Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-28T02:31:16.295Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Skip-Free M/G/1 and G/M/1 Type Markov Chains

Part of: Special processes Markov processes Computer system organization Operations research and management science

Published online by Cambridge University Press:  01 July 2016

H. R. Gail ,
S. L. Hantler  and
B. A. Taylor
H. R. Gail*
Affiliation:
IBM Thomas J. Watson Research Center
S. L. Hantler*
Affiliation:
IBM Thomas J. Watson Research Center
B. A. Taylor*
Affiliation:
The University of Michigan
*
∗ Postal address: IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, USA.
∗ Postal address: IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, USA.
∗∗ Postal address: The University of Michigan, Ann Arbor, MI 48109, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

For Markov chains of M/G/1 type that are not skip-free to the left, the corresponding G matrix is shown to have special structure and be determined by its first block row. An algorithm that takes advantage of this structure is developed for computing G. For non-skip-free M/G/1 type Markov chains, the algorithm significantly reduces the computational complexity of calculating the G matrix, when compared with reblocking to a system that is skip-free to the left and then applying usual iteration schemes to find G. A similar algorithm to calculate the R matrix for G/M/1 type Markov chains that are not skip-free to the right is also described.

Keywords

MATRIX ANALYTIC METHODSMARKOV CHAINS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 3 , September 1997 , pp. 733 - 758
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bini, D. and Meini, B. (1995) On cyclic reduction applied to a class of Toeplitz-like matrices arising in queueing problems. In Computations with Markov Chains. ed. Stewart, W. J.. Kluwer, Dordrecht. pp. 21–38.Google Scholar
[2] Bini, D. and Meini, B. (1996) On the solution of a nonlinear matrix equation arising in queueing problems. SIAM J. Matrix Anal. Appl. 17, 906–926.CrossRefGoogle Scholar
[3] Çinlar, E. (1967) Time dependence of queues with semi-Markovian services. J. Appl. Prob. 4, 356–364.Google Scholar
[4] Çinlar, E. (1967) Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365–379.CrossRefGoogle Scholar
[5] Gail, H. R., Hantler, S. L., Konheim, A. G. and Taylor, B. A. (1994) An analysis of a class of telecommunications models. Perf. Eval. 21, 151–161.Google Scholar
[6] Gail, H. R., Hantler, S. L. and Taylor, B. A. (1994) Solutions of the basic matrix equation for M/G/1 and G/M/1 type Markov chains. Commun. Statist.-Stoch. Models 10, 1–43.CrossRefGoogle Scholar
[7] Gail, H. R., Hantler, S. L. and Taylor, B. A. (1996) Spectral analysis of M/G/1 and G/M/1 type Markov chains. Adv. Appl. Prob. 28, 114–165.CrossRefGoogle Scholar
[8] Gohberg, I., Lancaster, P. and Rodman, L. (1982) Matrix Polynomials. Academic Press, New York.Google Scholar
[9] Grassmann, W. K. and Heyman, D. P. (1990) Equilibrium distribution of block-structured Markov chains with repeating rows. J. Appl. Prob. 27, 557–576.Google Scholar
[10] Grassmann, W. K. and Heyman, D. P. (1993) Computation of steady-state probabilities for infinite-state Markov chains with repeating rows. ORSA J. Computing 5, 292–303.CrossRefGoogle Scholar
[11] Horn, R. A. and Johnson, C. A. (1985) Matrix Analysis. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[12] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1966) Denumerable Markov Chains. Van Nostrand, New York.Google Scholar
[13] Latouche, G. (1993) Algorithms for infinite Markov chains with repeating columns. In Linear Algebra, Markov Chains, and Queueing Models. ed. Meyer, C. D. and Plemmons, R. J.. Springer, Berlin. pp. 231–265.Google Scholar
[14] Latouche, G. and Ramaswami, V. (1993) A logarithmic reduction algorithm for quasi-birth-death processes. J. Appl. Prob., 30, 650–674.CrossRefGoogle Scholar
[15] Meyer, C. D. (1989) Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev. 31, 240–272.Google Scholar
[16] Neuts, M. F. (1966) The single server queue with Poisson input and semi-Markov service times. J. Appl. Prob. 3, 202–230.CrossRefGoogle Scholar
[17] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
[18] Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
[19] Neuts, M. F. (1995) Matrix-analytic methods in queueing theory. In Advances in Queueing: Theory, Methods and Open Problems. ed. Dshalalow, J. H., CRC Press, New York. pp. 265–292.Google Scholar
[20] Ramaswami, V. (1988) A stable recursion for the steady state vector in Markov chains of M/G/1 type. Commun. Statist.-Stoch. Models 4, 183–188.Google Scholar