7. Comparison with statistics of real train traffic

Let us consider the following random variable: the deviation of the real moment of arrival at a certain station from the scheduled one. Denote it by ξ. Statistical analysis of data on this random variable, received from the Russian railways, has led to the conclusion that in many cases, they obey the modified exponential law with the distribution function of the form Eq. (33) with b ¼ 0: Using data on the suburban trains of the direction "Moscow-Tver" for the period: January, 11–15, February, 1–6, 2016, we obtained a sample from the distribution of ξ of the size n ¼ 50 with the sample mean 1.44 and sample variance 2.7. We tested the hypothesis that ξ obeys distribution (Eq. (33)) with λ ¼ 0:35 and a ¼ 0:64. To this end, we applied the Kolmogorov goodness-of-fit test with the significance level α ¼ 0:05 and obtained the fit between the hypothesis and the sample data (see Figure 16).

Remark 7. It should be noted that in considered example the deviation ξ is nonnegative. But in reality, it can frequently be both positive and negative. Positive values are due to arisen delay. Negative values occur due to the fact that sometimes early arrivals take place.

Figure 16. The empirical distribution function and the calculated function of the form Eq. (33).

7. Comparison with statistics of real train traffic

Figure 15. Behavior of distributions Gkð Þt (a) and densities gkð Þt (b), k ¼ 2, 3, 4 and α ¼ 3.

Figure 14. Behavior of distribution G4ð Þt when (a) α ¼ 0:5, 3, 8 and (b) β ¼ 0:1, 0:5, 1.

190 Probabilistic Modeling in System Engineering

between the hypothesis and the sample data (see Figure 16).

Let us consider the following random variable: the deviation of the real moment of arrival at a certain station from the scheduled one. Denote it by ξ. Statistical analysis of data on this random variable, received from the Russian railways, has led to the conclusion that in many cases, they obey the modified exponential law with the distribution function of the form Eq. (33) with b ¼ 0: Using data on the suburban trains of the direction "Moscow-Tver" for the period: January, 11–15, February, 1–6, 2016, we obtained a sample from the distribution of ξ of the size n ¼ 50 with the sample mean 1.44 and sample variance 2.7. We tested the hypothesis that ξ obeys distribution (Eq. (33)) with λ ¼ 0:35 and a ¼ 0:64. To this end, we applied the Kolmogorov goodness-of-fit test with the significance level α ¼ 0:05 and obtained the fit

Remark 7. It should be noted that in considered example the deviation ξ is nonnegative. But in reality, it can frequently be both positive and negative. Positive values are due to arisen delay.

Negative values occur due to the fact that sometimes early arrivals take place.

Remark 8. Although the hypothetical distribution function from Figure 16 is constructed for deviations without any details about the train number k, it is well correlated with the graph of the function G2ð Þt with α ¼ 0:5 from Figure 12.

This allows us to assume that the distribution of the deviation ξ is mainly determined by the distribution of the delay τ2.

Remark 9. It was verified that if the length of the random variables μ<sup>j</sup> have the same gamma distribution, any variation of the parameters of this distribution (α and β) has a rather small influence on behavior of output distribution (see Figures 12–15).

Remark 10. Since the primary delay has a great influence on formation of the output distribution of deviations from the schedule (τk), then a knowledge of the primary delay distribution in each particular situation allows to predict the distribution of knock-on delays.

One important practical effect of the considered model is that it enables us to estimate the standard deviation (SD) of the actual arrival delays at the destination station. As an example, we calculated this parameter for the suburban railway line. The data analyzed were collected at the Tver station in the period of January 2016 and February 2016.

Example 8. Due to statistical data, we can consider that τ has the exponential distribution with the parameter λ ¼ 0:25 (i.e., τ has the distribution function (Eq. (33)) with λ ¼ 0:25, a ¼ 1, b ¼ 0), and μ<sup>2</sup> has gamma distribution with the density function (Eq. (17)), where α ¼ 0:6, β ¼ 11:7. Using formulas (49) and (50) with k ¼ 2, we have:

$$SD^2 = \int\_{-\infty}^{\infty} \left(t - a\_2\right)^2 dG\_2(t) = \int\_{-\infty}^{\infty} t^2 dG\_2(t) - a\_2^2 = \int\_0^{\infty} t^2 g\_2(t) dt - a\_2^2 \approx 10.987.7$$

$$\text{Here } a\_2 = \int\_{-\infty}^{\infty} t \, d \, G\_2(t) = \frac{1}{\lambda} \left(\lambda \beta + 1\right)^{-0.6} \approx 1.763,\\ \int\_{-\infty}^{\infty} t^2 \, d \, G\_2(t) = \frac{2}{\lambda^2} \left(\lambda \beta + 1\right)^{-0.6} \approx 14.088,$$

$$\int\_0^{\infty} t^2 g\_2(t) \, dt - a\_2^2 = \frac{2}{\lambda^2} \left(\lambda \beta + 1\right)^{-0.6} - \frac{1}{\lambda^2} \left(\lambda \beta + 1\right)^{-1.2} \approx 10.987.$$

Thus, theoretical SD ≈ ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>10</sup>:<sup>987</sup> <sup>p</sup> <sup>≈</sup> <sup>3</sup>:315 min. This corresponds with the real statistics which shows the SD amount is 3.32 min for the mentioned station.

[5] Yuan J. Stochastic modeling of train delays and delay propagation in stations [thesis]. The

Probabilistic Model of Delay Propagation along the Train Flow

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

193

[6] Meester LE, Muns S. Stochastic delay propagation in railway networks and phase-type

[7] Berger A, Gebhardt A, Müller-Hannemann M, Ostrowski M. Stochastic delay prediction in large train networks. In: Proceedings of the 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'11); 8 September 2011;

[8] Büker T, Seybold B. Stochastic modelling of delay propagation in large networks. Journal

[9] Yuan J. Statistical analysis of train delays at The Hague HS. In: Hansen IA, editor. Train Delays at Stations and Network Stability (Workshop). Delft, The Netherlands: TRAIL;

[10] Yuan J, Goverde RMP, Hansen IA. Propagation of train delays in stations. In: Allan J, Hill RJ, Brebbia CA, Sciutto G, Sone S, editors. Computers in Railways VIII. WIT Press; 2002.

[11] Alexandrova NB. Distribution of the train delays duration of due to station malfunctions. In: Proceedings of the Regional Conference "Universities of Siberia and Far East for

[12] Fikhtengolts GM. Course of Differential and Integral Calculus. 8th ed. Vol. 3. M.:

[13] Davydov B, Dynkin B, Chebotarev V. Optimal rescheduling for the mixed passenger and freight line. In: Proceedings of the 14th International Conference on Railway Engineering

Design and Optimization (COMPRAIL 2014), Rome, Italy. 2014. pp. 649-661

Netherlands: Technische Universiteit Delft; 2006

Saarbrücken, Germany. pp. 100-111

2001

pp. 975-984

Fizmatlit; 2003. 728p

distributions. Transportation Research Part B. 2007;41:218-230

of Rail Transport Planning and Management. 2012;2(12):34-50

Transsib". Novosibirsk. 2002. pp. 20-21. (in Russian)
