Stability of Data Networks: Stationary and Bursty Models

This paper studies stability of network models that capture macroscopic features of data communication networks, including the Internet. The network model consists of a set of links and a set of possible routes that are fixed subsets of links. A connection is dynamically established along one of the routes to transmit data as requested and is terminated after the transmission is over. The transmission bandwidth of a link is dynamically allocated, according to specific bandwidth allocation policy, to ongoing connections that traverse the link. A network model is said to be stable under a given bandwidth allocation policy if, roughly, the number of ongoing connections in the network will not blow up over time. We consider a stationary and a bursty network model; the former assumes stochastically stationary arrival processes of connections as did many theoretical studies, while the latter allows more realistic bursty and correlated arrival processes. For both models under a necessary stability condition (i.e., the average offered transmission load on each link is within its bandwidth capacity), we show that the proportionally fair, the minimum potential delay, the max-min fair, and a class of utility-maximizing bandwidth allocation policies ensure network model stability, while some priority-oriented and maximum throughput policies do not. Interestingly, the bandwidth allocation policy that maximizes the arctan(·) utility ensures the stability of the stationary model butnot the bursty model. This raises a serious concern about the current practice in the Internet protocol design, since such a policy is thought of as a good approximation of one of the most widely used TCP in the Internet.

[1]  Hong Chen,et al.  Discrete Flow Networks: Bottleneck Analysis and Fluid Approximations , 1991, Math. Oper. Res..

[2]  A. Stolyar.  On the Stability of Multiclass Queueing Networks: A Relaxed SuÆcient Condition via Limiting Fluid Processes , .

[3]  Rene L. Cruz,et al.  A calculus for network delay, Part I: Network elements in isolation , 1991, IEEE Trans. Inf. Theory.

[4]  Gustavo de Veciana,et al.  Stability and performance analysis of networks supporting elastic services , 2001, TNET.

[5]  Frank Kelly,et al.  Mathematical modeling of the Internet , 1999 .

[6]  V. Jacobson,et al.  Congestion avoidance and control , 1988, CCRV.

[7]  Hong Chen.  Fluid Approximations and Stability of Multiclass Queueing Networks: Work-Conserving Disciplines , 1995 .

[8]  Laurent Massoulié,et al.  Impact of fairness on Internet performance , 2001, SIGMETRICS '01.

[9]  Jean C. Walrand,et al.  Fair end-to-end window-based congestion control , 2000, TNET.

[10]  Sally Floyd,et al.  Promoting the use of end-to-end congestion control in the Internet , 1999, TNET.

[11]  Laurent Massoulié,et al.  Best-effort networks: modeling and performance analysis via large networks asymptotics , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[12]  Allan Borodin,et al.  Adversarial queuing theory , 2001, JACM.

[13]  Zheng Wang,et al.  An Architecture for Differentiated Services , 1998, RFC.

[14]  David D. Yao,et al.  Fundamentals of Queueing Networks , 2001 .

[15]  Steven H. Low,et al.  A duality model of TCP and queue management algorithms , 2003, TNET.

[16]  Sally Floyd,et al.  Wide area traffic: the failure of Poisson modeling , 1995, TNET.

[17]  W. Rudin.  Real and complex analysis , 1968 .

[18]  Hong Chen,et al.  Fluid approximations for a processor-sharing queue , 1997, Queueing Syst. Theory Appl..

[19]  R. Srikant,et al.  End-to-end congestion control schemes: utility functions, random losses and ECN marks , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[20]  S. Jamaloddin Golestani,et al.  A class of end-to-end congestion control algorithms for the Internet , 1998, Proceedings Sixth International Conference on Network Protocols (Cat. No.98TB100256).

[21]  David Gamarnik.  Using fluid models to prove stability of adversarial queueing networks , 2000, IEEE Trans. Autom. Control..

[22]  Frank Kelly,et al.  Charging and rate control for elastic traffic , 1997, Eur. Trans. Telecommun..

[23]  J. Dai.  On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models , 1995 .

[24]  Mark H. A. Davis.  Piecewise-deterministic Markov processes , 1993 .

[25]  Matthew Mathis,et al.  The macroscopic behavior of the TCP congestion avoidance algorithm , 1997, CCRV.

[26]  Scott Shenker,et al.  Fundamental Design Issues for the Future Internet (Invited Paper) , 1995, IEEE J. Sel. Areas Commun..

[27]  Laurent Massoulié,et al.  Bandwidth sharing and admission control for elastic traffic , 2000, Telecommun. Syst..

[28]  Steven H. Low,et al.  Optimization flow control—I: basic algorithm and convergence , 1999, TNET.

[29]  Rene L. Cruz,et al.  A calculus for network delay, Part II: Network analysis , 1991, IEEE Trans. Inf. Theory.

[30]  R. J. Williams,et al.  Fluid model for a network operating under a fair bandwidth-sharing policy , 2004, math/0407057.

[31]  Frank Kelly,et al.  Rate control for communication networks: shadow prices, proportional fairness and stability , 1998, J. Oper. Res. Soc..

[32]  David L. Black,et al.  An Architecture for Differentiated Service , 1998 .

[33]  D. Yao,et al.  Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization , 2001, IEEE Transactions on Automatic Control.

[34]  J. Boudec,et al.  The Fairness of Additive Increase and Multiplicative Decrease , 1998 .

[35]  Maury Bramson,et al.  Stability of two families of queueing networks and a discussion of fluid limits , 1998, Queueing Syst. Theory Appl..

[36]  Hong Chen,et al.  Lyapunov method for the stability of fluid networks , 2001, Oper. Res. Lett..

[37]  Heng-Qing Ye,et al.  Stability of data networks under an optimization-based bandwidth allocation , 2003, IEEE Trans. Autom. Control..

[38]  H. Ye,et al.  Stability of A Bandwidth Allocation Model for TCP Internet , 2001 .

[39]  Gideon Weiss,et al.  Stability and Instability of Fluid Models for Reentrant Lines , 1996, Math. Oper. Res..

[40]  Sean P. Meyn,et al.  Stability and convergence of moments for multiclass queueing networks via fluid limit models , 1995, IEEE Trans. Autom. Control..

[41]  Laurent Massoulié,et al.  Bandwidth sharing: objectives and algorithms , 2002, TNET.

[42]  Amber L. Puha,et al.  THE FLUID LIMIT OF A HEAVILY LOADED PROCESSOR SHARING QUEUE , 2002 .

[43]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[44]  Gary L. Miller,et al.  Proceedings of the twenty-eighth annual ACM symposium on Theory of computing , 1996, STOC 1996.

[45]  Dimitri P. Bertsekas,et al.  Data Networks , 1986 .

[46]  Donald F. Towsley,et al.  Modeling TCP Reno performance: a simple model and its empirical validation , 2000, TNET.