A Simplified Model for Evaluating Software Reliability at the Developmental Stage

Shelbi Joseph, Shouri P.V & Jagathy Raj V. P
International Journal Software Engineering (IJSE), Volume (1): Issue (5) 125
A Simplified Model for Evaluating Software Reliability at the
Developmental Stage
Shelbi Joseph achayanshelbil@gmail.com
Division of Information Technology
School of Engineering
Cochin University of Science and Technology
Cochin, India
Shouri P.V pvshouri@gmail.com
Department of Mechanical Engineering
Model Engineering College
Cochin, India
Jagathy Raj V. P jagathy@cusat.ac.in
School of Management Studies,
Cochin University of Science and Technology
Cochin, India
Abstract
The use of open source software is becoming more and more predominant and it is important
that the reliability of this software are evaluated. Even though a lot of researchers have tried
to establish the failure pattern of different packages a deterministic model for evaluating
reliability is not yet developed. The present work details a simplified model for evaluating the
reliability of the open source software based on the available failure data. The methodology
involves identifying a fixed number of packages at the start of the time and defining the failure
rate based on the failure data for these preset number of packages. The defined function of
the failure rate is used to arrive at the reliability model. The reliability values obtained using
the developed model are also compared with the exact reliability values.
Key words: Bugs, Failure Density, Failure Rate, Open Source Software, Reliability
1. INTRODUCTION
Open Source Software (OSS) has attracted significant attention in recent years [1]. It is being accepted as a
viable alternative to commercial software [2]. OSS in general refers to any software whose source code is freely
available for distribution [3]. However the OSS development approach is still not fully understood [4]. Reliability
estimation plays a vital role during the developmental phase of the open source software. In fact, once the
package has stabilized (or developed) then chances of further failure are relatively low and package will be more
or less reliable. However, during the developmental stage failures or bug arrival are more frequent and it is
important that a model has to be developed to evaluate the reliability during this period. The bug arrivals usually
peak at the code inspection phase and get rather stabilized in the system test phase [5]. Software reliability
evaluation is an increasingly important aspect of software development process [6].
Reliability can be defined as the probability of failure free operation of a computer program in a specified
environment for a specified period of time [4,5]. It is evident from the definition that there are four key elements
associated with the reliability namely element of probability, function of the product, environmental conditions,
and time.
Reliability is nothing but the probability of success. As success and failure are complementary, a measure of the
failure is essential to arrive at the reliability. That is,

Reliability = Probability of success = 1- Probability of failure (1)
From equation (1), it is evident that the first step in reliability analysis is failure data analysis. This involves fixing
up a time interval and noting down the failures at different time intervals. The number of packages at the start of
the analysis is defined as the initial population and the survivors at any point of time is the difference of initial
population and the failures that have occurred till this point. Failure rate associated with a time interval can be
defined as the ratio of number of bugs reported during the
NOMENCLATURE
fd (t) failure density
N initial population
R(t) reliability
t time
Z(t) failure rate
λ constant failure rate
given time interval to the average population associated with the time interval. Once the variation of failure rate
with respect to time can be established an equation can be used to fit the variation which will be the failure
model for reliability estimation. Typical reliability models include Jelinski-Moranda [6], Littlewood [7] , Goel-
Okumoto [8], Nelson model [9], Mills model [10],Basin model [10],Halstead model [11] and Musa-Okumoto [4].
For software projects that have not been in operation long enough, the failure data collected may not be
sufficient to provide a decent picture of software quality, which may lead to anomalous reliability estimates [12,
13] Weibull function is also used for reliability analysis and the function has been particularly valuable for
situations for which the data samples are relatively small [14].
Concern about software reliability has been around for a long time [15,16 ] and as open source is a relatively
novel software development approach differing significantly from proprietary software waterfall model, we do not
yet have any mature or stable technique to assess open source software reliability [17].
It is clear from the above discussions that even though a variety of models are available for reliability prediction,
a deterministic model is presently not available. Or in other words, none of these models quantifies reliability.
The present work focuses on development of an algorithm and there by a simplified method of quantifying
reliability of a software.
2. MODEL DEVELOPMENT AND ALGORITHM
An open source program typically consists of multiple modules [18]. Attributes of the reliability models have been
usually defined with respect to time with four general ways to characterize [19, 20] reliability, time of failure, time
interval between failures, cumulative number of faults upto a period of time and failure found in a time interval.
The present methodology involves defining an equation for the pattern of failure based on the available bug
arrival rate and developing a generalized model for the reliability of the software. The following are the
assumptions involved in the analysis.
1. The software analyzed is an open source.
2. As the open source software is made up of a very large community the environmental changes are not
considered.
3. The total number of packages at the beginning of the analysis is assumed to remain constant and is
taken as the initial population.
4. The failures of various packages are assumed to be independent of each other.
5. The model is developed for evaluation of the software reliability at the developmental stage and the
packages that fail during this period are not further considered. It is further assumed that by the end of

developmental stage the bug associated with the failed packages would be eliminated and will be stable
further.
6. The reliability of the software is inversely proportional to the number of bugs reported at any point of
time.
7. The beginning of the time period after which the bug arrival or failure rate remains constant marks the
culmination of the developmental stage and the software will be stabilize.
Based on the above assumptions a 6- step algorithm is developed for the analysis as detailed below.
1. Indentify the total initial population. This corresponds to the total number of packages existing at the
beginning of the time period. That is, at the start of analysis.
2. Define a time period and find out the bugs reported during this time interval. As the failure would have
occurred anywhere between the time interval, the reported failures are indicated in between the time
interval.
3. Calculate the cumulative failures and thereby the survivors and different points in time.
4. Estimate the failure rate associated with the time intervals by dividing the number of failures associated
with the given unit time interval by average population associated with the time interval. Average
population associated with a given time interval is the average of survivors at the beginning and end of
the time period.
5. Plot the graphs defining the relation between failure rate and time and obtain the equation defining the
relation between failure rate and time.
6. Obtain the expression for reliability of the software by substituting the equation of failure rate in
equation(1) given as
∫−
=
t
dttZ
etR 0
)(
)( (1)
3. RESULTS AND DISCUSSION
A total of 1880 packages were available at the start of the analysis as per the details available from the
official website of Debian [21]. This is taken as the initial population. A time interval of 1 month is fixed and
the bug arrival rate during this interval is noted. The reported errors at different time intervals are given in the
Table 1. The observations are taken for 1 year after which the bug arrival is negligible indicating that the
software has more or less stabilized.
TABLE 1: Failure Data Analysis

F(t) = - 0.0004 t + 0.078
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9 10
Time (months)
Failurerate
FIGURE 1. Variation of failure rate with time
The variation of failure rate with respect to time is shown in Fig. 1. It can be seen that after the 8
th
month
onwards the software has somewhat stabilized indicating the completion of developmental phase. The
failure model corresponding to the failure rate can be expressed by the equation (2)
078.00004.0)( +−= ttZ (2)
The corresponding reliability can be expressed by the equation (3) as
dtt
t
etR
)078.00004.0(
0
)(
∫ +−−
=
That is,
t
t
etR
078.0
2
0004.0 2
)(
−
= (3)
Failure density associated with the time intervals is the ratio of number of failures associated with the given
unit time interval to the initial population. Failure density can be related with Reliability and failure rate using
the equation (4) as
)()()( tZtRtfd ×= (4)
Therefore, based on the developed model failure density can be expressed as
)078.00004.0()(
078.0
2
0004.0 2
+−×=
−
tetf
t
t
d (5)
The reliability of the software at different points in time is calculated using the equation (3). The actual
values of reliability obtained by dividing the survivors at the given point in time by the initial population are
also calculated. The Musa model assumes a constant value for the failure rate and by considering this as
the average value of failure rates the reliability values are calculated using the equation
t
etR λ−
=)( (6)

The reliability values calculated using the three different methods and the failure density values are shown in
table 2.
TABLE2. Reliability and failure density
Fig. 2 shows a comparison of reliability obtained using the developed simplified model and Musa model with
the actual reliability values. It can be seen that the simplified model and the Musa model nearly provides the
same results. Further, these two models very closely approximate the real situation. The variation of failure
density with time is also shown in Fig. 3
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
Time (months)
Reliability
Reliability (Model)
Reliability (Actual)
Reliability (Musa)
FIGURE2. Comparison of reliability obtained using different models

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
1 2 3 4 5 6 7 8 9 10
Time (months)
Failuredensity
FIGURE 3. Variation of Failure density with time
Fig 4 compares the reliability value obtained using the model with the theoretical value. It can be seen that the
percentage error is always within 10% of the actual value which is a reasonably good result for all engineering problems.
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8 9 10
Time (months)
%Error
FIGURE 4. Error Analysis
4. CONCLUSION
A simplified model for evaluation of software reliability was presented. This is a relatively simplified and a
totally new method of analysis of software reliability as the initial population is assumed to remain constant.
The method provides fairly good results and the related errors are negligible. It is hoped that this model will
prove to be a powerful tool for software reliability analysis.
5. REFERENCES
1. Ying ZHOU,Joseph Davis. Open Source Software reliability model: an empirical approach, ACM 2005
2. Sharifah Mashita Syed-Mohamad,Tom McBride. A comparison of the Reliability Growth of Open Source
and In-House Software. 2008 IEEE 15th Asia-Pacific Software Engineering Conference

3. Cobra Rahmani,Harvey Siy, Azad Azadmanesh. An Experimental Analysis of Open Source Software
Reliability. Department of Defense/Air Force Office of Scientific Research
4. Lars M.Karg, Michael Grottke, Arne Beckhausa . Conformance Quality and Failure Costs in the Software
Industry: An Empirical Analysis of Open Source Software. 2009 IEEE
5. Kan H.S. Metrics and models in software quality engineering, 2nd edition, Addison-Wesley(2003)
6. S.P. Leblanc, P.A.Roman Reliability Estimation of Hierarchical Software Systems. 2002 Proceedings
annual reliability and maintainability symposium
7. J D Musa and K. Okumoto. A logarithamic Poisson execution time model for software reliability
measurement 7th international conference on Software Engineering(ICSE),1984, pp. 230-238
8. H. Pham, Software Reliability Spriner-Veriag , 2000
9. E.C. Nelson, A Stastistical Basis for Software Reliability Assessment, TRW-SS-73-03. 1973.
10. ShaoPing Wang, Software Engineering. BEIJING BUAA PRESS.
11. M.H.Halstead, Elements of Software Science, North Holand, 1977.
12. Z. Jelinski and P.B. Moranda, Software Reliability research, in Stastistical Computer Performance
Evaluation, W. Freiberger, Ed. New York: Academic Press, 1972, pp.465-484
13. B. Littlewood and J.L. Verrall, A Bayesian reliability growth model for computer software, Applied
Stastistics, Vol 22, 1973, pp 332-346
14. A. L. Goel and K. Okumoto, A time-dependent error-detection rate model for software reliability and
other performance measure, IEEE Transactions on Reliability, vol. R-28, 1979, pp. 206-211.
15. Adalberto Nobiato Crespo, Alberto Pasquini ,”Applying Code Coverage Approach to an Infinite Failure
Software Reliabilit Model” , 2009 XXIII Brazilian Symposium on Software Engineering, 2009 IEEE.
16. Hudson, A, “Program Error as a British and Death Process”, Technical Report SP- 3011, Santa Monica,
Cal,:Systems development Corporation, 1967.
17. Fenghong Zou , Joseph Davis “Analysing and Modeling Open Source Software Bug Report Data’, 19th
Australian Confeence on Software Engineering.,2008 IEEE.
18. Fenghong Zou , Joseph Davis “ A Model of Bug Dynamics for Open Source ”, The second International
Conference on Secure System Integration and Reliability Improvement., 2008 IEEE.
19. Sharifah Mashita Syed-Mohamad, Tom McBride, ‘Reliability Growth of Open Source Software using
Defect Analysis’, 2008 International conference on Computer Science and Software Engineering, 2008
IEEE.
20. Musa, J.D., Iannino, A. and Okumoto, K. (1987),”Software Reliability: Measurement, Prediction,
Application’, pp. 621.
21. http://www.debian.orgs

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