Markov Chains Analysis

A Markov chain is a state-transition model in which the system occupies one state at a time - for example operational, degraded, failed safe, or failed unsafe - and moves between states at rates derived from failure rates, diagnostic coverage, and repair rates. The defining Markov property is that the next transition depends only on the current state, not on history, which makes the mathematics tractable. In ISO 26262 hardware evaluation, Markov models are recognized among the quantitative analysis techniques and are particularly suited to PMHF calculations for redundant and repairable systems where the order and timing of failures matter.

FMEA enumerates failure modes bottom-up and FTA combines failure probabilities top-down, but both are essentially static: they struggle when the system repairs itself, degrades through intermediate states, or when the probability of a dangerous outcome depends on the sequence and timing of events. Markov analysis shines exactly there - repairable redundancy, periodic diagnostics, degraded operating modes, and availability questions. The course positions the methods honestly: Markov modeling costs more effort and state spaces grow quickly, so use it for the architectures whose dynamics static methods cannot represent.

The transition matrix collects the rates (continuous time) or probabilities (discrete time) of moving from every state to every other state; together with an initial state it fully defines the model. Constructing it means translating each failure mode, diagnostic detection, and repair action into a rate entering the right cell. Validation checks the structural properties - rows behaving correctly, no unintended absorbing states, rates consistent with the underlying failure data - before the results are trusted. The course dedicates a full chapter to construction, steady-state solution, and these validation checks.

The PMHF target for a safety goal is a rate of dangerous failure over operating time, and for redundant or repairable architectures the honest way to compute it is to track how probability mass flows into the failed-unsafe state over the mission. A Markov model does exactly this: with failure rates, diagnostic coverage splitting detected from undetected faults, and repair rates as inputs, it yields the time-dependent probability of the dangerous state. The interactive calculator in chapter 4 demonstrates the workflow end to end, including SPFM output and an ASIL compliance check.

The course has 7 chapters: four in-depth theory and practice chapters (introduction with reliability versus availability, mathematical fundamentals, transition matrix construction, and the interactive calculator for a redundant two-component system), a thematic exam chapter with questions from beginner to expert level, and two roadmap chapters - modeling process workflows and ready-to-use templates - that are still in development. The platform also offers a separate standalone Markov simulator available to all registered users.