Discrete and Continuous Time High-Order Markov Models for Software Reliability Assessment

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1 Dscrete and Contnuous Tme Hgh-Order Markov Models for Software Relablty Assessment Vtaly Yakovyna and Oksana Nytrebych Software Department, Lvv Polytechnc Natonal Unversty, Lvv, Ukrane Abstract. Due to the crtcal challenges and complexty of modern software systems developed over the last decade, there has arsen an ever ncreasng attenton to look for products wth hgh relablty at reasonable costs. Software development process moves toward component-based desgn, and archtecture based approach n software relablty modelng s wdely used. However, n lots of models for software relablty assessment the assumpton of ndependent software runs s a smplfcaton of real software behavour. Ths paper descrbes two software relablty models that use hgh-order Markov chans thus takng nto account dependences among software component runs for more accurate software relablty assessment. The effcency and accuracy of developed models s nvestgated by the example of several software products. It s shown that usng the software relablty models based on the hgh-order Markov chans results n the software relablty assessment accuracy up to 10 20%. Keywords. Software relablty, archtecture-based model, hgh-order Markov chans. Key Terms. SoftwareComponent, SoftwareSystem, MathematcalModel. 1 Introducton Computer systems are wdely used n modern ndustry for control and automaton purposes. All of these systems are controlled by software. Thus, software s used n ar traffc control, nuclear power plants, automated patents montorng etc. Therefore hgh requrements for software relablty are demanded because falures of such systems can lead not only to sgnfcant fnancal losses, but also threaten human lfe and health. Although technques that allow assessng and ensurng the specfed hardware relablty requrements have been developed, there are no common approaches for software systems assessment. Accordng to STD-729, software relablty s defned as the probablty of falurefree software operaton for a specfed perod of tme n a specfed envronment [1, 2]. Although Software Relablty s defned as a probablstc functon, and comes wth the noton of tme, t should be noted that, contrast to tradtonal hardware rela-

2 blty, software relablty s not a drect functon of tme [1]. Electronc and mechancal parts may become "old" and wear out wth tme and usage, but software wll not rust or wear-out durng ts lfe cycle. Software wll not change over tme unless ntentonally changed or upgraded. The hstory of software relablty assessment methods and tools began n the 60s of last century. A number of researchers [3-7] worked on ssues of development and research of software relablty analyss and assessment models and methods that would allow reducng the costs requred for software testng stage. New software relablty assessment models that reflect the nternal structure of the applcaton and nteracton of ts components [6], called archtectural-based models, are beng developed because of ncreasng complexty of software systems and the result of expandng ther functonal purpose. In known models based on archtectural approach [6, 7] the theory of frst order Markov chans s used wth the assumpton that the software components runs are ndependent. Ths assumpton s not always true due to the complexty of modern software archtecture and huge set of usage scenaros ths assumpton [8, 9]. Therefore, for the development of adequate software relablty assessment models, whch wll mprove the testng process (e.g. allow to reduce the needed resources), one should consder the hgh-order Markov chans, whch allow to take nto account the nterdependence of components runs [10]. Ths paper descrbes the software relablty assessment models based on archtectural approach, whch use dscrete and contnuous tme hgh-order Markov chans and ther comparson based on real world data. Nowadays archtecture-based software relablty models have been well studed n theory, but there s lack of papers descrbng ther practcal applcatons. Developng of hgh-order models for software relablty assessment s not descrbed enough n lterature as well. Thus, the development of new archtecture-based software relablty assessment models that take nto account the dependences among software component runs along wth ther applcaton to real world data s stll a problem watng for soluton. 2 Software Relablty Model wth Dscrete Tme Hgh-Order Markov Chan Ths model consders absorbng Markov process that mples the exstence of one or more absorbng states,.e. states that, once entered, cannot be left. The developed software relablty assessment model [10] s herarchcal, that s ntally software archtecture parameters are calculated based on software usage model [11] usng the theory of Markov processes, and then the behavour of each component falures s taken nto account. The dscrete tme HOMC software relablty model can be descrbed usng the followng components C s the graph wth vertces correspondng to software components, whle edges ndcates the program control flow ( 1, N, where N s the number of program components); P p.. kl s the hgh-order transton proba-

3 blty matrx ( p.. kl transton probablty from component і to component l dependng on beng n prevous K components); Q q.. kl s the ntal probablty vector; t s the falure rate of -th software component. Accordng to ths model the relablty of the whole system s calculated as R N R l l1. In turn, the relabltes of each component ( R l ) usng hgh-order Markov chans are calculated by R exp( l V t.. k... kl... kl ( t) dt) kl 0. To calculate V.. the expected number of vsts a component l dependng on beng n prevous K components one has to solve the followng system of lnear equatons: N 1 V j.. kl q.. k V.. k p.. kl (3) 1, here t.. denotes the tme spent at the component l dependng on beng n prevous kl K components. l (1) (2) 3 Software Relablty Model wth Contnuous Tme Hgh- Order Markov Chan Usng dscrete-tme model has a number of sgnfcant smplfcatons and restrctons. Thus, ths model takes nto account only the number of vsts to -th component wthout takng nto account of the dstrbuton functon of ths random varable (whle takng t nto account the expected value should be used). In addton, t s clear that at any gven tme t the software falure s generally caused by the falure of the software component, whch s executed at a gven tme (n case of sequental connecton of software components n Relablty Block Dagram). Thus, to ncrease the degree of adequacy of the software relablty archtectural model the contnuous tme Markov chans should be used. The contnuous tme HOMC software relablty model can be descrbed usng the followng components C s the graph wth vertces correspondng to software components, whle edges ndcates the program control flow ( the number of program components); a 1, N, where N s A s the hgh-order transton probabl-

4 ty matrx, j 1, N (the values of matrx elements a depends on the way of gettng nto the state ); p s the probablty beng n state P p t s the probablty vector, where t t C at tme t ; s the falure rate of -th software component. In ths case, the falure rate of a software system consstng of N components can be wrtten as [12] here N t p t t 1 t s the falure rate of -th component, t, (4) p s the probablty of -th component executon at tme t. Components falure rates t can be obtaned from the results of unt testng usng known software relablty models, for example ones based on an nhomogeneous Posson process [13]. If the flow control between the components of the software s presented as a Markov process wth contnuous tme, assumng that the -th state of the process s the executon of -th component, the tme dependences of the probabltes beng n -th state ( t ) can be obtaned by solvng the system of equatons of the Kolmogorov p Chapman [14] for ths process: t dp ( t) p j ( t) j ( t) p j ( t), S (5) dt js js, here (t) s the transton ntensty from component і to component j at tme t, and S denotes the set of all system states. In general the transton ntenstes (t) depends on the transton probabltes a and could be calculated from latter. To take nto account the nterdependence of executon of software component (and consequently changes of transton probablty from -th state) dependng on the way of gettng to the current state, the hgh-order Markov chan should be used (the order K of the model determnes the accounted length of the path). In a case of hgh-order Markov chans the actual problem s to calculate the transton probabltes dependng on the program control flow background. It s well known that a hgh order Markov chan can be represented as a frst order chan by approprate redefnng the state space [15]. For software mplementaton of models that use hgh-order chans, t s necessary to have the formalzed algorthm for ths representaton. Usng the analogy wth the known Erlang phase method [16] a hgh-order Markov process can be represented as an equvalent frst-order process wth addtonal vrtual states. Each state of the orgnal graph C (geometrc da-

5 gram, showng the possble states of the system and the possble transtons of the system from one state to another) s splt nto such number of vrtual states as many dfferent paths of length K to ths state exst. Thus the problem soluton essentally s reduced to usng of graph theory. Therefore, to calculate the number m of chans of K-th order from state to state j the followng expresson based on the Floyd method can be used: m K S K 1 m el l j1 here e are the elements of the dentty matrx. Usng (6) one can buld an equvalent graph C whch s a representaton of the ntal graph C, takng nto account all K -th order paths to -th state. Ths allows to avod the dependences of the transton ntenstes (t) on the program control flow background. Then the tme dependence of the software component executon probablty s obtaned by solvng the system of Kolmogorov Chapman equatons (5) for the equvalent frst order Markov process. Usng ths dependence n (4) together wth the component falure rate, the falure rate of the whole software system can be calculated. For calculaton of software relablty measures, based on the obtaned relatonshp (4), one can use the followng relatons [14]: - relablty functon P(t) the probablty that no falure wll occur wthn the tme nterval [0, t] can be calculated as P( t) exp( t 0 m 1 lj, K (6) ( ) d ) (7) - mean operatng tme to falure T l 1 ) 0 T P( d (8) 4 Determnaton of the Markov Chan Order for Software Relablty Assessment In ths paper t s proposed to use AIC and BIC crtera [17] snce they are not a hypothess test and they don t use the sgnfcance level. Although these crtera gve consstent results and don t depend on the estmated model order. They are effcently

6 used for weather forecastng and selecton of adequate envronmental model [18]. But the usage of these crtera n software relablty modelng stll remans unexamned. In general AIC crteron [19] s calculated by the followng expresson: AIC 2k 2ln( L), (9) here k s the number of ndependent parameters n the model, and L s the model s maxmum lkelhood functon. If the value of ths software model s maxmum lkelhood functon s substtuted n the expresson (9), and nstead of parameter k one substtute the number of the K-th K order model parameters, whch contan N components ( k N ( N 1) ), then the followng expresson can be obtaned [20] here kl AIC ( K) 2N K ( N 1) 2ln( p n.. kl.. kl ), (10), j,.., k, l n.. s the number of transtons from component і to component l dependng on beng n prevous components ( j... k l ) n observed sequence and p.. s the transton probablty n ths sequence. kl As t can be seen from (10), ths crteron s ndependent of sample sze. Therefore, AIC crteron s used n the case of a large sample sze (observable sequence) when the software s tested many tmes and the sequence of software component runs s logged. An alternatve to AIC crteron s ВІС [21], whch takes nto account the number of elements n observable sequence and expressed as BIC ( K) 2ln( L) ln( n) k. (11) In the case of usng the BIC crteron for the Markov chan optmal order determnaton n the case of software relablty modelng, the expresson smlar to AIC crteron (10) s obtaned: BIC ( K) 2N K ( N 1)ln( n) 2 n.. kl ln p.. kl. (12), j,.., k, l It s worth noted that the BIC crteron should be used at the small sample sze of emprcal data concernng the software usage, because t mposes stronger penaltes even when the sample sze n>8. Thus, after determnng the order of Markov chan, the well-known classcal Markov tools can be used for software relablty assessment as t was descrbed above.

7 5 Verfcaton of the Models To study the effcency of the developed software relablty assessment model that uses the dscrete tme hgh-order Markov chans, the relabltes of the fve software systems, developed by one of the authors, were calculated usng the frst and the hgh order Markov chan models. The average amount of software components n tested software systems s 10. The results of relablty calculatons usng the frst and the hgh-order models are shown n Fg. 1 along wth the relablty obtaned from unt testng data. 0,90 0,85 Frst-order model Second-order model Testng data Relablty, R 0,80 0,75 0, Software Fg.1. The comparson dagram of the software relablty assessment accuracy usng frst and hgh order dscrete tme models. As shown n fg. 1, the usage of software relablty assessment model based on dscrete tme hgh-order Markov chans makes t possble to ncrease the software relablty accuracy to 6% even for software systems wth a small number of components. To llustrate the effcency of contnuous tme software relablty model, the relablty of the authors' developed program wth four components (n ths case, each component refers to one of the followng program classes Input, Calculaton, Output, Ext) was analyzed. The values of each component falures rates are presented n Table 1. Table 1. Falure detecton frequency for each component Component Falures detecton frequency Input 0.11 Calculaton 0.18 Output 0.09 Ext 0.01

8 The probablty transton matrx and the ntal probablty vector of the Markov chan were calculated and are summarzed n Table 2 and Table 3 correspondngly. Table 2. The frst order probablty transton matrx Component Input Calculaton Output Ext Input Calculaton Output Ext Table 3. The frst order ntal probablty vector Components Input Calculaton Output Ext Probablty The AIC crteron was used for optmal Markov chan order determnaton (the sample set contans 188 entres). The AIC values are lsted n Table 4. Table 4. The values of АІС for dfferent chan orders The process order AIC value The ntal graph C ndcatng the program components and control flow, as well as graph C of the equvalent second order (see Table 4) Markov process are shown n Fg. 2 and 3 correspondngly. The notatons n these fgures are as follows: I correspond to Input state, C to Calculaton, O to Output, and E to Ext state (see Table 2); ndexes ndcates the prevous state n control flow hstory (thus state OC means that the current state Output has been reached from prevous state Calculaton).

9 I C E O Fg.2. Intal graph, representng the components and control flows of the software used for contnuous tme relablty model verfcaton. E Io Ic I C Oc Cc Oo Fg.3. Graph representng an equvalent to the second order Markov chan process for software shown n Fg. 2. Fg. 4 shows the tme dependency of falure rate on tested software system (Fg. 2) obtaned from contnuous tme frst- and second-order Markov chans. It s worth notng that for contnuous tme model smulatons tme was counted as arbtrary tme unts (a.u). As t s seen from ths fgure the frst order model gves slghtly (1 5%) reduced falure rate value for small values of tme, whle t gves sgnfcantly (20%) overestmated falure rate value for mddle tme values range, and when the value of t ncreases by more than 60 a.u. the dfference between values of t obtaned from

10 both models almost tends to be neglgble. Reducng the dfference of t values, obtaned from the frst and the second-order models, to zero at hgh tmes can be explaned by decreasng of the t value tself and by the absence of dfferences n the software behavour descrpton by two models n ths case. In the case of t both models suggest that system s n the Ext state (see Fg. 2, 3) and, accordngly, ts falure rate s lmted by ths component falure rate. Dfferences of t behavour on the ntal evoluton stages of software system can be entrely explaned by dfferences n the software system behavour descrpton by dfferent models, where the frst-order model gnores the nterdependence of software component runs, and therefore the probabltes of components executng are dfferent at gven tme t for both models. Thus, t could be argued that software relablty model based on the hgh order Markov chan descrbes the software system behavour more adequately and determnes ts relablty measures more accurately. 0, frst order model 2 - second order model 0,10 (t) 0, , t, a.u. Fg.4. Tme dependence of the tested software falure rate, obtaned from the frst (1) and second-order (2) models. Ths concluson s confrmed by the analyss of software system relablty functon tme dependence obtaned usng equaton (7) as shown n Fg. 5. Note that the relablty functon s calculated usng Ukranan natonal standards [22] and, as t was ndcated n (7), represents the probablty of falure free operaton durng the tme nterval 0,t, but not exactly at the tme t. So, t s evdently that the relablty functon s tme-decreasng one. As t can be seen from Fg. 5, the dfference n the values P t ncreases up to 20% wth tme value ncreasng. Ths behavor s easly

11 understood f takng nto account the nterval estmaton of relablty for the range 0,t [15], and ts devaton evdently ncreases wth nterval length ncreasng. So, we can conclude that gnorng the nterdependence of software components runs could results n ncreasng naccuracy of software relablty measures estmaton up to 20%. Another measure of relablty, whch was used to determne the developed hghorder model effectveness and adequacy, s the mean operatng tme to falure T 1. The value of mean operatng tme to falure calculated by the expresson (8) s 20.9 tme unts for frst-order model, whle the value obtaned from the second-order model s 23.3 tme unts. Obvously the dfference between the frst and the second-order models s 11.5%, whch can be crucal n the relablty analyss of the complex techncal systems. 1,0 1 - frst order model 2 - second order model 0,8 P(t) 0,6 0,4 0, t, a.u. Fg.5. The tme dependence of the tested software relablty functon, obtaned from the frst (1) and second-order (2) models. Therefore, the usage of software relablty models based on the hgh-order Markov chans even wth a small components number (n ths case there was 4 component software, see Fg. 2) and complexty (n ths case the optmal model order s 2, see Table 4) results n the ncrease of model adequacy and the software relablty assessment accuracy on 10 20%. The value of such growth of relablty estmaton accuracy s especally mportant n the case of complex hardware and software systems n whch the mstake of software component relablty estmaton can repeatedly affect the accuracy of whole system relablty assessment. Usng the hgh-order models for more complex software systems could results n more sgnfcant mprovements of relablty assessment.

12 6 Conclusons In ths paper the software relablty models usng dscrete and contnuous tme hgh-order Markov chans, whch enable consderaton the dependences among software components runs, have been developed. Software relablty assessment based on the hgh-order dscrete tme Markov chans allows to ncrease the software relablty measures accuracy up to 6 %, and hgh-order contnuous tme Markov chan up to 10 20%. The advantage of hgh-order dscrete tme Markov chans model for software relablty analyss s small computng resources needed even for software wth a lot of components, whle the advantages of hgh-order contnuous tme Markov chans model are ndependence from the samplng step (n the case of dscrete tme model t may cause naccurate transton probabltes calculaton), and also automatc consderaton of transtons p that avods unnecessary computatons. Practcal aspects of estmatng nput parameters for the models as well as studyng the dependence of the software relablty on ts characterstcs (components number, the average number of varables n the component, components coheson etc.) wll be descrbed elsewhere. References 1. Lyu, M. R.: Handbook of Software Relablty Engneerng. McGraw-Hll, New York, U.S.A. and IEEE Computer Socety Press, Los Alamtos, Calforna, U.S.A. (1996) 2. Standard Glossary of Software Engneerng Termnology, STD (1991) 3. Musa, J.D.: Valdty of Executon Tme Theory of Software Relablty. IEEE Trans. on Relablty 3, (1979) 4. Goel, A. L., Okumoto K.: Tme-Dependent Error Detecton Rate Model for Software and other Performance Measures. IEEE Trans. on Relablty R-28, (1979) 5. Xe, M.: Software Relablty Modellng. World Scentfc, Sngapore (1991) 6. Goševa-Popstojanova, K., Trved Kshor S.: Archtecture-Based Approach to Relablty Assessment of Software Systems. Performance Evaluaton, 45, ( 2001) 7. Gokhale, S.S., Wong, W.E., Horgan, J.R., Trved Kshor, S.: An Analytcal Approach to Archtecture-Based Software Performance Relablty Predcton. Performance Evaluaton 58(4), (2004) 8. Takag, T., Furukawa, Z., Yamasak, T.: Accurate Usage Model Constructon Usng Hgh- Order Markov Chans. In: Supplementary Proc. 17th Int. Symposum on Software Relablty Engneerng, pp.1 2 (2006) 9. Burkhart, W., Fatha, Z. (Eds.): Testng Software and Systems. Proc. 23rd IFIP WG 6.1 Int. Conf. LNCS, vol (2011) 10. Yakovyna, V., Serdyuk, P., Nytrebych, O., Fedasyuk, D.: Hgh-Order Markov Chans Usage n Software Relablty Analyss. Bulletn of Lvv Polytechnc Natonal Unversty 771, (2013) (n Ukranan) 11. Fedasyuk, D., Yakovyna, V., Serdyuk, P., Nytrebych О.: Varable State-Based Software Usage Model Based on ts Varables. Econtechmod 3, (2014) 12. Yakovyna, V., Masyukevych, V.: The Model for Software Relablty Estmaton Usng Hgh-Order Contnuous-Tme Markov Chans. In: Proc.9th Int. Scentfc and Techncal Conference on Computer Scence and Informaton Technologes, pp (2014)

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