Software reliability is defined as the probability of failure

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1 Evolutoary Regresso Predcto for Software Cumulatve Falure Modelg: a comparatve study M. Beaddy, M. Wakrm & S. Aljahdal 2 : Dept. of Math. & Ifo. Equpe MMS, Ib Zohr Uversty Morocco. beaddym@yahoo.fr 2: Taf Uversty Saud Araba Abstract A evolutoary regresso modelg approach for software cumulatve falure predcto based o auto-regresso order 4, 7 ad 0 models are proposed. A real coded geetc algorthm s used to optmze the mea square of the error produced by trag the auto-regresso model. I ths paper, we preset a real coded geetc algorthm that uses the approprate operators for ths ecodg type to tra the autoregresso model. To evaluate the predctve capablty of the developed model data sets, varous projects were used. A comparso betwee auto-regresso order 4 model traed usg least square estmato [] ad real coded geetc algorthm trag s provded, also a comparso betwee the autoregresso order 7 ad 0 models traed usg the geetc algorthm s preseted. Expermetal results show that the trag of dfferet auto-regresso model by the real coded geetc algorthm has a good predctve capablty. to predct the umber of faults the software as show by Aljahdal []-[2]. Least square estmato s the most techque used to estmate lear models as observed the lterature []. I ths paper we have developed a real coded geetc algorthm (RCGA) as a alteratve to estmate the auto-regresso model proposed by Aljahdal []-[2], that optmzes the error made by the model. Evolvg the parameters set for the auto-regresso model wth the verted error as a ftess fucto. To show the effect of the order of the model, a auto-regresso model order 7 ad 0 s establshed by usg our proposed approach to estmate ts parameters. II. SOFTWARE RELIABILITY DATA SET: Keywords Geetc Algorthms, Real Coded Geetc Algorthms, Auto Regresso Model, Software Relablty. I. INTRODUCTION Software relablty s defed as the probablty of falure free software operato for a specfed perod of tme a specfed evromet [3]. Socety s relace o large complex systems madates hgh relablty. Relable software s a ecessary compoet. Cotrollg faults software requres that oe ca predct problems early eough to take prevetve acto. I the past 35 years more tha 00 software relablty models have bee developed to solve relablty models [2]. Most of these models as the models of software relablty growth deped o a certa a pror assumptos about the ature of software faults ad the stochastc behavor of software process [4]-[5]. As a result, dfferet models have dfferet predctve performace at dfferet testg phases across varous projects. A sgle uversal model that ca provde hghly accurate predctos uder all crcumstaces wthout ay assumptos s most desrable [5]-[]. Regresso models are the most popular method for buldg models ad are used to calbrate almost all of the models [2]. Regresso predcto models are oe of the proposed models The Software Relablty set was compled by Joh Musa of Bell Telephoe Laboratores [6]. Hs objectve was to collect falure terval data to assst software maagers motorg test status ad predctg schedules ad to assst software researchers valdatg software relablty models. These models are appled the dscple of Software Relablty Egeerg. The dataset cossts of software falure data o 6 projects. Careful cotrols were employed durg data collecto to esure that the data would be of hgh qualty. The data was collected throughout the md 970s. It represets projects from a varety of applcatos cludg real tme commad ad cotrol, word processg, commercal, ad mltary applcatos. I our case, we used data from three dfferet projects. They are Mltary, Real Tme Cotrol ad Operatg System. The falure data were tally stored arrays, ordered by day of occurrece so that t could be processed. III. REGRESSION MODELS Oe of the most famous regresso models s the Auto- Regressve models. Ths model has bee used may applcatos. The auto-regressve model ca be descrbed as followg form []: y( k) = a0 + a y( k τ ) () = y( k τ ) s the observed cumulatve falure days before the curret day, a s the tug parameter for the auto-

2 regresso model ad s referred to as the order of the model. IV. THE GENETIC ALGORITHM The geetc algorthm was developed ad formalzed by Hollad [0]. It was further developed ad show to have wde applcablty by Goldberg [8]. For a good troducto ad examato of GAs, see Mchalewcz [3]. Although there are may possble varetes o the basc GAs, the operatoal of every geetc algorthm s descrbed the followg steps:. Radomly create a tal populato of chromosomes. 2. Compute the ftess of every member of the curret populato. 3. If there s a member of the curret populato that satsfes the problem requremets the stop. Otherwse cotue to the ext step. 4. Create a termedate populato by extractg members from the curret populato usg a selecto operator. 5. Geerate a ew populato by applyg the geetc operators of crossover ad mutato to ths termedate populato. 6. Go back to step 2. V. REAL CODED GENETIC ALGORITHMS The most commo represetato GAs s bary [9]. The chromosomes cossts of a set of gees, whch are geerally characters belogg to a alphabet {0, }. Therefore, a chromosome s a vector C cosstg of l gees c : C=(c,c 2,,c l ), c ={0,}, Where l s the legth of the chromosome. However the optmzato problems of parameters wth varables cotuous domas, t s more atural to represet the gees drectly as a real umbers sce the represetato of soluto are very close to the atural formulato,.e. there are o dffereces betwee the geotype ad the pheotype. The use of ths real-codg umercal optmzato o cotuous domas appears Mchalewcz [3]. I ths case, a chromosome s a vector of floatg pot umbers. The chromosome legth s the vector legth of the soluto of the problem; thus, each gee represets a varable of the problem. The gee values are forced to rema the terval establshed by the varables they represet, so may geetc operators are developed for them, such as, Flat crossover [6], Arthmetc crossover [4] ad BLX-α crossover [7] for the crossover operators ad Radom mutato ad o-uform mutato for the mutato operators [4]. VI. THE REAL CODED GENETIC ALGORITHM TO ESTIMATE AUTO-REGRESSION PARAMETRS FOR SOFTWARE RELIABILITY PREDICTION As metoed above, real codg s the most sutable codg for cotuous domas. Sce our goal s auto-regresso model parameter s estmatg whch predcts the cumulatve future faults the software, t appears logcal to use ths codg ad geetc operators assocated to t. Amog the advatages of usg real-valued codg over bary codg s creased precso. Bary codg of real-valued umbers ca suffer loss of precso depedg o the umber of bts used to represet oe umber. Moreover, real-valued codg chromosome strg become much shorter. For real-valued optmzato problems, real-valued codg s smply much easer ad more effcet to mplemet, sce t s coceptually closer to the problem space. I partcular, our am s to tra a auto-regresso model to predct future faults the software from the prevous dscovered faults. A chromosome cossts of all the parameters. Oe gee of a chromosome represets a sgle parameter value. I our case there are; 5, 8 ad parameters for the auto-regresso order 4, 7 ad 0 respectvely. The legth of the chromosomes s l= 5, l= 8 ad l=. The parameters of the auto-regresso are placed o a chromosome as show fgure. θ 0 θθ 2... θ θ Fgure: The chromosomal represetato of the auto-regresso model. Ftess fucto: the ftess fucto should reflect the dvdual s performace the curret problem. We have chose /(+mse) as a ftess fucto Eq. (3), where mse s the mea squared error durg trag defed Eq (2). mse ˆ 2 = ( β β ) (2) Where s the umber of trag faults used durg the trag process. β Ad βˆ are the actual ad the predcted output respectvely durg the learg process. ftess = + mse Selecto mechasm: The roulette wheel selecto s used to create the termedate populato. For each chromosome C a populato P, the probablty p s (C ), of cludg a copy of ths chromosome the termedate populato calculated as Eq. (4) p j= ftess( C ) j (3) P s s ( C ) = P (4) ftess( C ) Where P s the umber of dvduals the populato P.

3 VII. THE REAL CODED GENETIC ALGORITHM TRAINING AND TESTING RESULTS The tal parameters were radomly chose the terval [0, ]. For each project we performed a umber of smulatos wth a populato of 200 dvduals ad a maxmum of geerato equal to Gmax. After the trag process the Normalze Root Square Error (NRMSE, see Eq. 6) s computed to compare the results obtaed by real coded geetc algorthm wth these obtaed by the least square estmato. NRMSE = = ( β ( ) ˆ( β )) = ( β ( )) The results of NRMSE obtaed, by the LSE test phase are gve a table. Table : Results for NRMSE obtaed usg Auto-regresso model order 4 model testg case [3]. Real Tme Operatg Mltary Cotrol System Trag Testg Regresso Model The results of MSE ad NRMSE obtaed, by the trag the Auto-regresso order 4 model wth our real coded geetc algorthm trag ad testg phases are gve table 2. Table 2: Results for the MSE ad NRMSE obtaed usg AR-4 traed by RCGA the trag ad testg phases. Real Tme Operatg Mltary Cotrol System Trag MSE NRMSE.6062E E E-5 Testg MSE NRMSE E E E-5 I fgure 2 to 7 we are showg the testg results ad error dfferece for varous projects usg the AR-4 traed by our real coded geetc algorthm. 2 2 (5) Fgure 2:Actual ad Predcted Testg phase: Mltary Fgure 3: Predcto Error testg phase: Mltary Fgure 4: Actual ad Predcted Testg phase: Real Tme Cotrol

4 VIII. RESULTS FOR THE AR-7 AND AR-0 TRAINED BY THE RCGA The results of MSE ad NRMSE obtaed, by the trag the Auto-regresso order 7 ad 0 models wth our real coded geetc algorthm trag ad testg phases are gve tables 3 ad 4. Fgure 5: Predcto Error testg phase:real Tme Cotrol Fgure 6: Actual ad Predcted Testg phase: Operatg System Table 3: Results for the MSE ad NRMSE obtaed usg AR-7 traed by RCGA the trag ad testg phases. Mltary Real Tme Cotrol Operatg System Trag MSE NRMSE.9752E E E-5 Testg MSE NRMSE 4.836E E E-5 Tableau 4: Results for the MSE ad NRMSE obtaed usg AR-0 traed by RCGA the trag ad testg phases. Mltary Real Tme Cotrol Operatg System Trag MSE NRMSE E E E-5 Testg MSE NRMSE E E E-5 I fgure 8 to 3 we are showg the testg results ad error dfferece for varous projects usg the AR-7 traed by our real coded geetc algorthm. Fgure 7: Predcto Error testg phase: Operatg System Fgure 8: Actual ad Predcted Testg phase: Mltary

5 Fgure 9: Predcto Error testg phase: Mltary Fgure 2: Actual ad Predcted Testg phase: Operatg System Fgure 0: Actual ad Predcted Testg phase: Real Tme Cotrol Fgure 3: Predcto Error testg phase: Operatg System I fgure 4 to 9 we are showg the testg results ad error dfferece for varous projects usg the AR-0 traed by our real coded geetc algorthm. Fgure : Predcto Error testg phase:real Tme Cotrol Fgure 4: Actual ad Predcted Testg phase: Mltary

6 Fgure 5: Predcto Error testg phase: Mltary Fgure 8: Actual ad Predcted Testg phase: Operatg System Fgure 6: Actual ad Predcted Testg phase: Real Tme Cotrol Fgure 9: Predcto Error testg phase: Operatg System IX. CONCLUSION I ths paper, a evolutoary Auto-regresso modelg approach for software cumulatve falure predcto s proposed. Geetc algorthm s used to lear the autoregresso models by optmzg the mea square error produced by these models. Expermetal results show that our proposed approach adapts well across dfferet projects, ad has a better performace compared to the results obtaed by Auto-regresso models for cumulatve falure, leared by the Lest Square Estmato. Fgure 7: Predcto Error testg phase:real Tme Cotrol The order of the auto-regresso model does t affect the model as show dfferet results for dstct projects. REFERENCES [] S. Aljahdal, A. Sheta ad D. Re, Predcto of Software Relablty: A Comparso betwee regresso ad eural

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