Uniformization (probability theory): Difference between revisions

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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, by approximating the process by a [[discrete time Markov chain]].<ref name="ibe" /> 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 ''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 Winfried Grassmann in 1977.<ref>{{cite jstor|172104}}</ref><ref>{{citeCite journal doi| last1 = Grassmann | first1 = W. K. | doi = 10.1016/0305-0548(77)90007-7 | title = Transient solutions in markovian queueing systems | journal = Computers & Operations Research | volume = 4 | pages = 47–00 | year = 1977 | pmid = | pmc = }}</ref><ref>{{citeCite journal doi| last1 = Grassmann | first1 = W. K.| title = Transient solutions in Markovian queues | doi = 10.1016/0377-2217(77)90049-2 | journal = European Journal of Operational Research | volume = 1 | issue = 6 | pages = 396–242 | year = 1977 | pmid = | pmc = }}</ref>

==Method description==

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==Implementation==

[[Pseudocode]] for the algorithm is included in Appendix A of Reibman and Trivedi's 1988 paper.<ref name="reibman">{{citeCite journal doi| last1 = Reibman | first1 = A. | last2 = Trivedi | first2 = K. | doi = 10.1016/0305-0548(88)90026-3 | title = Numerical transient analysis of markov models | journal = Computers & Operations Research | volume = 15 | pages = 19 | year = 1988 | pmid = | pmc = | url = http://people.ee.duke.edu/~kst/markovpapers/numerical.pdf}}</ref> Using a [[parallel algorithm|parallel]] version of the algorithm, chains with state spaces of larger than 10⁷ have been analysed.<ref>{{cite doi|10.1016/j.jpdc.2004.03.017}}</ref>

==Limitations==