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.^[1]

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 ${\displaystyle \scriptstyle {T=\{t_{1},t_{2},\ldots \}}}$ of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain ${\displaystyle \{X_{i}\}}$ 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,

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

where q(i,j) is the transition rate from state i to state j.^[2]

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,

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

where p(i,j) is the probability of moving from state i to state j.^[3]

Consider the matrix

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

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

${\displaystyle P_{t}={\begin{pmatrix}{\frac {7}{8}}&{\frac {1}{8}}\\{\frac {1}{16}}&{\frac {15}{16}}\end{pmatrix}}}$

and call P_t the lumped matrix of P on t.

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.^[5]

• Nearly completely decomposable Markov chain