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Method description

An M/G/1-type stochastic matrix is one of the form⁶

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 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.⁷⁸ If P is irreducible and positive recurrent then the stationary distribution is given by the solution to the equations⁹

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 that¹⁰

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

G is called the auxiliary matrix.¹¹ Matrices are defined¹²

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 solving¹³

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 πi are given by Ramaswami's formula,¹⁴ a numerically stable relationship first published by Vaidyanathan Ramaswami in 1988.¹⁵

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.}

Computation of G

There are two popular iterative methods for computing G,¹⁶¹⁷

• functional iterations
• cyclic reduction.

Tools

• MAMSolver¹⁸

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. 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. /wiki/Doi_(identifier) ↩

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. /wiki/Beatrice_Meini ↩

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. /wiki/Evgenia_Smirni ↩

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. 978-3-540-44252-3 ↩

6. 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. /wiki/Beatrice_Meini ↩

7. 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. 978-0471565253 ↩

8. 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. 978-3-540-78724-2 ↩

9. 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. /wiki/Beatrice_Meini ↩

10. 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. /wiki/Beatrice_Meini ↩

11. 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. /wiki/Evgenia_Smirni ↩

12. 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. /wiki/Beatrice_Meini ↩

13. 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. /wiki/Beatrice_Meini ↩

14. 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. /wiki/Beatrice_Meini ↩

15. 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. /wiki/Doi_(identifier) ↩

16. Bini, D. A.; Latouche, G.; Meini, B. (2005). Numerical Methods for Structured Markov Chains. doi:10.1093/acprof:oso/9780198527688.001.0001. ISBN 9780198527688. 9780198527688 ↩

17. 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. /wiki/Beatrice_Meini ↩

18. 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. 978-3-540-43539-6 ↩