Cost-benefit analysis of a one-server two-unit imperfect switch system with delayed repair

Microelectron.Reliab.,Vol.25,No. 5, pp. 865-867,1985.

0026-2714/8553.00+ .00

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COST-BENEFIT ANALYSIS OF A ONE-SERVER TWO-UNIT IMPERFECT SWITCH SYSTEM WITH DELAYED REPAIR M. N. GOPALA~ a n d S. S. WAGHMARE Department of Mathematics, Indian Institute of Technology, Powai, Bombay, 400 076, India

(Received for publication 27 February 1985) Abstract--This paper deals with the cost-benefit analysis of a one-server two-unit system with imperfect switch where the server is summoned upon failure of an item (i.e. unit or switch). The amount of time the server takes to arrive is a random variable, distributed arbitrarily, The server leaves when there is no item waiting for repair. The repair times are arbitrarily distributed whereas all failure rates are constant. Initially one unit is switched on (switch is working at t = 0) and the other is kept as cold standby. Explicit expressions for the expected uptime in (0, t) of the system, busy period of the server due to repair of a unit and that of the switch are obtained to carry out the cost-benefit analysis.

11 (wR, w, uR) 12 (wR, s, uR)

I. INTRODUCTION

Gopalan and Waghmare [I] have recently analysed a

13 (uR, wR, wR)

one-server two-unit imperfect switch system u n d e r the a s s u m p t i o n t h a t the server is available t h r o u g h o u t . In the present paper, it is a s s u m e d t h a t the server is not available t h r o u g h o u t b u t is s u m m o n e d w h e n e v e r a failure of a n item (unit/switch) occurs. T h e server takes a r a n d o m a m o u n t of time to arrive a n d starts • the repair work. T h e d i s t r i b u t i o n of the time t h a t elapses between the first failure of a n item a n d the arrival of the server is arbitrary. T h e server is retained till n o item is waiting for repair. T h e situation where the server is available t h r o u g h o u t is considered as a particular case.

14 (wR, wR, vR) AvJ(t) P(V(t) = I l X ( 0 ) - - j X j r 0 , 1 ..... 14 /~(t) expected up-time of the system in (0, t)

A(u) B.(u) C~(u) E,~(u) L.(u)

m(u

2s e -As° (1 - e "au') + 2v e -~uo (1 -e-as~)] do

Mh(u) h(u) e-ao ~ Nh(u) h(u)(1 - e -av") My(u) Ou (u) e -as~

Nv(u) gu(U)(1 -e-aS ")

2. NOTATION

Ms(U) gs(U)e-aU" Ns(u) gs(u)(1 -e-au").

The following notation will be used in addition to that given in [1]. V(t) dichotomous random variable = 1, if the system is up at the instant t = O, otherwise W working state of a unit or switch S standby WR waits for repair (unit/switch) WS waits for server (unit]switch) UR under repair (unit]switch) h(.) pdf of waiting time for the server 9v('),Os(') repair time density of a unit and the switch respectively J-v, 2s failure rates of a unit and the switch respectively

0 1 2 3 4

e-(,tu+as)u m(u)e -(au+as~', m = h,g u m(u)e -as~ (1--e-aU") re(u) e~ av" (1 - e -as')

3. EXPECTED

UP-TIME IN (0, t)

The system is up when it is in one of the states {0, 1, 2, 3, 5, 7, 9, 1 1}. Therefore

Av°(t) = e-t~v+~s)t + ~ o A * A v l ( t ) + AsA*Av2(t )

Avl(t) = B~Av3(t)+CfAvS(t)+ EfAvlt(t) + L*~Av14(t)+ e-~vtA(t) Av3(t) = G(t)e-~V t + B~vAv°(t)+C*vAv3(t) + E ' r a y 7(t) + L*gvAvl2(t )

(W,S,W) (ws, w,w) (W,S, WS) (UR, W,W) (WS,WR, W)

Av 2(t) = e-~Vt H(t) + M * A v 7(t) + N*AvI2(t)

Av8(t) = M~Av3(t)+N~AvI2(t) Av14(t) = gfAv8(t)

5 (WS, S, WR) 6 (wR, s, ws) 7 (W,S, UR)

Av11(t) = ~s(t) e-au t + M f A v 3 ( t ) + NfAv8(t)

8 (ug, WR, w) 9 (UR,w, wR)

Aol2(t) = o~Ao3(t) Av7(t) = ezs(t)e-aUt + M~Av°(t)+ N~Av3(t).

10 (WS, WR, WR) 865

866

M.N. GOPALAN and S. S. WAGHMARE

Upon taking Laplace transforms, we get

Av °. (s) = 1/(itu + its + s) + (ituAv ~"(s) + its Av2* (s))/(itv + its + s) Avl'(s) = h*O.u + its + s)Av~'(s) + H*(itu + s) + (h* (its + s) - h*(it u + its + s))Av s° (s) + (h*(s)-h*(its + s ) - h * ( i t u +s) + h*(itu +its +s))AvI4*(S)+ (h*(itu + s ) -- h*(itu + its + s)Av H*(s) A v2"(s) = / - / * (s + itu) + h*(itv + s) Ao7* (s) + (h*(s) -- h*(it v + s))Av ~2.(s)

AvS" (s) = OS(its + s)AvS" (s) + (05 (s)-o5 (its +s))Av~'(s) AVI4*(s) = g~(s)AvS'(s) AV11. (s) = d~.](itV + S) + g'~(itU + S)Av3. (S)

+ (9~(s)-g~(itv +s))Av~'(s) AvI2° (S) = Ot(s)AvS* (s) AoT" (s) = d~(itU + S) + O~](s)Av°* (S)

+ (9~(s)-o~(itu +s))Av~'(s) AvS" (s) = CS(itv + s) + oS (itv +its + s)Av°" (s) + (9 5 (its + s) -- 05 (its + itu + s))Ao 3*(s) + (95 (itu +S)--95 (its +itU +s))AvT'(s)

+ (95 (s)-oS(its +s)-oS(;.v +s)

Avg 1"(s) = h*(itv + its + s)Aonv3" (s) + (h*(its + s ) - h*(itv + its + s))Avf s" (s) + (h* (it v + s) - h*(itv + its +s))Av~n'(s) + (h*(s)- h*(it u + s) - h* (its + s)

+ h*(itv + its + s))Av§ 14"(s) Avnu2. (s) = (h*(s)- h*(ito + s))Avnu12. (s) + h*(it v +s)Ao~I7°(s) AD B3*(s) ~- Cr~(S)+ ~ (its+ itU + s)Ao~O*(-S)

+ (9~ (itS + S) -- g~ (itS + )tO + s))Av~ 3. (s)

+ (05 (itv + s ) - gS(its + itv + s)avf]" (s) + (95 (s)-g5 (its +s) - 0 5 (ito +s) + g5 (its +itv + s))Av~'(s) Avff" (s) = g~(itv + s)Avff" (s) -I- (9~(3) -- 0~(it U -{-S)) AI)B3. (S)

AV~Is° (S) = G5 (s) + 05 (its + s) AvB3* (s)

+ (05 ( s ) - o5 (its + s))Ao~ ~2"(s) Av~ 1~"(s) = (9t(s)- O~(itv + s))'avg8. (s) + O~(itv + s)Av~ s.(s) AV~12"(s) ----gf(s)Av~ 3. (s) AV~114°(s) = at (s)Av~fl* (s). The Laplace transform of expected busy period in (0, t) of the server due to repair of the u n i t / ~ " (s) and hence pnu (t) can be obtained as explained in Section 2.

+05 (itS + itU + s) )AoI2* (S). This system of equations can be solved for Av°'(s) in terms of Laplace transforms of known functions. The Laplace transform of expected up-time in (0, t) is #*(s) = Av°'(s)/s which, upon inversion, yields #(t).

5. EXPECTED BUSY PERIOD IN (0, t) OF THE SERVER

DUE TO REPAIR OF THE SWITCH The server is busy due to repair of the switch when the system is in one of the states {7, 11,12,14}.

A v~ (t) = itv A*Avg I (t) + itsA*Avf 2(t) 4. EXPECTED BUSY PERIOD IN (0,t) OF THE SERVER DUE TO REPAIR OF A UNIT

The server is busy with repair of a unit when the system is in one of the states {3, 8, 9, 13}. We have

Av~°(t) = itvA*Av~ ! (t)+ itsA*Av~2(t) Av~'(t) = BhAvv , 83 (t)+ChAvu * ns (t) + E'Avail(t) + L*Avb'(t) Avg 2(t) = MhAvu , n7 ( t ) + N h, A v sl2 u (t) Av~s(t) _- Gv - (t) + Boy * Avvno(t) + Co, * ,Avvns (t) , B7 * BI~ + EguAvv (t)+12gvAv e (t) Av~8 (t) -_ ~o(t) + M S Av~3(t) + NS Av~t2(t ) Av~'4(t) = g~Av~a(t) Av~ H (t) = M~Av~ 3(t)+ N~Av~ 8(t) Av~ 12(t) = g~Av~ ~(t) Ao~7(t) = M~Avnv°(t)+ N~Av~3(t). Upon taking Laplace transforms, we get

Av BO*(S) = (itoAv~1. (s) + itsAv~l 2° (S))/(it u + its + S)

Av~ 1(t) = B h*Avms(t) + C'AviS(t) + b"* A.BII t.~-t*AvOl4t~ ~hzavS U'I"r" h S ~t] Av~2(t)= MhAvs , ~7( t ) + N h*A v sBt2 (t) Av~ g (t) = M 5 Ao~ 3 (t) + N5 AVBs12(t) Av~ 3 (t) = 8~v AvBs0 (t) + ~gO f'* Av SB3ILl t.,

+ E*v Arts 7(t) + Z~u Avg 12(t) Av~14(t) = Gs(t)+g~Av~S(t) Av~ l' (t) = Gs(t) + M~Av~S (t) + N~ Av~S (t) Av~ 12(t) = Gs (t) + o~Av~ s (t) Av~ 7 (t) = Gs (t) + M~Avsn° (t) + N'~Av~ s (t).

Upon taking Laplace transforms, we get Av~ °. (s) = (itu Av~l* (t ) + its Av~ 2"(s) )/(itu + its + s) A v f 14° (S) = ( ~ ( S ) +g~(s)Av~ 8. (S)

Av~s12. (s) = d~(s) + 9~(s)Av~ 3. (s), etc. The expected busy period in (0, t) of the server due to repair of the switch /~sB* (s) and hence psn(t) can be obtained as explained in Section 2.

Reliability analysis 6. COST-BENEFITANALYSIS

We use the above results to carry out cost-benefit analysis. Expected net gain in (0, t)

= Clau (t)- Cu#~ (t)- Cs#f (t)

867

Av~2"(s) = Avff" (s) Av~3* (s) = G- ~ ( s ) + 0 ~ ( A s + , t v +s)AvvBO*(s) + (g~ (As + s ) - g ~ ( A v + As +'s))Av~j3. (S)

+ (g~ (Au + s,)- g~ (Au + ~s + s))Av~7"(s) + (g~ (s)-o~ (As + s)-gMAu + s) + g~ O-u +As + 5))Av~12" (s)

where C revenue per unit up-time of the system Cu cost per unit repair time of a unit Cs cost per unit repair time of the switch.

AV~12. (S) ---- g~(s)Av~3*(s)

Av~z7. (s) = g~(Au + 5)AvuBO*(5) + (g~(S) -g~(Au +5))Av~j3. (S).

7. PARTICULARCASE We now consider the case when the server is instantaneously available as soon as a failure takes place. We then have h*(s) = 1. Therefore from Section 3, we get

Av °" (s) = (AuAv 1"(s) + AsAv 2"(s))/(A v + AS + 5) + 1/(2u + s ) AO 1"(s) = AV 3. (S) AO2*(5) = AoT*(5)

Av v° (s) = 1/(Av + s) + g~l (s)Av °* (s)

+ (O~(s)-o~(s+Av))Av~'(s) Av 3° (S) = I/(Au + s) + g~ (Au + As + s)Av °* (s) + (.q~ (AS + 5) -- g~ (AS + A V + 5))Av3* (s)

+ (g~ (Av + s)-o~ (As + Av + s))Ad'(s) + (0~ (s)- o~ (Av + s)-o~ (As + s) -t-

The system of equations is exactly the same as the one derived in El] if we observe that state 12 here is state 5 of the model discussed in I-l]. Substituting h*(s) = 1 in S e c t i o n 5, we get

Avf O*(s) = (Av Av BI" (s) + AsAvg 2" (s) )/(A v + AS + 5) AV at* (5) = A v f 3. (5) A v f 2. (s) -- Av~7*(s) Av~ 3. (5) = g~ (~v + As + s)AvsBO*(s) + (g~ (As + S)-- g~ (As + A v + s))Avf 3" (s)

+ (g~(~v + s ) - g~ (~s'+ Au + s))Av~ 7. (s)

+ (g~ (s)-o~ (As + s)-o~(Av + s) + g~ (~s + ~v + s))Av~ ~2"(s) A v f 7. (s) = G~(5) "F (g~(5) -- o~(Au .4-5))Ao B3*(s)

+ o~(Av + s)avf °"(s) Av~ll2* (5) --_ ~'r~(s) + o~(5)Avf3* (5)

0~ (At/-{-AS-I- s))Av 12.(S)

Av 12"(s) = O~(s)Av s"(s) which is same as previously obtained results. F r o m Section 4 with h*(s) = I we get

AVuBO"(5) = (auAo~/l* (s) + AsAv~l2. (s))/(Au + As + 5) AV~ 1" (s) = AV~3" (s)

which agree with the equations derived in [1]. REFERENCE 1. M . N . Gopalan and S. S. Waghmare, Cost-benefit analysis

of one-server two-unit system with imperfect switch, Microelectron. Reliab. 25, 643-644.