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Asymptotic Behavior of a Generalized TCP Congestion Avoidance Algorithm

Part of: Stochastic processes Stochastic analysis Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Teunis J. Ott  and
Jason Swanson
Teunis J. Ott*
Affiliation:
Rutgers University
Jason Swanson*
Affiliation:
University of Wisconsin-Madison
*
∗Postal address: WINLAB, Rutgers University, New Brunswick, NJ 07930, USA. Email address: ott@winlab.rutgers.edu
∗∗Postal address: Mathematics Department, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706-1388, USA. Email address: swanson@math.wisc.edu
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Abstract

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The transmission control protocol (TCP) is a transport protocol used in the Internet. In Ott (2005), a more general class of candidate transport protocols called ‘protocols in the TCP paradigm’ was introduced. The long-term objective of studying this class is to find protocols with promising performance characteristics. In this paper we study Markov chain models derived from protocols in the TCP paradigm. Protocols in the TCP paradigm, as TCP, protect the network from congestion by decreasing the ‘congestion window’ (i.e. the amount of data allowed to be sent but not yet acknowledged) when there is packet loss or packet marking, and increasing it when there is no loss. When loss of different packets are assumed to be independent events and the probability p of loss is assumed to be constant, the protocol gives rise to a Markov chain {Wn}, where Wn is the size of the congestion window after the transmission of the nth packet. For a wide class of such Markov chains, we prove weak convergence results, after appropriate rescaling of time and space, as p → 0. The limiting processes are defined by stochastic differential equations. Depending on certain parameter values, the stochastic differential equation can define an Ornstein-Uhlenbeck process or can be driven by a Poisson process.

Keywords

Weak convergencestochastic differential equationstationary distributionTCP/IPcongestion avoidance

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60H10: Stochastic ordinary differential equations 60J05: Discrete-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 618 - 635
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Altman, E., Avrachenkov, K. and Barakat, C. (2000). A stochastic model of TCP/IP with stationary random losses. ACM SIGCOMM Comput. Commun. Rev. 30, 231–242.CrossRefGoogle Scholar
[2] Altman, E., Avrachenkov, K. and Barakat, C. (2002). TCP network calculus: the case of large delay-bandwidth product. In Proc. IEEE INFOCOM 2002, pp. 417–426.CrossRefGoogle Scholar
[3] Altman, E., Avrachenkov, K. and Prabhu, B. (2005). Fairness in MIMD congestion control algorithms. Proc. IEEE INFOCOM 2005, pp. 1350–1361.CrossRefGoogle Scholar
[4] Altman, E., Barakat, C. and Ramos, V. M. (2005). Analysis of AIMD protocols over paths with variable delay. Comput. Commun. 28, 1605–1617.Google Scholar
[5] Altman, E., Jimenez, T. and Kofman, D. (2004). DPS queues with stationary ergodic service times and the performance of TCP in overload. In Proc. IEEE INFOCOM 2004, pp. 975–983.CrossRefGoogle Scholar
[6] Altman, E., Avrachenkov, K., Barakat, C. and Nunez-Queija, R. (2001). TCP modeling in the presence of nonlinear window growth. In Proc. ITC-17 2001, pp. 883–894.CrossRefGoogle Scholar
[7] Altman, E., Avrachenkov, K., Kherani, A. and Prabhu, B. (2005). Performance analysis and stochastic stability of congestion control protocols. In Proc. IEEE INFOCOM 2005, pp. 1316–1327.CrossRefGoogle Scholar
[8] Altman, E. et al. (2004). Analysis of scalable TCP. In Proc. IEEE HSNMC 2004, pp. 51–62.CrossRefGoogle Scholar
[9] Baccelli, F., McDonald, D. and Reynier, J. (2002). A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation 49, 77–97.CrossRefGoogle Scholar
[10] Baccelli, F., Chaintreau, A., de Vleeschauwer, D. and McDonald, D. (2004). A mean-field analysis of short lived interacting TCP flows. In Proc. ACM SIGMETRICS 2004, ACM, New York, pp. 343–354.Google Scholar
[11] Baras, J., Misra, A. and Ott, T. J. (1999). The window distribution of multiple TCPs with random loss queues. In Proc. Globecomm'99, pp. 1714–1726.Google Scholar
[12] Baras, J., Misra, A. and Ott, T. J. (2000). Generalized TCP congestion avoidance and its effect on bandwidth sharing and variability. In Proc. Globecomm 2000, pp. 329–337.Google Scholar
[13] Baras, J., Misra, A. and Ott, T. J. (2000). Using drop-biasing to stabilize the occupancy of random drop queues with TCP traffic. In Proc. ICCS 2000.Google Scholar
[14] Baras, J., Misra, A. and Ott, T. J. (2002). Predicting bottleneck bandwidth sharing by generalized TCP flows. Comput. Networks 40, 557–576.Google Scholar
[15] Bohacek, S. (2003). A stochastic model of TCP and fair video transmission. In Proc. IEEE INFOCOM 2003, pp. 1134–1144.CrossRefGoogle Scholar
[16] Budhiraja, A., Hernandez-Campos, F., Kulkarni, V. G. and Smith, F. D. (2004). Stochastic differential equation for TCP window size: analysis and experimental validation. Prob. Eng. Inf. Sci. 18, 111–140.CrossRefGoogle Scholar
[17] Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of additive-increase, multiplicative decrease (AIMD) algorithms. Adv. Appl. Prob. 34, 85–111.CrossRefGoogle Scholar
[18] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
[19] Floyd, S. (1994). TCP and explicit congestion notification. ACM Comput. Commun. Rev. 21, 8–23.CrossRefGoogle Scholar
[20] Guillemin, F., Robert, P. and Zwart, B. (2004). AIMD algorithms and exponential functionals. Ann. Appl. Prob. 14, 90–117.CrossRefGoogle Scholar
[21] Hollot, C., Misra, V., Towsley, D. and Gong, W. B. (2001). A control theoretic analysis of RED. In Proc. IEEE INFOCOM 2001, pp. 1510–1519.CrossRefGoogle Scholar
[22] Kelly, C. T. (2003). Scalable TCP: improving performance in high speed wide area networks. ACM SIGCOMM Comput. Commun. Rev. 32, 83–91.CrossRefGoogle Scholar
[23] Kelly, C. T. (2004). Engineering flow controls in the internet. , Cambridge University.Google Scholar
[24] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Prob. 19, 1035–1070.CrossRefGoogle Scholar
[25] Lakshman, T. V. and Madhow, U. (1997). The performance of networks with high bandwidth-delay products and random loss. IEEE/ACM Trans. Networking 5, 336–350.CrossRefGoogle Scholar
[26] Marquez, R., Altman, E. and Sole-Alvarez, S. (2004). Modeling TCP and high speed TCP: a nonlinear extension to AIMD mechanisms. In Proc. IEEE HSNMC 2004, pp. 132–143.CrossRefGoogle Scholar
[27] Mathis, M., Semke, J., Mahdavi, J. and Ott, T. J. (1997). The macroscopic behavior of the TCP congestion avoidance algorithm. ACM SIGCOMM Comput. Commun. Rev. 27, 67–82.CrossRefGoogle Scholar
[28] Misra, A. and Ott, T. J. (1999). The window distribution of idealized TCP congestion avoidance with variable packet loss. In Proc. IEEE INFOCOM 1999, pp. 1564–1572.CrossRefGoogle Scholar
[29] Misra, A. and Ott, T. J. (2001). Effect of exponential averaging on the variability of a RED queue. In Proc. ICC 2001, pp. 1817–1823.CrossRefGoogle Scholar
[30] Misra, A. and Ott, T. J. (2001). Jointly coordinating ECN and TCP for rapid adaptation to varying bandwidth. In Proc. MILCOM 2001, pp. 719–725.CrossRefGoogle Scholar
[31] Misra, A. and Ott, T. J. (2003). Performance sensitivity and fairness of ECN-aware ‘modified TCP’. J. Performance Evaluation, 53, 255–272.CrossRefGoogle Scholar
[32] Misra, V., Gong, W. B. and Towsley, D. (1999). Stochastic differential equation modeling and analysis of TCP-windowsize behavior. In Proc. IFIP WG 7.3 Performance 1999.Google Scholar
[33] Ott, T. J. (1999). ECN protocols and the TCP paradigm. Preprint. Available at www.teunisott.com.Google Scholar
[34] Ott, T. J. (2005). Transport protocols in the TCP paradigm and their performance. Telecommun. Systems 30, 351–385.CrossRefGoogle Scholar
[35] Ott, T. J. (2006). On the Ornstein–Uhlenbeck process with delayed feedback. Preprint. Available at www.teunisott.com.Google Scholar
[36] Ott, T. J. (2006). Rate of convergence for the ‘square root formula’. Adv. Appl. Prob. 38, 1132–1154.Google Scholar
[37] Ott, T. J. and Kemperman, J. H. B. (2007). The transient behavior of processes in the TCP paradigm. Work in Progress. Available at www.teunisott.com.Google Scholar
[38] Ott, T. J. and Swanson, J. (2006). Stationarity of some processes in transport protocols. SIGMETRICS Perf. Eval. Rev. 34, 30–32.CrossRefGoogle Scholar
[39] Ott, T. J., Kemperman, J. H. B. and Mathis, M. (1996). The stationary behavior of idealized TCP congestion behavior. Preprint. Available at www.teunisott.com.Google Scholar
[40] Ott, T. J., Lakshman, T. V. and Wong, L. H. (1999). SRED: stabilized RED. In Proc. IEEE INFOCOM 1999, pp. 1346–1355.CrossRefGoogle Scholar
[41] Padhye, J., Firoiu, V., Towsley, D. and Kurose, J. (1998). Modeling TCP throughput: a simple model and its empirical validation. In Proc. ACM SIGCOMM 1998, pp. 303–314.CrossRefGoogle Scholar
[42] Padhye, J., Firoiu, V., Towsley, D. and Kurose, J. (2000). Modeling of TCP Reno performance: a simple model and its empirical validation. IEEE/ACM Trans. Networking 8, 133–145.CrossRefGoogle Scholar
[43] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
[44] Ramakrishnan, K. K., Floyd, S. and Black, D. (2001). The addition of explicit congestion control (ECN) to IP. In Proc. IETF RFC 3168 2001.Google Scholar