Gamma Distribution ◾ 295
NIST Formula for Gamma Parameter Extraction
Instead of the approach we used in plotting Figure 18.4 where we had used visual
judgment of histogram of data to decide on shape and the method of moment
formula given in Equation 18.2, to calculate scale, we can use the more rigorous
NIST [5] formulas presented in Equations 18.5 and 18.6 for parameter extraction:
Shape α =
x
s
2
(18.5)
Scale β =
s
x
2
(18.6)
where
x
is the mean and s is the standard deviation.
Applying Gamma Distribution to Software
Reliability Growth Modeling
Gamma distribution has been used in reliability studies. e gamma cumulative dis-
tribution function (CDF) is an S curve and can represent cumulative defects discov-
ered in software. An example of CDF, although on a different metric, may be found
in Figure 18.5. A CDF plotted with cumulative defects is known as the software
reliability growth curve, a subject treated in detail in Chapter 19 where the Weibull
distribution is used and in Chapter 21 where the Gompertz distribution is used.
Although the Weibull distribution is popularly used to fit failure data, the
gamma distribution is more suited in certain cases. In a research study, Sonia
Meskini [6] applies gamma distribution to construct a software reliability growth
model for smart phones. Sonia observes,
Working with a PDF such as gamma has further advantages. First, gaps in
data are filled by the equation. Second, one can extrapolate mathematically
and see beyond boundaries. ird, strategic planners can estimate risk.
ere are several options available while choosing a PDF for rainfall data.
Statistical techniques include the compound Poisson–exponential distribu-
tion; the log, square root, and cube root normal distributions; the gamma
distribution; various normalizing transforms; the kappa distribution, the
Weibull distribution, and the Box–Cox transformation.
However, the gamma distribution is frequently used to represent precipita-
tion because it provides a flexible representation of a variety of distribution
shapes while using only two parameters: shape and scale (based on Husak [4]).
296 ◾ Simple Statistical Methods for Software Engineering
In Skype Version I the Weibull distribution is the closest to the actual
behavior curve of the application, followed by the gamma distribution.
In Skype Version 2 it is to be noted that although the S-shaped distri-
bution is a particular case of the gamma distribution, it fits the data
slightly better. For Skype Version 3 it can be concluded that the gamma
distribution is the closest to the actual behavior curve of the application
failure data, followed by the Rayleigh distribution.
In her conclusion Sonia mentions,
We collected data from all over the world and divided them into differ-
ent versions, and grouped them into different time periods (days, weeks,
and months). Each application version failure data, when plotted in
time periods, shows the same pattern: an early ‘burst of failures,’ likely
due to the most evident defects, followed by a steep decrease in failure
rate. We first tried several non-linear distributions to better fit the fail-
ure data, and after numerous experiments, we found that the observed
behavior is better modeled by the Weibull or gamma distributions.
Sonia finally states that, in her observation,
It can be noted that the gamma distribution, along with its particular
S-shaped case, model more frequently the failure data.
In essence, gamma distribution is attractive because of its simplic-
ity, to perform as on Software reliability growth model. Gamma fits
elegantly to processes that are not in our direct control but dependent
on external factors like customer. With less data points we can do simu-
lation using gamma distribution.
Box 18.5 modelinG earthquake damaGe
with Gamma diStriBution
e application of the gamma distribution to failure data has been extended
to model damages produced by earthquakes. Repeated overloading of struc-
tures due to earthquake shocks cumulates damage on the hit structure. e
parameter “residual ductility to collapse” is used to measure this damage.
e cumulated level of deterioration can be modeled by gamma distribution.
e occurrence of earthquakes is a Poisson process (see Chapter 12, Law of
Rare Events), but cumulative wear is a gamma process. Structure reliability is
obtained by estimating the probability of cumulated damage exceeding the
threshold (based on Iervolino and Chioccarelli [7]).
Gamma Distribution ◾ 297
Review Questions
1. What is the relationship that connects the scale and shape parameters of
gamma distribution?
2. How will you estimate the scale parameter, if you can guess the shape param-
eter and have the value of mean of all observations?
3. How will you estimate the scale parameter directly from data?
4. Mention three applications of the gamma distribution.
5. What is the difference between a Poisson process and a gamma process?
Exercises
1. Plot a gamma curve using Excel function GAMMA.DIST for a shape of 3
and a scale of 10 units.
2. Plot a software reliability growth model using gamma distribution with a
scale of 30 days and a shape of 2.2.
3. For the SGRM you have drawn in Exercise 2, calculate the fraction of defects
remaining in the software on day 60. (Clue: use Excel function GAMMA.
INV to calculate the result.)
4. e mean value of a certain data set is 32.2. If the shape factor is estimated
as 3 by seeing the histogram of data, what would be the scale factor of the
gamma distribution of the data?
5. If the mean of data = 12 and the standard deviation is 2, estimate the gamma
shape and scale parameters to obtain a gamma model of data.
References
1. D. Kundu and A. Manglick, Discriminating between the Log-normal and Gamma
Distributions, Faculty of Mathematics and Informatics, University of Passau, Germany.
2. J. F. Lawless, Statistical Models and Methods for Lifetime Data, Wiley, New York, 1982.
3. H. Aksoy, Use of gamma distribution in hydrological analysis, Turkish Journal of
Engineering and Environmental Sciences, 24, 419–428, 2000.
4. G. J. Husak, J. Michaelsen and C. Funk, Use of the gamma distribution to represent
monthly rainfall in Africa for drought monitoring applications, International Journal of
Climatology, 27(7), 935–944, 2007.
5. NIST/SEMATECH Engineering Statistics Handbook, e National Institute of Standards
and Technology (NIST) is an agency of the U.S. Department of Commerce. Available
at http://www.nist.gov /itl/sed/gsg/handbook_project.cfm.
6. S. Meskini, Reliability Models Applied to Smartphone Applications (thesis), e School
of Graduate and Postdoctoral Studies, e University of Western Ontario London,
Ontario, 2013.
7. I. Iervolino and E. Chioccarelli, Gamma modeling of continuous deterioration and
cumulative damage in life-cycle analysis of earthquake-resistant structures, Proceedings
of the 11th Conference on Structural Safety and Reliability, New York, June 16–20, 2013.
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