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:

$$G(\mathbf{x}) \equiv \mathbf{P}(\tau < \mathbf{x}) = I(\mathbf{x} \ge b) \left(1 - a e^{-\lambda(\mathbf{x} - b)}\right) \tag{33}$$

where 0 ≤ a ≤ 1, b ≥ 0, and λ > 0 are some parameters. Such distribution function is considered, for example, in [8]. It is easy to see that G xð Þ¼ ð Þ 1 � a G0ð Þþ x � b aG xð Þ � b; λ , where G0ð Þx is the distribution function of the degenerate distribution concentrated at the point x ¼ 0, G xð Þ¼ ; <sup>λ</sup> I xð Þ <sup>≥</sup> <sup>0</sup> <sup>1</sup> � <sup>e</sup>�λ<sup>x</sup> � �.

Let us find out the form of the distribution functions (Eqs. (13) and (14)) in the case of Eq. (33), when the function Ψ is continuous. In what follows, we mean that n ≥ 3.

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

$$W\_2(t) = I(t > t\_0) \left(\Psi(t - t\_0 + b) + a e^{\lambda(t - t\_0 + b)} \int\_{t - t\_0 + b}^{\infty} e^{-\lambda z} d\Psi(z)\right),\tag{34}$$

$$W\_k(t) = I(t > t\_0) \left\{ \Psi(t - t\_0) + a e^{\lambda(t - t\_0 + b)} \left[ \int\_b^u e^{-\lambda u} d\Psi^{\* (k - 2)}(u) \int\_{t - t\_0}^u e^{-\lambda z} d\Psi(z) \right] \right. \\ \left. + \int\_{t - t\_0}^u e^{-\lambda z} d\Psi(z) \right\} $$

$$+ \int\_{-\infty}^b e^{-\lambda u} \left( \int\_{t - t\_0 - u + b}^u e^{-\lambda z} d\Psi(z) \right) d\Psi^{\* (k - 2)}(u) \Big|\_{t - t\_0} \tag{35}$$

$$+ \int\_{-\infty}^b (\Psi(t - t\_0 - u + b) - \Psi(t - t\_0)) d\Psi^{\* (k - 2)}(u) \Big), \quad k \ge 3.$$

Proof. According to Eq. (33), one may conclude that function G xð Þ has a unique discontinuity point <sup>x</sup> <sup>¼</sup> <sup>b</sup>. Hence, the integral <sup>Ð</sup> <sup>∞</sup> �<sup>∞</sup> G zð Þ � <sup>t</sup> <sup>þ</sup> <sup>t</sup><sup>0</sup> <sup>d</sup>Ψð Þ<sup>z</sup> exists for any continuous distribution function Ψ. Note that if Ψð Þz had a discontinuity point z ¼ t1, then the function G zð Þ � t þ t<sup>0</sup> would also be discontinuous at the point z ¼ t<sup>1</sup> for t ¼ t<sup>0</sup> þ t<sup>1</sup> � b, and then the considered integral would not exist (see Remark 1). Since

$$\overline{G}(\mathbf{x}) = I(\mathbf{x} < b) + I(\mathbf{x} \ge b)ae^{-\lambda(\mathbf{x}-b)},\tag{36}$$

then

Proof of Corollary 9. It follows from Corollary 8 that τ<sup>k</sup> ¼ 0 if τ ≤ μ<sup>k</sup> (see, e.g., Figure 8), and τ<sup>k</sup> ¼ τ � μ<sup>k</sup> if τ > μ<sup>k</sup> (see, e.g., Figure 9a). Using the law of total probability, we obtain the

� � <sup>þ</sup> <sup>Ρ</sup> <sup>τ</sup><sup>k</sup> <sup>&</sup>lt; <sup>t</sup>j<sup>τ</sup> <sup>&</sup>gt; <sup>μ</sup><sup>k</sup>

<sup>¼</sup> I tð Þ <sup>&</sup>gt; <sup>0</sup> <sup>Ρ</sup> <sup>τ</sup> � <sup>μ</sup><sup>k</sup> <sup>≤</sup> <sup>0</sup> � � <sup>þ</sup> I tð Þ <sup>&</sup>gt; <sup>0</sup> <sup>Ρ</sup> <sup>0</sup> <sup>&</sup>lt; <sup>τ</sup> � <sup>μ</sup><sup>k</sup> <sup>&</sup>lt; <sup>t</sup> � � <sup>¼</sup> I tð Þ <sup>&</sup>gt; <sup>0</sup> <sup>Ρ</sup> <sup>τ</sup> � <sup>μ</sup><sup>k</sup> <sup>&</sup>lt; <sup>t</sup> � �: □

� � � �

Proof of Corollary 12. Apply the well-known assertion to Eq. (22): if Y<sup>1</sup> and Y<sup>2</sup> are independent random variables, then for any function of two variables fð Þ �; � and any c∈R, the following

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

ð<sup>t</sup>þ<sup>y</sup> �∞

g zð Þdz � �

> ð∞ �∞

g zð Þdz � �

� �<sup>Ρ</sup> <sup>τ</sup> <sup>&</sup>gt; <sup>μ</sup><sup>k</sup>

�<sup>∞</sup> <sup>Ρ</sup>ð Þ f yð Þ ;Y<sup>2</sup> <sup>&</sup>lt; <sup>c</sup> dF1ð Þ<sup>y</sup> , where <sup>F</sup><sup>1</sup> is the distribution func-

�<sup>∞</sup> <sup>Ρ</sup>ð Þ <sup>τ</sup> � <sup>y</sup> <sup>&</sup>lt; <sup>t</sup> <sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>1</sup> ð Þ<sup>y</sup> . This implies Eq. (25). □

<sup>ψ</sup><sup>∗</sup>ð Þ <sup>k</sup>�<sup>1</sup> ð Þ<sup>y</sup> dy, (29)

g tð Þ <sup>þ</sup> <sup>y</sup> <sup>ψ</sup><sup>∗</sup>ð Þ <sup>k</sup>�<sup>1</sup> ð Þ<sup>y</sup> dy: (30)

<sup>ψ</sup><sup>∗</sup>ð Þ <sup>k</sup>�<sup>1</sup> ð Þ<sup>y</sup> dy, (31)

, (33)

g tð Þ <sup>þ</sup> <sup>y</sup> <sup>ψ</sup><sup>∗</sup>ð Þ <sup>k</sup>�<sup>1</sup> ð Þ<sup>y</sup> dy: (32)

ð Þy is the j-fold

following chain of equalities:

184 Probabilistic Modeling in System Engineering

equality holds: <sup>Ρ</sup>ðf Yð Þ <sup>1</sup>;Y<sup>2</sup> <sup>&</sup>lt; <sup>c</sup>Þ ¼ <sup>Ð</sup> <sup>∞</sup>

Gkð Þ¼ <sup>0</sup><sup>þ</sup> <sup>Ð</sup> <sup>∞</sup>

and we get from Eq. (30),

tion of <sup>Y</sup>1. Consequently, GkðÞ¼ <sup>t</sup> I tð Þ <sup>&</sup>gt; <sup>0</sup> <sup>Ð</sup> <sup>∞</sup>

GkðÞ¼ t Ρð Þ¼ τ<sup>k</sup> < t I tð Þ > 0 Ρð Þ τ<sup>k</sup> < t ¼ I tð Þ > 0 Ρ τ<sup>k</sup> < tjτ ≤ μ<sup>k</sup>

� �<sup>Ρ</sup> <sup>τ</sup> <sup>≤</sup> <sup>μ</sup><sup>k</sup>

Note that the function Gkð Þt has a jump at zero which is equal to:

GkðÞ¼ t I tð Þ > 0

convolution of the density ψð Þ� . In this case, we also have

gkð Þt ≔ I tð Þ > 0 G<sup>0</sup>

If we assume that τ ≥ 0, then we deduce from Eq. (29) that

GkðÞ¼ t I tð Þ > 0

mixture of exponential and one-point distributions

Consider the cumulative distribution function of the following type:

gkðÞ¼ t I tð Þ > 0

�<sup>∞</sup> <sup>G</sup>þð Þ<sup>y</sup> <sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>1</sup> ð Þ<sup>y</sup> , where <sup>G</sup>þð Þ¼ <sup>y</sup> lim<sup>t</sup>!0<sup>þ</sup> G tð Þ <sup>þ</sup> <sup>y</sup> .

In the case, when τ and μ<sup>j</sup> are absolutely continuous, it follows from Eq. (25) that

ð∞ �∞

where <sup>g</sup>ð Þ� and <sup>ψ</sup>ð Þ� are the density functions of <sup>τ</sup> and <sup>μ</sup>1, respectively, <sup>ψ</sup><sup>∗</sup><sup>j</sup>

<sup>k</sup>ðÞ¼ t I tð Þ > 0

ð∞ �t ð<sup>t</sup>þ<sup>y</sup> 0

ð∞ �t

6. Corollary of Theorem 2 when the distribution of primary delay is a

G xð Þ� <sup>Ρ</sup>ð Þ¼ <sup>τ</sup> <sup>&</sup>lt; <sup>x</sup> I xð Þ <sup>≥</sup> <sup>b</sup> <sup>1</sup> � ae�λð Þ <sup>x</sup>�<sup>b</sup> � �

$$\int\_{-\infty}^{\infty} \overline{G}(z - t + t\_0) d\Psi(z) = \Psi(t - t\_0 + b) + a e^{\lambda(t - t\_0 + b)} \int\_{t - t\_0 + b}^{\infty} e^{-\lambda z} d\Psi(z). \tag{37}$$

In accordance with Eq. (13), the relation (Eq. (34)) is proved. Let k ≥ 3. It follows from Eq. (14) that

$$\mathcal{W}\_k(t) = I(t > t\_0) \left[ \Psi(t - t\_0) + \int\_{-\infty}^{\infty} V(u) d\Psi^{\*(k-2)}(u) \right],\tag{38}$$

where V uð Þ¼ <sup>Ð</sup> <sup>∞</sup> <sup>t</sup>�t<sup>0</sup> G zð Þ <sup>þ</sup> <sup>u</sup> � <sup>t</sup> <sup>þ</sup> <sup>t</sup><sup>0</sup> <sup>d</sup>Ψð Þ<sup>z</sup> . Given Eq. (36), it is easy to see that

$$V(\mu) = V\_1(\mu) + V\_2(\mu),\tag{39}$$

<sup>V</sup>1ð Þ¼ <sup>u</sup> ae�λð Þ <sup>u</sup>�tþt0�<sup>b</sup> <sup>Ð</sup> <sup>∞</sup> <sup>t</sup>�t<sup>0</sup> I zð Þ <sup>þ</sup> <sup>u</sup> � <sup>t</sup> <sup>þ</sup> <sup>t</sup><sup>0</sup> <sup>≥</sup> <sup>b</sup> <sup>e</sup>�<sup>λ</sup>zdΨð Þ<sup>z</sup> , <sup>V</sup>2ð Þ¼ <sup>u</sup> <sup>Ð</sup> <sup>∞</sup> <sup>t</sup>�t<sup>0</sup> I zð þ <sup>þ</sup> <sup>u</sup> � <sup>t</sup> <sup>t</sup><sup>0</sup> <sup>&</sup>lt; <sup>b</sup>ÞdΨð Þ<sup>z</sup> . By using equalities

$$\begin{aligned} \{ (u, z) : u \ge b, z > t - t\_0, z \ge t - t\_0 - u + b \} &= \{ (u, z) : u \ge b, z > t - t\_0 \}, \\ \{ (u, z) : u < b, z > t - t\_0, z \ge t - t\_0 - u + b \} &= \{ (u, z) : u < b, z \ge t - t\_0 - u + b \} \end{aligned}$$

we receive

$$\begin{split} \int\_{-\infty}^{\infty} V\_1(u) d\Psi^{\*(k-2)}(u) &= a e^{\lambda(t-t\_0+b)} \left[ \int\_{b}^{\infty} \left( \int\_{t-t\_0}^{\infty} e^{-\lambda(z+u)} d\Psi(z) \right) d\Psi^{\*(k-2)}(u) \\ &+ \int\_{-\infty}^{b} \left( \int\_{t-t\_0-u+b}^{\infty} e^{-\lambda(z+u)} d\Psi(z) \right) d\Psi^{\*(k-2)}(u) \right]. \end{split} \tag{40}$$

<sup>D</sup>ν<sup>k</sup> <sup>¼</sup> <sup>I</sup>ð Þ <sup>0</sup> <sup>≤</sup> <sup>b</sup> <sup>≤</sup> ð Þ <sup>k</sup> � <sup>2</sup> <sup>T</sup> <sup>a</sup>

for the following parameters:

λ2 e

<sup>þ</sup> I k ð Þ ð Þ � <sup>2</sup> <sup>T</sup> <sup>&</sup>lt; <sup>b</sup> <sup>&</sup>lt; ð Þ <sup>k</sup> � <sup>1</sup> <sup>T</sup> <sup>a</sup>

�λð Þ ð Þ <sup>k</sup>�<sup>2</sup> <sup>T</sup>�<sup>b</sup> 2 1 � <sup>e</sup>

λ2 e

� ae�λð Þ ð Þ <sup>k</sup>�<sup>2</sup> <sup>T</sup>�<sup>b</sup> <sup>e</sup><sup>λ</sup>ð Þ ð Þ <sup>k</sup>�<sup>2</sup> <sup>T</sup>�<sup>b</sup> � <sup>e</sup>�λ<sup>T</sup> � �<sup>2</sup> � <sup>2</sup>λð Þ ð Þ <sup>k</sup> � <sup>1</sup> <sup>T</sup> � <sup>b</sup> <sup>e</sup>�λ<sup>T</sup>

�λð Þ ð Þ <sup>k</sup>�<sup>2</sup> <sup>T</sup>�<sup>b</sup> 2 e

Example 5. Figure 10 depicts the graphs of the functions Wkð Þt defined by Eq. (43) with k = 2, 3

Remark 4. It can be easily seen that the larger k, Wkð Þt from Eq. (43) is closer to W tð Þ ≔ I bð Þ ≥ 0 I tð Þ > t<sup>0</sup> þ T . This agrees with Figure 10 and the formulas (44) and (45) due to which we have Εν<sup>k</sup> ! t<sup>0</sup> þ T, Dν<sup>k</sup> ! 0 as k ! ∞, and also with the results of calculations inTable 1.

Let the random variable τ be distributed with the density (Eq. (33)) with parameters a ¼ 1, b ¼ 0. Now, we find the condition on the parameter T, under which the probability that at least

k ¼ 2 k ¼ 3 k ¼ 5 k ¼ 8 k ¼ 10

Εν<sup>k</sup> 7.77702 10.47779 10.98629 10.99994 10.99999 <sup>D</sup>ν<sup>k</sup> 5.68009 2.33067 0.068156 0.00029 7.63176 � <sup>10</sup>�<sup>6</sup>

Table 1. The behavior Εν<sup>k</sup> and Dν<sup>k</sup> with growth of the parameter k.

Figure 10. Behavior of the functions W2ð Þt and W3ð Þt .

We calculated the values of Εν<sup>k</sup> and Dν<sup>k</sup> using the formulas (44) and (45) (see Table 1).

�λ<sup>T</sup> � � � ae�λð Þ ð Þ <sup>k</sup>�<sup>2</sup> <sup>T</sup>�<sup>b</sup> <sup>1</sup> � <sup>e</sup>

� <sup>2</sup>λTe�λ<sup>T</sup> h i

i :

<sup>λ</sup>ð Þ ð Þ <sup>k</sup>�<sup>2</sup> <sup>T</sup>�<sup>b</sup> � <sup>e</sup> �λ<sup>T</sup> h � �

a ¼ 1, b ¼ 0, λ ¼ 0:26, t<sup>0</sup> ¼ 4, T ¼ 7: (46)

�λ<sup>T</sup> � �<sup>2</sup>

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

Probabilistic Model of Delay Propagation along the Train Flow

(45)

187

Since f g ð Þ u; z : u ≥ b; z > t � t0; z < t � t<sup>0</sup> � u þ b ¼ ∅,

$$\{(u, z) : u < b, z > t - t\_0, z < t - t\_0 - u + b\} = \{(u, z) : u < b, t - t\_0 < z < t - t\_0 - u + b\}.$$

then

$$\int\_{-\infty}^{\infty} V\_2(u) d\Psi^{\*(k-2)}(u) = \int\_{-\infty}^{b} \left( \int\_{t-t\_0}^{t-t\_0-u+b} d\, \Psi(z) \right) d\Psi^{\*(k-2)}(u). \tag{41}$$

It follows from Eqs. (39)–(41) that

$$\begin{split} \int\_{-\kappa}^{\kappa} V(u) d\Psi^{s(k-2)}(u) &= a e^{\lambda(t-t\_0+b)} \left[ \int\_{b}^{\kappa} e^{-\lambda u} d\Psi^{s(k-2)}(u) \int\_{t-t\_0}^{\kappa} e^{-\lambda z} d\Psi(z) \\ &+ \int\_{-\kappa}^{b} \left( \int\_{t-t\_0-u+b}^{\kappa} e^{-\lambda z} d\Psi(z) \right) e^{-\lambda u} d\Psi^{s(k-2)}(u) \right] \\ &+ \int\_{-\kappa}^{b} \left( \Psi(t-t\_0-u+b) - \Psi(t-t\_0) \right) d\Psi^{s(k-2)}(u). \end{split} \tag{42}$$

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 μ<sup>j</sup> are not random variables, and they are equal to the same constant.

Corollary 14. Let μ<sup>j</sup> ¼ T > 0, 2 ≤ j ≤ n, be a constant. Let function G be defined by Eq. (33). Then, for 2 ≤ k ≤ n, the following formula holds:

$$\begin{split} W\_{k}(t) &= I(0 \le b \le (k-2)T) \left[ I(0 < t - t\_{0} \le T) a e^{-\lambda((k-1)T - t + t\_{0} - b)} + I(t - t\_{0} > T) \right] \\ &+ I((k-2)T < b < (k-1)T) \left[ I(0 < t - t\_{0} \le (k-1)T - b) a e^{-\lambda((k-1)T - t + t\_{0} - b)} \\ &+ I(t - t\_{0} > (k-1)T - b) \right] + I(b \ge (k-1)T) I(t > t\_{0}). \end{split} \tag{43}$$

Furthermore,

$$\begin{split} \mathrm{E}\nu\_{k} &= I(0 \le b \le (k-2)T) \left[ t\_{0} + T - \frac{a}{\lambda} e^{-\lambda((k-2)T - b)} \left( 1 - e^{-\lambda T} \right) \right] + I(b \ge (k-1)T) t\_{0} \\ &+ I((k-2)T < b < (k-1)T) \left[ t\_{0} + (k-1)T - b + \frac{a}{\lambda} \left( e^{-\lambda((k-1)T - b)} - 1 \right) \right]. \end{split} \tag{44}$$

$$\begin{split} \mathbf{D}\nu\_{k} &= I(0 \le b \le (k-2)T) \frac{a}{\lambda^{2}} e^{-\lambda((k-2)T-b)} \Big[ 2\left(1 - e^{-\lambda T}\right) - a e^{-\lambda((k-2)T-b)} \left(1 - e^{-\lambda T}\right)^{2} - 2\lambda T e^{-\lambda T} \Big] \\ &+ I((k-2)T < b < (k-1)T) \frac{a}{\lambda^{2}} e^{-\lambda((k-2)T-b)} \Big[ 2\left(e^{\lambda((k-2)T-b)} - e^{-\lambda T}\right) \\ &- a e^{-\lambda((k-2)T-b)} \left(e^{\lambda((k-2)T-b)} - e^{-\lambda T}\right)^{2} - 2\lambda((k-1)T-b)e^{-\lambda T} \Big]. \end{split} \tag{45}$$

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

$$a = 1, \ b = 0, \ \lambda = 0.26, \ t\_0 = 4, \ T = 7. \tag{46}$$

We calculated the values of Εν<sup>k</sup> and Dν<sup>k</sup> using the formulas (44) and (45) (see Table 1).

Remark 4. It can be easily seen that the larger k, Wkð Þt from Eq. (43) is closer to W tð Þ ≔ I bð Þ ≥ 0 I tð Þ > t<sup>0</sup> þ T . This agrees with Figure 10 and the formulas (44) and (45) due to which we have Εν<sup>k</sup> ! t<sup>0</sup> þ T, Dν<sup>k</sup> ! 0 as k ! ∞, and also with the results of calculations inTable 1.

Let the random variable τ be distributed with the density (Eq. (33)) with parameters a ¼ 1, b ¼ 0. Now, we find the condition on the parameter T, under which the probability that at least

Figure 10. Behavior of the functions W2ð Þt and W3ð Þt .

By using equalities

ð∞ �∞

186 Probabilistic Modeling in System Engineering

we receive

then

ð∞ �∞ fð Þ u; z : u ≥ b; z > t � t0; z ≥ t � t<sup>0</sup> � u þ bg ¼ f g ð Þ u; z : u ≥ b; z > t � t<sup>0</sup> ,

<sup>V</sup>1ð Þ <sup>u</sup> <sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ¼ <sup>u</sup> ae<sup>λ</sup>ð Þ <sup>t</sup>�t0þ<sup>b</sup>

<sup>V</sup>2ð Þ <sup>u</sup> <sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ¼ <sup>u</sup>

ð∞ b e

t�t0�uþb

e �λz dΨð Þz

� �

�

ð∞

Since f g ð Þ u; z : u ≥ b; z > t � t0; z < t � t<sup>0</sup> � u þ b ¼ ∅,

ð∞ �∞

It follows from Eqs. (39)–(41) that

2 ≤ k ≤ n, the following formula holds:

Furthermore,

V uð ÞdΨ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ¼ <sup>u</sup> ae<sup>λ</sup>ð Þ <sup>t</sup>�t0þ<sup>b</sup>

þ ðb �∞

<sup>þ</sup> <sup>Ð</sup> <sup>b</sup>

variables, and they are equal to the same constant.

<sup>Ε</sup>ν<sup>k</sup> <sup>¼</sup> <sup>I</sup>ð Þ <sup>0</sup> <sup>≤</sup> <sup>b</sup> <sup>≤</sup> ð Þ <sup>k</sup> � <sup>2</sup> <sup>T</sup> <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>T</sup> � <sup>a</sup>

þ ðb �∞

f g ð Þ u; z : u < b; z > t � t0; z ≥ t � t<sup>0</sup> � u þ b ¼ f g ð Þ u; z : u < b; z ≥ t � t<sup>0</sup> � u þ b ,

ð∞ b

t�t0�uþb

f g ð Þ u; z : u < b; z > t � t0; z < t � t<sup>0</sup> � u þ b ¼ f g ð Þ u; z : u < b; t � t<sup>0</sup> < z < t � t<sup>0</sup> � u þ b ,

ðb �∞

�<sup>λ</sup>udΨ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup>

�

ð∞

ð∞ t�t<sup>0</sup> e

e

ð<sup>t</sup>�t0�uþ<sup>b</sup> t�t<sup>0</sup>

� �

ð∞ t�t<sup>0</sup> e �λz dΨð Þz

e

�<sup>∞</sup> ð Þ <sup>Ψ</sup>ð<sup>t</sup> � <sup>t</sup><sup>0</sup> � <sup>u</sup> <sup>þ</sup> <sup>b</sup>Þ � <sup>Ψ</sup>ð Þ <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup> :

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 μ<sup>j</sup> are not random

Corollary 14. Let μ<sup>j</sup> ¼ T > 0, 2 ≤ j ≤ n, be a constant. Let function G be defined by Eq. (33). Then, for

<sup>þ</sup> I k ð Þ ð Þ � <sup>2</sup> <sup>T</sup> <sup>&</sup>lt; <sup>b</sup> <sup>&</sup>lt; ð Þ <sup>k</sup> � <sup>1</sup> <sup>T</sup> <sup>I</sup>ð Þ <sup>0</sup> <sup>&</sup>lt; <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>≤</sup> ð Þ <sup>k</sup> � <sup>1</sup> <sup>T</sup> � <sup>b</sup> ae�λð Þ ð Þ <sup>k</sup>�<sup>1</sup> <sup>T</sup>�tþt0�<sup>b</sup> �

�λð Þ ð Þ <sup>k</sup>�<sup>2</sup> <sup>T</sup>�<sup>b</sup> <sup>1</sup> � <sup>e</sup> �λ<sup>T</sup> � � h i

WkðÞ¼ <sup>t</sup> <sup>I</sup>ð Þ <sup>0</sup> <sup>≤</sup> <sup>b</sup> <sup>≤</sup> ð Þ <sup>k</sup> � <sup>2</sup> <sup>T</sup> <sup>I</sup>ð Þ <sup>0</sup> <sup>&</sup>lt; <sup>t</sup> � <sup>t</sup><sup>0</sup> <sup>≤</sup> <sup>T</sup> ae�λð Þ ð Þ <sup>k</sup>�<sup>1</sup> <sup>T</sup>�tþt0�<sup>b</sup> <sup>þ</sup> I tð Þ � <sup>t</sup><sup>0</sup> <sup>&</sup>gt; <sup>T</sup> � �

λ e

þ I tð � t<sup>0</sup> > ð Þ k � 1 T � bÞ� þ I bð Þ ≥ ð Þ k � 1 T I tð Þ > t<sup>0</sup> :

þ I k ð Þ ð Þ � 2 T < b < ð Þ k � 1 T t<sup>0</sup> þ ð Þ k � 1 T � b þ

� �

�λð Þ <sup>z</sup>þ<sup>u</sup> <sup>d</sup>Ψð Þ<sup>z</sup> � �

�λð Þ <sup>z</sup>þ<sup>u</sup> <sup>d</sup>Ψð Þ<sup>z</sup>

dΨð Þz

�<sup>λ</sup>udΨ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup>

�

<sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup>

� :

<sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup> : (41)

þ I bð Þ ≥ ð Þ k � 1 T t<sup>0</sup>

�λð Þ ð Þ <sup>k</sup>�<sup>1</sup> <sup>T</sup>�<sup>b</sup> � <sup>1</sup>

:

a <sup>λ</sup> <sup>e</sup>

h i � �

(40)

(42)

(43)

(44)

<sup>d</sup>Ψ<sup>∗</sup>ð Þ <sup>k</sup>�<sup>2</sup> ð Þ <sup>u</sup>


Table 1. The behavior Εν<sup>k</sup> and Dν<sup>k</sup> with growth of the parameter k.

m of knock-on delays will occur would not exceed a given probability p. Note that the departure headway is equal to T þ t0.

According to Corollary 6, it is necessary to solve the inequality exp ð Þ �λmT ≤ p. As a result, we obtain the desired condition:

$$T \ge (1/(m\lambda))\ln\left(1/p\right) \tag{47}$$

Corollary 15 can be reformulated as follows.

the form Eq. (33) with a <sup>¼</sup> λβ <sup>þ</sup> <sup>1</sup> �ð Þ <sup>k</sup>�<sup>1</sup> <sup>α</sup>

3 ≤ k ≤ n. Hence, Ρð Þ! τ<sup>k</sup> ¼ 0 1 as k ! ∞.

λ is equal to 0.25 and αβ ¼ 7 as it observes in reality.

Figure 12. Behavior of distribution G2ð Þt when (a) α ¼ 0:5, 3, 8 and (b) β ¼ 0:1, 0:5, 1.

Figure 13. Behavior of distribution G3ð Þt when (a) α ¼ 0:5, 3, 8 and (b) β ¼ 0:1, 0:5, 1.

Corollary 15\*. Let primary delay τ is exponentially distributed with a parameter λ, and μk, 2 ≤ k ≤ n, have the same gamma distribution with the density (Eq. (17)). Then, τ<sup>k</sup> has the distribution function of

<sup>Ρ</sup>ð Þ¼ <sup>τ</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup> Gkð Þ¼ <sup>0</sup><sup>þ</sup> <sup>1</sup> � λβ <sup>þ</sup> <sup>1</sup> �ð Þ <sup>k</sup>�<sup>1</sup> <sup>α</sup>

Remark 6. Let <sup>Ρ</sup>ð Þ¼ <sup>τ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> <sup>p</sup>, 0 <sup>&</sup>lt; <sup>p</sup> <sup>&</sup>lt; <sup>1</sup>: Then by Corollary 15\*, <sup>Ρ</sup>ð Þ¼ <sup>τ</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup> <sup>1</sup> � ð Þ <sup>1</sup> � <sup>p</sup> <sup>k</sup>�<sup>1</sup>

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 τ<sup>k</sup> with various combinations of parameters: α, β, k. The results are presented in graphical form in Figures 12–15 The functions Gkð Þt are calculated by formula (49), and the functions gkð Þt by formula (50). Note that product αβ is the mean of μk. Parameter

, b = 0, and, consequently,

:

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,

189

(see also [13]). Denote by T mð Þ ; p; λ the minimal T satisfying the inequality (Eq. (47)).

Example 6. Let us fix λ ¼ 0:26. The behavior of T mð Þ ; p; λ as a function of the continuous parameter m with p ¼ 0:1 and p ¼ 0:05 is shown in Figure 11a. Obviously, T mð Þ ; p; λ is the decreasing function with respect to the argument p. Exact calculations can be made using the formula:

$$T(m, p, \lambda) = (1/(m\lambda)) \ln\left(1/p\right). \tag{48}$$

Let p ¼ 0:1. The behavior of T mð Þ ; p; λ as a function of the continuous parameter m with λ ¼ 0:26 and λ ¼ 0:15 is shown in Figure 11b. In accordance with Eq. (48), T mð Þ ; p; λ is the decreasing function with respect to the argument λ. In the case of exponential density g tð Þ, we have Ετ ¼ 1=λ. Therefore, the decrease of λ leads 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 μ<sup>j</sup> are distributed according to the gamma-law with the density (Eq. (17)).

Corollary 15. If primary delay <sup>τ</sup> has an exponential distribution g tðÞ¼ I tð Þ <sup>&</sup>gt; <sup>0</sup> <sup>λ</sup>e�λ<sup>t</sup> and <sup>μ</sup>k, <sup>2</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> n, has the density (Eq. (17)), then the following formulas are true:

$$G\_k(t) = I(t>0)\left(1 - e^{-\lambda t} \left(\lambda \beta + 1\right)^{-(k-1)a}\right) \tag{49}$$

$$\mathcal{g}\_k(t) = I(t>0) \left(\lambda \beta + 1\right)^{-(k-1)a} \lambda e^{-\lambda t}.\tag{50}$$

Remark 5. The function gkðÞ¼ t I tð Þ > 0 G<sup>0</sup> <sup>k</sup>ð Þt is not a density, in particular, because of Ð ∞ �<sup>∞</sup> gkð Þ<sup>t</sup> dt <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> gkð Þt dt ¼ Gkð Þ� ∞ Gkð Þ¼ 0þ 1 � hk 6¼ 1, where hk is the jump of the function Gkð Þ<sup>t</sup> at the origin. At the same time, the function <sup>~</sup>gkð Þ<sup>t</sup> <sup>≔</sup> <sup>1</sup> 1�hk gkð Þt is a density.

Figure 11. The behavior of the function T mð Þ ; p; λ : (a) λ ¼ 0:26, p ¼ 0:1, or p ¼ 0:05; (b) p ¼ 0:1, λ ¼ 0:26, or λ ¼ 0:15.

Corollary 15 can be reformulated as follows.

m of knock-on delays will occur would not exceed a given probability p. Note that the

According to Corollary 6, it is necessary to solve the inequality exp ð Þ �λmT ≤ p. As a result, we

Example 6. Let us fix λ ¼ 0:26. The behavior of T mð Þ ; p; λ as a function of the continuous parameter m with p ¼ 0:1 and p ¼ 0:05 is shown in Figure 11a. Obviously, T mð Þ ; p; λ is the decreasing function with respect to the argument p. Exact calculations can be made using the formula:

Let p ¼ 0:1. The behavior of T mð Þ ; p; λ as a function of the continuous parameter m with λ ¼ 0:26 and λ ¼ 0:15 is shown in Figure 11b. In accordance with Eq. (48), T mð Þ ; p; λ is the decreasing function with respect to the argument λ. In the case of exponential density g tð Þ, we have Ετ ¼ 1=λ. Therefore, the decrease of λ leads to increase in the average of primary delay

We also obtain the corollaries of Lemma 3 in the case when μ<sup>j</sup> are distributed according to the

Corollary 15. If primary delay <sup>τ</sup> has an exponential distribution g tðÞ¼ I tð Þ <sup>&</sup>gt; <sup>0</sup> <sup>λ</sup>e�λ<sup>t</sup> and <sup>μ</sup>k, <sup>2</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> n,

gkðÞ¼ <sup>t</sup> I tð Þ <sup>&</sup>gt; <sup>0</sup> λβ <sup>þ</sup> <sup>1</sup> � ��ð Þ <sup>k</sup>�<sup>1</sup> <sup>α</sup>

Figure 11. The behavior of the function T mð Þ ; p; λ : (a) λ ¼ 0:26, p ¼ 0:1, or p ¼ 0:05; (b) p ¼ 0:1, λ ¼ 0:26, or λ ¼ 0:15.

�λ<sup>t</sup> λβ <sup>þ</sup> <sup>1</sup> � ��ð Þ <sup>k</sup>�<sup>1</sup> <sup>α</sup> � �

<sup>0</sup> gkð Þt dt ¼ Gkð Þ� ∞ Gkð Þ¼ 0þ 1 � hk 6¼ 1, where hk is the jump of the function

λe �λt

1�hk

<sup>k</sup>ð Þt is not a density, in particular, because of

gkð Þt is a density.

(see also [13]). Denote by T mð Þ ; p; λ the minimal T satisfying the inequality (Eq. (47)).

and the departure headways (if we want to reduce the number of knock-on delays).

GkðÞ¼ t I tð Þ > 0 1 � e

T ≥ ð Þ 1=ð Þ mλ ln 1ð Þ =p (47)

T mð Þ¼ ; p; λ ð Þ 1=ð Þ mλ ln 1ð Þ =p : (48)

, (49)

: (50)

departure headway is equal to T þ t0.

gamma-law with the density (Eq. (17)).

Remark 5. The function gkðÞ¼ t I tð Þ > 0 G<sup>0</sup>

Ð ∞

�<sup>∞</sup> gkð Þ<sup>t</sup> dt <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup>

has the density (Eq. (17)), then the following formulas are true:

Gkð Þ<sup>t</sup> at the origin. At the same time, the function <sup>~</sup>gkð Þ<sup>t</sup> <sup>≔</sup> <sup>1</sup>

obtain the desired condition:

188 Probabilistic Modeling in System Engineering

Corollary 15\*. Let primary delay τ is exponentially distributed with a parameter λ, and μk, 2 ≤ k ≤ n, have the same gamma distribution with the density (Eq. (17)). Then, τ<sup>k</sup> has the distribution function of the form Eq. (33) with a <sup>¼</sup> λβ <sup>þ</sup> <sup>1</sup> �ð Þ <sup>k</sup>�<sup>1</sup> <sup>α</sup> , b = 0, and, consequently,

$$\mathbf{P}(\tau\_k = \mathbf{0}) = \mathbf{G}\_k(\mathbf{0} + ) = \mathbf{1} - \left(\lambda\beta + 1\right)^{-(k-1)\alpha}.$$

Remark 6. Let <sup>Ρ</sup>ð Þ¼ <sup>τ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> <sup>p</sup>, 0 <sup>&</sup>lt; <sup>p</sup> <sup>&</sup>lt; <sup>1</sup>: Then by Corollary 15\*, <sup>Ρ</sup>ð Þ¼ <sup>τ</sup><sup>k</sup> <sup>¼</sup> <sup>0</sup> <sup>1</sup> � ð Þ <sup>1</sup> � <sup>p</sup> <sup>k</sup>�<sup>1</sup> , 3 ≤ k ≤ n. Hence, Ρð Þ! τ<sup>k</sup> ¼ 0 1 as 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 τ<sup>k</sup> with various combinations of parameters: α, β, k. The results are presented in graphical form in Figures 12–15 The functions Gkð Þt are calculated by formula (49), and the functions gkð Þt by formula (50). Note that product αβ is the mean of μk. Parameter λ is equal to 0.25 and αβ ¼ 7 as it observes in reality.

Figure 12. Behavior of distribution G2ð Þt when (a) α ¼ 0:5, 3, 8 and (b) β ¼ 0:1, 0:5, 1.

Figure 13. Behavior of distribution G3ð Þt when (a) α ¼ 0:5, 3, 8 and (b) β ¼ 0:1, 0:5, 1.

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

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

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

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

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

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

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,

d G2ð Þ� <sup>t</sup> <sup>a</sup><sup>2</sup>

Ð ∞ �<sup>∞</sup> <sup>t</sup>

<sup>λ</sup><sup>2</sup> λ β <sup>þ</sup> <sup>1</sup> � ��0:<sup>6</sup> � <sup>1</sup>

<sup>2</sup> ¼ ð∞ 0 t

<sup>2</sup> d G2ðÞ¼ <sup>t</sup> <sup>2</sup>

<sup>2</sup> <sup>g</sup>2ð Þ<sup>t</sup> dt � <sup>a</sup><sup>2</sup>

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<sup>λ</sup><sup>2</sup> λ β <sup>þ</sup> <sup>1</sup> � ��0:<sup>6</sup>

<sup>λ</sup><sup>2</sup> λ β <sup>þ</sup> <sup>1</sup> � ��1:<sup>2</sup> <sup>≈</sup> <sup>10</sup>:987:

<sup>2</sup> ≈ 10:987:

≈ 14:088,

the function G2ð Þt with α ¼ 0:5 from Figure 12.

influence on behavior of output distribution (see Figures 12–15).

at the Tver station in the period of January 2016 and February 2016.

d G2ðÞ¼ t

<sup>2</sup> <sup>¼</sup> <sup>2</sup>

<sup>λ</sup> λ β <sup>þ</sup> <sup>1</sup> � ��0:<sup>6</sup>

ð∞ �∞ t 2

≈ 1:763,

β ¼ 11:7. Using formulas (49) and (50) with k ¼ 2, we have:

ð Þ t � a<sup>2</sup> 2

<sup>2</sup> <sup>g</sup>2ð Þ<sup>t</sup> dt � <sup>a</sup><sup>2</sup>

each particular situation allows to predict the distribution of knock-on delays.

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

distribution of the delay τ2.

SD<sup>2</sup> <sup>¼</sup>

Here <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> <sup>∞</sup>

ð∞ �∞

�<sup>∞</sup> tdG2ðÞ¼ <sup>t</sup> <sup>1</sup>

ð∞ 0 t

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