Matrix geometric method

In probability theory, the matrix geometric method is a method for the analysis of quasi-birth–death processes, continuous-time Markov chain whose transition rate matrices with a repetitive block structure.^[1] The method was developed "largely by Marcel F. Neuts and his students starting around 1975."^[2]

Method description

The method requires a transition rate matrix with tridiagonal block structure as follows

$Q=\begin{pmatrix} B_{00} & B_{01} \\ B_{10} & A_1 & A_2 \\ & A_0 & A_1 & A_2 \\ && A_0 & A_1 & A_2 \\ &&& A_0 & A_1 & A_2 \\ &&&& \ddots & \ddots & \ddots \end{pmatrix}$

where each of B₀₀, B₀₁, B₁₀, A₀, A₁ and A₂ are matrices. To compute the stationary distribution π writing π Q = 0 the balance equations are considered for sub-vectors π_i

$\begin{align} \pi_0 B_{00} + \pi_1 B_{10} &= 0\\ \pi_0 B_{01} + \pi_1 A_1 + \pi_2 A_0 &= 0\\ \pi_1 A_2 + \pi_2 A_1 + \pi_3 A_0 &= 0 \\ & \vdots \\ \pi_{i-1} A_2 + \pi_i A_1 + \pi_{i+1} A_0 &= 0\\ & \vdots \\ \end{align}$

Observe that the relationship

$\pi_i = \pi_1 R^{i-1}$

holds where R is the Neut's rate matrix,^[3] which can be computed numerically. Using this we write

$\begin{align} \begin{pmatrix}\pi_0 & \pi_1 \end{pmatrix} \begin{pmatrix}B_{00} & B_{01} \\ B_{10} & A_1 + RA_0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \end{pmatrix} \end{align}$

which can be solve to find π₀ and π₁ and therefore iteratively all the π_i.

Computation of R

The matrix R can be computed using cyclic reduction^[4] or logarithmic reduction.^[5][6]

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 matrices.^[7] Such models are harder because no relationship like π_i = π₁ R^i – 1 used above holds.^[8]

External links

• Performance Modelling and Markov Chains (part 2) by William J. Stewart at 7th International School on Formal Methods for the Design of Computer, Communication and Software Systems: Performance Evaluation

References

1. Harrison, Peter G.; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley. pp. 317–322. ISBN 0-201-54419-9.
2. Asmussen, S. R. (2003). "Random Walks". Applied Probability and Queues. Stochastic Modelling and Applied Probability 51. pp. 220–243. doi:10.1007/0-387-21525-5_8. ISBN 978-0-387-00211-8.
3. Ramaswami, V. (1990). "A duality theorem for the matrix paradigms in queueing theory". Communications in Statistics. Stochastic Models 6: 151–161. doi:10.1080/15326349908807141.
4. Bini, D.; Meini, B. (1996). "On the Solution of a Nonlinear Matrix Equation Arising in Queueing Problems". SIAM Journal on Matrix Analysis and Applications 17 (4): 906. doi:10.1137/S0895479895284804.
5. Latouche, Guy; Ramaswami, V. (1993). "A Logarithmic Reduction Algorithm for Quasi-Birth-Death Processes". Journal of Applied Probability (Applied Probability Trust) 30 (3): 650–674. JSTOR 3214773.
6. Pérez, J. F.; Van Houdt, B. (2011). "Quasi-birth-and-death processes with restricted transitions and its applications" (PDF). Performance Evaluation 68 (2): 126. doi:10.1016/j.peva.2010.04.003.
7. Alfa, A. S.; Ramaswami, V. (2011). "Matrix Analytic Method: Overview and History". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0631. ISBN 9780470400531.
8. 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. 259. ISBN 0471565253.