# 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]

## Method description

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

${\displaystyle P={\begin{pmatrix}B_{0}&B_{1}&B_{2}&B_{3}&\cdots \\A_{0}&A_{1}&A_{2}&A_{3}&\cdots \\&A_{0}&A_{1}&A_{2}&\cdots \\&&A_{0}&A_{1}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{pmatrix}}}$

where Bi and Ai 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]

${\displaystyle P\pi =\pi \quad {\text{ and }}\quad \mathbf {e} ^{\text{T}}\pi =1}$

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]

${\displaystyle G=\sum _{i=0}^{\infty }G^{i}A_{i}.}$

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

{\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]

{\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 πi are given by Ramaswami's formula,[3] a numerically stable relationship first published by Vaidyanathan Ramaswami in 1988.[9]

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

## Computation of G

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

## References

1. ^ Harchol-Balter, M. (2012). "Phase-Type Distributions and Matrix-Analytic Methods". Performance Modeling and Design of Computer Systems. p. 359. ISBN 9781139226424. doi:10.1017/CBO9781139226424.028.
2. ^ Neuts, M. F. (1984). "Matrix-analytic methods in queuing theory". European Journal of Operational Research. 15: 2–12. doi:10.1016/0377-2217(84)90034-1.
3. Meini, B. (1997). "An improved FFT-based version of Ramaswami's formula". Communications in Statistics. Stochastic Models. 13 (2): 223–238. doi:10.1080/15326349708807423.
4. ^ Stathopoulos, A.; Riska, A.; Hua, Z.; Smirni, E. (2005). "Bridging ETAQA and Ramaswami's formula for the solution of M/G/1-type processes". Performance Evaluation. 62: 331. doi:10.1016/j.peva.2005.07.003.
5. ^ Riska, A.; Smirni, E. (2002). "M/G/1-Type Markov Processes: A Tutorial". Performance Evaluation of Complex Systems: Techniques and Tools (PDF). Lecture Notes in Computer Science. 2459. p. 36. ISBN 978-3-540-44252-3. doi:10.1007/3-540-45798-4_3.
6. ^ Bolch, Gunter; Greiner, Stefan; de Meer, Hermann; Shridharbhai Trivedi, Kishor (2006). Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications (2 ed.). John Wiley & Sons, Inc. p. 250. ISBN 0471565253.
7. ^ Artalejo, Jesús R.; Gómez-Corral, Antonio (2008). "The Matrix-Analytic Formalism". Retrial Queueing Systems. pp. 187–205. ISBN 978-3-540-78724-2. doi:10.1007/978-3-540-78725-9_7.
8. ^ Riska, A.; Smirni, E. (2002). "Exact aggregate solutions for M/G/1-type Markov processes". ACM SIGMETRICS Performance Evaluation Review. 30: 86. doi:10.1145/511399.511346.
9. ^ Ramaswami, V. (1988). "A stable recursion for the steady state vector in markov chains of m/g/1 type". Communications in Statistics. Stochastic Models. 4: 183–188. doi:10.1080/15326348808807077.
10. ^ Bini, D. A.; Latouche, G.; Meini, B. (2005). "Numerical Methods for Structured Markov Chains". ISBN 9780198527688. doi:10.1093/acprof:oso/9780198527688.001.0001.
11. ^ Meini, B. (1998). "Solving m/g/l type markov chains: Recent advances and applications". Communications in Statistics. Stochastic Models. 14: 479–496. doi:10.1080/15326349808807483.
12. ^ Riska, A.; Smirni, E. (2002). "MAMSolver: A Matrix Analytic Methods Tool". Computer Performance Evaluation: Modelling Techniques and Tools. Lecture Notes in Computer Science. 2324. p. 205. ISBN 978-3-540-43539-6. doi:10.1007/3-540-46029-2_14.