John F. Shortle

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.

Simulating Collision Probabilities of Landing Airplanes at Nontowered Airports

Making greater use of smaller airports is one way that has been proposed to increase the capacity of the National Airspace System. A major difficulty is that many small airports do not have control towers, and thus capacity is severely limited during poor visibility. The authors consider a proposed system in which airplanes self-separate, so they are able to land at higher capacities without a control tower. Before such a system is implemented, it must first be shown to be safe. Safety is a difficult metric to measure and predict because accidents are so rare. Even computer simulation can be slow because of the long time to observe accidents. One methodology that has been successful in assessing aviation safety through simulation is TOPAZ (Traffic Organizer and Perturbation AnalyZer). In this study, the authors apply the methodology to assess the safety of the proposed nontowered system.

Transmission-Capacity Expansion for Minimizing Blackout Probabilities

The objective of this paper is to determine an optimal plan for expanding the capacity of a power grid in order to minimize the likelihood of a large cascading blackout. Capacity-expansion decisions considered in this paper include the addition of new transmission lines and the addition of capacity to existing lines. We embody these interacting considerations in a simulation optimization model, where the objective is to minimize the probability of a large blackout subject to a budget constraint. The probability of a large-scale blackout is estimated via Monte Carlo simulation of a probabilistic cascading blackout model. Because the events of interest are rare, standard simulation is often intractable from a computational perspective. We apply a variance-reduction technique within the simulation to provide results in a reasonable time frame. Numerical results are given for some small test networks including an IEEE 14-bus test network. A key conclusion is that the different expansion strategies lead to different shapes of the tails of the blackout distributions. In other words, there is a tradeoff between reducing the frequency of small-scale blackouts versus reducing the frequency of large-scale blackouts.

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.

Rare-event splitting simulation for analysis of power system blackouts

Severe blackouts due to cascading failures in electric grid are rare but catastrophic. The analysis of such blackouts may include evaluation of rare-event probabilities which is difficult to estimate. In this paper, we present an effective rare-event simulation technique to estimate the rare-event probability. We apply our technique to an IEEE-bus electric network and show that our technique can dramatically improve the simulation efficiency. We also demonstrate that we can effectively detect the most vulnerable link in the electric grid, which has the highest probability of leading to a blackout event.

Analytical Distribution of Waiting Time in the M/{iD}/1 Queue

We give an analytical formula for the steady-state distribution of queue-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.

An equivalent random method with hyper-exponential service

The Equivalent Random Method (ERM) has been widely used to predict blocking probabilities at overflow service stations. The method assumes that service times follow an exponential distribution. While this may be a reasonable assumption for voice traffic, it is not a good assumption for dial-up Internet traffic, where service times typically have a coefficient of variation (standard deviation/mean) greater than 1. In this paper, we give a modified ERM for two-term hyper-exponential service distributions. The method is based on an efficient algorithm to estimate the peakedness of the overflow process of an M/H 2 /S/S queue. Finally, we investigate the accuracy of the modified ERM using simulation and also compare systems with hyper-exponential service to systems with heavy-tailed service.

Runway landing safety analysis: a case study of Atlanta Hartsfield airport

According to historical data, aircraft are subject to a higher accident risk during the landing phase than during other flight phases. With the growth in air traffic volume evaluating safety during the landing phase is an important problem. This article presents an analysis and estimate of two safety metrics at ATL airport: probability of a simultaneous runway occupancy by two landing aircraft and probability of a collision on the runway. We begin with the first order analysis to estimate the simultaneous runway occupancy probability, based on field observations. To obtain a more accurate estimate and to evaluate the runway collision risk, we construct a stochastic model of the aircraft approaching and landing process. The result of Monte Carlo simulation gives an improved estimate for the simultaneous occupancy probability. We then numerically evaluate the runway collision risk using a generalization of the Reich collision model. Finally, we carry out sensitivity analysis to examine the impact on safety and capacity when the separation variance changes.

Optimal splitting for rare-event simulation

Simulation is a popular tool for analyzing large, complex, stochastic engineering systems. When estimating rare-event probabilities, efficiency is a big concern, since a huge number of simulation replications may be needed in order to obtain a reasonable estimate of the rare-event probability. The idea of splitting has emerged as a promising variance reduction technique. The basic idea is to create separate copies (splits) of the simulation whenever it gets close to the rare event. Some splitting methods use an equal number of splits at all levels. This can compromise the efficiency and can even increase the estimation variance. This article formulates the problem of determining the number of splits as an optimization problem that minimizes the variance of an estimator subject to a constraint on the total computing budget. An optimal solution for a certain class of problems is derived that is then extended to the problem of choosing the better of two designs, where each design is evaluated via rare-event simulation. Theoretical results for the improvements that are achievable using the methods are provided. Numerical experiments indicate that the proposed approaches are efficient and robust.