Elsevier

Performance Evaluation

Volume 67, Issue 11, November 2010, Pages 1030-1043
Performance Evaluation

Modeling TCP throughput: An elaborated large-deviations-based model and its empirical validation

https://doi.org/10.1016/j.peva.2010.08.016Get rights and content

Abstract

In today’s Internet, a large part of the traffic is carried using the TCP transport protocol. Characterization of the variations of TCP traffic is thus an important issue, both for resource provisioning and Quality of Service purposes. However, most existing models are limited to the prediction of the (almost-sure) mean TCP throughput and are unable to characterize deviations from this value.

In this paper, we propose a method to describe the deviations of a long TCP flow’s throughput from its almost-sure mean value. This method relies on an ergodic large-deviations result, which was recently proved to hold on almost every single realization for a large class of stochastic processes. Applying this result to a Markov chain modeling the congestion window’s evolution of a long-lived TCP flow, we show that it is practically possible to quantify and to statistically bound the throughput’s variations at different scales of interest for applications. Our Markov-chain model can take into account various network conditions and we demonstrate the accuracy of our method’s prediction in different situations using simulations, experiments and real-world Internet traffic. In particular, in the classical case of Bernoulli losses, we demonstrate: (i) the consistency of our method with the widely-used square-root formula predicting the almost-sure mean throughput, and (ii) its ability to additionally predict finer properties reflecting the traffic’s variability at different scales.

Introduction

A deep understanding of the properties of network traffic is important for Internet Service Providers to optimally control the traffic, to dimension the hardware and software, and eventually to offer users the best possible Quality of Service. In this context, a lot of research has been recently focusing on the mathematical modeling of network traffic, especially from a statistical viewpoint. However, comprehensive modeling of the traffic is a very arduous problem because it encompasses several difficulties of different natures, such as the transport protocols and their associated control mechanisms, the flows dependencies and the complexity due to the network’s topology. The design of models, simple yet rich enough to reproduce essential characteristics observed in the traffic, is then an important challenge for resource optimization, traffic control or prediction, with a real impact on industrial applications.

In particular, great attention has been devoted to the statistical modeling of the TCP protocol, the most widely used transport protocol in today’s Internet. Following Padhye’s paper [1] and its well-known square-root formula, many models have appeared in the last decade to predict the mean throughput achieved by a long-lived TCP connection in various network conditions (see next section). Such models are of great interest and have been extensively used because long-lived connections accurately model long TCP flows, whose ratio is constantly increasing in Internet traffic.

While it provides very useful information, prediction of the sole mean throughput may be insufficient for some purposes. For example, information about the throughput’s fluctuations can be needed to evaluate the risks of congestion. An important step towards a better comprehension of the traffic, to optimize resource utilization at several levels, is then to characterize the variations of TCP traffic around its mean.

In this paper, we consider the flow’s throughput averaged at scale n, that is throughput’s sample mean calculated over time windows of size n×RTTs, with n significantly larger than one but significantly smaller than the total flow length. For different values of the scale n, we demonstrate that it is possible to analytically derive the statistical distribution of the corresponding throughput, and hence to compute the probability that it deviates from its global mean value. We also prove that in practice, the throughput deviations that are observable on a finite-size realization of a TCP flow are statistically bounded.

These results complete the scope of Padhye’s result in many different situations. For instance, it can be the case when the scale of interest is imposed by a buffer–recall that the condition n1 precludes network-level buffers–and we look for the maximum traffic variations at this scale. Then, our method allows determining the probability of congestion or starvation, depending on the problem at stake. Conversely, if we assume that the maximum acceptable deviation of the averaged throughput is now fixed, we may want to know at which minimum scale the traffic should be averaged to remain within this tolerable range. As an example, consider a generic server sequentially dealing with several clients. Given the elapsed time between two consecutive services of a same client, our method allows dimensioning the minimum size of the client buffer to ensure zero data loss. More generally, characterizing traffic variations beyond the mean throughput presents clear advantages in congestion and starvation problems. Yet, we do not elaborate any further on specific practical uses, and we focus on the technical aspects of the proposed method.

Our approach relies on a large-deviations principle that was recently proved to hold for any particular realization of a wide class of stochastic processes [2]. Here, we adapt this general result to the specific case of one long-lived TCP flow, and show that it allows predicting the deviations of the flow throughput around its mean value.

More precisely, it is the so-called large-deviations spectrum, a scale-invariant function, that conveys information about the distribution of the throughput measured over a given time-scale horizon. The large-deviations spectrum is reminiscent of a peculiar scaling law exclusively generated by the Markov property of the AIMD mechanism. As a result, it is important to stress that the present scale invariance must not be confused with long-range dependence, another scaling law whose impact on performance has already been extensively studied. Moreover, while long-range dependence was shown to be independent of the used protocol [3], this new scaling directly originates in TCP’s control mechanisms, and the related large-deviations spectrum really is a straightforward fingerprint of the performance achieved by a TCP connection.

The present contribution is the methodological exploitation of a mathematical result we derived in [2], applied to the specific context of a TCP connection. We show that, in addition to the almost-sure mean value predicted by the existing models (such as the square-root formula), it is also possible to fully (statistically) characterize the deviation of the averaged throughput from its most frequent value.

More precisely, we show how to empirically estimate, on a TCP Reno connection, the so-called large-deviations spectrum of the throughput at different scales. In addition, we prove that a theoretical spectrum can analytically and consistently be derived from a Markov-chain model fitting the TCP time series. Practical validity of the proposed method is tested: (i) on simulations in the case of Bernoulli losses where an extensive comparison of our results with the results of Padhye’s square-root formula is performed; and (ii) on real TCP traffic from controlled experiments. Then, we experimentally demonstrate that our model remains valid under less restrictive assumptions. Notably, we treat the case of softened Markov conditions, we confront our model to real uncontrolled Internet traffic, and we address examples of other TCP variants.

The rest of the paper is organized as follows. In Section 2, we briefly review the works most closely related to our approach. We expose useful theoretical results in Section 3, in particular those of [2] on which the proposed method to describe the TCP throughput’s variability relies. In Section 4, we present illustrations of the method in controlled situations, in particular in the Bernoulli case, using simulations and experiments. In Section 5, we present experimental validations of the method in more complex situations, and in particular on real-word Internet traffic. We also show an extension of the method to TCP traffic using a variant other than Reno.

Section snippets

Related work

Markov chains and similar tools have been extensively used in the last decade to model TCP traffic. All these models basically obtain steady-state information on the throughput, most often the first-order statistics (the mean), under various loss assumptions and including various complex TCP mechanisms (e.g., timeout). The following is a non-exhaustive list of models that explicitly rely on the AIMD mechanism and are then conceptually the most related to our approach (bear in mind that we focus

Large deviations for Markov chains: Theoretical results

In this section, we expose the fundamental theoretical tools and results supporting the method proposed to describe the TCP throughput’s variability. Given the applied nature of the present contribution, simplicity has voluntarily been preferred to rigorous statement of the results developed in a companion mathematical article [2].

Throughout the paper, the congestion-window size in packets and the throughput in packets per RTT are considered equal, and both terms are used interchangeably.

One TCP Reno connection in controlled situations

The numerical and experimental results of this section aim at illustrating the potential and the versatility of our method in a large set of situations.

Further validation and extension

In this section, we demonstrate the accuracy of TCP Reno throughput prediction provided by our method in more complex environments, including long-range dependent cross-traffic and real Internet traffic. We also propose an extension to different TCP variants.

Conclusion

In this work, we proposed a method to predict TCP throughput’s variations. This method relies on a recent ergodic large-deviations result that we theoretically derived in [2], and that we applied here to a simple Markov-chain model of the evolution of TCP Reno’s congestion window. The striking accuracy of the model to characterize real traffic in terms of these elaborated large-deviations properties reinforces the adequacy between Markov models and TCP data that had been observed in prior works

Acknowledgements

Part of the work presented in this paper was done while the first author was with Université de Lyon/École Normale Supérieure de Lyon, Lyon, France. The authors are very grateful to the ALADDIN-G5K initiative whose developers and engineers provided us with full support to perform our experiments on the Grid’5000 infrastructure. They also would like to warmly thank Pr. Richard Baraniuk of the DSP department at RICE university (TX, US) for allowing us to use their computer resource to perform

Patrick Loiseau received a degree of Professeur-Agrégé de Sciences-Physiques (2005), a M.S. degree of physics (2006), and a Ph.D. degree of computer science (2009) from École Normale Supérieure de Lyon. He also received a M.S. degree of mathematics (2010) from Université Pierre et Marie Curie (Paris 6) and École Polytechnique. He is currently working as a post-doctoral fellow at INRIA Paris-Rocquencourt.

His main research interests are in probability and in statistical analysis and modeling of

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    Patrick Loiseau received a degree of Professeur-Agrégé de Sciences-Physiques (2005), a M.S. degree of physics (2006), and a Ph.D. degree of computer science (2009) from École Normale Supérieure de Lyon. He also received a M.S. degree of mathematics (2010) from Université Pierre et Marie Curie (Paris 6) and École Polytechnique. He is currently working as a post-doctoral fellow at INRIA Paris-Rocquencourt.

    His main research interests are in probability and in statistical analysis and modeling of network traffic and performance. This includes wavelet-based analysis of scaling phenomena, long-range dependent and multifractal models, large deviations for Markov processes, and queuing theory with correlated input processes. He is also interested in statistical estimation methods and their application to network measurement problems; and in the analysis and modeling of the heart-rate variability using control theory.

    Paulo Gonçalves graduated from the Signal Processing Department of CPE Lyon, France in 1993. He received the M.S. and Ph.D. degrees in signal processing from INPG, France, in 1990 and 1993 respectively. While working toward his Ph.D. degree, he was with ÉNS Lyon. In 1994–96, he was a Postdoctoral Fellow at Rice Univ., US. Since 1996, he is an associate researcher at INRIA, first with Fractales (1996–99), then with is2 (2000–2003) and now with team RESO at the Parallel Computing Lab. (LIP), ÉNS Lyon. From 2003 to 2005, he was on leave at IST Lisbon, Portugal.

    His research interests are in multiscale analysis and in wavelet-based statistical inference. His principal application is in metrology and deals with grid-traffic statistical characterization and modeling for protocol quality assessment and control.

    Julien Barral received the Professeur-agrégé degree in Mathematics in 1995, and the M.S., Ph.D. and Habilitation à Diriger des recherches degrees from Paris-Sud University respectively in 1995, 1997 and 2005.

    From 1999 to 2001, he was a Maître de conférences with Montpellier-II University and did his reserach in the Convex Analysis team. Then he was Associate Researcher with INRIA, from 2001 to 2005 in Fractales team, and from 2005 to 2009 in Sisyphe team. He is currently a Professeur in Mathematics with Paris-Nord University, where he is a member of Ergodic Theory and Dynamical Systems team.

    Pascale Vicat-Blanc Primet is senior researcher at the National Institute of Research in Computer Science (INRIA) since 2005. Since 2002, she has been leading the INRIA RESO team (22 researchers and engineers) within the LIP laboratory of École Normale Superieure de Lyon. Since the beginning of 2008, she is also leading the “Semantic Networking” research team of the INRIA-Bell Labs common laboratory.

    Her research interests include High-Speed and High-Performance Networks, Internet protocols’ design and architecture, Quality of Service, network and traffic measurement, Network programmability and virtualization, and Grid networking. She is a member of the scientifical committee of Grid5000’s/ALADDIN—French Computer Science Grid initiative. She has published more than 80 papers in International Journal and Conferences in Networking and Grid computing. She obtained her Habilitation à Diriger les Recherches from Université de Lyon in 2002, her Ph.D. (88) in Computer Science, MsC (84) and Engineer diploma (84) in CS from INSA de Lyon.

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