Role of Change Point in Non-Homogeneous Poisson Process Software Reliability Growth Model

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IJSTE - International Journal of Science Technology & Engineering | Volume 4 | Issue 5 | November 2017 ISSN (online): 2349-784X

Role of Change Point in Non-Homogeneous Poisson Process Software Reliability Growth Model Dr. Syed Faizul Islam Department of Statistics Mettu University

Abstract Software reliability is the probability that given software functions correctly under a given environment and during a specified period of time. The software reliability is highly related to the amount of testing-effort. Also, the reliability of software needs to be carefully assessed and analyzed during the software development process. Furthermore, the prediction of software reliability is also very important for failure free operation by software. To asses, analyze and predict a software reliability we need to develop a software reliability growth model. In the earlier research, it is found that the probability of fault detection is not constant. It can be changed at some point of time which is called a change point. In this paper, we are presenting the role of change point in nonhomogenous Poisson process software reliability growth model with testing-effort. Keywords: Change Point, Non-homogenous Poisson process software reliability growth model, and Testing-effort function ________________________________________________________________________________________________________ I.

INTRODUCTION

A change point is competent of finding one or more changes occur during a software testing phase. A change point analysis is very simple and flexible process to show the changes on a graph. We can easily detect the change point with the help of divergence point. In software testing process, the fault detection rate may not smooth and can be changed at some time moment which is called a change point (Huang, 2004). The change can take place due to some important factors like the skill of test teams, program size and software testability. Therefore, we incorporated change point along with testing-effort in the software reliability growth model. In this paper, we are discussing the role of change point in non-homogenous Poisson process software reliability growth model with testing-effort. We tried to apply the change point concept on three different testing-effort functions which are Burr type XII, Log-logistic and Exponentiated Weibull. We found that applying change point is more suitable to assess and predict the software reliability. It is much more realistic and fits the real project data fairly well. The graphs are also very smooth to evaluate the software reliability. In the remaining of this paper, there are five more sections. In Section 2, we provide an attention on research motivation. Also, the literature review on change point discussed in section 3. The Section 4 and 5 are responsible to confer estimation of model parameters and conclusion. Finally, references are given in Section 6. II. RESEARCH MOTIVATION During the past research, plenty of software reliability growth models were proposed to assess and predict the software reliability. Some of those are based on testing-effort functions, but a very few are devoted to the testing-efforts along with the change point. Our research presents a non-homogenous Poisson process software reliability growth model with change point. The main focus is to provide a method for software reliability modeling, which incorporates both testing-efforts and change points. In our research work we applied three different testing-effort functions these are Burr Type XII testing-effort function, exponentiated Weibull testing-effort function and the Log-logistic testing-effort function along with change point. In common, the fault detection rate assumes constant. But in practical, it can be changed at some time moment which is called a change point. Also, a change point shows a change graphically without any complex analysis. Therefore, a software reliability growth model with testing-effort function and change point evaluate and predict the software reliability professionally. The software reliability growth model parameters are estimated with the help of least square estimation and maximum likelihood estimation. (Zhao et al., 2006). III. LITERATURE REVIEW Yamada et al. (2013) proposed an extended hazard rate model for software reliability assessment with effect at change point. A software hazard rate model is known as one of the important and useful mathematical models for describing the software failure

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Role of Change Point in Non-Homogeneous Poisson Process Software Reliability Growth Model (IJSTE/ Volume 4 / Issue 5 / 028)

occurrence phenomenon observed in a testing phase. It is difficult to say that the testing environment always constant during a testing phase due to changing the specification and fault target and so forth. Therefore, taking into consideration of the effect of the change in software reliability growth modeling is expected to conduct more accurate software reliability assessment. In this model, the authors develop extended software hazard rate models based on well-known Jelinski – Moranda and Moranda models, by considering with a change of testing environment. Especially in this model, the authors incorporate the uncertainty of the effect of the change on the software reliability growth process into the software hazard rate modeling. Finally, they showed numerical examples for their model and results of model comparisons by using actual data. Zhang et al. (2012) discussed a stochastic SRGM with learning and change point. Over the past years, many SRGMs have been proposed and these models treated the software fault detection process as a discrete state space. However, as the size of software system becomes larger, the SRGMs based on a continuous state space were proposed. Although some researchers have investigated the continuous state space SRGMs, none of them considered learning phenomenon and the problem of change point in the fault detection process. In actually, the fault detection process is complex and can be affected by many different factors. The fault detection rate may be a change if the testing environment, strategy or resource allocation is changed. Thus, in order to describe the realistic situations, a continuous state space SRGM based on Stochastic Differential Equation (SDE) SRGMs with learning and change point have been proposed. Further, actual software reliability data have been used to demonstrate the proposed model. Shyur (2003) proposed a stochastic software reliability model with imperfect-debugging and change-point. In this model the author considered the software reliability growth model that incorporates with both imperfect debugging and change-point problem. The proposed model utilizes the failure data collected from software development projects to analyze the software reliability and the remaining errors of a released software program. The maximum likelihood approach is derived to estimate the unknown parameters of the new model. He investigated the new model and demonstrated its applicability in the software reliability engineering field. The analysis suggests that if a change-point exists in a testing process, it should be considered in creating a software reliability estimation model. Singh et al. (2011) discussed a unified framework for developing two dimensional SRGM with change point. In order to assure software quality and assess software reliability, many SRGMs have been proposed. In one dimension SRGMs researcher used one factor such as testing-time, testing-effort or coverage, etc for designing the model but in two-dimensional software reliability growth model, process depends on two-types of reliability growth factors like: testing-time and testing-effort or testing-time and testing-coverage or any combination between factors. Also in more realistic situations, the failure distribution can be affected by many factors, such as the running environment, testing strategy and resource allocation. Once these factors are changed during testing phase, it could result in failure intensity function that increases or decreases non-monotonically and the time point corresponding to abrupt fluctuations is called change point. In this model, the researchers discuss generalized framework for twodimensional SRGM with change point for software reliability assessment. The models developed have been validated on real data set. Inoue et al. (2011) explained a bivariate SRGM with change point. Testing-time when a change of a stochastic characteristic of the software failure-occurrence time or software failure-occurrence time-interval is observed is called change point. It is said that effect of the change point on the software reliability growth process influences on accuracy for software reliability assessment based on a SRGM. The authors proposed an SRGM with the effect of the change point based on a bivariate SRGM, in which the software reliability growth process is assumed to depend on the testing-time and testing-effort factors simultaneously, for accurate software reliability assessment. And the authors discuss an optimal software release problem for deriving optimal testing-effort expenditures based on our model. Further, they show numerical examples of software reliability assessment based on the bivariate SRGM and estimation of optimal testing-effort expenditures by using actual data. IV. ESTIMATION OF MODEL PARAMETERS The parameters of the SRGM are estimated by least square method and maximum likelihood method (Ahmad et. al, 2011). Least Square Estimation (LSE) The parameters Îą, β, θ and m in the testing-effort function can be estimated by the method of LSE. These parameters are determined for n observed data pairs in the form (tk, Wk) (k = 1, 2, . . . , n; 0 < t1 < t2< . . . , < tn), where Wk is the cumulative testing-effort consumed in time (0, tk]. The least squares estimators đ?›źĚ‚, đ?›˝Ě‚ , đ?œƒĚ‚, đ?‘Žđ?‘›đ?‘‘ đ?‘š Ě‚ can be obtained by minimizing: đ?‘›

đ?‘†(đ?›ź, đ?›˝, đ?œƒ, đ?‘š) = ∑(đ?‘Šđ?‘˜ − đ?‘Š(đ?‘Ąđ?‘˜ ))2

(1)

đ?‘˜=1

Maximum Likelihood Estimation (MLE) Once the estimates of Îą, β, θ and m are known, the parameters of SRGMs would be estimated through MLE method. The estimators for a, r1 and r2 are determined for the n observed data pairs in the form (tk, yk) (k = 1,2,‌.., n; 0 < t1 < t2 < ‌‌..<tn), where yk is the cumulative number of software errors detected up to time tk of (0, tk]. Then the likelihood function (Islam, S. F., 2015) for the unknown parameters a, r1 and r2 in the Non-Homogeneous Poisson Process model (Equation 1) is given by:

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Role of Change Point in Non-Homogeneous Poisson Process Software Reliability Growth Model (IJSTE/ Volume 4 / Issue 5 / 028) đ?‘›

đ??żâ€˛ (đ?‘Ž, đ?‘&#x;1 , đ?‘&#x;2 ) = âˆ? đ?‘˜=1

[đ?‘š(đ?‘Ąđ?‘˜ ) − đ?‘š(đ?‘Ąđ?‘˜ − 1)](đ?‘Śđ?‘˜âˆ’đ?‘Śđ?‘˜âˆ’1) −[đ?‘š(đ?‘Ą )−đ?‘š(đ?‘Ą )], đ?‘˜ đ?‘˜âˆ’1 .đ?‘’ (2) (đ?‘Śđ?‘˜ − đ?‘Śđ?‘˜âˆ’1 )!

where t0 = 0 and y0 = 0, The maximum likelihood estimates of SRGM parameters a, r 1 and r2 can be obtained by solving the following three equations: ∂L =0 (3) ∂a ∂L =0 (4) ∂r1 ∂L =0 (5) ∂r2 The above non-linear equations can be solved numerically. We are estimating the parameters using SPSS. We can describe the arithmetic appearance of non-homogeneous software reliability growth model with testing effort function (Log-logistic) and change point as follows: dm(t) 1 Ă— = r Ă— (a − m(t)) (6) dt w(t) and r 0≤t≤τ r(t) = { 1, r2, t>Ď„ The solution of equation (6) under the marginal condition of m(t) = 0, m(t) = a Ă— (1 − e−rW(t) ) (7) Solving equation (6) under the boundary conditions m(0) = 0 and W(0) = 0 and r(t) = r 2 for 0 ≤ t < Ď„, m(t) = a(1 − e−r1 W(t) ) (8) Plug in the value of W(t) from the equation (6), we get δ −1

m(t) = a (1 − e−r1Îą[1−{1+ βt) } ] ) (9) Again, solving equation (8) under the boundary condition m(0) = 0 and W(0) = 0 and r(t) = r 2 for t < Ď„, m(t) = a Ă— [1 − e−(r1W(Ď„)− r2 W(Ď„)+ r2 W(t)) ] (10) Plug in the value of W(t) from the equation (10), we get δ −1 m(t) = a Ă— [1 − e−(r1 W(Ď„)− r2W(Ď„)+ r2Îą[1−{1+ βt) } ]) ] (11) This is a non-homogeneous software reliability growth model which incorporates testing effort function (Log-logistic) and change point. V. CONCLUSIONS Software reliability is very important in the software development area. Unreliable software can causes failure in the computer system that would be major damage. Therefore, proper analysis of software reliability is most important. To achieve this purpose we incorporated a change point into different types of testing effort functions. A software reliability growth model testing-effort and change point is more flexible than other SRGMs. It shows widely fit and there is wider applicability of the model. It is very much realistic and suitable for describing the software reliability easily. REFERENCES [1] [2] [3] [4]

[5] [6] [7] [8] [9]

Ahmad, N., Khan, M.G.M and Rafi, L. S.(2011) “Analysis of an Inflection S-shaped Software Reliability Model Considering Log-logistic Testing-Effort and Imperfect Debugging�, IJCSNS International Journal of Computer Science and Network Security, Vol.11 No.1 pp. 161-171. Ahmad, N. and Islam, A. (1996), “Optimal Accelerated Life Test Designs for Burr Type XII Distributions under Periodic Inspection and Type Censoring�, Naval Research Logistics, Vol. 43, pp. 1049-1077. Ahmad, N., Bokhari, M.U., Quadri, S.M.K. and Khan, M.G.M. (2007), “The Exponentiated Weibull Software Reliability Growth Model with various TestingEfforts and Optimal Release Policy: a Performance Analysis�, International Journal of Quality and Reliability Management, Vol. 25 (2), pp. 211-235] Ahmad, N., Khan, M. G. M., and Islam, S. F(2012), “Optimal Allocation of Testing Resource for Modular Software based on Testing-Effort Dependent Software Reliability Growth�, Proceeding of 3rd International Conference on Computing Communications and Networking Technologies, 2012 (ICCCNT_2012) July 26-28, 2012, Coimbatore, Tamilnadu, India, Presented & Published by IEEE Computer Society. Huang, C.Y (2004), “Performance Analysis of Software Reliability Growth Models with Testing-Effort and Change point�, The Journal of System and Software, pp. 181-194. Inoue, S. and Yamada, S (2011),�A Bivariate Software Reliability Model with Change Point and It’s Applications�, American Journal of Operation Research, Vol. 1, pp. 1-7. Islam, S. F.(2015), “Analysis of Change Point based Software Reliability Growth Model with Log- Logistic Testing Effort Function�, International Research Journal of Computers and Electronics Engineering (IRJCEE) Vol. 3, Iss. 1, IRJCEE-021315-0604-SF-2015-S-1. Singh, O., and Kapur, P.K. (2011), “Unified Framework for Developing Two Dimensional Software Reliability Growth Models with Change Point�, IEEE Conference publication, pp. 570-574. Shyur, H.J., (2003), “A Stochastic Software Reliability Model with Imperfect-Debugging and Change point�. Journal of Systems and Software, Vol. 66, pp. 135-141.

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Role of Change Point in Non-Homogeneous Poisson Process Software Reliability Growth Model (IJSTE/ Volume 4 / Issue 5 / 028) [10] Yamada, S., Shino, H. and Inoue, S. (2013), “Extended Hazard Rate Models for Software Reliability Assessment with Effect at Change Point”, International Journal of Reliability, Quality and Safety Engineering, Vol. 20, pp. 0-11. [11] Zhang, N., Gang, C. and Hong (2012),”A Stochastic Software Reliability Growth Model with Learning and Change Point”, Conference Publication, pp. 399403] [12] Zhao, J., Liu, W. and Yang, Z. (2006),”Software Reliability Growth Model with Change Point and Environmental Function”, Journal, pp. 1-10.

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