Abstract
Analytical models are important tools for the performance investigation of the Internet. The literature shows that the fixed point algorithm (FPA) is one of the most useful ways of solving analytical models of Internet performance.
Apart from what is observed in experimental literature, no comprehensive proof of the convergence and uniqueness of the FPA is given. In this paper we show how analytical models of reliable Internet protocols (TCP) converge to a unique fixed point. Unlike previous work in the literature the basic principles of our proof apply to both single and multiple bottleneck networks, to short and long-lived TCP connections and to both Drop Tail and Active Queue Management (AQM) routers. Our proof of convergence is based on a well known fixed point theorem and our uniqueness proof exploits the feedback nature of the reliable protocol.
The paper specifies conditions under which the FPA of analytical models of TCP converges to a unique point. The concepts used in the proof can also be extended to analyze the equilibrium, stability and global uniqueness issues of TCP, other reliable protocols and the Internet as a whole.
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Baccelli, F., McDonald, D.: A square root formula for the rate of non-persistent TCP flows. In: First Conference on Next Generation Internet Networks (NGI 2005), Rome, Italy, April 2005, pp. 171–176 (2005)
Border, K.C.: Fixed Point Theorems with Applications to Economics and Game Theory. Press Syndicate of the University of Cambridge, The Pitt Building, Trumpington Street, Cambridge (1985)
Bu, T., Towsley, D.: Fixed point approximations for TCP behavior in an AQM network. In: Proceedings of ACM SIGMETRICS, Cambridge, Massachusetts, USA (June 2001)
Garetto, M., Lo Cigno, R., Meo, M., Marsan, M.A.: Closed queueing network models of interacting long-lived TCP flows. IEEE/ACM Transactions on Networking 12(2), 300–311 (2004)
Gibbens, R., Sargood, S., Van Eijl, C., Kelly, F., Azmoodeh, H., Macfadyen, R., Macfadyen, N.: Fixed-point models for the end-to-end performance analysis of IP networks. In: Proceedings of 13th ITC Special Seminar: IP Traffic Management, Modeling and Management, Monterey, California (September 2000)
Herings, J.J., van der Laan, G., Talman, D., Yang, Z.: A fixed point theorem for discontinuous functions. Tinbergen Institute Discussion Papers 05-004/1, Department of Econometrics and Tinbergen Institute, Vrije Universiteit, De Boelelaan 1105, 1081, HV, Amsterdam, The Netherlands (December 2004)
Kaj, I., Olsén, J.: Stochastic equilibrium modeling of the TCP dynamics in various AQM environments. In: International Symposium on Performance Evaluation of Computer and Telecommunication Systems (SPECTS 2002), San Diego, USA (July 2002)
Kelly, C.T.: Engineering flow controls for the Internet, A dissertation submitted to the Cambridge University for the degree of Doctor of Philosophy. PhD thesis, Laboratory for Communication Engineering, Department of Engineering, Cambridge University (2004)
Meo, M., Garetto, M., Marsan, M.A., Lo Cigno, R.: On the use of fixed point approximations to study reliable protocols over congested links. In: Globecom, San Francisco, USA (December 2003)
Olsén, J.: Stochastic Modeling and Simulation of the TCP Protocol. PhD thesis, Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden (2003)
Padhye, J., Firoiu, V., Towsley, D., Kurose, J.: Modeling TCP Reno performance: a simple model and its empirical validation. IEEE/ACM Transactions on Networking 8(2), 133–145 (2000)
Tang, A., Wang, J., Low, S.H., Chiang, M.: Network equilibrium of heterogeneous congestion control protocols. In: IEEE INFOCOM, Miami, FL USA (March 2005)
Wang, J., Li, L., Low, S.H., Doyle, J.C.: Cross-layer optimization in TCP/IP networks. IEEE/ACM Transactions on Networking 13(3), 568–582 (2005)
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Fesehaye Kassa, D., Wittevrongel, S. (2006). Convergence of the Fixed Point Algorithm of Analytical Models of Reliable Internet Protocols (TCP). In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science – ICCS 2006. ICCS 2006. Lecture Notes in Computer Science, vol 3994. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758549_9
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DOI: https://doi.org/10.1007/11758549_9
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