PDMP in Reliability

From a mathematical point of view, the notion of a PDMP is very intuitive and simple to describe. Starting from a point of the state space, the process follows a deterministic trajectory, namely a flow indexed by the mode, until the first jump time, which occurs either spontaneously in a random manner or when the trajectory hits the boundary of the state space. Between two jumps, the mode is assumed to be constant. In both cases, a new point and a new regime are selected by a random operator and the process restarts from this new point under this new mode. There exist two types of jump. The first one is deterministic. From the mathematical point of view, it is given by the trajectory hitting the boundary of the state space. From the physical point of view, it can be seen as a modification of the mode of operation when a physical parameter reaches a prescribed level, for example when the pressure of a tank reaches the critical value. The second one is stochastic. It models the random nature of failures or inputs that modify the mode of operation of the system.

This chapter is dedicated to the definition of PDMPs and statement of the main properties that we will use throughout the book. It is organized as follows. Section 1.2 gathers general notation. In Section 1.3, we give the formal definition of a PDMP. In Section 1.4 we state and comment the main assumptions that will be required in this book. In Section 1.5 we define a time-augmented PDMP and establish that the properties of the original PDMP transfer to the time-augmented one. In Section 1.6 we define and study a discrete-time Markov chain naturally embedded in a PDMP. This chain is at the heart of our numerical approximations. Section 1.7 provides some technical properties of the stopping times of a PDMP. Finally, we give in Section 1.8 some simple examples of PDMPs that will serve to illustrate our results throughout the book.