Proportional Lumpability

We deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the performance indices of large stochastic models using a state aggregation technique. The lumpability method applies to Markov chains exhibiting some structural regularity and allows one to efficiently compute the exact values of the performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case only upper and lower bounds on the performance indices can be derived. In this paper we introduce a novel notion of quasi lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently than the general definition of quasi lumpability, it allows one to derive exact performance indices for the original process.