Uniformization (probability theory)

In probability theory, uniformization (also known as Jensen's method^[1] or the method of randomization^[2]) is a method to transform a continuous time Markov chain to an analgous discrete time Markov chain.^[2] The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero).^[1]

For a continuous time Markov chain with transition rate matrix Q, the uniformized discrete time Markov chain has probability transition matrix P calculated by^[1]^[3]^[4]

p_{ij}={\begin{cases}q_{ij}/\gamma &{\text{ if }}i\neq j\\1+q_{ii}/\gamma &{\text{ if }}i=j\end{cases}}

with $\gamma$ chosen such that $\gamma \geq max_{i}|q_{ii}|$ .

Notes

^ ^a ^b ^c Stewart, William J. (2009). Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Princeton University Press. p. 361. ISBN 0691140626.
^ ^a ^b Ibe, Oliver C. (2009). Markov processes for stochastic modeling. Academic Press. p. 98. ISBN 0123744512.
^ Cassandras, Christos G.; Lafortune, Stéphane (2008). Introduction to discrete event systems. Springer. ISBN 0387333320.
^ Ross, Sheldon M. (2007). Introduction to probability models. Academic Press. ISBN 0125980620.

[stewart-1] Stewart, William J. (2009). Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Princeton University Press. p. 361. ISBN 0691140626.

[ibe-2] Ibe, Oliver C. (2009). Markov processes for stochastic modeling. Academic Press. p. 98. ISBN 0123744512.

[cass-3] Cassandras, Christos G.; Lafortune, Stéphane (2008). Introduction to discrete event systems. Springer. ISBN 0387333320.

[ross-4] Ross, Sheldon M. (2007). Introduction to probability models. Academic Press. ISBN 0125980620.

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