Lumpability

Definition

Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by t_i. This forms a partition T = { t 1 , t 2 , … } of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain { X i } is lumpable with respect to the partition T if and only if, for any subsets t_i and t_j in the partition, and for any states n,n’ in subset t_i,

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_i and t_j in the partition, and for any states n,n’ in subset t_i,

where p(i,j) is the probability of moving from state i to state j.

Example

Consider the matrix

and notice it is lumpable on the partition t = {(1,2),(3,4)} so we write

and call P_t the lumped matrix of P on t.

Successively lumpable processes

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.

Quasi–lumpability

Franceschinis and Muntz introduced quasi-lumpability, a property whereby a small change in the rate matrix makes the chain lumpable.