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28
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...
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 25 (8 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
Solving Probability Transform Functional Equations for Numerical Inversion
, 1991
"... Many methods for numerically inverting transforms require values of the transform at complex arguments. However, in some applications, the transforms are only characterized implicitly via functional equations. This is illustrated by the busyperiod distribution in the M/G/1 queue. In this paper we p ..."
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Cited by 16 (13 self)
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Many methods for numerically inverting transforms require values of the transform at complex arguments. However, in some applications, the transforms are only characterized implicitly via functional equations. This is illustrated by the busyperiod distribution in the M/G/1 queue. In this paper we provide conditions for iterative methods to converge for complex arguments. Moreover, we show that stochastic monotonicity properties can provide useful bounds.
An analytical model of temperature in microprocessors
, 2005
"... Temperature has become an important design constraint for highperformance microprocessors. Research on temperatureconstrained microarchitecture requires an efficient modeling of temperature. We propose an analytical model of temperature, based on solving a boundaryvalue problem of heat conduction ..."
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Cited by 12 (7 self)
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Temperature has become an important design constraint for highperformance microprocessors. Research on temperatureconstrained microarchitecture requires an efficient modeling of temperature. We propose an analytical model of temperature, based on solving a boundaryvalue problem of heat conduction. The model gives steadystate and transient temperature at any point on the dissipating plane, assuming rectangleshaped surface sources. The model can be used to reason about temperature. It can also be implemented in a performance/power microarchitecture simulator. We provide two examples illustrating these uses.
Analytical Model for Connectivity in Vehicular Ad Hoc Networks
 In Proceedings of IEEE Transactions on Vehicular Technology (VTC ’08
, 2008
"... We investigate connectivity in the ad hoc network formed between vehicles that move on a typical highway. We use the common model in vehicular traffic theory in which a fixed point on the highway sees cars passing it separated by times with exponentially distributed duration. We obtain the distribut ..."
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Cited by 11 (0 self)
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We investigate connectivity in the ad hoc network formed between vehicles that move on a typical highway. We use the common model in vehicular traffic theory in which a fixed point on the highway sees cars passing it separated by times with exponentially distributed duration. We obtain the distribution of the distances between cars, which allows us to use techniques from queuing theory for studying connectivity. We obtain the Laplace transform of the probability distribution of connectivity distance, explicit expressions for the expected connectivity distance as well as the probability distribution and expectation of the number of cars in a platoon. Then, we conduct extensive simulation studies in order to evaluate the obtained results. The analytical model we present is able to describe the effects of various system’s parameters, including road traffic parameters (i.e. speed distribution and traffic flow) and transmission range of vehicles, on the connectivity. In order to study the effect of speed on connectivity more precisely, we provide bounds obtained using stochastic ordering techniques. Our approach is based on the work of Miorandi and Altman [10] that transformed the problem of connectivity distance distribution into that of the distribution of the busy period of an equivalent infinite server queue. We use our analytical results along with common road traffic statistical data to understand connectivity in VANETs. 1
A unified framework for numerically inverting Laplace transforms
 INFORMS Journal on Computing
, 2006
"... We introduce and investigate a framework for constructing algorithms to numerically invert Laplace transforms. Given a Laplace transform ˆ f of a complexvalued function of a nonnegative realvariable, f, the function f is approximated by a finite linear combination of the transform values; i.e., w ..."
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Cited by 5 (1 self)
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We introduce and investigate a framework for constructing algorithms to numerically invert Laplace transforms. Given a Laplace transform ˆ f of a complexvalued function of a nonnegative realvariable, f, the function f is approximated by a finite linear combination of the transform values; i.e., we use the inversion formula f(t) ≈ fn(t) ≡ 1 t n� ωk ˆ f k=0 αk
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
Flow Simulation in Heterogeneous Reservoirs using the Dual Reciprocity Boundary Element Method and the Green Element Method
, 1998
"... Green's functions are established tools for solving petroleum engineering flow problems. Their utility and rigor was extended to arbitrarily shaped reservoirs using the Boundary Element Method (BEM). Traditional BEMs are limited to singlephase flow in homogeneous reservoirs. Earlier authors have de ..."
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Cited by 2 (1 self)
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Green's functions are established tools for solving petroleum engineering flow problems. Their utility and rigor was extended to arbitrarily shaped reservoirs using the Boundary Element Method (BEM). Traditional BEMs are limited to singlephase flow in homogeneous reservoirs. Earlier authors have developed techniques to handle heterogeneity. These methods are perturbationbased and computationintensive. The current work adapted the most recent developments in boundary element methods to reservoir engineering problems. The transient pressure (diffusion) and convectiondiffusion equations were solved in heterogeneous media using the Dual Reciprocity Boundary Element Method (DRBEM) and the Green Element Method (GEM). Numerical experiments showed that DRBEM is more accurate than a standard finite difference method. However like finite difference methods, DRBEM is subject to spurious oscillation at high Peclet numbers. DRBEM also requires the solution of a dense system of equations. GEM, wh...
Valuing American continuousinstallment options
 Hokkaido University
, 2007
"... Installment options are weakly pathdependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuousinstallment options written on dividendpaying assets. The s ..."
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Cited by 1 (1 self)
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Installment options are weakly pathdependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuousinstallment options written on dividendpaying assets. The setup is a standard BlackScholesMerton framework where the price of the underlying asset evolves according to a geometric Brownian motion. The valuation of installment options can be formulated as an optimal stopping problem, due to the flexibility of continuing or stopping to pay installments as well as the chance of early exercise. Analyzing cash flow generated by the optimal stop, we can characterize asymptotic behaviors of the stopping and early exercise boundaries close to expiry. Combining the PDE and Laplace transform approaches, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping and early exercise boundaries. Abelian theorems of Laplace transforms enable us to obtain a concise result for the perpetual case. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the boundaries.