In probability theory, lumpability is a method for reducing the size of the state space of some continuous-time Markov chains, first published by Kemeny and Snell.
[3] Consider the matrix and notice it is lumpable on the partition t = {(1,2),(3,4)} so we write and call Pt the lumped matrix of P on t. In 2012, Katehakis and Smit discovered the Successively Lumpable processes for which the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains.
Each of the latter chains has a (typically much) smaller state space and this yields significant computational improvements.
These results have many applications reliability and queueing models and problems.
[4] Franceschinis and Muntz introduced quasi-lumpability, a property whereby a small change in the rate matrix makes the chain lumpable.