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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. Such models are often described as M/G/1 type Markov chains because they can describe transitions in an M/G/1 queue. The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains.

Method description

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

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

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. If P is irreducible and positive recurrent then the stationary distribution is given by the solution to the equations

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

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

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

G is called the auxiliary matrix. Matrices are defined

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

then π₀ is found by solving

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

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

${\displaystyle \pi _{i}=(I-{\overline {A}}_{1})^{-1}\left,i\geq 1.}$

Computation of G

There are two popular iterative methods for computing G,

• functional iterations
• cyclic reduction.

Tools

• MAMSolver

References

1. Harchol-Balter, M. (2012). "Phase-Type Distributions and Matrix-Analytic Methods". Performance Modeling and Design of Computer Systems. pp. 359–379. doi:10.1017/CBO9781139226424.028. ISBN 9781139226424.
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 (1–4): 331–348. CiteSeerX 10.1.1.80.9473. doi:10.1016/j.peva.2005.07.003.
5. Riska, A.; Smirni, E. (2002). "M/G/1-Type Markov Processes: A Tutorial" (PDF). Performance Evaluation of Complex Systems: Techniques and Tools. Lecture Notes in Computer Science. Vol. 2459. pp. 36. doi:10.1007/3-540-45798-4_3. ISBN 978-3-540-44252-3.
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 978-0471565253.
7. Artalejo, Jesús R.; Gómez-Corral, Antonio (2008). "The Matrix-Analytic Formalism". Retrial Queueing Systems. pp. 187–205. doi:10.1007/978-3-540-78725-9_7. ISBN 978-3-540-78724-2.
8. Riska, A.; Smirni, E. (2002). "Exact aggregate solutions for M/G/1-type Markov processes". ACM SIGMETRICS Performance Evaluation Review. 30: 86. CiteSeerX 10.1.1.109.2225. 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. doi:10.1093/acprof:oso/9780198527688.001.0001. ISBN 9780198527688.
11. Meini, B. (1998). "Solving m/g/l type markov chains: Recent advances and applications". Communications in Statistics. Stochastic Models. 14 (1–2): 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. Vol. 2324. p. 205. CiteSeerX 10.1.1.146.2080. doi:10.1007/3-540-46029-2_14. ISBN 978-3-540-43539-6.

v t e Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson point process Markovian arrival process Rational arrival process
Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network
Service policies	FIFO LIFO Processor sharing Round-robin Shortest job next Shortest remaining time
Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method
Limit theorems	Fluid limit Mean-field theory Heavy traffic approximation Reflected Brownian motion
Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue
Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering
Category

• Markov processes
• Single queueing nodes
• Probability theory