Keywords- Software Reliability, SRGMs, Non Homogeneous Poisson Process, Calendar time, Execution time. Fig 1. Types of Software Reliability Models

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1 Volume 4, Issue 4, April 2014 ISSN: X Iteratioal Joural of Advaced Research i Computer Sciece ad Software Egieerig Research Paper Available olie at: A Study of Various Reliability Growth s Soia Deswal 1 Reu Dalal 2 1 Departmet of Computer Sciece 2 Departmet of Computer Sciece PDM College of Egieerig, MD Uiversity AIACT&R, IP Uiversity Bahadurgarh, HR, Idia Geeta Coloy, Delhi , Idia Abstract -I literature, we have various software reliability growth models (SRGM) which have bee developed to facilitate the developers i moitorig the reliability of the software durig the software developmet. Software reliability models ca be used to predict the behaviour of software systems. SRGMs are geerally classified ito two groups based o the differet sets of assumptios ad eviromets- cotiuous time models ad discrete time models. I this paper, we aalyze both the discrete as well as the cotiuous time models. Differet types of discrete ad cotiuous time models are compared, their assumptios ad applicatios are studied. Keywords- Software Reliability, SRGMs, No Homogeeous Poisso Process, Caledar time, Executio time. I. INTRODUCTION Now a days, almost i every field, computers affect the people i oe way or the other. Computers are used i various ways for may applicatios lie that of air traffic cotrol, uclear reactors, aircraft, real-time sesor etwors, idustrial process cotrol, ad hospital health care affectig millios of people. Now as the computer system perform fuctios for almost all tass ad fuctios performed by computer systems are becomig essetial ad complicated ad as the size ad complexity of critical applicatios icreases, the eed to quatify ad predict the reliability of computer system i various complex eviromets arises. [1] Software Reliability is defied as the probability of failure free operatio of software i a specified eviromet for a specified period of time [2]. With the icreasig eed of software with zero defects, predictig reliability of software systems is gaiig more ad more importace. Various SRGM models are frequetly used i the literature to estimate the reliability of a software product. A umber of software reliability growth models have bee developed uder differet sets of assumptios ad eviromet.srgms ca be classified ito two categories. The first category comprises of the cotiuous time models. Cotiuous time models are those models which uses the executio time (i.e. CPU time) or caledar time. The secod category comprises of the discrete time models. Discrete time models are those models which uses the test cases as a uit of fault removal period. Such models are called discrete time models, sice the uit of software fault removal period is coutable. [3] Till Now, there are so may SRGMs have bee developed that exist i the first category while oly a few SRGM are developed that exist i the secod category due to the problems ad difficulties ivolved. Fig 1. Types of Software Reliability s Software Reliability models are used to estimate the reliability of a software. We ca classified the software reliability models i 3 parts. Oe is for cotious time models, secod is as discrete ad the third as others category i which the other models ca be grouped other tha the cotious ad discrete models. But till ow, more tha 60% of the SRGMs are cotious time while some are discrete models. 2014, IJARCSSE All Rights Reserved Page 1213

2 Sales 20% 20% 60% Fig 2. Classificatio of Software Reliability s II. CONTINUOUS TIME MODES 1) Goel- Oumoto [1] The Goel- Oumoto model is based o the followig assumptios:_ a) From the failure detectio poit of view, faults i a program are mutually idepedet. b) The probability is costat of the failures for faults actually detected. This meas that the curret umber of faults i a program is c) Before further testig, the isolated faults are removed. d) The software error which caused a failure is removed immediately ad ew errors are ot itroduced. This is show by the followig equatio:- mt () ba m() t t Where 'a' is the expected total umber of faults that exist i the software before testig. ad 'b' is the failure detectio rate or the failure itesity of a fault. The mea value fuctio of the above equatio be give as:- m( t) a 1e bt 2) Yamada Delayed S-Shaped [1] A stochastic model based o NHPP for a detected software error i which the growth curve is s-shaped for the observed failure data, of the umber of detected software errors. This model ca be characterized as a learig process of the testig team. The delayed S-shaped model is based o the followig assumptios:- a) From the failure detectio poit of view, faults i a program are mutually idepedet. b) The probability is costat of the failures for faults actually detected. c) The curret umber of faults i a program is d) The software's iitial error cotet is a variable. e) Errors preset i the software system leads the system fail at radom times. f) The time betwee (i-1) th ad i th failures depeds o the time to the (i-1) th failure. g) The software error which caused a failure is removed immediately ad ew errors are ot itroduced. This is show by the followig equatio:- bt () 2 bt bt 1 Where b is the error detectio rate per error i the steady-state. The mea value fuctio is give by m( t) a 1 1bt e bt 3) Iflectio S-Shaped model [1] This model is based o the depedecy of faults. The iflectio S-Shaped model is based o the followig assumptios:- a) Before the removal of some faults, some other faults are ot detected. b) The curret umber of faults i a software program is is 2014, IJARCSSE All Rights Reserved Page 1214

3 c) Each detectable fault failure rate is costat ad idetical. d) Removal of isolated faults etirely. Assume b bt () 1 bt e Where 'b' represets the failure-detectio rate ad β represets the iflectio factor. The mea value fuctio is give by:- a bt m( t) 1e 4) Musa s Basic Executio Time [4] The Musa-Basic model, also termed as the expoetial model, is give by the followig mea value:- E 0 m( t) 1 e 1 E t E E Where 0 : is the expected ad is the hazard rate or i other words the amout that each fault 1 cotributes to the overall failure rate. 5) Musa Oumoto logarithmic Poisso [4] I software cost estimatio models with high accuracy, this Musa Oumoto logarithmic model is used. This model is also ow as the logarithmic model. The mea value fuctio of the model is give by:- The required data to build this model are the oe from the time betwee failures ad the time of failure. III. DISCRETE TIME MODES 1) Discrete logistic curve models [8] A logistic curve model is a determiistic model which has bee applied to may SRGMs.It ca be described as d t = α (t)(-(t)) dt K Where α ad are costat parameters which ca be ow oly by regressio aalysis, ad (t) is the cumulative umber of software failures 1.1) Discrete logistic curve model with Morishita's equatio Morishita's states the product of the cumulative umber of faults detected up to discrete time (+1) ad umber of remaiig faults at discrete time i the software is umber of faults detected. Morishita s gives the followig equatio as a discrete equatio as: = δ α +1(- ) It has a exact solutio:- Where = total umber of software failures, m = costat of itegratio 1.2) Discrete logistic curve model with Hirota's equatio This model states that the umber of faults detected durig time differece is product of the cumulative umber of faults detected up to discrete time ad the umber of remaiig faults at discrete time +1 i the software. Hirota gives the followig equatio as a discrete equatio +1 - = δ α (- +1 ) This has a exact solutio 1 Where = total umber of software faults m = costat of itegratio 2) Discrete Gompertz Curve [8] This model is from S- shaped SRGMs class which gave good approximatios to a cumulative umber of software faults observed i testig software. This model gives the followig equatio dg t 1 e bt ( t) 1 t 0 1 1m1 t 1 1 m t G t = (logb) G(t) log dt where G(t) is cumulative umber of software faults detectig up to testig time, ad = iitial fault cotet 2014, IJARCSSE All Rights Reserved Page 1215

4 By itegratig the above equatio ad assume G(0) = a, G(t) ca be writte as G(t) = a b' where represets the iitial fault cotet A exact solutio of this equatio is G a 1 logb IV. DIFFERENT MODES AONG WITH THEIR ASSUMPTIONS AND APPICATIONS Name Proposed Year Assumptios 1) Goel -Oumoto 1979 a) From the failure detectio poit of view, faults i a program are mutually idepedet. b) The probability is costat of the failures for faults actually detected. This meas that the curret umber of faults i a program is c) Before further testig, the isolated faults are removed. d) The software error which caused a failure is removed immediately ad ew errors are ot itroduced. 2) Yamada Delayed S-Shaped 1984 a) From the failure detectio poit of view, faults i a program are mutually idepedet. b) The probability is costat of the failures for faults actually detected. c) The curret umber of faults i a program is d) The software's iitial error cotet is a variable. e) Errors preset i the software system leads the system fail at radom times. f) The time betwee (i-1) th ad i th failures depeds o the time to the (i-1) th failure. g) The software error which caused a failure is removed immediately ad ew errors are ot Mea Value Fuctio(m(t)/ Exact Solutio(G(t)) m( t) a 1e bt m( t) a 1 1bt e bt Applicatios This model ca be used to predict the Mea Test Cases to Failures(MTCTF) ad ca also be used to estimate the Remaiig Software Defect Estimatio (RSDE) o actual software failure data. This model is used i the situatios i which the observed growth curve of the cumulative umber of detectio error is s-shaped. Yamada delayed s- shaped model ca also be used to estimate the Remaiig Software Defect Estimatio (RSDE) o actual software failure data. 2014, IJARCSSE All Rights Reserved Page 1216

5 itroduced. 3) Iflectio S- Shaped model 4) Musa s basic executio time 5) Musa- Oumoto logarithmic poisso model 1984 a) Before the removal of some faults, some other faults are ot detected. b) The curret umber of faults i a software program is proportioal to the is c) Each detectable fault failure rate is costat ad idetical. d) Removal of isolated faults etirely a) From the failure detectio poit of view, faults i a program are mutually idepedet. b) The probability is costat of the failures for faults actually detected. This meas that the curret umber of faults i a program is c) Before further testig, the isolated faults are removed. d) The software error which caused a failure is removed immediately ad ew errors are ot itroduced. The executio time, i.e., the actual processig time used i executig the program is the best time domai for expressig reliability. a m( t) 1e bt 1 e This model is best i situatio where we eed to fid out the expected umber of remaiig errors at a specified time. E 1 E t 0 e This model is used i m( t) 1 bt applicatios where the cost estimatio is required before the release of the software. m( t) 0 1 1t This model is used especially for the executio time data but it ca also be applied to caledar time data by applyig a coversio from caledar to executio time 6) Yamada- Osai Expoetial Growth 1985 The Failure itesity of faults withi differet modules are assumed to be differet while the failure itesity of faults withi the same module are assumed to be the same. Assume that the expected umbers of faults detected for each module are expoetial. bt i m( t) a p i 1e 2014, IJARCSSE All Rights Reserved Page 1217 i1 We may use Yamada- Osai Expoetial Growth where the failure itesity of faults withi differet modules are assumed to be differet while the failure itesity of faults withi the same module are assumed to be the same ad the expected umber of faults detected are expoetial.

6 7) Pham- Nordma- Zhag(PNZ) 1999 a) The itroductio rate is a liear fuctio timedepedet overall fault cotet fuctio. b) The fault detectio rate fuctio is o-decreasig time-depedet with a iflectio S-shaped model. a bt m( t) 1 e 1 t bt 1 e The PNZ model icorporates the imperfect debuggig pheomeo by assumig that faults ca be itroduced durig the debuggig phase at a costat rate of fault per detected fault 8) Pham Expoetial Imperfect Debuggig 2000 a) The itroductio rate is a expoetial fuctio of testig time. b) The error detectio rate fuctio is o-decreasig with a iflectio S- shaped model. bt b e 1 mt () bt b e c This model ca be used where the error detectio rate is a expoetial fuctio of testig time. 9) Discrete ogistic Curve s 9.1) Discrete ogistic Curve with Morishita s equatio 9.2) Discrete logistic curve model with Hirota's equatio Morishita's states the product of the cumulative umber of faults detected up to discrete time (+1) ad umber of remaiig faults at discrete time i the software is umber of faults detected. The umber of faults detected durig time differece is proportioal to the product of the cumulative umber of faults detected up to discrete time ad the umber of remaiig faults at discrete time +1 i the software. 1m 1 t t 1 1 m 1 This model is determiistic, eve though it ca be applied to may softwarereliability growth processes i software testig. This model is oe of the best ad simplest models for estimatig a S-shaped software reliability growth curve model. 10) Discrete Gompertz Curve Gave good approximatios to a cumulative umber of software faults observed i testig software. G a 1 logb Gompertz is from oe of the S- shaped SRGM models. This model give good approximatios to a cumulative umber of software faults observed i testig software. Table 1. Compariso of Differet Software Reliability s Based o Their Assumptios ad Applicatios. V. CONCUSION Reliability models are a powerful tool for predictig ad assessig software reliability. I studyig hardware ad software reliability problems, NHPP models have bee successfully used the SRGMs are especially used to describe processes of failure which possess certai treds, such as reliability growth, ad hece maig the applicatio of NHPP models to software reliability aalysis easily implemeted.i this paper, we first categorize the various SRGMs ito cotious time models ad discrete time models ad the studied the various SRGMs uder cotious time models ad discrete time models. A release of a software product is always be a trade-off betwee early release ad the product release deferral to ehace fuctioality. We coclude that the SRGMs ca help the software desiger to decide whe the software system is ready for release, if the reliability of the software has reached a give threshold. 2014, IJARCSSE All Rights Reserved Page 1218

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