Dennis C. Dietz

Scheduling customer arrivals to a stochastic service system

An efficient algorithm for determining the optimal arrival schedule for customers in a stochastic service system is developed. All customers arrive exactly when scheduled, and service times are modeled as iid Erlang random variables. Costs are incurred at a fixed rate per unit of time each customer waits for service, and an additional cost is incurred for every unit of time the server operates beyond a scheduled closing time. The objective is to minimize total operating cost. This type of problem arises in many operational contexts including transportation, manufacturing, and appointment-based services.

Scheduling and Sequencing Arrivals to an Appointment System

The authors develop methods for optimally scheduling and sequencing customer arrivals to a single-server appointment system. Customers are characterized by probabilistic service times with distinct distributions, and the server works according to a first-come, first-served discipline. A customer may fail to show for an appointment with known probability, but all arriving customers are assumed to be punctual. Costs are incurred at a specified rate per unit time that each customer waits for service, and an additional cost is incurred for every unit of time that the server operates beyond a scheduled closing time. The objective is to minimize the combined costs of customer waiting and server overtime. Possible applications include scheduling surgeons to operating suites, scheduling military aircraft to training ranges, and scheduling service activities for telecommunication technicians.

Series-system reliability-estimation using very small binomial samples

This investigation explored the effect of incorporating prior information into series-system reliability estimates, where the inferences are made using very small sets (less than 10 observations) of binomial test-data. To capture this effect, the performance of a set of Bayes interval estimators was compared to that of a set of classical estimators over a wide range of subsystem beta prior-distribution parameters. During a Monte Carlo simulation, the Bayes estimators tended to provide shorter interval estimators when the mean of the prior system-reliability differed from the true reliability by 20 percent of less, but the classical estimators dominated when the difference was greater. Based on these results, the authors conclude that there is no clear advantage to using Bayes interval estimation for sample sizes less than 10 unless the poor mean system reliability is believed to be within 20 percent of the true system reliability. Otherwise, the Lindstrom-Madden estimator, a useful classical alternative for very small samples, should be used.

Modified goodness-of-fit tests for the inverse Gaussian distribution

Abstract Modified Kolmogorov-Smirnov (KS), Kuiper (V), Cramer-von Mises (CV), Watson (W), Anderson-Darling (AD) and sequential goodness-of-fit tests are developed for the inverse Gaussian distribution with unknown parameters. A Monte Carlo procedure is employed to generate critical values for a wide range of sample sizes and shape parameters. Power studies indicate that the W test is most effective against alternate distributions that are very similar in shape to the null inverse Gaussian distribution. Otherwise, the modified AD test generally demonstrates the highest power among single tests. To eliminate the need for extensive critical value tables, functional relationships between critical values, sample sizes, and shape parameters are reported.

Analysis of aircraft sortie generation with the use of a fork‐join queueing network model

This article presents an approximate analytical method for evaluating an aircraft sortie generation process. The process is modeled as a closed network of multiserver queues and fork-join nodes that allow concurrent service activities. The model uses a variation of mean value analysis (MVA) to capture the effect of mean service times, resource levels, and network topology on performance measures including resource utilizations and the overall sortie generation rate. The quality of the analytical approximation is demonstrated through comparison with simulation results.

Moment matching using a family of phase-type distributions

An approach to matching the first three moments is presented for distributions with positive support and coefficient of variation greater than one Necessary and sufficient bounds are derived for the phase-type distribution defined by appending a Coxian stage to an Erlang distribution. This result is an extension of Altioks use of a Coxian-2 distribution for 3-moment matching

Optimization models for ATM network planning

This article describes mathematical programming models that have been developed and employed to evaluate configuration strategies for metropolitan ATM telecommunication networks. The models determine the optimal placement of ATM switch hardware and fiber optic transport. The objective is to satisfy point-to-point traffic demand at minimum cost while ensuring specified performance under a core failure contingency. Model results suggest that a simple architecture based on two core switch locations provides a robust, near-optimal solution.

Choosing the best approach to matrix exponentiation

Abstract There is no ideal single approach to matrix exponentiation; an application may have some characteristic that enables or precludes a specific approach. Even methods that theoretically yield precise answers can produce extremely large errors when implemented in floating point arithmetic, and simply utilizing double or quadruple precision representations may not ensure accuracy. An empirical method is employed here to examine the efficacy of selected methods of matrix exponentiation for a particular application. The method centers around parametrizing a sample matrix in order to determine the effects of specific characteristics. The matrices to be exponentiated are upper triangular and stochastic. They may have nearly confluent eigenvalues, as well as widely divergent eigenvalues. Such problems are common in queueing applications using phase-type distributions. Scope and purpose The solution procedures for many modeling problems involve the exponentiation of matrices. Many competing approaches are available, each with advantages and potentially catastrophic disadvantages in different applications. This article summarizes some common procedures, and presents a comparative evaluation in a particular problem context (determination of state probabilities in a transient queueing system). Practitioners may find the evaluation to be a useful model for comparing exponentiation methods in another problem domain.

Nested Fork-Join Queueing Network Model for Analysis of Airfield Operations

This article presents a nested fork-join queueing network model of the synchronized ground processing of aircraft transiting an aire eld. The queueing network is analyzed using a decomposition algorithm that provides approximatenetworkperformancemeasuressuchasthroughputandexpectedqueuelengths.Theresultsproduced are comparable in accuracy to those produced by simulation, but are generated in much less elapsed time. Using a case study of contingency operations at a military mobility aire eld, we demonstrate the model’ s utility for rapidly developing important insights into operational performance. HE success of a modern military campaign often depends on the rapid air transport of critical resources to a distant theater of operations. To accomplish this mission, the U.S. Air Force employs a e eet of large cargo aircraft to move people and equipment through a worldwide network of aire elds. To plan and execute large deployments, commanders and mobility planners rely on sophisticated modeling and analysis methods to provide meaningful estimatesofoperational capability. Particular attentionmustbepaid to throughput capacity and resource bottlenecks at key aire elds. To gain insight into aire eld performance, transportation analysts normally study the e ow of aircraft through a series of synchronized ground processing activities using high-resolution simulation modeling techniques. In this paper, however, we approach the problem through an analytical queueing network model. Our approach is not meant to completely replace high-resolution modeling, which may be necessary for studying complex aire eld processing or resource allocation schemes. Rather, the approach is designed to provide rapid insights into the relationship between aire eld resource levels, the e ow of mobility aircraft, and aire eld throughput capacity. The model is more general than an earlier method offered by Dietz, 1 which required restrictive assumptions about service time distributions and the aircraft arrival process.

Rapid, efficient analysis of the l( n )/ C k /r/N queue, with application to decomposition of closed queuing networks

Abstract A queuing network model (QNM) of a logistics or manufacturing process could conceivably contain service stations with multiple homogeneous servers, each with a general service time distribution. The λ ( n )/ C k / r / N queue, which forms at an N -capacity, r -server station where each server has a k -phase Coxian service distribution, plays an important role in the analysis of such models. In this paper, we propose a strategy for quickly and efficiently solving for the stationary probabilities of the related continuous time Markov chain (CTMC) of this queue. The proposed solution method makes more practical the analysis of queues with a much broader range of parameters than was previously possible, thereby making it easier to use a QNM to model industrial processes. In this paper, we exploit advances in computational linear algebra and computing power to solve for the stationary probability vector of the related continuous time Markov chain (CTMC) of a λ ( n )/ C k / r / N queue. The insights gained from studying the transition rate matrix of this chain are used to develop a list of candidate solution methods, each of which is used to calculate stationary probabilities for a set of 46 representative large-dimension problems. Based on this computational experience, a preferred solution approach for this type of CTMC is proposed. The effectiveness of the solution approach is demonstrated by using Maries method to decompose four queuing networks containing stations with multiple k -phase Coxian servers.