Matrix analytic method: Difference between revisions
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In [[probability theory]], '''Ramaswami's formula''' gives a numerically stable method to compute the stationary probability distribution of an [[M/G/1]] type model,<ref name="meini">{{cite doi|10.1080/15326349708807423}}</ref><ref>{{cite doi|10.1016/j.peva.2005.07.003}}</ref> first published by V. Ramaswami in 1988.<ref>{{cite doi|10.1080/15326348808807077}}</ref> |
In [[probability theory]], '''Ramaswami's formula''' gives a numerically stable method to compute the stationary probability distribution of an [[M/G/1]] type model,<ref name="meini">{{cite doi|10.1080/15326349708807423}}</ref><ref>{{cite doi|10.1016/j.peva.2005.07.003}}</ref> first published by V. Ramaswami in 1988.<ref>{{cite doi|10.1080/15326348808807077}}</ref> It is the classical solution method for M/G/1 chains.<ref>{{cite doi|10.1007/3-540-45798-4_3}}</ref> |
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==Formula== |
==Formula== |
Revision as of 01:06, 25 January 2013
In probability theory, Ramaswami's formula gives a numerically stable method to compute the stationary probability distribution of an M/G/1 type model,[1][2] first published by V. Ramaswami in 1988.[3] It is the classical solution method for M/G/1 chains.[4]
Formula
An M/G/1-type stochastic matrix is of the form[1]
where Bi and Ai are k x k matrices. (Note that unmarked matrix entries represent zeroes.) If P is irreducible and positive recurrent then the stationary distribution is given by the solution to the equations[1]
where e represents a vector of suitable dimension with all values equal to 1. Matching the structure of P, π is partitioned to π1, π2, π3, …. To compute these probabilities the column stochastic matrix G is computed such that[1]
and matrices are defined[1]
then π0 is found by solving[1]
and the πi are given by Ramaswami's formula[1]
Computation of G
There are two popular iterative methods for computing G,[5][6]
- functional iterations
- cyclic reduction.
If transitions up or down are restricted faster algorithms can take advantage of the simpler structure of the model.[7]
References
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