Journal of Applied Probability

Asymptotic fluid optimality and efficiency of the tracking policy for bandwidth-sharing networks

Konstantin Avrachenkov, Alexey Piunovskiy, and Yi Zhang

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Optimal control of stochastic bandwidth-sharing networks is typically difficult. In order to facilitate the analysis, deterministic analogues of stochastic bandwidth-sharing networks, the so-called fluid models, are often taken for analysis, as their optimal control can be found more easily. The tracking policy translates the fluid optimal control policy back to a control policy for the stochastic model, so that the fluid optimality can be achieved asymptotically when the stochastic model is scaled properly. In this work we study the efficiency of the tracking policy, that is, how fast the fluid optimality can be achieved in the stochastic model with respect to the scaling parameter. In particular, our result shows that, under certain conditions, the tracking policy can be as efficient as feedback policies.

Article information

J. Appl. Probab., Volume 48, Number 1 (2011), 90-113.

First available in Project Euclid: 15 March 2011

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Bandwidth-sharing network fluid model optimal control tracking policy rate of convergence


Avrachenkov, Konstantin; Piunovskiy, Alexey; Zhang, Yi. Asymptotic fluid optimality and efficiency of the tracking policy for bandwidth-sharing networks. J. Appl. Probab. 48 (2011), no. 1, 90--113. doi:10.1239/jap/1300198138.

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  • Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.
  • Bäuerle, N. (2000). Asymptotic optimality of tracking policies in stochastic networks. Ann. Appl. Prob. 10, 1065–1083.
  • Cantrell, P. (1986). Computation of the transient M/M/1 queue CDF, PDF, and mean with generalized Q-functions. IEEE Trans. Commun. 34, 814–817.
  • Chen, H. (1996). Rate of convergence of the fluid approximation for generalized Jackson networks. J. Appl. Prob. 33, 804–814.
  • Chen, H. and Mandelbaum, A. (1991). Discrete flow networks: bottleneck analysis and fluid approximations. Math. Operat. Res. 16, 408–446.
  • Chen, H. and Mandelbaum, A. (1994). Hierachical modeling of stochastic networks. Part I. Fluid models. In Stochastic Modeling and Analysis of Manufacturing Systems, ed. D. Yao, Springer, New York, pp. 47–105.
  • Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Prob. 5, 49–77.
  • Gajrat, A. and Hordijk, A. (2000). Fluid approximation of a controlled multiclass tandem network. Queueing Systems 35, 349–380.
  • Gajrat, A., Hordijk, A. and Ridder, A. (2003). Large-deviations analysis of the fluid approximation for a controllable tandem queue. Ann. Appl. Prob. 13, 1423–1448.
  • Gleissner, W. (1988). The spread of epidemics. Appl. Math. Comput. 27, 167–171.
  • Guo, X. and Hernández-Lerma, O. (2009). Continuous-Time Markov Decision Processes. Springer, Berlin.
  • Hernández-Lerma, O. (1994). Lectures on Continuous-Time Markov Control Processes. Sociedad Matemática Mexicana, Mexico City.
  • Maglaras, C. (2000). Discrete-review policies for scheduling stochastic networks: trajectory tracking and fluid-scale asymptotic optimality. Ann. Appl. Prob. 10, 897–929.
  • Mandelbaum, A. and Pats, G. (1995). State-dependent queues: approximations and applications. In Stochastic Networks (IMA Vol. Math. Appl. 71), eds F. Kelly and R. Williams, Springer, New York, pp. 239–282.
  • Massoulié, L. and Roberts, J. W. (2000). Bandwidth sharing and admission control for elastic traffic. Telecommun. Systems 15, 185–201.
  • Pang, G. and Day, M. V. (2007). Fluid limits of optimally controlled queueing networks. J. Appl. Math. Stoch. Anal. 2007, 19 pp.
  • Piunovskiy, A. (2009). Controlled jump Markov processes with local transitions and their fluid approximation. WSEAS Trans. Systems Control 4, 399–412.
  • Piunovskiy, A. B. (2009). Random walk, birth-and-death process and their fluid approximations: absorbing case. Math. Meth. Operat. Res. 70, 285–312.
  • Piunovskiy, A. B. and Clancy, D. (2008). An explicit optimal intervention policy for a determinisitc epidemic model. Optimal Control Appl. Meth. 29, 413–428.
  • Piunovskiy, A. and Zhang, Y. (2011). Accuracy of fluid approximations to controlled birth-and-death processes: absorbing case. Math. Meth. Operat. Res., 29 pp.
  • Piunovskiy, A. and Zhang, Y. (2011). On the fluid approximations of a class of general inventory level-dependent EOQ and EPQ models. To appear in Adv. Operat. Res.
  • Pullan, M. C. (1995). Forms of optimal solutions for separated continuous linear programs. SIAM J. Control. Optimization 33, 1952–1977.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.
  • Robert, P. (2003). Stochastic Networks and Queues (Appl. Math. 52). Springer, Berlin.
  • Roberts, J. and Massoulie, L. (1998). Bandwidth sharing and admission control for elastic traffic. In Proc. of ITC Specialist Seminar, Yokohama, Japan, pp. 185–201.
  • Sharma, O. P. and Tarabia, A. M. K. (2000). A simple transient analysis of an M/M/1/N queue. Sankhy$\overlinea$ A 62, 273–281.
  • Verloop, I. M. (2009). Scheduling in Stochastic Resource-Sharing Systems. Doctoral Thesis, Eindhoven University of Technology.
  • Verloop, I. M. and Núñez-Queija, R. (2009). Assessing the efficiency of resource allocations in bandwidth-sharing networks. Performance Evaluation 66, 59–77.
  • Ye, L., Guo, X. and Hernández-Lerma, O. (2008). Existence and regularity of a nonhomogeneous transition matrix under measurability conditions. J. Theoret. Prob. 21, 604–627.
  • Zhang, J. and Coyle, E. J., Jr. (1991). The transient solution of time-dependent M/M/1 queues. IEEE Trans. Inf. Theory 37, 1690–1696.