Page 175 - Fundamentals of Probability and Statistics for Engineers

158 Fundamentals of Probability and Statistics for Engineers

A

B

Figure 5.24 Parallel arrangement of components A and B, for Problem 5.20

5.20 Consider a system with a parallel arrangement, as shown in Figure 5.24, and let A
be the primary component and B its redundant mate (backup component). The
operating lives of A and B are denoted by T 1 and T 2 , respectively, and they follow
the exponential distributions
a 1 e ; for t 1 > 0;

a 1 t 1
f t 1 †ˆ
T 1
0; elsewhere;
a 2 e
a 2 t 2 ; for t 2 > 0;

f t 2 †ˆ
T 2
0; elsewhere:
Let the life of the system be denoted by T. Then T ˆ T 1 ‡ T 2 if the redundant part
comes into operation only when the primary component fails (so-called ‘cold
redundancy’) and T ˆ max (T 1 , T 2 ) if the redundant part is kept in a ready condi-
tion at all times so that delay is minimized in the event of changeover from the
primary component to its redundant mate (so-called ‘hot redundancy’).
(a) Let T C ˆ T 1 ‡ T 2 , and T H ˆ max (T 1 , T 2 ). Determine their respective prob-
ability density functions.
(b) Suppose that we wish to maximize the probability P(T t) for some t. Which
type of redundancy is preferred?
5.21 Consider a system with components arranged in series, as shown in Figure 5.25,
and let T 1 and T 2 be independent random variables, representing the operating
lives of A and B, for which the pdfs are given in Problem 5.20. Determine the pdf of
system life T ˆ min (T 1 , T 2 ). Generalize to the case of n components in series.
5.22 At a taxi stand, the number X 1 of taxis arriving during some time interval has a
Poisson distribution with pmf given by
k
e
p k†ˆ ; k ˆ 0; 1; 2; ... ;
k!
X 1

A B

Figure 5.25 Components A and B arranged in series, for Problem 5.21

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