service provider
company because it is considered to be most suitable for handling billing and other
inventory related data. The parameters a and b were estimated using Maximum
Likelihood Estimator (MLE) method. The non-linear equation for parameter b was
solved using a widely used numerical root-solving technique namely Newton-
Raphson. Initial approximation for b given in this method was a small value like 0.1
or 0.2. The resulting value of b was then used to compute parameter a. Parameters
were estimated for each week based on failure data of previous weeks and sub-
sequently reliability was calculated from the values of the parameters. A graph was
plotted between the reliability function and time to verify the concave nature of the
GO model. Graphs were also plotted for a versus time and b versus time which
showed the characteristic curves for total number of faults at end of each week
resulting from the testing process and failure occurrence rate respectively. We
observe:
Maximum Likelihood Estimation technique results in instable and non-existent
values for parameter a and b for certain weeks due to which reliability function
values are incorrect for those weeks.
Erroneous portions in the concave graph of reliability versus time.
Erroneous portions in graphs for a versus time and b versus time.
2
Simulation plots presented here take 11th week as their starting week. The
starting value is user de
flexibility.
Figure
3
shows how the graphs for weeks 11 and 12 have incorrect reliability
values due to NaN values of parameter b. This b value when substituted into the
equation for r(t|s) made the power of exponential term 0
ned for more
fl
finally making value of
reliability zero. For rest of the weeks, parameter values being stable resulted in
correct concave graph which is shown without errors in graph of Fig.
4
.
Parameter a represents the
final number of faults that can be detected during
testing. It increases as certain numbers of failures are detected in each succeeding
week and gets added up to the total count. This is shown correctly for weeks 13
28
as depicted in graph of Fig.
5
. However, MLE results in incorrect estimation of
values of a for week 11 and 12 which produce erroneous graph of Fig.
6
. Graph in
Fig.
6
excludes the values of a for weeks 11 and 12 to give correct characteristics.
Figure
7
demonstrates graph for parameter b which denotes the failure occur-
rence rate, versus time. The graph has neglected the NaN values for week 11 and
12. Comparing it with the software reliability revised bathtub curve, it can be found
that it resembles the useful life portion of the graph. As upgrades are made, failure
-
2
Graphs presented here take 11th week as their starting week. The starting value can be user
de
ned for more flexibility.
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