Software Reliability Estimation Based on Cubic Splines

Transcription

1 Software Relablty Estmaton Based on Cubc Splnes P.L.M. Kelan Bandara, G.N. Wramanayae and J.S.Goonethllae Abstract Software relablty s one of the most mportant software qualty attrbute and Software relablty estmaton s a hard problem to solve accurately. However for management of software qualty and standard practce of the organzaton, accurate relablty estmaton s mportant. Non-homogeneous Posson Process (NHPP) models and Artfcal Neural Networ (ANN) models are among the most mportant software relablty growth models. In ths paper we study an approach usng past fault-related data wth cubc splne Networ model to estmate relablty. A numercal example s shown wth smulated datasets. The example shows that the proposed model accurately estmate the software relablty. Index Terms Software Relablty, Cubc Splne Smoothng, Artfcal Neural Networ, Software Relablty Growth Model. I. INTRODUCTION Software engneerng s a well establshed dscplne focused on set of management and desgn actvtes of software development. The ey ssues of software engneerng are the management of tme, cost and qualty le n other engneerng dscplnes. Software qualty engneerng comes nto play promnently when the software systems grow and the mpact of the software systems affect almost all n the socety. Today a rapdly ncreasng dependency exsts on the software systems. Usage of software systems can be seen n varous actvtes rangng even to lfe crtcal systems and crtcal economc functons. One of the software qualty ssues s accuracy whch s affected by errors n the software. People are used to stre off the software systems due to software qualty problems. However, when the software usage becomes mature and when the system becomes safety crtcal, such an atttude wll not be accepted. Therefore software qualty engneers are responsble to provde measurements of the qualty attrbute of the software system. A. Software Qualty The term software qualty has several meanngs and the scope s also broadened. ISO926 defnes the software qualty as: the totalty of features and characterstcs of a Manuscrpt receved March 0, Software Relablty Estmaton Based on Cubc Splnes. P. L. M. Kelan Bandara s wth Vocatonal Tranng Authorty of Sr Lana, Colombo 00500, Sr Lana and s a postgraduate student at Unversty of Colombo School of Computng (e-mal: manohar@gmal.com). G. N. Wramanayae s wth Unversty of Colombo School of Computng, Colombo 00700, Sr Lana (e-mal: gnw@ucsc.cmb.ac.l). J. S. Goonethllae s wth Unversty of Colombo School of Computng, Colombo 00700, Sr Lana (e-mal: jsg@ucsc.cmb.ac.l). software product that bear on ts ablty to satsfy stated or mpled needs []. Software qualty s descrbed n the means of models whch are called software qualty models and these have ther own qualty attrbutes. ISO926 defnes software qualty wth sx software qualty attrbutes as functonalty, relablty, usablty, effectveness, mantanablty and portablty. Another famous and useful categorzaton of factors that affect the software qualty was proposed by McCall, Rchards, and Walters [2]. Accordng to that categorzaton, qualty factors are categorzed nto three categores as product transton, product operaton, and product revson. Software relablty s one of fve product operaton qualty attrbutes. Accordng to the fact that the software relablty s a qualty attrbute n most of the qualty models, t can be concluded that hgh qualty of the software s dependent on the software relablty too. Hence f a company s to develop hgh qualty software, t s mportant to employee the efforts on software relablty. In spte of ths, lterature state that the relablty has not been made use of wth regard to the qualty actvtes n the commercal software development [], [4]. The followng sectons descrbe the term software relablty and why the ndustry doesn t pay much attenton on assurance of software relablty for ther software products. In secton III, a model whch overcomes those problems has been proposed and fnally results of our model are presented. B. Software Relablty IEEE defnes software relablty as the ablty of a program to perform requred functons under stated condtons for a stated perod of tme []. Hence falures of the software reduce the relablty and to ensure the qualty of the software, the measurements of the software operatonal falures are mportant. The followng secton descrbes how the relablty of the software s measured. Software relablty s measured durng the software development and durng the operatons usng a software relablty model. There are two types of software relablty models, accordng to the phase whch the relablty s measured. ( Software relablty growth model (SRGM) or software relablty estmaton model estmates the software relablty based on the observed falure data durng the testng and operaton phase. (2) Software relablty predcton model predcts the software relablty based on the relablty matrces measured or calculated durng early stages of software development lfe cycle (pror to the ntegrated testng n the testng phase). Ths paper focuses on the frst type, the software relablty estmaton model.

2 II. EXISTING SOFTWARE RELIABILITY GROWTH MODELS The frst None-homogeneous Poson Process (NHPP) model, whch strongly nfluences the development of many other models, was proposed by Goel and Oumoto [4]. Huang et al. [5] have dscussed a unfed scheme of dscrete NHPP models by applyng the concepts of weghted arthmetc, weghted geometrc or weghted harmonc means. Ohba [6] presented a NHPP model wth S-shaped mean value functon. Lots of the generalzed SRGMs, ncludng the generalzed SRGMs mentoned above, have been dscussed n terms of contnuous-tme SRGMs, because the contnuous-tme SRGM s specfcally applcable to the relablty analyss [7]. S. Inoue et al. gve a Generalzed Dscrete Software Relablty Modelng based on program sze [7]. Oamura et al. [8] have dscussed a unfed parameter estmaton method based on the expectaton maxmzaton (EM) prncple and nvestgated the effectveness of the estmaton method based on the EM algorthm by comparng wth Newton s method [9]. Khoshgoftarr et al. [0] ntroduced the use of the neural networs as a tool for predctng software qualty. Ther model used doman metrcs derved from the complexty metrc data. Papers []-[6] have also adapted neural networs to software relablty ssues. Emanm and Melo [7] have performed to construct a logstc regresson model to predct whch classes n a future release of a commercal Java applcaton wll be faulty. A. Lmtatons of SRGMs Even though numerous models have been dscussed n lterature, none s worng fne n all the crcumstances [8]-[9]. The most promnent lmtatons of the models are as the software behavor changes because the software code changes durng the testng phase and hence, assumpton of estmated mean tme of SRGMs s volated by the dataset and more numbers of falure data are requred to estmate the relablty [4], [8], [20]. Due to these lmtatons n SRGMs the estmaton accuracy s mpared. Although ths affects dramatcally the relablty no attenton has been gven to t by the ndustry [8]. So t s necessary for a SRGM to overcome the above lmtatons n order to mprove the relablty technques used n the ndustry. B. The Accuracy of Software Relablty Estmaton Splnes are used to nterpolate a dataset to pecewse arbtrary functons whch consst of thrd order polynomals [2], [22]. Cubc splne nterpolaton doesn t assume any statstcal dstrbuton whch as shown n the secton II, s one of the most promnent lmtatons n most exstng models. The software relablty s an arbtrary event and our model employee arbtrarness n several places. Pattern recognton usng artfcal neural networ [2] maes use of randomness n ts archtecture when dentfyng an unnown pattern. That s when a new pattern s to be recognzed; the artfcal neural networ s traned usng nown patterns before dentfyng the new pattern. As dscussed n the Secton II A, the past falure dataset s usually small to mae the estmaton whch s another lmtaton and that we have overcome here. To mae use of our model only 9 recent falure data s needed. Ths solves the problem of past falure data do not show the future behavor problem dscussed n Secton II A. III. SRGM WITH CUBIC SPLINES Ths paper gves the Artfcal Neural Networ (ANN) based model to capture the nput output (I/O) relatonshps of software systems to correspondng falures and to mprove the accuracy of relablty predcton. Ths networ captures nput and output through an evolutonary algorthm. For the nput vector of X = [ xn 5, xn 4,..., xn ] (where n 9 ), the correspondng mappng of cubc splne networ can be wrtten as x ˆn = g( xn 5, xn 4,... xn ). Our model for software relablty predcton s desgned as a three-layer structure wth an nput layer, cubc splne layer, and an output layer. Each layer has fxed nodes such as Input layer has 5 nodes correspondng to each nput and cubc splne layer has nodes each correspondng to the boundary condtons. The nput data vector X s connected to the nput th nodes of the networ to predct the tme to n falure. We can derve the actvaton functons usng cubc splnes. Gven a functon f = S whch passes through the x n 5, xn 4,..., xn nodes, can be represented n splnes S defned on [ x, x ] where n 5 n. n = = S S n 5 A cubc splne nterpolant S, for f s a functon that satsfes the followng sx condtons,. The splne forms a contnuous functon.e... S + ( x + ) = S ( x+ ) for each = n 5, The splne forms a smooth functon.e. S ( ) + x+ = S ( x+ for each = n 5, The second dervatve s contnuous.e. S ( + x+ = S ( x+ for each = n 5, v. S s a cubc polynomal, denoted S on subnterval [ x, x +] for each = n 5, S ( x ) = f ( x for each = n 5, n 4,..., n v. The splne passes through each node ) v. One of followng the boundary condtons s satsfed a. S ( x 5 ) n = S ( xn = 0 b. Set boundary dervatves for specfed values. S ( x = n 5 ) S ( xn = α c. Set boundary dervatves for specfed values S ( x = n 5 ) S ( xn = β α, β are any real numbers,.e. < α, β < + The exstng dataset s consdered as ntervals (.e. th nterval s x X x ). In our relablty estmaton model, the cubc splnes are used for future predcton and hence the boundary condtons are mportant. Ths networ captures networ nodes n the cubc splne layer; each s correspondng to the each boundary condton above. Varyng the boundary condtons the results can be enhanced.

3 x n 5 0, S ω x n 4 α, S ω 2 xˆn ω x n β, S Input Layer Cubc Splne Layer Output Layer Fg. : cubc splne networs 2 S ( X ) = a + b ( X x ) + c ( X x ). + d ( X x ) for each = n 5, n 4,..., n and a, b, c and d are constants. Let h = x + x for all = n 5, The equatons are smplfed n fndng the coeffcents as follows. S ( x ) = 2 * c ( f ( x ) = a ( 2) h c + 2( h + h ) + h c + = ( / h )( a a ) ( ) c = c + * d * h ( 4) + b = ( / h )( a + a ) ( h / )(2c + c+ ) ( 5) The actvaton functon of the th cubc splne node for estmatng the n th falure s S ( x n ) = a + b ( x n xn 2 ) + 2 c ( x n x n 2 ) + d ( x n xn 2 ) =,2, for When =, the boundary condton S ( x n 5 ) = S ( x n ) = 0 s appled. When = 2, the boundary condton S ( x n 5 ) = S ( xn ) = α s appled When =, the boundary condton S = S ( x ) = β s appled ( x n 5 ) n The weght ω that connects the th weghtng node and the output node are ndcated by the weghtng vectors ω = [ ω, ω2, ω]. These parameters are determned by global bac propagaton algorthm [22]. The fnal output of the cubc splne networs summng layer s: y = S ( X n + * = ) ω The relablty estmaton of the n th falure s y IV. RESULTS To llustrate the proposed approach wth cubc splne networ model, numercal examples are studed n ths secton. Table I: Tme to falure data for x th falure taen from [8]. FNo Tme to Falure FNo Tme to Falure The example data of Table I s based on a project for a large Telecommuncaton software system [8] wth FNo ndcatng the Falure Number for x= Usng ( (5) and the boundary condtons n secton II, the estmaton of y has been done. Table II shows the estmated relablty values for our model. Valdty of the estmaton was checed usng estmated and actual tme to falure values for dataset (for x=9, 0 40). Here x commence from 9 as past 8 data s needed to estmate the parameters of our model.

4 Table II: Actual and Estmated tmes for each FNo, usng our model FNo Actual Tme Estmated Tme FNo Actual Tme Estmated Tme Ac CSE Fg. 2(a): Actual tme (Ac) and estmated tme usng our model (CSE- Cubc Splne model Estmaton) to falures Ac Ge JM LL MuB MuL Nh LG CSE Fg. 2(b): Comparson of estmaton models wth ours Fg. 2(a) compares the estmated values (Ac-Actual) usng our model (CSE- Cubc Splne model Estmaton) wth actual tme to falure data for the dataset n Table II. Fg. 2(b) compares estmaton ablty of software relablty usng famous models (Ac- Actual; Ge.-Geometrc Model; JM -Jelns/Morada Model; LL - Lttlewood Lnear Model; MuB - Musa's Basc Model; MuL - Musa's Log Model; NHPP None-homogeneous Poson Process Model; LG - Lttlewood Geometrc Model; CSE - Cubc Splnes Model). Accordng to the comparson, t can clearly be seen that our model and None-homogenous Posson Process software relablty estmaton model are more applcable. The dfference between actual and estmaton values s lower usng our model than usng None-homogenous Posson Process model. Our model s more sutable to estmate the relablty for a dataset whch doesnot have sudden devatons n the pattern (.e. no outlers among the data values). V. CONCLUSIONS AND FUTURE WORK In ths paper, we have studed an approach of relablty predcton by Cubc splne networs. Snce cubc splne nterpolaton assumes a random dstrbuton for the data, our model tae nto account the randomness. The analyss wth example shows that the proposed approach wors effectvely. Fg. 2 ndcates that our model estmaton s more accurate Cubc splnes nterpolate a polynomal. The estmaton usng cubc splnes can have errors upon the approxmaton. Snce the estmaton goes through weghtng functon n the output layer, the error component s mnmzed. Snce ths method requres less number of nput data, ths can be used n the early predcton accurately. Changng the falure behavor and the code of the software, the old data may not be vald when accountng them n relablty estmaton. Recent data are only consdered n our model and hence, our model mproves accuracy of software relablty predcton. However, when the dataset has outler data such as huge ncrement or decrement of relablty, the estmaton made usng our model s less accurate. It s an dentfed lmtaton of our model. Ths model contans more calculatons and hence the practce of the model s trcy unless t s automated. REFERENCES [] Scalet et al, 2000: ISO/IEC 926 and 4598 ntegraton aspects: A Brazlan vewpont. The Second World Congress on Software Qualty, Yoohama, Japan, [2] J.A. McCall, P.K. Rchards, and G.F. Walters, Software Qualty Assurance, 998. [] A. Fres and A. Sen, A survey of dscrete Relablty-growth models., IEEE Trans. Rel., 45(4), 996, pp [4] A. L. Goel and K. Oumoto, "Tme dependent error detecton rate Model for Software Relablty and Other Performance Measures, IEEE Trans. on Rel., 28(), 979, pp [5] C.Y. Huang, M. R. Lyu and S. Y. Kuo, A unfed Scheme of some Non-homogeneous Posson Process models for software relablty Estmaton, IEEE Trans. Soft. Eng., 29(), 200, pp [6] S. Inoue and S. Yamada Generalzed Dscrete Software Relablty Modelng Wth Effect of Program Sze. IEEE Trans. on Sys., Man, and Cybernetcs Part A: Systems and Humans, 7(2), 2007, pp [7] P.M. Khoshgoftaar, E.B. Allen, J.P. Hudepohl and S.J. Aud, Applcaton of Neural networs to software qualty modelng of a very large telecommuncatons system, IEEE Trans. on Neural Networ, vol. 8, 997, pp [8] H. Oamura, A. Murayama, and T. Doh, EM Algorthm for dscrete software relablty Models: A unfed parameter estmaton Method, n Proc. 8 th IEEE Int. Symp. HASE, 2004, pp [9] P. Deuflhard, Newton Methods for Nonlnear Problems. Affne Invarance and Adaptve Algorthms. Sprnger Seres n Computatonal Mathematcs, Vol. 5. Sprnger, Berln, 2004.

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