William G. Marchal

Bio: William G. Marchal is an academic researcher from College of Business Administration. The author has contributed to research in topics: Queueing theory & Markov chain. The author has an hindex of 3, co-authored 7 publications receiving 159 citations.

Robustness of Rootfinding in Single-Server Queueing Models

Abstract: There has been frequent controversy over the years regarding the use of numerical rootfinding for the solution of queueing problems. It has been said that such problems quite often present computational difficulties. However, it turns out that rootfinding in queueing is so well structured that problems rarely occur. There are fundamental properties possessed by the well-known queueing models that eliminate classical rootfinding problems. Most importantly, we show that distinctness of roots is common within simply determined regions in the complex plane and provide conditions under which the characteristic equations for the G/EK/1 and EK/G/1 models have easily found, distinct roots. Furthermore, we show that the characteristic equation for the more general G/GEK/1 model has a collection of real and complex roots which are effectively distinct and located in clearly defined regions of the complex domain. Extensive computational results are given to support our contentions. INFORMS Journal on Computing, ISSN...

An Approximate Formula for Waiting Time in Single Server Queues

Abstract: This note reports the development of a new approximation formula for the mean wait in queue of a single-server queue. Use of the formula requires knowledge of only the first two (2) moments of the interarrival and service time distributions. Comparison of the proposed approximation to existing techniques is included.

Distribution Estimation Using Laplace Transforms

Abstract: We propose two related methods for deriving probability distribution estimates using approximate rational Laplace transform representations Whatever method is used, the result is a Coxian estimate for an arbitrary distribution form or plain sample data, with the algebra of the Coxian often simplifying to a generalized hyperexponentia l or phase-type The transform (or, alternatively, the moment-generating function) is used to facilitate the computations and leads to an attractive algorithm For method one, the first 2N - 1 derivatives of the transform are matched with those of an approximate rational function; for the second method, a like number of values of the transform are matched with those of the approximation The numerical process in both cases begins with an empirical Laplace transform or truncation of the actual transform, and then requires only the solution of a relatively small system of linear equations, followed by root finding for a low-degree polynomial Besides the computationally attractive features of the overall procedure, it addresses the question of the number of terms, or the order, involved in a generalized hyperexponential, phase-type, or Coxian distribution, a problem not adequately treated by existing methods Coxian distributions are commonly used in the modeling of single-stage and network queueing problems, inventory theory, and reliability analyses They are particularly handy in the development of large-scale model approximations

Computing optimal server-vacation policies

Abstract: The server-vacation model describes many situations in complex manufacturing or communications environments in which a central “server” (such as a robot, materials handling system, or switch) must “service” several different classes of “customers.” The customers compete with each other for the attention of the server, and while the server is handling one class, it appears to be on “vacation” to the others. Management frequently can set operating policies by dynamically changing priorities among the waiting customers to determine who gets served next under different circumstances. This paper describes how to apply the Poisson input, single-server vacation model to derive optimal operating or control policies in these situations. A principal observation is that the determination of an optimal policy can be formulated as a linear program. Hence optimal policies will generally be of the simple switchpoint or retarded switchpoint (hysteretic) types. Moreover, since linear programming codes are widely ...

An Algorithm for Finding Characteristic Roots of Quasi-Triangular Markov Chains

Abstract: : Numerical root finding problems are quite common in stochastic modeling. However, many solutions stop at the presentation of a probability generating function for the state probabilities. But with increasing easy access to computing power, many problems whose answers were typically left in incomplete form or for which there has been a search for alternative solution methods are currently being reexamined. The class of Markov chains whose transition matrices have quasi-triangular layouts (i.e., those having sub- or super-triangular sets of zeros) is a good case in point. They have an especially nice structure which leads to a rather concise representation for the generating functions. But the complete solution then requires the finding of roots. Fortunately, these problems can be shown to have special properties that make accurate root finding quite feasible, and we thus supply an efficient numerical procedure for solution. Keywords: Applied probability; Computational analysis; Computational probability; Markov chains; Numerical methods; Probability; Queues; Stochastic models. (jhd)

Fundamentals of Queueing Theory

Abstract: Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

Design and Control of Efficient Order Picking Processes

Abstract: textBinnen een logistieke keten dienen producten fysiek te worden verplaatst van de ene locatie naar de andere, van producenten naar eindgebruikers Tijdens dit proces worden producten gewoonlijk opgeslagen op bepaalde plaatsen (magazijnen) voor een bepaalde periode Orderverzameling – het ophalen van producten uit de opslaglocatie in het magazijn naar aanleiding van een specifieke klantorder – is het meest kritieke magazijnproces Het is een arbeidsintensieve operatie in handmatig bestuurde systemen, en een kapitaalintensieve operatie in geautomatiseerde systemen Een niet optimaal functionerend orderverzamelingsproces kan leiden tot onbevredigende service en hoge operationele kosten voor het magazijn, en dientengevolge voor de hele keten Om efficient te kunnen functioneren dient het orderverzamelingsproces robuust te zijn ontworpen en optimaal te worden bestuurd Dit proefschrift heeft als doel analytische modellen te ontwerpen die het ontwerp en de besturing van efficiente orderverzamelingsprocessen ondersteunen Verschillende methoden worden voorgesteld voor het schatten van de route langs de locaties van de te verzamelen producten, het bepalen van de optimale grenzen van zones in het magazijn die bestemd zijn voor opslag, de indeling van het magazijn, het aantal producten die tegelijk (in een ronde) worden verzameld (de batch size) en het aantal zones in het magazijn die worden ingericht voor het verzamelen en gereedmaken van orders De methoden worden getest middels simulatie experimenten en worden inzichtelijk gemaakt met behulp van rekenexperimenten

Spectral analysis of M/G/ 1 and G / M /1 type Markov chains

Abstract: When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

Travel Time Estimation and Order Barching in a 2-Block Warehouse

Abstract: textThe order batching problem (OBP) is the problem of determining the number of orders to be picked together in one picking tour. Although various objectives may arise in practice, minimizing the average throughput time of a random order is a common concern. In this paper, we consider the OBP for a 2-block rectangular warehouse with the assumptions that orders arrive according to a Poisson process and the method used for routing the order-pickers is the well-known S-shape heuristic. We first elaborate on the first and second moment of the order-picker's travel time. Then we use these moments to estimate the average throughput time of a random order. This enables us to estimate the optimal picking batch size. Results from simulation show that the method provides a high accuracy level. Furthermore, the method is rather simple and can be easily applied in practice.