## Systems Failure And Reliability Structure

A branch of reliability theory, which studies reliability of an entire system given the reliability of its components and components' arrangement (reliability structure), is called system reliability theory (Rausand and Hoyland, 2003). System reliability involves the study of the overall performance of systems of interconnected components. The main objective of system reliability is the construction of a model that represents the times-to-failure of the entire system based on the life distributions of the components, from which it is composed. Consideration of some basic ideas and models of the system reliability theory is important because living organisms may be represented as structured systems comprised of organs, tissues and cells.

System reliability theory tells us that the component arrangement strongly affects the reliability of the whole system. The arrangement of components that are important for system reliability is also called reliability structure and is graphically represented by a schema of logical connectivity. It is important to understand that the model of logical connectivity is focused only on those components that are relevant for the functioning ability of the system. Components that do not play a direct role in the system's reliability usually are not included in the analyzed reliability structure (Rausand and Hoyland, 2003). For example, organs of vision are not included in the reliability structure of living organisms if death is the only type of failure to be analyzed (complete failure of vision does not cause an immediate death of the organism). On the other hand, if disability is the type of failure under consideration, then organs of vision should be included in the schema of reliability structure. Therefore, reliability structure does not necessarily reflect a physical structure of the object.

There are two major types of components arrangement (connection) in the system: components connected in series and components connected in parallel (Rausand and Hoyland, 2003). Here we consider a simple system of n statistically independent components, where failure of one component does not affect the failure rate of other components of the system.

Components connected in series For a system of n independent components connected in series, the system fails if any one of the components fails, as in electrical circuits connected in series. Thus, failure of any one component results in failure of the whole system as in the Christmas tree lighting chains. Figure 5.3a shows a schema of the logical connectivity of the system in series.

This type of system is also called a weakest-link system (Ayyub and McCuen, 2003). In living organisms many organs and tissues (heart, lung, liver, brain) are vital for the organism's survival—a good example of series-connected components. Thus, the series connection means a logical connectivity, but not necessarily a physical or anatomical one.

The reliability of a system in series (with independent failure events of the components), Ps, is a product of reliabilities of its components:

Ps = PiP2 ••• Pn where p1 ... pn are reliabilities of the system's components. This formula explains why complex systems with many critical components are so sensitive to early failures of their components.

For example, for a system built of 458 critical components, the initial period of the components' life when their cumulative risk of failure is only 1% corresponds to the end of the system's life when 99% of systems have already failed. This difference between the lifetimes of systems and the potential lifetimes of their components is increasing further with system complexity (numbers of critical components). Therefore, the early failure kinetics of components is so important in determining the failure kinetics of a complex system for its entire life.

The failure rate of a system connected in series is the sum of failure rates of its components (Barlow et al., 1965):

If the failure rates of all components are equal, then the failure rate of the system with n components is nM-It follows from this formula that if the system's