On the reliability of a cold standby system attended by a single repairman

Microelectron. Reliab., Vol. 35, No. 12, pp. 1511-1513, 1995 Copyright ~) 1995 ElsevierScienceLtd Printed in Great Britain.All rights reserved 0026-2714/95 $9.50+ ~00

Pergamon

0026-2714(94)00149-9

ON THE RELIABILITY OF A COLD STANDBY SYSTEM ATTENDED BY A SINGLE REPAIRMAN E. J. Vanderperre Department of Mathematics and Computer Science, King Mongkut's Institute of Technology, Ladkrabang, Chalongykrung Road, Ladkrabang, Bangkok 10520, Thailand (Received for publication 26 August 1994) Abstract--We present a reliability analysis of a multiple cold standby system satisfying the usual conditions (i.i.d. random variables, single repair facility, perfect repair, queueing). Each unit has a constant failure rate but an arbitrary repair time distribution. We introduce a basic set of measures related to a stochastic process defined on some filtered probability space. The set of measures constitutes a system of renewal integral equations. The solution is constructed by means of a Cauchy integral. A particular case (deterministic repair) provides some explicit results illustrated by a computer-plotted graph.

INTRODUCTION Apart from important but particular cases, such as state dependent exponential distributions, e.g. [1, 2-1, the reliability analysis of multiple cold standby systems--subject to arbitrary distributions--is far from complete. Our present paper deals with a reliability analysis of a K + l unit cold standby system satisfying the usual conditions (i.i.d. random variables, single repair facility, perfect repair, queueing). Each unit has a constant failure rate but an arbitrary repair time distribution. Thus, on the one hand, our assumptions are less general with respect to the assumptions proposed in Refs [1, 2] (state dependent parameters), but on the other hand more general since we allow arbitrary repair. We introduce a basic set of measures related to a stochastic process defined on some filtered probability space. The set constitutes a system of renewal integral equations. The solution is constructed by means of a Cauchy integral [3-1. A particular case (deterministic repair) provides some explicit results illustrated by a computer-plotted graph.

FORMULATION Consider a K + 1 unit cold standby system subject to the usual conditions. Each unit has a constant failure rate 2 and an arbitrary repair time distribution

R(t), R(0) = 0. Let X, be the number of units in the repair-shop at time t, so that X,~ {0, 1. . . . . K , K + 1}. The non-Markovian stochastic process {Xt, t/> 0} is defined on some filtered probability space, {f2, M , P , ~ } where the history ~ ' , = { 4 , t>~0} satisfies the Dellacherie conditions:

(1) ~o contains the P-null sets of ~ , (2) Vt ~> 0, ~ = n ~-u i.e. ~ is right-continuous. u>t

Define the o~-stopping times inf{t > 0:X~ = K + llXo = 1}, f OK ,= ~ if the set is non-empty, / t o o , otherwise. inf{t > 0: X~ = 0]Xo = 1}, ® ,=

if the set is non-empty, ~ , otherwise.

In reliability theory, OK is called the first systemdown time. The survival function I-4, 5] will be denoted by ~Kt(t):=P{Ox

> tlXo = I},

t~>0.

0 is called the busy period of the repairman. The origin of time is fixed at the instant of the first failure, so that a busy period starts at time t = 0 (Xo = 1, P - a.s). We introduce the notations r the repair time of a unit no the number of repairs completed during 0 tn the instant of the nth repair completion after t=0;n=l, 2. . . . Zk a gamma distributed random variable with parameters ~, k; k = 1, 2 , . . . {v,} a Poisson process with parameter ~.t, t >~ 0 (A) the indicator of an event A ~ 6~k the Kronecker function [t] the greatest integer <~ t r(s) the Laplace-Stieltjes transform of R(t), i.e. r(s) ,= E e -s', Re s >/0. Do a circle with center at the origin of the complex co-plane, with counter clockwise orientation and

1511

1512

E.J. Vanderperre radius [col < Ip(s)[, where p(s) is the smallest root (in absolute value) of the functional equation co - r(s + 2 - 2co) = 0, I-6].

The set {g~r)(s); k = 0 . . . . . K - 1} is then uniquely determined by the K equations

Oir)(s) = E{e-"(v, = k)} k + l--6k,K-i

+

PRELIMINARY RESULT

X

g~)(s)E{e-S'(v, = k + 1 -- m)}.

ra=l

Observe that state 0 is regenerative for the process {X,, t >~0}. Therefore the following renewal integral equation holds,

Clearly for K >/ 1, E Ie~ ~o(\0<~
1)}=g(or)(s),

whereas P{®K ~ 0} E{e-~°~(o mino X, > 0 ) } +

Ifl

dP{rl ~
= E{e-"~(rK < r)}

v+w+z<~t

K+I

t',w,z~> O

+ ~ gkK)_k(s)E{e - ' .... (zk+ l < r)}. k:l

× d P { O ~< w, 0
1} SOLUTION OF THE EQUATIONS

× dP{0K ~< z}.

Theorem

Whence for Re s ~> 0,

Fork=0,

Ete-~°~ ( min X ~ > 0)} ~ ,,0
Ee_SO ~ =

E e -~° max X , < K + Z+ S ( \o
1-7 ~-

1. . . . . K -

~fo g~K~(s) = g(°K~(S)

1

acg'(s) =

INTEGRAL EQUATIONS In order to compute the above functionals, we introduce the following set of measures: I ; K > ~ 1, let

G~X'~k(t)'=P{O<~t'n°>~n'X'"~=k' o<~1 1} is the restriction of {iX',, t/> 0} to the state space {0, 1 .....

G~K)k(t) = (K) G..~(t) =

P{v~ = k} dR(z),

k+ i -O~.K-I fO

y~

Lf

Gi"_)~,m(t- ~) I-re}dR(z).

In order to solve the system of integral equations, we define g.,k(S)'= ~ '

;o

e-S~dG~K~(t),

Rest>O,

dco

2zi Jo,., cot +1 r(s + 2 - 2co) - co The proof of the statement follows from direct substitution of the expression into the above equations, noting that for Do, D~: 0 < Izl < Icol < [p(s)l,

r(s+2-Az) _ 1 [ " dco r(s+A-Aw) r(s + 2 -- 2z) -- z 2~zi JD,,, m - z r(s + 2 - 2co) - co' whereas for k = 0, 1. . . . E{e-~r(vr = k)}

If 2~zi

dzkZT r(s + 2

There exist several methods to evaluate the Cauchy integrals. A very general method is based on Faa di Bruno's formula, e.g. Ref. 17]. However, it would not be handy to rely upon a single prescribed method. For instance, in the following case, we use Mertens' theorem [8]. DETERMINISTIC REPAIR

n~>l, Let

'=

g,.k(s). n=l

2z).

D~

Remark

m=l

× P{v z = k +

r(s+2-2o)

1 ~ dco r(s + 2 - 2co) 21ri - - L ,~ COKr(s + ,~ -- ; ~ - - co

K).

By the law of total probability, we obtain

dco r(s+;~-2co) ,COg+, r(s + ~2-~ ~o) ~ co

1 ~ do

Note that 0 K is finite P - a.s.

Forn>~ 1 ; k = 0 , . . . , K -

1;K~> 1,

1,

if t /> to /> O,

R(t)= O, i f t < t o .

Standby system

1513

W e fix to as time-unit, hence r = 1 P - a.s., so t h a t r(s) = e -~.

U s i n g M e r t e n s ' t h e o r e m , we o b t a i n r(s + ~. -- ).o9) r(s + ). - ).o) --

=

(-- 1) j J! k=O j=o

o

X e t~+ ~)(k- j) (k - j ) jok. W h e n c e for k / > 0,

1 ~ 27zi

do)

lIO)(t)

r(s+2-2o)

L

, (ok + l r(s + 2 -- 2 0 ) -- e) k

"

Fig. 1. Graph of ~ m ( t ) and

"

= ~ (-1)J/"!e(~+z'(t-J)(k-j) ~o j!

.~(2)(t).

j, REFERENCES

so t h a t E e -soK is completely d e t e r m i n e d , I n v e r t i n g the Laplace t r a n s f o r m 1 -

E e - so~

......

,

Res>0,

K=I,2

S

yields,

~(21(t) = e -z'

1+

ttl

(t -- k) k

k=o

k!

)k+l

C],

( E x p r e s s i o n s for K > 2 are r a t h e r lengthy a n d therefore omitted.) T h e following figure s h o w s the g r a p h of .~"~(t) c o m p a r e d with the g r a p h of ~(2)(t), t >~ O, 2 = 0.5.

1. V. Anisimov and J. Sztrik, Asymptotic analysis of some complex renewable systems operating in random environments, Eur. J. Opns Res. 41, 162-168 (1989). 2. L. R. Goel and P. Shrivastava, Transient analysis of a multiple-unit redundant system, Microelectron. Reliab. 32, 1361-1365 (1992). 3. L. H6rmander, An Introduction to Complex Analysis in Several Variables, North-Holland, Amsterdam (1991). 4. A. Birolini, On the use of stochastic processes in modelling reliability problems. In Lecture Notes in Economics and Mathematical Systems, p. 252. Springer, Berlin (1985). 5. M. Shaked and J. G. Shanthikumar, Reliability and maintainability, In Handbook in Operations Research and Management Science, Eds D. P. Heyman and M. J. Sobel, 2. North-Holland, Amsterdam (1990). 6. J. W. Cohen, The Single Server Queue, North-Holland, Amsterdam (1982). 7. L. Tak~cs, Combinatorial Methods in the Theor) of Stochastic Processes. Wiley, New York (1967). 8. T. M. Apostol, Mathematical Analysis. Addison-Wesley, Amsterdam (1978).