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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...
Asymptotics for M/G/1 lowpriority waitingtime tail probabilities
, 1997
"... We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptiveresume disciplines. We show that the lowpriority steadystate waitingtime can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waitingtime distribution. ..."
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Cited by 39 (6 self)
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We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptiveresume disciplines. We show that the lowpriority steadystate waitingtime can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waitingtime distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have nonexponential asymptotics. This phenomenon even occurs when both servicetime distributions are exponential. When nonexponential asymptotics holds, the asymptotic form tends to be determined by the nonexponential asymptotics for the highpriority busyperiod distribution. We obtain asymptotic expansions for the lowpriority waitingtime distribution by obtaining an asymptotic expansion for the busyperiod transform from Kendall’s functional equation. We identify the boundary between the exponential and nonexponential asymptotic regions. For the special cases of an exponential highpriority servicetime distribution and of common general servicetime distributions, we obtain convenient explicit forms for the lowpriority waitingtime transform. We also establish asymptotic results for cases with longtail servicetime distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the nonexponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waitingtime transform.
An operational calculus for probability distributions via Laplace transforms
 ADVANCES IN APPLIED PROBABILITY
, 1996
"... In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their LaplaceStieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting mode ..."
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Cited by 21 (17 self)
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In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their LaplaceStieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Le vy ´ processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulantmomenttransfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a powermixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steadystate waiting time in an M/G/1 queue is the difference of two EMIGs when the servicetime distribution is an EMIG. We consider several transforms related to first passage times, e.g., for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.
Transient Behavior of the M/G/1 Workload Process
, 1992
"... In this paper we describe the timedependent moments of the workload process in the M/G/1 queue. The k th moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the timedependent serveroccupation probability. For general initial ..."
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Cited by 17 (9 self)
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In this paper we describe the timedependent moments of the workload process in the M/G/1 queue. The k th moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the timedependent serveroccupation probability. For general initial conditions, we show that the first two moment functions can be represented as the difference of two nondecreasing functions, one of which is the moment function starting at zero. The two nondecreasing components can be regarded as probability cumulative distribution functions (cdf's) after appropriate normalization. The normalized moment functions starting empty are called moment cdf's; the other normalized components are called momentdifference cdf's. We establish relations among these cdf's using stationaryexcess relations. We apply these relations to calculate moments and derivatives at the origin of these cdf's. We also obtain results for the covariance function of the stationary workload process. It is interesting that these various timedependent characteristics can be described directly in terms of the steadystate workload distribution. Subject classification: queues, transient results: M/G/1 workload process. queues, busyperiod analysis: M/G/1 queue. In this paper, we derive some simple descriptions of the transient behavior of the classical M/G/1 queue. In particular, we focus on the workload process {W(t) : t 0} (also known as the unfinished work process and the virtual waiting time process), which is convenient to analyze because it is a Markov process. Our main results describe the timedependent probability that the server is busy, P(W(t) > 0), the timedependent moments of the workload process, E[W(t) k ], and the covariance function of the stationary ...
Computing Laplace transforms for numerical inversion via continued fractions
 INFORMS Journal on Computing
, 1998
"... Abstract — It is often possible to effectively calculate probability density functions (pdf’s) and cumulative distribution functions (cdf’s) by numerically inverting Laplace transforms. However, to do so it is necessary to compute the Laplace transform values. Unfortunately, convenient explicit expr ..."
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Cited by 14 (4 self)
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Abstract — It is often possible to effectively calculate probability density functions (pdf’s) and cumulative distribution functions (cdf’s) by numerically inverting Laplace transforms. However, to do so it is necessary to compute the Laplace transform values. Unfortunately, convenient explicit expressions for required transforms are often unavailable for component pdf’s in a probability model. In that event, we show that it is sometimes possible to find continuedfraction representations for required Laplace transforms that can serve as a basis for computing the transform values needed in the inversion algorithm. This property is very likely to prevail for completely monotone pdf’s, because their Laplace transforms have special continued fractions called S fractions, which have desirable convergence properties. We illustrate the approach by considering applications
Limits and approximations for the busyperiod distribution in singleserver queues
 Prob. Engr. Inf. Sci. 9
, 1995
"... This paper is an extension of Abate and Whitt (1988b), in which we studied the M/M/1 busyperiod distribution and proposed approximations for busyperiod distributions in more general singleserver queues. Here we provide additional theoretical and empirical support for two approximations proposed in ..."
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Cited by 10 (5 self)
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This paper is an extension of Abate and Whitt (1988b), in which we studied the M/M/1 busyperiod distribution and proposed approximations for busyperiod distributions in more general singleserver queues. Here we provide additional theoretical and empirical support for two approximations proposed in Abate and Whitt (1988b), the natural generalization of the asymptotic normal approximation in (4.3) there and the inverse Gaussian approximation in (6.6), (8.3) and (8.4) there. These approximations yield convenient closedform expressions depending on only a few parameters, and they help reveal the general structure of the busyperiod distribution. The busyperiod distribution is known to be important for determining system behavior.
Modeling servicetime distributions with nonexponential tails: beta mixtures of exponentials
 STOCHASTIC MODELS
, 1999
"... Motivated by interest in probability density functions (pdf’s) with nonexponential tails in queueing and related areas, we introduce and investigate two classes of beta mixtures of exponential pdf’s. These classes include distributions introduced by Boxma and Cohen (1997) and Gaver and Jacobs (1998) ..."
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Cited by 8 (3 self)
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Motivated by interest in probability density functions (pdf’s) with nonexponential tails in queueing and related areas, we introduce and investigate two classes of beta mixtures of exponential pdf’s. These classes include distributions introduced by Boxma and Cohen (1997) and Gaver and Jacobs (1998) to study queues with longtail servicetime distributions. When the standard beta pdf is used as the mixing pdf, we obtain pdf’s with an exponentially damped power tail, i.e., f(t) ∼ αt −q e −ηt as t → ∞. This pdf decays exponentially, but analysis is complicated by the power term. When the beta pdf of the second kind is used as the mixing pdf, we obtain pdf’s with a power tail, i.e., f(t) ∼ αt −q as t → ∞. We obtain explicit representations for the cumulative distributions functions, Laplace transforms, moments and asymptotics by exploiting connections to the Tricomi function. Properties of the powertail class can be deduced directly from properties of the other class, because the powertail pdf’s are undamped versions of the other pdf’s. The powertail class can also be represented as gamma mixtures of Pareto pdf’s. Both classes of pdf’s have simple explicit Laguerreseries expansions.
BirthDeath Processes and Associated Polynomials
, 2001
"... We consider birthdeath processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birthdeath polynomials. The sequence of associated polynomials linked with a sequence of birthdeath polynomials and its orthogonalizing measure can be used in the analysis ..."
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Cited by 7 (3 self)
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We consider birthdeath processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birthdeath polynomials. The sequence of associated polynomials linked with a sequence of birthdeath polynomials and its orthogonalizing measure can be used in the analysis of the underlying birthdeath process in several ways. We briefly review the known applications of associated polynomials, which concern transition and firstentrance time probabilities, and establish some new results in this vein. In particular, our findings indicate how the prevalence of recurrence or #recurrence in a birthdeath process can be recognized from certain properties of the orthogonalizing measure for the associated polynomials. Keywords and phrases: birthdeath process, spectral measure, firstentrance time, recurrence, #recurrence, orthogonal polynomials, associated polynomials 2000 Mathematics Subject Classification: Primary 60J80, Secondary 42C05 1
On MarkovKrein Characterization of Mean Sojourn Time in Queueing Systems
, 2011
"... We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multiserver system, (ii) queueing systems with fluctuating arrival and service rates, and (iii) the M/G/1 roundrobin queue. We argue that rather than looking for exact expre ..."
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Cited by 1 (1 self)
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We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multiserver system, (ii) queueing systems with fluctuating arrival and service rates, and (iii) the M/G/1 roundrobin queue. We argue that rather than looking for exact expressions for the mean response time as a function of the job size distribution, a more fruitful approach is to find distributions which minimize or maximize the mean response time given the first n moments of the job size distribution. We prove that for the M/G/k system in lighttraffic asymptote and given first n ( = 2, 3) moments of the job size distribution, analogous to the classical MarkovKrein Theorem, these ‘extremal ’ distributions are given by the principal representations of the moment sequence. Furthermore, if we restrict the distributions to lie in the class of Completely Monotone (CM) distributions, then for all the three queueing systems, for any n, the extremal distributions under the appropriate “light traffic ” asymptotics are hyperexponential distributions with finite number of phases. We conjecture that the property of extremality should be invariant to the system load, and thus our light traffic results should hold for general load as well, and propose potential strategies for a unified approach
SERIES JACKSON NETWORKS AND NONCROSSING PROBABILITIES
, 808
"... Abstract. This paper studies the queue length process in series Jackson networks with external input to the first station. We show that its Markov transition probabilities can be written as a finite sum of noncrossing probabilities, so that questions on timedependent queueing behavior are translat ..."
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Abstract. This paper studies the queue length process in series Jackson networks with external input to the first station. We show that its Markov transition probabilities can be written as a finite sum of noncrossing probabilities, so that questions on timedependent queueing behavior are translated to questions on noncrossing probabilities. This makes previous work on noncrossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the ‘spectral gap’) of a positive recurrent system equals the relaxation time of an M/M/1 queue with the same arrival and service rates as the network’s bottleneck station. This resolves a conjecture of Blanc [6], which he proved for two queues in series. 1.