Martin J. Fischer

An Algorithm to Compute the Waiting Time Distribution for the M/G/1 Queue

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.

Internet-Type Queues with Power-Tailed Interarrival Times and Computational Methods for Their Analysis

Internet traffic flows have often been characterized as having power-tailed (long-tailed, fat-tailed, heavy-tailed) packet interarrival times or service requirements. In this work, we focus on the development of complete and computationally efficient steady-state solutions of queues with power-tailed interarrival times when the service times are assumed exponential. The classical method for obtaining the steady-state probabilities and delay-time distributions for the G/M/1 (G/M/ c) queue requires solving a root-finding problem involving the Laplace-Stieltjes transform of the interarrival-time distribution function. Then the exponential service assumption is combined with the derived geometric arrival-point probabilities to get both the limiting general-time state and delay distributions. However, in situations where there is a power tail, the interarrival transform is typically quite complicated and never analytically tractable. In addition, it is possible that there is only a degenerate steady-state system-size probability distribution. Thus, an alternative approach to obtaining a steady-state solution is typically needed when power-tailed interarrivals are present. We exploit the exponentiality of the steady-state delay distributions for the G/M/1 and G/M/ c queues, using level-crossings and a transform approximation method, to develop an alternative root-finding problem when there are power-tailed interarrival times. Extensive computational results are given.

Difficulties in simulating queues with Pareto service

M/G/1 queues, where G is a heavy-tailed distribution, have applications in Internet modeling and modeling for insurance claim risk. The Pareto distribution is a special heavy-tailed distribution called a power-tailed distribution, and has been found to serve as adequate models for many of these situations. However, to get the waiting time distribution, one must resort to numerical methods, e.g., simulation. Many difficulties arise in simulating queues with Pareto service and we investigate why this may be so. Even if we are willing to consider truncated Pareto service, there still can be problems in simulating if the truncation point (maximum service time possible) is too large.

Approximation for a two-class weighted fair queueing discipline

This paper presents an approximating model for a 2-class weighted fair queueing (or random polling) model. The approximating system can be analyzed analytically to obtain mean performance measures such as expected delay. We show through a formal argument that the approximation works well when the overall utilization of the system @r is small. Based on simulation experiments, we develop a modified version of the approximation that is accurate for a wide range of @r. Finally, we extend the approximation to more complex queueing scenarios, such as the low-latency-queueing discipline and systems with more than 2 classes.

Performance modeling of distributed automatic call distribution systems

The number of businesses using Automatic Call Distribution (ACD) systems has grown significantly in the last five years. The industry shows all the signs of continued or greater growth in the foreseeable future. While ACD systems have proliferated they have also evolved from fundamentally local to distributed systems. An ACD manager can no longer optimize his traffic by using inputs from a simple set of queueing tables. The most common system is now a distributed network where subsystems interact with each other and cannot be analyzed in isolation. This paper examines the strengths and weaknesses of queueing models that have been used historically with ACD systems and develops modifications to these models (including agents wrap-up times) that are combined with queueing network theories to construct an original ACD network performance algorithm to work with distributed systems.

Blended call center performance analysis

The performance analysis of blended PSTN and IP call centers is likely to be in demand in the near future as the technology for these centers develops further. The authors did not find an analysis for a system of this type in the literature. The development of a user-friendly and portable tool based on their analysis methodology should be useful to organizations that have implemented, or are considering implementing, a blended call center. We have shown BCATs wide range of uses. In the future, we plan to enhance BCAT to allow for skills-based routing to, for example, agents who can handle PSTN calls only, IP calls only, or both call types. This is a much more complicated queueing problem to model, but will provide increased flexibility to call center supervisors in terms of workforce management planning.

Using quantile estimates in simulating internet queues with Pareto service times

It is readily apparent how important the Internet is to modern life. The exponential growth in its use requires good tools for analyzing congestion. Much has been written recently asserting that classical queueing models assuming Poisson arrivals or exponential service cannot be used for the accurate study of congestion in major portions of the Internet. Internet traffic data indicate that heavy-tailed distributions (e.g., Pareto) serve as better models in many situations for packet service lengths. But these distributions may not possess closed-form analytic Laplace transforms; hence, much standard queueing theory cannot be used. Simulating such queues becomes essential; however, previous research pointed out difficulties in obtaining the usual moment performance measures such as mean wait in queue. We investigate the use of quantile estimates of waiting times (e.g., median instead of mean), which appear to be considerably more efficient when service times are Pareto.

An Enhanced Extension to Wilkinson's Equivalent Random Technique with Application to Traffic Engineering

In this paper we present an enhanced extension to Wilkinsons equivalent random technique that allows one to easily compute the different levels of blocking that various streams of traffic see using the same trunk group. This empirical extension is based on a regression analysis and is superior to a similar extension by Katz. An application of our extension to system sizing problems in AUTOVON is presented. Previous sizing methods in the AUTOVON access area did not properly account for the peakedness of overflow traffic and thereby violated desired grades of service. The methods presented here overcome this problem, and yield a lower cost solution using fewer two-way trunks.

Modeling the performance of low latency queueing for emergency telecommunications

Event simulation and analytic modeling are used to evaluate the performance of low latency queueing (LLQ), a queueing discipline available in some Internet packet switching routers for integrated services performance assurance. LLQ combines priority queueing with class-based weighted fair queueing (CBWFQ). Priority queueing is used to ensure satisfying tight delay constraints for real-time traffic, whereas CBWFQ is used to ensure acceptable throughput for traffic classes that are less sensitive to delay. Simulations are developed both using a commercial product, OPNET Modeler, and also custom simulators that we developed. Our custom simulators model two different approaches to CBWFQ; and comparisons between the approaches and that of the commercial simulator are conducted. Our computational experiences (central processing unit [CPU] times for model execution and postprocessing) in using the simulators are described. This work is an important first step in the ability to model a proposed enhancement to LLQ which may be beneficial to emergency telecommunications services.

Waiting-Time Distribution of M/DN/1 Queues Through Numerical Laplace Inversion

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.