For a parallel system, the entire system would perform satisfactorily if any one or more of its components or modes of operation is functioning satisfactorily; the entire system would fail only if all its components or modes of operation fail.

In the framework of load-resistance interference for different modes of oper­ation, the failure probability of a parallel system, according to Eq. (7.9), is

which can be computed as the multivariate normal probability discussed in Sec. 2.7.2. The bounds for system failure probability also can be computed if the exact value of Pf, sys is not required. In the case that all performance variables W’s are independent, the system failure probability reduces to

M

Pf, sys = П ®(-Pm) (7.44)

m=1

Alternatively, the reliability of a parallel system can be expressed as

The second-order bounds for this system reliability, according to Eq. (7.27), are

in which L( jij, em 1 pjm) — P [Zj > ej, Zm > em — Ф( ej, em 1 pjm) +

Ф(в j) + Ф^)-!.

Example 7.8 Referring to Example 7.6, determine the system reliability by consid­ering that the system would fail if all three modes of operation fail.

Solution Since the system is in parallel, the system failure probability can be calculated as

Pf, sys — P(Wi < 0, W2 < 0, W3 < 0)

— P(Z1 < -2.68, Z2 < -3.46, Z3 < -2.68) — 0.0001556

which is obtained in Example 7.6. Hence the reliability of the system is 0.9998444.

In the framework of time-to-failure analysis, the unreliability of a parallel system involving M independent components can be computed as

M

pf, sys(t) — ^ pf, m(t)

m—1

in which pf, m(t) — P(Tm < t), the unreliability of the mth component within the specified time interval (0, t]. Hence the system reliability in time interval (0, t] is

M

Ps, sys(t) = 1 – Pf, sys(t) = 1 – П Pf, m(t) (7.48)

m=l

The failure density function f sys(t) and failure rate hsys(t) for a parallel system consisting of M independent components are

For each component having an exponential failure density function with the parameter Xm, for m — 1,2,…, M, the failure probability of a parallel system can be computed as

M

Pf ,sys(t) — П (1 – e~lmt) (7.51)

m—1

with the corresponding system reliability

M

Ps, sys(t) — 1 – П (1 – e-lmt) (7.52)

m—1

The system failure density function f sys(t) is

In the case that all components have an identical failure rate, that is, X1 — X2 — ■ ■ — XM — X, the MTTF of the system is

The unavailability of a parallel system involving M independent compo­nents is

M

Usys(t) —Ц Um(t) (7.55a)

m—1

and the corresponding system availability is

M

Asys(t) = 1 – Ц Um(t) (7.55b)

m=1

Under the condition of independent exponential repair functions for the M components, the unavailability of a parallel system is

and the stationary system unavailability is

Example 7.9 As an example of a parallel system, consider a pumping station con­sisting of two identical pumps operating in a redundant configuration so that either pump could fail, and the peak discharge could still be delivered. Both pumps have a failure rate of k = 0.0005 failures/h, and both pumps start operating at t = 0. The system reliability for a mission time of t = 1000 h, according to Eq. (7.52), is

ps, sys(t = 1000) = 1 – (1 – e-(0 0005>(1000>)(1 – e-(0-0005)(1000)) = 0.8452 The MTTF, according to Eq. (7.53), is

In this section the reliability of some simple systems will be discussed. In the framework of time-to-failure analysis, availability of such systems will be pre­sented. Information such as this is essential to serve as the building blocks for determination of reliability or availability of more complex systems.

7.3.1 Series systems

A series system requires that all its components or modes of operation perform satisfactorily to ensure a satisfactory operation of the entire system. In the context of load-resistance interference, the failure event associated with a mode of operation is

Fm = {Wm < 0} form = 1, 2,…, M

in which Wm is the random performance variable associated with the mth mode of operation. Referring to Chap. 4, the failure probability and reliability asso­ciated with the mth mode of operation, respectively, are

P (Fm) = P ( Wm < 0) = P (Zm < ~Pm) = ®(-fim) (7.28a)

P (Fm) = P ( Wm > 0) = P (Zm > – Pm) = 4Pm) (7.28b)

in which Zm is the standard normal random variable associated with Wm, and pm is the reliability index associated with the mth mode of operation.

The failure probability of a series system involving M modes of operation, according to Eq. (7.1), can be expressed as

Because all the standardized normal random variables Zm’s generally are cor­related, computation of the exact system failure probability using Eq. (7.29) may not be practical, especially when the number of modes of operation M is large. For this case, the second-order bounds for Pf, sys could be viable. According to Eq. (7.26), the bounds for system failure probability are

in which Ф(—вj, —em I Pjm) is the bivariate normal probability, which can be computed by procedures described in Sec. 2.7.2, with Pjm being the correlation coefficient between the performance variables Wj and Wm for the j th and mth modes of operation. Accordingly, the bounds on reliability of a series system can be obtained easily by using Eq. (7.11b).

Although computation of the exact bivariate normal probability can be ob­tained through numerical integration, sometimes information about its bounds is sufficient. Under a positively correlated case, narrow bounds of Ф(—вj, —em I Pjm > 0) that require evaluations of only univariate normal probabilities are

where ві Im = в, Ртв" (7.32a)

v/irpm

emu = em,— P’"’m’ (7.32b)

j

In the case that the pair of performance functions is negatively correlated, the bounds for joint failure probability are

0 < Ф(— Єі, —em I Pjm < 0) < minm—emm—ej | m), Ф( — в} m—em | j )] (7.33)

The derivations of Eqs.(7.31) and (7.33) are given in Appendix 7A. Ang and Tang (1984) pointed out that use of an approximation of Eq. (7.31) could improve (tighten) the second-order bound of Eq. (7.30) when the single-mode failure probabilities are small, say, on the order of 10—4. However, if the single-mode
failure probabilities are all large (e. g., 10-2), the bound of Eq. (7.31) will be wide.

In fact, the reliability of a series system can be computed, according to Eq. (7.2), as

It should be pointed out that, in general, P [n(Zm > —em)] = P [n(Zm < fim)] unless for the univariate case. As can be seen, the reliability of a series system is the multivariate normal probability whose determination can be made by Ditlevsen’s approach, described in Sec. 2.7.2, or by various bounding approaches discussed in Sec. 2.7.3.

Example 7.6 Consider a system consisting of three modes of operation, each of which is specified by the following linear performance functions:

W1( X) = X1 + 2 X2

W2( X) = X1 + X2 + X3

W3( X) = X2 + 2X3

in which the stochastic basic variables X1, X2, and X3 are multivariate normal ran­dom variables with the vector of means

Vx = (М1, М2, М3/ = (6, 6, 6/

and covariance matrix

The state of the system is such that if any of the three modes of operation fail, the system would fail. Calculate the system reliability.

Solution From the preceding covariance matrix Cx, it is understood that all three stochastic basic variables are uncorrelated, each with a variance of 9, that is, Var(X1) = Var(X2) = Var(X3) = 9. The vector of expected values of W1, W2, and W3 is

Vw = (Mw1, Mw2, Mw3) = [6 + 2(6), 6 + 6 + 6, 6 + 2(6)/ = (18, 18, 18/

The covariance matrix of the three performance functions W’s can be computed as

Cw = S1 Cx S

in which S, the sensitivity matrix, is an K x M matrix, with M and K being the number of performance functions and stochastic basic variables, respectively. The sensitivity matrix S contains, in each column, the vector of sensitivity coefficients for each performance function with respect to individual stochastic basic variable, that is,

S = [S1, 82,…, sm ]

for m = 1,2,…, M. In this example, since all performance functions are linear, the sensitivity matrix consists of coefficients in the performance functions, that is,

Hence the covariance matrix of the three performance functions can be obtained as

As can be seen, even though the three stochastic basic variables are uncorrelated, the three performance functions are correlated because they are defined by some stochastic basic variables common to the other performance functions. Hence the variances of W1, W2, and W3 appear on the diagonal of Cw, namely,

Var( W1) = 45 Var( W2) = 27 and Var( W3) = 47

The corresponding correlation matrix of random W’s can be obtained easily as

‘1.000 0.7746 0.4000’

0.7746 1.000 0.7746

0.4000 0.7746 1.000

The system failure probability is defined as

pf ,sys = P [(W1 < 0) U (W2 < 0) U (W3 < 0)]

= P [(Z1 < -2.68) U (Z2 < -3.46) U (Z3 < -2.68)]

The exact system failure probability can be obtained as pf sys = P [(Z1 < -2.68) U (Z2 < -3.46) U (Z3 < -2.68)]

= [P(Z1 < -2.68) + P(Z2 < -3.46) + P(Z3 < -2.68)]

– [P(Z1 < -2.68, Z2 < -3.46) + P(Z1 < -2.68, Z3 < -2.68)

+ P(Z2 < -3.46, Z3 < -2.68)] + P(Z1 < -2.68, Z2 < -3.46, Z3 < -2.68)

= (0.003681 + 0.0002701 + 0.003681) – (0.0001659 + 0.0001987 + 0.0001659) + 0.0001556 = 0.0072572

Hence the system reliability ps>sys = 1 – 0.0072572 = 0.9927428. Note that the preceding bivariate normal probabilities are calculated by Eq. (2.121), whereas the trivariate normal probability is computed according to Ditlevsen’s algorithm using Taylor expansion described in Sec. 2.7.2.

Alternatively, the second-order bounds for the system failure probability can be computed according to Eq. (7.30). The results are

0.007102 < pf, sys < 0.007268

and the corresponding bounds on the system reliability ps, sys are

0.992732 < ps, sys < 0.992898

In the framework of time-to-failure analysis, the reliability ps, m(t) and failure probability pf, m(t) of the mth component over the time interval (0, t ], according to Eqs. (5.1a) and (5.1b), are

TO

fm(r) dr (7.35a)

and P(Fm) = pf, m(t) — [ fт(т)dx (7.35b)

70

respectively, where f m(t) is the failure density function for the mth component.

In the case that the performance of individual components is independent of each other, the reliability of a series system is

Similarly, the availability of a series system involving M independent compo­nents is

M

Asys(t) — Ц Am(t) (7.37)

m—1

in which Aeys(t) and Am(t) are availabilities of the entire system and the mth component, respectively, at time t.

According to Eqs. (5.2) and (5.3), the failure density function f sys(t) and the failure rate hsys(t) for a series system involving M independent components can be derived, respectively, as

For the special case of an exponential failure density function such as

for t > 0, Xm > 0, m — 1,2,…, M

the reliability and unreliability of a series system with M independent compo­nents, respectively, are

Assuming an exponential repair function for each independent component, the availability and unavailability for a series system, according to Eqs. (7.37) and (5.59), are

and Usys(t) — 1 Asys(t)

in which nm is the constant repair rate for the mth component, and Usys(t) is the system unavailability at time t. The stationary system availability, by Eq. (5.60), can be expressed as

in which MTTRm and MTTFm are, respectively, the mean time to repair and mean time to failure of the mth component.

Example 7.7 As an example of a series system, consider a pumping station consisting of two different pumps in series, both of which must operate to pump the required quantity. The constant failure rates for the pumps are Л1 = 0.0003 failures/h and Л2 = 0.0002 failures/h. For a 2000-h mission time, the system reliability, according to Eq. (7.40a), is

ps, sys(t = 2000) = exp[-(0.0003 + 0.0002)(2000)] = 0.90484 and the MTTF of the system is

Despite the system under consideration being a series or parallel system, the evaluation of system reliability or failure probability involves probabilities of union or intersection of multiple events. Without losing generality, the deriva­tion of the bounds for P (A1U A2 U—U AM) and P (A1 n A2 П—П AM) are given below.

For example, consider P(A1 U A2 U-U AM). When Am = Fm, the probability is for the failure probability of a series system, whereas when Am = Fm, the probability is for the reliability of a parallel system. The bounds of system failure probability, that is, can be defined as follows:

Pf ,sys – Pf, sys – Pf, sys (7.11a)

with p f sys and p f, sys being the lower and upper bounds of system failure prob­ability, respectively. The corresponding bounds for system reliability can be obtained as

1 – Pf, sys = PS, SyS – Ps, sys – Ps, sys = 1 – Pf, sys (7.11b)

Similarly, after the bounds on system reliability are obtained, the bounds on system failure reliability can be computed easily.

First-order bounds. These bounds also are called unimodal bounds (Ang and Tang, 1984, p. 450). They can be derived as follows. Referring to Eq. (2.7), the probability of the joint occurrence of several events can be computed as

P ( П A^ = P (A1) x P (A2 | A1) x—x P ( Am | Am-1, Am-2, ■ ■■, A2, A1)

m=1 )

(7.12)

Under the condition that all events Aj and Am are positively correlated, the following inequality relationship holds:

P (Am | Aj ) > P (Am)

Hence

P (Aj, Am) = P (Am | Aj)P ( Aj ) > P ( Am)P (Aj )

This can be extended to a multiple-event case as

M П Am m=1
P	(7.13)
m1

As can be seen, the lower bound of the probability of an intersection is when all events are as if they are independent. Furthermore, it is also true that

П Am c Aj for any j = 1, 2, ■■■, M

m=1

Therefore,

c min{ A1, A2, ■■■, Am }

Consequently,

Based on Eqs. (7.13) and (7.14), the bounds on probability of joint occurrence of several positively correlated events are

Example 7.3 Consider three standardized normal random variables Z1, Z2, and Z3 with the following correlation matrix:

Compute the first-order bounds for P{(Z1 < —2.71) U (Z2 < —2.88) U (Z3 < —3.44)}.

Solution The three events corresponding to the preceding trivariate normal probability are

A1 = {Z1 < -2.71} A2 = {Z2 < -2.88} A3 = {Z3 < -3.44}

Since,

P(A1 U A2 U A3) = 1 — P(Aj П A П A3)

the first-order bounds for P( A1U A2 U A3) can be obtained from those of P(A1П A П A3).

To derive the first-order bounds for P(A1 П A П A3), individual probabilities are required, which can be obtained from Table 2.2 or Eq. (2.63) as

P(A[) = P(Z1 > -2.71) = 0.99664

P(A2) = P(Z2 > -2.88) = 0.99801

P(A3) = P(Z3 > -3.44) = 0.99971

Furthermore, because all Am’s are positively correlated, all Am’s are also positively correlated. According to Eq. (7.15), the first-order bounds for P(A1 П A^ П A3) is

(0.99664)(0.99801)(0.99971) < P[ n Am) < min{0.99664, 0.99801, 0.99971}

m=1

which can be reduced to

Therefore, the corresponding first-order bounds for P(A1 U A2 U A3) is

which can be reduced to

Referring to Eq. (7.2), the first-order bounds for reliability of a series system with positively correlated nonfailure events can be computed as

M

П P (Fm) < ps, SyS < min{P (Fm)} (7.16a)

m=1

or in terms of failure probability as

M

IT [1 – P (Fm)] < Ps, sys < min{1 – P (Fm)} (7.16b)

m

m=1

Similarly, referring to Eq. (7.7), by letting Am = Fm, the first-order bounds on the failure probability of a parallel system with positively correlated failure events can be immediately obtained as

M

TT P(Fm) < Pf, sys < min{P(Fm)} (7.17a)

m

m=1

and the corresponding bounds for the system reliability, according to Eq. (7.11b), is

Example 7.4 Consider that the M identical components in a system are positively correlated and the component reliability is ps. Determine the reliability bounds for the system.

Solution If the system is a series system, the bounds on system reliability, according to Eq. (7.16a), are

PM < Ps, sys < Ps

When the system is in parallel, the bounds on the system reliability, according to Eq. (7.17b), are

Ps < Ps, sys < 1 – (1 – Ps)M

The bounds for system reliabilities for different M, with the component reliability of 0.95, for series and parallel systems are shown in Fig. 7.6. As can be observed, the bounds for the system widen as the number of components increases.

In the case that all events Am’s are negatively correlated, the following rela­tionships hold:

P (Am | Aj) < P (Aj) for all j, m

P(Aj, Am) < P(Aj )P(Am)

Hence the first-order bounds for the probability of joint occurrence of several negatively correlated events is

M

< П P (Am)

m=1

The bounds for reliability of a series system, with Am = F’m, containing nega­tively correlated events are

M

0 < Ps, sys < [1 – P (Fm)]

m=1

whereas for a parallel system, with Am = Fm,

It should be pointed out that the first-order bounds for system reliability may be too wide to be meaningful. Tighter bounds sometimes are required and can be obtained at the expense of more computations.

Second-order bounds (Bimodal bounds). The second-order bounds are obtained by retaining the terms involving the joint probability of two events. By Eq. (2.4), the probability of the union of several events is

p(u Am) = E P (Am)-EE P (Ai, Aj) + Y, P (Ai, Aj, Ak)-•••

‘ ‘ m=1 i< j i< j < k

+ (-1) MP (A1, A2,…, Am ) (7.18)

Notice the alternating signs in Eq. (7.18) as the order of the terms increases. It is evident that the inclusion of only the first-order terms, that is, P(Am), produces an upper bound for P (A1 U A2 U ■ ■ ■ U AM). Consideration of the only the first two order terms yields a lower bound, the first three order terms again an upper bound, and so on (Melchers, 1999).

Simple bounds for the probability of a union are

It should be pointed out that these bounds produce adequate results only when the values of P(Am) and P(Am, Aj) are small. Equation (7.18) alternatively can be written as

P M Am) = [P (A1)] + [P (A2) – P (A1, A2)] + [ P (A3) – P (A1, A3)

m=1

— P (A2, A3) + P (A1, A2, A3)] + [P (A4) — P (A1, A4) — P (A2, A4)

— P (A3, A4) + P (A1, A2, A4) + P (A1, A3, A4) + P (A2, A3, A4)

— P (A1, A2, A3, A4)] + [P (A5) ••• (7.20)

To derive the lower bound, consider each of the terms in brackets in Eq. (7.20). For example, consider the bracket containing the terms associated with event A4. Note that apart from P(A4), the remaining terms in the bracket are

-P [(A1, A4) U (A2, A4) U (A3, A4)]

Furthermore, event A4 contains (A1, A4)U(A2, A4)U(A3, A4), which implies that P (A4) > P [(A1, A4) U (A2, A4) U (A3, A4)]. Consequently, each of the bracketed terms in Eq. (7.20) has a nonnegative probability value. Also notice that

and thus the following inequality holds:

3

> P(A4) – Y P (Aj, At)

j=1

This equation can be generalized as

m-1

> P(Am) – Y P (Aj, Am) (7.21)

j=1

It should be pointed out that the terms on the right-hand-side of Eq. (7.21) could be negative, especially when m is large. Owing to the fact that each of the

bracketed terms should be nonnegative, abetter lower bound to Eq. (7.20) can be obtained if the right-hand-side of Eq. (7.21) makes a nonnegative contribution to the lower bound (Ditlevsen, 1979), namely,

Earlier, Kounias (1968) proposed an alternative second-order lower bound by selecting only those combinations in Eq. (7.20) which give the maximum values of the lower bound:

pQ^ Am^j > P(A1) + max J ^[P(Am) – P(Aj, Am)](7.23)

j <m J

It should be pointed out that both lower bounds for the probability of a union depend on the order in which the events are labeled. Algorithms have been de­veloped for identifying the optimal ordering of events to obtain the best bounds (Dawson and Sankoff, 1967; Hunter, 1977). A useful rule of thumb is to order the events in the order of decreasing importance (Melchers, 1999). In other words, events are ordered such that P(A[1]) > P ( A[2]) > ••• > P(A[M]), with [m] representing the rank of the event according to its probability of occurrence. For a given ordering, Ramachandran (1984) showed that the lower bound pro­vided by Eq. (7.22) is better than Eq. (7.23), whereas both bounds are equal if all possible orderings are considered.

To derive the upper bound, attention is focused back to Eq. (7.20) and on each of the terms in brackets. For example, consider the bracket containing the terms associated with event A4. As discussed earlier, apart from P ( A4), the remaining terms in the bracket are

—P [(A1, A4) U (A2, A4) U (A3, A4)]

Using the fact that P(A U B) > max[P(A), P(B)], the following inequality holds:

U (Aj, A4) > max[P(Aj, A4)]

Lj <4 j j <4

Hence the probability in the fourth bracket involving event A4 satisfies

This inequality relation is true for all the bracketed terms of Eq. (7.22), and the upper bound can be obtained as

Example 7.5 Refer to Example 7.3. Compute the second-order bounds for the multi­variate normal probability.

Solution To compute the second-order bounds, the probabilities of individual events as well as the joint probabilities between two different event pairs must be computed. From Table 2.2 or Eq. (2.63), the probabilities of individual events are

P(A1) = P(Z1 < -2.71) = 0.003364

P(A2) = P(Z2 < -2.88) = 0.001988

P(A3) = P(Z3 < -3.44) = 0.0002909

The joint probabilities, according to the procedures described in Sec. 2.7.2, are P(A1, A2) = P(Z1 < -2.71, Z2 < -2.88 | p = 0.841) = 0.0009247

P(A1, A3) = P(Z1 < -2.71, Z3 < -3.44 | p = 0.014) = 0.000001142

P(A2, A3) = P(Z2 < -2.88, Z3 < -3.44 | p = 0.536) = 0.00004231 The lower bound of P(A1 U A2 U A3), according to Eq. (7.23), is

= P(A1) + max{[P(A2) – P(A1, A2)], ^ + max{[P(A3) – P(A1, A2) – P(A2, Ae)],0} = 0.003364 + max[[0.001988 – 0.0009247], 0} +max{[2.909 x 10-4 – 1.142 x 10-6) -4.231 x 10-5], 0} = 0.003364 + 0.00106633 + 0.00024736 = 0.004675

The upper bound of P(Ai U A2 U A3), according to Eq. (7.25), can be computed as

3 3

V P(Am) – У max[P(Aj, Am)]

m=1 m=2

3

= У P(Am) – {max[P(A1, A2)] + max[P(A1, A3), P(A2, A3)]}

m=i

= (0.003364 + 0.001988 + 0.002909) У max(0.0009247)

+ max[(1.142 x 10-6, 4.231 x 10-5)]} = 0.005641 – (0.000924 + 0.0000425)

= 0.004677

In summary, the second-order bounds for the trivariate normal probability P (A1 U A2 U A3) are

0.004675 < P(A1 U A2 U A3) < 0.004677

Comparing with the first-order bounds obtained in Example 7.3, the second-order bounds are definitely tighter.