Abstract
A Markovian model for intermittent connections of various classes in a communication network is established and investigated. Any connection alternates between being OFF (idle) or ON (active, with data to transmit), and evolves in a way depending only on its class and the state of the network, in particular for the routes it uses among the network nodes to transmit data. The congestion of a given node is a functional of the throughputs of all ON connections going through it, and causes losses to these connections. Any ON connection reacts to its losses by self-adapting its throughput in TCP-like fashion so as to control network congestion. The connections interact through this feedback loop. The system constituted of their states (either OFF, or ON with some throughput) evolves in Markovian fashion, and since the number of connections in each class is potentially huge, a mean-field limit result with an adequate scaling is proved so as to reduce dimensionality. The limit is a nonlinear Markov process given by a McKean–Vlasov equation, of dimension the number of classes. It is then proved that the stationary distributions of the limit can be expressed in terms of the solutions of a finite-dimensional fixed-point equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aldous, D.: Exchangeability and related topics. In: École d’été de Probabilités de Saint-Flour XIII, Lecture Notes in Mathematics, vol. 1117, pp. 1–198. Springer, Berlin (1985)
Asmussen, S.: Applied probability and queues. Applications of Mathematics (New York), vol. 51. 2nd edn. Springer, New York (2003). Stochastic Modelling and Applied Probability
Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics. 2nd edn. Wiley-Interscience, New York (1999)
Chafaï, D., Malrieu, F., Paroux, K.: On the long time behavior of the TCP window size process. Stoch. Process. Appl. 120(8), 1518–1534 (2010)
Davis, M.H.A.: Markov Models and Optimization. Chapman & Hall, London (1993)
Dieudonné, J.: Foundations of modern analysis. Pure and Applied Mathematics, vol. X. Academic Press, New York (1960)
Dumas, V., Guillemin, F., Robert, P.: A Markovian analysis of additive-increase multiplicative-decrease (AIMD) algorithms. Adv. Appl. Probab. 34(1), 85–111 (2002)
Graham, C.: Chaoticity for multi-class systems and exchangeability within classes. J. Appl. Probab. 45, 1196–1203 (2008)
Graham, C., Robert, P.: Interacting multi-class transmissions in large stochastic networks. Ann. Appl. Probab. 19(6), 2334–2361 (2009)
Graham, C., Robert, P., Verloop, M.: Stability properties of networks with interacting tcp flows. In: Network Control and Optimization, Proceedings NET-COOP 2009, pp. 1–15 (2009)
Guillemin, F., Robert, P., Zwart, B.: AIMD algorithms and exponential functionals. Ann. Appl. Probab. 14(1), 90–117 (2004)
Nummelin, E.: General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press, Cambridge (1984)
Robert, P.: Stochastic Networks and Queues. Stochastic Modelling and Applied Probability Series, vol. 52. Springer, New York (2003)
Sznitman, A.: Topics in propagation of chaos. In: École d’été de Saint-Flour. Lecture Notes in Maths, vol. 1464, pp. 167–243. Springer, Berlin (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Graham, C., Robert, P. Self-adaptive congestion control for multiclass intermittent connections in a communication network. Queueing Syst 69, 237–257 (2011). https://doi.org/10.1007/s11134-011-9260-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-011-9260-z
Keywords
- Flow control algorithm
- Multiclass system
- Mean-field limit
- Chaoticity
- McKean–Vlasov equation
- Nonlinear Markov process
- Stationary distribution