Quantified safety modeling of autonomous systems with hierarchical semi-Markov processes

Abstract: In quantified safety engineering, mathematical probability models are used to predict the risk of failure or hazardous events in systems. Markov processes have commonly been utilized to analyze the safety of systems modeled as discrete-state stochastic processes. In continuous time Markov models, transition time between states are exponentially distributed. Semi-Markov processes expand this modeling framework by allowing transition time between states to follow any distribution. This master thesis project seeks to extend the semi-Markov modeling framework even further by allowing hierarchical states, which further relaxes Markov-assumptions by allowing models to keep memory even in state transition. To achieve this, the master thesis proposes a method using the phase-type distribution to replace Markov-chains of states to a single state. For application purposes, it is shown how semi-Markov chains with phase-type distributed transitions can be evaluated by a method using the Laplace-Stieltjes transform. Furthermore, to replace semi-Markov chains, a method to approximate these by the phase-type distribution is presented. This is done by deriving the moments of the time to absorption in a semi-Markov process with a method using the Laplace-Stieltjes transform, and fitting a phase-type distribution with these moments. To evaluate the methods, some case studies are performed on appropriate models. Analytical results are compared with Monte-Carlo simulations and Laplace-transform inverse methods. The results are used to show how hierarchical semi-Markov models can be replaced in an exact manner, and how semi-Markov models can be replaced approximately with varying accuracy. An important conclusion is that by enabling hierarchical modeling, it is possible to predict the safety of systems which demand a more realistic model, as relaxing Markov assumptions allows for more complexity.