Markov chains hierarchical dependability models: Worst-case computations

Dependability models allow calculating the rate of an event leading to a hazard state - a situation, where safety of the modeled dependable system (e.g. railway station signaling and interlocking equipment, automotive systems, etc.) is violated, thus the system may cause material loss, serious injuries or casualties. A hierarchical dependability model allows expressing multiple redundancies made at multiple levels of a system decomposed to multiple cooperating blocks. A hierarchical dependability model based on Markov chains allows each block and relations between these blocks to be expressed independently by Markov chains. This allows a decomposition of a complex dependability model into multiple small models to be made. The decomposed model is easier to read, understand and modify. A hazard rate is calculated significantly faster using hierarchical model, because the decomposition allows exponential calculation-time explosion to be avoided. The paper shows a method how to reduce Markov chains and use them to create hierarchical dependability models. An example study is used to demonstrate the advantages of the hierarchical dependability models (the decomposition of the complex model into multiple simple models and the speedup of the hazard rate calculation).