Probabilistic Model of Delay Propagation along the Train Flow

Abstract

In this chapter, we propose a probabilistic model for train delay propagation. There are deduced formulas for the probability distributions of arrival headways and knock-on delays depending on distributions of the primary delay duration and the departure headways. We prove some key mathematical statements. The obtained formulas allow to predict the frequency of train arrival delays and to determine the optimal traffic adjustments. Several important special cases of initial probability distributions are considered. Results of the theoretical analysis are verified by comparison with statistical data on the train traffic at the Russian railways.

Stochastic modeling of train delays formation

“…where α > 0, β > 0, b 1 > 0. the histogram of the travel time given in table 2 (column 1) is fairly well approximated by shifted gamma distribution (14). note that for the convenience of displaying in travel time histogram, the vertical axis intersects the horizontal one not at the origin, but at the point (15, 0).…”

“…. by the previous study [14], the resulting cumulative distribution function of delay τ 2 has the following form:…”

“…Studies show that in the case of relatively free traffic (e.g., daytime hours), the graph of the probability density function of the travel time has a longer right branch. it is shown in [14] that such distribution is well approximated by the gamma distribution, i.e.…”

“…where l 1 (t) is determined the same as l (t) in (14), l 2 (t) = l 1 (t-b 2 ) with b 2 ˃ 0. bimodality of the density l (t) is determined by the influence of two random factors (e.g., weather conditions and speed mode). to simulate the degree of influence of each of them, the parameter P is used.…”