Vectorial addition chain

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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]

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 of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain 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.[2]

Similarly, for a stochastic matrix P, P is a lumpable matrix on a partition T if and only 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.[3]

Example

Consider the matrix

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

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.[4]

Quasi–lumpability

Franceschinis and Muntz introduced quasi-lumpability, a property whereby a small change in the rate matrix makes the chain lumpable.[5]

See also

References

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  2. Jane Hillston, Compositional Markovian Modelling Using A Process Algebra in the Proceedings of the Second International Workshop on Numerical Solution of Markov Chains: Computations with Markov Chains, Raleigh, North Carolina, January 1995. Kluwer Academic Press
  3. Peter G. Harrison and Naresh M. Patel, Performance Modelling of Communication Networks and Computer Architectures Addison-Wesley 1992
  4. Template:Cite doi
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