Analysis of a Repairable (n-1)-out-of-n: G System with Failure and Repair Times Arbitrarily Distributed

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1 Amerca Joural of Mathematcs ad Statstcs. ; (: -8 DOI:.593/j.ajms.. Aalyss of a Reparable (--out-of-: G System wth Falure ad Repar Tmes Arbtrarly Dstrbuted M. Gherda, M. Boushaba, Departmet of Mathematcs, Faculty of sceces, Uversty of Jjel, Algera Laboratory of mathematcal modelg ad smulato, Uversty of Costate Route daï-el-bey Costate, Algera Abstract A (--out-of-: G system s a system that cossts of compoets ad works f ad oly f (- compoets amog the work smultaeously. The system ad each of ts compoets ca oly oe of two states: workg or faled. Whe a compoet fals t s put uder repar ad the other compoets stay the "workg" state wth adjusted rates of falure. After repar, a compoet works as ew ad ts actual lfetme s the same as tally. If the faled compoet s repared before aother compoet fals, the (- compoets recover ther tal lfetme. The lfetme ad tme of repar are depedet. I ths paper, we propose a techque to calculate the mea tme of repar, the probablty of varous states of our system ad ts avalablty by usg the theory of dstrbuto. Keywords G System, System Avalablty, Reparable System, Falure Rates, Dstrbuto Theory. Itroducto The k-out-of- system structure s a very popular type of redudat fault-tolerat systems. It fds wde applcatos both dustral ad mltary systems. These systems clude the multdsplay system cockpts, the multege system a arplae, ad the multpurpose system a hydrau-lc cotrol system. I a commucatos system wth three trasmtters, the average message load may be such that at least two trasmtters must be operatoal at all tmes or crtcal messages may be lost. The trasmsso subsystem fuctos as a -out-of-3: G system. Systems wth spares may also be represeted by the k-out-of- system model. I the case of a automoble wth four tres, for example, usually oe addtoal spare tre s carred o the vehcle. Thus, the vehcle ca be drve as log as at least 4-out-of-5 tres are good codto. I ths paper, we cosder the reparable case where k-,.e. a reparable (--out-of-: G system whch works f ad oly f at least (- compoets amog the work smultaeously. Few papers aalyze ths kd of systems, Gaver[] ad Jack[] cosder a -ut parallel system, Gherda & Boushaba[3] aalyze a -out-of-3 system whe the dstrbutos of the tme of falure ad the tme of repar are geeral. I ths work, we geeralze the result of Gherda & Boushaba[3]: We suppose that all compoets ad the system have ether oe of two states: a "workg" state or a "faled" state. whe Correspodg author: mboushabafr@yahoo.fr (M. Boushaba ublshed ole at Copyrght Scetfc & Academc ublshg. All Rghts Reserved a compoet fals t s put uder repar ad the other compoets stay the "workg" state wth adjusted rates of falure. After repar, the compoet works as ew ad ts actual lfetme s the same as tally. If the faled compoet s repared before aother compoet fals, the (- compoets recover ther tal lfetme. The lfetme ad tme of repar are depedet. I ths paper, we propose a techque to calculate the mea tme of falure, the probablty of varous states of our system ad ts avalablty by usg the theory of dstrbuto. Fally, we gve some umercal examples.. Notato C : compoet of the system,,,..., E s : the state of system, s,,. We say that the system s the state E s at tme t f there are exactly s faled compoets at tme t. : rate of falure of compoet C whe all compoets j, ( j are workg. : rate of falure of compoet C whe compoet j,( j s "faled". X : tme of repar of compoet C. ; ( t;x : the desty of the probablty of evet: " oly the compoet C s fals at tme t ad t s uder repar sce a tme x" ; ( t;x : the desty of probablty of evet : "Two

2 M. Gherda et al.: Aalyss of a Reparable (--out-of-: G System wth Falure ad Repar Tmes Arbtrarly Dstrbuted compoets C j ad C k are "faled" ad compoets C are the "workg" state at tme t sce tme x; {,...,}/{j; k}. ( t : robablty of the evet: "No compoet s "faled" at tme t" ϕ ( s : The Laplace trasform of the dstrbuto of T A: The avalablty of the system G : the dstrbuto fucto of u : hazard rate fucto x X ad G G u( y dy u( y dy dg G e ad e u. lm t x lm t,x,, t,. t x lm t,x, t, G, G : the Laplace trasform of ( f : the verse Laplace trasform of f : Covoluto product 3. Model x G ad G Let N (t ad X (t be two stochastc process wth cotuous tme such that N (t s. If the system s the state E s ad X (t x f the compoet whch faled at tme t s uder repar sce date x: Cosder the evet N (t + dt ; t ca be obtaed ( + dfferet ways: - At tme t, N (t ad durg the terval of tme [t; t + dt] there are o falures, the probablty of ths evet s t dt Or at tme t, N (t ad durg the terval of tme [t; t + dt]; the faled compoet C s repared, the probablty of ths evet s t dt t,x u x + dt;,,...,., The the probablty of ths evet s ( t + dt r ( N ( t + dt ( t dt+ j t j dt dt, ( t, x u j j whe dt we obta t d ( t +, u dt ( Cosder the evet N (t + dt ; t ca be obtaed dfferet ways : - N (t + dt ad X (t + dt x + dt: f N (t, X (t x ad the repar of the faled compoet s ot fshed at tme t + dt: the probablty of ths evet oted by t + dt, x + dt s gve by:, t + dt,x + dt,, ( t, x j + u dt+ ( dt f t > x j j otherwse whe dt we obta: θ( t x, + θ( t x, t x ( j + u θ( t x, j j where θ s the dcator fucto. - Or N (t + dt ad X (t + dt dt: f N (t ad a falure occurs durg the terval of tme [t; t + dt]: Ths case s gve by the tal codtos:, ( t, ad (. Our system ca be represeted by the fgure. Fgure

3 Amerca Joural of Mathematcs ad Statstcs. ; (: Calculato of the State robabltes of a System 4.. Calculato of ( t ad, ( t,x Let ( s ad, ( t ad, fucto ( t x s,x be the Laplace trasform of t,x respectvely. Because the dcator θ s a regular dstrbuto, we ca use the dervato of ( the sese of dstrbuto: θ( t x, + θ( t x, t x j + u θ( t x, j j By takg the Laplace trasform of the two members of ths last equalty we obta s,x, x (3 ( s exp s + j x exp u ( y dy j j Now, takg the Laplace trasform of the two members of the equalty (:, ( s,x ( s exp s + j x G (4 j j we ote that, ( s,x ( s G s+ j x+ j j by applyg a proposto [4] page 5 to, ( s,x, whch verfes ths majorato, we obta d θ ( t x, f dt ( s,x where f s the verse trasform of ( s,x, We ote that s the Laplace trasform of the s covoluto product. By usg (4:, s, ( s,x sx exp x j G x s e j s j s exp x j G ( t δ ( t x t j j where δ s the Drac fucto. The verse Laplace trasform of, s,x s gve by: d exp x j G ( t δ ( t x t dt j j exp x j G ( t (5 j j Now, we obta: ( t ( t + ( t G j t j j By takg the Laplace trasform of the two members of ths equalty we obta: ( s s+ G j j j by usg the verse Laplace trasformed, we obta: ( t exp G j t j j ad ( t exp x j G f t x, > j j otherwse lm t ad Wth codtos t x lm t,x t, t t,

4 4 M. Gherda et al.: Aalyss of a Reparable (--out-of-: G System wth Falure ad Repar Tmes Arbtrarly Dstrbuted 4.. Calculato of ( x, The evet N(t + dt ca be obtaed two ways:. N(t, X(x+dt dt; X(t ad durg the terval of tme [t; t + dt], oe compoet amog the workg compoets fals ad the repar of the faled compoet s ot fshed. Ths evet has the probablty:,, j dt u dt+ ( dt j j, j dt f x < t j j otherwse. Or N(t, X(x + dt dt, ad ay repar are fshed ad o falure occurs durg the terval of tme [t; t + dt] :Ths evet has the probablty:, + j dt f x < t j j otherwse So, t + dt,x +, u j dt+ j j, + u j dt f x < t j j whe dt we obta:, t,x, t,x + t x u, u, f x < t j j Otherwse, by usg the Laplace trasform ad the tal codto (t; t, we obta: s + u, ( s,x +, ( s,x x, ( s,x j j j Oe soluto of ths equato s gve by:, ( s,x exp ( α k + j, ( s,x exp ( α j j where α s a prmtve of [ s u ] (6 + ad k a costat. Now by usg the covoluto product, the soluto of (6 s gve by, ( s,x exp ( sx exp u. j ( s j j K + exp s + j j j xg exp ( sx exp ( u fally, we obta :

5 Amerca Joural of Mathematcs ad Statstcs. ; (: -8 5 δ s,x K t x exp u x +, ( s δ ( t x exp u j j j exp j x j j by usg the verse Laplace trasform ad the tal codto (t; t (K, we obta, xe x G ( x G j j j exp t GI j j j ad t,x t,x,, exp t G j j j G j j j G j j j 5. Characterstcs of the System 5.. Mea Tme of Falure LetΦ be the Laplace trasform of, ( t,x : Φ ( s, ( s,x Φ ( s exp x j G j j s+ G j j j exp x j G j j s+ G j j j ( G j j j j Fally j j Φ E T s+ G j j j ( G j j j j j j G j j j 5.. Avalablty of the System Let A(t, x the avalablty of the system. The ( t A(t, x +, + exp j G exp G j t j j j j

6 6 M. Gherda et al.: Aalyss of a Reparable (--out-of-: G System wth Falure ad Repar Tmes Arbtrarly Dstrbuted 5.3. Case of No Idetcal Compoets I the stablty case, we obta the followg equatos for the system: u ( x, j d, j + u ( x, j j d, + u ( x, j, ( x j j Ths system becomes: + G (,, j j j j,, j j j exp x G x G x exp x,, j j j by usg ths hypothess: ( u x,, j j j (codtos at the lmts, we obta +, +, j Gj j j j j j ad +,j + j So,j + j exp x j G j j,j + G exp x j j j E( X,j I we ca wrte the soluto the followg form:, +, + - G j j j j j j -, E( X j fally, we derve:, B C + D Where Gj ke X j k j j k B G j j k k j k C Gj Gj k k j j k ad D ( j G j( + j k 4 j j j j 5.4. Case of N Idetcal Compoets I the stablty case ( t, we obta the followg equatos for the system: u ( x ( x

7 Amerca Joural of Mathematcs ad Statstcs. ; (: -8 7 d ( + u ( x d + u ( x ( by usg the tal codtos ad codtos at the lmts : ( ad ( + u x, we obta + ( -G (( + + By the same argumet as the o detcal case, we derve - G (( G (( E( X - - G ( G ( + [ -6] Remark : whe 3, we obta the result gve page [3] 6. Numercal Examples Cosder the case where ad the stablty case ( t. Whe the compoets are o detcal, the equatos of the system are: ( + u, ( x, 3 + u, x,, exp 3 x G x,, G x exp 3 x ( ( By combg these equatos, we obta: + x u x, 3, 3, A + where ( ( 3 ( + G ( ( G ( G G 3 G 3- G 3- ( + G ( ( G ( ( ( E( X ( G G G G G 3- I the detcal case, we have ( G G ( + E( X G ( G ( + E( X G ( + G ( A. G ( + E( X I the followg, we wll gve some umercal examples whe we cosder the cases: -X s costat, X E(X, G ( -exp (- -X follows the expoetal law G + - X follows the law 6 We ote m G + 6 f.375 ad.75, the values of E(X for dfferet values of m ad dfferet laws of X are gve Tables ad, respectvely: Table. The values of E(X for dfferet values of m ad dfferet laws of X.(.375 X Γ ( / 6 m Xct X exp Table. The values of E(X for dfferet values of m ad dfferet laws of X.(.75 X Γ ( / 6 m Xct X exp The values of the probablty states ad the avalablty, for varous laws of X, are gve the Tables 3, 4 ad 5.

8 8 M. Gherda et al.: Aalyss of a Reparable (--out-of-: G System wth Falure ad Repar Tmes Arbtrarly Dstrbuted Table 3. m The values of probablty y states ad avalablty for X costat Table 4. The values of probablty y states ad avalablty for X exp ( m Coclusos The work preseted here s a study of a (--out-of-: G reparable system whose compoets are depedet ad ot detcal. The durato of compoet falure ad that of the repars are radom varables followg expoetal laws ad arbtrary laws respectvely. The ovelty ths cotrbuto s the use of the followg, very realstc hypothess: whe a compoet fals, the falure rates of all other compoets chage; aturally, whe that compoet s repared, these same compoets recover ther tal falure rates. To the authors kowledge, prevous works dd ot tackle ths case from the same agle as ths study. The authors beleve that dfferetato of dstrbutos s the techque best suted for the calculato of the avalablty ad of the other characterstcs of ths kd of systems. Addtoally, the choce of the techque to use s beleved to be judcous ad has ever bee used before ths kd of stuatos. More geerally, the case k-out-of- systems remas a ope problem that deserves more explorato. A A Table 5. The values of probablty y states ad avalablty for X Γ 6 m ACKNOWLEDGEMENTS The authors are grateful for rofessor Fabrzo Rugger from IMATI (Italy for the valuable suggestos ad commets whch have cosderably mproved the presetato of the paper. We ackowledge, wth thaks, the costructve commets made by the referees whch helped to ehace the qualty of ths paper. REFERENCES [] D.. Gaver, Tme to falure of paralled systems wth repar, IEEE trasacto o relablty, vol R-, 96 [] N. Jack, Aalyss of a reparable -ut parallel redudat system wth repar, IEEE trasacto o relablty, vol R-35, 986 [3] M. Gherda ad M. Boushaba, Aalyss of a reparable -out-of-3 system wth falure ad repas tmes arbtrarly dstrbuted, proccedg of 3th ISSAT teratoal coferece, Seattle, Washgto, USA, august -4 7, eds: T. Nakagara, H. ham ad S. Yamada, pp 96- [4] L. Shwartz, Méthodes mathématques pour les sceces physques, 966, eds Herma ars [5] K. N. Fracs Leug, Y. L. Zhag, K. K. La, Aalyss for a two-dssmlar-compoet cold stadby reparable system wth repar prorty, Relablty Egeerg ad system safety, vol. 96 ssue, November, pp [6] Y. L. Zhag, G. J. Wag, A geometrc process repar model for a reparable cold stadby system wth prorty use ad repar, Relablty Egeerg ad system safety, vol. 94 ssue, No-vember 9, pp A