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Figure 5.3. Logical schemas of systems with different types of elements connectivity: (a) system connected in series, (b) system connected in parallel, (c) series-parallel system, (d) series-parallel system with distributed redundancy, Source: Gavrilov and Gavrilova, 2005.

components do not age = const), then the entire system connected in series does not age either.

Components connected in parallel A parallel system of a n independent components fails only when all the components fail (as in electrical circuits connected in parallel). The logical structure of parallel system is presented in Figure 5.3b.

An example of a parallel system is a system with components all performing an identical function. This function will be destroyed only in the case when all the components fail. The number of additional components in parallel structure with one and the same function is called the redundancy or reserve of the system. In living organisms vital organs and tissues (such as liver, kidney, or pancreas) consist of many cells performing one and the same specialized function.

For a parallel system with n independent components the probability of the system's failure, Qs, is the product of the probabilities of failure for its components, q{.

Hence the reliability of a parallel system, Ps, is related to the reliabilities of its components in the following way.

The reliability of a parallel system with components of equal reliability, p, is

What is very important is the emergence of aging in parallel systems—a parallel system is aging even if it is built of nonaging components with a constant failure rate (see more details in the section on causes of failure rate increase with age).

In a real world most systems are more complex than simply series and parallel structures, but in many cases they can be represented as combinations of these structures.

More complex types of reliability structure The simplest combination of the two reliability structures is a series-parallel system with equal redundancy shown in Figure 5.3c.

A general series-parallel system is a system of m subsystems (blocks) connected in series, where each block is a set of n components connected in parallel. It turns out that even if the components themselves are not aging, the system as a whole has an aging behavior— its failure rate grows with age according to the Weibull law and then levels off at advanced ages (Gavrilov and Gavrilova, 1991; 2001, 2003a). This type of system is important to consider, because a living organism can be presented as a system of critical organs and tissues connected in series, while each organ consists of specialized cells connected in parallel. A reliability model for this type of system is described in more detail in the section on causes of failure rate increase with age.

Another type of reliability structure, a series-parallel system with distributed redundancy, was introduced by Gavrilov and Gavrilova in 1991 (Gavrilov and Gavrilova, 1991; 2001). The series-connected blocks of this system have nonequal redundancy (different numbers of elements connected in parallel), and the elements are distributed between the system's blocks according to some particular distribution law (see schema in Figure 5.3d).

Gavrilov and Gavrilova (1991; 2001) studied the reliability and failure rate of series-parallel systems with distributed redundancy for two special cases. (1) the redundancy distributed within an organism according to the Poisson law or (2) according to the binomial law. They found that the failure rate of such systems initially grows according to the Gompertz law (in the case of the Poisson distributed redundancy) or binomial failure law in the case of the binomially distributed redundancy (Gavrilov and Gavrilova, 1991; 2001). At advanced ages the failure rate for both systems asymptotically approaches an upper limit (mortality plateau). Reliability models for this type of system are described in more detail in the section on theoretical models of systems failure in aging.

Now that the basic concepts of reliability theory have been discussed, we may proceed to linking them to empirical observations on aging and mortality.

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