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In this chapter, we propose a probabilistic model for train delay propagation. There are deduced formulas for the probability distributions of arrival headways and knockon 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.
*Address all correspondence to: kseniya0407@mail.ru
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.
A substantial volume of literature is devoted to study of the train delays effect on the railway functioning. Deterministic models for primary and knockon 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>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.
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 Tm of departure train m can be found as (as shown at Figure 1):
Tm=∑j=2mμj+m−1t0,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 knockon 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 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=0 and 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, νk is actual departure headway between the trains with numbers (k − 1) and k. It is required to determine the distribution functions Wkt of 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=3k−1, k=1,5¯. Figure 2 shows the process of headways νk forming, 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) Tk−Tk−1>t0, k = 2, 3, …, n.
Denote by Rk the real departure time of the train with number k, which depends on τ and t0.
We suppose that the departure times of trains satisfy the following two rules. Let k be fixed, 2≤k≤n. The first rule: if Rk−1≤Tk−t0, then Rk=Tk. The second rule: if Rk−1≥Tk−t0, then Rk=Rk−1+t0. Obviously, Rk≥Tk.
In what follows, we use the notation Ix∈A=1,ifx∈A,0,ifx∈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 νk=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 and will be published in our other work.
Let us introduce the notations, Gx=Ρτ<x, G¯x=Ρτ>x. Note that Gx+G¯x+Ρτ=x=1. We denote by gx the density function of τ in the case when it is absolutely continuous.
Further, some corollaries of Theorem 2 are formulated.
Corollary 1. Letμj,2≤j≤n, be arbitrary positive numbers, then for2≤k≤n
Wkt=It0<t≤μk+t0G¯∑j=2kμj−t+t0+It>μk+t0,E6
in particular,
W2t=It>t0G¯μ2−t+t0.E7
Example 2. Let the primary delay τ have exponential distribution, that is,
gt=It≥0λe−λt,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 W2t from Eq. (7), W3t and W4t 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: μk, Tk, t0, τ, τk, νk, T, b, Ενk (minutes, min); λ (1/min); Dνk (min^{2}). The product αβ (as mean of μk), where α is a shape parameter, β is a scale parameter (in min).
Corollary 2. Letμj=T,2≤j≤n, be a positive constant, then for2≤k≤n
Wkt=It0<t≤T+t0G¯k−1T−t+t0+It>T+t0,E10
in particular,
W2t=It>t0G¯T−t+t0.E11
Example 3. Let τ has density (Eq. (8)). As initial parameters, we take the following quantities:
Figures 3 and 4 show that in the case of constant μj, the primary delay τ practically does not affect the fourth train and all subsequent ones. This is consistent with the equality limk→∞Wkt=It>t0+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 F1 and F2 is determined by the formula F1∗F2x=∫−∞∞F1x−ydF2y, where the integral sign means the improper RiemannStieltjes integral. We consider exceptionally piecewisecontinuous distribution functions, then the indicated integral exists with the exception of the case when F1 and F2 have 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: F∗2≔F∗F, F∗m≔F∗F∗m−1, m≥2. By definition, we assume that F∗1≔F. The convolution f1∗f2x of densities f1 and f2 is defined as the improper Riemann integral ∫−∞∞f1x−yf2ydy.
Corollary 3. Letμj,2≤j≤n, be independent identically distributed random variables with a continuous distribution functionΨx. Letτbe independent ofμj,2≤j≤n. Then
Corollary 4. Letμj,2≤j≤n, be independent identically distributed random variables with a density functionψx. Letτbe independent of allμjand has a density functiongx. Then
Remark 2. The integration limit “−∞” can be replaced by 0 in Corollaries 3 and 4 if μj≥0. On the other hand, we may consider in these corollaries the case when μj 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 μj 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 μj have the same gamma density
ψt=It>0e−t/βtα−1Γαβα,E17
where α>0,β>0, Γα=∫0∞xα−1e−xdx is 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
where Γαy=∫y∞xα−1e−xdx is incomplete gamma function. Graphs of the distribution functions Wkt,2≤k≤5, with the parameters (Eq. (18)) are depicted in Figure 5.
It is not difficult to verify that for Wkt from Example 4 the following formula holds:
It can be seen from Figure 5, curves W4t,W5t 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 0fold convolution as a generalized function with the following property: the equality ∫−∞∞vtψ∗0tdt=v0 holds for any bounded continuous function vt. Then, Eq. (16) for k=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.
m = 1, 2, …, n – 2, N=n−1=τ+n−1t0>Tn. This implies that
ΡN≥m=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. Ifμj=Tis a constant value,2≤j≤n, then for every fixed integer m,1≤m≤n−1, we have the equalityΡN≥m=G¯mT.
Corollary 6. Ifμj=Tis a constant value,2≤j≤n, andτis exponentially distributed with parameterλ, then for every fixed integer m,1≤m≤n−1, the following equality holds,
ΡN≥m=e−λmT.
Corollary 7. If μ2, …,μnare independent identically distributed random variables with a density functionψ, then for every fixed integer m,1≤m≤n−1, we have the equality
ΡN≥m=∫−∞∞G¯uψ∗mudu.
In what follows, τ1=τ is the delay duration of the first train, τk, k = 2, …, n, is the knockon delay of the kth 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 a∨b instead of maxab.
Theorem 3. The following formula holds:
τk=τk−1−μk∨0,2≤k≤n.E20
Corollary 8. The following formula holds:
τk=τ−∑j=2kμj∨0,2≤k≤n.E21
It should be noted that within the framework of our model the deviation of the real arrival time from the planned one for kth train coincides with τk, 1≤k≤n. 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 τ and the train 2 on the τ2.
Denote μ¯k=∑j=2kμj, 2≤k≤n. As it follows from the assumption that the random variables μ2,…,μk have the same distribution function Ψt. They are mutually independent. The random variable μ¯k has the distribution function Ψ∗k−1t.
Corollary 9. The distribution function ofτkhas the following form:
Gkt=It>0Ρτ−μ¯k<t,2≤k≤n.E22
The next Corollaries 10 and 11 follow from Corollary 9 in an obvious way.
Corollary 10. Let μj>0,2≤j≤nbe some constant values. Then
Gkt=It>0Gt+μ¯k.E23
Corollary 11. Letμj=T>0,2≤j≤nbe a constant value. Then
Gkt=It>0Gt+k−1T.E24
Corollary 12. Letμj,2≤j≤nbe independent identically distributed random variables with a continuous distribution functionΨt. Letτbe independent ofμ2,…,μn. Then
Gkt=It>0∫−∞∞Gt+ydΨ∗k−1y.E25
Corollary 13. Letμj,2≤j≤nbe independent identically distributed random variables with a density functionψt. Letτbe independent ofμj,2≤j≤nand has a density functiongt. ThenGkt=It>0∫−∞∞∫−∞t+ygzdzψ∗k−1ydy.
Proof. Let t>0 be 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=0 holds under the condition τ≤μ2. The departure time of the train 1 after stopping is t+τ. The time point when train 2 reaches s can be written as μ2+t0+t. The knockon delay of train 2 will not occur, i.e., τ2=0, in the case, when the indicated time points are separated by the value r≥t0, i.e., r=μ2+t0+t−t+τ≥t0, or, which is the same thing, τ≤μ2. The considered case is illustrated in Figure 7a.
The knockon delay of the duration τ2=τ−μ2 will occur when τ>μ2. Indeed, since trains after a random stop depart simultaneously, then the equality t+τ=μ2+t0+t−t0+τ2 holds, 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=τk−μk+1∨0,2≤k+1≤n.E27
It follows from the inductive hypothesis that τk=0 under the condition τk−1≤μk. But if the delay of the kth 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 τk−1>μk, a knockon delay of the kth train occurs and equals to τk=τk−1−μk (according to the inductive hypothesis). Further, two cases are possible: either (1) a delay τk entails a delay τk+1, or (2) τk+1=0.
Case 1. If the kth train is delayed, then (k + 1)th one will be delayed only if τk>μk+1, and its delay duration is τk+1=τk−μk+1 (this fact follows from the equality of the moments of departure of the kth and (k + 1)th trains after an unscheduled stop: Tk+t−k−1t0+τk=Tk+1+t−kt0+τk+1). Case 1 is illustrated in Figure 9a.
Case 2. If the kth train is delayed, then (k + 1)th one will not be delayed (τk+1=0) only if τk≤μk+1. Case 2 is illustrated in Figure 9b. Note that if the knockon delay of the kth train occurs, a conflict of the kth 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=τ−μ2∨0,τ3=τ2−μ3∨0,τk=τk−1−μk∨0.E28
Using the method of mathematical induction and taking into account that μ¯k−1+μk=μ¯k, we obtain Eq. (21) from Eq. (28).□
Proof of Corollary 9. It follows from Corollary 8 that τk=0 if τ≤μ¯k (see, e.g., Figure 8), and τk=τ−μ¯k if τ>μ¯k (see, e.g., Figure 9a). Using the law of total probability, we obtain the following chain of equalities:
Proof of Corollary 12. Apply the wellknown assertion to Eq. (22): if Y1 and Y2 are independent random variables, then for any function of two variables f⋅⋅ and any c∈R, the following equality holds: ΡfY1Y2<c=∫−∞∞ΡfyY2<cdF1y, where F1 is the distribution function of Y1. Consequently, Gkt=It>0∫−∞∞Ρτ−y<tdΨ∗k−1y. This implies Eq. (25).□
Proof of Corollary 13. The assertion follows from Eq. (25).□
Note that the function Gkt has a jump at zero which is equal to:
Gk0+=∫−∞∞G+ydΨ∗k−1y, where G+y=limt→0+Gt+y.
In the case, when τ and μj are absolutely continuous, it follows from Eq. (25) that
Gkt=It>0∫−∞∞∫−∞t+ygzdzψ∗k−1ydy,E29
where g⋅ and ψ⋅ are the density functions of τ and μ1, respectively, ψ∗jy is the jfold convolution of the density ψ⋅. In this case, we also have
gkt≔It>0Gk′t=It>0∫−∞∞gt+yψ∗k−1ydy.E30
If we assume that τ≥0, then we deduce from Eq. (29) that
6. Corollary of Theorem 2 when the distribution of primary delay is a mixture of exponential and onepoint distributions
Consider the cumulative distribution function of the following type:
Gx≡Ρτ<x=Ix≥b1−ae−λx−b,E33
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 Gx=1−aG0x−b+aGx−bλ, where G0x is the distribution function of the degenerate distribution concentrated at the point x=0, Gxλ=Ix≥01−e−λx.
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 byEq. (33), andΨbe continuous. Then
Proof. According to Eq. (33), one may conclude that function G¯x has a unique discontinuity point x=b. Hence, the integral ∫−∞∞G¯z−t+t0dΨz exists for any continuous distribution function Ψ. Note that if Ψz had a discontinuity point z=t1, then the function G¯z−t+t0 would also be discontinuous at the point z=t1 for t=t0+t1−b, and then the considered integral would not exist (see Remark 1). Since
The behavior Ενk and Dνk with growth of the parameter k.
Remark 4. It can be easily seen that the larger k, Wkt from Eq. (43) is closer to Wt≔Ib≥0It>t0+T. This agrees with Figure 10 and the formulas (44) and (45) due to which we have Ενk→t0+T, Dνk→0 as k→∞, and also with the results of calculations in Table 1.
Let the random variable τ be 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 knockon 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 exp−λmT≤p. As a result, we obtain the desired condition:
T≥1/mλln1/pE47
(see also [13]). Denote by Tmpλ the minimal T satisfying the inequality (Eq. (47)).
Example 6. Let us fix λ=0.26. The behavior of Tmpλ as a function of the continuous parameter m with p=0.1 and p=0.05 is shown in Figure 11a. Obviously, Tmpλ is the decreasing function with respect to the argument p. Exact calculations can be made using the formula:
Tmpλ=1/mλln1/p.E48
Let p=0.1. The behavior of Tmpλ as a function of the continuous parameter m with λ=0.26 and λ=0.15 is shown in Figure 11b. In accordance with Eq. (48), Tmpλ is 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 λ leads to increase in the average of primary delay and the departure headways (if we want to reduce the number of knockon delays).
We also obtain the corollaries of Lemma 3 in the case when μj are distributed according to the gammalaw with the density (Eq. (17)).
Corollary 15. If primary delayτhas an exponential distributiongt=It>0λe−λtandμk,2≤k≤n, has the density (Eq. (17)), then the following formulas are true:
Gkt=It>01−e−λtλβ+1−k−1α,E49
gkt=It>0λβ+1−k−1αλe−λt.E50
Remark 5. The function gkt=It>0Gk′t is not a density, in particular, because of ∫−∞∞gktdt=∫0∞gktdt=Gk∞−Gk0+=1−hk≠1, where hk is the jump of the function Gkt at the origin. At the same time, the function g˜kt≔11−hkgkt is a density.
Corollary 15 can be reformulated as follows.
Corollary 15*. Let primary delayτis exponentially distributed with a parameterλ, andμk,2≤k≤n, have the same gamma distribution with the density (Eq. (17)). Then,τkhas the distribution function of the formEq. (33)witha=λβ+1−k−1α, b = 0, and, consequently,
Ρτk=0=Gk0+=1−λβ+1−k−1α.
Remark 6. Let Ρτ2=0=p, 0<p<1. Then by Corollary 15*, Ρτk=0=1−1−pk−1,3≤k≤n. Hence, Ρτk=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 τk with various combinations of parameters: α, β, k. The results are presented in graphical form in Figures 12–15 The functions Gkt are calculated by formula (49), and the functions gkt by formula (50). Note that product αβ is the mean of μk. Parameter λ is equal to 0.25 and αβ=7 as 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 “MoscowTver” for the period: January, 11–15, February, 1–6, 2016, we obtained a sample from the distribution of ξ of the size n=50 with the sample mean 1.44 and sample variance 2.7. We tested the hypothesis that ξ obeys distribution (Eq. (33)) with λ=0.35 and a=0.64. To this end, we applied the Kolmogorov goodnessoffit test with the significance level α=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.
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 G2t with α=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 μj 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 knockon 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=0.25 (i.e., τ has the distribution function (Eq. (33)) with λ=0.25, a=1, b=0), and μ2 has 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:
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.
The research is funded by the JSC Russian Railways (grant 2016 for development of the scientific school).
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Written By
Vladimir Chebotarev, Boris Davydov and Kseniya Kablukova
Submitted: 31 October 2017Reviewed: 15 February 2018Published: 26 September 2018