1. Introduction

The trains' movement is subject to a variety of random factors which leads to unplanned delays. This causes the scattering of the arrival times, hence, the inconvenience to passengers and consignees. Knowledge of the arrival times' distribution properties leads to the possibility of predicting the characteristics of the train traffic and making correct decisions on the transportation process management. This makes it possible to improve the punctuality of train traffic and save resources, in particular, electric power.

The properties of the arrival headways distributions allow us to estimate the probability of delays emergence and theirs characteristics, which are important from a practical point of

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view. Probabilistic modeling of the delay propagation process along the train flow is the main tool for solving this problem.

The models for the distribution of delays in a dense train flow are divided into two classes. These are deterministic and stochastic models. Stochastic models take into account the unpredictable nature of obstacles in the railway. A mathematical model, proposed in the present chapter, make it possible to determine the probability distributions of the arrival headways of two consecutive trains at the station. The distribution properties are analyzed for different scattering of input random variables (the primary delay and the initial headways). Comparison of theoretical distributions with real statistics of train traffic on the Russian railways is performed.

Let us also introduce the notations: μ<sup>j</sup> ¼ Xj=v0, t<sup>0</sup> ¼ s0=v0. Suppose that train 1 departs from station <sup>A</sup> at the time <sup>t</sup> <sup>¼</sup> 0. Then, the moment <sup>T</sup>ð Þ <sup>m</sup> of departure train <sup>m</sup> can be found as (as

Assume that at some point in time, train 1 makes unplanned stop. The duration of this stop is random value τ. The subsequent trains suffer knock-on delays, when the value τ is large enough. Following train stops when the distance to the front train is reduced to s0. It is assumed that as soon as the front train restore running, then the next one immediately follows it. The following problem is considered: to find out the probability distribution of the random arrival headway between the trains (k � 1) and k at the destination B (denote this headway as νk), assume that only the first train makes an unplanned stop. In other words, we need to find the (cumulative) distribution functions WkðÞ¼ t Ρð Þ ν<sup>k</sup> < t , k = 2, 3, …, n. Call this problem by the first

Suppose that train 1 was delayed at station A at the moment t ¼ 0 and waited for a random time <sup>τ</sup>. If <sup>τ</sup> <sup>&</sup>lt; <sup>μ</sup>2, then trains 2, 3, and so on, depart at the planned times: <sup>T</sup>ð Þ<sup>2</sup> , <sup>T</sup>ð Þ<sup>3</sup> , etc. If <sup>τ</sup> <sup>&</sup>gt; <sup>μ</sup>2, then train 2 will be delayed and will depart at the time <sup>τ</sup> <sup>þ</sup> <sup>t</sup><sup>0</sup> <sup>&</sup>gt; <sup>T</sup>ð Þ<sup>2</sup> : Train 3 departs according to the same rule depending on the delay time of train 2, and so on. In this formulation, ν<sup>k</sup> is actual departure headway between the trains with numbers (k � 1) and k. It is required to

Example 1. Let n = 5, μ<sup>k</sup> ¼ 2, k ¼ 2, 5, t<sup>0</sup> ¼ 1. The moments of planned departures of trains satisfy the equalities <sup>T</sup>ð Þ<sup>k</sup> <sup>¼</sup> <sup>3</sup>ð Þ <sup>k</sup> � <sup>1</sup> , <sup>k</sup> <sup>¼</sup> <sup>1</sup>, 5. Figure 2 shows the process of headways <sup>ν</sup><sup>k</sup> forming, k ¼ 2, 5, depending on the six values of the interval τ. The dots represent real train

The basic model assumptions are follows: (1) only train 1 is exposed to primary delay τ. (2)

Denote by Rð Þ<sup>k</sup> the real departure time of the train with number k, which depends on τ and t0.

determine the distribution functions Wkð Þt of random variables νk, k = 2, 3, …, n.

departure times that result from the primary delay τ.

<sup>T</sup>ð Þ<sup>k</sup> � <sup>T</sup>ð Þ <sup>k</sup>�<sup>1</sup> <sup>&</sup>gt; <sup>t</sup>0, <sup>k</sup> = 2, 3, …, <sup>n</sup>.

μ<sup>j</sup> þ ð Þ m � 1 t0, m ¼ 2, 3, …, n (1)

Probabilistic Model of Delay Propagation along the Train Flow

http://dx.doi.org/10.5772/intechopen.75494

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<sup>T</sup>ð Þ <sup>m</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup>

Figure 1. Departure times of trains 1, 2, and 3 from station A.

j¼2

shown at Figure 1):

problem.

3.2. The second model
