The common system has the following properties: the failure of any unit will not improve the system performance; The system does not contain redundant units that will not affect its performance. This system is called correlation system. This property can be expressed by structural function: let φ (φ) be the structural function of the system. For any state vector у ≤у, there is φ (у) ≤φ (у), where у ≤уmeans that each Xi ≤ yi; ; For any i( 1≤i≤n), there is a state vector such that φ(0i, ω)= 0, φ( 1i, ω)= 1, where (0i, ω) and (.
Typical related systems are: series system, that is, if any unit fails, the system will fail; Parallel system, that is, when all units fail, the system fails; K-out-of-n(F) system, that is, when K or more units fail, the system will fail, which is the generalization of series or parallel system. In practice, the commonly used 2-out-of-3 (F) system consists of three units, and voting is conducted according to the status of most units. The structural functions of these three systems are interrelated systems, and the problems studied are the expression of structural functions of complex systems, the solution of system reliability and its upper and lower bounds. In order to reflect the gradual change of unit and system functions, the research of polymorphic interconnected system has been paid attention to.
Network Reliability Many practical systems can be abstracted into networks. Such as computer Internet, communication network and oil and gas transmission network. It is assumed that the vertices and edges of a network (see graph theory) have only two states: normal and fault. The faults are independent of each other, and the probability of each vertex and edge being normal is known. The probability that information can be sent from one vertex to another (or k designated vertices) is called the reliability of the network. When calculating the network reliability, because of its complex structure, it is necessary to find methods and effective algorithms to simplify the network, and compare the advantages and disadvantages of different algorithms. In recent years, many good algorithms have appeared, and the computational complexity has also improved.
Fault tree analysis is referred to as FTA. Through deduction, according to the logical relationship between events, we can find out the possible combination of all events that lead to system failure or an unexpected event (called top event). For example, when studying the boiler explosion event T, the explosion is caused by various events such as excessive pressure, A, B, …, D, etc. If the occurrence of one of the events A, B, …, D will cause the occurrence of T, then the relationship between T and these events is represented by the logic gate or; If the simultaneous appearance of a and b leads to the appearance of t, then the relationship between t and a and b is represented by logic gate and. Then the events of A, B, …, D are analyzed one by one until the most basic fault cause (basic event) is found out. Where stands for OR gate; Represents and gate; Represents an event; ○ Represents a basic event.
When analyzing the fault tree of the top event T, the basic steps are: establishing the fault tree; Qualitative evaluation, that is, to find out the combination of all possible basic events that cause T; Quantitative evaluation, that is, according to the probability of basic events, find out the probability of T.
FTA, which originated in the early 1960s, has been applied in industrial fields such as space navigation and safety analysis of nuclear power plants. Because this method is intuitive, it is convenient for engineers and managers to use. The disadvantage of this method is that it takes a lot of time and manpower to establish fault tree, and it is inevitable to miss some important fault causes for complex systems. In addition, the evaluation of complex fault tree must be carried out with the help of computer.
At present, the evaluation method of fault tree, including NOT gates and other logic gates, and computer-aided fault tree establishment are the centers of FTA research.
Reliability analysis of complex systems It is common that the system consists of 1000 units. If the reliability of each unit is 0.999, each unit is independent of each other, and the failure of any unit makes the system fail, then the reliability of this system is obviously quite low. Therefore, in order to improve the reliability (availability) of the system, spare parts can work in parallel, or maintenance and replacement can be introduced into the system. The problems discussed include: knowing the structure of the system, the distribution of unit life and repair (or replacement) time, the number of repairmen and repair rules in the system, studying the quantitative index of system reliability or discussing how to reasonably determine the number of repairmen or repair rules to achieve an optimal objective function. Through the mathematical model, Markov process, updating process, Markov updating process, supplementary variable method and other analytical methods are used to study, and its processing method is similar to queuing theory.
For example, the simplest system consisting of one unit. If the life and repair time of the system have exponential distribution of parameters λ and μ and are independent of each other. When time t=0, the system is normal, and the system repaired after the failure is as good as ever. Then the time before the first failure of the system has an exponential distribution of the parameter λ. Using Markov process or update process, the availability at time t and the average number of failures in (0, t) can be obtained.