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The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy ..."
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Cited by 149 (51 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Timestep Stochastic Simulation of Computer Networks using Diffusion Approximation
, 1903
"... Timestep stochastic simulation (TSS) is a novel method for generating sample paths of computer networks, with low computation cost independent of packet rates. It has accuracy adequate to evaluate general network and flow configurations, including arbitrary flow start times and durations, droptail ..."
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Cited by 2 (0 self)
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Timestep stochastic simulation (TSS) is a novel method for generating sample paths of computer networks, with low computation cost independent of packet rates. It has accuracy adequate to evaluate general network and flow configurations, including arbitrary flow start times and durations, droptail queuing (i.e., does not require RED), and arbitrary statedependent control mechanisms for congestion control and routing. TSS generates the evolution of the system state S(t) on a sample path in time steps of size δ. At each step, S(t+δ) is randomly chosen according to S(t) and the probability distribution P r[S(t + δ)S(t)] obtained using the diffusion approximation. Because packet transmission and reception events are replaced by time steps, TSS generates sample paths at a fraction of the cost of packetlevel simulation. Because TSS generates sample paths, control feedback can be based on sample path metrics, rather than ensemble metrics, thereby accurately capturing the effects of statedependent control mechanisms. 1
MODELLING: A Tool for the Design and Optimization of Computer Systems
, 1976
"... In this paper we consider the behaviour of the slotted broadcast channel used by an ensemble of terminals for the transmission of packets of data. A mathematical model is used to prove that the channel is unstable, leading to zero effective throughput, if no comtrol is imposed on the channel behavio ..."
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In this paper we consider the behaviour of the slotted broadcast channel used by an ensemble of terminals for the transmission of packets of data. A mathematical model is used to prove that the channel is unstable, leading to zero effective throughput, if no comtrol is imposed on the channel behaviour. Two classes of control policies, acting on the input and on retrmnsmissions from blocked terminals, are then analyzed and stability and optimality conditions for the channel with these policies are derived. The theorem on instability of the uncontrolled channel can in fact be considered as being a corollary of the stability theorem for the retransmission 1 control policy. We show that e is a lower bound to the maximum achievable throughput with an input control policy. On the other hand, an optimal retransmission control policy must regulate retransmissions so that the probability of retransmission of an individual blocked terminal is of the form (l)n1 in each slot, where n is the total number of blocked terminals. Some simulation results are provided in order to illustrate the effect of this policy.
ANALYSIS OF INTEGRATED VOICE AND DATA COMMUNICATION NETWORK LihHsing
, 1977
"... This thesis studies the performance of an integrated voice and data communication network. Different characteristics from other communication networks and a different set of key parameters for its performance are addressed. The relationship of and the tradeoff between the key parameters are also di ..."
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This thesis studies the performance of an integrated voice and data communication network. Different characteristics from other communication networks and a different set of key parameters for its performance are addressed. The relationship of and the tradeoff between the key parameters are also discussed. Voice communication requires slow but continuous information e×change while data communication requires burst type of information exchange. A new integrated switch is designed to support both type of communications _ line switch for voice and packet switch for data. Class] traffic, voice or video, is modeled as an M/M/n queue and Class lI traffic, data or bulk, is modeled as an M/M/Y queue. A wild distribution of Class [I queue length is discovered and a significant tradeoff between communication and ccmputer facilities is implied. The study shows that the queue length grows very rapidly when the Class]/Class]] job size ratio increases. A small integrated switch with relatively small job size ratio is studied in details. However large switches with realistcal job size ratio are only approximated and detail quantitative results for such system require further study.