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Cost-benefit analysis of one-server two-unit replacement system with imperfect switch
Microelectron.Reliab.,Vol.25, No. 5, pp. 877-879,1985. Printed in Great Britain.
0026-2714/8553.00+ .00 © 1985PergamonPressLtd.
COST-BENEFIT ANALYSIS OF ONE-SERVER TWO-UNIT REPLACEMENT SYSTEM WITH IMPERFECT SWITCH M. N. GOPALAN and S. S. W A G H M A R E Department of Mathematics, Indian Instituteof Technology, Powai, Bombay 400 076,India
(Received for publication 11 March 1985) Abstract--This paper deals with the cost-benefit analysis of a one-server two-unit system with imperfect switch where the unit, if available, is preferred to the standby. The repair time of an item (unit/standby/switch is arbitrarily distributed while failure rate of an item is constant. Initially, the unit is switched on (switch is working at t -- 0). The repair of the switch is given preference after the current repair of the unit or standby is over. Explicit expressions for the expected up-time due to the unit, expected up-time due to the standby; expected busy period of the server due to the unit, due to the standby and that due to the switch are obtained to carry out the cost-benefit analysis. INTRODUCTION
8 9 l0 11 12 AvJ(t)
W e c o n s i d e r a o n e - s e r v e r t w o - u n i t s y s t e m w i t h imperfect switch w h e r e it m a y n o t b e p o s s i b l e / d e s i r a b l e to h a v e a s t a n d b y wffich is identical w i t h t h e basic unit. S u c h a s i t u a t i o n will arise w h e n basic u n i t is difficult to p r o c u r e or its shelf-life failure r a t e is very high. T h e n if a n o t h e r type o f u n i t (serving t h e s a m e p u r p o s e as t h e basic u n i t ) h a v i n g z e r o or negligible shelf-life failure r a t e if available, will h a v e to be used as a s t a n d b y . H o w e v e r , its h i g h cost of r e p a i r or h i g h o p e r a t i o n a l c h a r g e s will f o r b i d its use as a basic unit. So use of t h e s t a n d b y is a d v i s a b l e o n l y w h e n it is a b s o l u t e l y essential. T h e p u r p o s e of t h e p r e s e n t p a p e r is t o c a r r y o u t t h e c o s t - b e n e f i t analysis in s u c h cases.
(G, W, UR) (WR, UR, W) (W, UR, WR) (G, WR, UR) (WR, UR, WR)
P(U~(t) = l lX(0) = j ) , i = U , C ; j = 0 ..... 12 Av~i(t) P(K,(t) = I IX(O) =j), i *- U, C, S;j = 0 ..... 12 #~(t)
expected number of repair completions of item i in
#v(t )
~£Aoo~(U)du
E0. t2
#c(t )
2 AvC(u) du
/~s(t)
iAv~(u)du
A~(u) e -('~,+~ls)u, i = 0, U NOTATIONS
X(t)
state of the system at instant t = U if the item under consideration is unit = C if the item under consideration is standby = S if the item under consideration is switch 2~ failure rate of the item i, i = U, C = 0, S g~(. ) repair density o f t h e i t e m i, i = U, C, S i
U~(. )
= 1, if the system is up due to the item i, i = U, C
K~(. )
= 0, otherwise = 1, if the server is busy due to repair of the item i,
i = U,C,S = 0, otherwise W "working" state of the item i, i = U, C, S C standby W R "waits for repair" state of the item i, i = U, C, S UR "under repair" state of the item i, i = U, C, S G "repaired but cannot be switched on" state of U ( ..... ) status of the basic unit, the standby and the switch respectively
0 1 2 3 4
gl(u) (1 - e-~J")e -'ku, k,j, i = O, U, S Bl.~.k(u) gl(u )e-¢~'.~~')"; i,A k = O, U, S Es.,(u) gs(U)e-~,"; i = U,O Is. ,(u) gs(U)(1 -e-~'"), i = U,O
DI.j,k(u)
Io
Lkj(U)
O~(u)
[ 2 s e - ~ 4 ( l - e - ~ f ) + 2f-~J~(1-e-~f)]dv,
.,.(.)
i,j = U,O convolution of two functions.
E X P E C T E D UP-TIME IN [0, t] DUE TO THE BASIC UNIT
The system is up when it is in one of the states {0, 2, 6,10}. AvVo(t) = A ( t ) + 2uA~AvV~ (t)+ 2sA~AvV(t)
av~(t) = Bv, * o.sAvo(t)+Du, v , o,sAv6(t) u + D~, s, o Avu(t) + I_~, o Av~I (t)
Av~(t) = e-autC, s(t) + E*. vAv~(t) + It. vAv~(t)
(W,C,W) (UR, W,W) (w, c, UR) (UR, WR, W) (UR, W, WR)
Av~(t) = e-~utGo(t) + B* v.sAv~(t) + D*.u.sAvt~(t) + Dg.s. uAvV2(t) + L*o.vAv~(t) AvVlt(t) = gfAv~(t)
5 (WR, C, UR)
Avt~(t) = E~, o AvVo(t) + I~, o Av~(t)
6 (W, UR, W)
Avt~(t) = g'~ Av~(t).
7 (UR,WR, WR) 877
878
M.N. GOPALANand S. S. WAGHMARE
The Laplace transform (LT) of ltv(t) is given by ltU'(s) = Av~*(s)/s where Av~*(s) is obtained by solving the above system of equations by Laplace transform technique (LTT).
AvBsC(t) = E~.oAvSoc(t)+ I'~.oAvfC(t) AvB6C(t) = Go(t) + B*, v,sAv~C(t) + D*, u, sAvBxC(t) + D*. s, v Av~C(t) + L*o,v Av~C(t) Av~C(t) = g~Av~C(t)
EXPECTED UP-TIME IN [0. t] DUE TO STANDBY
The system is up due to standby when it is in one of the states {1,4, 8}.
Ave(t) = 2vA3AvC(t) + 2sA3AvC(t)
Av~C(t) = g~Av~C(t). The LT of pBc(t) is given by #BC*(s)= Av~C'(s)/s where Av~C'(s) is obtained from the above system of equations by LTT.
ArC(t) = e-~°'Gv(t) + B~. o.sAvC(t) + D3. o.sAvC(t) + D3.s.oAvC(t) + L*v.oAvC, (t) Ave(t) = Es.vAvo(t)+1s.vAv~(t) , c , c ArC(t) = Bo.v.sAvo(t)+Do.v.sAv~(t)+Do.s.uAv2(t) , c , c , c +L'eAves(t) AvC,(t) = ~'~AvC(t)
EXPECTED BUSY PERIOD IN [0, t] DUE TO REPAIR OF THE SWITCH
The server is busy due to repair of the switch if the system is in one of the states {2, 5, 8, 12}.
Av~S(t) = 2vAuAvl , Bs ( t ) + 2 s A v, A v Bs 2 (t) Avats(t) = Bv.o.sAv , , Bs oss (t)+Dv.o.sAv 6 (t)
ArC(t) = e-)'o' Gs(t) + E~. oAvC(t) + I~. oAvC(t) ArC(t) = g~AvC(t). The LT of #c(t) is given by I~C'(s) = AvC'(s)/s where AvCo'(s) is obtained by solving the above system of equations by LTT.
* kS * BS + Dv,s.oAvs (t)+ Rv, sAvll(t)
AvfS(t) = Gs(t) + E*, uAvgS(t) + I~, uAvfS(t) AyeS(t) = Bo.v.sAv * Bs , as o (t)+Do.v.sAv I (t) * * BS + Do.s, vAv BS 2 (t)+12o.sAv s (t)
Av~S(t) = Gs(t)+g~Av~S(t) EXPECTED BUSY PERIOD IN [0, t] DUE TO REPAIR OF A UNIT
The server is busy due to repair of a unit when the system is in one of the states { I, 3, 4, 7, 9}.
Aver(t) = 2vA~Av~V(t) + 2sA* Av~V(t)
Av~S(t) = Gs(t) + Es.oAv , oas (t)+ I*s.o Av as 6 (t) AviS(t) = Gs(t) + g'AviS(t). The LT of #Bs(t) is given by ItBS*(s)= AvaS'(s)/s where AvSS'(s) is obtained from the above system of equations by LTT.
Av~V(t) = Cru(t) + B~.o.sAv~oV(t) + D~. o.sAv~V(t) , BU , BU + Dv.s.oAvs (t)+13v.oAvll (t)
Av~U(t) = e~. vAv~U(t) + I*. vAvt~v(t) Av~V(t) = Bo.v.sAv , , B1u (t) o8u (t)+Do.v.sAv * BU * BU +Do.s.vAv 2 (t)+l?,o.vAv s (t)
Av~V~(t) = 9~av~V(t) AvfU(t) = E*. oAv~U(t) + l~.oAvfU(t) Av~V(t)
=
g'Aver(t).
The LT of izsv(t) is given by #~V*(s)= AvBoV*(s)/s where Av~V'(s) is obtained from the above system of equations by LTT. EXPECTED BUSY PERIOD IN [0, t] DUE TO REPAIR OF THE STANDBY
The server is busy due to repair of the standby when the system is in one of the states {6, 10, 11, 13}.
Av~C(t) = 2vA~Av~C(t) + 2sA~Av~C(t) Av~c(t) = Bv.o.sAvo , Bc(t)+Dv.o.sAv , ac ~ (t) , BC , BC + Du.s.oAvs (t)+ I?,v.oAvll(t) Av~C(t) = Es.vAv , Bc o (t)+ls., vAv ac 1 (t)
EXPECTED NUMBER OF REPAIR COMPLETIONS 1N [0. t] OF THE BASIC UNIT
We have , u l (t)+2sAvlz , su2 (t) ~.vAvpR #vl(t ) = G v ( u ) + A v, # suo (t)+Dv.o.s#R , 06 (t) /~o(t ) =
* U8 * U12 +Dv.s.o# R (t)+I3v.o# R (t) /.t~2(t) = Es., vll R VO(t)+ Is.v# , Vl a (t)
/~V6(t) = Bo.u.sltR * V0(t)+Do.v.s#s * Vl (t)+Do.s.v#a * U2(t)
+ L*o.v#[5(t)
#Rv8(t)
v6 = E s, . v # avo ( t ) + I s ,. o # R (t)
#~s(t) = g~#~'(t).
EXPECTED NUMBER OF REPAIR COMPLETIONS IN [0, (] OF THE STANDBY
~co(t ) = 2uAo# , cl a (t)+ 2 s A v, # c2 R (t) #el(t) = Bu.o.slZ , co , , ca ~ (t)+ Du.o.s# c6 R (t)+ Dv.s.o# R (t) + L*v,o# c' '(t)
Reliability analysis # c 2 ( t ) = Es, , ugRco (t)+ls,, vla~ cl (t) #cs(t )
,
co
,
COST-BENEFIT ANALYSIS
c6
= Es.o# R (t)+ls.o# ~ (t)
/zc6(t) = Go(t)+ B,.s.ulaCO(t) + Do.u.sl~ R , cl (t) ,
C2
,
C5
+ Do.s, ulaR (t)+/.3o.v/.t R (t) ~s(t)
= g*~'(t)
~.~'(t)
= g*~(t).
_
+L*
Cv, Cc: revenue per unit up-time due to unit and standby respectively
, s6 , ss R (t)+ Du, o.sla R (t)+ Dv, o.sla R (t)
so
,~(t) ~6(t)
, , S1 , = Bo.u.sl~ ~SO (t)+ Do.v.s# a (t)+ Do.s.vla ~S2 (t)
=
+ L*o,uljS'(t) Gs(t)+g~.ljs6(t)
i~ss(t) =
Gu(t) + E~',o~[°(t) + I~, o,S~(t)
#ss(t) =
Gs(t)+g*ljSl(t).
M.R. 2515----E
Cav, C Be, CBs: cost per unit repair time of the unit, standby and switch respectively.
If the server is paid according to the number of repairs completed, the net expected gain = Cu~'(t) + Cc#C(t)- C ~ , ~ ° ( t ) - C ~ ° ( t )
o/zsH (t)
's~tt~ = G u~q~+"*"s°tt~+~* J ~sPR t J ~sPR ~i
~s"(t)
= Cv#V(t) + Cc#C(t)- ~sv#Bv(t)- CSClx~c(t) --'CBSl~'S(t ) (1)
/~so(t) _- ~vAv#~ , st (t)+2sAu#R , s2 (t) , -Bv.o.s#
If the server is paid according to time spent in repair, the net expected gain
where
EXPECTED NUMBER O F REPAIR COMPLETIONS IN [0, t] O F THE SWITCH
i~s,(t)
879
-cs ~°(t)
(2)
where CRv, C c, C s: cost per repair completion of unit, standby and switch respectively. For any given system, (1) and (2) can be evaluated and compared to decide which policy is preferable, viz. to pay according to busy period or to pay according to number of repair completions.
0026-2714/8553.00+ .00 © 1985PergamonPressLtd.
COST-BENEFIT ANALYSIS OF ONE-SERVER TWO-UNIT REPLACEMENT SYSTEM WITH IMPERFECT SWITCH M. N. GOPALAN and S. S. W A G H M A R E Department of Mathematics, Indian Instituteof Technology, Powai, Bombay 400 076,India
(Received for publication 11 March 1985) Abstract--This paper deals with the cost-benefit analysis of a one-server two-unit system with imperfect switch where the unit, if available, is preferred to the standby. The repair time of an item (unit/standby/switch is arbitrarily distributed while failure rate of an item is constant. Initially, the unit is switched on (switch is working at t -- 0). The repair of the switch is given preference after the current repair of the unit or standby is over. Explicit expressions for the expected up-time due to the unit, expected up-time due to the standby; expected busy period of the server due to the unit, due to the standby and that due to the switch are obtained to carry out the cost-benefit analysis. INTRODUCTION
8 9 l0 11 12 AvJ(t)
W e c o n s i d e r a o n e - s e r v e r t w o - u n i t s y s t e m w i t h imperfect switch w h e r e it m a y n o t b e p o s s i b l e / d e s i r a b l e to h a v e a s t a n d b y wffich is identical w i t h t h e basic unit. S u c h a s i t u a t i o n will arise w h e n basic u n i t is difficult to p r o c u r e or its shelf-life failure r a t e is very high. T h e n if a n o t h e r type o f u n i t (serving t h e s a m e p u r p o s e as t h e basic u n i t ) h a v i n g z e r o or negligible shelf-life failure r a t e if available, will h a v e to be used as a s t a n d b y . H o w e v e r , its h i g h cost of r e p a i r or h i g h o p e r a t i o n a l c h a r g e s will f o r b i d its use as a basic unit. So use of t h e s t a n d b y is a d v i s a b l e o n l y w h e n it is a b s o l u t e l y essential. T h e p u r p o s e of t h e p r e s e n t p a p e r is t o c a r r y o u t t h e c o s t - b e n e f i t analysis in s u c h cases.
(G, W, UR) (WR, UR, W) (W, UR, WR) (G, WR, UR) (WR, UR, WR)
P(U~(t) = l lX(0) = j ) , i = U , C ; j = 0 ..... 12 Av~i(t) P(K,(t) = I IX(O) =j), i *- U, C, S;j = 0 ..... 12 #~(t)
expected number of repair completions of item i in
#v(t )
~£Aoo~(U)du
E0. t2
#c(t )
2 AvC(u) du
/~s(t)
iAv~(u)du
A~(u) e -('~,+~ls)u, i = 0, U NOTATIONS
X(t)
state of the system at instant t = U if the item under consideration is unit = C if the item under consideration is standby = S if the item under consideration is switch 2~ failure rate of the item i, i = U, C = 0, S g~(. ) repair density o f t h e i t e m i, i = U, C, S i
U~(. )
= 1, if the system is up due to the item i, i = U, C
K~(. )
= 0, otherwise = 1, if the server is busy due to repair of the item i,
i = U,C,S = 0, otherwise W "working" state of the item i, i = U, C, S C standby W R "waits for repair" state of the item i, i = U, C, S UR "under repair" state of the item i, i = U, C, S G "repaired but cannot be switched on" state of U ( ..... ) status of the basic unit, the standby and the switch respectively
0 1 2 3 4
gl(u) (1 - e-~J")e -'ku, k,j, i = O, U, S Bl.~.k(u) gl(u )e-¢~'.~~')"; i,A k = O, U, S Es.,(u) gs(U)e-~,"; i = U,O Is. ,(u) gs(U)(1 -e-~'"), i = U,O
DI.j,k(u)
Io
Lkj(U)
O~(u)
[ 2 s e - ~ 4 ( l - e - ~ f ) + 2f-~J~(1-e-~f)]dv,
.,.(.)
i,j = U,O convolution of two functions.
E X P E C T E D UP-TIME IN [0, t] DUE TO THE BASIC UNIT
The system is up when it is in one of the states {0, 2, 6,10}. AvVo(t) = A ( t ) + 2uA~AvV~ (t)+ 2sA~AvV(t)
av~(t) = Bv, * o.sAvo(t)+Du, v , o,sAv6(t) u + D~, s, o Avu(t) + I_~, o Av~I (t)
Av~(t) = e-autC, s(t) + E*. vAv~(t) + It. vAv~(t)
(W,C,W) (UR, W,W) (w, c, UR) (UR, WR, W) (UR, W, WR)
Av~(t) = e-~utGo(t) + B* v.sAv~(t) + D*.u.sAvt~(t) + Dg.s. uAvV2(t) + L*o.vAv~(t) AvVlt(t) = gfAv~(t)
5 (WR, C, UR)
Avt~(t) = E~, o AvVo(t) + I~, o Av~(t)
6 (W, UR, W)
Avt~(t) = g'~ Av~(t).
7 (UR,WR, WR) 877
878
M.N. GOPALANand S. S. WAGHMARE
The Laplace transform (LT) of ltv(t) is given by ltU'(s) = Av~*(s)/s where Av~*(s) is obtained by solving the above system of equations by Laplace transform technique (LTT).
AvBsC(t) = E~.oAvSoc(t)+ I'~.oAvfC(t) AvB6C(t) = Go(t) + B*, v,sAv~C(t) + D*, u, sAvBxC(t) + D*. s, v Av~C(t) + L*o,v Av~C(t) Av~C(t) = g~Av~C(t)
EXPECTED UP-TIME IN [0. t] DUE TO STANDBY
The system is up due to standby when it is in one of the states {1,4, 8}.
Ave(t) = 2vA3AvC(t) + 2sA3AvC(t)
Av~C(t) = g~Av~C(t). The LT of pBc(t) is given by #BC*(s)= Av~C'(s)/s where Av~C'(s) is obtained from the above system of equations by LTT.
ArC(t) = e-~°'Gv(t) + B~. o.sAvC(t) + D3. o.sAvC(t) + D3.s.oAvC(t) + L*v.oAvC, (t) Ave(t) = Es.vAvo(t)+1s.vAv~(t) , c , c ArC(t) = Bo.v.sAvo(t)+Do.v.sAv~(t)+Do.s.uAv2(t) , c , c , c +L'eAves(t) AvC,(t) = ~'~AvC(t)
EXPECTED BUSY PERIOD IN [0, t] DUE TO REPAIR OF THE SWITCH
The server is busy due to repair of the switch if the system is in one of the states {2, 5, 8, 12}.
Av~S(t) = 2vAuAvl , Bs ( t ) + 2 s A v, A v Bs 2 (t) Avats(t) = Bv.o.sAv , , Bs oss (t)+Dv.o.sAv 6 (t)
ArC(t) = e-)'o' Gs(t) + E~. oAvC(t) + I~. oAvC(t) ArC(t) = g~AvC(t). The LT of #c(t) is given by I~C'(s) = AvC'(s)/s where AvCo'(s) is obtained by solving the above system of equations by LTT.
* kS * BS + Dv,s.oAvs (t)+ Rv, sAvll(t)
AvfS(t) = Gs(t) + E*, uAvgS(t) + I~, uAvfS(t) AyeS(t) = Bo.v.sAv * Bs , as o (t)+Do.v.sAv I (t) * * BS + Do.s, vAv BS 2 (t)+12o.sAv s (t)
Av~S(t) = Gs(t)+g~Av~S(t) EXPECTED BUSY PERIOD IN [0, t] DUE TO REPAIR OF A UNIT
The server is busy due to repair of a unit when the system is in one of the states { I, 3, 4, 7, 9}.
Aver(t) = 2vA~Av~V(t) + 2sA* Av~V(t)
Av~S(t) = Gs(t) + Es.oAv , oas (t)+ I*s.o Av as 6 (t) AviS(t) = Gs(t) + g'AviS(t). The LT of #Bs(t) is given by ItBS*(s)= AvaS'(s)/s where AvSS'(s) is obtained from the above system of equations by LTT.
Av~V(t) = Cru(t) + B~.o.sAv~oV(t) + D~. o.sAv~V(t) , BU , BU + Dv.s.oAvs (t)+13v.oAvll (t)
Av~U(t) = e~. vAv~U(t) + I*. vAvt~v(t) Av~V(t) = Bo.v.sAv , , B1u (t) o8u (t)+Do.v.sAv * BU * BU +Do.s.vAv 2 (t)+l?,o.vAv s (t)
Av~V~(t) = 9~av~V(t) AvfU(t) = E*. oAv~U(t) + l~.oAvfU(t) Av~V(t)
=
g'Aver(t).
The LT of izsv(t) is given by #~V*(s)= AvBoV*(s)/s where Av~V'(s) is obtained from the above system of equations by LTT. EXPECTED BUSY PERIOD IN [0, t] DUE TO REPAIR OF THE STANDBY
The server is busy due to repair of the standby when the system is in one of the states {6, 10, 11, 13}.
Av~C(t) = 2vA~Av~C(t) + 2sA~Av~C(t) Av~c(t) = Bv.o.sAvo , Bc(t)+Dv.o.sAv , ac ~ (t) , BC , BC + Du.s.oAvs (t)+ I?,v.oAvll(t) Av~C(t) = Es.vAv , Bc o (t)+ls., vAv ac 1 (t)
EXPECTED NUMBER OF REPAIR COMPLETIONS 1N [0. t] OF THE BASIC UNIT
We have , u l (t)+2sAvlz , su2 (t) ~.vAvpR #vl(t ) = G v ( u ) + A v, # suo (t)+Dv.o.s#R , 06 (t) /~o(t ) =
* U8 * U12 +Dv.s.o# R (t)+I3v.o# R (t) /.t~2(t) = Es., vll R VO(t)+ Is.v# , Vl a (t)
/~V6(t) = Bo.u.sltR * V0(t)+Do.v.s#s * Vl (t)+Do.s.v#a * U2(t)
+ L*o.v#[5(t)
#Rv8(t)
v6 = E s, . v # avo ( t ) + I s ,. o # R (t)
#~s(t) = g~#~'(t).
EXPECTED NUMBER OF REPAIR COMPLETIONS IN [0, (] OF THE STANDBY
~co(t ) = 2uAo# , cl a (t)+ 2 s A v, # c2 R (t) #el(t) = Bu.o.slZ , co , , ca ~ (t)+ Du.o.s# c6 R (t)+ Dv.s.o# R (t) + L*v,o# c' '(t)
Reliability analysis # c 2 ( t ) = Es, , ugRco (t)+ls,, vla~ cl (t) #cs(t )
,
co
,
COST-BENEFIT ANALYSIS
c6
= Es.o# R (t)+ls.o# ~ (t)
/zc6(t) = Go(t)+ B,.s.ulaCO(t) + Do.u.sl~ R , cl (t) ,
C2
,
C5
+ Do.s, ulaR (t)+/.3o.v/.t R (t) ~s(t)
= g*~'(t)
~.~'(t)
= g*~(t).
_
+L*
Cv, Cc: revenue per unit up-time due to unit and standby respectively
, s6 , ss R (t)+ Du, o.sla R (t)+ Dv, o.sla R (t)
so
,~(t) ~6(t)
, , S1 , = Bo.u.sl~ ~SO (t)+ Do.v.s# a (t)+ Do.s.vla ~S2 (t)
=
+ L*o,uljS'(t) Gs(t)+g~.ljs6(t)
i~ss(t) =
Gu(t) + E~',o~[°(t) + I~, o,S~(t)
#ss(t) =
Gs(t)+g*ljSl(t).
M.R. 2515----E
Cav, C Be, CBs: cost per unit repair time of the unit, standby and switch respectively.
If the server is paid according to the number of repairs completed, the net expected gain = Cu~'(t) + Cc#C(t)- C ~ , ~ ° ( t ) - C ~ ° ( t )
o/zsH (t)
's~tt~ = G u~q~+"*"s°tt~+~* J ~sPR t J ~sPR ~i
~s"(t)
= Cv#V(t) + Cc#C(t)- ~sv#Bv(t)- CSClx~c(t) --'CBSl~'S(t ) (1)
/~so(t) _- ~vAv#~ , st (t)+2sAu#R , s2 (t) , -Bv.o.s#
If the server is paid according to time spent in repair, the net expected gain
where
EXPECTED NUMBER O F REPAIR COMPLETIONS IN [0, t] O F THE SWITCH
i~s,(t)
879
-cs ~°(t)
(2)
where CRv, C c, C s: cost per repair completion of unit, standby and switch respectively. For any given system, (1) and (2) can be evaluated and compared to decide which policy is preferable, viz. to pay according to busy period or to pay according to number of repair completions.