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Transient Moments of the TCP Window Size Process

Part of: Markov processes Special processes Computer system organization Stochastic processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Andreas H. Löpker  and
Johan S. H. van Leeuwaarden
Andreas H. Löpker*
Affiliation:
EURANDOM
Johan S. H. van Leeuwaarden*
Affiliation:
EURANDOM and Eindhoven University of Technology
*
∗Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands.
∗Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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Abstract

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The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.

Keywords

Piecewise-deterministic Markov processDynkin's martingaleoptional stoppingTCPwindow sizeAIMDstationary distributiontransient momentsexponential functionaldifference equationmixing time

MSC classification

Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents 60K10: Applications (reliability, demand theory, etc.) 90B20: Traffic problems 90B15: Network models, stochastic 90B18: Communication networks 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60G44: Martingales with continuous parameter 68M20: Performance evaluation; queueing; scheduling
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 163 - 175
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Altman, E., Avrachenkov, K. E., Barakat, C. and Núñez Queija, R. (2002). State-dependent M/G/1 type queueing analysis for congestion control in data networks. Comput. Networks 39, 789–808.CrossRefGoogle Scholar
[2] Altman, E., Avrachenkov, K. E., Kherani, A. A. and Prabhu, B. J. (2005). Performance analysis and stochastic stability of congestion control protocols. In Proc. IEEE INFOCOM.Google Scholar
[3] Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.Google Scholar
[4] Baccelli, F. and McDonald, D. R. (2006). Mellin transforms for TCP throughput with applications to cross layer optimization. In Proc. 40th Annual Conf. Inf. Sci. Systems, IEEE, pp. 32–37.Google Scholar
[5] Baccelli, F., Kim, K. B. and McDonald, D. R. (2007). Equilibria of a class of transport equations arising in congestion control. Queueing Systems 55, 1–8.Google Scholar
[6] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Prob. Surveys 2, 191–212.Google Scholar
[7] Boxma, O., Perry, D., Stadje, W. and Zacks, S. (2006). A Markovian growth-collapse model. Adv. Appl. Prob. 38, 221–243.Google Scholar
[8] Carmona, P., Petit, F. and Yor, M. (2001). Exponential functionals of Lévy processes. In Lévy Processes, eds Barndorff-Nielsen, O. et al., Birkhäuser, Boston, MA, pp. 41–55.Google Scholar
[9] Davis, M. (1993). Markov Models and Optimization. Chapman & Hall, London.Google Scholar
[10] Eliazar, I. and Klafter, K. (2004). A growth-collapse model: Lévy inflow, geometric crashes, and generalized Ornstein–Uhlenbeck dynamics. Physica A 334, 1–21.Google Scholar
[11] Gjessing, H. and Paulsen, J. (1997). Present value distributions with application to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144.CrossRefGoogle Scholar
[12] Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Prob. 34, 468–492.Google Scholar
[13] Guillemin, F., Robert, P. and Zwart, B. (2004). AIMD algorithms and exponential functionals. Ann. Appl. Prob. 14, 90–117.Google Scholar
[14] Maulik, K. and Zwart, B. (2006). Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177.CrossRefGoogle Scholar
[15] Milne-Thomson, L. M. (1933). The Calculus of Finite Differences. Macmillan, London.Google Scholar
[16] Ott, T. J. (2005). Transport protocols in the TCP paradigm and their performance. Telecommun. Systems 30, 351–385.Google Scholar
[17] Ott, T. J. and Kemperman, J. H. B. (2007). The transient behavior of processes in the TCP paradigm. To appear in Prob. Eng. Inf. Sci. Google Scholar
[18] Ott, T. J., Kemperman, J. H. B. and Mathis, M. (1996). The stationary behavior of ideal TCP congestion avoidance. Unpublished manuscript. Available at www.teunisott.com.Google Scholar
[19] Roberts, A. W. and Varberg, D. E. (1973). Convex Functions. Academic Press, New York.Google Scholar
[20] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, New York.Google Scholar
[21] Shanthikumar, J. and Sumita, U. (1983). General shock models associated with correlated renewal sequences. J. Appl. Prob. 20, 600–614.Google Scholar
[22] Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar