Stochastic Modeling of Train Delays Formation

Stochastic Modeling of Train Delays Formation

Vladimir Chebotarev Boris Davydov Kseniya Kablukova

CC FEB RAS, Russia

FESTU, Russia

Available online: 
| Citation



Two stochastic models of train interaction are considered in the paper. These models are designed to study the process of calculation the arrival deviation probability density function. The stochastic models allow you to receive an adequate forecast the development of traffic situation taking random fluctuations of the train run trajectory into account. The paper proves the equivalence of two train traffic models that reflect the mechanism of formation the arrival time distribution. Both models take the scattering of train departure deviations into account. The first type model describes the result of the run time random nature, while the second model reflects the impact of short-term unplanned train stops. The study also makes an attempt to outline the regularity of formation the standard and the abnormal arrival deviation distributions. The proposed models are verified by using the results of the historical data analysis obtained at the main railway line.


arrival deviation distribution, bimodal distribution, departure time distribution, random speed variations, running time distribution, stochastic model, train traffic


[1] Carey, M. & Kwiecinski, A., Stochastic approximation to the effects of headways on knock-on delays of trains. Transportation Research Part B, 28(4), pp. 251–267, 1994.

[2] Meester, L.E. & Muns, S., Stochastic delay propagation in railway networks and phasetype distributions. Transportation Research Part B, 41, pp. 218–230, 2007.

[3] Buker, T. & Seybold, B., Stochastic modelling of delay propagation in large networks. Journal of Rail Transport Planning and Management, 2(12), pp. 34–50, 2012.

[4] Boucherie, R.J. & Huisman, T., Running times on railway sections with heterogeneous train traffic. Transportation Research Part  B: Methodological, 35(3), pp.  271–292, 2001.

[5] Berger, A., Gebhardt, A., Muller-Hannemann, M. & Ostrowski, M., Stochastic Delay Prediction in Large Train Networks. Proceedings of the 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS’11), Saarbrucken, Germany, pp. 100–111, 2011.

[6] Muhlhans, E., Berechnung der Verspatungsentwicklung bei Zugfahrten/Eisenbahntechn. Rundschau ETR, 39(7/8), pp. 465–468, 1990.

[7] Shapkin, I.N., Yusipov, R.A. & Kozhanov, E.M., Simulation of train functioning on the basis of multivariate regulation of technological operations. Bulletin VNIIZhT, 4, pp. 30–36, 2006 (in Russian).

[8] Karetnikov,  A.D. & Vorobyev,  N.A., Schedule of Train Traffic. Transport, Moscow, 301 p., 1979 (in Russian).

[9] Salido,  M.A., Robustness in Railway Transportation Scheduling. Salido,  M.A., Barber, F. & Ingolotti, L. (eds.), Proceedings of the 7th World Congress on Intelligent Control and Automation (WCICA 2008), Chongqing, China, pp. 2833–2837, 2008.

[10] Al-Ibrahim, A., Dynamic Delay Management at Railways. A Semi-Markovian Decision Approach, PhD thesis, Universiteit van Amsterdam, 335 p., 2010.

[11] Goverde, R.M.P., Punctuality of Railway Operations and Timetable Stability Analysis, PhD thesis, Technical University of Delft, 165 p., 2005.

[12] Corman, F. & Kecman, P., Stochastic prediction of train delays in real-time using Bayesian networks. Transportation Research Part C: Emerging Technologies, 95, pp. 599–615, 2018. DOI:10.1016/j.trc.2018.08.003.

[13] Chebotarev, V., Davydov, B. & Kablukova, K., Random delays forming in the dense train flow. Proceedings of the 16th International Conference on Railway Engineering Design & Operation (COMPRAIL 2018), Lisbon, Portugal, pp. 435–445, 2018.

[14] Chebotarev, V., Davydov, B. & Kablukova, K., Probabilistic Model of Delay Propagation along the Train Flow. Probabilistic Modeling in System Engineering (ed. Kostogryzov, A.), IntechOpen, pp. 171–193, 2018. DOI: 10.5772/intechopen.75494.