Fundamentals of Probability and Statistics for Engineers

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

Let the life of the system be denoted by T. Then T T 1 T 2 if the redundant part

comes into operation only when the primary component fails (so-called ‘cold

redundancy’) and T 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 TC T 1 T 2 ,andTH 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 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

A

B

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

A B

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

158 Fundamentals of Probability and Statistics for Engineers

fT 1 …t 1 †ˆ

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