## Complex Systems

In this chapter we introduce a general class of binary systems, which contain the ones dealt with in Chapter 1 as special cases. The reliability structures of most real-life binary systems can be modeled by a member of this class. We use the same basic notation as the one proposed in Section 1.2.1, but consider the system at a …xed time point or in the stationary regime. The foundations of the theory developed in this chapter have been laid by Birnbaum et al. [52]. The indicator variables of the system S and its elements ei; i = 1; 2; :::; n;

are denoted as

zs =

1 if S is available 0 otherwise

; zi =

1 if ei is available 0 otherwise

:

For zs and the zi being random variables, there exist probabilities ps and pi with

zs =

1 with probability ps 0 with probability 1 ps ; zi =

1 with probability pi 0 with probability 1 pi ;

where ps is called the system availability (reliability) and the pi are the element availabilities (reliabilities). Since zs and zi are (0; 1)variables,

ps = Pr(zs = 1) = E(zs); pi = Pr(zi = 1) = E(zi):

The following assumption is essential in what follows: The states of the elements uniquely determine the state of the system. Hence, for di¤erent systems S there exist di¤erent functions ' with

zs = '(z1; z2; :::; zn): (2.1)

' is called the structure function or the system function of S, and n is the order of the system or the order of '. Function ' characterizes the structural

The system availability ps = Pr(' = 1) is the expected value of ':

ps = E('(z1; z2; :::; zn)): (2.2)

Thus, knowledge of the structure function is crucial for determining the system availability. In particular, if the z1; z2; :::; zn are independent random variables, then we will see that ps is a function of the element availabilities p1; p2; :::; pn: In this case, letting p = (p1; p2; :::; pn), formula (2.2) is written as

ps = h(p) or simply as ps = h(p) if p = p1 = p2 = = pn: (2.3) Given a reliability block diagram of a system (or its fault tree), there are computerized algorithms for determining its structure function. Unfortunately, the computation time for obtaining the structure function is generally exponentially increased with increasing order n. Thus, developing computationally e¢ cient algorithms for determining ' is a main problem in reliability theory. In achieving this goal, the formalism of Boolean algebra will play an important role. Hence, we will next summarize its terminology and basic rules. Let x and y be any two (0; 1)variables, i.e. two Boolean or binary vari-

ables, which assume values 0 or 1. The key relations between x and y are conjunction, disjunction, and negation.