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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.
Series Expansions of Lyapunov Exponents and Forgetful Monoids
, 2000
"... We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for ..."
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We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for Tetrislike heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lyapunov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special "forgetful" monoid, defined by generators and relations.
A central limit theorem for stochastic recursive sequences of topical operators
, 2007
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Finite Abstractions of Stochastic MaxPlusLinear Systems
"... Abstract. This work investigates the use of finite abstractions to study the finitehorizon probabilistic invariance problem over Stochastic MaxPlusLinear (SMPL) systems. SMPL systems are probabilistic extensions of discreteevent MPL systems that are widely employed in the engineering practice f ..."
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Abstract. This work investigates the use of finite abstractions to study the finitehorizon probabilistic invariance problem over Stochastic MaxPlusLinear (SMPL) systems. SMPL systems are probabilistic extensions of discreteevent MPL systems that are widely employed in the engineering practice for timing and synchronisation studies. We construct finite abstractions by reformulating the SMPL system as a discretetime Markov process, then tailoring formal abstraction techniques in the literature to generate a finitestate Markov Chain (MC), together with precise guarantees on the level of the introduced approximation. This finally allows to probabilistically model check the obtained MC against the finitehorizon probabilistic invariance specification. The approach is practically implemented via a dedicated software, and elucidated in this work over numerical examples.
Series Expansions of Lyapunov Exponents and
, 2000
"... 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.
On the stability of a batch clearing system with Poisson arrivals and subadditive service times
, 2001
"... We study a service system in which, in each service period, the server performs the current set B of tasks as a batch, taking time s(B), where the function s(\Delta) is subadditive. A natural definition of "traffic intensity under congestion" in this setting is ae := lim t!1 t \Gamma1 ..."
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We study a service system in which, in each service period, the server performs the current set B of tasks as a batch, taking time s(B), where the function s(\Delta) is subadditive. A natural definition of "traffic intensity under congestion" in this setting is ae := lim t!1 t \Gamma1 Es(all tasks arriving during time [0; t]): We show that ae ! 1, and finite mean of individual service times, are necessary and sufficient to imply stability of the system. A key observation is that the numbers of arrivals during successive service periods form a Markov chain fA n g, enabling us to apply classical regenerative techniques and to express the stationary distribution of the process in terms of the stationary distribution of fA n g.
The MIMS Secretary
, 2012
"... Critical path statistics of maxplus linear systems ..."
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unknown title
, 2003
"... www.elsevier.com/locate/laa A family of inequalities originating from coding of messages ..."
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www.elsevier.com/locate/laa A family of inequalities originating from coding of messages
Critical path statistics of maxplus linear systems with Gaussian noise
, 2012
"... The critical paths of a maxplus linear systems with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating ..."
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The critical paths of a maxplus linear systems with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating these new statistics as well as the previously studied maxplus exponent. We show that for low amplitude noise the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that as the noise amplitude is increased the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy.