Limiting distributions of the residual lifetimes of several repairable systems

Microelectron. Reliab., Vol. 33, No. 8, pp. 1069--1072,1993.

0026-271419356.00+ .00 Pergamon Press Ltd

Printed in Great Britain.

L I M I T I N G D I S T R I B U T I O N S OF THE R E S I D U A L LIFETIMES OF SEVERAL R E P A I R A B L E SYSTEMS WEI LI and JtNHUA CAO Institute of Applied Mathematics, Academia Sinica, Beijing, People's Republic of China

(Received for publication 24 March 1992) Al~tract--In this paper, the limiting distributions of the residual lifetimes of several repairable systems, conditional upon the event that the systems have not been down at any time in (0, t], are proved to be exponential irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.

1. INTRODUCTION

2.2. Notation

In certain reliability problems, where failure-system recovery is very expensive, the behaviour of the system after its first breakdown may not be of much interest, such as the control system of a space mission. In such a case, it would be of particular importance to study the time to the first system failure (i.e. the system lifetime or failure time) and the residual lifetime of the system conditional upon the event that the system has not been down at any time in (0, t]. Keilson [l] studied four failure times for the system of independent Markov units, and then K a l p a k a m and Hameed [2] considered the quasi-stationary distribution, which is similar to the limiting distribution of the residual lifetime distribution discussed in this paper, of a two-similar-unit warm standby redundant system. In this paper, we study the limiting distributions of the residual lifetimes of three special repairable systems, one of which extended the result of Ref. [2], by using Markov renewal theory [3]. It is shown that the limiting distributions of the residual lifetimes of such three systems are all exponential, irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.

State 0: epoch of entering this state is that unit 2 is beginning to operate while unit 1 is still good. State 1: epoch of entering this state is that unit 2 is beginning to be repaired while unit I is still good. T~ time to the first system failure given that the system enters state i at time t = 0 (i = 0, t) X(t) residual lifetime of the system at any instant t L[f(t)]orf*(s) Laplace transform of any arbitrary function f(t) f(t).g(t) convolution of any two arbitrary functions f(t) and g(t), i.e.

f(t) * g(t) = .(f f(x)g(t -- x) dx f~")(t) the n-fold convolution of any function f(t) with itself f(.) and g~(.) the p.d.f, of random variables X~ and Y~ (i = 1, 2), respectively Fi(') and G~(') the c.d.f, of random variables Xl and Yi (i = 1, 2), respectively F*(s) Laplace transform of F(t) = 1 - F(t) h0(x)= ~ P . e - ~ * g d x ) ] ~"~ nffil

q01(t) --f2 (t) [e -~' + h0(t) * e - ~

qlo(t ) ffi e-;~g2(t)

2. MODEL 1

61 the root of equation g~(s + ~.)q~l(s) = 1

2.1. Assumptions (a) The system is a two-dissimilar-unit parallel system with a single repair facility. The repair discipline is "first fail, first repaired". (b) The failure rate of unit 1 is constant 2, while the lifetime X2 of unit 2 and the repair, time Y~ of unit i (i = 1, 2) have arbitrary distributions satisfying the condition that the Laplace transforms of the distributions are all rational functions. (c) U p o n repair, a unit becomes as good as new. (d) The failure times and the repair times of two units are mutually independent.

that is situated closest to the right of the s -plane.

2.3. Conclusion

Lemma 1. Letting R i ( t ) = P{Ti> t} (i =0, 1), we have

1069

R*(s) =

R*(s) =

~ ( s ) + G*(s + &)q~,(s) l -- g~ (s + 2)q~ (s)

8~(s + 4) + ~ ( s ) g ~ ( s + 2)

1 -g*(s +,~)q~,(s)

1070

W. LI and J. CAO

Proof Let Y(t) be the state of the system at time t and So = 0, S l , 8 2 , . . . , be the successive transition time instants: by using the exponential distribution, it is easy to see that S,, n = 0, 1. . . . , are the regeneration points. Therefore, if Z. = Y(S, + 0), {Z,, S,} is a homogeneous Markov renewal process with the finite state space E = {0, 1}. Letting

by using Cauchy and residue theorems we know that q*l (s) is a rational function too. Denoting

O~(s) = g ( s ) N(s)'

ff'~2(s) = P(s) Q(s)

and q~l (S) --

Qo(t) = P { Z . + l = j , S . + l - S. ~< t, and since the

U(s) V(s)

system has not been down at any time in

then from Lemma 1 we can obtain

(0, tllZ. = i} (i,j = 0, 1),

R~(s) P(s)N(s)V(s) + M(s)U(s)Q(s) Q(s)[V(s)N(s) + (s + 2 )U(s)M(s) - U(s)N(s)]"

we have Qol (t) = ~o P X2 ~< t, i=0 [X~° + YI°] ~< X 2

< ~] [)to + r~o] + Xt" + l, t

)

t=O =

q01 ( X ) d x

Therefore R* (s) is also a rational function. Using the result in Ref. [4] and the definitions 601 and y, we can get the result (1). (2) By using Cauchy theorem again, we can get from (1) that

~(s) q~'~(s) = - -

l+pl

Qto(t)= P{Y~ < t, Y2 < X1}

+T~i1

= .f[ ql°(x) dx

for+,

-i® ( s - z ) [ 1 + 2 G i ~ ( s - z ) ] dz

Res
Qoo(t) = Oil(l) = 0

(where Pl = 2EYI). Thus q ' l ( ? ) = o~. But noting

and then by using the definition of R~(t) (i = 1, 2) and the transitional relations between states 0 and 1, we get

f2(t)[e-~' + h o ( t ) , e - ~ q d t < 1

q~'~(O) = and

Ro(t) = qol(t) * Rl(t) + ff2(t)

g*(2) ~< 1,

Rl(t) = qlo(t) * Ro(t) + ffl(t)Cz(t). By taking Laplace transforms on both sides of above equations, Lemma 1 is proved.

Lemma 2. Letting 301 and y be the poles of R*(s) and ff'~2(s) that are situated closest to the fight of the s-plane, respectively, we have

we know that there exists at least one zero fl of 1 - g ~ ( s + 2)q~'l(S) such that y < Re(fl) < O. Because 6o, is assumed to be the pole of R*(s) which is closest to the fight of the s-plane, we have 7 < 60, < 0.

Corollary 1. 601 is the zero of 1 - g * ( s + 2)q~'t(s) which is closest to the fight of s-plane and 6ol = 6l. Proof This is straightforward.

(1) 601 and y are all real and negative, (2) 7 < 6 0 1 < 0 .

Theorem 1. The limiting distribution of residual lifetime of a two-dissimilar-unit parallel system, lim .... P{X(t) ~< x 17",.> t}, is exponential with rate - 6 1 > 0.

Proof (1) Since q*l (s) = L [ f z ( x ) ( e - ~ + ho(x) * e-~)]

Proof Since R~(s) is a rational function, we know that E, the set of residues, is finite. Therefore, by using a residue theorem and noting Lemma 2, we get

l~s>o

×If:

(e-~+ho(x'*e-Z')e-°-~'~dx] dz

Ro(t) = ~ Res{R~(s)e s', i} lEE

i

1 f[ +'~

--2hi

~(z)P*(s-z)

-

_~ 1 - - f ~ ( s - - z ) g ~ ( s - - z ) dz

Res>a

1 ~'+'®

(m -- 1)! S ~ l

+

~(z)

dz,

2ni ,]~_~ s + 2 - z - 2g*(s - z) Res>a

and that (b) in 2.1. yields if*(.) and G*(.) to be all rational functions because of f *(s) + sff*(s) = 1, then

d m- 1

lim ~

E

(s - 61 )mRS*(s)e s'

Res{R*(s) es',i}

iE~-{,h} m--l

= ~ ako(61)tke~'t+o(e ~l') k=O

[where m is the multiple number of the pole 61 and ako(6,) is not relevant to t (k = O, 1. . . . . m - 1), while

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Limiting distributions

Re s{D(s), i} represents the residue of function D(s) in i (i e E)]. Then lim P { X ( t ) <~x lTo > t} = 1 - lira R°(t + x)

302 the zero of 1-q~2(s)q*l(S) which is closest to the fight of the s-plane ),~ the pole off,*(s) which is closest to the right of the s-plane (i = 1, 2). The other notation is the same as that in Model I.

= 1 - e a~x. By using the same reasoning, we can obtain Rl(t) = ~ akl(fl)tke~" + o(e 6't) k=O

(where ak~(3~) is not relevant to t, k = 0, 1. . . . . m - 1). Thus

3.3. Conclusion

Theorem 2. The limiting distribution of residual lifetime of a two-dissimilar-unit warm standby system, l i m , ~ P { X ( t ) <~x I T~> t}, is exponential with rate - 6 2 > 0, in which 62 = max{~02 , 71,72}" Proof. Similarly to Theorem 1, we have the following Markov renewal equations

lim P { X ( t ) <~x I T1 > t} = 1 -- e a'*.

Ro(t ) = qo2(t) * R2(t ) +

t~oo

g,(t)

Rl(t ) = ql2(t) * Rz(t) + ffl(t)

2.4. Example

R2(t) =

Suppose f:(x) = ;t e x p { - 2 x } and g(x) =/~ e x p { - p x } (i = 1, 2). In this

q2,(t) •

RI(0 +

Pdt)

and then

1 -- g~' (s + 2)q~St(s) = 0

R*(s) =

~(s) + )(s)q?ds) I - q*2(s)q* (s)

implies (s + 22)(s + 2 + p) -- 22p = 0. Therefore 6, = ½[-(32 + #) + ~/2 2 + #2 + 62U1. 3. M O D E L

R~(s) =

R*(s) = q*2(s)R*(s) + ff~l (s).

2

3.1. Assumptions (a) The system is a two-dissimilar-unit warm standby system supported by a single repair facility. The repair discipline is "first fail, first repaired". (b) The lifetime of the online unit i is r a n d o m variable X~ with p.d.f, f ( . ) , and the lifetime of the warm standby unit i has a constant failure rate ~t~ (i -- 1, 2). The repair times of unit i from operating and warm standby states are assumed to be r a n d o m variables W~ with p.d.f, gi(') and V~ with p.d.f, e~(.), respectively (i = 1, 2). (c) The Laplace transforms of the f ( . ) , g~(.) and e~(.) are all rational functions (i = 1, 2). (d) U p o n repair, a unit becomes as good as new. (e) The failure times and the repair times of two units are mutually independent. 3.2. Notation State 0: epoch of entering this state is that unit 1 is beginning to work and unit 2 is in warm standby. State 1: epoch of entering this state is that unit 1 is beginning to work and unit 2 is beginning to be repaired because of a failure online. State 2: epoch of entering this state is that unit 2 is beginning to work and unit I is beginning to be repaired because of a failure online. T~ time to the first system failure given that the system enters state i at time t = 0 (i = 0, 1, 2)

h~(t) = ~ [a~e.... .e~(t)] ("~, i = 1,2 n=l

qo2(t) =fl (t)[h2(t) * e .... + e - " ] ql2(t) =fl (t)[g2(t) * h2(t) * e .... + g2(t) • e-':q q2l(t) =f2(t)[gl (t) * hi (t) * e .... + gt (t) * e-"q

~ ( 8 ) + ~l (s)q~l(S) 1 -- q*2(s)q* (s)

By assumption (c) in 3.1., and similarly to the proof in Theorem 1, we know that q*(s) (i = 1, 2) are all rational functions. Letting

E,As) q* = ro(s )

and

ff'~ ( s ) = ~

i , j = l,2,

we have R~(s) =

F:~(s)[G(s)a,(s)F~:(s) + C,(s)a:(s)E~:(s)J C 1(s)C 2(s) [/712(s)r21 (s) - El2 (s)E21 (s)]

Supposing that the multiple n u m b e r of pole 62 = max{y~, 72,602} is m, we have din-- 1

1 1)-----~sl -i ~m: Rl(t) = (m --

+

~.,

(s - 6 2 ) " R * ( s ) e ~'

Res{R*(s)e*',i}

ie E- {~2}

= ~ b,(f2)tkea~'+ o(e '~2') k~0

[where E is the set of residues of R'{(s), and bk(t~2) is not relevant to t (k = 0, 1. . . . . m - 1)]. Thus lim P{X(t) > xlT~ > t} = lim Rt(t + x) . . . . . . Rl(t) e~i2x.

By the same reasoning, we easily get

lira P{X(t) > x]T 2 > t} = e ~2x.

1072

W. L! and J. CAO

Since qo2(t) ~ f l ( t ) , we know that the pole of q~'2(s) must be the pole of f~l (s). Then the pole of q*2(s), i.e. the zero of Fo2(S), that is closest to the right of the s-plane is negative and less than y~. Noting

ng(s) -~ E02 (s)FI2 ($) [el ($)B2 (s)F21 (s) + C2 ($)S 1(s)e21 (s)] Fo2(s )C I ($ )C2 (s ) [FI2 (s )F21 (s ) - EI2 (S )E21 (s)] we can easily get

m--| n o ( t ) = ~ Ck(62)tk e62t + o(e ~2') k-0

(where Ck(62) is

not relevant to t, k = 0 , 1. . . . .

m - 1). Therefore

4.3. Conclusion

Theorem 3. The limiting distribution of residual lifetime of a two-dissimilar-unit cold standby system, limt~ ® P { X ( t ) <, xlTi > t}, is exponential with rate - 6 3 > 0, in which 63 = max{603, Yl, Y2}. Proof Similarly to Theorem 1, we have the following Markov renewal equations

lim P { X ( t ) > xlTo > t} = lim no(t + x_.___~=) e~2~ ,.~

603 the zero of I -L[f2(x)GI(x)]L~(x)G2(x)] which is closest to the fight of the s-plane. V~ the pole ofJ~*~(s) which is closest to the right of the s-plane (i = 1, 2). T~ time to the first system failure given that the system enters state i at time t = 0 (i = 0, i, 2). The other notation is the same as that in Model I.

,.o~

n0(t)

no(t) = f l (t)* R 1(t) + ffl (t)

"

Remark. When f l ( t ) = f 2 ( t ) , g l ( t ) = g2(t) and el(t) = e2(t), it can be shown that the result is identical to the result in [2].

Rl(t) = q12(t) * R2(t) + ff2(t) R2(t) = q2i(t) *Rl(t) + ffl(t), where q12(t)=f2(t)Gl(t) Therefore

4. MODEL 3

4.1. Assumptions (a) The system is a two-dissimilar-unit cold standby system supported by a single repair facility. The repair discipline is "first fail, first repaired". (b) The lifetime of the unit i is r a n d o m variable X~ with p.d.f, f(.), and the repair time of unit i is assumed to be r a n d o m variable Y~ with p.d.f, gt(') (i ----l, 2). (c) The Laplace transforms of the f ( . ) and gi(') are all rational functions (i = 1, 2). (d) U p o n repair, a unit becomes as good as new. (e) The failure times and the repair times of two units are mutually independent. 4.2. Notation State 0: epoch of entering this state is that unit 1 is beginning to work and unit 2 is in cold standby. State 1: epoch of entering this state is that unit 1 is beginning to be repaired and unit 2 is beginning to work. State 2: epoch of entering this state is that unit 2 is beginning to be repaired and unit 1 is beginning to work.

R*(s) = R*(s) =

and q21(t)=fl(t)(G2)(t ).

-~2 (s) + ff-'~l(s)q~(s) 1 -- q*:(s)q~l (s)

~(s) + ~ (s)q~,(s) 1 - q ~2(s)q*, (s)

R* (s) = ~ (s)R* (s) + ~ (s). By using the same method as that in Model 2, we can complete the proof. REFERENCES

1. J. Keilson, Systems of independent Markov components and their transient behaviours, In Reliability and Fault Tree Analysis, pp. 351-464. SIAM, Philadelphia (1975). 2. S. Kalpakam and M. A. Shahul Hameed, Quasistationary distribution of a two-unit warm-standby redundant system. J. Appl. Prob. 20, 429-435 (1983). 3. J. Cao and K. Cheng, Introduction to Reliability Mathematics, Science Press, Beijing, China (1986). 4. D. V. Widder, An Introduction to Transform Theory, Academic Press, New York (1971).