Statistics of extremes, traffic jams, and natural disasters

Reinhart Kühne

doi:10.1117/12.724798

15 June 2007 Statistics of extremes, traffic jams and natural disasters

Reinhart Kühne

Proceedings Volume 6601, Noise and Stochastics in Complex Systems and Finance; 66010L (2007) https://doi.org/10.1117/12.724798
Event: SPIE Fourth International Symposium on Fluctuations and Noise, 2007, Florence, Italy

Abstract

Analyzing probability distributions from water level time series and calculating the first passage time distributions gives the probability of firstly exceeding a given threshold corresponding to a flood or general disaster event. The method will be applied to the water level recordings of the Danube river from which since 100 years very accurate notes exist. The method is transferred to time series of traffic volumes interpreting traffic breakdowns as extreme events. Three different traffic situations can be distinguished: (a) Stable traffic flow where any fluctuations decay over time (b) metastable traffic flow where fluctuations neither decay nor grow and (c) unstable traffic flow where a breakdown can be expected for sure if the observation time is long enough. The traffic dynamics is translated into a first passage time distribution. This describes the distribution of time periods observing for the first time the formation of a traffic jam of a certain length or number of vehicles. The distribution contains a time lag, a maximum corresponding to a time period of a Brownian motion drift reaching the critical jam length, and a tail describing exceptional long waiting times for jam formation. The cumulative first passage time distribution can be interpreted as breakdown probability distribution. It outlines when reaching a breakdown a given probability in an assumed observation time. It leads directly to the probabilistic definition of the capacity as a traffic volume leading to an unstable traffic pattern with a given probability within a given observation time. This definition can substitute the existing definitions and opens the possibility to quantitatively describing the influence of traffic control systems on the capacity.

Citation Download Citation

Reinhart Kühne "Statistics of extremes, traffic jams and natural disasters", Proc. SPIE 6601, Noise and Stochastics in Complex Systems and Finance, 66010L (15 June 2007); https://doi.org/10.1117/12.724798

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KEYWORDS

Natural disasters

Roads

Stochastic processes

Floods

Bistability

Control systems

Lead

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