Matrix geometric method: Difference between revisions

Revision as of 17:39, 4 September 2015

Revision as of 22:37, 22 June 2017

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In [[probability theory]], the '''matrix geometric method''' is a method for the analysis of [[quasi-birth–death process]]es, [[continuous-time Markov chain]] whose [[transition rate matrices]] with a repetitive block structure.<ref>{{cite book|first=Peter G.|last=Harrison|authorlink=Peter G. Harrison|first2=Naresh M.|last2=Patel|title=Performance Modelling of Communication Networks and Computer Architectures|publisher=Addison-Wesley|year=1992|pages=317-322317–322|isbn=0-201-54419-9}}</ref> The method was developed "largely by Marcel F. Neuts and his students starting around 1975."<ref>{{Cite book | first1 = S. R. | last1 = Asmussen| doi = 10.1007/0-387-21525-5_8 | chapter = Random Walks | title = Applied Probability and Queues | series = Stochastic Modelling and Applied Probability | volume = 51 | pages = 220–243 | year = 2003 | isbn = 978-0-387-00211-8 | pmid = | pmc = }}</ref>

==Method description==

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==Computation of ''R''==

The matrix ''R'' can be computed using [[cyclic reduction]]<ref>{{Cite journal | last1 = Bini | first1 = D. | last2 = Meini | first2 = B. | doi = 10.1137/S0895479895284804 | title = On the Solution of a Nonlinear Matrix Equation Arising in Queueing Problems | journal = SIAM Journal on Matrix Analysis and Applications | volume = 17 | issue = 4 | pages = 906 | year = 1996 | pmid = | pmc = }}</ref> or logarithmic reduction.<ref>{{cite journal | year = 1993 | title = A Logarithmic Reduction Algorithm for Quasi-Birth-Death Processes | journal = Journal of Applied Probability | volume = 30 | issue = 3 | pages = 650-674650–674 | publisher = Applied Probability Trust | jstor = 3214773 | url = | format = | accessdate = | first1 = Guy | last1 = Latouche | first2 = V. | last2 = Ramaswami}}</ref><ref>{{Cite journal | last1 = Pérez | first1 = J. F. | last2 = Van Houdt | first2 = B. | doi = 10.1016/j.peva.2010.04.003 | title = Quasi-birth-and-death processes with restricted transitions and its applications | journal = [[Performance Evaluation]]| volume = 68 | issue = 2 | pages = 126 | year = 2011 | url = http://www.doc.ic.ac.uk/~jperezbe/data/PerezVanHoudt_PEVA_2011.pdf| pmid = | pmc = }}</ref>

==Matrix analytic method==