Reliability and Sensitivity Analysis of a System with Warm Standbys and a Repairable Service Station

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1 Ieraoa Joura of Operaos Research Vo. 1, No. 1, 61 7 (4) Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao Kuo-Hsug Wag, u-ju a, ad Jyh-B Ke Deparme of Apped Mahemacs, Naoa Chug-Hsg Uversy, Tachug, 4, Tawa, R. O. C. Absrac We sudy he reaby ad sesvy aayss of a sysem wh M operag maches, S warm sadbys, ad a reparabe servce sao. Faure mes ad servce mes of each mache (operag or sadby) are assumed o be expoeay dsrbued. Whe he servce sao s workg, s subec o breakdows accordg o a Posso process. Whe he sao breaks dow, requres repar a a repar facy, he repar mes foow he egave expoea dsrbuo. The K ou of M + S sysem s aayzed K = 1,,, M. Ths paper preses dervaos for he sysem reaby, R y (), he mea me o sysem faure, MTTF, ad umerca usrao. Severa cases are aayzed o vesgae he effecs of varous parameers o he R y () ad he MTTF. Sesvy aayss for he R y () ad he MTTF s aso suded. Keywords reaby; sesvy aayss, sao breakdows 1. INTRODUCTION AND ITERATURE REVIEW I he ope eraure, mos of he papers aayze he queueg sysems he servce saos have ever faed. However, rea-fe suaos we ofe ecouer cases servce saos may break dow ad ca be repared. We sudy a sysem wh M + S deca maches ad a sge reparabe servce sao. As may as M of hese ca be operag smuaeousy parae, he res of he S maches are warm-sadby spares. A reparabe servce sao meas ha he servce sao s ypcay subec o upredcabe breakdows ad ca be repared. Severa researchers have vesgaed some queueg sysems whch a sge servce sao subec o breakdows s cosdered. Mos of he papers dea wh oy some queueg probems of he sysem, raher ha some reaby probems of he sysem. Pas work may be dvded o wo pars accordg o he sysem s suded from he vewpo of he queueg heory or from he vewpo of he reaby. I he frs caegory we revew prevous papers whch reae o a queueg heory vewpo oy. Ife source M/M/1 queue wh breakdows was frs proposed by Wag (1989). Wag (199) deveoped seady-sae aayc souos of he M/M/1 mache repar probem wh a sge servce sao subec o breakdows. The M/E k /1 mache repar probem wh a o-reabe servce sao was proposed by Wag (1997). The secod caegory of auhors dea wh papers whch reaes o a reaby vewpo oy. Cao ad Cheg (198) frs roduced reaby coceps o a queueg sysem wh a reparabe servce sao he fe me of he servce sao s expoeay dsrbued ad s repar me has a geera dsrbuo. Furher, he reaby aayss of a M/G/1 queueg sysem whch he servce sao has a m-u reaby seres srucure was aayzed by Cao (1994). Wag ad Svaza (1989) suded he reaby characerscs of a mupe-server (m + w)-u sysem wh w warm sadby us wh expoea faure ad expoea repar me dsrbuos. Cao (1985) derved he reaby quaes of a M/G/1 mache repar mode wh a reparabe servce sao whch cosss a sge u. u ad Cao (1995) exeded Cao s mode o a reparabe servce sao whose srucure coas a m-u reaby seres. e a. (1997) examed he reaby aayss of a M/G/1 queueg sysem wh server breakdows ad Berou vacaos. Tag (1997) vesgaed some reaby ad queueg probems of a sge-server M/G/1 queueg sysem subec o breakdows. Recey, he seady-sae avaaby ad he mea me o sysem faure of a reparabe sysem wh warm sadbys pus bakg ad reegg were suded by Ke ad Wag () ad Wag ad Ke (3). I hs paper, we sudy he reaby characerscs of a reparabe sysem o deerme how reaby ca be mproved by provdg suffce spares as sadbys. We aso perform a sesvy aayss for chages he reaby characerscs aog wh chages specfc vaues of he sysem parameers. Sysem faure s defed o be ess ha K maches acve operao, K = 1,,, M (K ou of M + S sysem). Tha s, he sysem faure s defed as: () he sysem fas whe a M + S maches fa; or () he sysem fas whe a eas oe of he M operag maches fas (.e. he sadby maches are emped). Ths paper shoud be dsgushed from prevous works ha: (a) he reaby probem wh sadbys has dsc characerscs whch are dffere from he mache repar probem wh sadbys; (b) cosders a arbrary umber of M maches operag smuaeousy, ad a arbrary umber of S Correspodg auhor s ema: khwag@amah.chu.edu.w

2 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 6 maches are preoperao (warm sadby); (c) cosders a reparabe servce sao whch s subec o breakdows; (d) performs a sesvy aayss. We frs deveop he expc expressos for he reaby, R (), ad he mea me o sysem faure, MTTF, by usg apace rasforms echques. Nex, we perform a paramerc vesgao whch provdes umerca resus o show he effecs of varous sysem parameers o he R (), ad o he MTTF. Fay, we perform a sesvy aayses for chages he R () ad he MTTF aog wh chages specfc vaues of he sysem parameers. 1.1 Noao M: umber of operag maches S: umber of warm sadby maches : umber of faed maches he sysem λ : faure rae of a operag mache η : faure rae of a warm sadby mache µ : servce rae of a faed mache α : breakdow rae of a servce sao β : repar rae of a servce sao λ : mea faure rae whe here are faed maches he sysem p( ): probaby ha he servce sao s workg ad here are faed maches he sysem a me P (): probaby vecor cossg of p() q( ): probaby ha he servce sao s broke dow ad here are faed maches he sysem a me Q() : probaby vecor cossg of q() s: apace rasform varabe p ( s ): apace rasform of P () P ( s ): apace rasform of P () P (): a vecor of P () whe = q ( s ): apace rasform of q() Q ( s ): apace rasform of Q () Q () : a vecor of Q() whe = : me o faure of he sysem R () : reaby fuco of he sysem MTTF: mea me o sysem faure. DESCRIPTION OF THE SSTEM We cosder a sysem wh M deca maches operag smuaeousy parae, S warm sadbys, ad a sge servce sao whch s subec o breakdows. I s assumed ha he swch s perfec ad ha he swchover me s saaeous. Each of he operag maches fas depedey of he sae of he ohers ad has a expoea me-o-faure dsrbuo wh parameer λ. If a operag mache fas, s mmedaey repaced by a spare f oe s avaabe. We assume ha each of he avaabe spares fas depedey of he sae of a he ohers ad has a expoea me-o-faure dsrbuo wh parameer η ( < η < λ ). The faed mache s se for servce, ad afer servce s reaed as a spare. I s assumed ha whe a spare moves o a operag sae, s faure characerscs w be ha of a operag mache. Wheever a operag mache or a spare fas, s mmedaey se o a servce sao s served order of breakdows, wh a me-o-servce whch s expoeay dsrbued wh parameer µ. Furher, he successo of faure mes ad he successo of servce mes are depedey dsrbued radom varabes. Suppose ha he servce sao ca break dow a ay me wh breakdow rae α. Wheever he servce sao breaks dow, s mmedaey repared a a repar rae β. Aga, breakdow mes ad repar mes of he servce sao are assumed o be expoeay dsrbued. We ow assume ha he servce sao ca serve oy oe faed mache a a me, ad ha he servce s depede of he faure of he maches. If he servce sao breaks dow, he a faed mache mus wa u he servce sao s repared. If servce of a faed mache s aowed o be erruped by a breakdow, resumpo akes pace as soo as he servce sao s avaabe or he repar compeo ermaes. If he servce sao breaks dow, he a faed mache mus wa u a servce sao s repared. Whe he repar of a servce sao s compeed, he servce sao mmedaey serves a faed mache. Ahough o servce occurs durg he repar perod of faed servce sao, faed maches coue o arrve accordg o a Posso process. If a operag mache fas(or spare fas) ad oe spare s avaabe a a sa whe he servce sao s avaabe, he faed mache a oce goes for servce, ad he spare s pu o operao. Oce a servce sao s repared, becomes as good as ew. Sysem reaby s suded accordg o he assumpos ha sysem faure s defed o be ess ha K maches acve operao, K = 1,,, M. Therefore, f deoes he umber of faed maches he sysem, he sysem s faed f ad oy f = M + S K REIABIIT ANASIS OF THE SSTEM A me =, he sysem has us sared operao wh o faed maches whe he servce sao s workg. The reaby fuco uder he expoea faure me, expoea servce me, expoea breakdow me, ad expoea repar me dsrbuos ca he be deveoped hrough he brh ad deah process. e p( ) probaby ha he servce sao s workg ad here are faed maches he sysem a me, q( ) probaby ha he servce sao s broke dow ad here are faed maches he sysem a me,

3 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 63 =,1,,...,, ad = M + S K + 1,( K = 1,,..., M). The mea faure rae Mλ + ( S ) η λ = ( M + S ) λ λ s gve by: f =,1,..., S 1; f = S, S + 1,..., 1; oherwse. The apace rasforms of p() ad q() are defed as: s p ( s) = e p ( ) d, =,1,...,, s q ( s) = e q ( ) d, =,1,...,. The foowg apace rasform expressos for ad q ( s ) are obaed erms of λ. p ( s ) () = ( λ + α + s) p ( s) µ p ( s) βq ( s) = p () (1a) 1 ()1 λ 1p 1( s) + ( λ + µ + α + s) p( s) µ p ( s) βq ( s) = p () + 1 (1b) + = (1d) λ 1p 1( s) sp( s) p() (v) = ( λ + β + sq ) ( s) α p( s) = q() (1e) (v)1 λ ( ) ( ) ( ) ( ) () 1q 1 s + λ + β + s q s α p s = q (1f) (v) = 1 λ q ( s) + ( λ 1 + β + s) q 1( s) α p ( s) = q () 1 1 (1g) (v) = 1q 1( s) + sq ( s) = q () (1h) λ = M + S K + 1, K = 1,,..., M, ad p () = 1, p() =, for = 1,,...,, q (), for =,1,,..., Equao(1) ca be wre foowg marx form () = 1 λ p ( s) + ( λ + µ + α + s) p ( s) 1 1 βq ( s) = p () 1 1 (1c) DsW ( ) ( s) = W() () D(s) = (v) = λ + α + s µ λ λ + µ + α + µ 1 s λ1 λ + µ + α + s λ + µ + α + s µ λ λ 1 + µ + α + s λ 1 s α α α α α

4 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 64 β β β β β λ + β + s λ λ + β + 1 s λ λ + β + 1 s λ + β + s λ λ 1 + β + s λ 1 s s a ( + 1) ( + 1) marx. W ( s ) s a coum vecor coag he se of eemes P ( s), Q ( s) T, P ( s ) = ( ), 1( ), ( ),, 1( ), ( ) T p s p s p s p s p s, Q ( s ) = ( ), 1( ), ( ),, 1( ), ( ) T q s q s q s q s q s, ad he symbos T deoes he raspose. W () s a coum vecor coag he se of eemes [P(), Q()] T, P () = [ p (), p (), p (),..., p (), p ()] T 1-1 = [ 1,,,...,] T, Q () = [ q (), q (), q (),..., q (), q ()] T 1 1 = [,,,...,] T Sovg () accordace wh Cramer s rue, we oba he expresso for p () s ad q ( s ) gve by [ + 1 s ] [ ( )] de N ( ) p ( s) =, (3) de D s de N ( 1) ( s ) + q ( s) =, (4) de D s [ ( )] de[d(s)] deoes he deerma of marx D(s), de[n +1 (s)] deoes he deerma obaed by repacg he ( + 1)h coum marx D(s) by he a vecor W()=[1,,,,,, ] T ad de[n (+1) (s)] s he deerma obaed by repacg he ( + 1)h coum marx D(s) by he a vecor W()=[1,,,,,, ]. I s oo compex o derve he expc souos p ( s ) ad q ( s ) of (3) ad (4), respecvey. Therefore, we use he compuer sofware MAPE o oba he souos p ( s ) ad q ( s ). We frs cosder he deomaor de[d(s)] (3) ad (4). I s easy o kow ha he equao de[d(s)] = has doube zero roos. e s = -r (r are ukow vaues), he we have D( r) = A ri, A = D() s a ( + 1) ( + 1) marx ad I s he dey marx. Thus () becomes ( A ri) W ( s) = W (). (5) We se he deerma of he marx A ri equa o zero, ad fd he correspodg dsc egevaues r (r ad =1,, 3,,) whch may be rea or compex. Suppose ha here are rea dsc egevaues (excudg zero) say r 1, r,, r, ad pars dsc cougae compex egevaues, say ( r+ 1, r+ 1), ( r+ 1, r+ ),, ( r, + 1 r+ ), ad sasfy + = I s o be oed ha = deoe a egevaues (excudg ) are compex, ad = represes a egevaues are rea. Nex, we cosder he umeraors de[n +1 (s)] ad de[n (+1) (s)] (3) ad (4), respecvey. The compuer sofware MAPE s used o evauae de[n +1 (s)] ad de[n (+1) (s)]. Thus, subsug de[d(s)] ad de[n +1 (s)] o (3) yeds a a1 a p ( s) = s s + r1 s + r bs 1 + c1 + s + ( r+ 1 + r+ 1) s + r+ 1r+ 1 bs + c + + s + r + r s + r r ( ) , 1,...,, 1, 1,,,...,, (6) a a a b c b c b c are ukow rea

5 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 65 umbers. kewse, subsug de[d(s)] ad de[n (+1) (s)] o (4), we oba q ( s) d d d s s + r s + r es+ f + 1 = s + ( r+ 1 + r+ 1) s + r+ 1r es+ f ( ) s r+ r+ s r+ r+, 1,...,, 1, 1,,,...,, d d d e f e f e f are ukow rea umbers. e u ad v represe he rea par ad he magary par of compex egevaue r + respecvey. Iverg he apace rasform (6) ad (7), we ge he expc expressos for p () = a + a e = 1 r u c bu u + be cosv ( ) + e sv ( ) = 1 v q () = d + d e = 1 r u f eu u + ee cosv ( ) + e sv ( ) = 1 v respecvey. Sce he sysem has faed durg he fe perod of me. Therefore we oba [ ] a + d = m p( ) + q( ) = 1 (1) 3.1 The reaby fuco R () e be he radom varabe ad represe he me o faure of he sysem. Sce p () s he probaby ha he sysem has faed o or before me whe he servce sao s workg, ad q () s he probaby ha he sysem has faed o or before me whe he servce sao s broke dow, we have he reaby fuco gve by R () = 1 p( ) q( ),. (11) 3. The mea me o sysem faure MTTF R If ( ) s ( ) () (7) (8) (9) R s = e R d s he apace rasform of ad aways fe, we have R () d = m R () s. (1) s Thus he MTTF s gve by MTTF = R () d, (13) or equvaey MTTF = m R ( s) s 1 a d a s = 1 s + r bs+ c d = m = + ( ) s 1 s r 1 r 1 s + r+ 1r+ = 1 s + r es + f = + ( ) + 1 s r 1 r 1 s r+ 1r+ a c d f = = 1 r = 1 r r = 1 r = 1 r r (14) SENSITIVIT ANASIS FOR R () AND MTTF I hs seco we frs perform a sesvy aayss for chages he R y () aog wh chages specfc vaues of he sysem parameers λ, µ, α, ad β. Numerca resus of he sesvy aayss for he R y () aog wh chages λ, µ, α, ad β are preseed. Dffereag () wh respec o λ, we oba Ds ( ) W ( s) W ( s) + D( s) =, or equvaey W ( s) 1 D( s) = D ( s) W ( s). (15) Usg he compuer sofware MAPE o sove (15), we ca oba he souos p ( S )/ ad q ( s )/. Afer verg he apace rasform souos, we ge p ()/ ad q ()/. Dffereag (11) wh respec o λ yeds R () = () () p q. (16) Subsug p ()/ ad q ()/ o (16), we oba R ()/ λ. Usg he same procedure sed above, we ca ge R ()/ µ, R ()/ α, ad R ()/ β. Nex, we perform a sesvy aayss for chages he MTTF aog wh chages specfc vaues of λ, µ, α, ad β. Numerca resus of he sesvy aayss for he MTTF aog wh chages λ, µ, α, ad β

6 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 66 are aso provded. Dffereag (13) wh respec o λ, we oba MTTF = R () d. (17) Subsug (16) o (17) yeds MTTF / λ. Usg he same procedure sed above, MTTF / µ, MTTF / α, ad MTTF / β ca be obaed. 5. NUMERICA IUSTRATION The purpose of hs seco s fourfod. The frs s o aayze graphcay o sudy he effecs of varous parameers o he sysem reaby. We fx λ =.6, η =.5, µ = 1., α =., β = 3., choose he umber of operag maches M = 3, ad cosder he cases whe he umber of warm sadbys S chages from 1 o 4 ad he vaues of K vary from 1 o 4. We ca easy see from Fgure 1 ha moderae mproveme he sysem reaby s obaed by addg he umber of warm sadbys. Moreover, Fgure shows ha he sysem reaby creases as K decreases. Obvousy, he vaues of K affec he sysem reaby sgfcay. We sha resrc ourseves o he reaby aayss of seecg fxed vaues M = 3, S =, K = 1, ad η =.5, for he foowg cases. Case 1: We fx µ = 1., α =., β = 3., ad vary he vaues of λ from. o.6. Case : We fx λ =.6, α =., β = 3., ad vary he vaues of µ from.5 o.. Case 3: We fx λ =.6, µ = 1., β = 3., ad vary he vaues of α from.1 o.4. Case 4: We fx λ =.6, µ = 1., α =., ad vary he vaues of β from 3. o 9.. I ca be easy observed from Fgure 3 ha he sysem reaby creases as λ decreases. Obvousy, he vaues of λ affec he sysem reaby sgfcay. Oe sees from Fgure 4 ha he sysem reaby creases wh creasg µ. Fgures 5-6 show ha he sysem reaby rarey chages whe α or β chages. Iuvey, he sysem reaby may be oo sesve o chages α or β. I appears from Fgures 3-6 ha he mos sgfca parameer o he sysem reaby s he parameer λ. Ry( ) M = 3, K = 1, λ =.6, η =.5, µ = 1., α =., β = 3. Fgure 1. Sysem reaby wh warm sadbys ad a reparabe servce sao. Sysem fas whe a M+S maches fa. R () y M = 3, S =, λ =.6, η =.5, µ = 1., α =., β = 3. Fgure. Sysem reaby wh warm sadbys ad a reparabe servce sao for dffere vaues of K.

7 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 67 () R y M = 3, S =, K = 1, η =.5, µ = 1., α =., β = 3. Fgure 3. Sysem reaby wh warm sadbys ad a reparabe servce sao. Sysem fas whe a M+S maches fa. () R y M = 3, S =, K = 1, η =.5, λ =.6, α =., β = 3. Fgure 4. Sysem reaby wh warm sadbys ad a reparabe servce sao. Sysem fas whe a M+S maches fa. R y () M = 3, S =, K = 1, η =.5, λ =.6, µ = 1., β = 3. Fgure 5. Sysem reaby wh warm sadbys ad a reparabe servce sao. Sysem fas whe a M+S maches fa. () R y M = 3, S =, K = 1, η =.5, λ =.6, µ = 1., α =. Fgure 6. Sysem reaby wh warm sadbys ad a reparabe servce sao. Sysem fas whe a M+S maches fa.

8 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 68 The secod purpose s o vesgae he effecs of varous parameers o he MTTF. We fx M = 3 ad choose η =.5. Varous vaues of λ are cosdered. Case 5: We fx K = 1, choose µ = 1., α =., β = 3., ad vary he umber of warm sadbys S from 1 o 4. Case 6: We fx S =, choose µ = 1., α =., β = 3., ad vary he vaues of K from 1 o 4. Case 7: We fx S =, K = 1, choose α =., β = 3., ad vary he vaues of µ from.5 o.. Case 8: We fx S =, K = 1, choose µ = 1., β = 3., ad vary he vaues of α from.1 o.4. Case 9: We fx S =, K = 1, choose µ = 1., α =., ad vary he vaues of β from 3. o 9.. The umerca resus of he MTTF are show Tabes 1-5. From Tabes 1-5, we ca easy see ha he MTTF decreases as λ creases. Obvousy, he MTTF ca moderaey decrease as λ creases for sma λ. Moreover, Tabes 1-5 show ha () he addo of warm sadbys S, he decrease K, ad he crease µ ca moderaey crease he MTTF for sma λ ; ad () he crease α or β rarey affecs he MTTF. The hrd purpose s o perform a sesvy aayss of he sysem reaby for chages he sysem parameers λ, µ, α, ad β. We fx M = 3, S =, K = 1, ad seec λ =.6, η =.5, µ =1., α =., β =3.. I Fgure 7, aog he me coordae, he sysem reaby w be affeced eve by mue chage of he sysem parameers λ, µ, α, ad β. Iuvey, creasg he vaues of µ ad β or decreasg he vaues of λ ad α w mprove he sysem reaby. I seems ha he order of mpacs of hese four parameers o he sysem reaby are: λ > µ > α > β. We observe ha he effecs of varyg α ad β o he sysem reaby ca be egeced whch maches he prevous cocusos show Fgures 5-6. Aso, he effecs of varous parameers o he sysem reaby occur oy he me erva < < 5, ad he mos sgfca effec occurs aroud = 8. The fourh purpose s o perform a sesvy aayss o he chage of he MTTF for varous parameers λ, µ, α, ad β. We fx M = 3, S =, K = 1, ad seec λ =.6, η =.5, µ =1., α =., β =3.. I ca be easy see from Tabe 6 ha he order of mpacs of hese four parameers o he MTTF are: λ > µ > α > β. The gross effec of β s eggbe whe comparg wh he gross effecs of λ, µ, ad α. I shoud be oed ha hese cocusos are oy vad for he above cases. We may reach oher cocusos for oher cases. Tabe 1. The MTTF for dffere vaues of λ ad S ( M = 3, K = 1, η =.5, µ = 1., α =., β = 3. ) λ S=1 S= S=3 S= Tabe. The MTTF for dffere vaues of λ ad K ( M = 3, S =, η =.5, µ = 1., α =., β = 3. ) λ K=1 K= K=3 K= Tabe 3. The MTTF for dffere vaues of λ ad µ ( M = 3, S =, K = 1, η =.5, α =., β = 3. ) λ µ =.5 µ =1. µ =1.5 µ =

9 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 69 Tabe 4. The MTTF for dffere vaues of λ ad α ( M = 3, S =, K = 1, η =.5, µ = 1., β = 3. ) λ α =.1 α =. α =.3 α = Tabe 5. The MTTF for dffere vaues of λ ad β ( M = 3, S =, K = 1, η =.5, µ = 1., α =. ) λ β =3. β =4. β =6. β = Tabe 6. Sesvy aayss for he MTTF wh case λ =.6, µ = 1., α =., β = 3. θ = λ θ = µ θ = α θ = β MTTF θ CONCUSIONS I hs paper, we have deveoped he expc expressos for he sysem reaby ad he MTTF. I shoud be frs oed from Fgures 1-6 ha α ad β rarey affec he sysem reaby, S has moderae effec, K, λ, ad µ affec he sysem reaby sgfcay. Nex, we shoud oe from Tabes 1-5 ha () α ad β rarey affec he MTTF; ad () S, K, ad µ affec he MTTF moderaey for sma λ. Fay, we have performed a sesvy bewee he sysem reaby, he MTTF ad specfc vaues of λ, µ, α, ad β. Our umerca vesgaos dcae ha he order of mpacs of hese four parameers o he sysem reaby ad he MTTF are: λ > µ > α > β. R y () θ θ = β θ = µ θ = α θ = λ M=3,S=,K=1,η =.5 Fgure 7. Sesvy aayss for he sysem reaby wh case λ =.6, µ = 1., α =., β = 3..

10 Wag, a, ad Ke: Reaby ad Sesvy Aayss of a Sysem wh Warm Sadbys ad a Reparabe Servce Sao IJOR Vo. 1, No. 1, 61 7 (4) 7 REFERENCES 1. Cao, J. ad Cheg, K. (198). Aayss of M/G/1 queueg sysem wh reparabe servce sao. Aca Mahemacae Appcae Sca, 5: Cao, J. (1985). Aayss of a mache servce mode wh a reparabe servce equpme. Joura of Mahemaca Research ad Exposo Egeerg, 5: Cao, J. (1994). Reaby aayss of M/G/1 queueg sysem wh reparabe servce sao of reaby seres srucure. Mcroeecrocs ad Reaby, 34: Ke, J.-C. ad Wag, K.-H. (). The reaby aayss of bakg ad reegg a reparabe sysem wh warm sadbys. Quay ad Reaby Egeerg Ieraoa, 18: , W., Sh, D.H., ad Chao, X.. (1997), Reaby aayss of M/G/1 queueg sysems wh server breakdows ad vacaos. Joura of Apped Probaby, 34: u, B. ad Cao, J. (1995). A mache servce mode wh a servce sao cossg of r ureabe us. Mcroeecrocs ad Reaby, 35: Tag,.H. (1997). A sge-server M/G/1 queueg sysem subec o breakdows-some reaby ad queueg probem. Mcroeecrocs ad Reaby, 37: Wag, K.-H. ad Svaza, B.D. (1989). Reaby of sysem wh warm sadbys ad reparme. Mcroeecrocs ad Reaby, 9: Wag, K.-H. (199). Ife source M/M/1 queue wh breakdow. Joura of he Chese Isue of Idusra Egeers, 7: Wag, K.-H. (199). Prof aayss of he mache repar probem wh a sge servce sao subec o breakdows. Joura of he Operaoa Research Socey, 41: Wag, K.-H. ad Kuo, M.-. (1997). Prof aayss of he M/E k /1 mache repar probem wh a o-reabe servce sao. Compuers ad Idusra Egeerg, 3: Wag, K.-H. ad Ke, J.-C. (3). Probabsc aayss of a reparabe sysem wh warm sadbys pus bakg ad reegg. Apped Mahemaca Modeg, 7: