Lumpability

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.

Definition
Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by ti. This forms a partition $$\scriptstyle{T = \{ t_1, t_2, \ldots \}}$$ 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 ti and tj in the partition, and for any states n,n’ in subset ti,


 * $$\sum_{m \in t_j} q(n,m) = \sum_{m \in t_j} q(n',m) ,$$

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 ti and tj in the partition, and for any states n,n’ in subset ti,


 * $$\sum_{m \in t_j} p(n,m) = \sum_{m \in t_j} p(n',m) ,$$

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

Example
Consider the matrix


 * $$P = \begin{pmatrix}

\frac{1}{2} & \frac{3}{8} & \frac{1}{16} & \frac{1}{16} \\ \frac{7}{16} & \frac{7}{16} & 0 & \frac{1}{8} \\ \frac{1}{16} & 0 & \frac{1}{2} & \frac{7}{16} \\ 0 & \frac{1}{16} & \frac{3}{8} & \frac{9}{16} \end{pmatrix}$$

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


 * $$P_t = \begin{pmatrix}

\frac{7}{8} & \frac{1}{8} \\ \frac{1}{16} & \frac{15}{16} \end{pmatrix}$$

and call Pt 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.