Reliability block diagrams

To discuss or clarify ideas, what engineer hasn't, at one time or another, quickly represented a system by drawing rectangles and lines to visualize its components and the relationships between said components? This popular and intuitive approach goes back to the dawn of time, and the literature doesn't seem to remember the name of its inventor. When used to model the logical relationships between the operational states of a system's components (called "blocks") and the operational state of the system itself, it is known as a block reliability diagram (BDF), or reliability diagram for short.

BDFs are part of the panoply of methods commonly used in the field of operational safety (fault trees, event trees, Markov graphs, Petri nets, etc.). Like fault trees (ADD), BDFs belong to the static and Boolean approaches, as they model logical structures that are independent of time. As a result, they are concerned with two-state components/systems (e.g.: on and off, working/faulting). ADD and BDF share the same underlying mathematics, but whereas ADD describes the failure of a system, BDF describes its correct operation. As a result, a BDF can always be translated into ADD and vice versa: the two approaches are said to be dual.

From a qualitative point of view, BDFs are the basis of the fundamental concept of minimum cut: a set of failed blocks necessary and sufficient to cause system failure.

From a quantitative point of view, BDFs essentially enable us to calculate the probability of the system operating correctly as a function of the probabilities of the blocks operating correctly, when the latter are constant. However, when the blocks evolve independently of each other over time, it is possible to calculate the probability of the system operating correctly at time t (i.e., its availability at time t) as a function of the probabilities of the blocks operating correctly at time t (i.e., the block availabilities). The same applies to the frequency of system failure at time t.

Paradoxically, given the name of the approach, it is not generally possible to calculate the reliability of the system (probability of correct operation over a period [0, t]) from the reliability of its blocks. However, in special cases, good approximations can be obtained.

For a long time, the use of BDFs, like that of ADDs, was limited by the combinatorial explosion in the number of minimal cuts, and the duration of calculations evolving exponentially with the number of blocks/events repeated several times in the same BDF/ADD. These limitations have been overcome since the introduction of binary decision diagrams (BDDs), which enable very rapid processing of BDFs or ADDs for large industrial systems (i.e. with several hundred...