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...

This paper studies alternative ways to manage a multi-server system such as a telephone call center. Three alternatives can be described succinctly by: (i) blocking, (ii) reneging and (iii) balk-ing. The first alternative – blocking – is to have no provision for waiting. The second alternative is to allow waiting, but neither inform customers about anticipated delays nor provide state infor-mation to allow arriving customers to predict delays. The second alternative tends to yield higher server utilizations. The first alternative tends to reduce to the second, without the first-come first-served service discipline, when customers can easily retry, as with automatic redialers in telephone access. The third alternative is to both allow waiting and inform customers about anticipated delays. The third alternative tends to cause balking when all servers are busy (abandonment upon arrival) instead of reneging (abandonment after waiting). Birth-and-death process models are pro-posed to describe the performance with each alternative. Algorithms are developed to compute the conditional distributions of the time to receive service and the time to renege given each outcome. Algorithms are also developed to help the service provider predict customer waiting times before beginning service, given estimated service-time distributions and the elapsed service times of the customers in service. Better predictions may be obtained by classifying customers and thereby obtaining better estimates of their service-time distributions.

The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 Tricomi-Widder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (1) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre gener-ating function, (2) developing systematic methods for scaling, and (3) using Wynn’s ɛ-algorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux’s theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero ge-ometrically fast. We illustrate the effectiveness of our algorithm through numerical examples. Subject classifications: Mathematics, functions: Laplace transforms. Probability, distributions: calculation by transform inversion. Queues, algorithms: Laplace transform inversion.

In present teletraffic applications of queueing theory service time distributions B(t) with a heavy tail occur, i.e. 1 \Gamma B(t) t \Gamma for t !1 with ? 1. For such service time distributions not much explicit information is available concerning the tail probabilities of the inherent waiting time distribution W (t). In the present study the waiting time distribution is studied for a stable M=G=1 model for a class of service time distributions with 1 ! ! 2. For = 1 1 2 the explicit expression for Q(t) is derived. For rational with 1 ! ! 2, an asymptotic series for the tail probabilities of W (t) is derived. 1991 Mathematics Subject Classification: 90B22, 60K25 Keywords and Phrases: M=G=1 model, stable, service time distribution, heavy-tails, waiting time distributions, asymptotic series for tail probabilities. Note: work carried out under project LRD. 1. Introduction In classical applications of teletraffic theory the service time distributions in queueing models are freq...

Passage time densities and quantiles are important performance and quality of service metrics, but their numerical derivation is, in general, computationally expensive. We present an iterative algorithm for the calculation of passage time densities in semi-Markov models, along with a theoretical analysis and empirical measurement of its convergence behaviour. In order to implement the algorithm efficiently in parallel, we use hypergraph partitioning to minimise communication between processors and to balance workloads. This enables the analysis of models with very large state spaces which could not be held within the memory of a single machine. We produce passage time densities and quantiles for very large semi-Markov models with over 15 million states and validate the results against simulation.

An algorithm is developed to rapidly compute approximations for all the standard steady-state performance measures in the basic call-center queueing model M/GI/s/r+GI, which has a Poisson arrival process, IID service times with a general distribution, s servers, r extra waiting spaces and IID customer abandonment times with a general distribution. Empirical studies of call centers indicate that the service-time and abandon-time distributions often are not nearly exponential, so that it is important to go beyond the Markovian M/M/s/r + M special case, but the general service-time and abandon-time distributions make the realistic model very difficult to analyze directly. The proposed algorithm is based on an approximation by an ap-propriate Markovian M/M/s/r + M(n) queueing model, where M(n) denotes state-dependent abandonment rates. After making an additional approximation, steady-state waiting-time dis-tributions are characterized via their Laplace transforms. Then the approximate distributions are computed by numerically inverting the transforms. Simulation experiments show that the approximation is quite accurate. The overall algorithm can be applied to determine desired staffing levels, e.g., the minimum number of servers needed to guarantee that, first, the aban-donment rate is below any specified target value and, second, that the conditional probability that an arriving customer will be served within a specified deadline, given that the customer eventually will be served, is at least a specified target value.

Abstract not found

Abstract not found

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace-Stieltjes 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 cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture 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 steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time 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.

In this paper, we investigate two numerical methods for pricing Asian options: Laplace transform inversion and Monte Carlo simulation. In attempting to numerically invert the Laplace transform of the Asian call option that has been derived previously in the literature, we point out some of the potential difficulties inherent in this approach. We investigate the effectiveness of two easy-to-implement algorithms, which not only provide a cross-check for accuracy, but also demonstrate superior precision to two alternatives proposed in the literature for the Asian pricing problem. We then extend the theory of Laplace transforms for this problem by deriving the double Laplace transform of the continuous arithmetic Asian option in both its strike and maturity. We contrast the numerical inversion approach with Monte Carlo simulation, one of the most widely used techniques, especially by practitioners, for the valuation of derivative securities. For the Asian option pricing problem, we show that this approach will be effective for cases when numerical inversion is likely