In [[probability theory]], '''uniformization''' method, (also known as '''Jensen's method'''<ref name="stewart" /> or the '''randomization method'''<ref name="ibe">{{cite book |title=Markov processes for stochastic modeling |last=Ibe |first=Oliver C. |year=2009 |publisher=[[Academic Press]] |isbn=0-12-374451-2 |page=98}}</ref>) is a method to compute transient solutions of finite state [[continuous-time Markov chain]]s. The, methodby involvesapproximating the constructionsprocess ofby an analogousa [[discrete time Markov chain]],.<ref name="ibe" /> whereThe transitionsoriginal occurchain accordingis toscaled an exponential distribution withby the samefastest parametertransition in every state. This parameter,rate ''γ'', isso that transitions occur at the same rate in allevery statesstate, hence the name ''uniform''isation. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero).<ref name="stewart" /> The method was first introduced by Grassman in 1977.<ref>{{cite jstor|172104}}</ref><ref>{{cite doi|10.1016/0305-0548(77)90007-7}}</ref><ref>{{cite doi|10.1016/0377-2217(77)90049-2}}</ref>
==Method description==
For a continuous time Markov chain with infinitesimal[[transition generatorrate matrix]] ''Q'' (sometimes also called "transition rate matrix"), the uniformized discrete time Markov chain has probability transition matrix <math>P:=(p_{ij})_{i,j}</math>, which is defined by<ref name="stewart">{{cite book |title=Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling|last=Stewart |first=William J. |year=2009 |publisher=[[Princeton University Press]] |isbn=0-691-14062-6 |page=361}}</ref><ref name="cass">{{cite book |title=Introduction to discrete event systems|last=Cassandras |first=Christos G. |last2=Lafortune| first2=Stéphane|year=2008 |publisher=Springer |isbn=0-387-33332-0}}</ref><ref name="ross">{{cite book |title=Introduction to probability models|last=Ross |first=Sheldon M. |year=2007 |publisher=Academic Press |isbn=0-12-598062-0}}</ref>
::<math>p_{ij} = \begin{cases} q_{ij}/\gamma &\text{ if } i \neq j \\ 1 - \sum_{j \neq i} q_{ij}/\gamma &\text{ if } i=j \end{cases}</math>
