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. 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 ti. This forms a partition T = { t 1 , t 2 , … } {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 { X i } {displaystyle {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, 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, where p(i,j) is the probability of moving from state i to state j.