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 þ s0, where j = 2, 3, …, n, s<sup>0</sup> > 0 is the minimal safe distance between trains, and X2, X3, …, Xn are the random variables (without any assumptions about their distributions). All trains have the same destination station.

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

view. Probabilistic modeling of the delay propagation process along the train flow is the main

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

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

Stochastic modeling takes the influence of random factors (e.g., see [3–8]) 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

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

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

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 þ s0, where j = 2, 3, …, n, s<sup>0</sup> > 0 is the minimal safe distance between trains, and X2, X3, …, Xn are the random variables (without any assumptions about

tool for solving this problem.

172 Probabilistic Modeling in System Engineering

performed.

2. Literature review

account the uncertainties that arise in reality.

correspond to the exponential distribution.

3.1. The first model

law. The paper [7] employs discretization of the delay distribution.

their distributions). All trains have the same destination station.

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 shown at Figure 1):

$$T^{(m)} = \sum\_{j=2}^{m} \mu\_j + (m-1)t\_{0\prime} \ m = 2, 3, \dots, n \tag{1}$$

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 problem.

#### 3.2. The second model

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

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 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) <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>.

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

ν<sup>m</sup> ¼ I mð Þ ≤ n μ<sup>m</sup> þ t<sup>0</sup>

k�1

1

j¼2 μj

W2ðÞ¼ t I tð Þ > t<sup>0</sup> Ρ τ þ t � t<sup>0</sup> > μ<sup>2</sup>

Let us introduce the notations, G xð Þ¼ Ρð Þ τ < x , G xð Þ¼ Ρð Þ τ > x . Note that G xð Þþ G xð Þþ Ρð Þ¼ τ ¼ x 1. We denote by g xð Þ the density function of τ in the case when it is absolutely

, 2 ≤ j ≤ n, be arbitrary positive numbers, then for 2 ≤ k ≤ n

0 @

k

j¼2

W2ðÞ¼ t I tð Þ > t<sup>0</sup> G μ<sup>2</sup> � t þ t<sup>0</sup>

Graphs of the functions W2ð Þt from Eq. (7), W3ð Þt and W4ð Þt from Eq. (6) with the parameters

It should be noted that in this and the subsequent examples, we use the following measures for

�λt

μ<sup>j</sup> � t þ t<sup>0</sup>

1

λ ¼ 0:4, t<sup>0</sup> ¼ 3, μ<sup>2</sup> ¼ 5, μ<sup>3</sup> ¼ 6, μ<sup>4</sup> ¼ 10 (9)

WkðÞ¼ t I tð Þ <sup>0</sup> < t ≤ T þ t<sup>0</sup> G k ðð Þ � 1 T � t þ t0Þ þ I tð Þ > T þ t<sup>0</sup> , (10)

A þ I t > μ<sup>k</sup> þ t<sup>0</sup>

� �: (7)

, λ > 0 (8)

� � G X

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

the values: <sup>μ</sup>k, <sup>T</sup>ð Þ<sup>k</sup> , <sup>t</sup>0, <sup>τ</sup>, <sup>τ</sup>k, <sup>ν</sup>k, <sup>T</sup>, <sup>b</sup>, <sup>Ε</sup>ν<sup>k</sup> (minutes, min); <sup>λ</sup> (1/min); Dν<sup>k</sup> (min<sup>2</sup>

(as mean of μ<sup>k</sup> ), where α is a shape parameter, β is a scale parameter (in min).

Corollary 2. Let μ<sup>j</sup> ¼ T, 2 ≤ j ≤ n, be a positive constant, then for 2 ≤ k ≤ n

g tðÞ¼ I tð Þ ≥ 0 λe

Theorem 2. Let n ≥ 2. For any k, 2 ≤ k ≤ n, the following formula holds

WkðÞ¼ <sup>t</sup> I tð Þ <sup>&</sup>gt; <sup>t</sup><sup>0</sup> <sup>Ρ</sup> <sup>μ</sup><sup>k</sup> <sup>&</sup>lt; <sup>t</sup> � <sup>t</sup>0; <sup>τ</sup> <sup>&</sup>lt; <sup>X</sup>

Further, some corollaries of Theorem 2 are formulated.

WkðÞ¼ t I t<sup>0</sup> < t ≤ μ<sup>k</sup> þ t<sup>0</sup>

As initial parameters, we take the following quantities.

(Eq. (9)) are depicted in Figure 3.

0 @

2 4

in particular,

continuous.

in particular,

Corollary 1. Let μ<sup>j</sup>

� �, m <sup>¼</sup> <sup>k</sup> <sup>þ</sup> <sup>2</sup>, …, n (3)

Probabilistic Model of Delay Propagation along the Train Flow

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

k

j¼2 μj ; τ ≥ X k�1

� � (5)

j¼2 μj

� �, (6)

). The product αβ

1 A

3

5, (4)

175

<sup>A</sup> <sup>þ</sup> <sup>Ρ</sup> <sup>τ</sup> <sup>þ</sup> <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>&</sup>gt; <sup>X</sup>

0 @

Figure 2. The headways ν<sup>k</sup> for some values τ.

We suppose that the departure times of trains satisfy the following two rules. Let k be fixed, <sup>2</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>n</sup>. The first rule: if <sup>R</sup>ð Þ <sup>k</sup>�<sup>1</sup> <sup>≤</sup> <sup>T</sup>ð Þ<sup>k</sup> � <sup>t</sup>0, then <sup>R</sup>ð Þ<sup>k</sup> <sup>¼</sup> <sup>T</sup>ð Þ<sup>k</sup> . The second rule: if <sup>R</sup>ð Þ <sup>k</sup>�<sup>1</sup> <sup>≥</sup> <sup>T</sup>ð Þ<sup>k</sup> � <sup>t</sup>0, then <sup>R</sup>ð Þ<sup>k</sup> <sup>¼</sup> <sup>R</sup>ð Þ <sup>k</sup>�<sup>1</sup> <sup>þ</sup> <sup>t</sup>0. Obviously, <sup>R</sup>ð Þ<sup>k</sup> <sup>≥</sup> <sup>T</sup>ð Þ<sup>k</sup> .

In what follows, we use the notation I xð Þ¼ ∈ A 1, if x ∈ A, <sup>0</sup>, if x <sup>∈</sup>R\A, � where A is an arbitrary set on the real line R.

Suppose that the total number of trains is equal to n ≥ 2. Formally, we set ν<sup>k</sup> ¼ 0 if 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

Theorem 1. 1. If τ < μ2, then ν<sup>2</sup> ¼ μ<sup>2</sup> þ t<sup>0</sup> � τ, ν<sup>k</sup> ¼ μ<sup>k</sup> þ t0, 3 ≤ k ≤ n.


and will be published in our other work.

$$\nu\_{k+1} = I(k+1 \le n) \left[ \sum\_{j=2}^{k+1} \mu\_j + t\_0 - \pi \right],\tag{2}$$

Probabilistic Model of Delay Propagation along the Train Flow http://dx.doi.org/10.5772/intechopen.75494 175

$$\nu\_m = I(m \le n) \left(\mu\_m + t\_0\right), \ m = k + 2, \ldots, n \tag{3}$$

Theorem 2. Let n ≥ 2. For any k, 2 ≤ k ≤ n, the following formula holds

$$\mathcal{W}\_k(t) = I(t > t\_0) \left[ \mathbf{P} \left( \mu\_k < t - t\_0, \tau < \sum\_{j=2}^{k-1} \mu\_j \right) + \mathbf{P} \left( \tau + t - t\_0 > \sum\_{j=2}^k \mu\_j, \tau \ge \sum\_{j=2}^{k-1} \mu\_j \right) \right], \tag{4}$$

in particular,

$$W\_2(t) = I(t > t\_0) \mathbf{P}(\tau + t - t\_0 > \mu\_2) \tag{5}$$

Let us introduce the notations, G xð Þ¼ Ρð Þ τ < x , G xð Þ¼ Ρð Þ τ > x . Note that G xð Þþ G xð Þþ Ρð Þ¼ τ ¼ x 1. We denote by g xð Þ the density function of τ in the case when it is absolutely continuous.

Further, some corollaries of Theorem 2 are formulated.

Corollary 1. Let μ<sup>j</sup> , 2 ≤ j ≤ n, be arbitrary positive numbers, then for 2 ≤ k ≤ n

$$\mathcal{W}\_k(t) = I\{t\_0 < t \le \mu\_k + t\_0\} \,\,\overline{G}\left(\sum\_{j=2}^k \mu\_j - t + t\_0\right) + I\{t > \mu\_k + t\_0\},\tag{6}$$

in particular,

We suppose that the departure times of trains satisfy the following two rules. Let k be fixed, <sup>2</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>n</sup>. The first rule: if <sup>R</sup>ð Þ <sup>k</sup>�<sup>1</sup> <sup>≤</sup> <sup>T</sup>ð Þ<sup>k</sup> � <sup>t</sup>0, then <sup>R</sup>ð Þ<sup>k</sup> <sup>¼</sup> <sup>T</sup>ð Þ<sup>k</sup> . The second rule: if <sup>R</sup>ð Þ <sup>k</sup>�<sup>1</sup> <sup>≥</sup> <sup>T</sup>ð Þ<sup>k</sup> � <sup>t</sup>0,

�

Suppose that the total number of trains is equal to n ≥ 2. Formally, we set ν<sup>k</sup> ¼ 0 if 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

kþ1

2 4

j¼2

<sup>j</sup>¼<sup>2</sup> <sup>μ</sup><sup>j</sup>

<sup>ν</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> I kð Þ <sup>þ</sup> <sup>1</sup> <sup>≤</sup> <sup>n</sup> <sup>X</sup>

1, if x ∈ A, 0, if x ∈R\A,

, then ν<sup>2</sup> ¼ … ¼ ν<sup>k</sup> ¼ t0.

μ<sup>j</sup> þ t<sup>0</sup> � τ

3

where A is an arbitrary set on

5, (2)

then <sup>R</sup>ð Þ<sup>k</sup> <sup>¼</sup> <sup>R</sup>ð Þ <sup>k</sup>�<sup>1</sup> <sup>þ</sup> <sup>t</sup>0. Obviously, <sup>R</sup>ð Þ<sup>k</sup> <sup>≥</sup> <sup>T</sup>ð Þ<sup>k</sup> .

Figure 2. The headways ν<sup>k</sup> for some values τ.

174 Probabilistic Modeling in System Engineering

and will be published in our other work.

2. Let k be a fixed integer, 2 ≤ k ≤ n. If τ ≥ P<sup>k</sup>

<sup>j</sup>¼<sup>2</sup> <sup>μ</sup><sup>j</sup>

, then

<sup>j</sup>¼<sup>2</sup> <sup>μ</sup><sup>j</sup> <sup>≤</sup> <sup>τ</sup> <sup>&</sup>lt; <sup>P</sup><sup>k</sup>þ<sup>1</sup>

Theorem 1. 1. If τ < μ2, then ν<sup>2</sup> ¼ μ<sup>2</sup> þ t<sup>0</sup> � τ, ν<sup>k</sup> ¼ μ<sup>k</sup> þ t0, 3 ≤ k ≤ n.

the real line R.

3. If P<sup>k</sup>

In what follows, we use the notation I xð Þ¼ ∈ A

$$\mathcal{W}\_2(t) = I(t > t\_0)\overline{G}(\mu\_2 - t + t\_0). \tag{7}$$

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

$$\log(t) = I(t \ge 0)\lambda e^{-\lambda t}, \ \lambda > 0\tag{8}$$

As initial parameters, we take the following quantities.

$$
\lambda = 0.4, \ t\_0 = 3, \ \mu\_2 = 5, \ \mu\_3 = 6, \ \mu\_4 = 10 \tag{9}
$$

Graphs of the functions W2ð Þt from Eq. (7), W3ð Þt and W4ð Þt from 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: <sup>μ</sup>k, <sup>T</sup>ð Þ<sup>k</sup> , <sup>t</sup>0, <sup>τ</sup>, <sup>τ</sup>k, <sup>ν</sup>k, <sup>T</sup>, <sup>b</sup>, <sup>Ε</sup>ν<sup>k</sup> (minutes, min); <sup>λ</sup> (1/min); Dν<sup>k</sup> (min<sup>2</sup> ). The product αβ (as mean of μ<sup>k</sup> ), where α is a shape parameter, β is a scale parameter (in min).

Corollary 2. Let μ<sup>j</sup> ¼ T, 2 ≤ j ≤ n, be a positive constant, then for 2 ≤ k ≤ n

$$\mathcal{W}\_k(t) = I(t\_0 < t \le T + t\_0)\overline{G}((k-1)T - t + t\_0) + I(t > T + t\_0),\tag{10}$$

Figure 3. Behavior of the functions Wkð Þt , k = 2, 3, 4.

in particular,

$$W\_2(t) = I(t > t\_0) \overline{G}(T - t + t\_0). \tag{11}$$

Corollary 3. Let μ<sup>j</sup>

Corollary 4. Let μ<sup>j</sup>

WkðÞ¼ t I tð Þ > t<sup>0</sup>

WkðÞ¼ t I tð Þ > t<sup>0</sup> Ψð Þþ t � t<sup>0</sup>

ous distribution function Ψð Þx . Let τ be independent of μ<sup>j</sup>

Figure 4. Behavior of the functions Wkð Þt , k = 2, 3, 4.

W2ðÞ¼ t I tð Þ > t<sup>0</sup>

ð∞ t�t<sup>0</sup>

function ψð Þx . Let τ be independent of all μ<sup>j</sup> and has a density function g xð Þ. Then

ð∞ �∞ ð∞ �∞

ð∞ t�t<sup>0</sup> ð∞

ð∞

3 ≤ k ≤ n:

Remark 2. The integration limit "�∞" can be replaced by 0 in Corollaries 3 and 4 if μ<sup>j</sup> ≥ 0. On the other hand, we may consider in these corollaries the case when μ<sup>j</sup> takes 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 μ<sup>j</sup> are normally distributed with

z�tþt<sup>0</sup>

zþu�tþt<sup>0</sup>

ð∞ �∞

W2ðÞ¼ t I tð Þ > t<sup>0</sup>

ψð Þz dz þ

ð<sup>t</sup>�t<sup>0</sup> �∞

ð∞ �∞

, 2 ≤ j ≤ n, be independent identically distributed random variables with a continu-

G zð Þ þ u � t þ t<sup>0</sup> dΨð Þz � �dΨ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup> � �, <sup>3</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>n</sup>: (14)

, 2 ≤ j ≤ n, be independent identically distributed random variables with a density

g xð Þdx � �ψð Þ<sup>z</sup> dz � �ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup> du � �, (16)

, 2 ≤ j ≤ n. Then

G zð Þ � t þ t<sup>0</sup> dΨð Þz , (13)

Probabilistic Model of Delay Propagation along the Train Flow

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177

g xð Þdx � �ψð Þ<sup>z</sup> dz, (15)

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

$$
\lambda = 0.4, \ t\_0 = 4, \ T = 8.\tag{12}
$$

Graphs of the functions W2ð Þt from Eq. (11), W3ð Þt and W4ð Þt from Eq. (10) with the parameters (Eq. (12)) are depicted in Figure 4.

Figures 3 and 4 show that in the case of constant μ<sup>j</sup> , the primary delay τ practically does not affect the fourth train and all subsequent ones. This is consistent with the equality lim<sup>k</sup>!<sup>∞</sup> WkðÞ¼ t I tð Þ > t<sup>0</sup> þ T which, 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 F<sup>1</sup> and F<sup>2</sup> is determined by the formula ð Þ <sup>F</sup>1∗F<sup>2</sup> ð Þ¼ <sup>x</sup> <sup>Ð</sup> <sup>∞</sup> �<sup>∞</sup> <sup>F</sup>1ð Þ <sup>x</sup> � <sup>y</sup> dF2ð Þ<sup>y</sup> , 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 F<sup>1</sup> and F<sup>2</sup> have at least one common discontinuity point (e.g., [12]). The convolution operation is permutable. In the case, when F<sup>1</sup> ¼ F<sup>2</sup> ¼ … ¼ Fm ¼ F, we shall use the following notations: F<sup>∗</sup><sup>2</sup> ≔ F∗F, F<sup>∗</sup><sup>m</sup> ≔ F∗F<sup>∗</sup>ð Þ <sup>m</sup>�<sup>1</sup> , m ≥ 2. By definition, we assume that F<sup>∗</sup><sup>1</sup> ≔ F. The convolution f <sup>1</sup>∗f <sup>2</sup> � �ð Þ<sup>x</sup> of densities <sup>f</sup> <sup>1</sup> and <sup>f</sup> <sup>2</sup> is defined as the improper Riemann integral Ð ∞ �<sup>∞</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>x</sup> � <sup>y</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>y</sup> dy.

Figure 4. Behavior of the functions Wkð Þt , k = 2, 3, 4.

in particular,

ters (Eq. (12)) are depicted in Figure 4.

Figure 3. Behavior of the functions Wkð Þt , k = 2, 3, 4.

176 Probabilistic Modeling in System Engineering

determined by the formula ð Þ <sup>F</sup>1∗F<sup>2</sup> ð Þ¼ <sup>x</sup> <sup>Ð</sup> <sup>∞</sup>

convolution f <sup>1</sup>∗f <sup>2</sup>

�<sup>∞</sup> <sup>f</sup> <sup>1</sup>ð Þ <sup>x</sup> � <sup>y</sup> <sup>f</sup> <sup>2</sup>ð Þ<sup>y</sup> dy.

Ð ∞

Figures 3 and 4 show that in the case of constant μ<sup>j</sup>

W2ðÞ¼ t I tð Þ > t<sup>0</sup> G Tð Þ � t þ t<sup>0</sup> : (11)

λ ¼ 0:4, t<sup>0</sup> ¼ 4, T ¼ 8: (12)

, the primary delay τ practically does

�<sup>∞</sup> <sup>F</sup>1ð Þ <sup>x</sup> � <sup>y</sup> dF2ð Þ<sup>y</sup> , where the integral sign means

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

Graphs of the functions W2ð Þt from Eq. (11), W3ð Þt and W4ð Þt from Eq. (10) with the parame-

not affect the fourth train and all subsequent ones. This is consistent with the equality

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 F<sup>1</sup> and F<sup>2</sup> is

the improper Riemann-Stieltjes integral. We consider exceptionally piecewise-continuous distribution functions, then the indicated integral exists with the exception of the case when F<sup>1</sup> and F<sup>2</sup> have at least one common discontinuity point (e.g., [12]). The convolution operation is permutable. In the case, when F<sup>1</sup> ¼ F<sup>2</sup> ¼ … ¼ Fm ¼ F, we shall use the following notations: F<sup>∗</sup><sup>2</sup> ≔ F∗F, F<sup>∗</sup><sup>m</sup> ≔ F∗F<sup>∗</sup>ð Þ <sup>m</sup>�<sup>1</sup> , m ≥ 2. By definition, we assume that F<sup>∗</sup><sup>1</sup> ≔ F. The

� �ð Þ<sup>x</sup> of densities <sup>f</sup> <sup>1</sup> and <sup>f</sup> <sup>2</sup> is defined as the improper Riemann integral

lim<sup>k</sup>!<sup>∞</sup> WkðÞ¼ t I tð Þ > t<sup>0</sup> þ T which, as it is not difficult to verify, follows from Eq. (10).

Corollary 3. Let μ<sup>j</sup> , 2 ≤ j ≤ n, be independent identically distributed random variables with a continuous distribution function Ψð Þx . Let τ be independent of μ<sup>j</sup> , 2 ≤ j ≤ n. Then

$$\mathcal{W}\_2(t) = I(t > t\_0) \int\_{-\infty}^{\infty} \overline{\mathcal{G}}(z - t + t\_0) d\Psi(z),\tag{13}$$

$$\mathcal{W}\_k(t) = I(t > t\_0) \left\{ \Psi(t - t\_0) + \int\_{-\infty}^{\infty} \left[ \int\_{t - t\_0}^{\infty} \overline{\mathbf{G}}(z + u - t + t\_0) d\Psi(z) \right] d\Psi^{\*(k-2)}(u) \right\}, \tag{14}$$

Corollary 4. Let μ<sup>j</sup> , 2 ≤ j ≤ n, be independent identically distributed random variables with a density function ψð Þx . Let τ be independent of all μ<sup>j</sup> and has a density function g xð Þ. Then

$$W\_2(t) = I(t > t\_0) \int\_{-\infty}^{\infty} \left( \int\_{z - t + t\_0}^{\infty} g(\mathbf{x}) d\mathbf{x} \right) \psi(z) dz,\tag{15}$$

$$\mathcal{W}\_k(t) = I(t > t\_0) \left\{ \int\_{-\infty}^{t-t\_0} \psi(z) dz + \int\_{-\infty}^{\infty} \left[ \int\_{t-t\_0}^{\infty} \left( \int\_{z+u-t+t\_0}^{\infty} g(x) dx \right) \psi(z) dz \right] \psi^{\*(k-2)}(u) du \right\}, \tag{16}$$
 
$$3 \le k \le n.$$

Remark 2. The integration limit "�∞" can be replaced by 0 in Corollaries 3 and 4 if μ<sup>j</sup> ≥ 0. On the other hand, we may consider in these corollaries the case when μ<sup>j</sup> takes 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 μ<sup>j</sup> are 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 τ has the density (Eq. (8)), and all μ<sup>j</sup> have the same gamma density

$$\psi(t) = I(t>0)\frac{e^{-t/\beta}t^{\alpha-1}}{\Gamma(\alpha)\beta^{\alpha}},\tag{17}$$

It can be seen from Figure 5, curves W4ð Þt , W5ð Þt and so on are practically merged. Hence, in the case under consideration, one can draw the following conclusion: primary delay τ affects

Remark 3. We define the 0-fold convolution as a generalized function with the following

We do not give proofs for the statements of Section 3 because of limitations on the volume. We

Denote by N the random number of knock-on delays (within the framework of the model

0 @

<sup>Ρ</sup>ð Þ¼ <sup>N</sup> <sup>≥</sup> <sup>m</sup> <sup>Ρ</sup> <sup>τ</sup> <sup>&</sup>gt; <sup>X</sup><sup>m</sup>þ<sup>1</sup>

Proof. Easily seen: f g <sup>N</sup> <sup>¼</sup> <sup>0</sup> <sup>¼</sup> <sup>t</sup><sup>0</sup> <sup>≤</sup> <sup>τ</sup> <sup>þ</sup> <sup>t</sup><sup>0</sup> <sup>≤</sup> <sup>T</sup>ð Þ<sup>2</sup> n o, Nf g <sup>¼</sup> <sup>m</sup> <sup>¼</sup> <sup>T</sup>ð Þ <sup>m</sup>þ<sup>1</sup> <sup>&</sup>lt; <sup>τ</sup> <sup>þ</sup> mt<sup>0</sup> <sup>≤</sup> <sup>T</sup>ð Þ <sup>m</sup>þ<sup>2</sup> � <sup>t</sup><sup>0</sup>

<sup>Ρ</sup>ð Þ¼ <sup>N</sup> <sup>≥</sup> <sup>m</sup> <sup>Ρ</sup> <sup>τ</sup> <sup>þ</sup> mt<sup>0</sup> <sup>&</sup>gt; <sup>T</sup>ð Þ <sup>m</sup>þ<sup>1</sup> � � <sup>¼</sup> <sup>Ρ</sup> <sup>τ</sup> <sup>&</sup>gt; <sup>X</sup><sup>m</sup>þ<sup>1</sup>

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

Corollary 5. If μ<sup>j</sup> ¼ T is a constant value, 2 ≤ j ≤ n, then for every fixed integer m, 1 ≤ m ≤ n � 1, we

Corollary 6. If μ<sup>j</sup> ¼ T is a constant value, 2 ≤ j ≤ n, and τ is exponentially distributed with parameter

Corollary 7. If μ2, …, μ<sup>n</sup> are independent identically distributed random variables with a density

�λmT:

Ρð Þ¼ N ≥ m e

<sup>m</sup> = 1, 2, …, <sup>n</sup> – 2, f g <sup>N</sup> <sup>¼</sup> <sup>n</sup> � <sup>1</sup> <sup>¼</sup> <sup>τ</sup> <sup>þ</sup> ð Þ <sup>n</sup> � <sup>1</sup> <sup>t</sup><sup>0</sup> <sup>&</sup>gt; <sup>T</sup>ð Þ <sup>n</sup> n o: This implies that

λ, then for every fixed integer m, 1 ≤ m ≤ n � 1, the following equality holds,

function ψ, then for every fixed integer m, 1 ≤ m ≤ n � 1, we have the equality

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

j¼2 μj 1

0 @

j¼2 μj 1

A: (19)

n o,

<sup>A</sup>: □

ð Þt dt ¼ vð Þ0 holds for any bounded continuous function v tð Þ.

Probabilistic Model of Delay Propagation along the Train Flow

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

179

to fifth and all successive trains approximately like on the fourth one.

�<sup>∞</sup> v tð Þψ<sup>∗</sup><sup>0</sup>

Then, Eq. (16) for k ¼ 2 coincides with Eq. (15).

4. Some results on the knock-on delays

Lemma 1. For each fixed integer m, 1 ≤ m ≤ n � 1,

property: the equality Ð <sup>∞</sup>

under consideration).

present them.

have the equality Ρð Þ¼ N ≥ m G mT ð Þ.

will make this in another work.

where <sup>α</sup> <sup>&</sup>gt; <sup>0</sup>, <sup>β</sup> <sup>&</sup>gt; 0, <sup>Γ</sup>ð Þ¼ <sup>α</sup> <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> x<sup>α</sup>�<sup>1</sup>e�xdx is gamma function. Put

$$
\lambda = 0.3, \ t\_0 = 5, \ \alpha = 14, \ \beta = 0.5. \tag{18}
$$

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

$$\mathcal{W}\_k(t) = I(t > t\_0) \left[ 1 - \frac{\Gamma\left(a, (t - t\_0 + b)/\beta\right)}{\Gamma(a)} + a e^{\lambda(t - t\_0 + b)} \left(\frac{1}{1 + \lambda\beta}\right)^{(k - 1)a} \frac{\Gamma\left(a, \left(1 + \lambda\beta\right)(t - t\_0 + b)/\beta\right)}{\Gamma(a)} \right] \beta$$

where <sup>Γ</sup>ð Þ¼ <sup>α</sup>; <sup>y</sup> <sup>Ð</sup> <sup>∞</sup> <sup>y</sup> x<sup>α</sup>�<sup>1</sup>e�<sup>x</sup> dx is incomplete gamma function. Graphs of the distribution functions Wkð Þt , 2 ≤ k ≤ 5, with the parameters (Eq. (18)) are depicted in Figure 5.

It is not difficult to verify that for Wkð Þt from Example 4 the following formula holds:

$$\begin{split} & \lim\_{k \to \infty} \mathcal{W}\_{k}(t)|\_{a=1,b=0} \\ & \quad = \lim\_{k \to \infty} \left[ I(t>t\_{0}) \left( 1 - \frac{\Gamma\left(a,(t-t\_{0})/\beta\right)}{\Gamma(a)} + e^{\lambda(t-t\_{0})} \left( \frac{1}{1+\lambda\beta} \right)^{(k-1)\alpha} \frac{\Gamma\left(a,\left(1+\lambda\beta\right)(t-t\_{0})/\beta\right)}{\Gamma(a)} \right) \right] \\ & \quad = \mathcal{W}\_{\alpha}(t) \coloneqq I(t>t\_{0}) \left[ 1 - \frac{\Gamma\left(a,(t-t\_{0})/\beta\right)}{\Gamma(a)} \right]. \end{split}$$

Figure 5. Behavior of the functions Wkð Þt , k = 2, 3, 4, 5.

It can be seen from Figure 5, curves W4ð Þt , W5ð Þt and so on are practically merged. Hence, in the case under consideration, one can draw the following conclusion: primary delay τ affects 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 Ð <sup>∞</sup> �<sup>∞</sup> v tð Þψ<sup>∗</sup><sup>0</sup> ð Þt dt ¼ vð Þ0 holds for any bounded continuous function v tð Þ. Then, Eq. (16) for k ¼ 2 coincides 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.
