Combining Decomposition and Reduction for State Space Analysis of a Self-Stabilizing System

Verifying fault tolerance properties of a distributed system can be achieved by state space analysis via Markov chains. Yet, the power of such exact analytic methods is confined by exponential growth of the chain´s state space in the size of the system modeled. We propose a method that alleviates this limit. Lumping is a well known reduction technique that can be applied to a Markov chain to prune redundant information. We propose a system decomposition to employ lumping piecewise on the considerably smaller Markov chains of the subsystems which are much more likely to be tractable. Recomposing the lumped Markov chains of the subsystems results in a state space that is likely to be considerably smaller. An example demonstrates how the limiting window availability (i.e. a fault tolerance property) can be computed for a system while exploiting the combination of lumping and decomposition.