Abstract
M/G/1-type processes are commonly encountered when modeling modern complex computer and communication systems. In this tutorial, we present a detailed survey of existing solution methods for M/G/1-type processes, focusing on the matrix-analytic methodology. From first principles and using simple examples, we derive the fundamental matrix-analytic results and lay out recent advances. Finally, we give an overview of an existing, state-of-the-art software tool for the analysis of M/G/1-type processes.
This work has been supported by National Science Foundation under grands EIA- 9974992, CCR-0098278, and ACI-0090221.
Chapter PDF
Similar content being viewed by others
References
D. A. Bini and B. Meini. Using displacement structure for solving non-skip-free M/G/1 type Markov chains. In A. Alfa and S. Chakravarthy, editors, Advances in Matrix Analytic Methods for Stochastic Models, pages 17–37, Notable Publications Inc. NJ, 1998.
D. A. Bini, B. Meini, and V. Ramaswami. Analyzing M/G/1 paradigms through QBDs: the role of the block structure in computing the matrix G. In G. Latouche and P. Taylor, editors, Advances in Matrix-Analytic Methods for Stochastic Models, pages 73–86, Notable Publications Inc. NJ, 2000.
L. Breuer. Parameter estimation for a class of BMAPs. In G. Latouche and P. Taylor, editors, Advances in Matrix-Analytic Methods for Stochastic Models, pages 87–97, Notable Publications Inc. NJ, 2000.
G. Ciardo, A. Riska, and E. Smirni. An aggregation-based solution method for M/G/1-type processes. In B. Plateau, W. J. Stewart, and M. Silva, editors, Numerical Solution of Markov Chains, pages 21–40. Prensas Universitarias de Zaragoza, Zaragoza, Spain, 1999.
G. Ciardo and E. Smirni. ETAQA: an efficient technique for the analysis of QBD processes by aggregation. Performance Evaluation, vol. 36—37, pages 71–93, 1999.
J. N. Daige and D. M. Lucantoni. Queueing systems having phase-dependent arrival and service rates. In J. W. Stewart, editor, Numerical Solution of Markov Chains, pages 179–215, Marcel Dekker, New York, 1991.
H. R. Gail, S. L. Hantler, and B. A. Taylor. Use of characteristic roots for solving infinite state Markov chains. In W. K. Grassmann, editor, Computational Probability, pages 205–255, Kluwer Academic Publishers, Boston, MA, 2000.
W. K. Grassmann and D. A. Stanford. Matrix analytic methods. In W. K. Grassmann, editor, Computational Probability, pages 153–204, Kluwer Academic Publishers, Boston, MA, 2000.
D. Green. Lag correlation of approximating departure process for MAP/PH/1 queues. In G. Latouche and P. Taylor, editors, Advances in Matrix-Analytic Methods for Stochastic Models, pages 135–151, Notable Publications Inc. NJ, 2000.
B. Haverkort, A. Van Moorsel, and A. Dijkstra. MGMtool: A Performance Analysis Tool Based on Matrix Geometric Methods. In R. Pooley, and J. Hillston, editors, ModellingT echniques and Tools, pages 312–316, Edinburgh University Press, 1993.
D. Heyman and A. Reeves. Numerical solutions of linear equations arising in Markov chain models. ORSA Journal on Computing, vol. 1 pages 52–60, 1989.
D. Heyman and D. Lucantoni. Modeling multiple IP traffic streams with rate limits. In Proceedings of the 17th International Teletraffic Congress, Brazil, Dec. 2001.
L. Kleinrock. Queueing systems. Volume I: Theory, Wiley, 1975.
G. Latouche. A simple proof for the matrix-geometric theorem. Applied Stochastic Models and Data Analysis, vol. 8, pages 25–29, 1992.
G. Latouche and G. W. Stewart. Numerical methods for M/G/1 type queues. In G. W. Stewart, editor, Computations with Markov chains, pages 571–581, Kluwer Academic Publishers, Boston, MA, 1995.
G. Latouche and V. Ramaswami. Introduction to Matrix Geometric Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia, PA, 1999.
D. M. Lucantoni. The BMAP/G/1 queue: A tutorial. In L. Donatiello and R. Nelson, editors, Models and Techniques for Performance Evaluation of Computer and Communication Systems, pages 330–358. Springer-Verlag, 1993.
D. M. Lucantoni. An algorithmic analysis of a communication model with retransmission of flawed messages. Pitman, Boston, 1983.
B. Meini. An improved FFT-based version of Ramaswami’s formula. Comm. Statist. Stochastic Models, vol. 13, pages 223–238, 1997.
B. Meini. Solving M/G/1 type Markov chains: Recent advances and applications. Comm. Statist. Stochastic Models, vol. 14(1&2), pages 479–496, 1998.
B. Meini. Fast algorithms for the numerical solution of structured Markov chains. Ph.D. Thesis, Department of Mathematics, University of Pisa, 1998.
C. D. Meyer. Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Review, vol. 31(2) pages 240–271, June 1989.
R. Nelson. Matrix geometric solutions in Markov models: a mathematical tutorial. Research Report RC 16777 (#742931), IBM T.J. Watson Res. Center, Yorktown Heights, NY, Apr. 1991.
R. Nelson. Probability, Stochastic Processes, and Queueing Theory. Springer-Verlag, 1995.
M. F. Neuts. Matrix-geometric solutions in stochastic models. Johns Hopkins University Press, Baltimore, MD, 1981.
M. F. Neuts. Structured stochastic matrices of M/G/1 type and their applications. Marcel Dekker, New York, NY, 1989.
B. F. Nielsen. Modeling long-range dependent and heavy-tailed phenomena by matrix analytic methods. In Advances in Matrix-Analytic Methods for Stochastic Models, G. Latouche and P. Taylor, editors, Notable Publications, pages 265–278, 2000.
V. Ramaswami and G. Latouche. A general class of Markov processes with explicit matrix-geometric solutions. OR Spektrum, vol. 8, pages 209–218, Aug. 1986.
V. Ramaswami. A stable recursion for the steady state vector in Markov chains of M/G/1 type. Comm. Statist. Stochastic Models, vol. 4, pages 183–263, 1988.
V. Ramaswami and J. L. Wang. A hybrid analysis/simulation for ATM performance with application to quality-of-service of CBR traffic. Telecommunication Systems, vol. 5, pages 25–48, 1996.
A. Riska and E. Smirni. An exact aggregation approach for M/G/1-type Markov chains. In the Proceedings of the ACM International Conference on Measurement and Modelingof Computer Systems (ACM SIGMETRICS’ 02), pages 86–96, Marina Del Rey, CA, 2002.
A. Riska and E. Smirni. MAMSolver: a Matrix-analytic methods tools. In T. Field et al. (editors), TOOLS 2002, LNCS 2324, pages 205–211, Springer-Verlag, 2002.
A. Riska, M. S. Squillante, S.-Z. Yu, Z. Liu, and L. Zhang. Matrix-analytic analysis of a MAP/PH/1 queue fitted to web server data. 4th Conference on Matrix-Analytic Methods (to appear), Adelaide, Australia, July 2002.
H. Schellhaas. On Ramaswami’s algorithm for the computation of the steady state vector in Markov chains of M/G/1 type. Comm. Statist. Stochastic Models, vol. 6, pages 541–550, 1990.
M. S. Squillante. Matrix-analytic methods: Applications, results and software tools. In G. Latouche and P. Taylor, editors, Advances in Matrix-Analytic Methods for Stochastic Models, Notable Publications Inc. NJ, 2000.
M. S. Squillante. MAGIC: A computer performance modeling tool based on matrixgeometric techniques. In G. Balbo and G. Serazzi, editors, Computer Performance Evaluation: Modeling Techniques and Tools, North-Holland, Amsterdam, pages 411–425, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Riska, A., Smirni, E. (2002). M/G/1-Type Markov Processes: A Tutorial. In: Calzarossa, M.C., Tucci, S. (eds) Performance Evaluation of Complex Systems: Techniques and Tools. Performance 2002. Lecture Notes in Computer Science, vol 2459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45798-4_3
Download citation
DOI: https://doi.org/10.1007/3-540-45798-4_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44252-3
Online ISBN: 978-3-540-45798-5
eBook Packages: Springer Book Archive