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Interaction of TCP Flows as Billiards
, 2003
"... The aim of this paper is to analyze the performance of a large number of long lived TCP controlled flows sharing many routers (or links), from the knowledge of the network parameters (capacity, buffer size, topology) and of the characteristics of each TCP flow (RTT, route etc.) when taking synchroni ..."
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Cited by 39 (3 self)
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The aim of this paper is to analyze the performance of a large number of long lived TCP controlled flows sharing many routers (or links), from the knowledge of the network parameters (capacity, buffer size, topology) and of the characteristics of each TCP flow (RTT, route etc.) when taking synchronization into account. It is shown that the dynamics of such a network can be described in terms of iterate of random piecewise affine maps, or geometrically as a billiards in the Euclidean space with as many dimensions as the number of flow classes and as many reflection facets as there are routers. This class of billiards exhibits both periodic and nonperiodic asymptotic oscillations, the characteristics of which are extremely sensitive to the parameters of the network. It is also shown that for large populations and in the presence of synchronization, aggregated throughputs exhibit fluctuations that are due to the network as a whole, that follow some complex fractal patterns, and that come on top of other and more classical flow or packet level fluctuations. The consequences on TCP's fairness are exemplified on a few typical cases of small dimension.
Comparison of TCP Reno and TCP Vegas via Fluid Approximation
, 1998
"... We compare the efficiency of the flow control of two versions of TCP, the transmission control protocol of the Internet: the current version called Reno, and a recently proposed version called Vegas. By means of a fluid approximation, we show that due to the use of roundtrip times measurement, the ..."
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Cited by 36 (0 self)
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We compare the efficiency of the flow control of two versions of TCP, the transmission control protocol of the Internet: the current version called Reno, and a recently proposed version called Vegas. By means of a fluid approximation, we show that due to the use of roundtrip times measurement, the window dynamics of TCP Vegas are much more stable than those of TCP Reno, resulting in a much more efficient utilization of the network resources. In addition, whereas TCP Reno discriminates against users with long propagation delays, TCP Vegas fairly shares the available bandwidth between the users, whatever their propagation delays.
TCP is MaxPlus Linear and what it tells us on its throughput
 in &quot;Proceedings of the conference on Applications, Technologies, Architectures, and Protocols for Computer Communication
, 2000
"... We give a representation of the packetlevel dynamical be havior of the Reno and Tahoe variants of TCP over a sin gle endtoend connection. This representation allows one to consider the case when the connection involves a net work made of several, possibly heterogeneous, deterministic or random ..."
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Cited by 26 (3 self)
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We give a representation of the packetlevel dynamical be havior of the Reno and Tahoe variants of TCP over a sin gle endtoend connection. This representation allows one to consider the case when the connection involves a net work made of several, possibly heterogeneous, deterministic or random routers in series. It is shown that the key features of the protocol and of the network can be expressed via a linear dynamical system in the so called maxplus algebra. This opens new ways of both analytical evaluation and fast simulation based on products of matrices in this algebra. This also leads to closed form formulas for the throughput allowed by TCP under natural assumptions on the behavior of the routers and on the detection of losses and timeouts; these new formulas are shown to refine those obtained from earlier models which either assume that the network could be reduced to a single bottleneck router and/or approximate the packets by a fluid.
Impact of Network Delay Variations on Multicast Sessions with TCPlike Congestion Control
, 2001
"... We study the impact of random noise (queueing delay) on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max,plus) formalism together with method ..."
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Cited by 16 (2 self)
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We study the impact of random noise (queueing delay) on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max,plus) formalism together with methods based on stochastic comparison (association and convex ordering) and on the theory of extremes (Lai and Robbins' notion of maximal characteristics) to prove various properties of the throughput. We first prove that the throughput obtained from Golestani's deterministic model [1] is systematically optimistic. In presence of light tailed random noise, we show that the throughput decreases like the inverse of the logarithm of the number of receivers. We find analytically an upper and a lower bound for the throughput degradation. Within these bounds, we characterize the degradation which is obtained for various tree topologies. In particular, we observe that a class of trees commonly found in IP multicast sessions [9] (which we call umbrella trees) is significantly more sensitive to network noise than other topologies. I.
Impact of TCPlike Congestion Control on the Throughput of Multicast Group
 IEEE/ACM Transactions on Networking
, 2002
"... Abstract—We study the impact of random queueing delays stemming from traffic variability on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max, ..."
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Cited by 11 (3 self)
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Abstract—We study the impact of random queueing delays stemming from traffic variability on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max,plus) formalism together with methods based on stochastic comparison (association and convex ordering) and on the theory of extremes to prove various properties of the throughput. We first prove that the throughput predicted by a deterministic model is systematically optimistic. In the presence of lighttailed random delays, we show that the throughput decreases according to the inverse of the logarithm of the number of receivers. We find analytically an upper and a lower bound for the throughput degradation. Within these bounds, we characterize the degradation which is obtained for various tree topologies. In particular, we observe that a class of trees commonly found in IP multicast sessions is significantly more sensitive to traffic variability than other topologies. Index Terms—Classification of tree topology, (max,+) linearity, multicast congestion control, stochastic comparison.
Multifractal Analysis of a Class of Additive Processes with Correlated NonStationary Increments, in "Electronic
 Journal of Probability
"... We consider a family of stochastic processes built from infinite sums of independent positive random functions on R+. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on R+. The motivation for studying these pro ..."
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Cited by 1 (0 self)
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We consider a family of stochastic processes built from infinite sums of independent positive random functions on R+. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on R+. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic on the Internet. Such processes bear some analogy with Lévy processes, but they are more complex in the sense that their increments are neither stationary nor independent. Nevertheless, we show that their multifractal behavior is very much the same as that of certain Lévy processes. More precisely, we compute the Hausdorff multifractal spectrum of our processes, and find that it shares the shape of the spectrum of a typical Lévy process. This result yields a theoretical basis to the empirical discovery of the multifractal nature of TCP traffic.
INRIA & Ecole Normale
"... Abstract—We study the impact of random queuing delays stemming from traffic variability, on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max ..."
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Abstract—We study the impact of random queuing delays stemming from traffic variability, on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max,plus) formalism together with methods based on stochastic comparison (association and convex ordering) and on the theory of extremes (Lai and Robbins ’ notion of maximal characteristics) to prove various properties of the throughput. We first prove that the throughput predicted by Golestani’s deterministic model is systematically optimistic. In presence of light tailed random delays, we show that the throughput decreases like the inverse of the logarithm of the number of receivers. We find analytically an upper and a lower bound for the throughput degradation. Within these bounds, we characterize the degradation which is obtained for various tree topologies. In particular, we observe that a class of trees commonly found in IP multicast sessions (which we call umbrella trees) is significantly more sensitive to traffic variability than other topologies. I.
Impact of Network Delay Variations on Multicast Sessions with TCPlike Congestion Control
"... Abstract—We study the impact of random noise (queueing delay) on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max,plus) formalism together w ..."
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Abstract—We study the impact of random noise (queueing delay) on the performance of a multicast session. With a simple analytical model, we analyze the throughput degradation within a multicast (onetomany) tree under TCPlike congestion and flow control. We use the (max,plus) formalism together with methods based on stochastic comparison (association and convex ordering) and on the theory of extremes (Lai and Robbins ’ notion of maximal characteristics) to prove various properties of the throughput. We first prove that the throughput obtained from Golestani’s deterministic model [1] is systematically optimistic. In presence of light tailed random noise, we show that the throughput decreases like the inverse of the logarithm of the number of receivers. We find analytically an upper and a lower bound for the throughput degradation. Within these bounds, we characterize the degradation which is obtained for various tree topologies. In particular, we observe that a class of trees commonly found in IP multicast sessions [9] (which we call umbrella trees) is significantly more sensitive to network noise than other topologies. I.
Collaborating queues: large service network and a limit order book
, 2012
"... We analyse the steadystate behaviour of two different models with collaborating queues: that is, models in which “customers ” can be served by many types of “servers”, and “servers ” can process many types of “customers”. The first example is a largescale service system, such as a call centre. Col ..."
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We analyse the steadystate behaviour of two different models with collaborating queues: that is, models in which “customers ” can be served by many types of “servers”, and “servers ” can process many types of “customers”. The first example is a largescale service system, such as a call centre. Collaboration is the result of crosstrained staff attending to several different types of incoming calls. We first examine a loadbalancing policy, which aims to keep servers in different pools equally busy. Although the policy behaves orderoptimally over fixed time horizons, we show that the steadystate distribution may fail to be tight on the diffusion scale. That is, in a family of everlarger networks whose arrival rates grow as O(r) (where r is a scaling parameter growing to ∞), the sequence of steadystate deviations from equilibrium scaled down by r −1/2 is not tight. We then propose a different policy, for which we show that the sequence of invariant distributions is tight on the r 1/2+ɛ scale, for any ɛ> 0. For this policy we conjecture that tightness holds on the diffusion scale as well. The second example models a limit order book, a pricing mechanism for a singlecommodity market in which buyers (respectively sellers) are prepared to wait for the
IS
, 2002
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.