SPC for Software Reliability: Imperfect Software Debugging Model

Transcription

1 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 8, Issue 3, o., May 0 ISS (Olie: SPC for Software Reliability: Imperfect Software Debuggig Model Dr. Satya Prasad Ravi,.Supriya ad G.Krisha Moha 3 Associate Professor, Dept. of Computer Sciece & Egg., Acharya agrjua Uiversity agarjua agar, Gutur, Adhrapradesh, Idia Assistat Professor, Dept. of Computer Sciece, Adiavi aaya uiversity Rajahmeddry-53305, Adhrapradesh,Idia 3 Reader, Dept. of Computer Sciece, P.B.Siddhartha college Vijayawada, Adhrapradesh, Idia Abstract Software reliability process ca be moitored efficietly by usig Statistical Process Cotrol (SPC. It assists the software developmet team to idetify failures ad actios to be tae durig software failure process ad hece, assures better software reliability. I this paper, we cosider a software reliability growth model of o-homogeous Poisso Process (HPP based, that icorporates imperfect debuggig problem. The proposed model utilizes the failure data collected from software developmet projects to aalyze the software reliability. The maximum lielihood approach is derived to estimate the uow poit estimators of the model. We ivestigate the model ad demostrate its applicability i the software reliability egieerig field. Keywords: Statistical Process Cotrol, Software reliability, mea value fuctio, imperfect debuggig, Probability limits, Cotrol Charts.. Itroductio Software reliability assessmet is importat to evaluate ad predict the reliability ad performace of software system, sice it is the mai attribute of software. To idetify ad elimiate huma errors i software developmet process ad also to improve software reliability, the Statistical Process Cotrol cocepts ad methods are the best choice. SPC cocepts ad methods are used to moitor the performace of a software process over time i order to verify that the process remais i the state of statistical cotrol. It helps i fidig assigable causes, log term improvemets i the software process. Software quality ad reliability ca be achieved by elimiatig the causes or improvig the software process or its operatig procedures [6]. The most popular techique for maitaiig process cotrol is cotrol chartig. The cotrol chart is oe of the seve tools for quality cotrol. Software process cotrol is used to secure the quality of the fial product which will coform to predefied stadards. I ay process, regardless of how carefully it is maitaied, a certai amout of atural variability will always exist. A process is said to be statistically i-cotrol whe it operates with oly chace causes of variatio. O the other had, whe assigable causes are preset, the we say that the process is statistically out-of-cotrol. The cotrol charts ca be classified ito several categories, as per several distict criteria. Depedig o the umber of quality characteristics uder ivestigatio, charts ca be divided ito uivariate cotrol charts ad multivariate cotrol charts. Furthermore, the quality characteristic of iterest may be a cotiuous radom variable or alteratively a discrete attribute. Cotrol charts should be capable to create a alarm whe a shift i the level of oe or more parameters of the uderlyig distributio or a o-radom behavior occurs. ormally, such a situatio will be reflected i the cotrol chart by poits plotted outside the cotrol limits or by the presece of specific patters. The most commo o-radom patters are cycles, treds, mixtures ad stratificatio [7]. For a process to be i cotrol the cotrol chart should ot have ay tred or oradom patter. SPC is a powerful tool to optimize the amout of iformatio eeded for use i maig maagemet decisios. SPC provides real time aalysis to establish cotrollable process baselies; lear, set, ad dyamically improves process capabilities; ad focus busiess areas

2 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 8, Issue 3, o., May 0 ISS (Olie: which eed improvemet. The early detectio of software failures will improve the software reliability. The selectio of proper SPC charts is essetial to effective statistical process cotrol implemetatio ad use. The SPC chart selectio is based o data, situatio ad eed [4]. The cotrol limits ca the be utilized to moitor the failure times of compoets. After each failure, the time ca be plotted o the chart. If the plotted poit falls betwee the calculated cotrol limits, it idicates that the process is i the state of statistical cotrol ad o actio is warrated. If the poit falls above the UCL, it idicates that the process average, or the failure occurrece rate, may have decreased which results i a icrease i the time betwee failures. This is a importat idicatio of possible process improvemet. If this happes, the maagemet should loo for possible causes for this improvemet ad if the causes are discovered the actio should be tae to maitai them. If the plotted poit falls below the LCL, It idicates that the process average, or the failure occurrece rate, may have icreased which results i a decrease i the failure time. This meas that process may have deteriorated ad thus actios should be tae to idetify ad causes may be removed. It ca be oted here that the parameter a, b should ormally be estimated with the data from the failure process. The cotrol limits for the chart are defied i such a maer that the process is cosidered to be out of cotrol whe the time to observe exactly oe failure is less tha LCL or greater tha UCL. Our aim is to moitor the failure process ad detect ay chage of the itesity parameter. Whe the process is ormal, there is a chace for this to happe ad it is commoly ow as false alarm. The traditioal false alarm probability is to set to be 0.7% although ay other false alarm probability ca be used. The actual acceptable false alarm probability should i fact deped o the actual product or process [9].. Literature Survey This sectio presets the theory that uderlies a distributio ad maximum lielihood estimatio for complete data. If t is a cotiuous radom variable with pdf: f ( t ;,,,. where,,, are uow costat parameters which eed to be estimated, ad cdf: F t. Where, The mathematical relatioship betwee the pdf ad cdf is give by: d F t f ( t. Let dt a deote the expected umber of faults that would be detected give ifiite testig time i case of fiite failure HPP models. The, the mea value fuctio of the fiite failure HPP models ca be writte as: m ( t af ( t, where F(t is a cumulative distributio fuctio. The failure itesity fuctio (t i case of the fiite failure HPP models is give by: ( t af ' ( t.. HPP model The o-homogeous Poisso Process (HPP based software reliability growth models (SRGMs are proved quite successful i practical software reliability egieerig [3]. The mai issue i the HPP model is to determie a appropriate mea value fuctio to deote the expected umber of failures experieced up to a certai time poit. Model parameters ca be estimated by usig Maximum Lielihood Estimate (MLE. Various HPP SRGMs have bee built upo various assumptios. May of the SRGMs assume that each time a failure occurs, the fault that caused it ca be immediately removed ad o ew faults are itroduced. Which is usually called perfect debuggig. Imperfect debuggig models have proposed a relaxatio of the above assumptio [8]. Let t, t 0 be the cumulative umber of software failures by time t. m(t is the mea value fuctio, represetig the expected umber of software failures by time t. t is the failure itesity fuctio, which is proportioal to the residual fault cotet. Thus bt mt a( e ad dm ( t t b ( a m ( t. dt Where a deotes the iitial umber of faults cotaied i a program ad b represets the fault detectio rate. I software reliability, the iitial umber of faults ad the fault detectio rate are always uow. The maximum lielihood techique ca be used to evaluate the uow parameters. I HPP SRGM t ca be expressed i a more geeral way as dm ( t t b t a t m t. dt Where a t is the time-depedet fault cotet fuctio which icludes the iitial ad itroduced faults i the program ad b t is the time-depedet fault detectio rate. A costat a t implies the perfect debuggig assumptio, i.e o ew faults are itroduced durig the debuggig process. If we assume that, whe the detected faults are removed, the there is a possibility to itroduce ew faults with a costat rate. The the mea value fuctio is [] give as a bt m ( t e.. ML (Maximum Lielihood Parameter Estimatio The idea behid maximum lielihood parameter estimatio is to determie the parameters that maximize

3 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 8, Issue 3, o., May 0 ISS (Olie: the probability (lielihood of the sample data. The method of maximum lielihood is cosidered to be more robust (with some exceptios ad yields estimators with good statistical properties. I other words, MLE methods are versatile ad apply to may models ad to differet types of data. Although the methodology for maximum lielihood estimatio is simple, the implemetatio is mathematically itese. Usig today's computer power, however, mathematical complexity is ot a big obstacle. If we coduct a experimet ad obtai idepedet observatios, t, t,, t. The the lielihood fuctio is give by[] the followig product: L, f ( ti ;,,, i t t,, t,,, L Liely hood fuctio by usig λ(t is: L = ( t i i The logarithmic lielihood fuctio is give by: l L l f ( t ;,,, i Log L = log ( ( t i i i which ca be writte as log ( t i i m ( t The maximum lielihood estimators (MLE of,,, are obtaied by maximizig L or, where is l L. By maximizig, which is much easier to wor with tha L, the maximum lielihood estimators (MLE of,,, are the simultaeous solutios of equatios such that: 0, j=,,, j The parameters a ad b are estimated usig iterative ewto Raphso Method, which is give as g ( x x x g ' ( x 3. Illustratig the MLE Method 3. parameter estimatio To estimate a ad b, for a sample of uits, first obtai the lielihood fuctio: assumig L i abe bt Tae the atural logarithm o both sides, The Log Lielihood fuctio is give as: Log L = log[ ( ] = log[ abe i t i i bt a bt = log( abe e i Taig the Partial derivative w.r.t a ad equatig to 0. (i.e log L 0 a bt a e Taig the Partial derivative w.r.t b ad equatig to 0. (i.e ( log L g b b 0 t g b 0 bt t i b e i Taig the partial derivative agai w.r.t b ad equatig to 0. (i.e log L g ' ( b 0 b bt t e g ' b 0 bt e b The parameter b is estimated by iterative ewto Raphso Method usig g ( b b b. which is g ' ( b substituted i fidig a. 3. Distributio of Time betwee failures Based o the iter failure data give i Table, we compute the software failures process through Mea Value Cotrol chart. We used cumulative time betwee failures data for software reliability moitorig usig Expoetial distributio. Table: Time betwee failures of a software Failure umber Time betwee failure(h Failure umber Time betwee failure(h bt ]

4 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 8, Issue 3, o., May 0 ISS (Olie: Assumig a acceptable probability of false alarm of 0.7%, the cotrol limits ca be obtaied as [5]: bt T U e bt 0. 5 T C e bt T L e a ad b are Maximum Liely hood Estimates (MLEs of parameters ad the values ca be computed usig iterative method for the give cumulative time betwee failures data show i table. Usig a ad b values we ca compute m (t. These limits are coverted to m ( t U, m ( t C ad m ( t L form. They are used to fid weather the software process is i cotrol or ot by placig the poits i Mea value chart show i Figure. A poit below the cotrol limit m ( t L idicates a alarmig sigal. A poit above the cotrol limit m ( t U idicates better quality. If the poits are fallig withi the cotrol limits it idicates the software process is i stable coditio [6]. The values of cotrol limits are as follows. m ( t U m ( t C m ( t L Table: successive differeces of cumulative mea values Cum Successive Cum Successive o m(t o m(t Failures differeces failures differeces Figure is obtaied by placig the differeces betwee cumulative failure data show i Table o y axis, failure umber o x axis ad the values of cotrol limits are placed o Mea Value chart. The Mea Value chart shows that the 0 th failure data has falle below m ( t L which idicates the failure process. It is sigificatly early detectio of failures usig Mea Value Chart. The software quality is determied by detectig failures at a early stage. The remaiig failure data are show i Figure are i stable. o failure data fall outside the m t. It does ot idicate ay alarm sigal. ( U

5 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 8, Issue 3, o., May 0 ISS (Olie: Mea Value Chart successive differeces UCL CL LCL Date/Time/Period Figure: Mea Value Chart 4. Coclusio The give 30 iter failure times are plotted through the estimated mea value fuctio agaist the failure serial order. The parameter estimatio is carried out by ewto Raphso Iterative method for Expoetial model. The graphs have show out of cotrol sigals i.e below the LCL. Hece we coclude that our method of estimatio ad the cotrol chart are givig a +ve recommedatio for their use i fidig out preferable cotrol process or desirable out of cotrol sigal. By observig the Mea value Cotrol chart we idetified that the failure situatio is detected at 0 th poit of Table- for the correspodig m ( t, which is below m t ( L. It idicates that the failure process is detected at a early stage compared with Xie et a, (00 cotrol chart [5], which detects the failure at 3 rd poit for the iter failure data above the UCL. Hece our proposed Mea Value Chart detects out of cotrol situatio at a earlier istat tha the situatio i time cotrol chart. The early detectio of software failure will improve the software Reliability. Whe the time betwee failures is less tha LCL, it is liely that there are assigable causes leadig to sigificat process deterioratio ad it should be ivestigated. O the other had, whe the time betwee failures has exceeded the UCL, there are probably reasos that have lead to sigificat improvemet. Refereces [] Hoag Pham, Hadboo Of Reliability Egieerig, Spriger: Apr 003, editio. [] Hua-Jyh Shyur A stochastic software reliability model with imperfect-debuggig ad chage-poit - The joural of systems ad software 66 ( [3] J.D. Musa., A. Iaio., K. Oumoto., 987. Software Reliability: Measuremet Predictio Applicatio. McGraw- Hill, ew Yor. [4] J. F. MacGregor ad T. Kourti Statistical process cotrol of multivariate processes. [5] M Xie, T. Goh, P.Raja Some effective cotrol chart procedures for reliability moitorig -Reliability egieerig ad System Safety [6] M. Kimura, S. Yamada, S. Osai. Statistical Software reliability predictio ad its applicability based o mea time betwee failures. [7] M.V.Koutras, S.Bersimis, P.E.Maravelais Statistical process cotrol usig shewart cotrol charts with supplemetary Rus rules Spriger Sciece + Busiess media 9:07-4, 007. [8] M.Ohba Software Reliability Aalysis Models IBM Joural Research Developmet Vol.9,o. 4, pp , July 984. [9] Swapa S. Gohale ad Kishore S.Trivedi, 998. Log- Logistic Software Reliability Growth Model. The 3rd IEEE Iteratioal Symposium o High-Assurace Systems Egieerig. IEEE Computer Society. Author s profile: First Author Dr. R. Satya Prasad received Ph.D. degree i Computer Sciece i the faculty of Egieerig i 007 from Acharya agarjua Uiversity, Adhra Pradesh. He received gold medal from Acharya agarjua Uiversity for his out stadig performace i Masters Degree. He is curretly worig as Associate Professor ad H.O.D, i the Departmet of Computer Sciece & Egieerig, Acharya agarjua Uiversity. His curret research is focused o Software Egieerig. He has published several papers i atioal & Iteratioal Jourals.

6 IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 8, Issue 3, o., May 0 ISS (Olie: Secod Author Mrs..Supriya Worig as Assistat professor i the departmet of Computer Sciece, Adiavi aaya uiversity, Rajahmeddry. She is at preset pursuig Ph.D at Acharya agarjua Uiversity. Her research iterests lies i Softwrare Egieerig ad Data Miig. Third Author Mr. G. Krisha Moha is worig as a Reader i the Departmet of Computer Sciece, P.B.Siddhartha College, Vijayawada. He obtaied his M.C.A degree from Acharya agarjua Uiversity i 000, M.Tech from JTU, Kaiada, M.Phil from Madurai Kamaraj Uiversity ad pursuig Ph.D at Acharya agarjua Uiversity. His research iterests lies i Data Miig ad Software Egieerig.