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.

Fundamentals of Queueing Theory

Real and Complex Analysis

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

An Introduction to Probability Theory and its Applications, Volume I

/pdf/reversibility-and-stochastic-networks-17kl9p5ino.pdf

Reversibility and Stochastic Networks

https://pure.tue.nl/ws/files/2342512/590819.pdf

On the application of Rouché's theorem in queueing theory

We apply a new corrected diffusion approximation for the Erlang C formula to determine staffing levels in cost minimization and constraint satisfaction problems. These problems are motivated by large customer contact centers that are modeled as an M/M/s queue with s the number of servers or agents. The proposed staffing levels are refinements of the celebrated square-root safety-staffing rule and have the appealing property that they are as simple as the conventional square-root safety-staffing rule. In addition, we provide theoretical support for the empirical fact that square-root safety-staffing works well for moderate-sized systems.

/pdf/refining-square-root-safety-staffing-by-expanding-erlang-c-dfo7g75lu4.pdf

Refining Square-Root Safety Staffing by Expanding Erlang C

In commonly used root-finding approaches for the discrete-time bulk service queue, the stationary queue length distribution follows from the roots inside or outside the unit circle of a characteristic equation. We present analytic representations of these roots in the form of sample values of periodic functions with analytically given Fourier series coefficients, making these approaches more transparent and explicit. The resulting computational scheme is easy to implement and numerically stable. We also discuss a method to determine the roots by applying successive substitutions to a fixed point equation. We outline under which conditions this method works, and compare these conditions with those needed for the Fourier series representation. Finally, we present a solution for the stationary queue length distribution that does not depend on roots. This solution is explicit and well-suited for determining tail probabilities up to a high accuracy, as demonstrated by some numerical examples.

https://www.univie.ac.at/nuhag-php/janssen/data/p130.pdf

Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

This paper presents new Gaussian approximations for the cumulative distri- bution function P(A� ≤ s) of a Poisson random variable Awith mean �. Using an integral transformation, we first bring the Poisson distribution into quasi- Gaussian form, which permits evaluation in terms of the normal distribution function �. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(A� ≤ s). The results for P(A� ≤ s) differ from most classical results related to the central limit theorem in that the leading term �(�), with � = (s − �)/ √ �, is replaced by �(�), whereis a simple function of s that converges to � as s → ∞. Changingintoturns out to increase precision for small and moderately large values of s. The results for P(A� ≤ s) lead to similar results related to the Erlang B for- mula. The asymptotic expansion for Erlang's B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.

/pdf/gaussian-expansions-and-bounds-for-the-poisson-distribution-1mn9sc2u84.pdf

Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

We consider a system of N parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among d randomly selected servers otherwise (1≤d≤N). This load balancing scheme subsumes the so-called join-the-idle queue policy (d=1) and the celebrated join-the-shortest queue policy (d=N) as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and establish that it does not depend on the value of d, implying that assigning tasks to idle servers is sufficient for diffusion level optimality.

J.S.H. van Leeuwaarden

Papers

On the application of Rouché's theorem in queueing theory

Refining Square-Root Safety Staffing by Expanding Erlang C

Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

Universality of load balancing schemes on the diffusion scale