Uniformization (probability theory): Difference between revisions

Revision as of 00:06, 15 December 2020

Latest revision as of 14:39, 2 September 2024

(3 intermediate revisions by one other user not shown)

Line 1:

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 |url=https://archive.org/details/markovprocessesf00ibeo |url-access=limited |last=Ibe |first=Oliver C. |year=2009 |publisher=[[Academic Press]] |isbn=0-12-374451-2 |page=[https://archive.org/details/markovprocessesf00ibeo/page/n112 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. 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 journal | last1 = Gross | first1 = D. | last2 = Miller | first2 = D. R. | title = The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes | journal = Operations Research | volume = 32 | issue = 2 | pages = 343–361 | doi = 10.1287/opre.32.2.343 | year = 1984 }}</ref><ref>{{Cite journal | 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 }}</ref><ref>{{Cite journal | 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 = 396–402| year = 1977 }}</ref>

==Method description==

Line 5:

For a continuous-time [[Markov chain]] with [[transition rate matrix]] ''Q'', 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|url=https://archive.org/details/probabilitymarko00stew|url-access=limited|last=Stewart |first=William J. |year=2009 |publisher=[[Princeton University Press]] |isbn=0-691-14062-6 |page=[https://archive.org/details/probabilitymarko00stew/page/n379 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_{jk \neq i} q_{ijik}/\gamma &\text{ if } i=j \end{cases}</math>

with ''γ'', the uniform rate parameter, chosen such that

Line 15:

::<math>P=I+\frac{1}{\gamma}Q.</math>

For a starting distribution π{{pi}}(0), the distribution at time ''t'', π{{pi}}(''t'') is computed by<ref name="stewart" />

::<math>\pi(t) = \sum_{n=0}^\infty \pi(0) P^n \frac{(\gamma t)^n}{n!}e^{-\gamma t}.</math>

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 &nbsp;''γt''.

In practice this [[series (mathematics)|series]] is terminated after finitely many terms.