Mathematical Aspects of Modeling of Discrete, Periodic and Aperiodic Events on the Continuous Basis in the Vniief 'Risk Assessment' Program

摘要

Till now two basic methods of the analysis of reliability (or probability of a failure) complex systems was usually used: the Fault Tree Analysis (with Event Tree Analysis) and the Analysis of Markov States. The Method of Fault Tree is graphic expression of top failure of system as a whole via its separate element failures. The Fault Tree method is applicable usually when between elements of systems there is no dependence (on time or on consequences) misses probability of a failure when an element is to be in a reserve, there is no dependence of failures of system as a whole from sequence of failures of elements of system. Besides it was supposed, that elements in the Fault Tree have a stationary fault rate that did not allow to take into account such important effects as "ageing" and a wear of devices, influence on failure rate of exterior factors varying in time (for example, thermodynamic requirements of a surrounding medium). Many of these problems are solved within the framework of the Markov Analysis. The method of the analysis of Markov States is graphical representation of system states and possible transitions between these states which are characterized by intensities (frequencies) of transition of system from one state in another. The queuing equations are put in correspondence to these graphical objects so, that as a result of their solution distribution functions on time for probability of each of states are obtained. The given method allows describing more precisely fault probabilities of systems and in cases when the Fault Tree method can give only approximate result [1, 2]. However the Markov analysis has essential weakness in comparison with a Fault Tree method. The number of analyzed conditions in the Markov analysis grows as a square of element number of elements. Therefore, its application is effective for systems with small number of elements that considerably lowers its practical value. Because of unhandiness and not clearness of the Markov Analysis analyzers prefer a Fault Tree method to the detriment of precision [1, 2]. Naturally there is an idea to join these two techniques to use all their advantages and to avoid disadvantages which each of methods has. In VNIIEF the "Continuous Risk Assessment" program was designed. The combined technique has been put in a basis of this program. The majority of limitations presenting in a generally accepted method of the Fault Tree analysis have been avoided in new technique. In the "Continuous Risk Assessment" program designed in VNIIEF the new technique of the Fault Tree Analysis is a special case of the Markov Analysis, analytical solution of Markov equations on failure and simple repair of independent elements of the system. For this reason in the "Continuous Risk Assessment" program the Markov analysis and the new Fault Tree analysis has been united naturally. In the methodology of the joined analysis it is possible to save all advantages of both techniques and to avoid their disadvantages. And though two techniques of the risk analysis are organically interlaced in the new methodology of the joined analysis, I shall divide them for convenience in the further reasoning. The new Fault Tree technique is remained visual and easy. Therefore at usage of the joined technique the analyzer should prefer whenever possible to new Fault Tree technique, and resort to the Markov analysis only then when dependence of events and elements of the system is tracked. Here I shall not stop particularly on ideas of new Fault Tree technique. Its bases explicitly were stated in the report on ISSC 19. Let's remind only, that in the report [3] the mathematical exposition of events characterized by continuous functions of fault rate (probabilities of failures to take place in unit of time) and intensities of prime repair was considered. In [3] PRA tasks ВАБ in view of aging inventory and a modification of its reliability as time function were basically considered. However not all events