Predicting the Evolution of Controlled Systems Modeled by Finite Markov Processes

The operation and maintenance of infrastructure components and systems can be modeled as a Markov process, partially or fully observable. Information about the current condition can be summarized by the “inner” state of a finite state controller. When a control policy is assigned, the stochastic evolution of the system is completely described by a Markov transition function. This article applies finite state Markov chain analyses to identify relevant features of the time evolution of a controlled system. We focus on assessing if some critical conditions are reachable (or if some actions will ever be taken), in identifying the probability of these critical events occurring within a time period, their expected time of occurrence, their long-term frequency, and the probability that some events occur before others. We present analytical methods based on linear algebra to address these questions, discuss their computational complexity and the structure of the solution. The analyses can be performed after a policy is selected for a Markov decision process (MDP) or a partially observable MDP. Their outcomes depend on the selected policy and examining these outcomes can provide the decision makers with deeper understanding of the consequences of following that policy, and may also suggest revising it.