Skip to main content
SpringerLink
Log in

The Finite Element Method for Computing the Stationary Distribution of an SRBM in a Hypercube with Applications to Finite Buffer Queueing Networks

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper proposes an algorithm, referred to as BNAfm (Brownian network analyzer with finite element method), for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. The SRBM serves as an approximate model of queueing networks with finite buffers. Our BNAfm algorithm is based on the finite element method and an extension of a generic algorithm developed by Dai and Harrison [14]. It uses piecewise polynomials to form an approximate subspace of an infinite-dimensional functional space. The BNAfm algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to the BNAsm algorithm (Brownian network analyzer with spectral method) of Dai and Harrison [14], which uses global polynomials to form the approximate subspace and which sometimes fails to produce meaningful estimates of these stationary probabilities. Extensive computational experiences from our implementation are reported, which may be useful for future numerical research on SRBMs. A three-station tandem network with finite buffers is presented to illustrate the effectiveness of the Brownian approximation model and our BNAfm algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Ashcraft, Ordering sparse matrices and transforming front trees, Working paper (1999).

  2. C. Ashcraft and R. Grimes, SPOOLES: An object-oriented sparse matrix library, in: Proc. of the 1999 SIAM Conf. on Parallel Processing for Scientific Computing, 22-27 March 1999.

  3. I. Bardhan and S. Mithal, Heavy traffic limits for an open network of finite buffer overflow queues: The single class case, Preprint (1993).

  4. E.B. Becker, G.F. Carey and J. Tinsley Oden, Finite Elements: An Introduction (Prentice-Hall, Englewood Cliffs, NJ, 1981).

    Google Scholar 

  5. D. Bertsekas and R. Gallager, Data Networks (Prentice-Hall, Englewood Cliffs, NJ, 1992).

    Google Scholar 

  6. J. Buzacott and J.G. Shanthikumar, Design of manufacturing systems using queueing models, Queueing Systems 12 (1992) 135-213.

    Google Scholar 

  7. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer, Berlin, 1988).

    Google Scholar 

  8. G.F. Carey and J. Tinsley Oden, Finite Elements: A Second Course (Prentice-Hall, Englewood Cliffs, NJ, 1981).

    Google Scholar 

  9. H. Chen and A. Mandelbaum, Hierarchical modelling of stochastic networks, Part II: Strong approximations, in: Stochastic Modeling and Analysis of Manufacturing Systems, ed. D.D. Yao (Springer, Berlin, 1994) pp. 107-131.

    Google Scholar 

  10. H. Chen and X. Shen, Computing the stationary distribution of SRBM in an orthant, Preprint (2000).

  11. H. Chen and D.D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization (Springer, Berlin, 2001).

    Google Scholar 

  12. J.G. Dai, Steady-state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications, Ph.D. thesis, Stanford University (1990).

  13. J.G. Dai and W. Dai, A heavy traffic limit theorem for a class of open queueing networks with finite buffers, Queueing Systems 32 (1998) 5-40.

    Google Scholar 

  14. J.G. Dai and J.M. Harrison, Steady-state analysis of RBM in a rectangle: Numerical methods and a queueing application, Ann. Appl. Probab. 1 (1991) 16-35.

    Google Scholar 

  15. J.G. Dai and J.M. Harrison, Reflected Brownian motion in an orthant: Numerical methods for steadystate analysis, Ann. Appl. Probab. 2 (1992) 65-86.

    Google Scholar 

  16. J.G. Dai and T.G. Kurtz, Characterization of the stationary distribution for a semimartingale reflecting Brownian motion in a convex polyhedron, Preprint (1997).

  17. J.G. Dai, V. Nguyen and M.I. Reiman, Sequential bottleneck decompositions: An approximation method for generalized Jackson networks, Oper. Res. 42 (1994) 119-136.

    Google Scholar 

  18. J.G. Dai and R. Williams, Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons, Theory Probab. Appl. 40 (1995) 1-40.

    Google Scholar 

  19. W. Dai, Brownian approximations for queueing networks with finite buffers: Modeling, heavy traffic analysis and numerical implementations, Ph.D. dissertation, Georgia Institute of Technology (1996).

  20. J. Dongarra, A. Lumsdaine, R. Pozo and K. Remington, A sparse matrix library in C++ for high performance architectures, in: Proc. of the 2nd Object Oriented Numerics Conf., 1994, pp. 214-218.

  21. J. Dongarra, A set of level 3 basic linear algebra subprograms, ACM Trans. Math. Software 16 (1990) 1-17.

    Google Scholar 

  22. S.N. Ethier and T.G. Kurtz, Markov Process: Characterization and Convergence (Wiley, New York, 1986).

    Google Scholar 

  23. P.W. Glynn, Diffusion approximations, in: Handbooks in Operations Research and Management Science, II: Stochastic Models, eds. D.P. Heyman and M.J. Sobel (North-Holland, Amsterdam, 1990) pp. 145-198.

    Google Scholar 

  24. J.M. Harrison, H. Landau and L.A. Shepp, The stationary distribution of reflected Brownian motion in a planar region, Ann. Probab. 13 (1985) 744-757.

    Google Scholar 

  25. J.M. Harrison, Brownian Motion and Stochastic Flow Systems (Wiley, New York, 1985).

    Google Scholar 

  26. J.M. Harrison and V. Nguyen, The QNET method for two-moment analysis of open queueing networks, Queueing Systems 6 (1990) 1-32.

    Google Scholar 

  27. J.M. Harrison and V. Nguyen, Brownian models of multiclass queueing networks: Current status and open problems, Queueing Systems 13 (1993) 5-40.

    Google Scholar 

  28. J.M. Harrison and R.J. Williams, Multidimensional reflected Brownian motions having exponential stationary distributions, Ann. Probab. 15 (1987) 115-137.

    Google Scholar 

  29. J.R. Jackson, Job shop-like queueing systems, Managm. Sci. 10 (1963) 131-142.

    Google Scholar 

  30. F.P. Kelly, Reversibility and Stochastic Networks (Wiley, New York, 1979).

    Google Scholar 

  31. L. Kleinrock, Queueing Systems II: Computer Applications (Wiley, New York, 1976).

    Google Scholar 

  32. A.J. Lemoine, Network of queues-a survey of weak convergence results, Managm. Sci. 24 (1978) 1175-1193.

    Google Scholar 

  33. M.J. Maron, Numerical Analysis: A Practical Approach, 2nd ed. (Macmillan, New York, 1987).

    Google Scholar 

  34. J.T. Oden and J.N. Reddy, An Introduction to the Mathematical Theory of Finite Elements (Wiley-Interscience, New York, 1976).

    Google Scholar 

  35. E. Schwerer, A linear programming approach to the steady-state analysis of Markov process, Ph.D. dissertation, Stanford University (1997).

  36. X. Shen, Performance evaluation of multiclass queueing networks via Brownian motions, Ph.D. dissertation, Faculty of Commerce and Business Administration, University of British Columbia (2001).

  37. L. Trefethen and R.J. Williams, Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions, J. Comput. Appl. Math. 14 (1985) 227-249.

    Google Scholar 

  38. J. Walrand, An Introduction to Queueing Networks (Prentice-Hall, Englewood Cliffs, NJ, 1988).

    Google Scholar 

  39. W. Whitt, Heavy traffic theorems for queues: A survey, in: Mathematical Methods in Queueing Theory, ed. A.B. Clarke (Springer, Berlin, 1974) pp. 307-350.

    Google Scholar 

  40. W. Whitt, The queueing network analyzer, Bell System Tech. J. 62(9) (1983) 2779-2815.

    Google Scholar 

  41. R.J. Williams, On the approximation of queueing networks in heavy traffic, in: Stochastic Networks: Theory and Applications, eds. F.P. Kelly, S. Zachary and I. Ziedins (Oxford Univ. Press, Oxford, 1996) pp. 35-56.

    Google Scholar 

  42. D.D. Yao, ed., Stochastic Modeling and Analysis of Manufacturing Systems (Springer, New York, 1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, Canada

    Xinyang Shen & Hong Chen

  2. School of Industrial and Systems Engineering and School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0205, USA

    J.G. Dai

  3. Department of Mathematics, Nanjing University, Nanjing, China

    Wanyang Dai

Authors
  1. Xinyang Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.G. Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wanyang Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shen, X., Chen, H., Dai, J. et al. The Finite Element Method for Computing the Stationary Distribution of an SRBM in a Hypercube with Applications to Finite Buffer Queueing Networks. Queueing Systems 42, 33–62 (2002). https://doi.org/10.1023/A:1019942711261

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019942711261

Search

Navigation