Open access peer-reviewed chapter

# Probabilistic Model of Delay Propagation along the Train Flow

By Vladimir Chebotarev, Boris Davydov and Kseniya Kablukova

Submitted: October 31st 2017Reviewed: February 15th 2018Published: September 26th 2018

DOI: 10.5772/intechopen.75494

## 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.

### Keywords

• train traffic
• stochastic model
• train delay propagation
• probabilistic modeling
• operative management

## 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 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.

## 2. Literature review

A substantial volume of literature is devoted to study of the train delays effect on the railway functioning. Deterministic models for primary and knock-on delays description were proposed in [1, 2]. These models based on the application of graph theory allow adjust the train traffic schedule. However, such approach considering the different characteristics of train traffic (e.g., travel and dwell times, headways, etc.) as deterministic values does not take into account the uncertainties that arise in reality.

Stochastic modeling takes the influence of random factors (e.g., see [3, 4, 5, 6, 7, 8]) into account. Authors of [7] determine a probabilistic distribution of the arrival times. The problem of finding a distribution of arrival train delays is examined in [8]. It should be noted that in these papers, special cases of primary delay distribution are considered. It is supposed in [8] that the random duration of the primary delay corresponds to some generalization of the exponential law. The paper [7] employs discretization of the delay distribution.

Some of the researchers have analyzed statistical data on deviations of the train arrival times from the planned ones. In particular, the papers [9, 10, 11] show that scattering of these deviations correspond to the exponential distribution.

## 3. Description of models and analysis of the arrival headways distribution

### 3.1. The first model

Trains follow one path one after another in one direction from station A to station B with the same average speed v0. Let the total number of trains is n. The distance from the train j to the train (j − 1) is denoted by Xj+s0, where j = 2, 3, …, n, s0>0is the minimal safe distance between trains, and X2, X3, …, Xnare the random variables (without any assumptions about their distributions). All trains have the same destination station.

Let us also introduce the notations: μj=Xj/v0, t0=s0/v0. Suppose that train 1 departs from station A at the time t=0. Then, the moment Tmof departure train m can be found as (as shown at Figure 1):

Tm=j=2mμj+m1t0, m=2,3,,nE1

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 τis 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 Wkt=Ρνk<t, k = 2, 3, …, n. Call this problem by the first problem.

### 3.2. The second model

Suppose that train 1 was delayed at station A at the moment t=0and waited for a random time τ. If τ<μ2, then trains 2, 3, and so on, depart at the planned times: T2, T3, etc. If τ>μ2, then train 2 will be delayed and will depart at the time τ+t0>T2.Train 3 departs according to the same rule depending on the delay time of train 2, and so on. In this formulation, νkis actual departure headway between the trains with numbers (k − 1) and k. It is required to determine the distribution functions Wktof random variables νk, k = 2, 3, …, n.

Example 1. Let n = 5, μk=2, k=2,5¯, t0=1. The moments of planned departures of trains satisfy the equalities Tk=3k1, k=1,5¯. Figure 2 shows the process of headways νkforming, k=2,5¯,depending on the six values of the interval τ. The dots represent real train departure times that result from the primary delay τ.

The basic model assumptions are follows: (1) only train 1 is exposed to primary delay τ. (2) TkTk1>t0, k = 2, 3, …, n.

Denote by Rkthe real departure time of the train with number k, which depends on τand t0.

We suppose that the departure times of trains satisfy the following two rules. Let k be fixed, 2kn. The first rule: if Rk1Tkt0, then Rk=Tk. The second rule: if Rk1Tkt0, then Rk=Rk1+t0. Obviously, RkTk.

In what follows, we use the notation IxA=1,ifxA,0,ifxR\A,where A is an arbitrary set on the real line R.

Suppose that the total number of trains is equal to n2. Formally, we set νk=0if k>n. Let us proceed to the formulation of the obtained results. We note that the proofs of the majority of the assertions are not given here due to the condition on the size. They take up a lot of space and will be published in our other work.

Theorem 1. 1. If τ<μ2, then ν2=μ2+t0τ, νk=μk+t0, 3kn.

2. Let k be a fixed integer, 2kn. If τj=2kμj, then ν2==νk=t0.

3. If j=2kμjτ<j=2k+1μj, then

νk+1=Ik+1nj=2k+1μj+t0τ,E2

νm=Imnμm+t0, m=k+2,,nE3

Theorem 2. Let n2. For any k, 2kn, the following formula holds

Wkt=It>t0Ρμk<tt0τ<j=2k1μj+Ρτ+tt0>j=2kμjτj=2k1μj,E4
in particular,
W2t=It>t0Ρτ+tt0>μ2E5

Let us introduce the notations, Gx=Ρτ<x, G¯x=Ρτ>x. Note that Gx+G¯x+Ρτ=x=1. We denote by gxthe density function of τin the case when it is absolutely continuous.

Further, some corollaries of Theorem 2 are formulated.

Corollary 1. Let μj, 2jn, be arbitrary positive numbers, then for 2kn

Wkt=It0<tμk+t0G¯j=2kμjt+t0+It>μk+t0,E6
in particular,
W2t=It>t0G¯μ2t+t0.E7

Example 2. Let the primary delay τhave exponential distribution, that is,

gt=It0λeλt, λ>0E8

As initial parameters, we take the following quantities.

λ=0.4, t0=3, μ2=5, μ3=6, μ4=10E9

Graphs of the functions W2tfrom Eq. (7), W3tand W4tfrom Eq. (6) with the parameters (Eq. (9)) are depicted in Figure 3.

It should be noted that in this and the subsequent examples, we use the following measures for the values: μk, Tk, t0, τ, τk, νk, T, b, Ενk(minutes, min); λ(1/min); Dνk(min2). The product αβ(as mean of μk), where αis a shape parameter, βis a scale parameter (in min).

Corollary 2. Let μj=T, 2jn, be a positive constant, then for 2kn

Wkt=It0<tT+t0G¯k1Tt+t0+It>T+t0,E10
in particular,
W2t=It>t0G¯Tt+t0.E11

Example 3. Let τhas density (Eq. (8)). As initial parameters, we take the following quantities:

λ=0.4, t0=4, T=8.E12

Graphs of the functions W2tfrom Eq. (11), W3tand W4tfrom Eq. (10) with the parameters (Eq. (12)) are depicted in Figure 4.

Figures 3 and 4 show that in the case of constant μj, the primary delay τpractically does not affect the fourth train and all subsequent ones. This is consistent with the equality limkWkt=It>t0+Twhich, as it is not difficult to verify, follows from Eq. (10).

Remark 1. It is known that the distribution of sum of the independent random variables is the convolution of their distributions. The convolution of distribution functions F1and F2is determined by the formula F1F2x=F1xydF2y, where the integral sign means the improper Riemann-Stieltjes integral. We consider exceptionally piecewise-continuous distribution functions, then the indicated integral exists with the exception of the case when F1and F2have at least one common discontinuity point (e.g., [12]). The convolution operation is permutable. In the case, when F1=F2==Fm=F, we shall use the following notations: F2FF, FmFFm1, m2. By definition, we assume that F1F. The convolution f1f2xof densities f1and f2is defined as the improper Riemann integral f1xyf2ydy.

Corollary 3. Let μj, 2jn, be independent identically distributed random variables with a continuous distribution function Ψx. Let τbe independent of μj, 2jn. Then

W2t=It>t0G¯zt+t0dΨz,E13
Wkt=It>t0Ψtt0+tt0G¯z+ut+t0dΨzdΨk2u, 3kn.E14

Corollary 4. Let μj, 2jn, be independent identically distributed random variables with a density function ψx. Let τbe independent of all μjand has a density function gx. Then

W2t=It>t0zt+t0gxdxψzdz,E15
Wkt=It>t0tt0ψzdz+tt0z+ut+t0gxdxψzdzψk2udu,E16
3kn.

Remark 2. The integration limit “” can be replaced by 0 in Corollaries 3 and 4 if μj0. On the other hand, we may consider in these corollaries the case when μjtakes values of different signs. From a practical point of view, such an approach is acceptable if the probability that these random quantities take negative values is small enough. This assumption allows to consider, for example, models in which the random variables μjare normally distributed with a variance small enough and to use the property that the class of normal distributions is closed with respect to the convolution operation.

Example 4. Let τhas the density (Eq. (8)), and all μjhave the same gamma density

ψt=It>0et/βtα1Γαβα,E17

where α>0,β>0, Γα=0xα1exdxis gamma function. Put

λ=0.3, t0=5, α=14, β=0.5.E18

One can show that in the example under consideration it follows from Eqs. (15) and (16) that

Wkt=It>t01Γαtt0+b/βΓα+aeλtt0+b11+λβk1αΓα1+λβtt0+b/βΓα,

where Γαy=yxα1exdxis incomplete gamma function. Graphs of the distribution functions Wkt,2k5,with the parameters (Eq. (18)) are depicted in Figure 5.

It is not difficult to verify that for Wktfrom Example 4 the following formula holds:

limkWkta=1,b=0=limkIt>t01Γαtt0/βΓα+eλtt011+λβk1αΓα1+λβtt0/βΓα=WtIt>t01Γαtt0/βΓα.

It can be seen from Figure 5, curves W4t,W5tand so on are practically merged. Hence, in the case under consideration, one can draw the following conclusion: primary delay τaffects to fifth and all successive trains approximately like on the fourth one.

Remark 3. We define the 0-fold convolution as a generalized function with the following property: the equality vtψ0tdt=v0holds for any bounded continuous function vt. Then, Eq. (16) for k=2coincides with Eq. (15).

We do not give proofs for the statements of Section 3 because of limitations on the volume. We will make this in another work.

## 4. Some results on the knock-on delays

Denote by N the random number of knock-on delays (within the framework of the model under consideration).

Lemma 1. For each fixed integer m, 1mn1,

ΡNm=Ρτ>j=2m+1μj.E19

Proof. Easily seen: N=0=t0τ+t0T2,N=m=Tm+1<τ+mt0Tm+2t0,

m = 1, 2, …, n – 2, N=n1=τ+n1t0>Tn.This implies that

ΡNm=Ρτ+mt0>Tm+1=Ρτ>j=2m+1μj.

Here and below, the sign □ denotes the end of the proof.

The corollaries of this lemma are given below. Their proofs are simple and therefore we do not present them.

Corollary 5. If μj=Tis a constant value, 2jn, then for every fixed integer m, 1mn1, we have the equality ΡNm=G¯mT.

Corollary 6. If μj=Tis a constant value, 2jn, and τis exponentially distributed with parameter λ, then for every fixed integer m, 1mn1, the following equality holds,

ΡNm=eλmT.

Corollary 7. If μ2, …, μnare independent identically distributed random variables with a density function ψ, then for every fixed integer m, 1mn1, we have the equality

ΡNm=G¯uψmudu.

In what follows, τ1=τis the delay duration of the first train, τk, k = 2, …, n, is the knock-on delay of the k-th train. The problem is to find the distribution functions Gkt=Ρτk<t, k = 2, 3, …, n. Note that the solution of this problem, which we call by the second problem, allows us to find the distribution of the deviations of the real arrival times from the planned ones.

In what follows, we will use the notation abinstead of maxab.

Theorem 3. The following formula holds:

τk=τk1μk0, 2kn.E20

Corollary 8. The following formula holds:

τk=τj=2kμj0, 2kn.E21

It should be noted that within the framework of our model the deviation of the real arrival time from the planned one for k-th train coincides with τk, 1kn. Figure 6 illustrates this statement.

The dotted lines (lines 1¯and 2¯) represent the scheduled trajectories of trains 1 and 2, solid lines (1 and 2) depict the real trajectories taking into account the delays. It can be seen that the arrival time of the train 1 differs from the schedule at τand the train 2 on the τ2.

Denote μ¯k=j=2kμj, 2kn. As it follows from the assumption that the random variables μ2,,μkhave the same distribution function Ψt. They are mutually independent. The random variable μ¯khas the distribution function Ψk1t.

Corollary 9. The distribution function of τkhas the following form:

Gkt=It>0Ρτμ¯k<t, 2kn.E22

The next Corollaries 10 and 11 follow from Corollary 9 in an obvious way.

Corollary 10. Let μj>0, 2jnbe some constant values. Then

Gkt=It>0Gt+μ¯k.E23

Corollary 11. Let μj=T>0, 2jnbe a constant value. Then

Gkt=It>0Gt+k1T.E24

Corollary 12. Let μj, 2jnbe independent identically distributed random variables with a continuous distribution function Ψt. Let τbe independent of μ2,,μn. Then

Gkt=It>0Gt+ydΨk1y.E25

Corollary 13. Let μj, 2jnbe independent identically distributed random variables with a density function ψt. Let τbe independent of μj, 2jnand has a density function gt. Then Gkt=It>0t+ygzdzψk1ydy.

## 5. Proof of Theorem 3 and its corollaries

Lemma 2. The following formula is valid:

τ2=τμ20.E26

Proof. Let t>0be the time spent by the train on the path length (distance to the place, where an unplanned stop of the train 1 occurred). We show the equality τ2=0holds under the condition τμ2. The departure time of the train 1 after stopping is t+τ. The time point when train 2 reaches scan be written as μ2+t0+t. The knock-on delay of train 2 will not occur, i.e., τ2=0, in the case, when the indicated time points are separated by the value rt0, i.e., r=μ2+t0+tt+τt0, or, which is the same thing, τμ2. The considered case is illustrated in Figure 7a.

The knock-on delay of the duration τ2=τμ2will occur when τ>μ2. Indeed, since trains after a random stop depart simultaneously, then the equality t+τ=μ2+t0+tt0+τ2holds, i.e., τ=μ2+τ2. The case under consideration is illustrated in Figure 7b. Thus, the validity of Eq. (26) is shown. □

Proof of Theorem 3. We shall use the method of mathematical induction. The equality (Eq. (20)) for k = 2 is established by Lemma 2. Let Eq. (20) be satisfied. We show that:

τk+1=τkμk+10,2k+1n.E27

It follows from the inductive hypothesis that τk=0under the condition τk1μk. But if the delay of the k-th train is 0, then the next train does not undergo any delay, that is, τk+1=0. The present case is illustrated in Figure 8.

In the case, when τk1>μk, a knock-on delay of the k-th train occurs and equals to τk=τk1μk(according to the inductive hypothesis). Further, two cases are possible: either (1) a delay τkentails a delay τk+1, or (2) τk+1=0.

Case 1. If the k-th train is delayed, then (k + 1)-th one will be delayed only if τk>μk+1, and its delay duration is τk+1=τkμk+1(this fact follows from the equality of the moments of departure of the k-th and (k + 1)-th trains after an unscheduled stop: Tk+tk1t0+τk=Tk+1+tkt0+τk+1). Case 1 is illustrated in Figure 9a.

Case 2. If the k-th train is delayed, then (k + 1)-th one will not be delayed (τk+1=0) only if τkμk+1. Case 2 is illustrated in Figure 9b. Note that if the knock-on delay of the k-th train occurs, a conflict of the k-th train with (k + 1)-th is described similar to the interaction of trains 1 and 2 (see Lemma 2). All described cases lead to Eq. (20).□

Proof of Corollary 8. We indicate that Eq. (21) is similar to Eq. (20). According to the statement of Theorem 3, we have:

τ2=τμ20, τ3=τ2μ30, τk=τk1μk0.E28

Using the method of mathematical induction and taking into account that μ¯k1+μk=μ¯k, we obtain Eq. (21) from Eq. (28).□

Proof of Corollary 9. It follows from Corollary 8 that τk=0if τμ¯k(see, e.g., Figure 8), and τk=τμ¯kif τ>μ¯k(see, e.g., Figure 9a). Using the law of total probability, we obtain the following chain of equalities:

Gkt=Ρτk<t=It>0Ρτk<t=It>0Ρτk<tτμ¯kΡτμ¯k+Ρτk<tτ>μ¯kΡτ>μ¯k=It>0Ρτμ¯k0+It>0Ρ0<τμ¯k<t=It>0Ρτμ¯k<t.

Proof of Corollary 12. Apply the well-known assertion to Eq. (22): if Y1and Y2are independent random variables, then for any function of two variables fand any cR, the following equality holds: ΡfY1Y2<c=ΡfyY2<cdF1y, where F1is the distribution function of Y1. Consequently, Gkt=It>0Ρτy<tdΨk1y. This implies Eq. (25).□

Proof of Corollary 13. The assertion follows from Eq. (25).□

Note that the function Gkthas a jump at zero which is equal to:

Gk0+=G+ydΨk1y, where G+y=limt0+Gt+y.

In the case, when τand μjare absolutely continuous, it follows from Eq. (25) that

Gkt=It>0t+ygzdzψk1ydy,E29

where gand ψare the density functions of τand μ1, respectively, ψjyis the j-fold convolution of the density ψ. In this case, we also have

gktIt>0Gkt=It>0gt+yψk1ydy.E30

If we assume that τ0, then we deduce from Eq. (29) that

Gkt=It>0t0t+ygzdzψk1ydy,E31
and we get from Eq. (30),
gkt=It>0tgt+yψk1ydy.E32

## 6. Corollary of Theorem 2 when the distribution of primary delay is a mixture of exponential and one-point distributions

Consider the cumulative distribution function of the following type:

GxΡτ<x=Ixb1aeλxb,E33

where 0a1, b0, and λ>0are some parameters. Such distribution function is considered, for example, in [8]. It is easy to see that Gx=1aG0xb+aGxbλ, where G0xis the distribution function of the degenerate distribution concentrated at the point x=0, Gxλ=Ix01eλx.

Let us find out the form of the distribution functions (Eqs. (13) and (14)) in the case of Eq. (33), when the function Ψis continuous. In what follows, we mean that n3.

Lemma 3. Let the function G be defined by Eq. (33), and Ψbe continuous. Then

W2t=It>t0Ψtt0+b+aeλtt0+btt0+beλzdΨz,E34
Wkt=It>t0Ψtt0+aeλtt0+bbeλudΨk2utt0eλzdΨz+beλutt0u+beλzdΨzdΨk2u+bΨtt0u+bΨtt0dΨk2u,k3.E35

Proof. According to Eq. (33), one may conclude that function G¯xhas a unique discontinuity point x=b. Hence, the integral G¯zt+t0dΨzexists for any continuous distribution function Ψ. Note that if Ψzhad a discontinuity point z=t1, then the function G¯zt+t0would also be discontinuous at the point z=t1for t=t0+t1b, and then the considered integral would not exist (see Remark 1). Since

G¯x=Ix<b+Ixbaeλxb,E36
then
G¯zt+t0dΨz=Ψtt0+b+aeλtt0+btt0+beλzdΨz.E37

In accordance with Eq. (13), the relation (Eq. (34)) is proved.

Let k3. It follows from Eq. (14) that

Wkt=It>t0Ψtt0+VudΨk2u,E38

where Vu=tt0G¯z+ut+t0dΨz. Given Eq. (36), it is easy to see that

Vu=V1u+V2u,E39

V1u=aeλut+t0btt0Iz+ut+t0beλzdΨz, V2u=tt0Iz+ut+t0<bdΨz.

By using equalities

uz:ubz>tt0ztt0u+b=uz:ubz>tt0,uz:u<bz>tt0ztt0u+b=uz:u<bztt0u+b,
V1udΨk2u=aeλtt0+bbtt0eλz+udΨzdΨk2u+btt0u+beλz+udΨzdΨk2u.E40

Since uz:ubz>tt0z<tt0u+b=,

uz:u<bz>tt0z<tt0u+b=uz:u<btt0<z<tt0u+b,
then
V2udΨk2u=btt0tt0u+bdΨzdΨk2u.E41

It follows from Eqs. (39)(41) that

VudΨk2u=aeλtt0+bbeλudΨk2utt0eλzdΨz+btt0u+beλzdΨzeλudΨk2u+bΨtt0u+bΨtt0dΨk2u.E42

The equalities Eq. (38) and Eq. (42) entail Eq. (35).□

Below we give without a proof a corollary of Lemma 3 in the case when μjare not random variables, and they are equal to the same constant.

Corollary 14. Let μj=T>0, 2jn, be a constant. Let function Gbe defined by Eq. (33). Then, for 2kn, the following formula holds:

Wkt=I0bk2TI0<tt0Taeλk1Tt+t0b+Itt0>T+Ik2T<b<k1TI0<tt0k1Tbaeλk1Tt+t0b+Itt0>k1Tb+Ibk1TIt>t0.E43

Furthermore,

Ενk=I0bk2Tt0+Taλeλk2Tb1eλT+Ibk1Tt0+Ik2T<b<k1Tt0+k1Tb+aλeλk1Tb1.E44

Dνk=I0bk2Taλ2eλk2Tb21eλTaeλk2Tb1eλT22λTeλT+Ik2T<b<k1Taλ2eλk2Tb2eλk2TbeλTaeλk2Tbeλk2TbeλT22λk1TbeλT.E45

Example 5. Figure 10 depicts the graphs of the functions Wktdefined by Eq. (43) with k = 2, 3 for the following parameters:

a=1, b=0, λ=0.26, t0=4, T=7.E46

We calculated the values of Ενkand Dνkusing the formulas (44) and (45) (see Table 1).

k=2k=3k=5k=8k=10
Ενk7.7770210.4777910.9862910.9999410.99999
Dνk5.680092.330670.0681560.000297.63176 × 10−6

### Table 1.

The behavior Ενkand Dνkwith growth of the parameter k.

Remark 4. It can be easily seen that the larger k, Wktfrom Eq. (43) is closer to WtIb0It>t0+T. This agrees with Figure 10 and the formulas (44) and (45) due to which we have Ενkt0+T, Dνk0as k, and also with the results of calculations in Table 1.

Let the random variable τbe distributed with the density (Eq. (33)) with parameters a=1, b=0. Now, we find the condition on the parameter T, under which the probability that at least m of knock-on delays will occur would not exceed a given probability p. Note that the departure headway is equal to T+t0.

According to Corollary 6, it is necessary to solve the inequality expλmTp. As a result, we obtain the desired condition:

T1/ln1/pE47

(see also [13]). Denote by Tmpλthe minimal T satisfying the inequality (Eq. (47)).

Example 6. Let us fix λ=0.26. The behavior of Tmpλas a function of the continuous parameter m with p=0.1and p=0.05is shown in Figure 11a. Obviously, Tmpλis the decreasing function with respect to the argument p. Exact calculations can be made using the formula:

Tmpλ=1/ln1/p.E48

Let p=0.1. The behavior of Tmpλas a function of the continuous parameter m with λ=0.26and λ=0.15is shown in Figure 11b. In accordance with Eq. (48), Tmpλis the decreasing function with respect to the argument λ. In the case of exponential density gt, we have Ετ=1/λ. Therefore, the decrease of λleads to increase in the average of primary delay and the departure headways (if we want to reduce the number of knock-on delays).

We also obtain the corollaries of Lemma 3 in the case when μjare distributed according to the gamma-law with the density (Eq. (17)).

Corollary 15. If primary delay τhas an exponential distribution gt=It>0λeλtand μk, 2kn, has the density (Eq. (17)), then the following formulas are true:

Gkt=It>01eλtλβ+1k1α,E49
gkt=It>0λβ+1k1αλeλt.E50

Remark 5. The function gkt=It>0Gktis not a density, in particular, because of gktdt=0gktdt=GkGk0+=1hk1, where hkis the jump of the function Gktat the origin. At the same time, the function g˜kt11hkgktis a density.

Corollary 15 can be reformulated as follows.

Corollary 15*. Let primary delay τis exponentially distributed with a parameter λ, and μk, 2kn, have the same gamma distribution with the density (Eq. (17)). Then, τkhas the distribution function of the form Eq. (33) with a=λβ+1k1α, b = 0, and, consequently,

Ρτk=0=Gk0+=1λβ+1k1α.

Remark 6. Let Ρτ2=0=p, 0<p<1.Then by Corollary 15*, Ρτk=0=11pk1,3kn. Hence, Ρτk=01as k.

Example 7. Let μ2,μ3,be independent random variables having the same density function (Eq. (17)). We perform three series of experiments and investigate a behavior of distribution of the arrival time deviations τkwith various combinations of parameters: α, β, k. The results are presented in graphical form in Figures 12–15 The functions Gktare calculated by formula (49), and the functions gktby formula (50). Note that product αβis the mean of μk. Parameter λis equal to 0.25 and αβ=7as it observes in reality.

## 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=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 ξof the size n=50with the sample mean 1.44 and sample variance 2.7. We tested the hypothesis that ξobeys distribution (Eq. (33)) with λ=0.35and a=0.64. To this end, we applied the Kolmogorov goodness-of-fit test with the significance level α=0.05and 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 ξis 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 G2twith α=0.5from Figure 12.

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

Remark 9. It was verified that if the length of the random variables μjhave the same gamma distribution, any variation of the parameters of this distribution (αand β) has a rather small influence on behavior of output distribution (see Figures 1215).

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 τhas the exponential distribution with the parameter λ=0.25(i.e., τhas the distribution function (Eq. (33)) with λ=0.25, a=1, b=0), and μ2has gamma distribution with the density function (Eq. (17)), where α=0.6, β=11.7. Using formulas (49) and (50) with k=2, we have:

SD2=ta22dG2t=t2dG2ta22=0t2g2tdta2210.987.

Here a2=tdG2t=1λλβ+10.61.763,t2dG2t=2λ2λβ+10.614.088,

0t2g2tdta22=2λ2λβ+10.61λ2λβ+11.210.987.

Thus, theoretical SD10.9873.315min. This corresponds with the real statistics which shows the SD amount is 3.32 min for the mentioned station.

## 8. Conclusions

The mathematical model of train traffic proposed in the chapter allows us to find conditions on initial headways, which provide a smallness of frequency of a large number of delays. In other words, the formulas for the distributions of arrival headways obtained in the chapter enable to optimize the frequency of arriving train delays.

## Acknowledgments

The research is funded by the JSC Russian Railways (grant 2016 for development of the scientific school).

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Vladimir Chebotarev, Boris Davydov and Kseniya Kablukova (September 26th 2018). Probabilistic Model of Delay Propagation along the Train Flow, Probabilistic Modeling in System Engineering, Andrey Kostogryzov, IntechOpen, DOI: 10.5772/intechopen.75494. Available from:

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