Skip to main content

Advertisement

Springer Nature Link
Log in

In Search of Sensitivity in Network Optimization

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

This paper concerns the following questions regarding policy synthesis in large queueing networks: (i) It is well known that an understanding of variability is important in the determination of safety stocks to prevent unwanted idleness. Is this the only value of high-order statistical information in policy synthesis? (ii) Will a translation of an optimal policy for the deterministic fluid model (in which there is no variability) lead to an allocation which is approximately optimal for the stochastic network? (iii) What are the sources of highest sensitivity in network control? A sensitivity analysis of an associated fluid-model optimal control problem provides an exact dichotomy in (ii). If an optimal policy for the fluid model is ‘maximally non-idling’, then variability plays a small role in control design. If this condition does not hold, then the ‘gap’ between the fluid and stochastic optimal policies is exactly proportional to system variability. Furthermore, under mild assumptions, we find that the optimal policy for the stochastic model is closely approximated by an affine shift of the fluid optimal solution. However, sensitivity of steady-state performance with respect to perturbations in the policy vanishes with increasing variability.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. S. Asmussen, Queueing simulation in heavy traffic, Math. Oper. Res. 17 (1992) 84–111.

    Google Scholar 

  2. F. Avram, D. Bertsimas and M. Ricard, An optimal control approach to optimization of multiclass queueing networks, in: Proc. of Workshop on Queueing Networks of the Mathematical Institute, eds. F. Kelly and R. Williams, Minneapolis, 1994, IMA Volumes in Mathematics and its Applications, Vol. 71 ( Springer, New York, 1995).

  3. F.L. Baccelli, G. Cohen and G.J. Olsder, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1992).

  4. S.L. Bell and R.J. Williams, Dynamic scheduling of a system with two parallel servers: Asymptotic optimality of a continuous review threshold policy in heavy traffic, in: Proc. of the 38th Conf. on Decision and Control, Phoenix, AZ, 1999, pp. 1743–1748.

  5. P. Billingsley, Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, 2nd ed. (Wiley-Interscience, New York, 1999).

  6. V.S. Borkar and S.P. Meyn, Value functions and simulation in stochastic networks, in:42th IEEE Conf. on Decision and Control, 2003, 2003, submitted.

  7. M. Bramson, State space collapse with application to heavy traffic limits for multiclass queueing networks, Queueing Systems 30 (1998) 89–148.

    Google Scholar 

  8. M. Bramson and R.J. Williams, On dynamic scheduling of stochastic networks in heavy traffic and some new results for the workload process, in: Proc. of the 39th Conf. on Decision and Control, 2000.

  9. H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximations, Math. Oper. Res. 16(2) (1991) 408–446.

    Google Scholar 

  10. H. Chen and D.D. Yao, Dynamic scheduling of a multiclass fluid network, Oper. Res. 41(6) (1993) 1104–1115.

    Google Scholar 

  11. H. Chen and D.D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization, Stochastic Modelling and Applied Probability (Springer, New York, 2001).

  12. M. Chen, R. Dubrawski and S.P. Meyn, Management of demand-driven production systems (2002) submitted for publication.

  13. R.L. Cruz, A calculus for network delay, part I: Network elements in isolation, IEEE Trans. Inform. Theory 31 (1991) 114–131.

    Google Scholar 

  14. J.G. Dai, On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models, Ann. Appl. Probab. 5(1) (1995) 49–77.

    Google Scholar 

  15. J.G. Dai and S.P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit models, IEEE Trans. Automat. Control 40 (November 1995) 1889–1904.

    Google Scholar 

  16. D. Down, S.P. Meyn and R.L. Tweedie, Exponential and uniform ergodicity of Markov processes, Ann. Probab. 23(4) (1995) 1671–1691.

    Google Scholar 

  17. P. Dupuis and H. Kushner, Numerical Methods for Stochastic Control Problems in Continuous Time, Applications of Mathematics, Vol. 24 (Springer, New York, 2001).

  18. J.D. Eng, Humphrey and S.P. Meyn, Fluid network models: Linear programs for control and performance bounds, in: Proc. of the 13th IFAC World Congress, eds. J. Cruz, J. Gertler and M. Peshkin, Vol. B, San Francisco, CA, 1996, pp. 19–24.

  19. S.B. Gershwin, Manufacturing Systems Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1993).

    Google Scholar 

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

  21. J.M. Harrison, Brownian models of queueing networks with heterogeneous customer populations, in: Stochastic Differential Systems, Stochastic Control Theory and Applications, Minneapolis, MN, 1986 (Springer, New York, 1988) pp. 147–186.

  22. J.M. Harrison, Brownian models of open processing networks: Canonical representations of workload, Ann. Appl. Probab. 10 (2000) 75–103.

    Google Scholar 

  23. J.M. Harrison, Stochastic networks and activity analysis, in: Analytic Methods in Applied Probability, In Memory of Fridrih Karpelevich, ed. edY. Suhov (Amer. Math. Soc., Providence, RI, 2002).

  24. J.M. Harrison and L.M. Wein, Scheduling networks of queues: Heavy traffic analysis of a two-station closed network, Oper. Res. 38(6) (1990) 1052–1064.

    Google Scholar 

  25. S.G. Henderson, Variance reduction via an approximating Markov process, Ph.D. thesis, Stanford University, Stanford, CA, USA (1997).

  26. S.G. Henderson and P. W. Glynn, Approximating martingales for variance reduction inMarkov process simulation, Math. Oper. Res. 27(2) (2002) 253–271.

    Google Scholar 

  27. S.G. Henderson and S.P. Meyn, Variance reduction for simulation in multiclass queueing networks, IIE Trans. Oper. Engrg. (2003) to appear.

  28. S.G. Henderson, S.P. Meyn and V. Tadic, Performance evaluation and policy selection in multiclass networks, Discrete Event Dynamic Systems: Theory and Applications, Special Issue on Learning and Optimization Methods in Discrete Event Dynamic Systems (2002) to appear.

  29. F.C. Kelly and C.N. Laws, Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling, Queueing Systems 13 (1993) 47–86.

    Google Scholar 

  30. I. Kontoyiannis and S.P. Meyn, Spectral theory and limit theorems for geometrically ergodic Markov processes, Ann. Appl. Probab. 13 (2003) 304–362; presented at The INFORMS Applied Probability Conference, New York, July 2001.

    Google Scholar 

  31. S. Kumar and M. Muthuraman, A numerical method for solving singular Brownian control problems, in: Proc. of the 39th Conf. on Decision and Control, 2000.

  32. H.J. Kushner, Heavy Traffic Analysis of Controlled Queueing and Communication Networks, Stochastic Modelling and Applied Probability (Springer, New York, 2001).

  33. H.J. Kushner and K.M. Ramchandran, Optimal and approximately optimal control policies for queues in heavy traffic, SIAM J. Control Optim. 27 (1989) 1293–1318.

    Google Scholar 

  34. X. Luo and D. Bertsimas, A new algorithm for state-constrained separated continuous linear programs, SIAM J. Control Optim. 37 (1998) 177–210.

    Google Scholar 

  35. C. Maglaras, Dynamic scheduling in multiclass queueing networks: Stability under discrete-review policies, Queueing Systems 31 (1999) 171–206.

    Google Scholar 

  36. C. Maglaras, Discrete-review policies for scheduling stochastic networks: Trajectory tracking and fluid-scale asymptotic optimally, Ann. Appl. Probab. 10 (2000).

  37. L.F. Martins and H.J. Kushner, Routing and singular control for queueing networks in heavy traffic, SIAM J. Control Optim. 28 (1990) 1209–1233.

    Google Scholar 

  38. L.F. Martins, S.E. Shreve and H.M. Soner, Heavy traffice convergence of a controlled, multiclass queueing system, SIAM J. Control Optim. 34(6) (1996) 2133–2171.

    Google Scholar 

  39. S.P. Meyn, The policy iteration algorithm for average reward Markov decision processes with general state space, IEEE Trans. Automat. Control 42 (1997); also presented at The 35th IEEE Conf. on Decision and Control, Kobe, Japan, December 1996.

  40. S.P. Meyn, Stability and optimization of queueing networks and their fluid models, in:Mathematics of Stochastic Manufacturing Systems, Williamsburg, VA, 1996 (Amer. Math. Soc., Providence, RI, 1997) pp. 175–199.

  41. S.P. Meyn, Sequencing and routing in multiclass queueing networks. Part I: Feedback regulation, SIAM J. Control Optim. 40(3) (2001) 741–776.

    Google Scholar 

  42. S.P. Meyn, Stability, performance evaluation, and optimization, in: Markov Decision Processes: Models, Methods, Directions, and Open Problems, eds. E. Feinberg and A. Shwartz (Kluwer, Dordrecht, 2001) pp. 43–82.

  43. S.P. Meyn, Sequencing and routing in multiclass queueing networks. Part II: Workload relaxations, SIAM J. Control Optim. (2003) to appear; also presented at The 2000 IEEE Internat. Symposium on Information Theory, Sorrento, Italy, June 25–June 30 2003.

  44. S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability (Springer, London, 1993).

  45. J. Perkins, Control of push and pull manufacturing systems, Ph.D. thesis, University of Illinois, Urbana, IL (1993), Technical Report No. UILU-ENG-93-2237 (DC-155).

    Google Scholar 

  46. M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9 (1984) 441–458.

    Google Scholar 

  47. M.I. Reiman, A multiclass queue in heavy traffic, Adv. in Appl. Probab. 20 (1988) 179–207.

    Google Scholar 

  48. L.C.G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, Vol. I: Foundations, 2nd ed. (Cambridge Univ. Press, Cambridge, 2000).

    Google Scholar 

  49. A. Shwartz and A. Weiss, Large Deviations for Performance Analysis: Queues, Communication and Computing (Chapman and Hall, London, UK, 1995).

    Google Scholar 

  50. M.H. Veatch, Using fluid solutions in dynamic scheduling (2001) submitted for publication.

  51. G. Weiss, Optimal draining of a fluid re-entrant line, in: IMA Volumes in Mathematics and its Applications, Vol. 71, eds. F. Kelly and R. Williams (Springer, New York, 1995) pp. 91–103.

    Google Scholar 

  52. G. Weiss, A simplex based algorithm to solve separated continuous linear programs, Technical Report, Department of Statistics, University of Haifa, Israel (2001).

    Google Scholar 

  53. W. Whitt, Planning queueing simulations, Managm. Sci. 35 (1994) 1341–1366.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering and the Coordinated Sciences Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Mike Chen, Charuhas Pandit & Sean Meyn

Authors
  1. Mike Chen
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Charuhas Pandit
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Sean Meyn
    View author publications

    You can also search for this author inPubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, M., Pandit, C. & Meyn, S. In Search of Sensitivity in Network Optimization. Queueing Systems 44, 313–363 (2003). https://doi.org/10.1023/A:1025130105303

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1025130105303