Theorem 29.1. A system with monotone structure function having com- ponents with IFRA survival functions similarly has an IFRA survival function
30 Survival Functions for Series and Parallel Structures
t, by the very definition of the IFRA property for a system. It is therefore between L(0) and L(co), but
(29.99)
is the failure rate of the system at the origin. One may show by using expres- sion (22.5) for the structure function and the notion of random structure function (Section 25) that:
(1) the failure rate at the origin is zero if the width of the structure is greater than 1 (see Section 17, p. 66); that is, there does not exist a cut of just one component. Indeed, in this case failure of the system involves failure of at least two components, an event whose probability is of second order for t + 0, by virtue of the independence of the lifetimes of the components (hy- pothesis (4) of Section 24, p. 11 5) ;
(2) if there exist cuts of just one component, the failure rate of the system at the origin is the sum of the failure rates at the origin of the com- ponents that occur in these cuts.
Calculation of the superior limit L(co) of the " mean " failure rate is more delicate. Esary and co-workers [20] have shown that, in the particular case of all components having an exponential lifetime, and therefore a constant failure rate, the limit for t + 00 of the mean failure rate L(t) is equal to the minimum, calculated on the set of links of the structure, of the sum of the failure rates of all the components occurring in a link.
To conclude, we note that it is the " mean " failure rate L(t) of the system that is in the interval [L(O), L(co)]; the instantaneous failure rate of the system may, for some values of t , exceed the value L(m) since the system is IFRA and not, in general, IFR.
30 Survival Functions for Series and Parallel Structures.
Asymptotic Results for a Large Number of Components Series Structure. Consider a series structure of n components; its struc- ture function is
(30.1)
the reliability function is (30.2)
and the survival function is
cp(.Y) = .Y1 .Y2 ... .Y" , M P ) = cp(P) = PI P2 ... P n 7
1 60 I Y S T U D Y O F T H E R E L I A B I L I T Y O F S Y S T E M S
where vl(t), v2(t),
... ,
v,(t) are the survival functions of its components. This relation may also be written as(30.4) Log r ( t ) = Log r , ( f )
+
Log ~ . ~ ( f )+
...+
Log r n ( f ).
It follows that the cumulative failure rate A(?) of the system (cf. Section 4, or (29.60) and (29.61)) is
(30.5)
from which it follows by differentiation that (30.6)
The failure rate (instantaneous or cumulative) of the system is therefore the sum of the failure rates of its components. As a consequence, if all the com- ponents have an IFR survival function (respectively, DFR; see Sections 10 and ll), the system similarly has an IFR (respectively, DFR) survival function.
In particular, if all the components have an exponential survival function
(30.7) ui(t) = e-"'
with a constant failure rate Ai(t) = A i , then it is the same for the system:
(30.8) o(t) = e-''
where A is given by (30.6). The series structure thus presents particularly simple properties. It is at the same time the most common; it corresponds to the case where the failure of any single component entails failure of the system.
Parallel Structure. Consider a parallel structure of n components; the structure function is
(30.9) cp(.u) = I - i =
n
n 1 ( I - Xi) ,and the survival function is
(30.10) r ( f ) = 1 -
n
n ( 1 - V i ( f ) ) i = ILet @(t) = 1 - u(t) be the distribution law of the lifetime of the structure, and Qi(t) = 1 - ui(t) be the distribution of the lifetime of component i. We obtain
(30.1 I ) a t ) =
n
n Q i ( f ).
i = 1
30 S U R V I V A L F U N C T I O N S F O R S E R I E S A N D P A R A L L E L S T R U C T U R E S 161 We now calculate the failure rate A(t) of the structure
By noting that (30.13) we obtain (30.14)
If the components have exponential survival laws, Ai(t) = l i , (30.15)
with
(30.16) o ( t ) = 1 -
n
n (1 - e-"").
i = 1
Remark. If the components of a system S have an exponential survival law and S does not have a series structure, then the survival function of S is not exponential. Indeed, we shall see in Example 1 below that the failure rate given by (30.15) is not monotone, which shows that a system with monotone structure having components with IFR survival functions (which is the case with the exponential law) does not necessarily have an IFR survival function (we have seen however in Theorem 29.1 that it is IFRA).
Examples.
the survival laws are exponential :
(1) Consider a system composed of two components in parallel, for which
Set A,
+ I,
= 1 and take different values for ,Il/&; we obtain the graph of Fig. 30.1.The failure rate tends, for very large t , toward a limit that is equal to the failure rate of the best of the components.
(2) In order to increase reliability, one places in parallel with a piece of equipment A a second piece of identical equipment B. A switch C automati- cally assures exchange of the equipment to be used in case of failure.
1 62 I V S T U D Y O F T H E R E L I A B I L I T Y O F S Y S T E M S
0.51
= 0.3 &= 0.7
o.2 hl=0.2 h2=0.8
0.1
FIG. 30. I .
We suppose that the fact of being used or not does not in..Jence the rate of failure for the equipment7 and that any failure of the switch interrupts functioning of the system. We shall examine whether this " redundance" in fact increases the reliability, as we would hope.
Equipment 'R"
Switch "C"
Equipment "6"
FIG. 30.2.
The reliability network is shown in Fig. 30.2; it is a series-parallel struc- ture, C being in series with the two pieces of equipment A and B in parallel.
The links are { C, A } and { C, B }; the structure function is therefore (cf. Eq. (22.1))
(30.19)
'
If this hypothesis is not made, then the probability of failure of one piece of equip- ment would depend on the state of the other. This would contradict hypothesis (4) of Section 24, p. 115, and the theory developed here would not be applicable; see also Chapter V, Section 32.30 S U R V I V A L F U N C T I O N S F O R S E R I E S A N D P A R A L L E L S T R U C T U R E S 163 (since x i = xc) and the reliability function is
(30.20) h(p) = PC(2 P E - Pi)
where h(p) is the reliability of the system; p c is the reliability of the switch;
and p E is the reliability of the equipment (A or B).
Redundance is therefore useful if
(30.21) h(P) > P E
or
(30.22)
P d 2
- P E ) >.
In Fig. 30.3 we have traced the curvepc(2 - p E ) = 1 ; this defines two zones:
upper zone: redundance is useful, lower zone: redundance is detrimental.
If we can vary the time of functioning f, the point with coordinates { pE(t), p J t ) } will describe a curve like the dashed curve in the figure. If t is small (t -+ 0), p c and p E are near to 1 and the redundance will be useful if p c > p E (the reliability of the switch is greater than the reliability of the equipment). If t is large ( t
-,
a), p c and p E are small, and the redundance will always be detrimental.Useful
0.5
/ Detrimental
c
FIG. 30.3.
P E
Study of Systems Composed of a Large Number of Identical Components.
Consider a system composed of n components and let TI, T 2 ,
... ,
T. be the lifetimes of these components. If all the components are in series, the first failure of a component will entail failure of the system, and the lifetime of the system is(30.23) T, = min ( T , , T,, ..., T.)
.
164 I V S T U D Y O F T H E R E L I A B I L I T Y O F S Y S T E M S
If, on the other hand, all the components are in parallel, the system lifetime will be
(30.24) T, = max (TI, T,, ..., Tn) .
If the number n of components is very large, the distribution law of Ts and Tp is studied by the theory of extreme values (cf. Gumbel [26]).
We present here, in a simplified way, certain results of this theory allow- ing applications to the study of reliability.
The survival function us(t) of a system of n series components is
(30.25) u,(t) = u(t)”
where u(t) is the survival function of each component. If n tends toward infinity and if u(t)
-=
1, then u,(t) tends toward 0.We shall concern ourselves only with the case where u(t) is near to 1, that is, t sufficiently small (t + 0) in order that the components have excellent reliabilities. Then
(30.26) v,(t) = en Log ~ ( 0 ‘v 1 -u(f)l
The survival function up(t) of a system of IZ parallel components is (30.27) u,(t) = 1 - (1 - u ( r ) ) ” .
If n tends toward infinity and if v(t) > 0, then vp(t) tends toward 1. We restrict ourselves to the case where v(t) is near 0 (t sufficiently large so that the components have very weak reliabilities). Then
(30.28) o p ( t ) = 1 - ,nLog[I -u(t)l ~ 1 - ,-nu(?)
.
Examples.
(1) Let
1 if t Q t , , 0 if t > t , ,
(30.29) u(t) = 1 - /l(t - to>” [l
+
E ( t - to)] if to Q t Q t , ,where t , is defined by
(30.30) 1 - P(t1 - to)” [l
+
E ( t l - to)] = 0 ,and where E ( t - to) is a function that tends toward 0 when t tends toward
t o . This survival law has the form given in Fig. 30.4.
(30.31)
We have for t
-,
to ( t > to),Log o(t) ‘v - /l(t - t o ) ” .
30 S U R V I V A L F U N C T I O N S FOR SERIES A N D P A R A L L E L S T R U C T U R E S 165
f'"'
FIG. 30.4.
The survival law of a system of n components in series is thus (30.32)
Therefore v,(t) is near a Weibull law ((Eq. 6.25')) for t + t , . If to = 0 and cc = 1, we obtain the exponential law (cf. Eq. (30.8))
(30.33) u,(t)
-
,-"Or.
(2) If
(30.34) u(t)
-
e-"' as t + c owe obtain for n components in parallel,
(30.35) u p ( t ) ,., 1 - e-ne-""
and, in the particular case where k = 1, we have
(30.36) u(t)
-
e-ar, u p ( t )-
1 - e-ne-"'.(3) If
(30.37) u(t)
-
- U as t + 0 0 , u, rn positive,we obtain for n components in parallel, t"
(30.38)
We give two particular cases of this formula:
I66 1V S T U D Y O F T H E R E L I A B I L I T Y O F S Y S T E M S
(4) Let
0 if t 2 t ,
p(tl - t)" [I
+
&(t, - t ) ] if 0<
t ,< t ,(30.41) v ( t ) =
where &(ti - t ) tends toward 0 if t, - t tends toward 0, and where a and
B
are positive.
We obtain for n components in parallel
This is, afresh, a Weibull law.