Uniformization (probability theory)
In probability theory, uniformization method, (also known as Jensen's method or the randomization method) is a method to compute transient solutions of finite state continuous-time Markov chains, by approximating the process by a discrete time Markov chain. The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name uniformisation. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero). The method was first introduced by Winfried Grassmann in 1977.
with γ, the uniform rate parameter, chosen such that
In matrix notation:
For a starting distribution π(0), the distribution at time t, π(t) is computed by
This representation shows, that a continuous time Markov Chain can be described by a discrete Markov Chain with transition matrix P as defined above where jumps occur according to a Poisson Process with intensity γt.
In practice this series is terminated after finitely many terms.
Pseudocode for the algorithm is included in Appendix A of Reibman and Trivedi's 1988 paper. Using a parallel version of the algorithm, chains with state spaces of larger than 107 have been analysed.
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