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

Stochastic Models, an Algorithmic Approach , by Henk C. Tijms (Chichester: Wiley, 1994), 375 pages, paperback.

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate l (>0) all demand the first ‘essential’ service, whereas only some of them demand the second ‘optional’ service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/m_2 (m_2>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/E_{k}/1} and M/M/1 have been derived as particular cases.

An M/G/1 queue with second optional service

We have divided this review into two parts. The first part is concerned with the optimal design of queueing systems and the second part deals with the optimal control of queueing systems. The second part, which has the lion’s share of the review since it has received the most attention, focuses mainly on the modelling aspects of the problem and describes the different kinds of threshold (control) policy models available in the literature. To limit the scope of this survey, we decided to limit ourselves to research on papers dealing with the three policies (N, T, and D), where a cost function is designed specifically and optimal thresholds that yield minimum cost are sought.

Optimal design and control of queues

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.

https://repub.eur.nl/pub/1810/ERS 2004 098 LIS.pdf

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

A cyclic queueing system with three servers and optional two-way feedback

A bulk queueing system with random failures and two phase repairs

On M/D/1 queue with deterministic server vacations

We analyze a single server queue with modified Bernoulli server vacations based on exhaustive service. Unlike other vacation policies, we assume that only at the completion of service of the last customer in the system, the server has the option to take a vacation or to remain idle in the system waiting for the next customer to arrive. The service times of the customers have been assumed to be exponential and vacations are phase type exponential. We obtain explicit steady state results for the probability generating functions of the queue length, the expected number of customers in the queue and the expected waiting time of the customer. Some earlier known results of the single server have been derived as a particular case.

Transient and steady state solution of a single server queue with modified bernoulli schedule vacations based on exhaustive service and a single vacation policy

We study two models of a single server queue with vacations. In model A, we study an M ⁄ M ⁄1 queue with deterministic vacations and designate it as M⁄M⁄D⁄l queue and in model B we study an M⁄D⁄1 queue with exponential vacations and designate it as M⁄D⁄M⁄1 queue. We obtain time dependent as well as steady state probability generating functions for both the models. For the steady state, we obtain explicit results for the mean number in the system and the mean waiting times for both models. All known results of the M ⁄ M ⁄1 and the M⁄D⁄1 queues are derived as special cases.

Kailash C. Madan

Papers

A cyclic queueing system with three servers and optional two-way feedback

A bulk queueing system with random failures and two phase repairs

On M/D/1 queue with deterministic server vacations

Transient and steady state solution of a single server queue with modified bernoulli schedule vacations based on exhaustive service and a single vacation policy

On Single Server Vacation Queues with Deterministic Service or Deterministic Vacations