International Journal of Mathematical Archive-5(11), 2014, Available online through ISSN

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1 Inernaional Journal of Mahemaical Archive-5(), 4, 64-7 Available online hrough IN TATITICAL MODELING OF A COMPUTER YTEM UNDER PRIORITY IN REPAIR DICIPLINE OF COMPONENT Jyoi Anand *, Geea ** and. C. Malik ** *Dronacharyra College of Engineering, Deparmen of Applied ciences and umaniies. ** AP Insiue, Delhi, India. **Deparmen of aisics, M. D. Universiy, Rohak-4, India. (Received On: --4; Revised & Acceped On: 5--4) ABTRACT A aisical model for a Compuer sysem is developed by considering he aspecs of prioriy in repair disciplines, cold sandby redundancy and independen failures of componens. There is a single server who visis he sysem immediaely for conducing h/w repair and s/w up-gradaion. erver inspecs he h/w componens a heir failure o see he feasibiliy of repair. If repair of he h/w componens is no feasible, hey are replaced by is idenical componens. owever, only up-gradaion of he s/w is made by new one a heir failure. And, prioriy for he s/w Up-gradaion in one compuer sysem is given over inspecion and repair of he h/w componens in anoher compuer sysem. The sysem works as new afer repair aciviies. All he failure ime disribuions are assumed o be negaive exponenial while ha of inspecion, repair and replacemen imes are aken as arbirary wih differen probabiliy densiy funcions (pdf). The expression for some imporan performance measures of he sysem are derived in he seady sae using semi-markov process and regeneraive poin echnique. The graphical behavior of MTF, availabiliy and profi funcion has been observed for paricular values o various parameers and coss. The profi of he presen model has also been compared wih ha of he model Malik and Anand []. Key Words: Compuer ysem, ardware Failure and Prioriy for he ofware Up-gradaion, Inspecion, Feasibiliy of Repair and aisical Modeling. INTRODUCTION Compuers can be designaed as one of he mos creaive innovaions of human beings. In coming days compuers are even going o be more pervasive, because echnology is geing advanced day by day. Bu breakdown of such sysems may be cosly, dangerous and may cause confusion in our sociey. For example, when he requiremens and dependencies on compuer increase, he possibiliy of poenial crises also increase. The impac of hese failures ranges from hard ship o economic sysem o loss he human life. Mos of he research work in he area of h/w and s/w has been limied o consideraion of eiher h/w subsysem alone or s/w subsysem alone. A few researchers including Freedman and Tran (99) and Welke e al. (995) have proposed reliabiliy models on compuer sysems wih hardware and sofware failures. Recenly, Malik, ureria and Anand [] analyzed a cold sandby compuer sysem giving prioriy o he replacemen of he s/w is given over repair of he h/w componens. Malik and Anand [] proposed a reliabiliy model of a compuer sysem wih cold sandby considering independen failure of h/w and s/w. The reliabiliy and availabiliy of muli componen repairable sysem can be enhanced using he rules of prioriy o repair aciviies of he componens. Also, no much work relaed o he reliabiliy modeling and cos analysis of compuer sysems wih inegraed hardware and sofware componens has been repored so far in he lieraure. Corresponding Auhor: Jyoi Anand * *Dronacharyra College of Engineering, Deparmen of Applied ciences and umaniies. Inernaional Journal of Mahemaical Archive- 5(), Nov. 4 64

2 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. Obvious he above and considering he pracical imporance of he compuer sysems in our daily rouine maers, a reliabiliy model for a compuer sysem having wo idenical unis in which h/w and s/w componens work ogeher is developed. Iniially one uni is operaive and oher is kep as spare in cold sandby. In each uni, h/w and s/w componens fail independenly from normal mode. There is a single server who visis he sysem immediaely o do inspecion, replacemen and repair of he h/w componens. The server inspecs he uni a is h/w failure o see he feasibiliy of repair. If repair of he h/w componens is no feasible, hey are replaced by new one immediaely. owever, prioriy for s/w Up-gradaion in one compuer sysem is given over inspecion and repair of he h/w componens of he oher. The swich devices and repairs are considered as perfec. The random variables are saisically independen o each oher. The failure ime of he componens are exponenially disribued while disribuions of inspecion, repair and replacemen imes of he componens are aken as arbirary wih differen probabiliy densiy funcions (pdf). The swich devices are considered as perfec and h/w componens work as new afer repair. The sysem is observed a suiable regeneraive epochs using semi-markov process and regeneraive poin echnique o derive he expressions for some reliabiliy and economic measures such as mean ime o sysem failure (MTF), availabiliy, busy period analysis, expeced number of replacemens due o hardware failure and sofware Upgradaion, expeced number of visis by he server and finally he economic funcion. The numerical resuls for a paricular case are obained o depic he behavior of MTF, availabiliy and profi of he sysem model. The profi of he presen model has also been compared wih ha of he model Malik and Anand []. NOTATION E O Cs a/b λ /λ p/q FUr/FUR FUi/FUI FWi / FWI FURp/FURP FWRp/FWRP h() / () f() / F() g() / G() q ij / Q ij () q ij.kr /Q ij.kr m ij : The se of regeneraive saes : The uni is operaive and in normal mode : The uni is cold sandby : Probabiliy ha he sysem has hardware / sofware failure : Consan hardware / sofware failure rae : Probabiliy ha repair of he uni due o hardware failure is no feasible / feasible : The uni is failed due o hardware and is under repair / under repair coninuously from previous sae : The uni is failed due o hardware and is under inspecion / under inspecion coninuously from previous sae : The uni is failed due o hardware and is waiing for inspecion/ waiing for inspecion coninuously from previous sae : The uni is failed due o he hardware and is under replacemen/ under replacemen coninuously from previous sae : The uni is failed due o he sofware and is waiing for Up-gradaion / waiing for replacemen coninuously from previous sae : pdf / cdf of inspecion ime of uni due o hardware failure : pdf / pdf of Up-gradaion ime of he sofware : pdf / cdf of repair ime of he uni due o hardware failure : pdf / cdf of passage ime from regeneraive sae i o a regeneraive sae j or o a failed sae j wihou visiing any oher regeneraive sae in (, ] : pdf/cdf of direc ransiion ime from regeneraive sae i o a regeneraive sae j or o a failed sae j visiing sae k, r once in (, ] : Conribuion o mean sojourn ime (µ i ) in sae i when sysem ransis direcly o sae j so ha * µ = m and m ij = dq ( ) = q '() / : ymbol for Laplace-ieljes convoluion/laplace convoluion ~ / * : ymbol for Laplace eiljes Transform (LT)/ Laplace Transform (LT) ' (desh) : Used o represen alernaive resul The following are he possible ransiion saes of he sysem: = (O, cs), = (O, FUi), = (O, FURp), 3 = (O, FUr), 4 = (FURP, FWRp), 5 = (FUI, FWI), 6 = (FUR, FWi), 7 = (FUr, FWI), 8 = (FWi, FURp), 9 = (FURP, FWi), = (FWr, FURp) i j ij ij ij 4, IJMA. All Righs Reserved 65

3 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. The sae 3, 8, and are regeneraive saes while he saes 4 7, 9 are non-regeneraive as shown in fig.. Up-sae Failed sae Regeneraive poin Fig. : ae Transiion Diagram TRANITION PROBABILITIE AND MEAN OJOURN TIME imple probabilisic consideraions yield he following expressions for he non-zero elemens p ij =Q ij ( )= qq iiii () dddd aaaa p = aλ, p =, p = ph* ( aλ+ ), aλ+ aλ+ p 3 =, p 5 = aλ qh* ( aλ, p =, + bλ ) h* ( aλ ) aλ bλ + f * ( aλ+ ) + p 8 = h* ( aλ, p 36 =, ) aλ bλ + aλ g* ( aλ ) + aλ+ bλ + p 3 = g* ( aλ+ ), p 3, = g* ( aλ ) () aλ + bλ + Take h () = θ, f () = and g () =, we have e θ θe θ αe α p.5 = paλ, p.57 = qaλ aλ+ + θ aλ+ + θ p.4 =, p.9 = θλ, p 3.6 = aλ () aλ + bλ + θ aλ + bλ + θ aλ + bλ + α I can be easily verified ha p +p = p +p 3 +p 5 +p 8 = p +p 4 +p, = p 3 +p 36 +p 3, = p +p 3 +p.5 +p.57 +p.8 +p.89 = p 3 +p 3.6 +p 3. = (3) 4, IJMA. All Righs Reserved 66

4 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. The mean sojourn imes (µ i ) is he sae i are µ =/ (aλ +bλ ), µ =/(aλ +bλ +θ ), µ =/(aλ +bλ +θ ), µ 3 =/(aλ +bλ +α) (4) Also m +m =μ m + m3 + m5 + m 8 = µ, m +m.4 +m.9 =μ, m3 + m36 + m 3, = µ 3 (5) and m +m 3 +m 5 +m 8 =μ (say), m +m.4 +m.9 =μ ( say) m 3 +m 3.6 +m 3. =μ 3 ( say) (6) Take h () = θ, f () = and g () =, we have e θe θ αe α (7) µ = [α (θ +aλ ) +qaλ θ ]/ (αθ (aλ +bλ +θ )) µ, = µ 3 = θ α (8) RELIABILITY AND MEAN TIME TO YTEM FAILURE (MTF) Le φ i () be he cdf of firs passage ime from regeneraive sae i o a failed sae. Regarding he failed sae as absorbing sae, we have he following recursive relaions for φ i (): φ () = Q () φ () + Q () φ () φ φ () = Q () () + Q 3 () 3 () + Q 5 () + Q 8 () φ φ φ φ φ ()=Q () ()+Q 4 ()+Q, (), 3()=Q 3 () ()+Q 36 ()+Q 3, () (9) Taking LT of above relaions (9) and solving for ϕ ( s). ~ φ(s) R*(s)= s We have () The reliabiliy of he sysem model can be obained by aking Laplace inverse ransform of (). The mean ime o sysem failure (MTF) is given by ~ φ(s) MTF= = N () s D N = µ p µ p µ p p µ and D = p p + p p p p ( ) 3 3 TEADY TATE AVAILABILITY Le A i () be he probabiliy ha he sysem is in up-sae a insan given ha he sysem enered regeneraive sae i a =. The recursive relaions for A i () are given as A () = M () + q () A () + q () A () A () = M () + q () A () + A () [q.5 () + q.57 ()] + q 3 () A 3 () + q 8 () A 8 () A () =M () + q () A () + q.9 () A () + q.4 () A () A 3 () =M 3 () + q 3 () A () + q 3.6 () A () + q 3, () A () 3 3 A 8 () =q 8 () A 8 (), A () = q, 3 () A 3 () () M i () is he probabiliy ha he sysem is up iniially in sae i E is up a ime wihou visiing o any regeneraive sae, we have 4, IJMA. All Righs Reserved 67

5 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. () ( a b ) M e λ + λ ( aλ+ ) =, M () = e (), ( aλ+ ) ( aλ+ ) M() = e F (), M() = e G () (3) 3 Taking LT of above relaions () and solving for A * () s, he seady sae availabiliy is given by * A( ) = lim sa( s) N = s D N = [p 3 p 3 +p (- p 3, )] [(- p.4 )µ +p µ ] + (-p 3, ) (-p.4 - p p )µ + [(-p.4) (p 3 p p 8) - p p 3 p ]µ 3 and D = [p 3 p 3 +p (- p 3, )] [(- p.4 )µ + p µ' ]+ [(-p.4 - p p )p 3 ](µ' 3 +p 3, µ' ) +(-p 3, ) (-p.4 - p p ) (µ' + p 8 µ 8 ) (4) BUY PERIOD ANALYI FOR ERVER (a) DUE TO ARDWARE FAILURE Le B i () be he probabiliy ha he server is busy in repairing he uni due o hardware failure a an insan given ha he sysem enered sae i a =. The recursive relaions for B i () are as follows: B () = q () B () + q () B () B () = W () + q () B () + (q.5 () + q.57 ()) B () + q 3 B 3 () + q 8 ()) B 8 () B () = q () B () + q.9 () B () + q.4 () B () B 3 () = W 3 () + q 3 () B () + q 3.6 () B () + q 3, () B () B 8 () = q 8 () B (), B () = q, 3 () B 3 () (5) W i () be he probabiliy ha he server is busy in sae i due o hardware failure up o ime wihou making any ransiion o any oher regeneraive sae or reurning o he same via one or more non-regeneraive saes and so ( aλ ) ( aλ ) W () = e + () + aλe + qh( ) G () ( aλ bλ ) ( aλ bλ ) W3 () = e + G () + aλe + G () (6) (b) DUE TO OFTWARE UP-GRADATION Le B i () be he probabiliy ha he server is busy due o sofware up-gradaion a an insan given ha he sysem enered he regeneraive sae i a =. We have he following recursive relaions for B i (): B () = q () B () + q () B () B () = q () B () + (q.5 () + q.57 ()) B () + q 3 B 3 () + q 8 ()) B 8 () B () = W () +q () B () + q.9 () B () + q.4 ()) B () B 3 () = q 3 () B () + q 3.6 () B () + q 3, () B 3 () B 8 ()=W 8 () +q 8 () B 8 (), B () = W () +q, 3 () B 3 () (7) W i () be he probabiliy ha he server is busy in sae i due o replacemen of he sofware up o ime wihou making any ransiion o any oher regeneraive sae or reurning o he same via one or more non-regeneraive saes and so ( a λ + b λ ) ( a λ b λ ) ( a λ b λ ) + + (8) W () = e F () + aλe F () + e F () 4, IJMA. All Righs Reserved 68

6 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. * * Taking LT of above relaions (5) and (7). And, solving for B and, he ime for which server is busy () s B () s due o repair and replacemens respecively is given by * N3 B = (9) = lim sb ( s) s D and * N = 3 B () = lim sb ( s) s D N 3 = (-p.4 -p p ) ((-p 3, ) W+p 3 W 3 ) N 3 =p (p (-p 3, )+p 3 p 3 )w +(-p.4 -p p ) [p 8 (-p 3, )W 8 +p 3 p 3, W ] and D is already menioned. EXPECTED NUMBER OF REPLACEMENT OF TE UNIT (a) DUE TO ARDWARE FAILURE Le R i () be he expeced number of replacemens of he failed hardware componens by he server in (, ] given ha he sysem enered he regeneraive sae i a =. The recursive relaions for R i () are given as R () = Q () R () + Q () R () R ()=Q ()[+R ()]+Q.57 ()R () + Q.5 ()[+R ()] + Q 3 ()R 3 () + Q 8 () R 8 () R () = Q ()R () + Q.9 ()R () + Q.4 ()R () R 3 () = Q 3 ()R () + Q 3.6 ()R () + Q 3, ()R () R 8 () = Q 8 () R (), R () = Q, 3 () R 3 () () (b) DUE TO OFTWARE UP-GRADATION Le R i () be he expeced number of replacemens of he sofware up-gradaion by he server in (, ] given ha he sysem enered he regeneraive sae i a =. The recursive relaions for R i () are given as R () = Q () R () + Q () R () R ()= Q ()[+R ()] +Q.57 ()R () + Q.5 ()[+R ()] + Q 3 ()R 3 () + Q 8 () R 8 () R () = Q ()R () + Q.9 ()R () + Q.4 ()R () R 3 () = Q 3 ()R () + Q 3.6 ()R () + Q 3, ()R () R 8 () = Q 8 ()(+R ()), R () = Q,3 ()(+R 3 ()) () Taking LT of relaions () and (). And, solving for R () and. The expeced numbers of replacemens per s R () s uni ime o he hardware and sofware failures are respecive of given by R = (3) ( ) = lim sr N4 ( s) s D R = (4) ( ) = lim sr N4 ( s) s D N = (p +p.5 ) (-p.4 p p ) (- p 3, ) 4 N 4 = (-p.4) [p 3 p 3, + p 8 (-p 3, )] +p [(-p 3, ) (p -p 8 p ) -p 3 (p p 3, - p 3 )] and D is already menioned. 4, IJMA. All Righs Reserved 69

7 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. EXPECTED NUMBER OF VIIT BY TE ERVER Le N i () be he expeced number of visis by he server in (, ] given ha he sysem enered he regeneraive sae i a =. The recursive relaions for N i () are given as N () = Q () [+N ()] + Q () [+N ()] N () = Q () N ()+[Q.57 ()+Q.5 ()]N () + Q 3 () N 3 () + Q 8 () N 8 () N () = Q () N () + Q.9 () N () + Q.4 () N () N 3 () = Q 3 () N () + Q 3.6 () N () + Q 3, () N () N 8 () = Q 8 () N (), N () = Q, 3 () N 3 () (5) Taking LT of relaion (4.8.) and solving for N () s. The expeced numbers of visis per uni ime by he server are given by N = (6) ( ) = lim sn N5 ( s) s D N 5 = (-p.4 ) (p (- p 3, ) +p 3 p 3 ) and D is already specified. ECONOMIC ANALYI The profi incurred o he sysem model in seady sae can be obained as P = KA KB KB KR KR KN (7) K = Revenue per uni up-ime of he sysem K = Cos per uni ime for which server is busy due o hardware failure K = Cos per uni ime for which server is busy due o sofware up-gradaion K 3 = Cos per uni replacemen of he failed hardware componen K 4 = Cos per uni replacemen of he sofwareup-gradaion K 5 = Cos per uni visi by he server and A, B, B, R, R, N are already defined. Figure : MTF V/ Failure Rae (λ ) 4, IJMA. All Righs Reserved 7

8 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. Figure 3: Availabiliy V/ Failure Rae (λ ) CONCLUION Figure 4: Profi V/ Failure Rae (λ ) There is a sudden decrease in mean ime o sysem failure (MTF) of he sysem model wih he increase of hardware and sofware up-gradaion raes for fixed values for oher parameers. owever, MTF goes on increasing as repair rae (α), replacemen rae (θ) and replacemen rae (θ ) of he uni increase. The sudy also revealed ha MTF becomes more by aking a < b. And, i is seen ha MTF goes on increasing furher by making replacemen of he uni a hardware failure insead of is repair. Figures 3 and 4 highligh he behavior of availabiliy and profi by he sysem model wih respec o hardware failure rae (λ ). I is observed ha availabiliy and profi decrease wih he increase of hardware and sofware up-gradaion raes λ and λ respecively. Bu heir values increase if repair rae (α) and replacemen raes θ and θ increase. Thus on he basis of he resuls obained for a paricular, i is concluded ha he concep of immediae replacemen of he uni a is hardware failure is useful o enhance he reliabiliy and profi of he sysem. 4, IJMA. All Righs Reserved 7

9 aisical Modeling of a Compuer ysem Under Prioriy in Repair Disciplines of Componens / IJMA- 5(), Nov.-4. The profi of he presen model is comparaively less han ha of he model discussed by Malik and Anand []. ence, i is concluded ha he concep of prioriy o he replacemen of s/w up-gradaion in one uni over inspecion and repair of h/w componens in he oher uni is no much economically beneficial. The profi comparison is shown in he following figure 5: REFERENCE Figure 5: Profi V/ Failure Rae (λ ). Malik,.C., ureria,j.k and Anand, Jyoi; Cos- Benefi Analysis of a Compuer ysem wih Prioriy o /W Replacemen over /W repair,pp.vol. 6, []. Friedman, M.A. and Tran, P.: Reliabiliy Techniques for Combined ardware/ofware ysems. Proceedings of Annual Reliabiliy and Mainainabiliy ymposium, pp [99] 3. Welke,.R.; Johnson, B.W. and Aylar, J..: Reliabiliy Modeling of ardware ofware ysems. IEEE Transacions on Reliabiliy, Vol. 44(3), pp ,[995] 4. Lia,C.D.; M.; K.L.; Dai,Y.. and Yang, P.: A Model for availabiliy analysis of disribued sofware/hardware. Informaion and sofware Technology, Vol.44:343-35,[]. 5. Malik,.C. and Anand, Jyoi.: Reliabiliy And Economic Analysis of a Compuer ysem Wih Independen /W and /W Failures. Bullein of Pure and Applied ciences (BPA), Vol.9E (No.),pp.4-53, []. ource of suppor: Nil, Conflic of ineres: None Declared [Copy righ 4. This is an Open Access aricle disribued under he erms of he Inernaional Journal of Mahemaical Archive (IJMA), which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied.] 4, IJMA. All Righs Reserved 7