Refinable Bounds for Large Markov-chains

A method to bound the steady-state solution of large Markov chains is presented. It integrates the concepts of eigenvector polyhedron and of aggregation and is iterative in nature. The bounds are obtained by considering a subset only of the system state space. This makes the method specially attractive for problems which are too large to be dealt with by traditional methods. The quality of the bounds depends on the locality of the system which is studied: when the system spends most of its time in a small subset of states, tight bounds can be obtained by considering this subset only. Finally, the bounds are refinable in the sense that the tightness of the bounds can be improved by enlarging the subset of states which is considered. The method is illustrated on a model of a repairable fault-tolerant system with 16 million states. Tight bounds on its availability are obtained by considering less than 0.1 percent of its state space