Matrix analytic method

In probability theory, the matrix analytic method is a technique to compute the stationary probability distribution of a Markov chain which has a repeating structure (after some point) and a state space which grows unboundedly in no more than one dimension.^[1][2] Such models are often described as M/G/1 type Markov chains because they can describe transitions in an M/G/1 queue.^[3][4] The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains.^[5]

An M/G/1-type stochastic matrix is one of the form^[3]

P = ( B 0 B 1 B 2 B 3 ⋯ A 0 A 1 A 2 A 3 ⋯ A 0 A 1 A 2 ⋯ A 0 A 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ) {\displaystyle P={\begin{pmatrix}B_{0}&B_{1}&B_{2}&B_{3}&\cdots \\A_{0}&A_{1}&A_{2}&A_{3}&\cdots \\&A_{0}&A_{1}&A_{2}&\cdots \\&&A_{0}&A_{1}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{pmatrix}}}

where B_i and A_i are k × k matrices. (Note that unmarked matrix entries represent zeroes.) Such a matrix describes the embedded Markov chain in an M/G/1 queue.^[6][7] If P is irreducible and positive recurrent then the stationary distribution is given by the solution to the equations^[3]

P π = π  and  e T π = 1 {\displaystyle P\pi =\pi \quad {\text{ and }}\quad \mathbf {e} ^{\text{T}}\pi =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^[3]

G = ∑ i = 0 ∞ G i A i . {\displaystyle G=\sum _{i=0}^{\infty }G^{i}A_{i}.}

G is called the auxiliary matrix.^[8] Matrices are defined^[3]

A ¯ i + 1 = ∑ j = i + 1 ∞ G j − i − 1 A j B ¯ i = ∑ j = i ∞ G j − i B j {\displaystyle {\begin{aligned}{\overline {A}}_{i+1}&=\sum _{j=i+1}^{\infty }G^{j-i-1}A_{j}\\{\overline {B}}_{i}&=\sum _{j=i}^{\infty }G^{j-i}B_{j}\end{aligned}}}

then π₀ is found by solving^[3]

B ¯ 0 π 0 = π 0 ( e T + e T ( I − ∑ i = 1 ∞ A ¯ i ) − 1 ∑ i = 1 ∞ B ¯ i ) π 0 = 1 {\displaystyle {\begin{aligned}{\overline {B}}_{0}\pi _{0}&=\pi _{0}\\\quad \left(\mathbf {e} ^{\text{T}}+\mathbf {e} ^{\text{T}}\left(I-\sum _{i=1}^{\infty }{\overline {A}}_{i}\right)^{-1}\sum _{i=1}^{\infty }{\overline {B}}_{i}\right)\pi _{0}&=1\end{aligned}}}

and the π_i are given by Ramaswami's formula,^[3] a numerically stable relationship first published by Vaidyanathan Ramaswami in 1988.^[9]

π i = ( I − A ¯ 1 ) − 1 [ B ¯ i + 1 π 0 + ∑ j = 1 i − 1 A ¯ i + 1 − j π j ] , i ≥ 1. {\displaystyle \pi _{i}=(I-{\overline {A}}_{1})^{-1}\left[{\overline {B}}_{i+1}\pi _{0}+\sum _{j=1}^{i-1}{\overline {A}}_{i+1-j}\pi _{j}\right],i\geq 1.}

There are two popular iterative methods for computing G,^[10][11]

• functional iterations
• cyclic reduction.

• MAMSolver^[12]