Testing Domain Dependent Software Reliability Growth Models

Transcription

1 Inernaional Journal of Mahemaical, Engineering and Managemen Sciences Tesing Domain Dependen Sofware Reliabiliy Growh Models Deepika 1, Ompal Singh 2, Adarsh Anand * Deparmen of Operaional Research Universiy of Delhi, Delhi , India 1 deepika.sre@gmail.com, 2 drompalsingh1@gmail.com * Corresponding auhor: adarsh.anand86@gmail.com Jyoish N. P. Singh Ramjas College Universiy of Delhi, Delhi , India jyoishdu@gmail.com (Received Ocober 31, 2016; Acceped January 1, 2017) Absrac Sofware Reliabiliy Growh Models (SRGMs) are supporing sofware indusries in expecing and scruinizing qualiy of sofware. Numerous SRGMs have been proposed; majoriy of which concenrae on esing period of sofware. For esing, domain specific knowledge plays a very crucial role. Based on necessiy condiion, a se of programmes are in esing phase of sofware developmen. Domain esing is a sofware echnique in which small number of es cases is seleced for rial. These ses of esing pahs, all of which are o be evenually influenced by designed es cases are called he esing domain which expands wih he progress of esing. Keeping his concep in mind, we propose SRGMs wih he concep of esing domain wih exponenial coverage. Uiliy of proposed framework has been emphasized in his paper hrough some models peraining o differen disribuion i.e Exponenial, Logisic, Weibull and Rayleigh. Moreover, he daa analysis is performed o find he esimaes of parameers by fiing he models on auhenic daa ses. Keywords: SRGMs, Deecion rae of fauls, Disribuion funcion, Densiy funcion, Tesing domain. 1. Inroducion Sofware is soliary essenial medium which is moivaing almos all elecronics and indusrial suffs. In various fields, enormous sofware sysems have been developed as compuer sysems have been uilized a he same ime (Rafi e al., 2012). Mos significan sage of Sofware Developmen Life Cycle (SDLC) is esing. One imporan ingredien for sofware is reliabiliy where i is defined by qualiy (Fujiwara and Yamada, 2001). As he esing goes on, bugs are observed and disconneced from he sofware o make he qualiy enhancemen. In order o perform sofware reliabiliy assessmen quaniaively, SRGM (Musa e al., 1987; Pham, 2000; Yamada, 1994) is well known as an effecive ool. Mahemaical models are consruced which are useful in describing he real ime esing environmen. Pleny of models are proposed o measure he sofware failure process successfully (Kapur e al., 2011). Some of hem based on Non-Homogenous Poisson Process (NHPP) models are proposed o predic he fuure failures (Anand e al., 2016). In he esing phase of sofware developmen, here is a se of modules and funcions in a sofware sysem o be influenced by execued cases (Ohera e al., 1990). The es cases execued on he sofware in he esing phase are developed o influence he fauls lying dorman in various modules/funcions implemened in he sofware based on he requiremen specificaions. These es cases indeed influence a se of esing pahs of hese modules and funcions. This se of esing pahs, all of which are o be evenually influenced by designed es cases is called he esing domain (Kapur e al., 2011). The domain of esing which ges influenced by he es cases execued by any ime in he esing phase is called he isolaed esing domain by ha ime. An 140

2 Inernaional Journal of Mahemaical, Engineering and Managemen Sciences imperaive role of esing domain is in reliabiliy consideraion and judgmen relaed o he esing and operaional phase (Yamada and Takahashi, 1993). The Measuremen of sofware reliabiliy wih respec o he isolaed esing domain is a region in SRGMs, which has no been grealy reconnoired by he researchers. There are hree ypes reliabiliy examinaion of esing domain funcion sudied in he prior research i.e. he basic esing domain, esing domain wih skill facor and esing domain wih imperfec debugging (Yamada and Fujiwara, 2001); and hen hese ypes of domain were very much conneced o behaviour of faul recogniion mehod of sofware esing (Kapur e al., 2011). In his manuscrip, SRGMs relaed o exponenial growh curve have been discussed wih esing domain. The quaniy of bugs diverges wih ime and esing domain coninues o srech in sofware during he developmen. Furher, we have aken faul deecion rae dependen on ime using four ypes of disribuions i.e. Exponenial, Logisic, Weibull and Rayleigh. Respie of he manuscrip is prearranged as follow: Segmen 2 comprises noaions relaed o SRGMs. In segmen 3, modeling framework has been discussed. The parameers have been endorsed on facual daa ses; consequenly, obained resuls and conclusion is accumulaed in segmen 4 and 5 respecively. 2. Noaions z() Toal number of bugs exising in he isolaed esing domain a he esing ime Tesing domain growh rae Toal faul conen () Toal faul conen dependen on ime Consan parameer () Mean value funcion () Faul deecion rae dependen on ime F () Cumulaive disribuion funcion for faul removal/ correcion imes Probabiliy densiy funcion for faul removal/ correcion imes Consan faul deecion rae Learning parameer 3. Modeling Framework 3.1 Assumpions These are following posulaes which provide illusraive descripion of he proposed SRGMs (Kapur e al., 2011). In he esing domain, he dorman fauls are disseminaed evenly. The debugging process is perfec. The escalaing rae of number of deecable fauls is sraigh comparaive o he number of fauls remaining in he sofware ouside of he isolaed domain a randomly ime. Wih above assumpion differenial equaion for esing domain can be wrien as dz() ( z ( )) (1) d 141

3 Inernaional Journal of Mahemaical, Engineering and Managemen Sciences wih he iniial condiion z(0) 0, we simplify Eq n (1) hen following expression for basic esing domain funcion is z( ) (1 e ) (2) where z () represen basic esing domain wih he exponenial growh curve and in equaion (2), he quaniy (1 e ) signifies he isolaed esing domain raio in he sofware sysem a ime. () represening he rae of he isolaed esing domain growh in he sofware sysem is given as d ( ) z( ) e (3) d 3.2 Dynamic Tesing Domain: Need and Imporance The definiion of basic esing domain has been consruced under he posulaion of perfec debugging. Inroducion of new fauls during he debugging progression is recurrenly knowledgeable. In he early sage of esing, correcion of deeced fauls is simple and he influenced region of modules and funcions by he faul correcion is very conical (Yamada and Fujiwara, 2001). Wih he evoluion of esing, he exclusion of deeced bugs becomes convolued and prejudiced area exends broadly. This period requires much cauious and capable debugging deed for he recificaion of addiional complicaed fauls as compared o he early sage (Kapur e al., 2011). While developing he basic isolaed esing domain funcion, we assume ha fauls are consan over SDLC. However, in pracical scenario i is also possible ha he number of fauls diverge wih ime and he esing domain coninues o increase in sofware during he developmen (Kapur e al., 2011). In his circumsances of equaion (1) is replaced by (). dz() ( ( ) z( )) (4) d An exponenial form of faul conen is used o incarcerae he slow preface rae in he early phase and higher in he laer phase i.e. The following funcional form of () is used (Kapur e al., 2011) () e, 0 (5) using equaion (5) in equaion (4), we see dz() ( e z( )) (6) d wih iniial condiion z(0) 0, we ge 142

4 Inernaional Journal of Mahemaical, Engineering and Managemen Sciences e z( ) 1 e ( ) ( ) (7) clearly basic esing domain is a special case of dynamic esing domain for 0. The growh rae (From equaion (7) shows as d ( ) z( ) e e d. 3.3 Tesing Domain Dependen Model Developmen These are following assumpions for SRGMs (Fujiwara and Yamada, 2001): The observed fauls are considered o survive in he isolaed domain. The proporion of isolaed esing domain progresses wih he ime. Rae of faul deecion is proporional o he number of bugs remaining domain ime. Based on above assumpion, he differenial equaion wih respec o () and esing domain funcion is d( ) f ( ) z( ) ( ) d 1 F( ) where is he hazard rae funcion. 1 F ( ) if () hen equaion (8), 1 F ( ) (8) d () ( ) z( ) ( ) (9) d In above equaion (8), F () follows differen kind of disribuion: SRGM-1 F () Exponenial disribuion i.e. 1 f ( ) Thus p( ) p consan 1 F( ) e using above calculaion, we can wrie equaion (8) as d () z( ) ( ) (10) d Now from he equaion (7) value of z () pu in equaion (10), we ge 143

5 Inernaional Journal of Mahemaical, Engineering and Managemen Sciences d() e d ( ) ( ) (1 e ) ( ) (11) using he iniial condiion (0) 0, using esing domain funcion, is 1 ( ) 1 ( ) e ( ) e ( )( ) ( ) ( ) ( ) (12) equaion (12) shows expeced number of fauls using esing domain funcion wih exponenial form and consan rae (using hazard rae echnique). SRGM-2 (1 e ) F () Logisic disribuion i.e. (1 e ) () Logisic rae 1 F( ) 1e using above logisic rae, equaion (8), d() z( ) ( ) d (1 e ) (13) from he equaion (7) and equaion (13), we ge d() e d (1 e ) ( ) ( ) (1 e ) ( ) (14) Under iniial condiion (0) 0, e 1 ( ) 1 ( ) e ( ) e ( )( )(1 e ) ( ) ( ) ( ) (15) Above equaion (15) inegraes learning phenomenon in esing domain. SRGM-3 b F () Weibull disribuion i.e. (1 e ) ; 0 where is shape parameer (or slope). 1 () 1 F ( ) using above rae, equaion (8) can be wrien as 144

6 Inernaional Journal of Mahemaical, Engineering and Managemen Sciences d() 1 z( ) ( ) (16) d by equaion (7), value of z () pu in equaion (16) d() 1 e ( ) (1 e ) ( ) d ( ) Solving equaion (17) using he seed value (0) 0, we ge e e () ( ) (1 ) (1 ) (17) (18) above equaion shows mean value funcion using esing domain funcion wih Weibull Faul deecion rae. SRGM-4 2 e F () Rayleigh disribuion i.e. 2 (1 ) () 1 F ( ) above discussed Rayleigh rae pu in equaion (8) d () z( ) ( ) (19) d puing he value of z () in equaion (19) d() e ( ) (1 e ) ( ) d ( ) (20) using iniial condiion (0) 0, below equaion (21) for Rayleigh rae e 1 e 1 b 2 e () ( ) ( ) ( ) ( ) ( ) (21) 145

7 Inernaional Journal of Mahemaical, Engineering and Managemen Sciences Following differen disribuions are used in all SRGMs (Anand e al., 2014): a) Exponenial Disribuion: This disribuion has a consan rae and i is broadly used in modeling of sofware reliabiliy. I designaes he uniform disribuion of fauls. b) Logisic Disribuion: I has an S-shaped represenaion ha is widely used in reliabiliy. I looks like he normal disribuion in conour. c) Weibull Disribuion: This disribuion is much used in reliabiliy engineering. I is a versaile disribuion in ha i can ake on he characerisics of oher ype of disribuions, based on he value of shape parameers. We can say ha Weibull disribuion is a generalizaion of he exponenial disribuion due o is flexible environmen. d) Rayleigh Disribuion: I is he disribuion of he magniude of a wo-dimensional random vecor whose coordinaes are independen, idenically disribued. 4. Daa Analysis, Validaion and Comparison Crieria To illusrae he esimaion, we have aken ou he daa analysis of auhenic sofware daa ses. The parameers have been evaluaed using Saisics Analyical Sofware (SAS). Daa se:1 (DS-1) has been compiled for 19 weeks in which 42 bugs were observed (Wood, 1996). In second daa se (DS-2), he sofware size was abou 300KB and i was wrien in assembly language (Kanoun e al., 1991; Anand e al., 2017). In he enire execuion period, a oal of 461 fauls have been removed in 81 weeks. SRGMs are measured hrough he capabiliy of fiing he previous failure daa and expecing he coming performance of fauls (as shown in Fig. 1 and 2). Esimaion of parameers and judgmen crieria resuls for DS-1and DS-2 of all models under deliberaion can be sigh hrough Table 1, Table 2, Table 3 and Table 4. I is clear from he ables ha he value of R 2 for SRGM-1 is higher and value of SSE, MSE and Roo MSE is lower in comparison wih oher models and provides beer goodness of fi for DS-1. Similarly we can see ha he value of R 2 for SRGM-3 is higher and value of SSE, MSE and Roo MSE is lower in comparison wih oher models and provides beer goodness of fi for DS-2. Models SRGM SRGM SRGM SRGM Table 1. Esimaed resuls for daase 1 Models SSE MSE Roo MSE R 2 Adj. R 2 SRGM SRGM SRGM SRGM Table 2. Comparison resuls for daa se 1 146

8 Cumulaive No. of Fauls Inernaional Journal of Mahemaical, Engineering and Managemen Sciences Models SRGM SRGM SRGM SRGM Table 3. Esimaed resuls for daa se 2 Models SSE MSE Roo MSE R 2 Adj. R 2 SRGM SRGM SRGM SRGM Table 4. Comparison resuls for daa se Acual SRGM 1 SRGM 2 SRGM 3 SRGM 4 Time Fig. 1. Goodness of fi curve for daa se 1 147

9 Cumulaive No. of Fauls Inernaional Journal of Mahemaical, Engineering and Managemen Sciences Acual SRGM 1 SRGM 2 SRGM 3 SRGM 4 Time Fig. 2. Goodness of fi curve for daa se 2 5. Conclusion In his aricle, SRGMs based on he concep of esing domain wih exponenial coverage have been discussed. In he mahemaical formulaion, faul deecion rae dependen on ime has been aken. Sandard disribuions have used in his paper e.g. Exponenial, Logisic, Weibull, Rayleigh for faul removal phenomenon. These SRGMs have been auhenicaed and validaed on wo sofware failures daa ses. As can be viewed hrough Table 1 and Table 3, he obained resuls are quie encouraging. For comparison analysis, we have differen kind of comparison crieria as can be seen in Table 2 and Table 4. In fuure imes he same concep can be applied o muli-release framework for sofware reliabiliy growh modelling. References Anand, A., Bha, N., Aggrawal, D., & Papic, L. (2017). Sofware reliabiliy modelling wih impac of bea esing on release decision. In Advances in Reliabiliy and Sysem Engineering (pp ). Springer Inernaional Publishing. Anand, A., Deepika, Singh, N. & Du, P. (2016). Sofware reliabiliy growh modeling based on in house esing and field esing. Communicaion in Dependabiliy and Qualiy Managemen: An Inernaional Journal, 19(1), Anand, A., Kapur, P. K., Agarwal, M., & Aggrawal, D. (2014, Ocober). Generalized innovaion diffusion modeling & weighed crieria based ranking. In Reliabiliy, Infocom Technologies and Opimizaion (ICRITO) (Trends and Fuure Direcions), rd Inernaional Conference on (pp. 1-6). IEEE. Fujiwara, T., & Yamada, S. (2001). Sofware reliabiliy growh modeling based on esing skill characerisics: Model and applicaion. Elecronics and Communicaions in Japan (Par III: Fundamenal Elecronic Science), 84(6),

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