Shortle, J.F., M.J. Fischer, and P. H. Brill. 2006. " Waiting Time Distributions of the M/DN/1 Queues Through Numerical Laplace Inversion ." Accepted by the INFORMS Journal on Computing. 

ABSTRACT
This paper considers an M/G/1 queue where the service time for each customer is a discrete random variable taking one of N values. We call this an M/DN/1 queue. There are potential numerical problems inverting Laplace transforms associated with this queue because the service distribution is discontinuous. The purpose of this paper is to investigate the performance of numerical transform inversion methods in analyzing this queue. We first derive continuity properties of the steady-state distribution of wait in the M/DN/1 queue. Then, we show analytically how continuity properties affect the performance of the Fourier method for inverting transforms. In particular, continuity is not required in all derivatives for best performance of the method. We also present a new inversion method specifically for the M/DN/1 queue. Finally, we give numerical experiments comparing these and four other inversion methods. Although no method clearly dominates, the recursion method performs well in most examples.

Shortle, J.F., M.J. Fischer, D. Gross, and D.M. Masi. 2005. “ Using the Pareto Distribution in Queueing Modeling .” Submitted to the Journal of Probability and Statistical Science. March 2005. 

ABSTRACT
The Pareto distribution was first formulated in the late 1800s by the Italian economist Vilfredo Pareto. He argued that the distribution of income could be described by the formula log(N) = log(A) + m log(x), where N is the proportion of income earners who receive incomes higher than x, and A and m are constants. Over the years Pareto’s Law has held up in empirical studies. The Pareto distribution has recently been used as a model for file sizes in the Internet and insurance losses. It has various forms; here, we consider a 1-parameter form and a 2-parameter form. The purpose of this paper is to show that using different forms of the Pareto distribution can result in drastically different congestion measures in a queueing system. This is true even if the means and variances between two Pareto distributions are the same. In other words, even within the same family of distributions, a matching of the first two moments is not sufficient to guarantee similar performance measures. In addition, we show that different forms of the Pareto can yield qualitatively different behavior in a queueing system, with respect to analogous Markovian models. We specifically consider P/M/1 and M/P/1 queueing systems.

Shortle, J.F. 2005. " Piecewise-Polynomial Approximations for Heavy-Tailed Distributions in Queueing Analysis ." Stochastic Models. Vol. 21, No. 1. p. 215-234. 2005. 

ABSTRACT
A basic difficulty in dealing w/ heavy-tailed distributions is that they may not have closed-form Laplace transforms. This makes numerical methods which use the Laplace transform more challenging. This paper generalizes an existing method for approximating heavy-tailed distributions, for use in queueing analysis. The generalization involves fitting Chebyshev polynomials to a probability density function g(t) at specified points t1, t2, …, tN. By choosing points ti which rapidly get far out in the tail, it is possible to capture the tail behavior with relatively few points, and to control the relative error in the approximation. We give numerical examples to evaluate the performance of the method in simple queueing problems.

Shortle, J.F., and Brill, P.H. 2005. " Analytical Distribution of Waiting Time in the M/{iD}/1 Queue ."  Queueing Systems. Vol. 50, p. 185-197  

ABSTRACT
We give an analytical formula for the steady-state distribution of wait in the M/G/1 queue, where the service time for each customer is a positive integer multiple of a constant D > 0. We call this an M/{iD}/1 queue. We give numerical algorithms to calculate the distribution. In addition, in the case that the service distribution is sparse, we give revised algorithms that can compute the distribution more quickly.

Hsiung, H., M.J. Fischer, and J.G. Leigh. 2005. " Internet Protocol Multi-Class Service Performance Analysis ." Telecommunications Review. 2005. Noblis, Falls Church, Va. 

ABSTRACT
In a circuit-switched network which carries multiple services, bandwidth is allocated for data service to ensure the minimum data delivery. When data demand exceeds the reserved bandwidth, data traffic tends to overflow and use resources allocated for voice, if voice resources are not used. In a packet-switched network, bandwidth is reserved for voice service to ensure the Service Level Agreement is met. This paper studies voice and data performance (i.e., delay and jitter) using four resource management and queueing rules.

Shortle, J. F., M.J. Fischer, and D.M. Masi. 2005. " Estimating Delay in a Data Network Through Numerical Laplace Inversion ." Telecommunications Review. 2005. Noblis, Falls Church, Va. 

ABSTRACT
In place, at present, is a simplified analytical queueing model to estimate the end-to-end delay of packets in a data network. [1] The key modeling assumption is that the end-to-end path can be divided into a series of independent queues. In other words, the total travel time is approximated by the sum of the waiting times in a series of individual queues that are independent of each other. The simplified model gives approximately the same mean performance metrics as the detailed OPNET Modeler simulation (less than 10 percent error for the problem considered), but takes only a fraction of the time.

This paper gives an even faster numerical method to solve the simplified analytical model. [1] The new method performs all computations initially using Laplace transforms. This avoids using numerical integration to convolve distribution functions. The new method is about 180 times faster than the previously implemented solution.

The main challenge with the new method is that the distribution functions of interest have discontinuities or atoms. These are known to cause problems for numerical transform inversion. In this paper, we present methods to remove these atoms prior to numerical inversion and then to re-insert the atoms into the resulting distribution function, thus preserving the accuracy and speed of the inversion methods.

Shortle, J. F., M.J. Fischer, D. Gross, and D.M. Masi. 2005. " One-Parameter Pareto, Two-Parameter Pareto, Three-Parameter Pareto: Is There A Modeling Difference? " Telecommunications Review. 2005. Noblis, Falls Church, Va. 

ABSTRACT
The Pareto distribution was first formulated in the late 1800s by the Italian economist Vilfredo Pareto. He presented the argument that in all countries and times, the distribution of income and wealth could be described by the formula log(N) = log(A) + mlog(x), where N is the proportion of income earners who receive incomes higher than x, and A and m are constants. Over the years, Pareto’s Law has held up in empirical studies. The Pareto distribution has recently been used as a model for file sizes on the Internet, insurance losses, financial behavior of the stock market, and in telecommunications systems. It has various forms; here, we consider a one-parameter form and a two-parameter form. Thus, we question if using one form of the Pareto gives different results than using another form. In this paper, we numerically address this question by studying queueing systems with either Pareto arrivals or service times. The two Pareto forms are studied in detail: Case 1, both Pareto forms have equal means and variances; and Case 2, both Pareto forms have equal mean and shape parameters. For both cases, our numerical results, substantiated by simulation studies, show that using the two-parameter Pareto results in lower congestion than the comparable one-parameter Pareto.

Fischer, M.J., Masi, D.M.B., Gross, D., and Shortle, J.F. 2004. " Loss Systems with Heavy-Tailed Arrivals ." Telecommunications Review. 2004. Noblis, Falls Church, Va. 

ABSTRACT
R. I. Wilkinson [1] was one of the first researchers to study traffic peakedness in circuit-switched systems. His pioneering work, called the Equivalent Random Method (ERM), gave engineers an approximation for a trunk group’s blocking probability when presented with a peaked offered load. ERM was based on the mean and variance of the offered load to the trunk group and did not assume the form of the underlying arrival distribution. With the advent of the Internet, traffic studies have indicated the existence of heavy- or power-tailed probability distributions that characterized these traffics. The Pareto, Weibull, and LogNormal distributions most frequently appear in these studies. A fact that seems to be overlooked is that different congestion results may be obtained depending upon which probability distribution is used in the congestion model, even when the mean and variance of the probability distributions are equal. In this paper, we study the effect of different heavy- or power-tailed arrival distributions on the blocking probability in loss systems.

Shortle, J.F., Brill, P.H., Fischer, M.J., Gross, D., and Masi, D.M.B. 2004.
"An Algorithm to Compute the Waiting Time Distribution for the M/G/1 Queue." INFORMS Journal on Computing . Vol. 16, p. 152-161. PDF file provided by INFORMS. Copyright 2004 INFORMS.  

ABSTRACT
In many modern applications of queueing theory, the classical assumption of exponentially decaying service distributions does not apply. In particular, Internet and insurance risk problems may involve heavy-tailed distributions. A difficulty with heavy-tailed distributions is that they may not have closed-form, analytic Laplace transforms. This makes numerical methods, which use the Laplace transform, challenging. In this paper, we develop a method for approximating Laplace transforms. Using the approximation, we give algorithms to compute the steady state probability distribution of the waiting time of an M/G/1 queue to a desired accuracy. We give several numerical examples, and we validate the approximation with known results where possible or with simulations otherwise. We also give convergence proofs for the methods.

Masi, D.M.B., Fischer, M.J., and Garbin, D. A., " Modeling Internet Protocol Networks: OPNET and Analytics ." Sigma: Mitretek Technology Summaries. Winter 2004. p. 52-63. 

ABSTRACT
With the advent of the events surrounding September 11, it is apparent that the protection of critical federal telecommunications networks needs to be ensured. This paper focuses on analytic queueing and simulation capabilities that have been developed to analyze the performance of federal private Internet Protocol (IP) networks.

We have developed an initial version of an analytic model called the IP Network Performance and Analysis Tool (IP-NPAT) and an initial OPNET simulation model. Calibration of the analytic model and the simulation for all link queues based on a realistic packet distribution indicate close agreement in the two analyses. Link loads, mean link delays, link jitter, and delay distribution were compared for each link, as well as end-to-end delay distributions.

Shortle, J.F., M.J. Fischer, and D. Gross. 2003. “ Using the Transform Approximation Method to Analyze Queues with Heavy-Tailed Service .” Journal of Probability and Statistical Science. February.  

ABSTRACT
Many modern queueing problems involve distributions which are heavy-tailed. This means their distribution functions decay more slowly than any exponential function. Analyzing queues with these distributions is difficult since they do not have closed-form, analytic Laplace transforms. This paper investigates a recently proposed method for numerically approximating Laplace transforms, called the Transform Approximation Method (TAM). While TAM can be used to approximate the Laplace transform of a heavy-tailed distribution, one must still invert a Laplace transform to recover the desired probability distribution. This paper investigates using TAM with two numerical methods for inverting Laplace transforms. In particular, we compare the well-known Fourier-series method with a recursion method specifically adapted for TAM. We give several benchmark problems and algorithms to compare the methods. In general, the Fourier method is better, though the recursion method is better for a specific class of problems.