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=317-322|isbn=0-201-54419-9}}</ref> The method was developed "largely by Marcel F. Neuts and his students starting around 1975."<ref>{{citeCite book doi| 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==
Line 29:
::<math>\pi_i = \pi_1 R^{i-1}</math>
holds where ''R'' is the Neut's rate matrix,<ref>{{citeCite journal doi| last1 = Ramaswami | first1 = V. | doi = 10.1080/15326349908807141 | title = A duality theorem for the matrix paradigms in queueing theory | journal = Communications in Statistics. Stochastic Models | volume = 6 | pages = 151–161 | year = 1990 | pmid = | pmc = }}</ref> which can be computed numerically. Using this we write
::<math>\begin{align}
Line 41:
==Computation of ''R''==
The matrix ''R'' can be computed using [[cyclic reduction]]<ref>{{citeCite journal doi| 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 jstor|3214773}}</ref><ref>{{cite doi|10.1016/j.peva.2010.04.003}}</ref>
==Matrix analytic method==
{{Main|Matrix analytic method}}
The matrix analytic method is a more complicated version of the matrix geometric solution method used to analyse models with block [[M/G/1 queue|M/G/1]] matrices.<ref>{{citeCite book doi| last1 = Alfa | first1 = A. S. | last2 = Ramaswami | first2 = V. | doi = 10.1002/9780470400531.eorms0631 | chapter = Matrix Analytic Method: Overview and History | title = Wiley Encyclopedia of Operations Research and Management Science | year = 2011 | isbn = 9780470400531 | pmid = | pmc = }}</ref> Such models are harder because no relationship like ''π''<sub>''i''</sub> = ''π''<sub>1</sub> R<sup>''i'' – 1</sup> used above holds.<ref>{{cite book|first1=Gunter|last1= Bolch|first2=Stefan |last2= Greiner |first3=Hermann |last3=de Meer |first4= Kishor |last4= Shridharbhai Trivedi | year=2006| edition=2 | page=259| title=Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications |publisher= John Wiley & Sons, Inc|isbn=0471565253}}</ref>
