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

Human mobility: Models and applications

With growing numbers of intelligent autonomous systems in human environments, the ability of such systems to perceive, understand, and anticipate human behavior becomes increasingly important. Spec...

Human motion trajectory prediction: a survey:

A Charter of Fundamental Rights for the European Union

The study of crowd dynamics is interesting because of the various self-organization phenomena resulting from the interactions of many pedestrians, which may improve or obstruct their flow. Besides formation of lanes of uniform walking direction and oscillations at bottlenecks at moderate densities, it was recently discovered that stop-and-go waves [D. Helbing et al., Phys. Rev. Lett. 97, 168001 (2006)] and a phenomenon called "crowd turbulence" can occur at high pedestrian densities [D. Helbing et al., Phys. Rev. E 75, 046109 (2007)]. Although the behavior of pedestrian crowds under extreme conditions is decisive for the safety of crowds during the access to or egress from mass events as well as for situations of emergency evacuation, there is still a lack of empirical studies of extreme crowding. Therefore, this paper discusses how one may study high-density conditions based on suitable video data. This is illustrated at the example of pilgrim flows entering the previous Jamarat Bridge in Mina, 5 kilometers from the Holy Mosque in Makkah, Saudi-Arabia. Our results reveal previously unexpected pattern formation phenomena and show that the average individual speed does not go to zero even at local densities of 10 persons per square meter. Since the maximum density and flow are different from measurements in other countries, this has implications for the capacity assessment and dimensioning of facilities for mass events. When conditions become congested, the flow drops significantly, which can cause stop-and-go waves and a further increase of the density until critical crowd conditions are reached. Then, "crowd turbulence" sets in, which may trigger crowd disasters.

From Crowd Dynamics to Crowd Safety: A Video-Based Analysis

Is there a way to describe pedestrian movement with simple rules, as in a cellular automaton, but without being restricted to a cellular grid? Inspired by the natural stepwise movement of humans, we develop a model that uses local discretization on a circle around virtual pedestrians. This allows for movement in arbitrary directions, only limited by the chosen optimization algorithm and numerical resolution. The radii of the circles correspond to the step lengths of pedestrians and thus are model parameters, which must be derived from empirical observation. Therefore, we conducted a controlled experiment, collected empirical data for step lengths in relation with different speeds, and used the findings in our model. We complement the model with a simple calibration algorithm that allows reproducing known density-velocity relations, which constitutes a proof of concept. Further validation of the model is achieved by reenacting an evacuation scenario from experimental research. The simulated egress times match the values reported for the experiment very well. A new normalized measure for space occupancy serves to visualize the results.

Natural discretization of pedestrian movement in continuous space.

Inspired by doubts from social scientists on the validity of computer models that see a crowd as a pure aggregation of individuals, we develop a mathematical model for group formation within crowds. It is based on a few simple characteristics. Most importantly, small groups stick together as they thread their way through a crowd. Additionally, groups have a tendency to walk abreast to ease communication. Through simulation, we establish that the occurrence of groups significantly impacts crowd movement, namely evacuation times. Further, we complement and validate the simulations by a small experiment: a classroom egress. The simulation results match the measurements qualitatively. We get a good quantitative match after calibrating the supposed desire to communicate while walking—and hence to walk abreast. We conclude that it is one of the crucial parameters to calibrate the group model against reality. While working on a mathematically complete model, a new gap between the mathematical modelling and the s...

On modelling the influence of group formations in a crowd

Pedestrians adjust both speed and stride length when they navigate difficult situations such as tight corners or dense crowds They try to avoid collisions and to preserve their personal space State-of-the-art pedestrian motion models automatically reduce speed in dense crowds simply because there is no space where the pedestrians could go The stride length and its correct adaptation, however, are rarely considered This leads to artefacts that impact macroscopic observation parameters such as densities in front of bottlenecks and, through this, flow Hence modelling stride adaptation is important to increase the predictive power of pedestrian models To achieve this we reformulate the problem as an optimisation problem on a disk around the pedestrian Each pedestrian seeks the position that is most attractive in a sense of balanced goals between the search for targets, the need for individual space and the need to keep a distance from obstacles The need for space is modelled according to findings from psychology defining zones around a person that, when invaded, cause unease The result is a fully automatic adjustment that allows calibration through meaningful social parameters and that gives visually natural results with an excellent fit to measured experimental data

Dynamic stride length adaptation according to utility and personal space

We present a microscopic ordinary differential equation (ODE)-based model for pedestrian dynamics: the gradient navigation model. The model uses a superposition of gradients of distance functions to directly change the direction of the velocity vector. The velocity is then integrated to obtain the location. The approach differs fundamentally from force-based models needing only three equations to derive the ODE system, as opposed to four in, e.g., the social force model. Also, as a result, pedestrians are no longer subject to inertia. Several other advantages ensue: Model-induced oscillations are avoided completely since no actual forces are present. The derivatives in the equations of motion are smooth and therefore allow the use of fast and accurate high-order numerical integrators. At the same time, the existence and uniqueness of the solution to the ODE system follow almost directly from the smoothness properties. In addition, we introduce a method to calibrate parameters by theoretical arguments based on empirically validated assumptions rather than by numerical tests. These parameters, combined with the accurate integration, yield simulation results with no collisions of pedestrians. Several empirically observed system phenomena emerge without the need to recalibrate the parameter set for each scenario: obstacle avoidance, lane formation, stop-and-go waves, and congestion at bottlenecks. The density evolution in the latter is shown to be quantitatively close to controlled experiments. Likewise, we observe a dependence of the crowd velocity on the local density that compares well with benchmark fundamental diagrams.

Gerta Köster

Papers

Natural discretization of pedestrian movement in continuous space.

On modelling the influence of group formations in a crowd

Dynamic stride length adaptation according to utility and personal space

Gradient navigation model for pedestrian dynamics.

Towards automatic and robust adjustment of human behavioral parameters in a pedestrian stream model to measured data