Equivalence, Insensitivity and Monotonicity Properties

  • Simonetta Balsamo
  • Vittoria de Nitto Personé
  • Raif Onvural
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 31)

Abstract

Equivalence, insensitivity and monotonicity are important properties of queueing network models that can be used in system performance comparison and evaluation. Insensitivity and equivalence properties provide the basis for comparing the performance of system models with different parameters and with different blocking mechanisms. These results can be applied, for example, to study the impact of the blocking type on system performance, by considering a given set of performance indices and network parameters. In particular, equivalences allow us to define more efficient algorithms to evaluate queueing networks with blocking and to extend the class of models that can be analyzed through analytical methods. Insensitivity properties lead to the identification of the factor that affect system performance and in certain cases it allow a generalization of solution methods. Some important consequence of these properties is that solution methods and algorithms already defined for a certain class of networks could be extended to other classes of networks with different blocking types and/or network parameters. For example, equivalence between networks with and without blocking immediately leads to the extension of efficient computational solution algorithms defined for BCMP networks such as MVA and Convolution algorithm to queueing networks with finite capacity queues. Monotonicity provides insights in the system behavior represented by the queueing network models.

Keywords

Monotonicity Property Blocking Type Equivalence Property Queueing Network Closed Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adan, I., and J. Van Der Wal “Monotonicity of the throughput in single server production and assembly networks with respect to the buffer size” in Queueing Networks with Blocking (H.G. Perros and T. Altiok Eds.), Elsevier, 1989, 325–344.Google Scholar
  2. Akyildiz, I.F., and N. Van Dijk “Exact Solution for Networks of Parallel Queues with Finite Buffers” in Performance ’90 (P.J. King 40, I. Mitrani and R.J. Pooley Eds.) North-Holland, 1990, 35–49.Google Scholar
  3. Akyildiz, I.F., and H. Von Brand “Central Server Models with Multiple Job Classes, State Dependent Routing, and Rejection Blocking” IEEE Trans. on Softw. Eng., Vol. 15 (1989) 1305–1312.CrossRefGoogle Scholar
  4. Akyildiz, I.F., and H. Von Brand “Exact solutions for open, closed and mixed queueing networks with rejection blocking” J. Theor. Computer Science, 64 (1989) 203–219.CrossRefGoogle Scholar
  5. Ammar, M.H., and S.B. Gershwin “Equivalence Relations in Queueing Models of Fork/Join Networks with Blocking” Performance Evaluation, Vol. 10 (1989) 233–245.CrossRefGoogle Scholar
  6. Balsamo, S., and V. De Nitto Personè “A survey of Product-form Queueing Networks with Blocking and their Equivalences” Annals of Operations Research, Vol. 48 (1994) 31–61.CrossRefGoogle Scholar
  7. Balsamo, S., and L. Donatiello “On the Cycle Time Distribution in a Two-stage Queueing Network with Blocking”, IEEE Transactions on Software Engineering, Vol. 13 (1989) 1206–1216.CrossRefGoogle Scholar
  8. Balsamo, S., and L. Donatiello “Two-stage Queueing Networks with Blocking: Cycle Time Distribution and Equivalence Properties”, in Modelling Techniques and Tools for Computer Performance Evaluation, (R. Puigjaner, D. Potier Eds.), Plenum Press, 1989.Google Scholar
  9. Balsamo, S., L. Donatiello and N. Van Dijk “Bounded performance analysis of parallel processing systems” IEEE Trans. on Par. and Distr. Systems, Vol. 9 (1998) 1041–1056.CrossRefGoogle Scholar
  10. Balsamo, S., and G. Iazeolla “Some Equivalence Properties for Queueing Networks with and without Blocking”, in Performance ’83 (A.K. Agrawala, S.K. Tripathi Eds.) North Holland, 1983.Google Scholar
  11. Baskett, F., K.M. Chandy, R.R. Muntz, and G. Palacios “Open, closed, and mixed networks of queues with different classes of customers” J. of ACM, Vol. 22 (1975) 248–260.CrossRefGoogle Scholar
  12. Chandy, K.M., A.J. Martin “A characterization of product-form queueing networks” J. ACM, Vol.30 (1983) 286–299.CrossRefGoogle Scholar
  13. Cheng, D.W. “Analysis of a tandem queue with state dependent general blocking: a GSMP perspective” Performance Evaluation, Vol. 17 (1993) 169–173.CrossRefGoogle Scholar
  14. Christofides, N. Graph Theory — An Algorithmic Approach. Academic Press, London, 1975.Google Scholar
  15. Dallery, Y., and S.B. Gershwin “Manufacturing flow line systems: A review of models and analytical results” Queueing Systems, Vol. 12 (1992) 3–94.CrossRefGoogle Scholar
  16. Dallery, Y., Z. Liu, and D.F. Towsley “Equivalence, reversibility, symmetry and concavity properties in fork/join queueing networks with blocking” Techn, Report, MASI.90.32, Université Pierre et Marie Curie, France, June, 1990 and J. of the ACM, Vol. 41 (1994) 903–942.Google Scholar
  17. Dallery, Y., Z. Liu, and D.F. Towsley “Properties of fork/join queueing networks with blocking under various blocking mechanisms” Techn, Report, MASI.92, Université Pierre et Marie Curie, France, April, 1992.Google Scholar
  18. Dallery, Y., and D.F. Towsley “Symmetry property of the throughput in closed tandem queueing networks with finite buffers” Op. Res. Letters, Vol. 10 (1991) 541–547.CrossRefGoogle Scholar
  19. De Nitto Personè, V. “Topology related index for performance comparison of blocking symmetrical networks” European J. of Oper. Res., Vol. 78 (1994) 413–425.CrossRefGoogle Scholar
  20. De Nitto Personè, V., and D. Grillo “Managing Blocking in Finite Capacity Symmetrical Ring Networks” Third Int. Conf. on Data Comm. Systems and their Performance, Rio de Janeiro, Brazil, June 22–25 (1987) 225–240.Google Scholar
  21. De Nitto Personè, V., and V. Grassi “An analytical model for a parallel fault-tolerant computing system” Performance Evaluation, Vol. 38 (1999) 201–218.CrossRefGoogle Scholar
  22. Gordon, W.J., and G.F. Newell “Cyclic queueing systems with restricted queues” Oper. Res., Vol. 15 (1967) 286–302.CrossRefGoogle Scholar
  23. Heidelberger, P., and S.K. Trivedi “Queueing Network models for parallel processing with synchronous tasks” IEEE Trans. on Comp., Vol. 31 (1982) 1099–1109.CrossRefGoogle Scholar
  24. Lam, S.S. “Queueing networks with capacity constraints” IBM J. Res. Develop., Vol. 21 (1977) 370–378.CrossRefGoogle Scholar
  25. Liu, X.G., and J.A. Buzacott “A decomposition related throughput property of tandem queueing networks with blocking” Queueing Systems, Vol. 13 (1993) 361–383.CrossRefGoogle Scholar
  26. Melamed, B. “A note on the reversibility and duality of some tandem blocking queueing systems” Manage. Sci., Vol. 32 (1986) 1648–1650.CrossRefGoogle Scholar
  27. Muth, E.J. “The reversibility property of production lines” Manage. Sci., Vol. 25 (1979) 152–158.CrossRefGoogle Scholar
  28. Nelson, R., D. Towsley and A. Tantawi “Performance analysis of parallel processing systems” IEEE Trans. on Softw. Eng., Vol. 14 (1988) 532–540.CrossRefGoogle Scholar
  29. Onvural, R.O. “A Note on the Product Form Solutions of Multiclass Closed Queueing Networks with Blocking” Performance Evaluation, Vol.10 (1989) 247–253.CrossRefGoogle Scholar
  30. Onvural R.O. “Survey of Closed Queueing Networks with Blocking” ACM Computing Surveys, Vol. 22 (1990) 83–121.CrossRefGoogle Scholar
  31. Onvural, R.O., and H.G. Perros “On equivalences of blocking mechanisms in queueing networks with blocking” Oper. Res. Letters, Vol. 5 (1986) 293–297.CrossRefGoogle Scholar
  32. Papadopoulus, H.T., and C. Heavey “Queueing Theory in manufacturing systems analysis and design: A classification of models for production and transfer lines” Europ. J. of Oper. Res., Vol. 92 (1996) 1–27.CrossRefGoogle Scholar
  33. Reed D.A., and D.C. Grunwald “The performance of Multicomputer Interconnection Networks” Computer, (1987) 63–73.Google Scholar
  34. Shantikumar, G.J., and D.D. Yao “Monotonicity properties in cyclic networks with finite buffers” Proc. First Int. Workshop on Queueing Networks with Blocking, (H.G. Perros and T. Altiok Eds.) North Holland, 1989.Google Scholar
  35. Van Dijk, N. “On ‘stop = repeat’ servicing for non-exponential queueing networks with blocking” J. Appl. Prob., Vol. 28 (1991) 159–173.CrossRefGoogle Scholar
  36. Van Dijk, N. “Stop = recirculate’ for exponential product form queueing networks with departure blocking” Oper. Res. Lett., Vol. 10 (1991) 343–351.CrossRefGoogle Scholar
  37. Van Dijk, N.M., and H.C. Tijms “Insensitivity in Two Node Blocking Models with Applications” in Teletraffic Analysis and Computer Performance Evaluation (Boxma, Cohen and Tijms Eds.), Elsevier Science Publishers, North Holland, 1986, 329–340.Google Scholar
  38. Yamazaki, G., and H. Sakasegawa “Problems of duality in tandem queueing networks” Ann. Inst. Math. Stat., Vol. 27 (1975) 201–212.CrossRefGoogle Scholar
  39. Yao, D.D., and J.A. Buzacott “Modeling a class of state-dependent routing in flexible manufacturing systems” Annals of Oper. Res., Vol. 3 (1985) 153–167.CrossRefGoogle Scholar
  40. Yao, D.D., and J.A. Buzacott “Modeling a class of flexible manufacturing systems with reversible routing” Oper. Res., Vol. 35 (1987) 87–93.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Simonetta Balsamo
    • 1
  • Vittoria de Nitto Personé
    • 2
  • Raif Onvural
    • 3
  1. 1.Universita’ di VeneziaItaly
  2. 2.Universita’ di Roma “Tor Vergata”Italy
  3. 3.IBMUSA

Personalised recommendations