Availability Analysis of Repairable Computer Systems and Stationarity Detection

1166 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 11, NOVEMBER 1999 Availabiliy Analysis of Rpairabl Compur Sysms and Saionariy Dcion Bruno Sricola AbsracÐPoin availabiliy and xpcd inrval availabiliy ar dpndabiliy masurs rspcivly dfind by h probabiliy ha a sysm is in opraion a a givn insan and by h man prcnag of im during which a sysm is in opraion ovr a fini obsrvaion priod. W considr a rpairabl compur sysm and w assum, as usual, ha h sysm is modld by a fini Markov procss. W propos in his papr a nw algorihm o compu hs wo availabiliy masurs. This algorihm is basd on h classical uniformizaion chniqu in which a s o dc h saionary bhavior of h sysm is usd o sop h compuaion if h saionariy is rachd. In ha cas, h algorihm givs no only h ransin availabiliy masurs, bu also h sady sa availabiliy, wih significan compuaional savings, spcially whn h im a which masurs ar ndd is larg. In h cas whr h saionariy is no rachd, h algorihm provids h ransin availabiliy masurs and bounds for h sady sa availabiliy. I is also shown how h nw algorihm can b xndd o h compuaion of prformabiliy masurs. Indx TrmsÐRpairabl compur sysms, dpndabiliy, availabiliy, prformabiliy, Markov procsss, saionariy dcion. æ 1 INTRODUCTION IN h dpndabiliy analysis of rpairabl compuing sysms, hr is an incrasing inrs in valuaing ransin masurs, in paricular, h poin availabiliy and h availabiliy ovr a givn priod. This papr dals wih h compuaion of h poin availabiliy and of h xpcd inrval availabiliy rspcivly dfind by h probabiliy ha h sysm is in opraion a a givn insan and by h man prcnag of im during which h sysm is in opraion ovr a fini obsrvaion priod. Formally, h sysm is modld by a Markov procss. Is sa spac is dividd ino wo disjoin ss which rprsn h up sas in which h sysm dlivrs h spcifid srvic and h down sas in which hr is no mor srvic dlivrd. Transiions from h up (rsp. down) sas o h down(rsp. up) sas ar calld failurs (rsp. rpairs). Th inrval availabiliy ovr ; is hn h fracion of h inrval ; during which h procss is in h up sas. This random variabl has bn sudid in prvious paprs as, for insanc, in [1], [], and [3], whr is disribuion is valuad using h uniformizaion chniqu. This approach is inrsing bcaus i has good numrical propris and i allows h usr o prform h compuaion wih an rror as small as dsird. An approach o dc h saionariy of Markov procsss has bn proposd in [4], [5]. This approach is basd on h uniformizaion mhod. Th sa probabiliy vcors of h uniformizd Markov chain ar succssivly compud and h iras ha ar spacd m iraions apar ar compard. Whn h diffrnc bwn wo such iras is small nough, h compuaion is soppd. Th main problm wih his mhod is ha, unlik h sandard uniformizaion, hr is no abiliy o spcify rror bounds asily compuabl. In his papr, w dvlop a nw mhod o compu h poin availabiliy and h xpcd inrval availabiliy which is also basd on h uniformizaion chniqu and on h saionary rgim dcion. In pracic, on usually dos no know whhr h im horizon h/sh is considring is larg nough for a sady sa analysis. Th main advanag of our algorihm is ha h compuaion is soppd whn h sady sa availabiliy of h sysm is rachd, giving boh ransin and sady sa masurs wih an rror olranc spcifid in advanc. Whn h saionariy is no rachd, h algorihm givs h ransin masurs and bounds for h sady sa availabiliy. Th rmaindr of h papr is organizd as follows In h following scion, w rcall h classical way o compu h poin availabiliy and w driv nw rsuls o sop h compuaion whn h saionary rgim is rachd. W also giv in his scion h psudocod of boh algorihms. In Scion 3, w considr h xpcd inrval availabiliy and w show how i can b compud using h saionariy dcion. In Scion 4, w show, by mans of a numrical xampl, ha our nw algorihm can considrably rduc h compuaion im of h availabiliy masurs considrd hr whn h im a which masurs ar ndd is sufficinly larg. I is also shown ha compuaional savings can b obaind vn whn h im horizon is small. In Scion 5, w show how h rsuls obaind for h availabiliy masurs can b asily xndd o h corrsponding prformabiliy masurs. Th las scion is dvod o som conclusions.. Th auhor is wih IRISA-INRIA, Campus d Bauliu, 354 Rnns CÂdx, Franc. E-mail sricola@irisa.fr. For informaion on obaining rprins of his aricl, plas snd -mail o c@compur.org, and rfrnc IEEECS Log Numbr 1137. POINT AVAILABILITY ANALYSIS Considr an irrducibl coninuous-im homognous Markov procss X ˆfX ; g, ovr a fini sa spac 18-934/99/$1. ß 1999 IEEE

SERICOLA AVAILABILITY ANALYSIS OF REPAIRABLE COMPUTER SYSTEMS AND STATIONARITY DETECTION 1167 dnod by S. Th sas of S ar dividd ino wo disjoin subss U, h s of h opraional sas (or h up sas), and D, h s of h unopraional sas (or h down sas). For a sysm modld by such a procss, h poin availabiliy a im is dnod by PAV and dfind by TABLE 1 Classical Algorihm for h Compuaion of PAV PAV ˆPrfX Ug Th procss X is, as usual, givn by is infinisimal gnraor, dnod by A, in which h ih diagonal nry A i; i vrifis A i; i ˆ P j6ˆi A i; j. Is iniial probabiliy disribuion is dnod by h row vcor. Th uniformizd Markov chain associad o h procss X is characrizd by is uniformizaion ra and by is ransiion probabiliy marix P [6]. Th uniformizaion ra vrifis max A i; i ; i S and P is rlad o A by P ˆ I A=, whr I dnos h idniy marix. Using his noaion, w g PAV ˆ A 1 U ˆ X 1 P n 1 U ; whr 1 U is a column vcor whos ih nry is 1 if i U and if i D. W dno by V n h column vcor dfind by V n ˆ P n 1 U. I follows ha, for vry n, w hav V n ˆ PV n 1 and V ˆ 1 U. In h following, w dfin for vry n, v n ˆ P n 1 U ˆ V n..1 Th Classical Uniformizaion Mhod Th classical way o compu h poin availabiliy a im is basd on (1). L " b a givn spcifid rror olranc and N b dfind as ( ) N ˆ min n IN Xn j 1 " j! Thn, w obain PAV ˆXN jˆ v n N ; whr h rs of h sris N vrifis N ˆ ˆ 1 XN v n " Th compuaion of ingr N can b mad wihou any numrical problms, vn for larg valus of, by using h mhod dscribd in [7]. Th runcaion lvl N is, in fac, a funcion of, say N. For a fixd valu of ", N is an incrasing funcion of. I follows ha if w wan o compu PAV for J disinc valus of, dnod by 1 < < J, w only nd o compu v n for n ˆ 1;...;N J sinc h valus of v n ar indpndn of h paramr. Th psudocod of h classical uniformizaion mhod can hn b wrin as shown in Tabl 1. 1. Saionariy Dcion Th saionariy dcion ha w considr is basd on h conrol of h squnc of vcors V n ˆ P n 1 U. L h row vcor dno h saionary probabiliy disribuion of h Markov procss X. This vcor vrifis A ˆ and P ˆ. Th sady sa availabiliy is givn by PAV 1 ˆ 1 U.To nsur h convrgnc of h squnc of vcors V n,w rquir ha h uniformizaion ra vrifis > max A i; i ; i S sinc his guarans ha h ransiion probabiliy marix P is apriodic. W hn hav, for vry i S, lim V n i ˆ1 U 1 W dscrib now h s usd o dc ha, for a givn valu of n, h nris of vcor V n ar clos o 1 U. For vry n, w dfin m n ˆ min V n i and M n ˆ max V n i is is No ha, sinc V ˆ 1 U, w hav M ˆ 1 and m ˆ. Th following rsul givs bounds of h sady sa availabiliy PAV 1 ˆ 1 U. Lmma.1. Th squncs m n and M n ar, rspcivly, nondcrasing and nonincrasing and, for vry n, w hav v n M n m n M n m n and 1 U M n m n M n m n Morovr, boh squncs m n and M n convrg o 1 U. Proof. For vry i S, w hav V n 1 i ˆPjS P i; j V n j. I follows ha m n V n 1 i M n and, so, w g m n m n 1 and M n 1 M n, which shows ha h squncs m n and M n ar, rspcivly, nondcrasing and nonincrasing. Sinc v n ˆ PjS j V n j, w g m n v n M n, which is quivaln o

1168 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 11, NOVEMBER 1999 v n M n m n M n m n Wriing now 1 U ˆ P n 1 U ˆ V n ˆ PjS j V n j, w g, in h sam way, m n 1 U M n, which is quivaln o 1 U M n m n M n m n Th sa spac S bing fini and h fac ha, for vry i S, V n i convrgs o 1 U show ha boh squncs m n and M n convrg o 1 U. u Rmark. W hav assumd ha h Markov procss X is irrducibl. If h Markov procss X is no irrducibl, bu conains an absorbing sa dnod by a wih a D, hn w hav 1 U ˆ and, for vry i S, w asily g V n i! whn n!1. Now, sinc a D, w hav V n a ˆ for vry n and so w also hav m n ˆ for vry n. Thus, in his cas, i sufficis o considr h squnc M n which is nonincrasing and convrgs o. This lmma shows ha h diffrnc M n m n convrgs o, ha is, for a fixd rror olranc ">, hr xiss an ingr k such ha, for n k, w hav M n m n ". Sinc m n M n, w hav m n m n 1 M n 1 M n and, so, h squnc M n m n is nonincrasing. W can hn dfin h following ingr K ˆ inffn jm n m n "=g Using h ingr K, (1) can b wrin as PAV ˆ XK whr 1 K ˆ M K m K v n 1 XK v n M K m K! 1 K ; X 1 Using Lmma.1, h rs 1 K vrifis j 1 K j v n M K m K "=4 This las inqualiy follows from h fac ha, for n K, w hav, from Lmma.1, m K m n v n M n M K and, so, v n M K m K M K m K "=4. Th im K can b inrprd as h discr im o saionariy wih rspc o h subs U. For vry and for vry ingr l, w dno by F l h funcion dfind by F l ˆXl M n m n I is asy o chck ha, for a fixd valu of l, h funcion F l dcrass from 1 o ovr h inrval ; 1. W can 3 4 hn dfin for vry ingr l and for vry ">, h im T l as T l ˆ inff ; F l "=4g W hn hav h following horm Thorm.. For vry ">, for vry T K w hav jpav 1 U j3"=4 1 U M K m K "=4 PAV M K m K " Proof. Firs no ha, from Lmma.1, w hav m n v n M n and m n 1 U M n, for vry n. I follows ha jv n 1 U jm n m n for vry n. W hn hav jpav 1 U jˆ v n 1 U jv n 1 U j M n m n ˆ F K M n m n Sinc T K, w hav F K "=4. In h scond rm, sinc n K, w hav M n m n M K m K "= and, so, w g (5). Rlaion (6) is immdia from Lmma.1. Finally, combining (5) and (6), w g (7). u Th im T K can b inrprd as h coninuous im o saionariy wih rspc o h subs U..3 Th Nw Algorihm Using hs rsuls, w obain h following nw algorihm (shown in Tabl ). To simplify h wriing of his algorihm, w dfin G l ˆXl H l ˆ1 Xl v n ; S l ˆ Ml m l ; No ha i is no ncssary o compu h coninuous im o saionariy T K wih a high prcision. I is sufficin o obain an uppr bound of T K such as, for insanc, dt K, which is h smalls ingr grar or qual o T K. I mus b also nod ha, in his algorihm, h runcaion sp N is a funcion of h im J as in h classical unformizaion algorihm, bu h ims o 5 6 7

SERICOLA AVAILABILITY ANALYSIS OF REPAIRABLE COMPUTER SYSTEMS AND STATIONARITY DETECTION 1169 TABLE Algorihm for h Compuaion of PAV Using Saionariy Dcion Z EIAV ˆ1 PAV s ds Using (1) and by ingraion ovr ;, w obain EIAV ˆX 1 1 P k 1 U W dno by Vn h column vcor dfind by Vn ˆ 1 P k 1 U ; and w dfin v n ˆ V n. By dfiniion of V n and v n in h prvious scion, w g, for vry n, Vn ˆ 1 V k and v n ˆ 1 I follows ha Vn and v n ar rcursivly givn, for n 1, by Vn ˆ n V n 1 1 V n; and v n ˆ n v n 1 1 v n; 8 wih V ˆ V ˆ 1 U and, hus, v ˆ v. For vry n, w hav v n 1. I follows ha, using h runcaion sp N dfind in (), w g h classical algorihm o compu h xpcd inrval availabiliy by wriing v k whr EIAV ˆXN v n N ; saionariy K and T K ar indpndn of h im paramr, whn h discr im K is rachd. Th compuaional im complxiy of boh algorihms is ssnially du o h compuaion of h vcors V n.to compu hs vcors, h classical algorihm rquirs N marix-vcor producs and our nw algorihm rquirs only min K;N marix-vcor producs. 3 EXPECTED INTERVAL AVAILABILITY ANALYSIS W show in his scion how h nw algorihm proposd abov for h poin availabiliy compuaion can b adapd o compu h xpcd inrval availabiliy aking accoun of h saionariy dcion. Th xpcd inrval availabiliy rprsns h man prcnag of im during which h sysm is in opraion ovr a fini obsrvaion priod ;. Th inrval availabiliy ovr ; is dnod by IAV and is xpcaion is givn by N ˆ v n ˆ 1 XN " This algorihm is basically as h on dpicd in Tabl 1. Mor prcisly h compuaion of v n in Tabl 1 mus b followd by h h rcursion (8), wih v ˆ v, and, in h las loop ovr j, v n mus b rplacd by v n in ordr o g EIAV j insad of PAV j. 3.1 Saionariy Dcion for h Expcd Inrval Availabiliy Using h rsuls obaind for h poin availabiliy, w can driv a nw mhod o obain h xpcd inrval availabiliy using h saionariy dcion. This mhod is basd on h wo following horms. Boh horms will b usd in h cas whr h discr im o saionariy K is such ha K N. Th firs horm sas ha, in ordr o compu h xpcd inrval availabiliy, EIAV, w only nd h valus of v n for n K. Th scond horm sas ha, in ordr o compu h xpcd inrval availabiliy, EIAV, for T K, w only nd h valu EIAV a a im such ha T K. W dno by G K h funcion

117 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 11, NOVEMBER 1999 and rcall ha H K ˆ1 XK G K ˆXK v n ; and ˆ MK m K Thorm 3.1. For vry, w hav EIAV G K K 1 v K S K H K 1 H K "=4 9 Proof. For vry, w hav whr EIAV ˆG K ; ˆ For n K 1, w hav v n ˆ 1 v k " # ˆ 1 X K v k Xn v k ˆ K 1 v K 1 ˆ K 1 v K 1 v k v n v k ˆ K 1 n K v K x n ; whr jx n jˆ 1 v k 1 jv k j n K n K " 4 "=4 Th inqualiy jv k j"=4, for k K, follows from Lmma.1; i has alrady bn usd o bound h rror 1 K in (4). If is h funcion dfind by ˆ x n ; w obain j j "=4. W hn hav ˆ K 1 v K n K By wriing n K ˆ K 1 in his las xprssion, w g ˆK 1 v K H K 1 H K W hn obain EIAV G K K 1 v K S K H K 1 H K ˆ ; which compls h proof sinc j j "=4. Thorm 3.. For vry ">, for vry and such ha T K w hav EIAV EIAV 1 " 1 Proof. For vry and such ha T K, w hav Z EIAV ˆ1 PAV s ds " ˆ 1 Z Z # PAV s ds PAV s ds " ˆ 1 Z PAV s ds Z PAV s Šds ˆ EIAV 1 1 Z PAV s Šds Using (7), w hav, sinc T K Z 1 PAV s Šds 1 Z jpav s jds 1 " "; which compls h proof. u No ha Thorm 3. is sill valid if w rplac T K by dt K. So, as for h poin availabiliy, w can us dt K insad of T K o mak asir h compuaion of h xpcd inrval availabiliy. Using hs wo horms, w obain a nw algorihm o compu h xpcd inrval availabiliy which is similar o h on dscribd in Tabl for h poin availabiliy. I suffics o prform h following changs in h algorihm givn in Tabl Th compuaion of v n givn by (8) mus b addd jus afr h compuaion of v n, wih v ˆ v. Th rlaion PAV j ˆG K j mus b rplacd by EIAV j ˆ G K j and h compuaions of PAV j in h cas whr K N mus b rplacd by hos of EIAV j givn in (9) for j T K and in (1) for j >T K. To us (1), w nd EIAV for on valu of such ha T K. Such a valu can b obaind by using (9) on mor im for h smalls valu of j such ha j T K. No ha w hav h wllknown saionary rlaion PAV 1 ˆ EIAV 1. u

SERICOLA AVAILABILITY ANALYSIS OF REPAIRABLE COMPUTER SYSTEMS AND STATIONARITY DETECTION 1171 Fig. 1. Sa-ransiion diagram for an n-procssor sysm. 4 NUMERICAL EXAMPLE W considr a faul-olran muliprocssor sysm wih fini buffr sags. This sysm was firs considrd in [8] for wo procssors wihou rpair and has bn xndd in [9] o includ rpair for h compuaion of h momns of prformabiliy. Is has bn also usd in [1] o obain h disribuion of prformabiliy. W us h sam modl hr for h compuaion of h poin availabiliy wih our nw mhod. I consiss of n idnical procssors and b buffr sags. Procssors fail indpndnly a ra and ar rpaird singly wih ra. Buffrs sags fail indpndnly a ra and ar rpaird wih ra. Procssor failurs caus a gracful dgradaion of h sysm and h numbr of opraional procssors is dcrasd by on. Th sysm is in a faild sa whn all h procssors hav faild or any of h buffr sags has faild. No addiional procssor failurs ar assumd o occur whn h sysm is in a faild sa. Th modl is rprsnd by a Markov procss wih sa ransiion diagram shown in Fig. 1. Th sa spac of h sysm is S ˆf i; j ; i n; j ˆ ; 1g. Th componn i of a sa i; j mans ha hr ar i opraional procssors and h componn j is zro if any of h buffr sags fails; ohrwis, i is on. I follows ha h s U of opraional sas is U ˆf i; 1 ;1 i ng. W valua h poin availabiliy givn ha h sysm sard in sa n; 1. Th numbr of procssors is fixd o 16, ach wih a failur ra ˆ 1 pr wk and a rpair ra ˆ 1666 pr hour. Th individual buffr sag failur ra is ˆ pr wk and is rpair ra is ˆ 1666 pr hour. Th rror olranc is " ˆ 1. In Fig., w plo h poin availabiliy, PAV, asa funcion of for diffrn valus of h numbr of buffr sags b. Th largs valu of, ha is, h valu of J in h algorihm, has bn chosn qual o 1; hours. For h largs valu of, w show, in Fig. 3, h runcaion sp N ˆ N 1, h discr im o saionariy K, and h coninuous im o saionariy T K (in fac, w Fig.. From op o h boom PAV for b ˆ ; 4; 8; 16; 3. giv dt K ) for diffrn valus of h numbr of buffr sags b. This figur shows, for xampl, ha whn b ˆ 16 h classical algorihm nds 3; 581 marix-vcor producs and our nw algorihm nds only 18 marix-vcor producs, h coninuous im o saionariy bing qual o 77. Whn b ˆ 1; 4 h classical algorihm nds 15; 616 marix-vcor producs and our nw algorihm nds only 86 marix-vcor producs, h coninuous im o saionariy bing qual o 6. Morovr, our algorihm also compus h sady sa poin availabiliy wih a prcision qual o "=4. Fig. 3 also shows ha boh siuaions, K<T K and K>T K, ar possibl. W considr in Fig. 4 smallr valus of J. Th numbr of buffr sags is fixd o b ˆ 8. For J 1, w g N 1 14 and h discr im o saionariy K is no rachd. This mans ha K>14. For J w g N J and h discr im o saionariy is rachd. Is valu is K ˆ 18 and h coninuous im o saionariy is dt K ˆ8. Fig. 4 shows ha, vn for small valus of J ( J <T K ), our algorihm can rduc h compuaion im wih rspc o h classical algorihm. For insanc, whn J ˆ 6, h classical algorihm nds 4 marix-vcor producs and our nw algorihm nds only 18 marix-vcor producs. 5 EXTENSION TO THE PERFORMABILITY ANALYSIS Th mhod proposd for h compuaion of h poin availabiliy and h xpcd inrval availabiliy using h sady sa availabiliy dcion can b xndd o mor gnral masurs such as h poin prformabiliy and h xpcd inrval prformabiliy. Fig. 3. Saionariy dcion for diffrn numbrs of buffr sags.

117 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 11, NOVEMBER 1999 Fig. 4. Saionariy dcion for small valus of h im. In prformabiliy modling (s, for insanc, [8], [9], [1], [11], [1], [13], [14], [15] and h rfrncs hrin), rward ras ar associad wih sas of h modl o quanify h abiliy of h sysm o prform in h corrsponding sas. W dno by i h rward ra associad o h sa i S. Th rward ras i ar assumd o b nonngaiv ral numbrs. Th poin prformabiliy a im, dnod by PP, and h xpcd inrval prformabiliy, dnod by EIP, ar dfind by PP ˆX is i PrfX ˆ ig and EIP ˆ1 Z PP s ds W dfin ˆ max is i and r i ˆ i = and w dno by r h column vcor whos ih nry is qual o r i. W hn hav PP ˆf and EIP ˆg, whr f ˆ A r and g ˆ1 f s ds Sinc, for vry i S, w hav r i 1, all h rsuls and algorihms obaind for h compuaion of h availabiliy masurs can b asily xndd o h compuaion of f and g. To do ha, i suffics o rplac h column vcor 1 U by h column vcor r. Th valus M and m bcom M ˆ max is r i and m ˆ min is r i. Morovr, w hav f 1 ˆ g 1 ˆ r. Z [3] G. Rubino and B. Sricola, ªInrval Availabiliy Analysis Using Dnumrabl Markov Procsss Applicaion o Muliprocssor Subjc o Brakdowns and Rpair,º IEEE Trans. Compurs, vol. 44, no., pp. 86-91, Fb. 1995. [4] G. Ciardo, A. Blakmor, P.F. Chimno, J. K. Muppala, and K.S. Trivdi, ªAuomad Gnraion and Analysis of Markov Rward Modl Using Sochasic Rward Ns,º Linar Algbra, Markov Chains, and Quuing Modls, C.D. Myr and E.R.J. Plmmons, ds., pp. 145-191, Springr-Vrlag, 1993. [5] M. Malhora, J.K. Muppala, and K.S. Trivdi, ªSiffnss-Tolran Mhods for Transin Analysis of Siff Markov Chains,º Microlcronics and Rliabiliy, vol. 34, no. 11, pp. 1,85-1,841, 1994. [6] S.M. Ross, Sochasic Procsss. John Wily & Sons, 1983. [7] P.N. Bowrman, R.G. Noly, and E.M. Schur, ªCalculaion of h Poisson Cumulaiv Disribuion Funcion,º IEEE Trans. Rliabiliy, vol. 39, pp. 158-161, 199. [8] J.F. Myr, ªClosd-Form Soluions for Prformabiliy,º IEEE Trans. Compurs, vol. 31, no. 7, pp. 648-657, July 198. [9] B.R. Iyr, L. Donaillo, and P. Hidlbrgr, ªAnalysis of Prformabiliy for Sochasic Modls of Faul-Tolran Sysms,º IEEE Trans. Compurs, vol. 35, no. 1, pp. 9-97, Oc. 1986. [1] V.G. Kulkarni, V.F. Nicola, R.M. Smih, and K.S. Trivdi, ªNumrical Evaluaion of Prformabiliy and Job Complion Tim in Rpairabl Faul-Tolran Sysms,º Proc. IEEE 16h Faul- Tolran Compuing Symp., pp. 5-57, Vinna, Ausria, July 1986. [11] G. Ciardo, R. Mari, B. Sricola, and K.S. Trivdi, ªPrformabiliy Analysis Using Smi-Markov Rward Procsss,º IEEE Trans. Compurs, vol. 39, no. 1, pp. 1,51-1,64, Oc. 199. [1] R.M. Smih, K.S. Trivdi, and A.V. Ramsh, ªPrformabiliy Analysis Masurs, an Algorihm, and a Cas Sudy,º IEEE Trans. Compurs, vol. 37, no. 4, pp. 46-417, Apr. 1988. [13] H. Nabli and B. Sricola, ªPrformabiliy Analysis A Nw Algorihm,º IEEE Trans. Compurs, vol. 45, no. 4, pp. 491-495, Apr. 1996. [14] E. d Souza Silva and H.R. Gail, ªCalculaing Availabiliy and Prformabiliy Masurs of Rpairabl Compur Sysms Using Randomizaion,º J. ACM, vol. 36, pp. 171-193, Jan. 1989. [15] J.F. Myr, ªOn Evaluaing h Prformabiliy of Dgradabl Compuing Sysms,º IEEE Trans. Compurs, vol. 9, no. 8, pp. 7-731, Aug. 198. 6 CONCLUSIONS A nw algorihm has bn dvlopd o compu h poin availabiliy and h xpcd inrval availabiliy of rpairabl compur sysms modld by Markov procsss. This nw algorihm is basd on h uniformizaion chniqu and on h dcion of h sady sa availabiliy. I compars favorably wih h classical uniformizaion algorihm whn h im horizon is larg and i is shown hrough a numrical xampl ha compuaional savings can b obaind vn whn h im horizon is small. Morovr, our algorihm givs h sady sa availabiliy if h saionariy is rachd and bounds of h sady sa availabiliy ohrwis. Finally, his mhod can b asily xndd o h compuaion of mor gnral masurs such as h poin prformabiliy and h xpcd inrval prformabiliy. Bruno Sricola rcivd h PhD dgr in compur scinc from h Univrsiy of Rnns I in 1988. H has bn wih INRIA (Insiu Naional d Rchrch n Informaiqu n Auomaiqu, (a public Frnch laboraory) sinc 1989. His main rsarch aciviy is in compur and communicaion sysms prformanc valuaion, dpndabiliy and prformabiliy of faulolran archicurs, and applid sochasic procsss. REFERENCES [1] E. d Souza Silva and H.R. Gail, ªCalculaing Cumulaiv Opraional Tim Disribuions of Rpairabl Compur Sysms,º IEEE Trans. Compurs, vol. 35, no. 4, pp. 3-33, Apr. 1986. [] G. Rubino and B. Sricola, ªInrval Availabiliy Disribuion Compuaion,º Proc. IEEE 3rd Faul-Tolran Compuing Symp., pp. 49-55, Toulous, Franc, Jun 1993.