The reliability of the system is the probability that the number of functional components is greater than or equal to k. Thus, the system reliability, Rs, is calculated by
If the components are not identical, the system reliability should be calculated by enumerating all combinations of working components.
1.3.5 Network System
There are systems that can be represented by network diagrams, for example, gas networks, telecommunications networks, and power networks. A network system consists of a set of nodes and links. All the nodes and links have a probability of failure.
1.4 System Reliability Assessment Methods
There are many reliability assessment approaches developed to compute the reliability of complex systems, e.g. networks. Path-set and cut-set methods, decomposition and factorization methods, and binary decision diagram (BDD) are four commonly used methods, and we will introduce them in this section.
1.4.1 Path-set and Cut-set Method
A path set P is a set of components, which by functioning ensures that the system is functioning. A path set is said to be minimal if it cannot be reduced without losing its status as a path set. A cut set K is a set of components, which by failing causes the system to fail. A cut set is said to be minimal if it cannot be reduced without losing its status as a cut set. We refer to these minimal sets as minimal path and cut sets or vectors (MPSs, MPVs and MCSs, MCVs).
Consider the minimal path sets of the system, P1, P2,…, Pp, and the minimal cut sets of the system, K1, K2,…, Kk. The reliability of the system is given by the union of all minimal path sets. The unreliability is given by the probability that at least one minimal cut set occurs.
Example 1.4
Consider a bridge structure with five edges, E1,…, E5, as shown in Figure 1.11:
1 Find the minimal path sets and the minimal cut sets of the system.
2 Calculate the reliability of the system if the reliability of each component is p.
Figure 1.11 Bridge system.
Solution
1 The minimal path sets areThe minimal cut sets are
2 The reliability of the system is calculated by the union of the path sets:
1.4.2 Decomposition and Factorization
The decomposition method begins by selecting a critical component, denoted by x, which is an important component of the complex system structure. The reliability of the system can be calculated by the conditional probability:
The factorization method is developed based on the decomposition method, which is used in a network system. Denote e as a critical edge in the network G. The reliability of the network is
1.4.3 Binary Decision Diagram
Binary decision diagram (BDD) is used to represent a Boolean function. A Boolean function can be represented as a rooted, directed, acyclic graph, which consists of several nodes and two terminal nodes. The two terminal nodes are labeled 0 (FALSE) and 1 (TRUE). Each node u is labeled by a Boolean variable xi and has two child nodes called low child and high child. The edge from a node to a child represents an assignment of the value FALSE (or TRUE, respectively) to variable xi. The advantage of BDD in reliability assessment is that its accuracy and efficiency are high [6]. The algorithm to compute the probability of a gate from a BDD is based on the Shannon Decomposition, which is defined by recursive equations.
Example 1.5
Calculate the reliability of the bridge system in Figure 1.11, if the reliability of each component is p.
Solution
The block decision diagram of the bridge system is shown in Figure 1.12.
Figure 1.12 Block decision diagram of the bridge system.
The reliability of the system is
1.5 Exercises
1 Consider an electrical generating system with two engines, E1, E2, and three generators, G1, G2, G3, each one with rate equal to 30 kVA. The system fails when the generators fail to supply at least 60 kVA. The structure of the system is shown in Figure 1.13.Figure 1.13 Electrical generating system.Find the minimal cut sets of the system.Estimate the unreliability of the system for one-month operation, given that the failure rate for each engine is 5×10−6h−1 and that for each generator is 10−5h−1.
2 Consider the reliability of the following system consisting of five components in Figure 1.14. All the components are identical and independent from each other. The reliability of components i is Ri. Let Rs be the reliability of the system. Give the reliability formulation of the system.Figure 1.14 Reliability block diagram of the system.
3 The system has N = 4 components. Each component has three states: (M∈{0,1,2}).