![ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011
Software Reliability Growth Model with Logistic-
Exponential Testing-Effort Function and Analysis of
Software Release Policy
Shaik.Mohammad Rafi 1, Shaheda Akthar 2
1
Dept. of Computer Science, Sri Mittapalli Institute of Technology for women, Guntur, A.P, India
E-mail:mdrafi.527@gmail.com
2
Dept. of Computer Science, Sri Mittapalli College of Engineering, Guntur, A.P, India
E-mail:shaheda.akthar@yahoo.com
Abstract- Software reliability is one of the important factors of I. INTRODUCTION
software quality. Before software delivered in to market it is
thoroughly checked and errors are removed. Every software Software becomes crucial in daily life. Computers,
industry wants to develop software that should be error free. commutation devices and electronics equipments every place
Software reliability growth models are helping the software we find software. The goal of every software industries is
industries to develop software which is error free and reliable. develop software which is error and fault free. Every industry
In this paper an analysis is done based on incorporating the is adopting a new testing technique to capture the errors
logistic-exponential testing-effort in to NHPP Software during the testing phase. But even though some of the faults
reliability growth model and also observed its release policy. were undetected. These faults create the problems in future.
Experiments are performed on the real datasets. Parameters
Reliability is defined as the working condition of the software
are calculated and observed that our model is best fitted for
the datasets. over certain time period of time in a given environmental
conditions. Large numbers of papers are presented in this
Keywords- Software Reliability, Software Testing, Testing context. Testing effort is defined as effort needed to detect
Effort, Non-homogeneous Poisson Process (NHPP), Software and correct the errors during the testing. Testing-effort can
Cost. be calculated as person/ month, CPU hours and number of
ACRONYMS test cases and so on. Generally the software testing consumes
a testing-effort during the testing phase [20 21].SRGM
NHPP : Non Homogeneous Poisson Process
proposed by several papers incorporated traditional effort
SRGM : Software Reliability Growth Model
curves like Exponential, Rayleigh, and Weibull. The TEF
MVF : Mean Value Function
which gives the effort required in testing and CPU time the
MLE : Maximum Likelihood Estimation
software for better error tracking. Many papers are published
TEF : Testing Effort Function
based on TEF in NHPP models [4, 5, 8, 11, 120, 12, 20, 21].
LOC : Lines of Code
All of them describe the tracking phenomenon with test
MSE : Mean Square fitting Error
expenditure.
NOTATIONS This paper we used logistic-exponential testing-effort
m (t) : Expected mean number of faults detected curve and incorporated in the SRGM. The result shows that
in time (0,t] the SRGM with logistic-exponential
ë (t) : Failure intensity for m(t)
n (t) : Fault content function II. SOFTWARE TESTING EFFORT FUNCTIONS
md (t) : Cumulative number of faults detected upto t
Several software testing-effort functions are defined in
mr (t) : Cumulative number of faults isolated up to t.
literature. w(t) is defined as the current testing effort and
W (t) : Cumulative testing effort consumption at timet.
W(t) describes the cumulative testing effort. The following
W*(t) : W (t)-W (0)
equation shows the relation between the w(t) and W(t)
A : Expected number of initial faults
r (t) : Failure detection rate function (1)
r : Constant fault detection rate function. The following are some of them
r1 : Constant fault detection rate in the Delayed S-
shaped model with logistic-Exponential TEF A. Exponential Testing effort function
r2 : Constant fault isolated rate in the Delayed S- The cumulative testing effort consumed in the time (0,t]
shaped model with logistic-Exponential TEF is [20]
B. Rayleigh Testing effort curve:
(2)
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![ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011
The cumulative testing effort consumed in the time (0,t]
is [12,20]
(3)
The Rayleigh curve increases to the peak and descends
gradually with decelerating rate.
C. Logistic-exponential testing-effort:
It has a great flexibility in accommodating all the forms
of the hazard rate function, can be used in a variety of B. Yamada Delayed S-shaped model with logistic-
problems for modeling software failure data. exponential testing-effort function
The logistic-exponential cumulative TEF over time The delayed ‘S’ shaped model originally proposed by
period (0,t] can be expressed as [27] Yamada [24] and it is different from NHPP by considering
that software testing is not only for error detection but error
isolation. And the cumulative errors detected follow the S-
shaped curve. This behavior is indeed initial phase testers
are familiar with type of errors and residual faults become
more difficult to uncover [1, 6, 15, 16]. From the above steps
described section 3.1, we will get a relationship between
m(t) and w(t). For extended Yamada S-shaped software
reliability model.The extended S-shaped model [24] is
modeled by
III. SOFTWARE RELIABILITY GROWTH MODELS
A. Software reliability growth model with logistic-
exponential TEF
The following assumptions are made for software
reliability growth modeling [1, 8, 11, 20, 21, 22]
(i) The fault removal process follows the Non-
Homogeneous Poisson process (NHPP)
(ii) The software system is subjected to failure at
random time caused by faults remaining in the
system.
(iii) The mean time number of faults detected in the time
interval (t, t+Ät) by the current test effort is
proportional for the mean number of remaining
faults in the system.
(iv) The proportionality is constant over the time.
(v) Consumption curve of testing effort is modeled by
a logistic-exponential TEF.
(vi) Each time a failure occurs, the fault that caused it
is immediately removed and no new faults are
introduced.
(vii) We can describe the mathematical expression of a
testing-effort based on following
IV. EVALUATION CRITERIA
A. The goodness of fit technique
Here we used MSE [5, 11, 17, 23 ]which gives real
measure of the difference between actual and predicted
values. The MSE defined as
A smaller MSE indicate a smaller fitting error and better
performance.
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![ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011
B. Coefficient of multiple determinations (R2)
Which measures the percentage of total variation about
mean accounted for the fitted model and tells us how well a
curve fits the data. It is frequently employed to compare
model and access which model provies the best fit to the
data. The best model is that which proves higher R2. that is
closer to 1.
C. The predictive Validity Criterion
The capability of the model to predict failure behavior
from present & past failure behavior is called predictive
Fig 1. Observed/estimated logistic-exponential and Rayleigh TEF for
validity. This approach, which was proposed by [26], can be DS1.
.
represented by computing RE for a data set
All parameters of other distribution are estimated through
LSE. The unknown parameters of Logistic-exponential TEF
are á=72(CPU hours), ë=0.04847, and k=1.387.
Correspondingly the estimated parameters of Rayleigh TEF
N=49.32 and b=0.00684/week. Fig.1 plots the comparison
between observed failure data and the data estimated by
Logistic-exponential TEF and Rayleigh TEF. The PE, Bias,
Variation, MRE and RMS-PE for Logistic-exponential and
Rayleigh are listed in Table I. From the TABLE I we can see
that Logistic-exponential has lower PE, Bias, Variation, MRE
and RMS-PE than Rayleigh TEF. We can say that our
proposed model fits better than the other one. In the TABLE
II we have listed estimated values of SRGM with different
testing-efforts. We have also given the values of SSE, R2
and MSE. We observed that our proposed model has smallest
MSE and SSE value when compared with other models. The
95% confidence limits for the all models are given in the
Table III.
V. MODEL PERFORMANCE ANALYSIS
A. DS1:
The first set of actual data is from the study by Ohba
1984 [15].the system is PL/1 data base application software B. DS2:
,consisting of approximately 1,317,000lines of code .During The dataset used here presented by wood [2] from
nineteen weeks of experiments, 47.65 CPU hours were a subset of products for four separate software releases at
consumed and about 328 software errors are removed. Tandem Computer Company. Wood Reported that the
Fitting the model to the actual data means by esti- specific products & releases are not identified and the test
mating the model parameter from actual failure data. Here data has been suitably transformed in order to avoid
we used the LSE (non-linear least square estimation) and Confidentiality issue. Here we use release 1 for illustrations.
MLE to estimate the parameters. Calculations are given in Over the course of 20 weeks, 10000 CPU that SRGM with
appendix A logistic-exponential TEF have less MSE than other models.
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![ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011
as Minimize C(T) subjected to R(t+Ät/t)e” R0 for C2 > C1, C3
>0, Ät>0, 0 < R0 <1.
Differentiate the equation (30) with respect to T and
setting it to zero, we obtain
VI. OPTIMAL SOFTWARE RELEASE POLICY
A. Software Release-Time Based on Reliability Criteria
Generally software release problem associated with the
reliability of a software system. Here in this first we discuss
the optimal time based on reliability criterion. If we know
software has reached its maximum reliability for a particular
time. By that we can decide right time for the software to be
delivered out. Goel and Okumoto [1] first dealed with the
software release problem considering the software cost-
benefit. The conditional reliability function after the last
failure occurs at time t is obtained by
B. Optimal release time based on cost-reliability
criterion
This section deals with the release policy based on the
cost-reliability criterion. Using the total software cost
evaluated by cost criterion, the cost of testing-effort
expenditures during software testing/development phase and
the cost of fixing errors before and after release are: [9, 13,
25]
Where C1 the cost of correcting an error during testing,
C2 is the cost of correcting an error during the operation, C2 we can easily get the required testing time needed to reach
> C 1, C 3 is the cost of testing per unit testing effort the reliability objective R0 . here our goal is to minimize the
expenditure and TLC is the software life-cycle length. total software cost under desired software reliability and then
From reliability criteria, we can obtain the required the optimal software release time is obtained. That is can
testing time needed to reach the reliability objective R0. Our minimize the C(T) subjected to R(t+Ät/t)e” R0 where 0< R0
aim is to determine the optimal software release time that <1 [9,25]
minimizes the total software cost to achieve the desired T* =optimal software release time or total testing time
software reliability. Therefore, the optimal software release =max{T 0, T 1}.Where T 0 =finite and unique solution T
policy for the proposed software reliability can be formulated satisfying Eq.(31) T1 =finite and unique T satisfying R(t+Ät/
t)=R0
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![ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011
By combining the above analysis and combining the cost
and reliability requirements we have the following theorem.
Theorem 1: Assume C2 <C1<0, C3<0, Ät>0, and 0<R0
<1. Let T*be the optimal software release time
CONCLUSION
In this paper, we proposed a SRGM incorporating the
Logistic-exponential testing effort function that is completely
different from the logistic type Curve. We Observed that most
of software failure is time dependent. By incorporating
testing-effort into SRGM we can make realistic assumptions
about the software failure. The experimental results indicate
that our proposed model fits fairly well.
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Software Reliability Growth Model with Logistic- Exponential Testing-Effort Function and Analysis of Software Release Policy
- 1. ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011 Software Reliability Growth Model with Logistic- Exponential Testing-Effort Function and Analysis of Software Release Policy Shaik.Mohammad Rafi 1, Shaheda Akthar 2 1 Dept. of Computer Science, Sri Mittapalli Institute of Technology for women, Guntur, A.P, India E-mail:[email protected] 2 Dept. of Computer Science, Sri Mittapalli College of Engineering, Guntur, A.P, India E-mail:[email protected] Abstract- Software reliability is one of the important factors of I. INTRODUCTION software quality. Before software delivered in to market it is thoroughly checked and errors are removed. Every software Software becomes crucial in daily life. Computers, industry wants to develop software that should be error free. commutation devices and electronics equipments every place Software reliability growth models are helping the software we find software. The goal of every software industries is industries to develop software which is error free and reliable. develop software which is error and fault free. Every industry In this paper an analysis is done based on incorporating the is adopting a new testing technique to capture the errors logistic-exponential testing-effort in to NHPP Software during the testing phase. But even though some of the faults reliability growth model and also observed its release policy. were undetected. These faults create the problems in future. Experiments are performed on the real datasets. Parameters Reliability is defined as the working condition of the software are calculated and observed that our model is best fitted for the datasets. over certain time period of time in a given environmental conditions. Large numbers of papers are presented in this Keywords- Software Reliability, Software Testing, Testing context. Testing effort is defined as effort needed to detect Effort, Non-homogeneous Poisson Process (NHPP), Software and correct the errors during the testing. Testing-effort can Cost. be calculated as person/ month, CPU hours and number of ACRONYMS test cases and so on. Generally the software testing consumes a testing-effort during the testing phase [20 21].SRGM NHPP : Non Homogeneous Poisson Process proposed by several papers incorporated traditional effort SRGM : Software Reliability Growth Model curves like Exponential, Rayleigh, and Weibull. The TEF MVF : Mean Value Function which gives the effort required in testing and CPU time the MLE : Maximum Likelihood Estimation software for better error tracking. Many papers are published TEF : Testing Effort Function based on TEF in NHPP models [4, 5, 8, 11, 120, 12, 20, 21]. LOC : Lines of Code All of them describe the tracking phenomenon with test MSE : Mean Square fitting Error expenditure. NOTATIONS This paper we used logistic-exponential testing-effort m (t) : Expected mean number of faults detected curve and incorporated in the SRGM. The result shows that in time (0,t] the SRGM with logistic-exponential ë (t) : Failure intensity for m(t) n (t) : Fault content function II. SOFTWARE TESTING EFFORT FUNCTIONS md (t) : Cumulative number of faults detected upto t Several software testing-effort functions are defined in mr (t) : Cumulative number of faults isolated up to t. literature. w(t) is defined as the current testing effort and W (t) : Cumulative testing effort consumption at timet. W(t) describes the cumulative testing effort. The following W*(t) : W (t)-W (0) equation shows the relation between the w(t) and W(t) A : Expected number of initial faults r (t) : Failure detection rate function (1) r : Constant fault detection rate function. The following are some of them r1 : Constant fault detection rate in the Delayed S- shaped model with logistic-Exponential TEF A. Exponential Testing effort function r2 : Constant fault isolated rate in the Delayed S- The cumulative testing effort consumed in the time (0,t] shaped model with logistic-Exponential TEF is [20] B. Rayleigh Testing effort curve: (2) © 2011 ACEEE 38 DOI: 01.IJNS.02.0.254
- 2. ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011 The cumulative testing effort consumed in the time (0,t] is [12,20] (3) The Rayleigh curve increases to the peak and descends gradually with decelerating rate. C. Logistic-exponential testing-effort: It has a great flexibility in accommodating all the forms of the hazard rate function, can be used in a variety of B. Yamada Delayed S-shaped model with logistic- problems for modeling software failure data. exponential testing-effort function The logistic-exponential cumulative TEF over time The delayed ‘S’ shaped model originally proposed by period (0,t] can be expressed as [27] Yamada [24] and it is different from NHPP by considering that software testing is not only for error detection but error isolation. And the cumulative errors detected follow the S- shaped curve. This behavior is indeed initial phase testers are familiar with type of errors and residual faults become more difficult to uncover [1, 6, 15, 16]. From the above steps described section 3.1, we will get a relationship between m(t) and w(t). For extended Yamada S-shaped software reliability model.The extended S-shaped model [24] is modeled by III. SOFTWARE RELIABILITY GROWTH MODELS A. Software reliability growth model with logistic- exponential TEF The following assumptions are made for software reliability growth modeling [1, 8, 11, 20, 21, 22] (i) The fault removal process follows the Non- Homogeneous Poisson process (NHPP) (ii) The software system is subjected to failure at random time caused by faults remaining in the system. (iii) The mean time number of faults detected in the time interval (t, t+Ät) by the current test effort is proportional for the mean number of remaining faults in the system. (iv) The proportionality is constant over the time. (v) Consumption curve of testing effort is modeled by a logistic-exponential TEF. (vi) Each time a failure occurs, the fault that caused it is immediately removed and no new faults are introduced. (vii) We can describe the mathematical expression of a testing-effort based on following IV. EVALUATION CRITERIA A. The goodness of fit technique Here we used MSE [5, 11, 17, 23 ]which gives real measure of the difference between actual and predicted values. The MSE defined as A smaller MSE indicate a smaller fitting error and better performance. © 2011 ACEEE 39 DOI: 01.IJNS.02.02.254
- 3. ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011 B. Coefficient of multiple determinations (R2) Which measures the percentage of total variation about mean accounted for the fitted model and tells us how well a curve fits the data. It is frequently employed to compare model and access which model provies the best fit to the data. The best model is that which proves higher R2. that is closer to 1. C. The predictive Validity Criterion The capability of the model to predict failure behavior from present & past failure behavior is called predictive Fig 1. Observed/estimated logistic-exponential and Rayleigh TEF for validity. This approach, which was proposed by [26], can be DS1. . represented by computing RE for a data set All parameters of other distribution are estimated through LSE. The unknown parameters of Logistic-exponential TEF are á=72(CPU hours), ë=0.04847, and k=1.387. Correspondingly the estimated parameters of Rayleigh TEF N=49.32 and b=0.00684/week. Fig.1 plots the comparison between observed failure data and the data estimated by Logistic-exponential TEF and Rayleigh TEF. The PE, Bias, Variation, MRE and RMS-PE for Logistic-exponential and Rayleigh are listed in Table I. From the TABLE I we can see that Logistic-exponential has lower PE, Bias, Variation, MRE and RMS-PE than Rayleigh TEF. We can say that our proposed model fits better than the other one. In the TABLE II we have listed estimated values of SRGM with different testing-efforts. We have also given the values of SSE, R2 and MSE. We observed that our proposed model has smallest MSE and SSE value when compared with other models. The 95% confidence limits for the all models are given in the Table III. V. MODEL PERFORMANCE ANALYSIS A. DS1: The first set of actual data is from the study by Ohba 1984 [15].the system is PL/1 data base application software B. DS2: ,consisting of approximately 1,317,000lines of code .During The dataset used here presented by wood [2] from nineteen weeks of experiments, 47.65 CPU hours were a subset of products for four separate software releases at consumed and about 328 software errors are removed. Tandem Computer Company. Wood Reported that the Fitting the model to the actual data means by esti- specific products & releases are not identified and the test mating the model parameter from actual failure data. Here data has been suitably transformed in order to avoid we used the LSE (non-linear least square estimation) and Confidentiality issue. Here we use release 1 for illustrations. MLE to estimate the parameters. Calculations are given in Over the course of 20 weeks, 10000 CPU that SRGM with appendix A logistic-exponential TEF have less MSE than other models. © 2011 ACEEE 40 DOI: 01.IJNS.02.02.254
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- 5. ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011 as Minimize C(T) subjected to R(t+Ät/t)e” R0 for C2 > C1, C3 >0, Ät>0, 0 < R0 <1. Differentiate the equation (30) with respect to T and setting it to zero, we obtain VI. OPTIMAL SOFTWARE RELEASE POLICY A. Software Release-Time Based on Reliability Criteria Generally software release problem associated with the reliability of a software system. Here in this first we discuss the optimal time based on reliability criterion. If we know software has reached its maximum reliability for a particular time. By that we can decide right time for the software to be delivered out. Goel and Okumoto [1] first dealed with the software release problem considering the software cost- benefit. The conditional reliability function after the last failure occurs at time t is obtained by B. Optimal release time based on cost-reliability criterion This section deals with the release policy based on the cost-reliability criterion. Using the total software cost evaluated by cost criterion, the cost of testing-effort expenditures during software testing/development phase and the cost of fixing errors before and after release are: [9, 13, 25] Where C1 the cost of correcting an error during testing, C2 is the cost of correcting an error during the operation, C2 we can easily get the required testing time needed to reach > C 1, C 3 is the cost of testing per unit testing effort the reliability objective R0 . here our goal is to minimize the expenditure and TLC is the software life-cycle length. total software cost under desired software reliability and then From reliability criteria, we can obtain the required the optimal software release time is obtained. That is can testing time needed to reach the reliability objective R0. Our minimize the C(T) subjected to R(t+Ät/t)e” R0 where 0< R0 aim is to determine the optimal software release time that <1 [9,25] minimizes the total software cost to achieve the desired T* =optimal software release time or total testing time software reliability. Therefore, the optimal software release =max{T 0, T 1}.Where T 0 =finite and unique solution T policy for the proposed software reliability can be formulated satisfying Eq.(31) T1 =finite and unique T satisfying R(t+Ät/ t)=R0 © 2011 ACEEE 42 DOI: 01.IJNS.02.02.254
- 6. ACEEE Int. J. on Network Security , Vol. 02, No. 02, Apr 2011 By combining the above analysis and combining the cost and reliability requirements we have the following theorem. Theorem 1: Assume C2 <C1<0, C3<0, Ät>0, and 0<R0 <1. Let T*be the optimal software release time CONCLUSION In this paper, we proposed a SRGM incorporating the Logistic-exponential testing effort function that is completely different from the logistic type Curve. We Observed that most of software failure is time dependent. By incorporating testing-effort into SRGM we can make realistic assumptions about the software failure. The experimental results indicate that our proposed model fits fairly well. REFERENCES [1] A.L. Goel and K. Okumoto, A time dependent error detection rate model for a large scale software system, Proc. 3rd USA- Japan Computer Conference, pp. 3540, San Francisco, CA (1978). From the dataset one estimated values of SRGM with [2] A.Wood, Predicting software reliability, IEEE computers 11 Logistic-exponential TEF á=72(CPU hours), ë=0.04847 / (1996) 69–77. week, k=1.387, a=578.8 and r=0.01903 when Ät=0.1 R0 [3] Bokhari, M.U. and Ahmad, N. (2005), “Software reliability =0.85 and we let C1=2, C2 =50, C3 =150 and TLC =100 the growth modeling for exponentiated Weibull functions with estimated time T1=37.1 weeks and release time from eq 30 actual software failures data”, in Proceedings of 3rd T0 =39.5 weeks. Now optimal Release Time max (37.1, 39.5) International Conference on Innovative Applications of is T*=39.5 weeks. Fig 10 shows the change in software cost Information Technology for Developing World (AACC’2005), Nepal. during the time span. Now total cost of the software at optimal [4] Bokhari, M.U. and Ahmad, N. (2006), “Analysis of a software time 8354. reliability growth models: the case of log-logistic test-effort From the dataset two estimated values of SRGM with function”, in Proceedings of the 17th International Conference Logistic-exponential TEF á=12600(CPU hours), ë=0.06352 on Modelling and Simulation (MS’2006), Montreal, Canada, /week, k=1.391, a=135.6 and r=0.0001432 when Ät=0.1 R0 pp. 540-5. =0.85 and we let C1=1, C2 =200, C3 =2 and TLC =100 the [5] C.-Y. Huang, S.-Y. Kuo, J.Y. Chen, Analysis of a software estimated time T1=18.1 weeks and release time from Eq 31 reliability growth model with logistic testing effort function T0 =8.05 weeks. Now optimal Release Time max (8.05, 18.1) proceeding of Eighth International Symposium on Software is T*=18.1 weeks. Fig 11 shows the change in software cost Reliability Engineering, 1997, pp. 378–388. [6] Goel, A.L., “Software reliability models: Assumptions, during the time span. Now total cost of the software at optimal limitations, and applicability”, IEEE Transactions on Software time 20,100. Engineering SE-11 (1985) 1411-1423. [7] Huang, C.Y. and Kuo, S.Y. (2002), “Analysis of incorporating logistic testing-effort function into software reliability modeling”, IEEE Transactions on Reliability, Vol. 51 No. 3, pp. 261-70. [8] Huang, C.Y., Kuo, S.Y. and Chen, I.Y. (1997), “Analysis of software reliability growth model with logistic testing-effort function”, in Proceeding of 8th International Symposium on Software Reliability Engineering (ISSRE’1997), Albuquerque, New Mexico, pp. 378-88. © 2011 ACEEE 43 DOI: 01.IJNS.02.02.254
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