On Latouche-Ramaswami's logarithmic reduction algorithm for quasi-birth-and-death processes

Stochastic Models

Volumn 18, Issue 3, 2002, Pages 449-467

  Ye, Qiang ^a  

^a UNIVERSITY OF KENTUCKY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; MARKOV PROCESSES; MATRIX ALGEBRA; NONLINEAR EQUATIONS;

ROUNDING ERRORS;

QUEUEING THEORY;

EID: 0042427252     PISSN: 15326349     EISSN: None     Source Type: Journal    
DOI: 10.1081/STM-120014221     Document Type: Article

References (24)

1. Alfa, A.S.; Xue, J.; Ye, Q. Accurate Computation of the Smallest Eigenvalue of a Diagonally Dominant M-matrix. Math. Comp. 2002, 71, 217-236.
2. Bai, Z.-Z. A Class of Iteration Methods Based on the Moser Formula for Nonlinear Equations in Markov Chains. Linear Alg. Appl. 1997, 265, 219-241.
3. Berman, A.; Plemmons, R. Nonnegative Matrices in Mathematical Science; Academic Press: New York, 1979.
4. Bini, D.A.; Meini, B. On the Solution of a Nonlinear Matrix Equation Arising in Queueing Problems. SIAM J. Matrix Anal. Appl. 1996, 17, 906-926.
5. Bini, D.A.; Meini, B. Improved Cyclic Reduction for Solving Queueing Problems. Numer. Alg. 1997, 15, 57-74.
6. Bini, D.A.; Meini, B.; Ramaswami, V. Analyzing M/G/1 Paradigms Through QBDs: The Role of the Block Structure in Computing the Matrix G. In Advances in Algorithmic Methods for Stochastic Models - Proceedings of MAM3; Latouche, G., Taylor, P.O., Eds.; Notable Publications Inc.: New Jersey, 2000; 73-86.
7. Bright, L.; Taylor, P.G. Calculating the Equilibrium Distribution in Level Dependent Quasi-Birth-and-Death Processes. Commun. Stat. Stochastic Models 1995, 11, 497-525.
8. Daigle, J.; Lucantoni, D. Queueing Systems Having Phase-Dependent Arrival and Service Rates. In Numerical Solution of Markov Chains; Stewart, W.J., Ed.; Marcel Dekker: New York, 1991; 161-202.
9. Favati, P.; Meini, B. Relaxed Functional Iteration Techniques for the Numerical Solution of M/G/1 Type Markov Chains. BIT 1998, 38, 510-526.
10. Golub, G.; Van Loan, C. Matrix Computations, 2nd Ed.; The Johns Hopkins University Press: Baltimore, MD, 1989.
11. Grassmann, W.K.; Taksar, M.I.; Heyman, D.P. Regenerative Analysis and Steady-State Distributions for Markov Chains. Oper. Res. 1985, 33, 1107-1116.
12. Guo, C.-H. On the Numerical Solution of a Nonlinear Matrix Equation in Markov Chains. Linear Alg. Appl. 1999, 288, 175-186.
13. Guo, C.-H. Convergence Analysis of the Latouche-Ramaswami Algorithm for Null Recurrent Quasi-Birth-Death Processes. SIAM J. Matrix Anal. Appl. 2002, 23, 744-760.
14. Hartfiel, D.J. Lecture Notes in Math. 1695; Springer: Berlin, 1998.
15. Latouche, G. Newton's Iteration for Nonlinear Equations in Markov Chains. IMA J. Numer. Anal. 1994, 14, 583-598.
16. Latouche, G.; Ramaswami, V. A Logarithmic Reduction Algorithm for Quasi-Birth-Death Processes. J. Appl. Probab. 1993, 30, 650-674.
17. Latouche, G.; Ramaswami, V. Introduction to Matrix Analytic Methods in Stochastic Modeling; SIAM: Philadelphia, 1999.
18. Lucantoni, D.M.; Ramaswami, V. Efficient Algorithms for Solving the Nonlinear Matrix Equations Arising in Phase Type Queues. Commun. Stat. Stochastic Models 1985, 1, 29-51.
19. Meini, B. New Convergence Results on Functional Iteration Techniques for the Numerical Solution of M/G/1 Type Markov Chains. Numer. Math. 1997, 78, 39-58.
20. O'Cinneide, C. Entrywise Perturbation Theory and Error Analysis for Markov Chains. Numer. Math. 1993, 65, 109-120.
21. Ramaswami, V. Nonlinear Matrix Equations in Applied Probability - Solution Techniques and Open Problems. SIAM Rev. 1988, 30, 256-263.
22. Ramaswami, V.; Taylor, P.G. Some Properties of the Rate Operators in Level Dependent Quasi-Birth-and-Death Processes with a Countable Number of Phases. Commun. Stat. Stochastic Models 1996, 12, 143-164.
23. Neuts,M.F. Matrix Geometric Solutions in Stochastic Models; The Johns Hopkins University Press: Baltimore, 1981.
24. Ye, Q. High Accuracy Algorithms for Solving Nonlinear Matrix Equations in Queueing Models. In Advances in Algorithmic Methods for Stochastic Models - Proceedings of MAM3; Latouche, G., Taylor, P.G., Eds.; Notable Publications Inc.: New Jersey, 2000; 401-415.