Lumpability
# 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 $t_i.$ 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 $t_i$ and $t_j$ in the partition, and for any states n,n’ in subset $t_i,$

    $\sum _{m\in t_{j}}q(n,m)=\sum _{m\in t_{j}}q(n',m),$

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

    $\sum _{m\in t_{j}}p(n,m)=\sum _{m\in t_{j}}p(n',m),$

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 $P_t$ 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.

