Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T07:37:31.105Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite- and finite-buffer Markov fluid queues: a unified analysis

Part of: Special processes Probabilistic methods, simulation and stochastic differential equations Computer system organization General theory of linear operators Numerical linear algebra

Published online by Cambridge University Press:  14 July 2016

Nail Akar  and
Khosrow Sohraby
Nail Akar*
Affiliation:
Bilkent University
Khosrow Sohraby*
Affiliation:
University of Missouri-Kansas City
*
∗ Postal address: Electrical and Electronic Engineering Department, Bilkent University, 06800 Bilkent, Ankara, Turkey.
∗∗ Postal address: School of Interdisciplinary Computing and Engineering, University of Missouri–Kansas City, Kansas City, MO 64110, USA. Email address: sohrabyk@umkc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

Keywords

Stochastic fluid modelMarkov fluid queueperformance analysiscomputer networkspectral divide-and-conquer problemgeneralized Newton iteration

MSC classification

Primary: 60K25: Queueing theory
Secondary: 65C40: Computational Markov chains 65F15: Eigenvalues, eigenvectors 68M20: Performance evaluation; queueing; scheduling 47A15: Invariant subspaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 557 - 569
Copyright
Copyright © Applied Probability Trust 2004 

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

Akar, N., and Sohraby, K. (1997). An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. Stoch. Models 13, 381–416.CrossRefGoogle Scholar
Anick, D., Mitra, D., and Sondhi, M. M. (1982). Stochastic theory of a data handling system with multiple sources. Bell Syst. Tech. J. 61, 1871–1894.CrossRefGoogle Scholar
Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian motion. Stoch. Models 11, 21–49.CrossRefGoogle Scholar
Bai, Z., and Demmel, J. W. (1993). Design of a parallel nonsymmetric eigenroutine toolbox, Part I. In Proc. 6th SIAM Conf. Parallel Processing for Scientific Computing, Vol. 1, eds Sincovec, R. F. et al., Society for Industrial and Applied Mathematics, Philadelphia, PA, pp. 391–398.Google Scholar
Bai, Z., Demmel, J., and Gu, M. (1997). Inverse free parallel spectral divide and conquer algorithms for nonsymmetric eigenproblems. Numer. Math. 76, 389–396.Google Scholar
Chan, T. F. (1987). Rank revealing QR factorizations. Linear Algebra Appl. 88/89, 67–82.CrossRefGoogle Scholar
Demmel, J. and Kågström, B. (1987). Computing stable eigendecompositions of matrix pencils. Linear Algebra Appl. 88/89, 139–186.CrossRefGoogle Scholar
Demmel, J. and Kågström, B. (1993). The generalized Schur decomposition of an arbitrary pencil A-λB: robust software with error bounds and applications. I. Theory and algorithms. ACM Trans. Math. Software 19, 160–174.CrossRefGoogle Scholar
Fiedler, M., and Voos, H. (2000). New results on the numerical stability of the stochastic fluid flow model analysis. In Proc. NETWORKING 2000, Broadband Communications, High Performance Networking, and Performance of Communication Networks, Springer, Paris, pp. 446–457.Google Scholar
Gardiner, J. D., and Laub, A. J. (1986). A generalization of the matrix-sign-function solution for algebraic Riccati equations. Internat. J. Control 44, 823–832.Google Scholar
Kågström, B., and Dooren, P. V. (1992). A generalized state-space approach for the additive decomposition of a transfer matrix. J. Numer. Linear Algebra Appl. 1, 165–181.Google Scholar
Kosten, L. (1984). Stochastic theory of data-handling systems with groups of multiple sources. In Performance of Computer Communication Systems, eds Ruding, H. and Bux, W., Elsevier, Amsterdam, pp. 321–331.Google Scholar
Kulkarni, V. G. (1997). Fluid models for single buffer systems. In Frontiers in Queuing: Models and Applications in Science and Engineering, ed. Dshalalow, J. H., CRC Press, Boca Raton, FL, pp. 321–338.Google Scholar
Mitra, D. (1988). Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Prob. 20, 646–676.Google Scholar
Nagarajan, R., Kurose, J. F., and Towsley, D. (1991). Approximation techniques for computing packet loss in finite-buffered voice multiplexers. IEEE J. Selected Areas Commun. 9, 368–377.Google Scholar
Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. App. Prob. 4, 390–413.Google Scholar
Schwartz, M. (1994). Broadband Integrated Networks. Prentice-Hall, Upper Saddle River, NJ.Google Scholar
Sun, X., and Quintana-Orti, E. S. (2004). Spectral division methods for block generalized Schur decompositions. To appear in Math. Computation.Google Scholar
Sun, X., and Quintana-Orti, E. S. (2003). The generalized Newton iteration for the matrix sign function. SIAM J. Sci. Comput. 24, 669–683.CrossRefGoogle Scholar
Tijms, H. C. (1986). Stochastic Modelling and Analysis: A Computational Approach. John Wiley, New York.Google Scholar
Tucker, R. (1988). Accurate method for analysis of a packet speech multiplexer with limited delay. IEEE Trans. Commun. 36, 479–483.Google Scholar