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Approximations of a two-unit cold standby system's reliability behaviour
Microelectron. Reliah., Vol. 36, No. 5, pp. 627-630, 1996
(~' Pergamon
Copyright ~' 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/96 $15.00+.00
0026-2714(95)00170-0
APPROXIMATIONS OF A TWO-UNIT COLD STANDBY SYSTEM'S RELIABILITY BEHAVIOUR C H E N G K A N and W A N G W E N Y I Institute of Applied Mathematics, Academia Sinica, and APORC, Beijing 100080, China
(Received lbr publication 18 August 1995) Abstract--In this paper we study the reliability behaviour of a two-unit cold standby system with priority. At time t = 0, the priority unit (p) begins to work and the standby unit (s) is in cold standby. The p-unit has priority whether it is working or being repaired. A single repairman is available, and the repaired unit is as good as new. We assume that the p-unit's working time X~ has a general life distribution F(x), and its repair time }I1 has a general distribution G(x). The s-unit's working time X z has an exponential distribution with mean 1/2, and its repair time has an exponential distribution with mean 1//1. Using the Markov renewal process and stochastic comparison, we give bounds of the mean time to the first failure of the system, and the system's availability.
Let
I. I N T R O D U C T I O N
There are many papers on the system's reliability analysis. However, due to the complexity of systems, it is not easy to obtain exact expressions of the system's reliability indices. Usually, only the Laplace (L) or Laplace-Stieltjes ( L - S ) transforms of reliability characteristics can be obtained. These results are difficult to use in engineering and technology practice. in this paper, we use the Markov renewal process and stochastic comparison to discuss reliability indices of a two-unit cold standby system with priority. We assume that the system has two types of units: a priority unit p and a standby unit s. At time t = 0, the p-unit starts to operate, and the s-unit is in cold standby. The p-unit has priority for operation (repair) over the s-unit while it is in normal mode (failed mode). A single repairman is available to repair a failed unit, and the repaired unit is as good as new. We assume that the p-unit's working time Xt has a general life distribution F(x), and its repair time ?'1 has a general distribution G(x). The s-unit's working time Xz has an exponential distribution with mean 1/)., and its repair time has an exponential distribution with mean 1/I~. Using the Markov renewal process and stochastic comparison, we give bounds of the mean time to the first failure of the system, and the system's availability. A practical example that suits our model is that of a room-heating system consisting of two units: blower and simple room heater. The blower may be taken as a priority unit and the room heater as a standby unit. In Section 2, properties of the distribution function of the system's first failure time are discussed. The system's availability is discussed in Section 3. In Section 4, we give other reliability indices of the system. 627
S(t) =
0,
if system is operating at time t
1,
ifsystemis failed at timet
First, we denote N(t) as the system's state at time t. State 0: starts at the epoch when the p-unit begins to work, while the s-unit is in standby. The state lasts until the p-unit fails. State 1: starts at the epoch when the p-unit begins to repair, while the s-unit begins to work. The state lasts until the p-unit is repaired. State 2: starts at the epoch when the p-unit begins to work, while the s-unit begins to repair. The state lasts until the p-unit fails. State 3: starts at the epoch when the p-unit begins to repair, while the repair of the s-unit is interrupted, and waiting begins. The state lasts until the p-unit is repaired. Let Z = {0 = Z o, Z~ . . . . . Z . . . . . ~ be transition epochs of the system's states, and N,, = N(Z,, + 0) the nth sojourning state of the system, n = 0, 1. . . . . Then by assumptions on the model, (N, Z) = {IV,,, Z,,, n = O, 1. . . . } is a time homogeneous Markov renewal process on state space {0, 1, 2, 3} with semi-Markov kernel Oij(t) (see, e.g. [1])
Q,i(t) = P{N,,~, = j , Z , , + , - Z, ~ tiN',, = i}, t>~O,i,j=0,1,2,3
.....
Then we have
Qol(t) = P{X1 ~ tl = F(t) 01o(0 = P{Xz >~ Y~, Y, <~ t} = f l e Z.~ dG(x) Qt2(t)=P{X2<~
Y~,Y~<~tI=
£,
(1 - e-)'-~) dG(x)
)
628
C. Kan and W. Wenyi
Q2.(I) = P { X 1 ~ t, ~ ~ x1} = J¢~ (1 - e-- ~tx) dF(x)
From Lemma 2, after some calculation, we can obtain Lemma 3.
Qz3( t ) = PIXI <~ Y3, x1 ~ t l = f l e
Lemma 3
~'~dF(x)
Q32(t) = P', }I1 <~ t I = G(t).
Mo=
For other i, j, Qii = O. We use F*(s) and fi(s) to denote the corresponding L and L S transforms of F(t), i.e.
M, =
1
+
it F
i-5(;0
l ' v S ( ; . ) + (1/2311 1-
-- (~(2)]
- Mo - Pr
5(2)
1 F(s)=
e-~'dF(t),
s/>0
--
)
;, + i L 8 i ; . ) j [ ' - f ( ' ) ]
= Mo -
F*(s) =
e ~'F(t) dt,
s > O.
(Mo
-
+
~p)P(,).
We rewrite Lemma I as:
Thus corresponding L - S transforms of Q~j(t) are
Ro(t) = H * Ro(t) + Ho(t)
O.o,(S) = P(s)
Ri(t) = H * Ro(t) + Hi(t), where
OIo(S) = 5 ( 2 -F S)
O,2(s) = ~(s) - 5(;. + s)
H(t) = Qol * Qlo(t) =
;/
F(t - x)e - ;~x dG(z)
)
O,~(s) = P(s) - P(l~ + s) Ho(t ) = F(t) + f l e ~'" X~G(t - X) dF(x)
O_~_~(s) = f'(i~ + s) O3~(s) = d(s).
Hdt)= 2. D I S T R I B U T I O N OF SYSTEM'S FIRST F A I L U R E TIME
e x'G(t) +
Noting that H ( ~ ) = G(2) < 1, thus H(t) is a defective distribution. There exists a unique 0 < k < ~, satisfying
In this section, we study properties of the system's first failure time. Let T be the system's first failure time. For i = 0, 1, 2, denote
R~(T) = PIT>>- tIN(O) = i I M, = E(TI
N(O) =
i).
In this paper, we use an asterisk to denote convolution, i.e. A * B(t) is defined as
A
• B(t) = fl B(t
X) dA(x).
o" ekx dH(x) = 1. By renewal theory (see [2]), we have Proposition 1.
Proposition 1 f f e k ' H i ( t ) at limek~Ro(t)= -, ....
Using the theory of the Markov renewal process, we have the following Markov renewal equations.
Lemma 1
e-~'xF(t - x) dG(x).
fo tek'
limP(Ti>~t+x[T~>,t)=
i=0,1.
dH(t)
e k~,
i=0,1,
where T,.is the system's first failure time when N(0) = i.
Ro(t) = Qol * Rl(t) + F(t) Rl(t) =
Qlo *
Ro(t) + e-2t6(t)
R2(t) = Q21 * Rl(t) + F(t).
Definition 1 ([3) Suppose that F(t) is a life distribution means #r < ~ . F is in the L(L) class if
with
Then the L transforms of R~(t) are as follows.
fi(s)= Lemma 2 (l/s)[1 -- f(s)]Qto(S)
R*(s) =
+ [1/(s + ;3][! - 5(s + ;3] 1 - O,o(S)Oo~(S)
R~(s) = R*(s)Qol(S) + (l/s)[1 -- fi(s)] R*(s) = R*(s)Q2 ~(s) + (l/s)[1 - F(s)].
e S~dF(t)~<(/>)__ ,s/>0. 1 + sit r
The last inequality is equivalent to
~
~e
s>~O.
Note that L is a very large distribution class. Actually, F 6 L is only a mild restriction in applications.
Two-unit cold standby system's reliability behaviour
629
Definition 2 ([3, 4])
Proposition 2 (a) For an arbitrary life distribution G, denoting 1
#r
,'1=
+
/.
- - - - ,
1 -
e
f~-e-~'dF,(t)>, f ;
z.,,
we have
ttr
1
M o >1 + ).
1 -- e -
Ft(t), F2(t) are two distribution functions, Fl(t) is said to be smaller than F2(t) in Laplace ordering if e-S'dF2(t),
We denote it as F~(t) ~LFz(t). The above inequality is equivalent to
: =a """
e ~tFl(t) dt ~ 1
1
e-
-
1
M2 )
e "F2(t) dt,
s ~> 0.
o
It F e - " " "
).
s>~O.
)'~'';
We now investigate under which conditions for the two systems the first time to failure and the distributions of working units have the same Laplace order. For system-(rj), suppose that F, is the working distribution of the p-unit, and G~ is the repair distribution. We use R~j to denote corresponding R~ for system-(rj), i = 0, 1, j, r = 1, 2. Then we have Proposition 3.
l'r e-~"" ][1
). + 1--~ e_ ~.~,,j -/O(U)] + gr = a - (a - [tr)Y(/O.
In addition, if F ~ L, then
[l .eT;:_] ,,.
M, /> ) +
+/~r
Proposition 3 (a) If Fl(t) <~LFz(t), then
FIF
= (1 + ap) . . . . . . . .
R~J(t)<~rRZJ(t),
itltv
1 +
i=0,1,
j=
1,2.
(b) If Gl(t) <~LGz(t), then (b) If G E L(L), denoting 1
R~l(t) >>.L R~i2(t), i = O, 1, r = 1,2. Proof. After some calculation, we find that R*°(s) is increasing in (~(s + 2), and decreasing in F(s). By the definition of Laplace ordering, we can obtain the results.
ltt:
b =
4;-
/-
-~- jtlF '
)-go
then Mo ~< (/>)
. z
b
3. AVAILABILITY
/.I16
Denote
1 gr M1 <~(>>-) + =h-l~. z ~.ito
M2 4 ( ~>) [-1 /+ -
I,
i=0,1,2,3.
Then we have following Markov renewal equations of Ai(t):
gV][l -F(10] +Itv
L~.
= b
Ai(t)=P{S(t)=o[m(o)=i
).ll6J
Ao(t) = F(t) + Qol * Al(t)
(b - llr)F(lO.
Al(t) = I - Q,4(t) - Qlo(t) + Q~o * Ao(t) + Q12" Az(t)
In addition, if G ~ L, F ~ L,
II ] , , it F M,>~_ ) + l!F.,.l,~jt +#ltr/,l,~ + I t r = ( l + ' l O ] + l ' g r
.
A2(t) = F(t) + Q21(t) • Al(t) + Q23(t) • A3(t) A3(t) = Q32 * A2(t). The Laplace transforms of Ai(t) are
If G 6 L, F is a general distribution, then A*(s) =
1
(1 - F(s)) + A*(s)O_,o(S)
S
.< ILo. + # r . I. 1
M 2 ~
--
e-j,,,)
+
,tl v, =
b
--
(b
--
Ftr)e -~'~'.
!
A*(s) =
Xll(;
~ [1 - (~(s + ,;.)] + A*(s)Olo(S) S+/.
Proof. Since e- ~' is convex in s, then by Jensen's
+ A*(s)O.~2(s)
inequality
A'~(s) = (~(s) = I " e-~' dG(t) ~> e -~'".
Jo Result (a) is evident. By the definition of the L(L) class and Proposition 2, we can obtain result (b).
1
[1 -- F(s)] + A*(s)Q2,(s) + A*(s)Qz3(s)
s
A*(s) = A*(s)Q32(s) Therefore, by Tauber's theorem [5], the limiting availability is given by Proposition 4.
630
C. Kan and W. Wenyi
Proposition 4
Let J = l i m , ~ l~(t)/t be the limiting failure frequency of a system. Using Tuber's theorem [5] (see, e.g. [2]) we have
A = lim Ai(t) = lira sAi(s) t ~,J
s~O
~,~
=
1
+
/~F + /t~
(l - f(~))O
- d(~))
~ ( ~ +/~G)(1 -- f(~)d(2)) i=0,
1,2,3.
Proposition 6 1
J=
-
6(,~)
(~ + ~)(1 - f(~)d0.))
Similar to Proposition 3, we have Proposition 5. Proposition 5 (a) For arbitrary distribution F(t) and G(t), #r
A~<
+
/it + #c,
1
(1 - e ""f)(l - e -a"~)
A>~
+
(a) For an arbitrary distribution F, if G ~ L
2 (/~F + #~)(1 - e -(uuf+a"~))
(b) If F e L(L), G E L(L), then ]lF
Proposition 7
1
J>>,
2p~ (#v + ~G)(1 + 2pG - e - " " 0
2#[AF# G
(b) For an arbitrary distribution G, if F ~ L 4. E X P E C T E D N U M B E R O F S Y S T E M ' S F A I L U R E I N (0, t]
In this section, we investigate mean number in (0, t]. Denote di(t) as the number of system failures in (0, t] when N(0) = i, i = 0, 1, 2, 3. Let
li(t)
J~<
(1
-
#/IF)(1
--
e -~"<~)
(/~r + / ~ ) ( 1 + ##~ - e -a"~)
The proof is similar to Proposition 3.
= EEJ,(t)].
Using the Markov renewal process method, we have lo(t) = Qol * ll(t) 1t(0 = Q12" (1 + 12(t)) + Q l o * Io(t)
Acknowledgement--This paper was supported by the National Natural Science Foundation of China and AsianPacific Operation Research Centre (APORC).
+ (Ql,,(t) - Ql2(t)) 12(0 = Q21 * It(t) + Q 2 3 . (1 + I3(t)) I3(t) = Qa2 * 12(0The corresponding Laplace-Stieltjes transforms of li(t) are lo(S)
= 001(S)II(S)
it(S) = 0t2(s)i2(s) + 010(S)I0(s) -~- 014(S) iz(S) = OEt(S)/l(S) + ~23(S)i3(S) -{--023(S) I3(S) = 032(S)12(S).
REFERENCES
1. E. Cinlar, Introduction to Stochastic Processes. PrenticeHall, Englewood Cliffs, NJ (1975). 2. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. Wiley, New York (1971). 3. B. Klefsjo, A useful ageing property based on Laplace transforms, J. appl. Prob. 20, 615-626 0983). 4. D. Stoyan, Comparison Models Queues and Other Stochastic Models. Wiley, New York (1983). 5. D.V. Widder, The Laplace Transform. Princeton University Press.
(~' Pergamon
Copyright ~' 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/96 $15.00+.00
0026-2714(95)00170-0
APPROXIMATIONS OF A TWO-UNIT COLD STANDBY SYSTEM'S RELIABILITY BEHAVIOUR C H E N G K A N and W A N G W E N Y I Institute of Applied Mathematics, Academia Sinica, and APORC, Beijing 100080, China
(Received lbr publication 18 August 1995) Abstract--In this paper we study the reliability behaviour of a two-unit cold standby system with priority. At time t = 0, the priority unit (p) begins to work and the standby unit (s) is in cold standby. The p-unit has priority whether it is working or being repaired. A single repairman is available, and the repaired unit is as good as new. We assume that the p-unit's working time X~ has a general life distribution F(x), and its repair time }I1 has a general distribution G(x). The s-unit's working time X z has an exponential distribution with mean 1/2, and its repair time has an exponential distribution with mean 1//1. Using the Markov renewal process and stochastic comparison, we give bounds of the mean time to the first failure of the system, and the system's availability.
Let
I. I N T R O D U C T I O N
There are many papers on the system's reliability analysis. However, due to the complexity of systems, it is not easy to obtain exact expressions of the system's reliability indices. Usually, only the Laplace (L) or Laplace-Stieltjes ( L - S ) transforms of reliability characteristics can be obtained. These results are difficult to use in engineering and technology practice. in this paper, we use the Markov renewal process and stochastic comparison to discuss reliability indices of a two-unit cold standby system with priority. We assume that the system has two types of units: a priority unit p and a standby unit s. At time t = 0, the p-unit starts to operate, and the s-unit is in cold standby. The p-unit has priority for operation (repair) over the s-unit while it is in normal mode (failed mode). A single repairman is available to repair a failed unit, and the repaired unit is as good as new. We assume that the p-unit's working time Xt has a general life distribution F(x), and its repair time ?'1 has a general distribution G(x). The s-unit's working time Xz has an exponential distribution with mean 1/)., and its repair time has an exponential distribution with mean 1/I~. Using the Markov renewal process and stochastic comparison, we give bounds of the mean time to the first failure of the system, and the system's availability. A practical example that suits our model is that of a room-heating system consisting of two units: blower and simple room heater. The blower may be taken as a priority unit and the room heater as a standby unit. In Section 2, properties of the distribution function of the system's first failure time are discussed. The system's availability is discussed in Section 3. In Section 4, we give other reliability indices of the system. 627
S(t) =
0,
if system is operating at time t
1,
ifsystemis failed at timet
First, we denote N(t) as the system's state at time t. State 0: starts at the epoch when the p-unit begins to work, while the s-unit is in standby. The state lasts until the p-unit fails. State 1: starts at the epoch when the p-unit begins to repair, while the s-unit begins to work. The state lasts until the p-unit is repaired. State 2: starts at the epoch when the p-unit begins to work, while the s-unit begins to repair. The state lasts until the p-unit fails. State 3: starts at the epoch when the p-unit begins to repair, while the repair of the s-unit is interrupted, and waiting begins. The state lasts until the p-unit is repaired. Let Z = {0 = Z o, Z~ . . . . . Z . . . . . ~ be transition epochs of the system's states, and N,, = N(Z,, + 0) the nth sojourning state of the system, n = 0, 1. . . . . Then by assumptions on the model, (N, Z) = {IV,,, Z,,, n = O, 1. . . . } is a time homogeneous Markov renewal process on state space {0, 1, 2, 3} with semi-Markov kernel Oij(t) (see, e.g. [1])
Q,i(t) = P{N,,~, = j , Z , , + , - Z, ~ tiN',, = i}, t>~O,i,j=0,1,2,3
.....
Then we have
Qol(t) = P{X1 ~ tl = F(t) 01o(0 = P{Xz >~ Y~, Y, <~ t} = f l e Z.~ dG(x) Qt2(t)=P{X2<~
Y~,Y~<~tI=
£,
(1 - e-)'-~) dG(x)
)
628
C. Kan and W. Wenyi
Q2.(I) = P { X 1 ~ t, ~ ~ x1} = J¢~ (1 - e-- ~tx) dF(x)
From Lemma 2, after some calculation, we can obtain Lemma 3.
Qz3( t ) = PIXI <~ Y3, x1 ~ t l = f l e
Lemma 3
~'~dF(x)
Q32(t) = P', }I1 <~ t I = G(t).
Mo=
For other i, j, Qii = O. We use F*(s) and fi(s) to denote the corresponding L and L S transforms of F(t), i.e.
M, =
1
+
it F
i-5(;0
l ' v S ( ; . ) + (1/2311 1-
-- (~(2)]
- Mo - Pr
5(2)
1 F(s)=
e-~'dF(t),
s/>0
--
)
;, + i L 8 i ; . ) j [ ' - f ( ' ) ]
= Mo -
F*(s) =
e ~'F(t) dt,
s > O.
(Mo
-
+
~p)P(,).
We rewrite Lemma I as:
Thus corresponding L - S transforms of Q~j(t) are
Ro(t) = H * Ro(t) + Ho(t)
O.o,(S) = P(s)
Ri(t) = H * Ro(t) + Hi(t), where
OIo(S) = 5 ( 2 -F S)
O,2(s) = ~(s) - 5(;. + s)
H(t) = Qol * Qlo(t) =
;/
F(t - x)e - ;~x dG(z)
)
O,~(s) = P(s) - P(l~ + s) Ho(t ) = F(t) + f l e ~'" X~G(t - X) dF(x)
O_~_~(s) = f'(i~ + s) O3~(s) = d(s).
Hdt)= 2. D I S T R I B U T I O N OF SYSTEM'S FIRST F A I L U R E TIME
e x'G(t) +
Noting that H ( ~ ) = G(2) < 1, thus H(t) is a defective distribution. There exists a unique 0 < k < ~, satisfying
In this section, we study properties of the system's first failure time. Let T be the system's first failure time. For i = 0, 1, 2, denote
R~(T) = PIT>>- tIN(O) = i I M, = E(TI
N(O) =
i).
In this paper, we use an asterisk to denote convolution, i.e. A * B(t) is defined as
A
• B(t) = fl B(t
X) dA(x).
o" ekx dH(x) = 1. By renewal theory (see [2]), we have Proposition 1.
Proposition 1 f f e k ' H i ( t ) at limek~Ro(t)= -, ....
Using the theory of the Markov renewal process, we have the following Markov renewal equations.
Lemma 1
e-~'xF(t - x) dG(x).
fo tek'
limP(Ti>~t+x[T~>,t)=
i=0,1.
dH(t)
e k~,
i=0,1,
where T,.is the system's first failure time when N(0) = i.
Ro(t) = Qol * Rl(t) + F(t) Rl(t) =
Qlo *
Ro(t) + e-2t6(t)
R2(t) = Q21 * Rl(t) + F(t).
Definition 1 ([3) Suppose that F(t) is a life distribution means #r < ~ . F is in the L(L) class if
with
Then the L transforms of R~(t) are as follows.
fi(s)= Lemma 2 (l/s)[1 -- f(s)]Qto(S)
R*(s) =
+ [1/(s + ;3][! - 5(s + ;3] 1 - O,o(S)Oo~(S)
R~(s) = R*(s)Qol(S) + (l/s)[1 -- fi(s)] R*(s) = R*(s)Q2 ~(s) + (l/s)[1 - F(s)].
e S~dF(t)~<(/>)__ ,s/>0. 1 + sit r
The last inequality is equivalent to
~
~e
s>~O.
Note that L is a very large distribution class. Actually, F 6 L is only a mild restriction in applications.
Two-unit cold standby system's reliability behaviour
629
Definition 2 ([3, 4])
Proposition 2 (a) For an arbitrary life distribution G, denoting 1
#r
,'1=
+
/.
- - - - ,
1 -
e
f~-e-~'dF,(t)>, f ;
z.,,
we have
ttr
1
M o >1 + ).
1 -- e -
Ft(t), F2(t) are two distribution functions, Fl(t) is said to be smaller than F2(t) in Laplace ordering if e-S'dF2(t),
We denote it as F~(t) ~LFz(t). The above inequality is equivalent to
: =a """
e ~tFl(t) dt ~ 1
1
e-
-
1
M2 )
e "F2(t) dt,
s ~> 0.
o
It F e - " " "
).
s>~O.
)'~'';
We now investigate under which conditions for the two systems the first time to failure and the distributions of working units have the same Laplace order. For system-(rj), suppose that F, is the working distribution of the p-unit, and G~ is the repair distribution. We use R~j to denote corresponding R~ for system-(rj), i = 0, 1, j, r = 1, 2. Then we have Proposition 3.
l'r e-~"" ][1
). + 1--~ e_ ~.~,,j -/O(U)] + gr = a - (a - [tr)Y(/O.
In addition, if F ~ L, then
[l .eT;:_] ,,.
M, /> ) +
+/~r
Proposition 3 (a) If Fl(t) <~LFz(t), then
FIF
= (1 + ap) . . . . . . . .
R~J(t)<~rRZJ(t),
itltv
1 +
i=0,1,
j=
1,2.
(b) If Gl(t) <~LGz(t), then (b) If G E L(L), denoting 1
R~l(t) >>.L R~i2(t), i = O, 1, r = 1,2. Proof. After some calculation, we find that R*°(s) is increasing in (~(s + 2), and decreasing in F(s). By the definition of Laplace ordering, we can obtain the results.
ltt:
b =
4;-
/-
-~- jtlF '
)-go
then Mo ~< (/>)
. z
b
3. AVAILABILITY
/.I16
Denote
1 gr M1 <~(>>-) + =h-l~. z ~.ito
M2 4 ( ~>) [-1 /+ -
I,
i=0,1,2,3.
Then we have following Markov renewal equations of Ai(t):
gV][l -F(10] +Itv
L~.
= b
Ai(t)=P{S(t)=o[m(o)=i
).ll6J
Ao(t) = F(t) + Qol * Al(t)
(b - llr)F(lO.
Al(t) = I - Q,4(t) - Qlo(t) + Q~o * Ao(t) + Q12" Az(t)
In addition, if G ~ L, F ~ L,
II ] , , it F M,>~_ ) + l!F.,.l,~jt +#ltr/,l,~ + I t r = ( l + ' l O ] + l ' g r
.
A2(t) = F(t) + Q21(t) • Al(t) + Q23(t) • A3(t) A3(t) = Q32 * A2(t). The Laplace transforms of Ai(t) are
If G 6 L, F is a general distribution, then A*(s) =
1
(1 - F(s)) + A*(s)O_,o(S)
S
.< ILo. + # r . I. 1
M 2 ~
--
e-j,,,)
+
,tl v, =
b
--
(b
--
Ftr)e -~'~'.
!
A*(s) =
Xll(;
~ [1 - (~(s + ,;.)] + A*(s)Olo(S) S+/.
Proof. Since e- ~' is convex in s, then by Jensen's
+ A*(s)O.~2(s)
inequality
A'~(s) = (~(s) = I " e-~' dG(t) ~> e -~'".
Jo Result (a) is evident. By the definition of the L(L) class and Proposition 2, we can obtain result (b).
1
[1 -- F(s)] + A*(s)Q2,(s) + A*(s)Qz3(s)
s
A*(s) = A*(s)Q32(s) Therefore, by Tauber's theorem [5], the limiting availability is given by Proposition 4.
630
C. Kan and W. Wenyi
Proposition 4
Let J = l i m , ~ l~(t)/t be the limiting failure frequency of a system. Using Tuber's theorem [5] (see, e.g. [2]) we have
A = lim Ai(t) = lira sAi(s) t ~,J
s~O
~,~
=
1
+
/~F + /t~
(l - f(~))O
- d(~))
~ ( ~ +/~G)(1 -- f(~)d(2)) i=0,
1,2,3.
Proposition 6 1
J=
-
6(,~)
(~ + ~)(1 - f(~)d0.))
Similar to Proposition 3, we have Proposition 5. Proposition 5 (a) For arbitrary distribution F(t) and G(t), #r
A~<
+
/it + #c,
1
(1 - e ""f)(l - e -a"~)
A>~
+
(a) For an arbitrary distribution F, if G ~ L
2 (/~F + #~)(1 - e -(uuf+a"~))
(b) If F e L(L), G E L(L), then ]lF
Proposition 7
1
J>>,
2p~ (#v + ~G)(1 + 2pG - e - " " 0
2#[AF# G
(b) For an arbitrary distribution G, if F ~ L 4. E X P E C T E D N U M B E R O F S Y S T E M ' S F A I L U R E I N (0, t]
In this section, we investigate mean number in (0, t]. Denote di(t) as the number of system failures in (0, t] when N(0) = i, i = 0, 1, 2, 3. Let
li(t)
J~<
(1
-
#/IF)(1
--
e -~"<~)
(/~r + / ~ ) ( 1 + ##~ - e -a"~)
The proof is similar to Proposition 3.
= EEJ,(t)].
Using the Markov renewal process method, we have lo(t) = Qol * ll(t) 1t(0 = Q12" (1 + 12(t)) + Q l o * Io(t)
Acknowledgement--This paper was supported by the National Natural Science Foundation of China and AsianPacific Operation Research Centre (APORC).
+ (Ql,,(t) - Ql2(t)) 12(0 = Q21 * It(t) + Q 2 3 . (1 + I3(t)) I3(t) = Qa2 * 12(0The corresponding Laplace-Stieltjes transforms of li(t) are lo(S)
= 001(S)II(S)
it(S) = 0t2(s)i2(s) + 010(S)I0(s) -~- 014(S) iz(S) = OEt(S)/l(S) + ~23(S)i3(S) -{--023(S) I3(S) = 032(S)12(S).
REFERENCES
1. E. Cinlar, Introduction to Stochastic Processes. PrenticeHall, Englewood Cliffs, NJ (1975). 2. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. Wiley, New York (1971). 3. B. Klefsjo, A useful ageing property based on Laplace transforms, J. appl. Prob. 20, 615-626 0983). 4. D. Stoyan, Comparison Models Queues and Other Stochastic Models. Wiley, New York (1983). 5. D.V. Widder, The Laplace Transform. Princeton University Press.