Denise M. Bevilacqua Masi

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.

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.

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.

Simulating network cyber attacks using splitting techniques

As a result of potential damage to our national infrastructure due to cyber attacks, a number of cyber-security bills have been introduced in Congress and a National Strategy for Trusted Identities in Cyber-space has been developed by the White House; a component of this strategy is the development of models to assess risks due to cyber incidents. A worm attack on a network is one type of attack that is possible. The simulation of rare events, such as the occurrence of a catastrophic worm attack, is impractical without special simulation techniques. In this paper we present an application of splitting methods to estimate rare-event probabilities associated with the propagation of a worm through a network. We explore the sensitivity of the benefits of splitting methods, as compared to standard simulation, to the rarity of the event and the level function used.

Simulating non-stationary congestion systems using splitting with applications to cyber security

According to the former counterterrorism czar, Richard A. Clarke (2010), our national infrastructure could be severely damaged in 15 minutes by a cyber attack. A worm attack on an Internet Protocol (IP) network is one type of attack that is possible. Such an attack would result in a non-stationary arrival process of packets on a link in the network. In this paper we present an initial use of our Optimal Splitting Technique for Rare Events (OSTRE) to simulate the congestion imposed by the worm on the link. This initial application is oriented to testing the technique in this dynamic environment and report on its use as compared with conventional simulations.

Simulating the performance of a class-based weighted fair queueing system

Class Based Weighted Fair Queueing (CBWFQ) is a very important router discipline that allows different types of Internet Protocol (IP) traffic like voice, video, and best effort data to receive the required quality of service measures they individually need. CBWFQ dynamically allocates the available bandwidth to each traffic class based on the class¿s weight. This discipline is playing a vital role as IP brings these traffic classes together in a truly converged network. Under stress and in extreme emergencies, it is critical to be able to determine how the CBWFQ discipline will perform. In this paper, we present and discuss the critical role simulation has played in our development of performance analysis tools for the CBWFQ discipline.

Analyzing internet packet traces using Lindley's Recursion

Internet trace packet data for a given network link contains information on each packets arrival time and size. An important problem is to model the congestion packets experienced over the collection period. Recent research has utilized a relationship from queueing theory known as Lindleys recursion to model packet congestion. This relationship has existed for 50 years and has been quite beneficial in analyzing these traces. We report on our use of Lindleys recursion to analyze publicly-available link data from the Abilene network, an Internet2 backbone network. We extend the use of Lindleys Recursion and include a discussion of the computational problems, numerical evaluation of trace packet performance and potential modeling issues, and a statistical investigation of the independence of packet interarrival times. In addition, we show how Lindleys recursion can be used to extend the baseline analysis to interject Voice over Internet Protocol (VoIP) packets into the trace

Computation of steady-state probabilities for resource-sharing call-center queueing systems

Two routing rules for a queueing system of two stations are considered as alternative models for modeling a call-center network. These routing rules allow customers to switch queues under certain server and other resource availability conditions, either external to the system upon arrival to the network, or internal to the system after arrival to a primary call center. Under the assumption of Poisson arrivals and exponentially distributed service times, these systems are analyzed using matrix-geometric techniques, yielding a non-trivial set of ergodicity conditions and the steady-state joint probability distribution for the number of customers at each station. An extensive numerical analysis is conducted, yielding some physical insight into these systems and related generalizations.

Approximating Low Latency Queueing Buffer Latency

Low latency queueing (LLQ) is an internet protocol (IP) router discipline that is being used to ensure that performance-sensitive high priority traffic, such as voice and video, receive their high level of performance, while allowing less performance-sensitive traffic, such as e-mail or best-effort IP, to receive some portion of the bandwidth. In this paper, we develop a simple analytic approximation for the buffer latency (expected buffer delay) for each traffic class using the LLQ system. The approximation is validated via a simulation model.

Numerical Methods for Analyzing Queues with Heavy-Tailed Distributions

In many queues associated with data traffic (for example, a buffer at a router), arrival and service distributions are heavy-tailed. A difficulty with analyzing these queues is that heavy-tailed distributions do not generally have closed-form Laplace transforms. A recently proposed method, the Transform Approximation Method (TAM), overcomes this by numerically approximating the transform. This paper investigates numerical issues of implementing the method for simple queueing systems. In particular, we argue that TAM can be used in conjunction with the Fourier-series method for inverting Laplace transforms, even though TAM is a discrete approximation and the Fourier method requires a continuous distribution. We give some numerical examples for an M/G/1 priority queue.