Skip to main content
×
×
Home

Optimal control of queueing networks: an approach via fluid models

  • Nicole Bäuerle (a1)
Abstract

We consider a general control problem for networks with linear dynamics which includes the special cases of scheduling in multiclass queueing networks and routeing problems. The fluid approximation of the network is used to derive new results about the optimal control for the stochastic network. The main emphasis lies on the average-cost criterion; however, the β-discounted as well as the finite-cost problems are also investigated. One of our main results states that the fluid problem provides a lower bound to the stochastic network problem. For scheduling problems in multiclass queueing networks we show the existence of an average-cost optimal decision rule, if the usual traffic conditions are satisfied. Moreover, we give under the same conditions a simple stabilizing scheduling policy. Another important issue that we address is the construction of simple asymptotically optimal decision rules. Asymptotic optimality is here seen with respect to fluid scaling. We show that every minimizer of the optimality equation is asymptotically optimal and, what is more important for practical purposes, we outline a general way to identify fluid optimal feedback rules as asymptotically optimal. Last, but not least, for routeing problems an asymptotically optimal decision rule is given explicitly, namely a so-called least-loaded-routeing rule.

Copyright
Corresponding author
Postal address: Department of Mathematics VII, University of Ulm, D-89069 Ulm, Germany. Email address: baeuerle@mathematik.uni-ulm.de
Footnotes
Hide All

Work supported in part by a grant from the University of Ulm.

Footnotes
References
Hide All
Alanyali, M. and Hajek, B. (1998). Analysis of simple algorithms for dynamic load balancing. Math. Operat. Res. 22, 840871.
Altman, E., Koole, G. and Jiménez, T. (2001). On the comparison of queueing systems with their fluid limits. Preprint. Prob. Eng. Inf. Sci. 15, 165178.
Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.
Atkins, D. and Chen, H. (1995). Performance evaluation of scheduling control of queueing networks: fluid model heuristics. Queueing Systems 21, 391413.
Avram, F., Bertsimas, D. and Ricard, M. (1995). Fluid models of sequencing problems in open queueing networks: an optimal control approach. In Stochastic Networks (IMA Vol. Math. Appl. 71), eds Kelly, F. P. and Williams, R. J., Springer, New York, pp. 199234.
Bäuerle, N., (1999). How to improve the performance of ATM multiplexers. Operat. Res. Lett. 24, 8189.
Bäuerle, N., (2000). Asymptotic optimality of tracking-policies in stochastic networks. Ann. Appl. Prob. 10, 10651083.
Bäuerle, N., (2001). Discounted stochastic fluid programs. To appear in Math. Operat. Res. 26, 401420.
Bäuerle, N. and Rieder, U. (2000). Optimal control of single-server fluid networks. Queueing Systems 35, 185200.
Chen, H. (1995). Fluid approximations and stability of multiclass queueing networks: work-conserving disciplines. Ann. Appl. Prob. 5, 637665.
Chen, H. and Xu, S. (1993). On the asymptote of the optimal routing policy for two service stations. IEEE Trans. Automatic Control 38, 187189.
Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Prob. 5, 4977.
Dai, J. G. (1999). Stability of Fluid and Stochastic Processing Networks (MaPhySto Miscellanea 9). Centre for Mathematical Physics and Stochastics, Aarhus.
Dai, J. G. and Williams, R. J. (1995). Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons. Theory Prob. Appl. 40, 140.
Gajrat, A. and Hordijk, A. (2000a). Fluid approximation of a controlled multiclass tandem network. Preprint. Queueing Systems 35, 349380.
Gajrat, A. and Hordijk, A. (2000b). Optimal service control of multiclass fluid tandem networks. Preprint, Leiden University.
Gajrat, A. and Hordijk, A. (2000c). Synthesis of the optimal control problem for fluid networks. Preprint, Leiden University.
Hajek, B. (1984). Optimal control of two interacting service stations. IEEE Trans. Automatic Control 29, 491499.
Harrison, J. M. (1996). The BIGSTEP approach to flow management in stochastic processing networks. In Stochastic Networks: Stochastic Control Theory and Applications (R. Statist. Soc. Lecture Notes 4), eds Kelly, F., Zachary, S. and Ziedins, I., Clarendon Press, Oxford, pp. 5790.
Hordijk, A. and Koole, G. (1992). On the assignment of customers to parallel queues. Prob. Eng. Inf. Sci. 6, 495511.
Kitaev, M. and Rykov, V. (1995). Controlled Queueing Systems. CRC Press, Boca Raton, FL.
Luo, X. and Bertsimas, D. (1998). A new algorithm for state-constrained separated continuous linear programs. SIAM J. Control Optimization 37, 177210.
Maglaras, C. (1999). Dynamic scheduling in multiclass queueing networks: stability under discrete-review policies. Queueing Systems 31, 171206.
Maglaras, C. (2000). Discrete-review policies for scheduling stochastic networks: trajectory tracking and fluid-scale asymptotic optimality. Preprint. Ann. Appl. Prob. 10, 897929.
Meyn, S. P. (1997a). The policy iteration algorithm for average reward Markov decision processes with general state space. IEEE Trans. Automatic Control 42, 16631679.
Meyn, S. P. (1997b). Stability and optimization of queueing networks and their fluid models. In Mathematics of Stochastic Manufacturing Systems (Lectures Appl. Math. 33), eds Yin, G. G. and Zhang, Q., American Mathematical Society, Providence, RI, pp. 175199.
Meyn, S. P. (2001). Sequencing and routing in multiclass queueing networks I: Feedback regulation. Preprint. SIAM J. Control Optimization 40, 741776.
Ott, T. J. and Shanthikumar, J. G. (1996). Discrete storage processes and their Poisson flow and fluid flow approximations. Queueing Systems 24, 101136.
Pullan, M. C. (1995). Forms of optimal solutions for separated continuous linear programs. SIAM J. Control Optimization 33, 19521977.
Seierstad, A. and Sydsæter, K. (1987). Optimal Control Theory with Economic Applications. North-Holland, Amsterdam.
Sennott, L. I. (1989). Average cost optimal stationary policies in infinite state Markov decision processes with unbounded costs. Operat. Res. 37, 626633.
Sennott, L. I. (1998). Stochastic Dynamic Programming and the Control of Queues. John Wiley, New York.
Sethi, S. P. and Zhang, Q. (1994). Hierarchical Decision Making in Stochastic Manufacturing Systems. Birkhäuser, Boston, MA.
Stidham, S. and Weber, R. (1993). A survey of Markov decision models for control of networks of queues. Queueing Systems 13, 291314.
Veatch, M. (2001). Fluid analysis of arrival routing. Preprint. IEEE Trans. Automat. Control 46, 12541257.
Weiss, G. (1996). Optimal draining of fluid re-entrant lines: some solved examples. In Stochastic Networks: Stochastic Control Theory and Applications (R. Statist. Soc. Lecture Notes 4), eds Kelly, F., Zachary, S. and Ziedins, I., Clarendon Press, Oxford, pp. 1934.
Williams, R. J. (1998). Some recent developments for queueing networks. In Probability Towards 2000 (Lecture Notes Statist. 128), eds Accardi, L. and Heyde, C. C., Springer, New York, pp. 340356.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed