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.
We present a comparison of nucleation in an isothermal-isochoric container with traffic congestion on a one-lane freeway. The analysis is based, in both cases, on the probabilistic description by stochastic master equations. Further we analyze the characteristic features of traffic breakdowns. To describe this phenomenon we apply the stochastic model regarding the jam emergence to the formation of a large car cluster on the highway.