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Generalised availability measures for a two-unit intermittently used repairable system
Microelectron. Reliab., Vol. 24, No. 1, pp. 29-34, 1984.
0026-2714/8453.00 + .00 © 1984 Pergamon Press Ltd.
Printed in Great Britain.
GENERALISED AVAILABILITY MEASURES FOR A TWO-UNIT INTERMITTENTLY USED REPAIRABLE SYSTEM P. K. K~d'UR Department of Operational Research, University of Delhi, India N. L. BUTANI C.H.M. College, Ulhasnagar-3, Thana, Maharastra, India and DEBASIS DUTTA Department of Operational Research, University of Delhi, Delhi-110007, India
(Received for publication 2 August 1983) Abstract--This paper obtains generalised availability measures as defined by Baxter, J. appl. Prob. 18, 227-235 (1981), for a two-unit cold stand-by intermittently used redundant repairable system. Regenerative point technique of Markov Renewal Process has been employed to obtain these measures. A particular case is also considered.
INTRODUCTION
G(t) cdf of the repair time of the system with
In a recent contribution to this journal, K a p u r et al. [5] obtained Generalised Availability Measures for a simple (one unit) intermittently used repairable system generalising the earlier results due to B a x t e r [ l ] . Availability analysis of such a system has earlier been done by K a p u r [-2, 3]. The purpose of this paper is to obtain the Generalised Availability Measures for a Two-Unit Intermittently Used Redundant Repairable System. Extension is non-trivial in nature. It unifies an earlier treatment [4, 5] to a study of this kind.
A(t) B(t) V(t) U(t) *
pdfg(t) ( = l - e -~t) cdf of occurrence time of need (= 1 - e -pt) cdf of holding time of need Pr{need occurs at time t/it occurred at time O} Pr{need does not occur at time t/it did not occur at time O} convolution, e.g.
A(t)*B(t) = f l A(p)B(t-#)du and
A(t)*B(x + t) = .t~ A(p)B(t + x - I~)du
ASSUMPTIONS
complement, e.g. (1) System has two identical units. (2) A standby unit never fails. (3) Once a unit is repaired, it is like new and goes into standby state or operating state. (4) All the switchings operations are 100 ~o reliable, instantaneous and never damage anything. (5) The cdf's of"failure time" and "repair time" are both arbitrary. (6) The cdf's of occurrence time and holding time of a need are exponential. (7) The system is not needed at time O. (8) All the random variables are mutually independent.
[1 -A(t)] *t-l) LT
F(t) = 1 -F(t) 1 + A(t)A(t) * A(t) +... Laplace Transformation, e.g.
f(s) = LT If(t)] =
e-Stf(t)dt
Qij(t) cdf of MRP (Markov Renewal Process) from state i into state j, i.e. probability that after transiting into state i, the process next transits into state j in time ~
NOTATIONS
F(t) cumulative distribution function (cdf) of the failure time of the system with probability density function (pdf)f(t) 29
et al.
30
P.K. KAPUR
Rj(x,t)
Pr{system is operating at time t and continues to operate without failing in starts from state j} j = - 1, 1, 2 = interval reliability {joint availability at t and starts from statej at time t = O} ..... t.] {joint availability at n distinct points h, t2,.--, t/system starts from statej at time t = 0} R}"~[(x~,t~), (x~,t~)..... (xo, to)] {joint interval reliability at n disjoint intervals +x2], It,, t, + x J/system starts from state j at time t = 0}.
Ql,(t)
=
l d(x)O(x) dF(x)
[t,t+x]/it
Pj(x,t) P}"~[tl,t2
t+x/system
[tt,h
DERIVATION
OF
(22o(t)
ir(x) dG(x)
=
Q23(t)
=
Qv,(t)
=
Q),~(t)
=
Q~(t)
=
l G(x)r/(x)dF(x) f'oG(X)V(x)dF(x)
[te,t2+x2]...
V(t) A N D U(t)
V(t+At) = V ( t ) ( 1 - f l A t ) + ~(t)~At, V(t) = [c~+fle I~+~!,]/[=+fl].
At--, 0
Similarly,
l G(x)U(x)dF(x) l G(x)O(x)dF(x)
Q~(t) =
l G(x)V(x)dF(x)
Q~(t) =
l G(x)V(x)dF(x)
V~(t) = fl[1-e-(~+~)t]/[m+fl]
and
U(t) = [fl + ~ e - (~ +~)~]/[~ + fl] U(t) = a[1 - e -(=+B}']/[~ + fl].
It may be noted that the occurrence of a need has the memoryless property because of the exponential assumptions. Next, we define the following time instants at which the system enters into State - 1
State 0
State 1 State 2
State 3 State 4 State 5
one unit goes on operation and the other goes on standby, while the system is not needed (i.e. initial state) the repair of the failed unit is completed (i.e. the unit goes on standby) and the other is operating, regardless of a need one unit goes on operation and the other goes on repair, while the system is needed one unit goes on operation and the other goes on repair, while the system is not needed system goes down while needed system goes down when it is not needed system is needed while it is down.
It may be noted that the time instants for states 1 and 2 are regeneration points.
DERIVATION
OF
O;3,ft) = OAf(t) = 0 O~(t) = Q~(t)
=
Oj,~(t) =
0
f][{f]
[7(y)dF(y)} *
Q~(t) = fl d(x,d [ { f~
U(y)dF(y)}.A(x)]
Q~(t) =
QA~(t) =
f2 G(x)d[{ff V{y)dF(y)}*A(x)]
f]I{;]ff(y)df(y)}*A(x)]dG(x) These
Qij(t),Q}'j)(t)and QlT'"~(t)can be found in [3].
Q~j(t) A N D Ql~)(t) DERIVATION OF
Q-11(t)
= fl V(x)dF(x)
Q - 12(0
= fl V(x)dF(x)
A.(x)]dG(x)
p~j(t)
Using Renewal Theoretic arguments, one can write expressions for
pij(t)'s;i=-l,l,2
and
j=0,1,2,3,4,5
as given below
Q'°(t) = fl F(x) dG(x) Q13(t)
= fl G(x)U(x)dF(x)
p_ lo(t) = q_ xl(t)*Plo(t)+q_12(t)*P2o(t)+F(t) p _ l l ( t ) = q_ll(t)*pll(t)+q_i2(t)*p21(t) p_12(t) = q_11(t)*P12(t)+q_12(t)*P22(t)
Reliability analysis P - is(t) = q_ il(t) * P l s ( t ) + q - i2(t) * P23(t)
31
Similarly,
P - 14(t) = q - l i (t) * Pl , ( t ) + q_ 12 (t) * P2,(t)
Rz(x, t) = F(x + t) + [qt2°)(t ) + q2t~)(t)] • R i (x, t) + [qt2°2)(t) + q ~ ( t ) ] * R2(x , t)
p_ ls(t) = q_ l l ( t ) * pls(t) + q_12(t)* p25(t ) where
+
Plo(t) = [1 - h l i (t)] *(- 1), [Qlo(t ) _ Q~O)(t) _ Q~O2)(t)] + hl2(t) * [1 - h 2 2 ( t ) ] *(- 1), [Q20(t ) - Q~20)(t)- Qt202)(t)] pll(t) = [1-hll(t)]*(-1),
+
t+X
f I
q~2°)(u)Rl(t+x-u,O)du
t+x
(2)
q~2°)2(u)R2(t+x-u)du.
Solving (1) and (2), we get:
[1-Qlo(t)
- g l s(t) - Q14(t)]
R l ( x , t ) = [1 - h 1 1 ( t ) ] * ( - 1) , F ( X "t- t) + hl2(t ) * [1 - h22(t)] *(- 1), F ( x + t)
Pi2(t) = h12(t ) * [1 - h22(t) *(- 1), [1 - Q20(t)
+ [1 - h i 1 (t)] *(-1) * C i
- Q z a ( t ) - QE4(t)]
+ h12(O * [1 - h22(t)] *~- 1), C2
pie(t) = [1 - h i l ( t ) ] *(- 1), [Qla(t)_Qtl~(t)] + hi2(t) * [1 - h 2 2 ( t ) ] *t- 1), [QEa(t)_Q~a)(t)]
and
R2(X, t) = [1 - h 2 2 ( t ) ] . 1 - 1 ) , ff(X + t)
pl4(t) = [i - h i l ( t ) ] *(-l) * [Qi4(t)-Q(l~(t)-Qtl~(t)]
+h21(t)* [ 1 - h l l ( t ) ] * l - 1 ) * F ( x + t )
+ h12(t ) * [1 - h22(t)] *(- 1), [Q24(t )
+ [1 - h22(t)] *(- 1), C2
- O ~ ( t ) - Qt2~(t)]
+ h21 (t) * [1 - h i 1(t)] *(- 1), C1
pls(t) = [1 - h 1 1 ( t ) ] *(- 1), [Q(l~(t)_Q~,s)(t)]
where,
+ hi2(t ) * [1 - h22(t)] *(- ') * [Qt2~(t ) - g~'~)(t)]
~ttt + x q~°)(u)Ri(t + x - u , O ) d u
Ci
t+x
and
+
p z 0 ( t ) = [1 - h 2 2 ( t ) ] * ( - 1 ) , [Q2o( t ) - O 2 1~o) ( t ) - O 2 2 (o) (t)]
and
+ h21(t ) * [1 - h i l ( t ) ] *(- 1), [Qlo(t ) - Q ~ ( t ) - Q~°2)(t)] P2i(t) = h21(t)* [ 1 - h l i ( t ) ] * ( - 1 ) * --
C 2 = ~ttt + x q t 2 ° ) ( u ) R l ( t + x - u , O ) d u
[1-Qlo(t )
Q13(t)- Q14(t)]
+ also
P22(t) = [1 - h22(t)] *(- 1), [1 - Q2o(t) - g 2 a ( t ) - Q24(t)]
p2a(t) = [1 - h22(t)] *~- 1), [Q2a(t ) _ Q~z~(t)- Qt2~(t)] + h21(t ) * [1 - h l i ( t ) ] *(-1) * [QI s(t) - Q~al)(t) - Q~32)(t)]
1
-
O~(t)]
- [1-q~2~(s)--qt2~(s)]/6
h21(s)
(0)
- - - -
p2s(t) = [1 - h22(t)] *(- 1), [Qt2~(t ) _ Qt2~.5)(t)]
(3)
(4,5)
[q2i(s)+q21(s)+q2 i (S)]/6
1-hll(s)
+ h 2 1 ( t ) * [1 - h , l ( t ) ] *(-1) • [ Q ~ ( t ) -- O~,s)(t)]. DERIVATION OF
1
(0) U
q22( ) R 2 ( t + x - u ) d u
1 - -- [1 - q ~ ( s ) -- q~3)(s) -- q~%'S)(s)]/b 1 -h22(s )
+h21(t)* [1 - h i l ( t ) ] *(-1) • [Qi4(t) Q~(t)
-
-hll(s )
I
t+x
hi2(s) - -- [q~°2)(s)+ q ~ ( s ) ] / 6 1 -h22(s )
p2a(t) = [1 -- h22(t)] *(- 1), [Q24(t ) _ Qg2(t)_ Q(2~(t)] -
q~°)2(u)R2(t+x-u)du
where, 6 = [1 -- qt2°)2(s) -- qt2~(s)] [1 -- q i(0) l (s) -- q i(3) , (s) -- ql(4.1'5) (s)] (0) (4) (0) (3) -[q12(s)+qi2(s)][q=l(s)+q21(s)+q=i
R j ( x , t)
(4 s)
(s)]
and By Renewal Theoretic argument, one can write:
Ri(t, O) = Ri(t), i = 1, 2 are as given below.
g 1(x, t) = F ( x + t) + q(l°) (t) * g 1 (x, t) + q ~)(t) * R 1 (x, t) + rq~°2) (t) + qtz~(t )] • R2 (x, t)
Therefore,
R _ l(x,t) = F(x + t ) + q _ l l ( t ) * R l ( x , t ) + q - 12(0 * R2(x, t)
+ .tlt+ ~ qtl°)(u)gl (t + x - u, O) du
~t + x
+ + ft+~q~°)2(u)R2(t+x-u)du where, ~ = 3 w 5.
(1)
t
q-ll(u)Rl(t+x-u)du
t
+ ~t t + x q _ 1 2 ( u ) R 2 ( t + x - u ) d u .
(3)
P.K. KAPUR et al.
32
Substituting the values of R l ( x , t ) and R2(x,t), one can obtain the explicit expression for R_ l(x, t). POINT AVAILABILITY
and
P_l(x,t) = F(x+t)+q il(t)*Pi(x,t) +q_12(t)*P2(x,t) +
Putting x = 0 in Rs(x, t),j = - 1, l, 2 we get
Rl(O,t ) = [1 - h l l ( t ) ] *~ l ) * F ( t ) +
+ h l 2 ( t ) * [ 1 - h 2 2 ( t ) ] *~ l)*F(t)
=_ P l o ( t ) + p l l ( t ) + p i z ( t )
f
q-il(u)Pl(t+x-u)du
t+x
q_12(U)Pz(t+x-u)du.
Solving (4) and (5), one can obtain
and
Pi(x,t) = R i ( x , t ) + [ 1 - h i l ( t ) ] * t - i ) *
R2(0, t) = [1 - h22(t)] *~- l~. F(t)
-[1-hli(t)]
*t 1)*C 1
- h l 2 ( t ) * [ 1 - h 2 2 ( t ) ] * t - l ) * C2
=- P20(t)+p21(t)+P22(t) similarly,
and
R l ( O , t ) = F ( t ) + q _ x l ( t ) * R l ( O , t ) + q a2(t)*R2(O,t) -- p l o ( t ) + P - l i ( t ) + P - 1 2 ( t ) •
PE(X, t) = RE(X, t) + [1 -- h22(t)] *c- i ) , D2 + h 2 i ( t ) * [ 1 - h a l ( t ) ] *t i ) * D 1 -
RELIABILITY
[1 - h 2 2 ( t ) ] *(- l ) , C2
- h 2 i ( t ) * [1 - h li(t)] *t-i~* Ci
Putting t = 0 in (1), (2) and (3) we get,
where,
Ri(x, 0 ) = [1 - K l l ( X ) ] *(-1~ * F(x) + K 12(x) * [1 - K2 : (x)] *c- 1), F(x)
D1 =
R2(x,0 ) = [1-K22(x)]*c-i)*F(x)
i
t+x
[qtl~(u)+q~l(u)]Pl(t + x - u ) d u + f t +x
+ K : i ( x ) * [1 - K l l(x)] *<- a), F(x)
D2 =
where,
I
+ x - u)
du
+ ~t +~ [q~z~(u) + q~2~(u)]P2( t + x - u ) d u J,
K1E(X) = q ~ ( x ) * [1 - q~°~(x)]*~- 1~ and
K 2 1 ( x ) = q21(x) CO) , [ 1 - q 2 zCO) (X)] ,(l) ,
+ q~(u)]P2(t
[q~2~(u)+q~)(u)]Pl(t+x-u)du
K ~, (x) = q~°)(x) + q ~ ( x ) * q(2°)(x) * [1 - q ~ ( x ) ] * ' - ' '
CO) (0) q22(Xj'+-q21(X)
[q~(u)
t+x
R_ 1 (X, 0 ) = q_ a l(x) * Rl(x, O)+q_ ~2(x) * R2(x, 0)
=
Oi
+ h i z ( t )* [ 1 - h z z ( t ) ] *(-1)*O2
+ h 2 1 ( t ) * [ 1 - / h l ( t ) ] *t ~)* F(t)
K22(x)
~t l + x
Pl(t) = P l o ( t ) + P l l ( t ) + P 1 2 ( t )
q]~(X)* [1--q]01'(X)]*~-l'
Pz(t) - P20(t) + P21(t) + P22(t) JOINT AVAILABILITY
Again, by Renewal Theoretic argument, one can write,
Pi (x, t) = F(x + t) + [q]°}(t) + q ~/(t)] * P1 (x, t) + [q~°2)(t) + q]~(t)] * Pz(x, t)
substituting the values of Pl(x, t) and P2(x, t) one can o b t a i n P _ l(x, t).
GENERAL CASES
Using Renewal Theoretic similarly write expressions for
+ .i t+x [q]°)(u)q [~/(u)]P 1(t + x - u) du
t
R(d)[(X 1, tl), (X2, t2) . . . . . (x., tn) ]
[q~l~(u)+q]~(u)]P2(t+x-u)du.
t
(4)
Similarly,
and R~)l[(xl, tl), (x2, t2) . . . . . (x., t.)] P]")(ti, t 2..... t,)
P2(x, t) = F(x + t)+ [q~°~(t)+ q~)(t)] * n l (x, t) + [q~2°)2(t)+q~(t)] * n2(x, t)
n 1
= Rx(t.-tl,ti)+[1-h,l(tl)]
*c-1~* ~ Dl~ i-I n
+ .f '+ x [q~2~(u) + q~2~)(u)]Pl(t + x -- u) du
+hlz(tl)*[1--h22(t1)]
+ ['t+. [q~z~(u) + q~2~(u)]P2 (t + x -- u) du ,jr
one can
P]n)(tl,t 2. . . . . t.); P~)i(tl,t 2. . . . . t.);
*t + x
+
arguments,
-[1-hll(tl)]
(5)
--hl2(fl)*
*c-l)]*
1
~ D2~ i=1
*~ L)*E 1
[1 - h 2 2 ( / 1 ) ] *(-1) * E 2
Reliability analysis
33
where, DI, =
D E R I V A T I O N O F R~')[(xl, tl)..... (x~, t.)]
f tti+l [ q ~ ( u ) + q [ ~ ) ( u ) ] n ~ - ° ( t i + , - u . . . . . t ~ - u ) 1
R~")[(xl, tl) . . . . . (x~, t.)] n--1
= Rdt~+x.-tl,
t l ) + [ 1 - h s l ( t s ) ] *~-1~* ~, Kl,
x du + It tl+l [q~(u)+q~(u)]P~2"-°(t~+~
i=1 n-I
+ h l 2 ( t l ) * [1-hE2(tl)]*(-1)* ~ K2, --u . . . . . t n - u ) d u
i=1
- [1 - h 11 (tl)3 *(- 1 ) , J1
D2, = ft tt+I [qt2~(u)+q(2~l)(u)]p(ln-O(ti+l-u . . . . . t n - u )
X du "~ f ttl÷l [q22(u)+q22(u)]P2 (o) (4) (n--i)(ti+l
- h12(tl) * [1 - h22(tl)] *t- t ) . J2 where, K1, =
It ti÷l [q~°l)(U)
-u ..... t.-u)du E1 =
I? I?
q(l°l)(u)Rl(t.-u)du +
!
and E2 =
f? t
+ I t`*' [qi~(u )
•Jti+.i + q~(u)]R~n- O[(xi +1, ti +1 - u) ..... (x.,t n - u)] du
q(2~)(u)Rt(t~-u)du +
1
+ q~)(u)] R~"-°[(x i + 1, ti +1 - u) ..... (x~, t. - u)] du
q~(u)R2(t.-u)du
1
q~(u)R2(t.-u)du K2, =
P~)(tl, t2 . . . . . t~)
.--1
ft ti+l [q~2°~(u) + q~21(u)]~"-ORl[(xi* 1, ti+ 1 - u ) . . . . . (x., t ~ - u)]
= R 2 ( t ~ - t l , t ~ ) + [ 1 - h 2 2 ( t l ) ] *(-1)* Y~ DE, i=l
x du + f tt~+l [q~2~(u)
n-1
+h21(t~)*[1-hH(t~)] *~-1~* ~ DI,
i+x~
i=1
+ q~2~(u)] ~"- ilR2 [(xi + t, ti + 1 - u) ..... (x., t. - u)] du
- [ 1 - h22(t~)] *(- u . E2
f ttn+xn J2 = q~z~(u)Rl(t.+ x . - u ) d u ! f ttn+2¢n
- h 2 1 ( t l ) * [1-h~l(t~)]*l- ~) * E~ similarly,
+
P~)~(t~, t 2. . . . . t.) and
= R_l(t.-tl,tl)
n--I
1
i=1 n-1
+
+q_~1(t~)*ht2(t~)* [1-h22(t~)]*t-l~* ~ D2, i=1
- q _ ~ l ( t ~ ) * [ 1 - h ~ ( t l ) ] * t - 1 ) * E1 - q - l d t ~ ) * h~2(tl)* [1 -h22(t~)] *t-~) *E 2
i
n--1
A(t) = B(t) = 1
i=1 n-1
+q_~2(t~)*hEl(t~)* [ 1 - h l ~ ( t l ) ] *t-l~* ~ D~,
i=1
--q_ 12(tl ) * [I -- h22(t~)] *t- 1). E2 - q - ~2 ( t l ) * h21 (t~) * [1 - h I ~ (t 1 ) ] * t - 1 ) , E~
i=l
f
1
M R 24/I~
R~I1 [(xD tl) . . . . . (x., t~)].
When the system is in constant usage throughout the process, the
+q-~2(t~)*[1-h22(t~)] *t-~)* ~. D2,
-
R~2")[(xl, tl ). . . . . (x., t~)] and
Particular case
-- ftt"q-ll(U)Rl(tn-u,O)du
+ ~.
i
f tt"+xn q~2~(u)R2(t.+x.-u)du 1
similarly, one can obtain expressions for
q_,l(U)P~"-O(ti+~-u . . . . . t ~ - u ) d u
i=1
q~2~(u)R2(t.+x.-u)du
J2 = f tt~+X~q~2~(u)Rt(t. + x . - u ) d u
+ q - t ~ ( t l ) * [ 1 - h ~ l ( t t ) ] *~-1)* ~. D1,
+ ~
1
U(t)= 1
and
V ( t ) = O , Vt.
As a result, the states 2, 4 and 5 of the model vanish and the model reduces to a constantly used 2-unit standby redundant repairable system. The results as given in [4] can be derived readily.
q_12(u)P~n-O(ti÷l-u . . . . . t ~ - u ) d u
DISCUSSION At the outset, we note that the solution of
q_~2(u)R2(t-u,O)du. P~)l(xl . . . . . x.)
and
R~)l[(xl, tl) . . . . . (x.,t.)]
34
P.K. KAPUR et al.
is n o t obtainable in closed f o r m [ l ] excepting for Poisson Renewal Process. O n e m a y calculate P _ l(x) directly by assuming t h a t b o t h failure a n d repair conform to Poisson Process. This calculation clearly shows that the process is n o n - M a r k o v i a n in nature. However, for the sake of simplicitly, we m a y assume that the process conforms to a five state M a r k o v chain in c o n t i n u o u s time ( M a r k o v Process). O n e can, then calculate P~)l(xi . . . . . x.) and R~_")1[(xl, tl) . . . . . (x., t.)] on the lines as described in [4]. Acknowledgement Authors are grateful to Dr. B. Biswas, Department of Mathematical Statistics, University of Delhi for constant inspiration.
REFERENCES
1. L. A. Baxter, Availability measures for a two-state system, J. appl. Prob. 18, 227-235 (1981). 2. K. R. Kapoor and P. K. Kapur, First uptime and disappointment time joint distributions of an intermittently used system, Microelectron. Reliab. 19, 91-93 (1979). 3. P. K. Kapur and K. R. Kapoor, Intermittently used redundant system, Microelectron. Reliab. 19, 593 96 (1978). 4. P. K. Kapur, Y. K. Mehta and S. Biswas, Generalised availability measures for repairable systems, J. Indian Ass. Prod. Quality Reliab. (in press). 5. P. K. Kapur, T. V. Natrajan and D. Chakrabarty, Generalised availability measures for an intermittently used repairable system, Microelectron. Reliab. (in press).
0026-2714/8453.00 + .00 © 1984 Pergamon Press Ltd.
Printed in Great Britain.
GENERALISED AVAILABILITY MEASURES FOR A TWO-UNIT INTERMITTENTLY USED REPAIRABLE SYSTEM P. K. K~d'UR Department of Operational Research, University of Delhi, India N. L. BUTANI C.H.M. College, Ulhasnagar-3, Thana, Maharastra, India and DEBASIS DUTTA Department of Operational Research, University of Delhi, Delhi-110007, India
(Received for publication 2 August 1983) Abstract--This paper obtains generalised availability measures as defined by Baxter, J. appl. Prob. 18, 227-235 (1981), for a two-unit cold stand-by intermittently used redundant repairable system. Regenerative point technique of Markov Renewal Process has been employed to obtain these measures. A particular case is also considered.
INTRODUCTION
G(t) cdf of the repair time of the system with
In a recent contribution to this journal, K a p u r et al. [5] obtained Generalised Availability Measures for a simple (one unit) intermittently used repairable system generalising the earlier results due to B a x t e r [ l ] . Availability analysis of such a system has earlier been done by K a p u r [-2, 3]. The purpose of this paper is to obtain the Generalised Availability Measures for a Two-Unit Intermittently Used Redundant Repairable System. Extension is non-trivial in nature. It unifies an earlier treatment [4, 5] to a study of this kind.
A(t) B(t) V(t) U(t) *
pdfg(t) ( = l - e -~t) cdf of occurrence time of need (= 1 - e -pt) cdf of holding time of need Pr{need occurs at time t/it occurred at time O} Pr{need does not occur at time t/it did not occur at time O} convolution, e.g.
A(t)*B(t) = f l A(p)B(t-#)du and
A(t)*B(x + t) = .t~ A(p)B(t + x - I~)du
ASSUMPTIONS
complement, e.g. (1) System has two identical units. (2) A standby unit never fails. (3) Once a unit is repaired, it is like new and goes into standby state or operating state. (4) All the switchings operations are 100 ~o reliable, instantaneous and never damage anything. (5) The cdf's of"failure time" and "repair time" are both arbitrary. (6) The cdf's of occurrence time and holding time of a need are exponential. (7) The system is not needed at time O. (8) All the random variables are mutually independent.
[1 -A(t)] *t-l) LT
F(t) = 1 -F(t) 1 + A(t)A(t) * A(t) +... Laplace Transformation, e.g.
f(s) = LT If(t)] =
e-Stf(t)dt
Qij(t) cdf of MRP (Markov Renewal Process) from state i into state j, i.e. probability that after transiting into state i, the process next transits into state j in time ~
NOTATIONS
F(t) cumulative distribution function (cdf) of the failure time of the system with probability density function (pdf)f(t) 29
et al.
30
P.K. KAPUR
Rj(x,t)
Pr{system is operating at time t and continues to operate without failing in starts from state j} j = - 1, 1, 2 = interval reliability {joint availability at t and starts from statej at time t = O} ..... t.] {joint availability at n distinct points h, t2,.--, t/system starts from statej at time t = 0} R}"~[(x~,t~), (x~,t~)..... (xo, to)] {joint interval reliability at n disjoint intervals +x2], It,, t, + x J/system starts from state j at time t = 0}.
Ql,(t)
=
l d(x)O(x) dF(x)
[t,t+x]/it
Pj(x,t) P}"~[tl,t2
t+x/system
[tt,h
DERIVATION
OF
(22o(t)
ir(x) dG(x)
=
Q23(t)
=
Qv,(t)
=
Q),~(t)
=
Q~(t)
=
l G(x)r/(x)dF(x) f'oG(X)V(x)dF(x)
[te,t2+x2]...
V(t) A N D U(t)
V(t+At) = V ( t ) ( 1 - f l A t ) + ~(t)~At, V(t) = [c~+fle I~+~!,]/[=+fl].
At--, 0
Similarly,
l G(x)U(x)dF(x) l G(x)O(x)dF(x)
Q~(t) =
l G(x)V(x)dF(x)
Q~(t) =
l G(x)V(x)dF(x)
V~(t) = fl[1-e-(~+~)t]/[m+fl]
and
U(t) = [fl + ~ e - (~ +~)~]/[~ + fl] U(t) = a[1 - e -(=+B}']/[~ + fl].
It may be noted that the occurrence of a need has the memoryless property because of the exponential assumptions. Next, we define the following time instants at which the system enters into State - 1
State 0
State 1 State 2
State 3 State 4 State 5
one unit goes on operation and the other goes on standby, while the system is not needed (i.e. initial state) the repair of the failed unit is completed (i.e. the unit goes on standby) and the other is operating, regardless of a need one unit goes on operation and the other goes on repair, while the system is needed one unit goes on operation and the other goes on repair, while the system is not needed system goes down while needed system goes down when it is not needed system is needed while it is down.
It may be noted that the time instants for states 1 and 2 are regeneration points.
DERIVATION
OF
O;3,ft) = OAf(t) = 0 O~(t) = Q~(t)
=
Oj,~(t) =
0
f][{f]
[7(y)dF(y)} *
Q~(t) = fl d(x,d [ { f~
U(y)dF(y)}.A(x)]
Q~(t) =
QA~(t) =
f2 G(x)d[{ff V{y)dF(y)}*A(x)]
f]I{;]ff(y)df(y)}*A(x)]dG(x) These
Qij(t),Q}'j)(t)and QlT'"~(t)can be found in [3].
Q~j(t) A N D Ql~)(t) DERIVATION OF
Q-11(t)
= fl V(x)dF(x)
Q - 12(0
= fl V(x)dF(x)
A.(x)]dG(x)
p~j(t)
Using Renewal Theoretic arguments, one can write expressions for
pij(t)'s;i=-l,l,2
and
j=0,1,2,3,4,5
as given below
Q'°(t) = fl F(x) dG(x) Q13(t)
= fl G(x)U(x)dF(x)
p_ lo(t) = q_ xl(t)*Plo(t)+q_12(t)*P2o(t)+F(t) p _ l l ( t ) = q_ll(t)*pll(t)+q_i2(t)*p21(t) p_12(t) = q_11(t)*P12(t)+q_12(t)*P22(t)
Reliability analysis P - is(t) = q_ il(t) * P l s ( t ) + q - i2(t) * P23(t)
31
Similarly,
P - 14(t) = q - l i (t) * Pl , ( t ) + q_ 12 (t) * P2,(t)
Rz(x, t) = F(x + t) + [qt2°)(t ) + q2t~)(t)] • R i (x, t) + [qt2°2)(t) + q ~ ( t ) ] * R2(x , t)
p_ ls(t) = q_ l l ( t ) * pls(t) + q_12(t)* p25(t ) where
+
Plo(t) = [1 - h l i (t)] *(- 1), [Qlo(t ) _ Q~O)(t) _ Q~O2)(t)] + hl2(t) * [1 - h 2 2 ( t ) ] *(- 1), [Q20(t ) - Q~20)(t)- Qt202)(t)] pll(t) = [1-hll(t)]*(-1),
+
t+X
f I
q~2°)(u)Rl(t+x-u,O)du
t+x
(2)
q~2°)2(u)R2(t+x-u)du.
Solving (1) and (2), we get:
[1-Qlo(t)
- g l s(t) - Q14(t)]
R l ( x , t ) = [1 - h 1 1 ( t ) ] * ( - 1) , F ( X "t- t) + hl2(t ) * [1 - h22(t)] *(- 1), F ( x + t)
Pi2(t) = h12(t ) * [1 - h22(t) *(- 1), [1 - Q20(t)
+ [1 - h i 1 (t)] *(-1) * C i
- Q z a ( t ) - QE4(t)]
+ h12(O * [1 - h22(t)] *~- 1), C2
pie(t) = [1 - h i l ( t ) ] *(- 1), [Qla(t)_Qtl~(t)] + hi2(t) * [1 - h 2 2 ( t ) ] *t- 1), [QEa(t)_Q~a)(t)]
and
R2(X, t) = [1 - h 2 2 ( t ) ] . 1 - 1 ) , ff(X + t)
pl4(t) = [i - h i l ( t ) ] *(-l) * [Qi4(t)-Q(l~(t)-Qtl~(t)]
+h21(t)* [ 1 - h l l ( t ) ] * l - 1 ) * F ( x + t )
+ h12(t ) * [1 - h22(t)] *(- 1), [Q24(t )
+ [1 - h22(t)] *(- 1), C2
- O ~ ( t ) - Qt2~(t)]
+ h21 (t) * [1 - h i 1(t)] *(- 1), C1
pls(t) = [1 - h 1 1 ( t ) ] *(- 1), [Q(l~(t)_Q~,s)(t)]
where,
+ hi2(t ) * [1 - h22(t)] *(- ') * [Qt2~(t ) - g~'~)(t)]
~ttt + x q~°)(u)Ri(t + x - u , O ) d u
Ci
t+x
and
+
p z 0 ( t ) = [1 - h 2 2 ( t ) ] * ( - 1 ) , [Q2o( t ) - O 2 1~o) ( t ) - O 2 2 (o) (t)]
and
+ h21(t ) * [1 - h i l ( t ) ] *(- 1), [Qlo(t ) - Q ~ ( t ) - Q~°2)(t)] P2i(t) = h21(t)* [ 1 - h l i ( t ) ] * ( - 1 ) * --
C 2 = ~ttt + x q t 2 ° ) ( u ) R l ( t + x - u , O ) d u
[1-Qlo(t )
Q13(t)- Q14(t)]
+ also
P22(t) = [1 - h22(t)] *(- 1), [1 - Q2o(t) - g 2 a ( t ) - Q24(t)]
p2a(t) = [1 - h22(t)] *~- 1), [Q2a(t ) _ Q~z~(t)- Qt2~(t)] + h21(t ) * [1 - h l i ( t ) ] *(-1) * [QI s(t) - Q~al)(t) - Q~32)(t)]
1
-
O~(t)]
- [1-q~2~(s)--qt2~(s)]/6
h21(s)
(0)
- - - -
p2s(t) = [1 - h22(t)] *(- 1), [Qt2~(t ) _ Qt2~.5)(t)]
(3)
(4,5)
[q2i(s)+q21(s)+q2 i (S)]/6
1-hll(s)
+ h 2 1 ( t ) * [1 - h , l ( t ) ] *(-1) • [ Q ~ ( t ) -- O~,s)(t)]. DERIVATION OF
1
(0) U
q22( ) R 2 ( t + x - u ) d u
1 - -- [1 - q ~ ( s ) -- q~3)(s) -- q~%'S)(s)]/b 1 -h22(s )
+h21(t)* [1 - h i l ( t ) ] *(-1) • [Qi4(t) Q~(t)
-
-hll(s )
I
t+x
hi2(s) - -- [q~°2)(s)+ q ~ ( s ) ] / 6 1 -h22(s )
p2a(t) = [1 -- h22(t)] *(- 1), [Q24(t ) _ Qg2(t)_ Q(2~(t)] -
q~°)2(u)R2(t+x-u)du
where, 6 = [1 -- qt2°)2(s) -- qt2~(s)] [1 -- q i(0) l (s) -- q i(3) , (s) -- ql(4.1'5) (s)] (0) (4) (0) (3) -[q12(s)+qi2(s)][q=l(s)+q21(s)+q=i
R j ( x , t)
(4 s)
(s)]
and By Renewal Theoretic argument, one can write:
Ri(t, O) = Ri(t), i = 1, 2 are as given below.
g 1(x, t) = F ( x + t) + q(l°) (t) * g 1 (x, t) + q ~)(t) * R 1 (x, t) + rq~°2) (t) + qtz~(t )] • R2 (x, t)
Therefore,
R _ l(x,t) = F(x + t ) + q _ l l ( t ) * R l ( x , t ) + q - 12(0 * R2(x, t)
+ .tlt+ ~ qtl°)(u)gl (t + x - u, O) du
~t + x
+ + ft+~q~°)2(u)R2(t+x-u)du where, ~ = 3 w 5.
(1)
t
q-ll(u)Rl(t+x-u)du
t
+ ~t t + x q _ 1 2 ( u ) R 2 ( t + x - u ) d u .
(3)
P.K. KAPUR et al.
32
Substituting the values of R l ( x , t ) and R2(x,t), one can obtain the explicit expression for R_ l(x, t). POINT AVAILABILITY
and
P_l(x,t) = F(x+t)+q il(t)*Pi(x,t) +q_12(t)*P2(x,t) +
Putting x = 0 in Rs(x, t),j = - 1, l, 2 we get
Rl(O,t ) = [1 - h l l ( t ) ] *~ l ) * F ( t ) +
+ h l 2 ( t ) * [ 1 - h 2 2 ( t ) ] *~ l)*F(t)
=_ P l o ( t ) + p l l ( t ) + p i z ( t )
f
q-il(u)Pl(t+x-u)du
t+x
q_12(U)Pz(t+x-u)du.
Solving (4) and (5), one can obtain
and
Pi(x,t) = R i ( x , t ) + [ 1 - h i l ( t ) ] * t - i ) *
R2(0, t) = [1 - h22(t)] *~- l~. F(t)
-[1-hli(t)]
*t 1)*C 1
- h l 2 ( t ) * [ 1 - h 2 2 ( t ) ] * t - l ) * C2
=- P20(t)+p21(t)+P22(t) similarly,
and
R l ( O , t ) = F ( t ) + q _ x l ( t ) * R l ( O , t ) + q a2(t)*R2(O,t) -- p l o ( t ) + P - l i ( t ) + P - 1 2 ( t ) •
PE(X, t) = RE(X, t) + [1 -- h22(t)] *c- i ) , D2 + h 2 i ( t ) * [ 1 - h a l ( t ) ] *t i ) * D 1 -
RELIABILITY
[1 - h 2 2 ( t ) ] *(- l ) , C2
- h 2 i ( t ) * [1 - h li(t)] *t-i~* Ci
Putting t = 0 in (1), (2) and (3) we get,
where,
Ri(x, 0 ) = [1 - K l l ( X ) ] *(-1~ * F(x) + K 12(x) * [1 - K2 : (x)] *c- 1), F(x)
D1 =
R2(x,0 ) = [1-K22(x)]*c-i)*F(x)
i
t+x
[qtl~(u)+q~l(u)]Pl(t + x - u ) d u + f t +x
+ K : i ( x ) * [1 - K l l(x)] *<- a), F(x)
D2 =
where,
I
+ x - u)
du
+ ~t +~ [q~z~(u) + q~2~(u)]P2( t + x - u ) d u J,
K1E(X) = q ~ ( x ) * [1 - q~°~(x)]*~- 1~ and
K 2 1 ( x ) = q21(x) CO) , [ 1 - q 2 zCO) (X)] ,(l) ,
+ q~(u)]P2(t
[q~2~(u)+q~)(u)]Pl(t+x-u)du
K ~, (x) = q~°)(x) + q ~ ( x ) * q(2°)(x) * [1 - q ~ ( x ) ] * ' - ' '
CO) (0) q22(Xj'+-q21(X)
[q~(u)
t+x
R_ 1 (X, 0 ) = q_ a l(x) * Rl(x, O)+q_ ~2(x) * R2(x, 0)
=
Oi
+ h i z ( t )* [ 1 - h z z ( t ) ] *(-1)*O2
+ h 2 1 ( t ) * [ 1 - / h l ( t ) ] *t ~)* F(t)
K22(x)
~t l + x
Pl(t) = P l o ( t ) + P l l ( t ) + P 1 2 ( t )
q]~(X)* [1--q]01'(X)]*~-l'
Pz(t) - P20(t) + P21(t) + P22(t) JOINT AVAILABILITY
Again, by Renewal Theoretic argument, one can write,
Pi (x, t) = F(x + t) + [q]°}(t) + q ~/(t)] * P1 (x, t) + [q~°2)(t) + q]~(t)] * Pz(x, t)
substituting the values of Pl(x, t) and P2(x, t) one can o b t a i n P _ l(x, t).
GENERAL CASES
Using Renewal Theoretic similarly write expressions for
+ .i t+x [q]°)(u)q [~/(u)]P 1(t + x - u) du
t
R(d)[(X 1, tl), (X2, t2) . . . . . (x., tn) ]
[q~l~(u)+q]~(u)]P2(t+x-u)du.
t
(4)
Similarly,
and R~)l[(xl, tl), (x2, t2) . . . . . (x., t.)] P]")(ti, t 2..... t,)
P2(x, t) = F(x + t)+ [q~°~(t)+ q~)(t)] * n l (x, t) + [q~2°)2(t)+q~(t)] * n2(x, t)
n 1
= Rx(t.-tl,ti)+[1-h,l(tl)]
*c-1~* ~ Dl~ i-I n
+ .f '+ x [q~2~(u) + q~2~)(u)]Pl(t + x -- u) du
+hlz(tl)*[1--h22(t1)]
+ ['t+. [q~z~(u) + q~2~(u)]P2 (t + x -- u) du ,jr
one can
P]n)(tl,t 2. . . . . t.); P~)i(tl,t 2. . . . . t.);
*t + x
+
arguments,
-[1-hll(tl)]
(5)
--hl2(fl)*
*c-l)]*
1
~ D2~ i=1
*~ L)*E 1
[1 - h 2 2 ( / 1 ) ] *(-1) * E 2
Reliability analysis
33
where, DI, =
D E R I V A T I O N O F R~')[(xl, tl)..... (x~, t.)]
f tti+l [ q ~ ( u ) + q [ ~ ) ( u ) ] n ~ - ° ( t i + , - u . . . . . t ~ - u ) 1
R~")[(xl, tl) . . . . . (x~, t.)] n--1
= Rdt~+x.-tl,
t l ) + [ 1 - h s l ( t s ) ] *~-1~* ~, Kl,
x du + It tl+l [q~(u)+q~(u)]P~2"-°(t~+~
i=1 n-I
+ h l 2 ( t l ) * [1-hE2(tl)]*(-1)* ~ K2, --u . . . . . t n - u ) d u
i=1
- [1 - h 11 (tl)3 *(- 1 ) , J1
D2, = ft tt+I [qt2~(u)+q(2~l)(u)]p(ln-O(ti+l-u . . . . . t n - u )
X du "~ f ttl÷l [q22(u)+q22(u)]P2 (o) (4) (n--i)(ti+l
- h12(tl) * [1 - h22(tl)] *t- t ) . J2 where, K1, =
It ti÷l [q~°l)(U)
-u ..... t.-u)du E1 =
I? I?
q(l°l)(u)Rl(t.-u)du +
!
and E2 =
f? t
+ I t`*' [qi~(u )
•Jti+.i + q~(u)]R~n- O[(xi +1, ti +1 - u) ..... (x.,t n - u)] du
q(2~)(u)Rt(t~-u)du +
1
+ q~)(u)] R~"-°[(x i + 1, ti +1 - u) ..... (x~, t. - u)] du
q~(u)R2(t.-u)du
1
q~(u)R2(t.-u)du K2, =
P~)(tl, t2 . . . . . t~)
.--1
ft ti+l [q~2°~(u) + q~21(u)]~"-ORl[(xi* 1, ti+ 1 - u ) . . . . . (x., t ~ - u)]
= R 2 ( t ~ - t l , t ~ ) + [ 1 - h 2 2 ( t l ) ] *(-1)* Y~ DE, i=l
x du + f tt~+l [q~2~(u)
n-1
+h21(t~)*[1-hH(t~)] *~-1~* ~ DI,
i+x~
i=1
+ q~2~(u)] ~"- ilR2 [(xi + t, ti + 1 - u) ..... (x., t. - u)] du
- [ 1 - h22(t~)] *(- u . E2
f ttn+xn J2 = q~z~(u)Rl(t.+ x . - u ) d u ! f ttn+2¢n
- h 2 1 ( t l ) * [1-h~l(t~)]*l- ~) * E~ similarly,
+
P~)~(t~, t 2. . . . . t.) and
= R_l(t.-tl,tl)
n--I
1
i=1 n-1
+
+q_~1(t~)*ht2(t~)* [1-h22(t~)]*t-l~* ~ D2, i=1
- q _ ~ l ( t ~ ) * [ 1 - h ~ ( t l ) ] * t - 1 ) * E1 - q - l d t ~ ) * h~2(tl)* [1 -h22(t~)] *t-~) *E 2
i
n--1
A(t) = B(t) = 1
i=1 n-1
+q_~2(t~)*hEl(t~)* [ 1 - h l ~ ( t l ) ] *t-l~* ~ D~,
i=1
--q_ 12(tl ) * [I -- h22(t~)] *t- 1). E2 - q - ~2 ( t l ) * h21 (t~) * [1 - h I ~ (t 1 ) ] * t - 1 ) , E~
i=l
f
1
M R 24/I~
R~I1 [(xD tl) . . . . . (x., t~)].
When the system is in constant usage throughout the process, the
+q-~2(t~)*[1-h22(t~)] *t-~)* ~. D2,
-
R~2")[(xl, tl ). . . . . (x., t~)] and
Particular case
-- ftt"q-ll(U)Rl(tn-u,O)du
+ ~.
i
f tt"+xn q~2~(u)R2(t.+x.-u)du 1
similarly, one can obtain expressions for
q_,l(U)P~"-O(ti+~-u . . . . . t ~ - u ) d u
i=1
q~2~(u)R2(t.+x.-u)du
J2 = f tt~+X~q~2~(u)Rt(t. + x . - u ) d u
+ q - t ~ ( t l ) * [ 1 - h ~ l ( t t ) ] *~-1)* ~. D1,
+ ~
1
U(t)= 1
and
V ( t ) = O , Vt.
As a result, the states 2, 4 and 5 of the model vanish and the model reduces to a constantly used 2-unit standby redundant repairable system. The results as given in [4] can be derived readily.
q_12(u)P~n-O(ti÷l-u . . . . . t ~ - u ) d u
DISCUSSION At the outset, we note that the solution of
q_~2(u)R2(t-u,O)du. P~)l(xl . . . . . x.)
and
R~)l[(xl, tl) . . . . . (x.,t.)]
34
P.K. KAPUR et al.
is n o t obtainable in closed f o r m [ l ] excepting for Poisson Renewal Process. O n e m a y calculate P _ l(x) directly by assuming t h a t b o t h failure a n d repair conform to Poisson Process. This calculation clearly shows that the process is n o n - M a r k o v i a n in nature. However, for the sake of simplicitly, we m a y assume that the process conforms to a five state M a r k o v chain in c o n t i n u o u s time ( M a r k o v Process). O n e can, then calculate P~)l(xi . . . . . x.) and R~_")1[(xl, tl) . . . . . (x., t.)] on the lines as described in [4]. Acknowledgement Authors are grateful to Dr. B. Biswas, Department of Mathematical Statistics, University of Delhi for constant inspiration.
REFERENCES
1. L. A. Baxter, Availability measures for a two-state system, J. appl. Prob. 18, 227-235 (1981). 2. K. R. Kapoor and P. K. Kapur, First uptime and disappointment time joint distributions of an intermittently used system, Microelectron. Reliab. 19, 91-93 (1979). 3. P. K. Kapur and K. R. Kapoor, Intermittently used redundant system, Microelectron. Reliab. 19, 593 96 (1978). 4. P. K. Kapur, Y. K. Mehta and S. Biswas, Generalised availability measures for repairable systems, J. Indian Ass. Prod. Quality Reliab. (in press). 5. P. K. Kapur, T. V. Natrajan and D. Chakrabarty, Generalised availability measures for an intermittently used repairable system, Microelectron. Reliab. (in press).