5.1 Introduction. Chapter-5: Warranty Cost Analysis of Software Reliability

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1 Chapter-5: Warranty Cost Analysis of Softare Reliability he optimal release policy for constructing the softare reliability groth model (SRGM) that incorporates both imperfect debugging and change-point concepts has been investigated. In any realistic situation the change-point problem may be realized due to the change in many factors like testing strategies, environment, resource allocation, etc.. he cost of the softare before release and the costs of removing errors before and after release are knon as softare cost factors. It is important to ensure hen to stop testing or hen to release the softare so that the total system development cost can be minimized subject to reliability constraint. o evaluate the total expected cost, a arranty cost model using non-homogeneous process along ith change-point phenomenon is discussed. Optimal release policies based on cost and reliability criteria are constructed for determining fairly accurate optimal softare release time. he phenomenon and applicability of the proposed model are established via numerical results. 5. Introduction Softare reliability engineering is an exponentially groing field and day by day their importance and needs have been increasing. Softare developers need to produce softare that should be more reliable; this can be possible by expanding high cost for fixing failures and taking care of safety concerns and legal liabilities. It is very difficult for developer to produce a completely fault free softare because many faults may be introduced during its development process. So the goal of softare development organization is to evaluate realistic softare reliability and provide useful guidance that product can meet market reliability expectation. Softare reliability groth models (SRGMs) facilitate a mathematical relationship beteen the number of faults removed and the testing time. hese models also provide the useful information that ho e can improve the reliability of the system softare. A large number of softare reliability 99

2 Chapter-5: Warranty Cost Analysis of Softare Reliability groth models (SRGMs) have been proposed by many researchers, still ne models are to be developed in this field so that they could fit a great number of reliability groth curves depending upon a variety of applications in different scenario. Mostly softare reliability groth models are based on the assumption that the debugging is perfect. But in real time system, during the testing phase debugging process are not alays performed perfectly. he softare is developed by human, so it is possible to introduce ne errors into the softare during development/testing phase. Sometimes, due to the complexity of fault, it could not be removed perfectly. his phenomenon is called imperfect debugging. Some studies of the effect of imperfect debugging on softare cost model ere made by Xie and Yang (3) and Pham (3), Williams (7). Chen et al. () and Shyur (3) analyzed the softare reliability groth model ith imperfect debugging and change-point. Jain and Priya (5) discussed various softare reliability issues under operational and testing phase. Recently, discrete softare reliability groth modeling as studied by Gosami et al. (7) by incorporating the concept of errors of different severity and change-point. Chiu et al. (8) studied the softare reliability groth model from the perspective effects. Zio (9) suggested the old problems and ne challenges related to reliability engineering. Recently, Jin et al. () considered reliability groth modeling for electronic systems considering latent failure modes. A recent trend in the development of NHPP based SRGMs is the incorporation of some additional information including the change-point concept. In more realistic situations, expenditures on softare testing based upon testing strategy, resource allocation, environment criterion etc., may be changed at some point of time. When any of these factors change during the softare testing, the failure intensity function increases or decreases non-monotonically. In 993, change-point problems in softare and hardare reliability ere discussed by Zhao. Recently, a fe researchers (cf. Kang, ; Zou, 3; Huang, 5b; Kapur et al., 6; Lin and Huang, 8) proposed that the softare testing may be changed at some time point. hey noticed that SRGMs ith change-point achieve a great improvement in the accuracy of evaluation of softare

3 Chapter-5: Warranty Cost Analysis of Softare Reliability reliability. he estimation of parameters for NHPP based SRGM ith change-point as done by Chang (6). Zhao et al. (6) examined the environmental effects on the softare reliability groth model ith change-point. he notable orks in this field as done by Kapur et al. (7, 8b). hey analyzed the effect of change-point on the errors on different severity and used the testing time as poer function. It is very important for softare developer to control a softare development process in terms of cost, reliability and optimal release time. he softare is released in the market for operational use hen it is tested perfectly. But a major problem for softare developers is to decide at hat time, the testing should be stopped and softare should be released so that the total cost can be minimized subject to reliability constraint. he softare release time problem of softare is also of vital importance from the vie point of economics. his problem has been discussed from time to time by many researchers under different constraints/policy. In literature, some research papers on optimization problem have also appeared. Kimura et al. (999) illustrated the economic analysis of softare release problem ith arranty cost and reliability requirement. Zheng () suggested release policies by considering the reliability constraint for softare testing time. One of the key issues of softare engineering is to manage the process of testing under a limited budget in a specified time interval. In this chapter, e investigate ho to integrate change-point concept into imperfect debugging softare reliability groth model (SRGM) ith arranty cost. We suggest the optimal release policies under cost and reliability constraints. he remaining part of this chapter is organized as follos; Section 5. is devoted to the mathematical formulation of non-homogeneous softare reliability groth model (SRGM) ith imperfect debugging and change-point concepts. In section 5.3, e study the arranty cost model. o measure the parameters of the proposed model, maximum likelihood function is given in section 5.4. We discuss optimum release policies based on cost and reliability criteria in section 5.5. o examine the behavior of parameters hich have most significant influence on optimal results, the

4 Chapter-5: Warranty Cost Analysis of Softare Reliability sensitivity analysis is performed in section 5.6. Finally concluding remarks are given in section NHPP Model We develop softare reliability model hich follos a non-homogeneous poisson process (NHPP) for the behavior of the mean of the cumulative number of detected faults. Let {N(t), t } be a counting process that represents the cumulative number of faults detected up to testing time t ith mean value function m(t). he SRGM based on the NHPP can be formulated as: prob. [ N( t) = n] = n m( t ) [ m( t) ] e n!, n =,,,3,... (5.) he mean value function is assumed to be a non decreasing function in testing time ith boundary conditions m() = and m( ) = a, here a is the initial expected error content function. 5.. SRGM ith Imperfect Debugging and Change-Point Most of the softare reliability groth models focus on the softare testing phase, here softare defects are detected, isolated and removed and then softare tends to gro. he point at hich fault detection/introduction rate is changed, is called changepoint. In this section, a SRGM is developed hich incorporates both imperfect debugging and change-point concepts. It is assumed that during the softare testing, fault detection rate and fault introduction rate change at some instant τ. he fault detection rate function ith change-point is defined as b ( t) b, = b, for t τ for t > τ (5.)

5 Chapter-5: Warranty Cost Analysis of Softare Reliability he fault introduction rate during testing ith change-point is defined as β ( t) β, = β, for t τ for t > τ (5.3) he proposed model is based on the folloing assumptions: he fault removal phenomenon is modeled by non-homogeneous Poisson process (NHPP). he softare system is subject to failures at random times caused by the manifestation of remaining faults in the system. he fault detection rate is constant over time. he time beteen (i-) th and i th failures depends on the time to the (i-) th failure. On the detection/removal of a softare failure, it may be possible that the effort to remove the failure may not be perfect i.e. due to imperfect debugging a ne fault may be introduced during removal of the error. he total number of faults at the beginning of the testing phase is finite i.e. the imperfect error debugging does not increase the initial error content. hese are the notations hich have been used to formulate the model: m(t) λ(t) a b β C C t C : Mean value function in the NHPP model. : Failure intensity function. : Initial number of errors in the softare before starting of the testing phase. : Fault detection rate. : Fault introduction rate. : Initial testing cost. : esting cost per unit time. : Maintenance cost per fault during the arranty period. 3

6 Chapter-5: Warranty Cost Analysis of Softare Reliability : Release time of the softare. * : Optimal release time of the softare. α EC() C () : Warranty period. : Discount rate of the cost. : Expected total maintenance cost of the softare. : Maintenance cost during the arranty period. he mean value function m(t) can be obtained as [cf. Shyur, 3]: ( ) m t a β = a β [ exp( ( β ) b t) ] [ exp( ( β ) b τ ( β ) b ( t τ) )], m + ( τ)( β β ) β, t τ t > τ (5.4) he corresponding failure intensity function is λ ( t) dm t = dt ( ) ab[ exp( ( β ) bt) ], = ab [ exp( ( β ) b τ ( β ) b ( t τ) )] 5.3 Warranty Cost Model, t τ t > τ (5.5) Softare testing is divided into the testing phase and operational/arranty phase. It is evident that the cost of remaining an error in operational phase is more expensive than the testing phase because during the debugging process some ne type of faults may be introduced into the softare in the operational phase. When the cost of the softare development is estimated, the softare developers have to consider the cost of after-sales support. his is knon as the arranty cost. he computation of this arranty cost is dependent on the release time of the softare. 4

7 Chapter-5: Warranty Cost Analysis of Softare Reliability No, e discuss the optimal testing time by considering the development and arranty cost ith the discount rate. he discount maintenance cost is considered to take care of the present value of the money. First of all e construct cost model for the softare by assuming that there are three types of costs i.e., (i) an initial testing cost, (ii) testing cost per unit time and (iii) the maintenance cost during the arranty period. Hence the total expected softare maintenance cost is given by ( ) = C + C exp( α) dt C ( ) EC t + (5.6) here are to cases for maintenance cost during arranty period: Case I : In this case, it is assumed that during the arranty period, the softare reliability groth does not occur. Here e assume that only minor errors that ill not affect the reliability of the softare, can be corrected. hen maintenance cost during arranty is obtained as C ( ) C = C + λ + λ ( ) exp( αt) ( ) exp( αt) dt, dt, < t τ t > τ C = C ab exp α ab exp α ( ( β ) b ) exp( α) [ exp( α )] t τ ( ( β ) b τ ( β ) b ( τ) ) exp( α) [ exp( α )],, t > τ No, using eqs (5.6)-(5.7), e get (5.7) (i) EC = t ( ) C + C exp( αt) dt + C exp( ( β ) b ) exp ab α ( α) [ exp( α )], for t τ (5.8) 5

8 Chapter-5: Warranty Cost Analysis of Softare Reliability (ii) EC = t ( ) C + C exp( αt) dt + C exp( ( β ) bτ ( β ) b ( τ) ) exp ab α ( α) [ exp( α )], for t >...(5.9) Differentiating equation (5.8) ith respect to and equating to zero, here ( t τ), e get ab C = = ln...(5.) [ α + b ( β )][ exp( α )] b( β) α Ct And differentiating equation (5.9) ith respect to and equating to zero, here (t > τ), e have = ab C ( τ[ ( β ) b ( β ) b ])[ α + b ( β )][ exp( α )] = ln exp b( β ) α Ct ( ) d EC Since d = >, ( ) d EC and d = >, EC() has a minimum value at = *, < t τ; and = *, t > τ. (5.) Case II : Here e assume that during the arranty period, the softare reliability groth occurs, even after the testing phase. In this case only major errors that ill improve the reliability of the softare can be corrected. hen C W () is given by C ( ) C = C + + λ λ ( t) exp( αt) ( t) exp( αt) dt, dt, for t τ for t > τ 6

9 Chapter-5: Warranty Cost Analysis of Softare Reliability C = C ab [( β ) b + α] ab [( β ) b + α] exp exp ( ( β ) b + α) [ exp( ( β ) b + α) ] [ z (( β ) b + α) ] [ exp[ (( β ) b + α) ] Substituting above values from eq. (5.) in eq. (5.6), e get, for t τ, for t > τ (5.) (i) EC( ) = C exp( t) dt C + Ct α + [( β ) b ] + α [ exp( ( β ) b + α) ], ab exp ( ( β ) b + α) for t τ...(5.3) (ii) EC( ) = C exp( t) dt C + Ct α + [( β ) b ] + α [ exp( ( β ) b + α) ], ab exp [(( β ) b + α) ] for t > τ...(5.4) Differentiating equation (5.) ith respect to and equating to zero, here ( t τ), e get C = (5.5) [ exp[ (( β ) b + α) ] 3 = ln ab b( β ) Ct Again, differentiating equation (5.3) ith respect to and equating to zero, here (t > τ), e find = C ( z) [ exp[ (( β ) b + α) ] 4 = ln ab exp b( β ) Ct (5.6) here ( β ) b τ + ( β ) τ z. = b ( ) d EC Again d = 3 >, ( ) d EC and d = 4 >. 7

10 Chapter-5: Warranty Cost Analysis of Softare Reliability hus EC() has a minimum value at 3 = * for t τ, and 4 = * for t > τ. 5.4 Warranty Cost Model ith Reliability Constraint he softare reliability of the NHPP model is defined as the probability that a softare failure ill not occur during the testing time interval (, +x]. he softare reliability function is formulated as: R ( x ) = exp m( + x) m( ) [ { }] exp = exp [ exp( ( β ) b ).m ( x) ], [ exp( ( β ) b τ ( β ) b ( τ) ).m ( x) ], for t τ for t > τ (5.7) here m i a = i i, i =,. β ( x) [ exp( ( β ) b x) ] i Let R be the optimal release time for the case (i) for testing time ( t τ) and (t > τ) satisfying the relation R x = R x, R R =, respectively. Similarly R be the optimal release time for the case (ii) for testing time 3 ( t τ) and 4 (t > τ) satisfying the relation R x = R and R x = R 3 respectively. No, 4 e get and ( ( )) ( ) R = ln m x ln ln (5.8) b β R ( ( )) (( ) ( ) ) ( ) R = ln m x + β b β b τ ln ln (5.9) b β R 8

11 Chapter-5: Warranty Cost Analysis of Softare Reliability Let R ( < R ) be the desired level of reliability and = * (optimum release time) hich minimizes the expected cost of the softare ith reliability objective R. hus, the optimization problem can be formulated as Minimize EC(), subject to R(x/) R (5.) 5.5 Optimal Softare Release Policies he softare release time problem is basically concerned ith the cost of the softare. he goal of the softare developers is to determine an optimum softare release time satisfying both cost and reliability requirements. he optimal softare release policies can be established by minimizing the total expected softare cost under the specified constraints so that the achieved softare reliability is not less than a prespecified level of reliability. Such cost-reliability optimization problem can be solved to predict the optimal softare release policies. he optimal softare release policies for the NHPP SRGM based on cost and reliability criteria are as follos: Optimal esting ime ith Cost (A) Optimal esting Policy (OP ) for case I: P A. : λ() > λ( ) λ(τ) and λ(τ) < λ( ), then * =. P A. : λ() < λ( ) λ(τ) and λ(τ) > λ( ), then * =. P A.3 : λ() < λ( ) λ(τ) and λ(τ) < λ( ), then * =. 9

12 Chapter-5: Warranty Cost Analysis of Softare Reliability P A.4 : λ() > λ( ) λ(τ) and λ(τ) > λ( ), then * = max(, ). (B) Optimal esting Policy (OP ) for case (ii): P B. : λ() > λ( 3 ) λ(τ) and λ(τ) < λ( 4 ), then * = 3. P B. : λ() < λ( 3 ) λ(τ) and λ(τ) > λ( 4 ), then * = 4. P B.3 : λ() < λ( 3 ) λ(τ) and λ(τ) < λ( 4 ), then * =. P B.4 : λ() > λ( 3 ) λ(τ) and λ(τ) > λ( 4 ), then * = max( 3, 4 ). Optimal esting ime ith Cost and Reliability Constraints (C) Optimal esting Policy 3 (OP 3) for case I: P C. : If λ() > λ( ) λ(τ) and λ(τ) < λ( ) and R(x/) > R, then * =. P C. : If λ() < λ( ) λ(τ) and λ(τ) > λ( ) and R(x/) > R, then * =. P C.3 : If λ() > λ( ) λ(τ) and λ(τ) > λ( ) and R(x/) > R, then * = max{, }.

13 Chapter-5: Warranty Cost Analysis of Softare Reliability P C.4 : If λ() > λ( ) λ(τ) and λ(τ) < λ( ) and R(x/) > R, then * =. P C.5 : If λ() > λ( ) λ(τ) and λ(τ) < λ( ) and R(x/) < R, then * = max{, R }. P C.6 : If λ() < λ( ) λ(τ) and λ(τ) > λ( ) and R(x/) > R, then * = max{, R }. P C.7 : If λ() > λ( ) λ(τ) and λ(τ) > λ( ) and R(x/) < R, then * = max{,, R }. (D) Optimal esting Policy 4 (OP 4) for case II: P D. : If λ() > λ( 3 ) λ(τ) and λ(τ) < λ( 4 ) and R(x/) > R, then * = 3. P D. : If λ() < λ( 3 ) λ(τ) and λ(τ) > λ( 4 ) and R(x/) > R, then * = 4. P D.3 : If λ() > λ( 3 ) λ(τ) and λ(τ) > λ( 4 ) and R(x/) > R, then * = max{ 3, 4 }. P D.4 : If λ() > λ( 3 ) λ(τ) and λ(τ) < λ( 4 ) and R(x/) > R, then * =. P D.5 : If λ() > λ( 3 ) λ(τ) and λ(τ) < λ( 4 ) and R(x/) < R, then * = max{ 3, R }.

14 Chapter-5: Warranty Cost Analysis of Softare Reliability P D.6 : If λ() < λ( 3 ) λ(τ) and λ(τ) > λ( 4 ) and R(x/) > R, then * = max{ 4, R }. P D.7 : If λ() > λ( 3 ) λ(τ) and λ(τ) > λ( 4 ) and R(x/) < R, then * = max{ 3, 4, R }. 5.6 Sensitivity Analysis o verify the proposed softare reliability groth model that incorporates both imperfect debugging and change-point concept, e perform the sensitivity analysis by computing the expected maintenance cost and softare reliability. he program has been coded using MALAB to check the validity of the analytical results. he effect of various parameters on the expected cost and reliability are explored by varying the parameters namely a (initial number of errors), b and b (error detection rates) and α (discount rate of cost) for both cases. Some other default parameters are fixed as C =, C t = 5, C = 5, = 5, β =.4, x =.9. here are to cases for describing the expected maintenance cost EC() and reliability R(). he graphical representation of the expected maintenance cost EC() has been done in figs (5.)-(5.4). he reliability R() has also been displayed in figs (5.5)- (5.7). Figs 5.(i)-5.(iii) exhibit the effects of a, b and α on EC() for the cases hereas figs 5.(i)-5.(iii) depict the effects of same parameters for case before the change-point. Figs 5.(i) and 5.(ii) reveal that EC() first decreases (approximately up to t = 4) and then after it increases for the testing time and finally it attains constant behavior. We notice the same trend in figs 5.(i) and 5.(ii) hich have been dran for case before change-point. Figs 5.(ii) and 5.(ii) also sho the same pattern for b ith respect to time for case and case, respectively. We also notice that EC() increases ith respect to a and b for both cases. In figs 5.(iii) and 5.(iii), EC() first decreases

15 Chapter-5: Warranty Cost Analysis of Softare Reliability gradually up to t = 6 and after that it increases sharply. It is also seen that after t = 6, EC() decreases ith respect to α. In figs 5.3(i)-5.3(iv) and figs 5.4(i)-5.4(iv), the effect of variations of parameters a, b, b and α on EC() for both cases and respectively, after change-point are plotted. We observe the similar trend for both cases as noted in figs 5.(i)-5.(iii) i.e. EC() first decreases and after some time it gros rapidly ith respect to testing time. We also see that, EC() increases for high number of initial faults a and high value of error detection rate b. On the other hand, for higher values of b and α, the expected cost EC() decreases. Figs (5.5)-(5.7) illustrate the pattern of the reliability of the softare by varying the parameters a, b, b and τ for both cases i.e. before change-point and after change-point. Figs 5.5(i) and 5.6(i) sho the trend of reliability by varying the parameter a before change-point and after change-point, respectively. We notice that after the change-point, the reliability increases as compare to before change-point. We also see the same pattern of reliability in figs 5.5(ii) and 5.6(ii). It can be easily seen that as initial number of faults (a) increases, the reliability increases. We notice the same effect of parameters b and τ on EC() in figs 5.7(i) and 5.7(ii) and in figs 5.5(ii) and 5.6(ii). Also, e found that EC() increases as error detection rate increases. In figs 5.7(i) and 5.7(ii), reliability is plotted by varying the parameters b and τ, respectively, for after the change-point case. In these graphs, the reliability is increasing sharply as compared to before change-point case. From the above results, it is over all concluded that there is a steep decreasing trend in the expected maintenance cost EC() for the loer values of ; and after some time, the EC() shos the linear increment for the further increased values of. It is demonstrated that the expected maintenance cost EC() is greatly affected by the testing time. he reason behind the high value of E{C()} in the beginning may be the initial testing cost, hich is taken significantly high. 3

16 Chapter-5: Warranty Cost Analysis of Softare Reliability 5.7 Concluding Remarks he softare reliability groth model developed in this investigation includes the imperfect debugging phenomenon as ell as change-point concepts. he softare cost subject to reliability constraint discussed may provide an insight to achieve maximum reliability ithin a given budget. It is noticed that if the cost could be less than the given budget and the customers are satisfied, then it is a great profit for an organization. he proposed model and findings have been validated by taking numerical illustration, hich demonstrates the applications of investigation done for different types of softare and large-scale real time embedded systems. 4

17 Chapter-5: Warranty Cost Analysis of Softare Reliability E C( ) E C() E C ( ) (i) (ii) alpha=. alpha=. alpha=.3 a=3 a=5 a= b=.4 b=.6 b= E C( ) E C() E C() (i) (ii) alpha=. alpha=. alpha=.4 a=3 a=4 a=5 b=.3 b=.5 b= (iii) Fig. 5.: Expected maintenance cost Vs for case before change-point for different values of (i) a (ii) b (iii) α. (iii) Fig. 5.: Expected maintenance cost Vs for case before change-point for different values of (i) a (ii) b (iii) α. 5

18 Chapter-5: Warranty cost analysis of softare reliability EC() a=3 a=5 a= EC() b=.3 b=.5 b= (i) (ii) b=.4 b=.5 b= alpha=.3 alpha=.5 alpha=.7 EC() 5 EC() (iii) (iv) Fig. 5.3: Expected maintenance cost Vs for case after change-point for different values of (i) a (ii) b (iii) b and (iv) α. 6

19 Chapter-5: Warranty cost analysis of softare reliability EC() a=5 a=7 a= (i) EC() b=.3 b=.5 b= (ii) EC() b=.4 b=.6 b=.8 EC() alpha=.3 alpha=.6 alpha= (iii) (iv) Fig. 5.4: Expected maintenance cost Vs for case after change-point for different values of (i) a (ii) b (iii) b and (iv) α. 7

20 Chapter-5: Warranty cost analysis of softare reliability R a=5 a=7 a= R a=5 a=7 a= (i) (i) R (ii) b=.5 b=.6 b= R Fig. 5.5: Softare reliability Vs Fig. 5.6: Softare reliability Vs before change point for different after change point for different values of (i) a (ii) b. values of (i) a (ii) b (ii) b=.3 b=.5 b=, b=.4.3 b=.6. b= R (i) R Fig. 5.7: Softare reliability Vs after change-point for different values of (i) b (ii) τ (ii) tau= tau=4 tau=6