Correlations between states and events in the simulation of dependability using Markov processes
Authors: Zelentsov B. P. 
Published: March 2022

The paper examines the correlations between states and events that are used in the construction of process diagrams that describe the dependability of items. Based on the constructed state and event diagram, input data is generated and the mathematical method is selected that is implemented in accordance with the problem at hand. The distinctive features and advantages of the matrix method are presented. Aim. To improve the simulation methods by clarifying the correlation between states and events and using matrix methods of calculation. Methods. The examined causal relationships between states and events allowed establishing correlations between them, i.e., an event can be the cause of a state change, then a state change is a consequence; a state can be the cause of an event, then an event is a consequence of a state. Under this approach, an event can cause a state change, while at the same time an event is a consequence of a state. The situation with states is similar. A state can be the cause of an event, while at the same time a state is the consequence of an event. It is also noted that a single state may cause a number of events, while an event can also cause a number of states. Examples of such correlations are given. It is noted that the duration of a state can be constant, random or zero. The examined correlations between states and events enable a substantiated construction of a diagram of states and transitions. A substantiated construction of a diagram of states and transitions results from a conceptual model, in which all states and events are given a physical and technical interpretation that transforms into a formal state-transition diagram. A special attention is given to the matrix methods that have a number of advantages, i.e., compactness and simplicity of converting the input characteristics into output characteristics, availability of standard software, use of verification procedures, feasibility of implementation using standard computer-based tools. The input data is also generated in matrix form. The paper indicates the characteristics of a state-transition diagram that can be calculated from the input data. Note is made of the use of methods based on semi-Markov processes. The author points out that, while using matrix methods, cycles should be generated. A relevant matter associated with the large number of states and the consequent problem of aggregation of states is touched upon. Two approaches to the aggregation of states are set forth that allow keeping the system’s output characteristics unchanged. Results. A proposal is formulated for the construction of a dependability model involving a number of stages, i.e., definition of the goal of simulation with the indication of the used dependability indicators, description of the conceptual model, construction of a substantiated state-transition diagram, selection of the mathematical method, calculations, discussing the findings, conclusions and suggestions based on the performed simulation. Discussion and conclusions. A dependability model should take into consideration the causal relationships between states and events that are established based on the physical, as well as the engineering and technical features of the item. Taking these relationships into account, a state diagram is generated that enables initial data compilation. The matrix method is efficient and has a number of useful features. The above considerations are methodological in their nature. They can be helpful for generating dependability models of technical systems and studying the dependability theory in educational institutions.