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25
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 remar ..."
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Cited by 153 (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 ...
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.
Maximum values in queueing processes
 Prob. Engrg. and Info. Sci
, 1995
"... Motivated by extremevalue engineering in service systems, we develop and evaluate simple approximations for the distributions of maximum values of queueing processes over large time intervals. We provide approximations for several different processes, such as the waiting times of successive custome ..."
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Cited by 10 (2 self)
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Motivated by extremevalue engineering in service systems, we develop and evaluate simple approximations for the distributions of maximum values of queueing processes over large time intervals. We provide approximations for several different processes, such as the waiting times of successive customers, the remaining workload at an arbitrary time, and the queue length at an arbitrary time, in a variety of models. All our approximations are based on extremevalue limit theorems. Our first approach is to approximate the queueing process by onedimensional reflected Brownian motion (RBM). We then apply the extremevalue limit for RBM, which we derive here. Our second approach starts from exponential asymptotics for the tail of the steadystate distribution. We obtain an approximation by relating the given process to an associated sequence of i.i.d. random variables with the same asymptotic exponential tail. We use estimates of the asymptotic variance of the queueing process to determine an approximate number of variables in this associated i.i.d. sequence. Our third approach is to simplify GI/G/1 extremevalue limiting formulas in Iglehart (1972) by approximating the distribution of an idle period by the stationaryexcess distribution of an interarrival time. We use simulation to evaluate the quality of these approximations for the maximum workload. From the simulations, we obtain a rough estimate of the time when the extreme value limit theorems begin to yield good approximations.
Transient analysis of the M/M/1 queue
 ADV. APPL. PROB
, 1993
"... A new approach is given to obtain the transient probabilities of the M=M=1 queueing system. A first step of this approach deals with the generating function of the transient probabilities of the uniformized Markov chain associated with this queue. The second step consists of the inversion of this ge ..."
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Cited by 9 (1 self)
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A new approach is given to obtain the transient probabilities of the M=M=1 queueing system. A first step of this approach deals with the generating function of the transient probabilities of the uniformized Markov chain associated with this queue. The second step consists of the inversion of this generating function. A new analytical expression of the transient probabilities of the M=M=1 queue is then obtained.
Limits and approximations for the M/G/1 LIFO waitingtime distribution, submitted
 Operations Research Letters
, 1997
"... We provide additional descriptions of the steadystate waitingtime distribution in the M/G/1 queue with the lastin firstout (LIFO) service discipline. We establish heavytraffic limits for both the cumulative distribution function (cdf) and the moments. We develop an approximation for the cdf tha ..."
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Cited by 5 (1 self)
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We provide additional descriptions of the steadystate waitingtime distribution in the M/G/1 queue with the lastin firstout (LIFO) service discipline. We establish heavytraffic limits for both the cumulative distribution function (cdf) and the moments. We develop an approximation for the cdf that is asymptotically correct both as the traffic intensity ρ → 1 for each time t and as t → ∞ for each ρ. We show that in heavy traffic the LIFO moments are related to the FIFO moments by the Catalan numbers. We also develop a new recursive algorithm for computing the moments.
and P.Romano, “Approximate Analytical Models for Networked Servers Subject to
 MMPP Arrival Processes”, Proc. 6th IEEE International Symposium on Network Computing and Applications (NCA
, 2007
"... ..."
Periodic load balancing
 Queueing Systems
, 1998
"... Multiprocessor load balancing aims to improve performance by moving jobs from highly loaded processors to more lightly loaded processors. Some schemes allow only migration of new jobs upon arrival, while other schemes allow migration of jobs in progress. A difficulty with all these schemes, however, ..."
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Cited by 2 (1 self)
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Multiprocessor load balancing aims to improve performance by moving jobs from highly loaded processors to more lightly loaded processors. Some schemes allow only migration of new jobs upon arrival, while other schemes allow migration of jobs in progress. A difficulty with all these schemes, however, is that they require continuously maintaining detailed state information. In this paper we consider the alternative of periodic load balancing, in which the loads are balanced only at each T time units for some appropriate T. With periodic load balancing, state information is only needed at the balancing times. Moreover, it is often possible to use slightly stale information collected during the interval between balancing times. In this paper we study the performance of periodic load balancing. We consider multiple queues in parallel with unlimited waiting space to which jobs come either in separate independent streams or by assignment (either random or cyclic) from a single stream. Resource sharing is achieved by periodically redistributing the jobs or the work in the system among the queues. The performance of these systems of queues coupled by periodic load balancing depends on the transient behavior of a single queue. We focus on useful approximations obtained by considering