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Method description

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

Q = ( 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 ⋱ ⋱ ⋱ ) {\displaystyle 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 πi

π 0 B 00 + π 1 B 10 = 0 π 0 B 01 + π 1 A 1 + π 2 A 0 = 0 π 1 A 2 + π 2 A 1 + π 3 A 0 = 0 ⋮ π i − 1 A 2 + π i A 1 + π i + 1 A 0 = 0 ⋮ {\displaystyle {\begin{aligned}\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{aligned}}}

Observe that the relationship

π i = π 1 R i − 1 {\displaystyle \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

( π 0 π 1 ) ( B 00 B 01 B 10 A 1 + R A 0 ) = ( 0 0 ) {\displaystyle {\begin{aligned}{\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{aligned}}}

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

Computation of R

The matrix R can be computed using cyclic reduction⁴ or logarithmic reduction.⁵⁶

Matrix analytic method

Main article: 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 πi = π1 Ri – 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. 0-201-54419-9 ↩

2. Asmussen, S. R. (2003). "Random Walks". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 220–243. doi:10.1007/0-387-21525-5_8. ISBN 978-0-387-00211-8. 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. /wiki/Doi_(identifier) ↩

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. /wiki/Beatrice_Meini ↩

5. Latouche, Guy; Ramaswami, V. (1993). "A Logarithmic Reduction Algorithm for Quasi-Birth-Death Processes". Journal of Applied Probability. 30 (3). Applied Probability Trust: 650–674. JSTOR 3214773. /wiki/JSTOR_(identifier) ↩

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. hdl:10067/859850151162165141. http://www.doc.ic.ac.uk/~jperezbe/data/PerezVanHoudt_PEVA_2011.pdf ↩

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. 9780470400531 ↩

8. Bolch, Gunter; Greiner, Stefan; de Meer, Hermann; Trivedi, Kishor Shridharbhai (2006). Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications (2 ed.). John Wiley & Sons, Inc. p. 259. ISBN 0471565253. 0471565253 ↩