This paper reviews the Fourier-series 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 Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series 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 Fourier-series 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...

We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptive-resume disciplines. We show that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting-time 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 non-exponential asymptotics. This phenomenon even occurs when both service-time distributions are exponential. When non-exponential asymptotics holds, the asymptotic form tends to be determined by the non-exponential asymptotics for the high-priority busy-period distribution. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall’s functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For the special cases of an exponential high-priority service-time distribution and of common general service-time distributions, we obtain convenient explicit forms for the low-priority waiting-time transform. We also establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waiting-time transform.

In this paper we describe the time-dependent 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 time-dependent server-occupation 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 moment-difference cdf's. We establish relations among these cdf's using stationary-excess 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 time-dependent characteristics can be described directly in terms of the steady-state workload distribution. Subject classification: queues, transient results: M/G/1 workload process. queues, busy-period 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 time-dependent probability that the server is busy, P(W(t) > 0), the time-dependent moments of the workload process, E[W(t) k ], and the covariance function of the stationary ...

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

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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 busy-period distributions in more general single-server 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 closed-form expressions depending on only a few parameters, and they help reveal the general structure of the busy-period distribution. The busy-period distribution is known to be important for determining system behavior.

One of the most important features of multimedia applications is the integration of multiple media streams that have to be presented in a synchronized fashion. In this paper we consider a distributed multimedia system where the communication between two nodes involve two media. Arrivals consist of two types of media packets, and the packets are processed for pairs of one packet from each media. We view this model as a two-dimensional nite birth-death process by considering the arrivals of the packets, following Poisson distribution, as births and the departures of the impatient packets, after waiting in the network for an exponential period, as deaths. We analyze the time-dependent behaviour of our model numerically. We study the various system characteristics like, time-dependent probabilities of the number of packets in each media, their averages, variances and the busy period. They are illustrated through tables and graphs. Keywords and phrases: Two-dimensional birth-dea...