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.
Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by t<sub>i</sub>. This forms a partition of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain is lumpable with respect to the partition T if and only if, for any subsets t<sub>i</sub> and t<sub>j</sub> in the partition, and for any states n,nâ in subset t<sub>i</sub>,
where q(i,j) is the transition rate from state i to state j.
Similarly, for a stochastic matrix P, P is a lumpable matrix on a partition T if and only if, for any subsets t<sub>i</sub> and t<sub>j</sub> in the partition, and for any states n,nâ in subset t<sub>i</sub>,
where p(i,j) is the probability of moving from state i to state j.
Consider the matrix
and notice it is lumpable on the partition t = {(1,2),(3,4)} so we write
and call P<sub>t</sub> 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.
Franceschinis and Muntz introduced quasi-lumpability, a property whereby a small change in the rate matrix makes the chain lumpable.