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.

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The article explores the problem of train rescheduling based on the actual situation. The proposed stochastic model uses specific distributions of operating times which are dependent on the current traffic conditions. The arrival time distribution is considered as a result of adjusting the train trajectory by speed control. The results of modelled arrival distributions correspond well with the experimental data received at the russian railways. The proposed model is used for prevention of sequence-of-trains conflicts and violations of connections. The basis of deviation prediction is two-train model of mesalevel which uses actual features of scattering of the operation times both at sites and at stations. The article also proposed a new measure of arrival delay which considers the share of satisfied passengers.

The schedule of the double-track railway line with mixed traffic has two sections, such as passenger and freight segments. Punctuality is a measure of the quality of the passenger traffic. Local traffic punctuality in freight train traffic is an insignificant indicator. The level of the railway operator's expenditure is a more important criterion when the flow of freight trains has passed. We propose a new approach to the problem of operational management, which is based on the direct economic assessment of the dispatcher adjustments.The method of the operative rescheduling of freight train flow prescribes that the intraday period is divided into the economic and the intensive segments. The criterion for traffic separation into segments is dependent on the level of knock-on delays, respectively, the amount of energy and additional expenditure losses. Cost-effective technology of the local management is used when the number of freight trains is reduced.In this paper, we consider the total number of secondary unscheduled stops as a measure of the quality of the freight trains flow. This criterion is used together with a specialized stochastic model. The model is based on the composition of the probability distributions of departure times from the original station and the duration of the primary train delays.

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