Nonhomogeneous Matrix Products

Hartfiel, D. J.

doi:10.1142/4707

Cited by 42 publications

(32 citation statements)

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“…The property that is established in lemma 1 is referred to in the literature as the paracontracting property; see [24] chapter 8 or [25]. We have thus shown that the matrices M i are paracontractive in S in L 1 norm.…”

Section: Remarkmentioning

confidence: 59%

Tools for the analysis and design of communication networks with Markovian dynamics

Leizarowitz

Stanojevic

Shorten

2006

IEE Proceedings - Control Theory and Applications

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Abstract:The stochastic properties of a class of communication networks whose dynamics are Markovian are analysed. The asymptotic behaviour of such a network in terms of the first and second moments of a stochastic process that describes the network dynamics is characterised and tools for their calculation are provided. Specifically, computation techniques for the calculation of these statistics are provided and that these algorithms converge exponentially fast is shown. Finally, how the results may be used for the design of network routers to realise networks with desired statistical properties is suggested. Introduction General remarksThe study of communication networks that carry transmission control protocol (TCP) traffic has been subject of intense interest in the computer science, network engineering, and applied mathematics literature [1 -9]. The principal motivation for much of this work has been to understand network behaviour, and to characterise important network properties with a view to developing analytic tools for the design of such networks. In particular, much of this work has focussed on understanding the manner in which the network allocates available bandwidth among competing network flows (network fairness) and the speed at which this bandwidth allocation takes place (network convergence rate). Recently, a very accurate random matrix model of TCP network dynamics was proposed [10]. This model was shown to be capable of capturing many essential features of networks in which TCP-like network flows compete for bandwidth via a bottleneck router. By making some simplifying assumptions concerning stochastic behaviour of the network, the authors demonstrate that this model may also be used as a basis to design networks in which bandwidth can be allocated in an arbitrary manner among competing flows. This may be achieved by redesigning the manner in which individual sources respond to network congestion, or by redesigning the manner in which network routers respond to network congestion (or both).The objective of this paper is to pursue further this line of research. However, rather than using the model as a basis for adjusting the behaviour of the individual flows to achieve desired network behaviour, we concentrate here on using this model to redesign the manner in which the bottleneck router drops packets when the network is congested; in particular, we analyse the properties of such networks when the bottleneck router drops packets according to some Markovian rules. Redesigning the manner in which network routers operate to allocate bandwidth is very important for a number of reasons related to network quality of service issues. Whereas the results in [10] are interesting from a theoretical perspective, router redesign along the lines suggested by this work would place an impossible computational burden on the network routers; on the other hand, dropping packets according to some Markovian rule could possibly be implemented using far fewer computational resources. Our principal contribution in t...

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Section: Remarkmentioning

confidence: 59%

Tools for the analysis and design of communication networks with Markovian dynamics

Leizarowitz

Stanojevic

Shorten

2006

IEE Proceedings - Control Theory and Applications

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“…Such discrete linear inclusions are unstable in the case when at least one of the matrices A l , l ∈ {0, 1, · · · , 2 n −1} has at least one eigenvalue larger than 1 (see [Dau92,Dau01,Har02]) and they satisfy various stability properties when all eigenvalues, of all matrices, are strictly smaller than 1. For the systems that we model, however, all matrices are oblique projections and, thus, have eigenvalues which are 0 or 1 making our systems live in the borderline between stable and unstable.…”

Section: Chapter 5 Stability Conditionsmentioning

confidence: 99%

Stability of Adaptive Distributed Real-TimeSystems with Dynamic Resource Management

Rafiliu¹

2013

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T oday's embedded distributed real-time systems, are exposed to large variations in resource usage due to complex software applications, sophisticated hardware platforms, and the impact of their run-time environment. As efficiency becomes more important, the applications running on these systems are extended with on-line resource managers whose job is to adapt the system in the face of such variations. Distributed systems are often heterogeneous, meaning that the hardware platform consists of computing nodes with different performance, operating systems, and scheduling policies, linked through one or more networks using different protocols.In this thesis we explore whether resource managers used in such distributed embedded systems are stable, meaning that the system's resource usage is controlled under all possible run-time scenarios. Stability implies a bounded worst-case behavior of the system and can be linked with classic real-time systems' properties such as bounded response times for the software applications. In the case of distributed systems, the stability problem is particularly hard because software applications distributed over the different resources generate complex, cyclic dependencies between the resources, that need to be taken into account. In this thesis we develop a detailed mathematical model of an adaptive, distributed real-time system and we derive conditions that, if satisfied, guarantee its stability. vii viii Popupärvetenskaplig sammanfattning V iär omgivna av ett ständigtökande antal inbyggda datorsystem. De finns exempelvis i våra bilar, telefoner, fotokameror och tvättmaskiner. Vår förväntningär att dessa produkter ska vara säkra, effektiva samt hålla hög kvalitet på de tjänster de tillhandahåller. Detta innebär bland annat att de datorsystem som finns inbyggda i våra bilar inte får leda till att passagerare skadas eller i värsta fall mister sina liv. Av en telefon förväntar vi oss lång batteritidäven om vi använder dessa dagligen för att prata och skicka SMS, lyssna på musik, ta bilder, läsa nyheter och skicka e-post. Vi förväntar oss samtidigt att de tjänster och funktioner som finns i våra mobiltelefoner håller hög kvalitet. Alla våra krav och förväntningar kan endast uppfyllas om systemens inbyggda resurser används på ett effektivt sätt. Exempel på sådana resurserär processorer, minnen, batterier och kommunikationsmedier.I denna avhandling beskrivs hur resurser i inbyggda datorsystem bör användas och kontrolleras för att optimera dess tjänster. Vidare tas hänsyn till att ett modernt system har stora variationer i resursanvändning. Detta beror delvis på att det mångfald av tjänster som finns i dagens moderna datorsystem i allra högsta grad består av komplex programvara. Detta beroräven på den komplicerade och sofistikerade hårdvara som behövs för att stödja all inbyggd programvara ix x samt påverkan från systemets omgivning.För att kunna förlita oss på dessa inbyggda systemär det nödvändigt att studera huruvida dess resurshanterareär stabila. Denna avhandling presenterar ...

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“…Since this limit is independent of the sequence {G(s)} ∞ s=0 , we have proved (18). In summary, at each step, every node can compute a local, unbiased, weighted least-square estimateθ i (t) whenever P i (t) is invertible.…”

Section: B Properties Of Intermediate Estimatesmentioning

confidence: 72%

“…(See also the book [18], which puts together much of the basic work on the convergence of nonhomogeneous matrix products.) Theorem 2 ( [16]): Suppose that a finite set of square matrices {W 1 , .…”

Section: A Infinite Products Of Paracontracting Matricesmentioning

confidence: 99%

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A scheme for robust distributed sensor fusion based on average consensus

Xiao¹,

Boyd

Lall

IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005.

303

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Abstract-We consider a network of distributed sensors, where each sensor takes a linear measurement of some unknown parameters, corrupted by independent Gaussian noises. We propose a simple distributed iterative scheme, based on distributed average consensus in the network, to compute the maximum-likelihood estimate of the parameters. This scheme doesn't involve explicit point-to-point message passing or routing; instead, it diffuses information across the network by updating each node's data with a weighted average of its neighbors' data (they maintain the same data structure). At each step, every node can compute a local weighted least-squares estimate, which converges to the global maximum-likelihood solution. This scheme is robust to unreliable communication links. We show that it works in a network with dynamically changing topology, provided that the infinitely occurring communication graphs are jointly connected.

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Nonhomogeneous Matrix Products

Cited by 42 publications

References 0 publications

Tools for the analysis and design of communication networks with Markovian dynamics

Tools for the analysis and design of communication networks with Markovian dynamics

Stability of Adaptive Distributed Real-TimeSystems with Dynamic Resource Management

A scheme for robust distributed sensor fusion based on average consensus

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