## Journal of Applied Probability

### Asymptotic behavior of a generalized TCP congestion avoidance algorithm

#### Abstract

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.

#### Article information

Source
J. Appl. Probab. Volume 44, Number 3 (2007), 618-635.

Dates
First available in Project Euclid: 13 September 2007

https://projecteuclid.org/euclid.jap/1189717533

Digital Object Identifier
doi:10.1239/jap/1189717533

Mathematical Reviews number (MathSciNet)
MR2355580

Zentralblatt MATH identifier
1233.60016

#### Citation

Ott, Teunis J.; Swanson, Jason. Asymptotic behavior of a generalized TCP congestion avoidance algorithm. J. Appl. Probab. 44 (2007), no. 3, 618--635. doi:10.1239/jap/1189717533. https://projecteuclid.org/euclid.jap/1189717533.

#### References

• 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.
• 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.
• Altman, E., Avrachenkov, K. and Prabhu, B. (2005). Fairness in MIMD congestion control algorithms. Proc. IEEE INFOCOM 2005, pp. 1350--1361.
• Altman, E., Barakat, C. and Ramos, V. M. (2005). Analysis of AIMD protocols over paths with variable delay. Comput. Commun. 28, 1605--1617.
• 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.
• 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.
• 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.
• Altman, E. \et (2004). Analysis of scalable TCP. In Proc. IEEE HSNMC 2004, pp. 51--62.
• 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.
• 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.
• 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.
• 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.
• 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.
• Baras, J., Misra, A. and Ott, T. J. (2002). Predicting bottleneck bandwidth sharing by generalized TCP flows. Comput. Networks 40, 557--576.
• Bohacek, S. (2003). A stochastic model of TCP and fair video transmission. In Proc. IEEE INFOCOM 2003, pp. 1134--1144.
• 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.
• Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of additive-increase, multiplicative decrease (AIMD) algorithms. Adv. Appl. Prob. 34, 85--111.
• Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.
• Floyd, S. (1994). TCP and explicit congestion notification. ACM Comput. Commun. Rev. 21, 8--23.
• Guillemin, F., Robert, P. and Zwart, B. (2004). AIMD algorithms and exponential functionals. Ann. Appl. Prob. 14, 90--117.
• 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.
• Kelly, C. T. (2003). Scalable TCP: improving performance in high speed wide area networks. ACM SIGCOMM Comput. Commun. Rev. 32, 83--91.
• Kelly, C. T. (2004). Engineering flow controls in the internet. Doctoral Thesis, Cambridge University.
• Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Prob. 19, 1035--1070.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Misra, A. and Ott, T. J. (2003). Performance sensitivity and fairness of ECN-aware modified TCP'. J. Performance Evaluation, 53, 255--272.
• 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.
• Ott, T. J. (1999). ECN protocols and the TCP paradigm. Preprint. Available at www.teunisott.com.
• Ott, T. J. (2005). Transport protocols in the TCP paradigm and their performance. Telecommun. Systems 30, 351--385.
• Ott, T. J. (2006). On the Ornstein--Uhlenbeck process with delayed feedback. Preprint. Available at www.teunisott.com.
• Ott, T. J. (2006). Rate of convergence for the square root formula'. Adv. Appl. Prob. 38, 1132--1154.
• 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.
• Ott, T. J. and Swanson, J. (2006). Stationarity of some processes in transport protocols. SIGMETRICS Perf. Eval. Rev. 34, 30--32.
• 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.
• Ott, T. J., Lakshman, T. V. and Wong, L. H. (1999). SRED: stabilized RED. In Proc. IEEE INFOCOM 1999, pp. 1346--1355.
• 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.
• 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.
• Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.
• Ramakrishnan, K. K., Floyd, S. and Black, D. (2001). The addition of explicit congestion control (ECN) to IP. In Proc. IETF RFC 3168 2001.