J. Software Engineering & Applications, 2011, 4, 596-601
doi:10.4236/jsea.2011.410070 Published Online October 2011 (http://www.SciRP.org/journal/jsea)
Using Artificial Neural-Networks in Stochastic
Differential Equations Based Software Reliability
Growth Modeling
Sunil Kumar Khatri1, Prakriti Trivedi2, Shiv Kant2, Nisha Dembla3
1Amity Institute of Information Technology, Amity University, Noida, India; 2Government Engineering College, Ajmer, India;
3Department of CSE & IT, NGF College of Engineering & Technology, Palwal, India.
Received July 4th, 2011; revised July 31st, 2011; accepted September 8th, 2011.
ABSTRACT
Due to high cost of fixing failures, safety concerns, and legal liabilities, organizations need to produce software that is
highly reliable. Software reliability growth models have been developed by software developers in tracking and meas-
uring the growth of reliability. Most of the Software Reliability Growth Models, which have been proposed, treat the
event of software fault detection in the testing and operational phase as a counting process. Moreover, if the size of
software system is large, the number of software faults detected during the testing phase becomes large, and the change
of the number of faults which are detected and removed through debugging activities becomes sufficiently small com-
pared with the initial fault content at the beginning of the testing phase. Therefore in such a situation, we can model the
software fault detection process as a stochastic process with a continuous state space. Recently, Artificial Neural Net-
works (ANN) have been applied in software reliability growth prediction. In this paper, we propose an ANN based
software reliability growth model based on ˆ
I
to type of stochastic differential equation. The model has been validated,
evaluated and compared with other existing NHPP model by applying it on actual failure/fault removal data sets cited
from real software development projects. The proposed model integrated with the concept of stochastic differential
equation performs comparatively better than the existing NHPP based model.
Keywords: Software Reliability Growth Model, Artificial Neural Network, Stochastic Differential Equation (SDE),
Stochastic Process
1. Introduction
1.1. Software Reliability Growth Modeling
There are numerous instances where failures of computer-
controlled systems have led to colossal loss of human
lives and money. With increased complexity of products
design, shortened development cycles and highly destruc-
tive consequences of software failures, a major responsi-
bility lies in the areas of Software Debugging, Testing
and Verification. As software systems have become more
and more complex, the importance of effective, well
planned testing has increased many folds.
The Software Reliability Growth Model (SRGM) is a
tool, which can be used to evaluate the software reliabil-
ity, develop test status, schedule status and monitor the
changes in reliability performance. Several Software Re-
liability models have been discussed in the literature.
Most of these are based upon historical failure data col-
lected during the testing phase. These models have been
utilized to evaluate the quality of the software and for
future reliability predictions. Software reliability engi-
neering (SRE) addresses all these issues, from design to
testing to maintenance phases.
The Software Reliability Growth Model (SRGM) is a
tool of SRE that can be used to evaluate the software
quantitatively, develop test status, schedule status and
monitor the changes in reliability performance [1,2]. In
the last two decades several Software Reliability models
have been developed in the literature showing that the
relationship between the testing time and the corre-
sponding number of faults removed is either Exponential
or S-Shaped or a mix of the two [1,2]. The software in-
cludes different types of faults and each fault requires
different strategies and different amounts of testing effort

Using Artificial Neural-Networks in Stochastic Differential Equations Based Software Reliability Growth Modeling597
to remove it.
A number of faults are detected and removed during
the long testing period before the system is released to
the market. However; the users then find number of faults
and the software company then release an updated ver-
sion of the system. Thus in this case the number of faults
that remain in the system can be considered to be a sto-
chastic process with continuous state space [3]. Yamada
et al. [4] proposed a simple software reliability growth
model to describe the fault detection process during the
testing phase by applying ˆ
I
to type Stochastic Differen-
tial Equation (SDE) and obtain several software reliabil-
ity measures using the probability distribution of the sto-
chastic process. Later on they proposed a flexible Sto-
chastic Differential Equation Model describing a fault-
detection process during the system-testing phase of the
distributed development environment [4]. Lee et al. [5]
used SDE to represent a per-fault detection rate that in-
corporate an irregular fluctuation instead of an NHPP,
and consider a per-fault detection rate that depends on the
testing time t.
1.2. Artificial Neural Networks
Many papers are published in the literature addressing
that neural networks offer promising approaches to soft-
ware reliability estimation and prediction. Karunanithi et
al. [6-8] first applied neural network architecture to esti-
mates the software reliability. They also illustrated the
usefulness of connectionist models for software reliabil-
ity growth predictions. Cai et al. [9] used the recent 50
inter-failure times as the multiple-delayed-inputs to pre-
dict the next failure time and found the effect of the num-
ber of input neurons, the number of neurons in the hidden
layer and the number of hidden layers by independently
varying the network architecture. They advocated the
development of fuzzy software reliability growth models
in place of probabilistic software reliability models.
Sherer [10] has applied neural networks for predicting
software faults in several NASA projects. Khoshgoftar et
al. [11] used the neural network as a tool for predicting
the number of faults in a program and concluded that the
neural networks produce models with better qua lity of fit
and predictive quality.
Su et al. [12] have proposed a neural network based
approach to software reliability assessment combining
various existing models into a Dynamic Weighted Com-
binational Model (DWCM). Kapur et al. [13] have pro-
posed an ANN based Dynamic Integrated Model (DIM),
which is an improvement over DWCM given by Su et al.
[12]. Kapur et al. [14] have proposed a Generalized Dy-
namic Integrated Model (GDIM) using ANN approach,
which incorporates the concept of n types of faults. Kha-
tri et al. [15] have proposed an artificial neural-network
based SRGM considering two types of imperfect debug-
ging during fault removal phenomenon. They considered
that during a removal attempt a fault might be removed
imperfectly. Such a situation results in number of failures
being more than number of removals and is known as
imperfect fault debugging. Otherwise it may happen that
a fault is generated while removing some fault and exis-
tence of a generated fault is known only after the perfect
removal of original fault. Due to error generation the total
fault content increases.
The paper is organized as follows. Section 2 presents
the model formulation for the proposed model. Model is
validated in Sections 3 and 4 based on two data sets cited
in the literature. Section 5 concludes the paper.
2. Framework for Modeling
2.1. Notations for the Proposed SRGM Using
SDE




Nt : The number of faults detected during the test-
ing time t and is a random variable;




ENt : Expected number of faults detected in the
time interval (0, t] during testing phase;
a: Total fault content;
b: Fault detection rates for simple, hard and complex
faults;

: Positive constant that represents the magnitude of
the irregular fluctuations for faults;


t

: Standardized Gaussian White Noise for faults.
2.2. Assumptions for the Proposed SRGM Using
SDE
1) The Software fault-detection process is modeled as
a stochastic process with a continuous state space.
2) The number of faults remaining in the software sys-
tem gradually decreases as the testing procedure goes on.
3) Software is subject to failures during execution
caused by faults remaining in the software.
4) During the fault isolation/removal, no new fault is
introduced into the system and the faults are debugged
perfectly.
2.3. Framework for Modeling for Proposed
SRGM
Several SRGM are based on the assumption of NHPP,
treating the fault detection process during the testing
phase as a discrete counting process. Recently Yamada et
al. [16] observed that if the size of the software system is
large, the number of software faults detected during the
testing phase becomes large and the change of the num-
ber of faults which are detected and removed through
debugging activities becomes sufficiently small com-

Using Artificial Neural-Networks in Stochastic Differential Equations Based Software Reliability Growth Modeling
598
pared with the initial fault conten t at the beginning of the
testing phase. So, in order to describe the stochastic be-
havior of the fault detection process, we can use a Sto-
chastic Model with continuous state space. Since the la-
tent faults in the software system are detected and elimi-
nated during the testing phase, the number of faults re-
maining in the software system gradually decreases as the
testing progresses. Therefore, it is reasonable to assume
the following differential equation
  
d
d
Nt rta Nt
t



 (1)
where is a fault-detection rate per remaining fault
at testing time t.

rt
However, the behavior of


rt is not completely
known since it is subject to random effects such as the
testing effort expenditu re, the skill lev el of the testers, the
testing tools and so on and thus might have irregular
fluctuati on. Thus , we have
 
noisert bt (2)
Let

t

be a standard Gaussian white noise and

a positive constant representing a magnitude of the ir-
regular fluctuations. So Equation (2) can b e written as
 
rt btt

  (3)
Hence, Equation (1) becomes [17]
  
d
d
Nt btta Nt
t
   



(4)
Equation (4) can be extended to the following stochas-
tic differential equation of an ˆ
I
to Type [3 and 16 ]
 
 
2
1
d
2
d
Ntbta Ntt
aNt wt



 
 

 

d

(5)
where is a one-dimensional Wiener process, which
is formally defined as an integration of the white noise

Wt


t

with respect to time t. Using ˆ
I
to formula solution
to Equation (5); using initial condition

0N

0; as
follows [3,16]
 
0
1exp d
t
Ntabx xWt 



 






 (6)
For, Goel-Okumoto Model b(x) = b (constant). Hence
Equation (6) reduces to
 

() 1()Nt aebtWt  
 
 
(7)
As N(t) is a random variable, its expected value will be
a useful measure.
 

 
2
1exp 1/2ENt abt    
 (8)
The Wiener process


Wt, is a Gaussian process and
it has the following prop erties:



  
Pr00 1,
0,
'min,
w
Ewt
Ewtwt tt





 
'
ANN Architecture of SRGM Based on Stoch astic
Differential Equation
We can apply the neural network based approach to build
a SRGM based on stochastic differential equations to
predict and estimate the software reliability of software.
There will be two hidden layers. Neural Network is de-
picted in Figure 1.
In practice, we can design different activation func-
tions on differen t neuron s in the h idden layer. The activ a-
tion functions for the units in hidden layer in Figure 1 are
defined as:


1,
x
x 

(9)

2 and
2
x
x   (10)


1e
x
x 

 (11)
The activation function for the unit in output layer is
defined as


x
x   (12)
w1, w2 and w3 are the weights of the network. Note that
w1 is the fault detection rate and w2 is equal to 2


. w5
is the proportion of total fault conten t in the software. We
assume that there is no bias in units of hidden layers and
output layer.
Input to the first unit of first hidden layer is


1_ 1in
htwt
Output from the first unit of hidden layer is







111_ 11in
hthtwt wt  
1

Figure 1. Network architecture of SRGM based on stochas-
tic differential equation.

Using Artificial Neural-Networks in Stochastic Differential Equations Based Software Reliability Growth Modeling599
t
Input to the second unit of first hidden layer is

2_ 2in
htw
Output from the second unit of first hidden layer is
 


2
222_ 22
2
in
wt
hth twt  
Input to the single unit of second hidden layer is

2
3_3 142
in
wt
htwwtw 
Output from the single unit of second hidd en layer is
 

2
31431 4
2
33_314
2
2
1e1e
in
wt w
wwt wwwwt
wt
hthtwwtw 

 






 
2
2





Input to the single unit of output layer is

2
31 42
51e
w
ww wt
in
yt w









Output from the single unit of output layer is
  
2
31 4
2
31 4
2
5
2
5
(1 e)
1e
w
ww wt
in
w
ww wt
yty tw
w























(13)
Using w1 = b, w2 = –σ2, w3 = 1, w4 = 1 and w5 = a,
Equation (13) is same as Equation (8), which is th e mean
value function for SRGM based on stochastic differential
equation.
3. Model Validation
To assess the performance of the proposed neural-net-
work model, we have carried out the parameter estima-
tion on two real software failure datasets.
3.1. Data Set 1 (DS-1)
The first data set (DS-1) had been collected during 38
weeks and 231 faults were detected during testing. This
data is cited from Misra [17].
3.2. Data Set 2 (DS-2)
The second data set (DS-2) had been collected during 21
weeks of testing and 26 faults were detected during test-
ing. This data is cited from Pham [18].
3.3. Comparison Criteria for SRGM
The performance of SRGM are judged by their ability to
fit the past software fault data (goodness-of-fit). The term
goodness-of-fit denotes the question of “How good does
a model fit to the data?”
1) The Mean Square Fitting Error (MSE):
The model under comparison is used to simulate the
fault data, the difference between the expected values,


ˆi
mt and the observed data yi is measured by MSE as
follows.



2
1
ˆ
MSE kii
i
mt y
k


 (14)
where k is the number of observations. The lower MSE
indicates less fitting error, thus better goodness-of-fit [2].
2) Bias:
The difference between the observation and prediction
of number of failures at any instant of time i is known as
PEi.(prediction error). The average of PEs is known as
bias. Lower the value of Bias better is th e goodness of fit
[19]. Prediction error is the difference between the actual
and the estimated number of faults.
3) Variation:
The standard deviation of prediction error is known as
variation.
 
2
Variation11PE Bias
i
N 
 (15)
Lower the value of Variation better is the goodness of
fit [19].
4) Root Mean Square Prediction Error:
It is a measure of closeness with which a model pre-
dicts the observation.

2
RMSPEBias Variation 2 (16)
Lower the value of Root Mean Square Prediction Error
better is the goodness of fit [19].
4. Data Analyses and Model Comparison
To judge the accuracy of the proposed model (equation
13) we had used MSE, Bias, Variation and RMSPE as the
performance measures. The comparison criteria results
are shown in Tables 1 and 2. Goodness-of-fit curves are
shown in Figures 2 and 3.
Table 1. For DS-1.
Models Compared MSE Variation BiasRMSPE
Goel Okumoto Model [1] 96.848 6.759 7.2369.902
Proposed Model (Equat ion ( 13))20.709 4.611 –0.0794.612
Table 2. For DS-2.
Models Compared MSE Variation BiasRMSPE
Goel Okumoto Model [1] 7.446 2.138 1.7582.768
Proposed Model ( Equatio n (13))5.721 2.390 0.5302.448

Using Artificial Neural-Networks in Stochastic Differential Equations Based Software Reliability Growth Modeling
600
Goodness-of-fit for DS-1
0
50
100
150
200
250
13579111315 17192123 25272931333537
Time (in weeks)
Actual DataGoel-Okumoto [1]Propose d Model (E quation 13)
Cumulative Faults
Figure 2. Goodness of fit curve for DS-1.
Goodness-of-fit for DS-2
0
5
10
15
20
25
30
12345678910 111213 1415 16 1718 192021
Time (in weeks)
Actual DataGoel-Okumoto Model [1]Proposed Model ( E quation 13)
Cumulative Faults
Figure 3. Goodness of fit curve for DS-2.
5. Conclusions
This paper presents an SRGM based on ˆ
I
to type Sto-
chastic Differential Equations using ANN approach. The
goodness of the fit analysis has been done on two real
software failure datasets. The goodness-of-fit of the pro-
posed Model is compared with Goel Okumoto model [1].
The results obtain ed show better fit and wider applicab il-
ity of the model to different types of failure datasets. From
the numerical illustrations, we see that the Proposed
Model provides improved results because of lower MSE,
Variation, RMSPE and Bias. The usability of SDE is not
only restricted to the model described in this paper but it
can also be extended to improve the results of any other
SRGM. For further research the Proposed Model can be
used along with error generation.
REFERENCES
[1] A. L. Goel and K. Okumoto, “Time Dependent Error De-
tection Rate Model for Software Reliability and Other
Performance Measure,” IEEE Transactions on Reliability,
Vol. 3, 1992, pp. 206-211. doi:10.1109/TR.1979.5220566
[2] P. K. Kapur, R. B. Garg and S. Kumar, “Contributions to
Hardware and Software Reliability,” World Scientific,
Singapore, 1999.
[3] B. Oksendal, “Stochastic Differential Equations—An
Introduction with Applications,” Springer, Berlin, 2003.
[4] S. Yamada and Y. Tamura, “A Flexible Stochastic Dif-
ferential Equation Model in Distributed Development En-
vironment,” European Journal of Operational Research,
Vol. 168, No. 1, 2006, pp. 143-152.
doi:10.1016/j.ejor.2004.04.034
[5] C. H. Lee, Y. T. Kim and D. H. Park, “S-Shaped Software
Reliability Growth Models Derived from Stochastic Dif-
ferential Equations,” IIE Transactions, Vol. 36, No. 12,
2004, pp. 1193-1199. doi:10.1080/07408170490507792
[6] N. Karunanithi and Y. K. Malaiya, “The Scaling Problem
in Neural Networks for Software Reliability Prediction,”
Proceedings of the 3rd International IEEE Symposium of
Software Reliability Engineering, Los Alamitos, 7-10 Oc-
tober 1992, pp. 76-82.
doi:10.1109/ISSRE.1992.285856
[7] N. Karunanithi, Y. K. Malaiya and D. Whitley, “Predic-
tion of Software Reliability Using Neural Networks,”
Proceedings of the 2nd IEEE International Symposium on
Software Reliability Engineering, Los Alamitos, 17-18
May 1991, pp. 124-130.
[8] N. Karunanithi, D. Whitley and Y. K. Malaiya, “Using
Neural Networks in Reliability Prediction,” IEEE Soft-
ware, Vol. 9, No. 4, 1992, pp. 53-59.
doi:10.1109/52.143107
[9] K. Y. Cai, L. Cai, W. D. Wang, Z. Y. Yu and D. Zhang,
“On the Neural Network Approach in Software Reliability
Modeling,” The Journal of Systems and Software, Vol. 58,
No. 1, 2001, pp. 47-62.
doi:10.1016/S0164-1212(01)00027-9
[10] S. A. Sherer, “Software Fault Prediction,” Journal of Sys-
tems and Software, Vol. 29, No. 2, 1995, pp. 97-105.
doi:10.1016/0164-1212(94)00051-N
[11] T. M. Khoshgoftar and R. M. Szabo, “Using Neural Net-
works to Predict Software Faults during Testing,” IEEE
Transactions on Reliability, Vol. 45, No. 3, 1996, pp. 456-
462. doi:10.1109/24.537016
[12] Y. S. Su, C. Y. Huang and Y. S. Chen, “An Artificial
Neural-Network Based Approach to Software Reliability
Assessment,” Proceedings of IEEE Region 10 Conference,
Melbourne, 21-24 November 2005, pp. 1-6.
[13] P. K. Kapur, S. K. Khatri, M. Basirzadeh and N. Dembla,
“Modeling Software Reliability Growth in Distributed
Environment Using Artificial Neural-Networks,” In: S. K.
Khatri and B. Kumar, Eds., Proceedings of International
Conference on Reliability, Infocom Technology and Op-
timization, Faridabad, 1-3 November 2010, pp. 372-382.
[14] P. K. Kapur, S. K. Khatri and D. N. Goswami, “A Gener-
alized Dynamic Integrated Software Reliability Growth
Model Based on Neural-Network Approach,” Proceed-

Using Artificial Neural-Networks in Stochastic Differential Equations Based Software Reliability Growth Modeling
601
ings of International Conference on Reliability, Safety
and Quality Engineering, 5-7 January 2008, pp. 831-838.
[15] S. K. Khatri, R. Kapur, P. Johri and P. Sharma, “Artificial
Neural-Networks Based Software Reliability Growth Mo-
deling with Two types of Imperfect Debugging,” In: S. K.
Khatri and B. Kumar, Eds., Proceedings of International
Conference on Reliability, Infocom Technology and Op-
timization, Faridabad, 1-3 November 2010, pp. 122-133.
[16] S. Yamada, A. Nishigaki and M. Kimura, “A Stochastic
Differential Equation Model for Software Reliability As-
sessment and Its Goodness of Fit,” International Journal
of Reliability and Applications, Vol. 4, No. 1, 2003, pp.
1-11.
[17] P. N. Misra, “Software Reliability Analysis,” IBM System
Journal, Vol. 22, No. 3, 1983, pp. 262-270.
doi:10.1147/sj.223.0262
[18] H. Pham, “Software Reliability,” Springer-Verlag, Singa-
pore City, 2000.
[19] K. Pillai and V. S. S. Nair, “A Model for Software Devel-
opment effort and Cost Estimation,” IEEE Transactions on
Software Engineering, Vol. 23, No. 8, 1997, pp. 485-497.
doi:10.1109/32.624305