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William A. Massey

Other affiliations: New York University, Bell Labs, AT&T  ...read more
Bio: William A. Massey is an academic researcher from Princeton University. The author has contributed to research in topics: Queueing theory & Queue. The author has an hindex of 29, co-authored 69 publications receiving 3651 citations. Previous affiliations of William A. Massey include New York University & Bell Labs.


Papers
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Journal ArticleDOI
TL;DR: This work develops limit theorems for a large class of stochastic service network models where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system.
Abstract: Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M_t/M_t/n_t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative.

279 citations

Journal ArticleDOI
TL;DR: It is significant that the well known insensitivity property of the stationary M/G/∞ model does not hold for the nonstationary Mt/G/, and the time-dependent mean function m depends on the service-time distribution beyond its mean.
Abstract: We establish some general structural results and derive some simple formulas describing the time-dependent performance of the Mt/G/∞ queue (with a nonhomogeneous Poisson arrival process). We know that, for appropriate initial conditions, the number of busy servers at time t has a Poisson distribution for each t. Our results show how the time-dependent mean function m depends on the time-dependent arrival-rate function λ and the service-time distribution. For example, when λ is quadratic, the mean m(t) coincides with the pointwise stationary approximation λ(t)E[S], where S is a service time, except for a time lag and a space shift. It is significant that the well known insensitivity property of the stationary M/G/∞ model does not hold for the nonstationary Mt/G/∞ model; the time-dependent mean function m depends on the service-time distribution beyond its mean. The service-time stationary-excess distribution plays an important role. When λ is decreasing before time t, m(t) is increasing in the service-time...

252 citations

Journal ArticleDOI
TL;DR: An approximate procedure based on a time-dependent normal distribution, where the mean and variance are determined by infinite-server approximations is developed, which is effective by making comparisons with the exact numerical solution of the Markovian M t /M/s t model.
Abstract: We consider a multiserver service system with general nonstationary arrival and service-time processes in which s(t), the number of servers as a function of time, needs to be selected to meet projected loads. We try to choose s(t) so that the probability of a delay (before beginning service) hits or falls just below a target probability at all times. We develop an approximate procedure based on a time-dependent normal distribution, where the mean and variance are determined by infinite-server approximations. We demonstrate that this approximation is effective by making comparisons with the exact numerical solution of the Markovian Mt/M/st model.

231 citations

Journal ArticleDOI
William A. Massey1, Ward Whitt1
TL;DR: A more general Poisson-arrival-location model (PALM) is introduced in which arrivals move independently through a general state space according to a location stochastic process after arrivingaccording to a nonhomogeneous Poisson process.
Abstract: In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.

211 citations

01 Jan 2005
TL;DR: In this article, a flexible simulation-based iterative-staffing algorithm (ISA) is proposed for the Mt/G/st + G model with nonhomogeneous Poisson arrival process (the Mt) and customer abandonment (the +G).
Abstract: This is a longer version of a paper with the same title, which has been submitted to Management Science. Abstract from the Journal Versionfrom the Journal Version This paper develops methods to determine appropriate staffing levels in call centers and other many-server queueing systems with time-varying arrival rates. The goal is to achieve targeted time-stable performance, even in the presence of significant time-variation in the arrival rates. The main contribution is a flexible simulation-based iterative-staffing algorithm (ISA) for the Mt/G/st + G model with nonhomogeneous Poisson arrival process (the Mt) and customer abandonment (the +G). For Markovian Mt/M/st + M special cases, the ISA is shown to converge. For that Mt/M/st+M model, simulation experiments show that the ISA yields timestable delay probabilities across a wide range of target delay probabilities. With ISA, other performance measures such as agent utilizations, abandonment probabilities and average waiting times are stable as well. The ISA staffing and performance agree closely with the modified-offered-load (MOL) approximation, which was previously shown to be an effective staffing algorithm without customer abandonment. While the ISA algorithm so far has only been extensively tested for Mt/M/st + M models, it can be applied much more generally, to Mt/G/st + G models and beyond. What is Contained Here? This longer version presents more examples; e.g., it treats the Mt/M/st model (without customer abandonment) and the Mt/M/st + M model with θ > μ and θ < μ, where θ is the abandonment rate and μ is the service rate. It treats the challenging example from Jennings et al. (1996). There is extra detail for the previous examples; there are 47 figures here, but only 10 in the journal version. This longer version also provides additional theoretical support.

179 citations


Cited by
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Journal ArticleDOI
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

Journal ArticleDOI
01 May 1975
TL;DR: The Fundamentals of Queueing Theory, Fourth Edition as discussed by the authors provides a comprehensive overview of simple and more advanced queuing models, with a self-contained presentation of key concepts and formulae.
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.

2,562 citations

01 Jan 1978
TL;DR: This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.), and is a "must-have" reference for every serious programmer's digital library.
Abstract: This ebook is the first authorized digital version of Kernighan and Ritchie's 1988 classic, The C Programming Language (2nd Ed.). One of the best-selling programming books published in the last fifty years, "K&R" has been called everything from the "bible" to "a landmark in computer science" and it has influenced generations of programmers. Available now for all leading ebook platforms, this concise and beautifully written text is a "must-have" reference for every serious programmers digital library. As modestly described by the authors in the Preface to the First Edition, this "is not an introductory programming manual; it assumes some familiarity with basic programming concepts like variables, assignment statements, loops, and functions. Nonetheless, a novice programmer should be able to read along and pick up the language, although access to a more knowledgeable colleague will help."

2,120 citations

Journal ArticleDOI
TL;DR: For wireless cellular communication systems, one seeks a simple effective means of power control of signals associated with randomly dispersed users that are reusing a single channel in different cells, and the authors demonstrate exponentially fast convergence to these settings whenever power settings exist for which all users meet the rho requirement.
Abstract: For wireless cellular communication systems, one seeks a simple effective means of power control of signals associated with randomly dispersed users that are reusing a single channel in different cells. By effecting the lowest interference environment, in meeting a required minimum signal-to-interference ratio of rho per user, channel reuse is maximized. Distributed procedures for doing this are of special interest, since the centrally administered alternative requires added infrastructure, latency, and network vulnerability. Successful distributed powering entails guiding the evolution of the transmitted power level of each of the signals, using only focal measurements, so that eventually all users meet the rho requirement. The local per channel power measurements include that of the intended signal as well as the undesired interference from other users (plus receiver noise). For a certain simple distributed type of algorithm, whenever power settings exist for which all users meet the rho requirement, the authors demonstrate exponentially fast convergence to these settings. >

1,831 citations