Dynamically Resolving and Abstracting Markov Models for System Resilience Analysis

Regarding the modeling of quasi-static systems with minor failures for failure prediction and maintenance, Markov models have shown to be very successful. Finite discrete state models can be considered as best practice in this domain, often even assumed to be homogeneous. The question arises if Markov models are also capable to model resilience of systems including major disruptions, where great fractions of the system and its functionality fail. To this end, analytical propositions are made that define model extensions. An initial scalable system is defined, including expected refinements and abstractions. In further phases, major disruptions occur. The disruptions can cause branching points opening routes to model extensions or abstractions. Also independent of disruptions, new states and transitions are introduced or merged for model granularity adoption. Overall system behavior can be interpreted in terms of system improvement with or without new system states or functionalities and corresponding transitions, reaching the ex-ante system state as before the disruption, reaching a deteriorated system state, or finally various degraded and failed overall system states. Definitions such as states, absorbing states and critical transitions are reinterpreted or extended to allow for dynamically resolving or abstracting the Markov model. The main results are extended definitions and derivations when compared to traditional Markov models. Based on the analytical expressions, an example is provided where the formalism could be applied with advantage for autonomous driving safety assessment by considering increasing or decreasing levels of resolution of subsystems or subfunctions.

Keywords: Resilience quantification, Dynamic Markov model extension, Analytical system assessment, Branching point, Hierarchical Markov model, Abstraction and refinement.

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