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65
Random Early Detection Gateways for Congestion Avoidance
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 1993
"... This paper presents Random Early Detection (RED) gateways for congestion avoidance in packetswitched networks. The gateway detects incipient congestion by computing the average queue size. The gateway could notify connections of congestion either by dropping packets arriving at the gateway or by ..."
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Cited by 2177 (31 self)
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This paper presents Random Early Detection (RED) gateways for congestion avoidance in packetswitched networks. The gateway detects incipient congestion by computing the average queue size. The gateway could notify connections of congestion either by dropping packets arriving at the gateway or by setting a bit in packet headers. When the average queue size exceeds a preset threshold,the gateway drops or marks each arriving packet with a certain probability, where the exact probability is a function of the average queue size. RED gateways keep the average queue size low while allowing occasional bursts of packets in the queue. During congestion, the probability that the gateway notifies a particular connection to reduce its window is roughly proportional to that connection's share of the bandwidth throughthe gateway. RED gateways are designed to accompany a transportlayer congestion control protocol such as TCP.The RED gateway has no bias against bursty traffic and avoids the global synchronization of many connectionsdecreasing their window at the same time. Simulations of a TCP/IP network are used to illustrate the performance of RED gateways.
On The Rapid Computation of Various Polylogarithmic Constants”, manuscript
, 1996
"... We give algorithms for the computation of the dth digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the d ..."
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Cited by 104 (31 self)
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We give algorithms for the computation of the dth digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log (2) or on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of, the billionth hexadecimal digits of 2 2 log(2) and log (2), and the ten billionth decimal digit of log(9=10). These calculations rest on the observation that very special types of identities exist for certain numbers like, 2,log(2) and log 2 (2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we deriveinthiswork appear to be new, for example the critical identity for:
Algorithmic Manipulations and Transformations of Univariate Holonomic Functions and Sequences
, 1996
"... Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be ..."
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Cited by 51 (0 self)
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Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be performed "algorithmically". As a consequence, we can prove any identity of holonomic functions or sequences automatically. Based on this theory, the author implemented a package that contains procedures for automatic manipulations and transformations of univariate holonomic functions and sequences within the computer algebra system Mathematica. This package is introduced in detail. In addition, we describe some different techniques for proving holonomic identities.
Diffraction of Random Tilings: Some Rigorous Results
 J. STAT. PHYS
, 1999
"... The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar rando ..."
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Cited by 26 (16 self)
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The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.
Continued Fractions, Comparison Algorithms, and Fine Structure Constants
, 2000
"... There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems ..."
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Cited by 10 (2 self)
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There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems (symbolic dynamics), number theory (continued fractions), special functions (multiple zeta values), functional analysis (transfer operators), numerical analysis (series acceleration), and complex analysis (the Riemann hypothesis). These domains all eventually contribute to a detailed characterization of the complexity of comparison and sorting algorithms, either on average or in probability.
An Iterative Solution Of The ThreeColour Problem On A Random Lattice,” Nucl
 Phys. B
, 1998
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Gaussian Integration of Chebyshev Polynomials and Analytic Functions
 Numer. Alg
"... . Explicit bounds for the quadrature error of the n th GaussLegendre quadrature rule applied to the m th Chebyshev polynomial are derived. They are precise up to the order O(m 4 n 6 ) . As an application, error constants for classes of functions, which are analytic in the interior of an ell ..."
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Cited by 8 (6 self)
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. Explicit bounds for the quadrature error of the n th GaussLegendre quadrature rule applied to the m th Chebyshev polynomial are derived. They are precise up to the order O(m 4 n 6 ) . As an application, error constants for classes of functions, which are analytic in the interior of an ellipse, are estimated. The location of the maxima of the corresponding kernel function is investigated. AMS classication: 65D30, 41A55 Keywords: Gaussian integration, Chebyshev polynomials, error bounds, analytic functions Running title: Gaussian integration of Chebyshev polynomials 1. Introduction Chebyshev expansions are very useful tools for numerical analysis. Their convergence is guaranteed under rather general conditions, they often converge fast compared with other polynomial expansions, and each summand of the series may easily be estimated. Considering functionals on certain function spaces it is therefore important to know, how they operate on the Chebyshev polynomials Tm of the ...
Explicit evaluations and reciprocity theorems for finite trigonometric sums
 ADV. IN APPL. MATH
, 2002
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