Analyzing Congestion Propagation on Urban Rail Transit Oversaturated Conditions: a Framework Based on SIR Epidemic Model

Urban Rail Transit (2018) 4(3):130–140 https://doi.org/10.1007/s40864-018-0084-6 http://www.urt.cn/

ORIGINAL RESEARCH PAPERS

Analyzing Congestion Propagation on Urban Rail Transit Oversaturated Conditions: A Framework Based on SIR Epidemic Model

1 1 Ziling Zeng • Taixun Li

Received: 24 August 2017 / Revised: 12 July 2018 / Accepted: 27 July 2018 / Published online: 16 August 2018 Ó The Author(s) 2018

Abstract Simulating the congestion propagation of urban 1 Introduction rail transit system is challenging, especially under oversaturated conditions. This paper presents a congestion Congestion, along with the expansion of urban transit propagation model based on SIR (susceptible, infected, networks, is becoming one of the major concerns of the recovered) epidemic model for capturing the congestion transportation system agencies, especially under the over- prorogation process through formalizing the propagation saturated conditions. The initiatives in the transportation by a congestion susceptibility recovery process. In addi- system consist of various management strategies and tion, as congestion propagation is the key parameter in effective restrictive measures which aim to reduce con- the congestion propagation model, a model for calculating gestion and avoid accidents. The analysis of congestion in congestion propagation rate is constructed. A gray system road networks includes macroscopic [1, 2], medium [3, 4] model is also introduced to quantify the propagation rate and microscopic [5, 6] levels. The rearrangement of under the joint effect of six influential factors: passenger demand in the network once an element fails is a particular flow, train headway, passenger transfer convenience, time focus of some studies [7–9]. Generally, there is an overlook of congestion occurring, initial congested station and sta- on the dynamic properties of traffic in previous papers, tion capacity. A numerical example is used to illustrate the such as the propagation of congestion. The congestion congestion propagation process and to demonstrate the propagation process on highway has been studied by improvements after taking corresponding measures. Sundara [10] and Zhang [11], whom both depicted the characteristics of congestion propagation under a highway Keywords SIR epidemic model Á Oversaturated traffic incident. Based on this research, Zang [12] proposed conditions Á Congestion propagation model Á Congestion a model to calculate the boundary and recovery rate of the propagation rate Á Gray system model congestion in the light of the traffic flow theory. Never- theless, research in the propagation of the congestion in rail transit networks is limited in the following two categories: the propagation mechanism and the congestion simulation. Research on the accident delay of the high-speed railway can be classified into two aspects: emergency rescheduling and delay propagation. Emergency rescheduling methods manage to minimize operation con- flicts and train delay to optimize rescheduling strategies & Ziling Zeng and predict influence according to accident characteristics. [email protected] Meng and Zhou [13] developed an innovative integer

1 programming model for train dispatching on an N-track School of Traffic and Transportation, Beijing Jiaotong network by means of simultaneously rerouting and University, Beijing 100044, People’s Republic of China rescheduling trains. Wang and Goverde [14] have deter- Communicated by Xuesong Zhou. mined a timetable constraints set with accessibility and 123 Urban Rail Transit (2018) 4(3):130–140 131 non-conflict under delay accidents of single-track railway, structure and station distribution). Liu [25] proposed a rail upon which a train rescheduling model was established transit model for the oversaturated conditions. aiming to reduce global delay time and energy consump- The details of the influential factors of the congestion tion. Carey and Kwiecin´ski [15] conducted a stochastic propagation are clearly analyzed, and the models for the simulation of trains on line section to establish an congestion propagation are constructed with brevity. approximation model between scheduled headway and However, the quantitative research in the congestion secondary delay for the coefficient modification and propagation process and the influential factors like the schedule optimization of current operation plan. Louwerse propagation rate is limited. Wu [26] calculated the and Huisman [16] studied the rescheduling methods to seek approximate rate of propagation based on time algorithm, a higher service level based on event-activity network but Wu has applied an assumed value of the congestion using integer programming. Cadarso and A´ ngel [17] pre- propagation rate in the simulation instead of calculating the sented an approach to optimize timetable and rolling stock value with accuracy. assignment with special consideration of passenger demand In previous research, however, there is an overlook of under large-scale disruptions in rapid transit network. the dynamic properties of traffic, such as the propagation of Delay propagation models help to combine micro-propa- congestion [2]. Furthermore, quantitative research in the gation model with mathematical statistics to predict and congestion propagation process is very limited [3, 4]. This estimate the service performance and transport efficiency, paper aims to provide extensive analyses and simple for- and max-plus algebra [18] and stochastic distribution [19] mulations to help understand the behavior of the conges- are often used in this kind of research. In the max-plus tion propagation in urban rail networks. We propose a algebra, by viewing the operation plan under periodic congestion propagation model in the urban rail networks by timetable as a discrete event dynamic system, a recursion making some simplified assumptions about the traffic function considering buffer time and recovery time can be behavior. It is important to reflect the decision-making formulated to describe the train operating status and further process of traffic controllers who need simple and appli- reveal the time–space propagation mechanism. In the cable formulations to help react in real time. stochastic distribution model, activity graph theory is This paper is organized as follows: The problem state- applied to describe primary delay and secondary delay and ment is delineated in Sect. 2 to illustrate the purpose of the construct relevant cumulative distribution function by research. The congestion propagation model is dedced and considering delay as a random variable. constructed in Sect. 3, and it is modeled as a susceptible- Some of the current studies emphasized on how the congested-recovered process. The method for calculating congestion spreads and grows in the urban rail networks. the congestion propagation rate is described in Sect. 4. Zhou [20] proposed the concept of the propagation of Numerical simulation on a real-world network with the passenger flow during peak hours for the first time, and field measurement data is presented in Sect. 5. Comparing Zhou analyzed the mechanism and influential factors of the the proposed model with the previous model presented in congestion propagation. Stephen [21] described the trans- the literature survey part is discussed in Sect. 6. The con- mission under the oversaturated conditions in rail transit clusion of this study is summarized in Sect. 7. based on the combustion theory, and he proposed that the congestion propagation is similar to the ripple effect. Other researchers focused on the congestion simulation and the 2 Problem Statement model construction. Duan [22] classified the urban railway stations into 3 categories: the starting, the intermediate and The classical SIR model [27] for disease outbreaks con- the terminal stations, and he extracted certain features of siders a population split into three compartments: the sus- the oversaturated conditions. He then proposed the model ceptible individuals S, the infected individuals I and the of the influence of passenger densities in the station waiting recovered individuals R. The model is commonly called the areas. Li [23] proposed a model based on the transmission susceptible-infected-recovered (SIR) model. The model dynamics of the complex network and compared the contains two parameters: the transmission rate k, i.e., the influencing factors of the congestion propagation, includ- contact rate times the probability of infection [28], and the ing the capacity of the stations, the number of the initial recovery rate r, i.e., the rate at which infective individuals congested stations, the propagation rate and the dissipation recover. The model equations are defined as follows. rate. The two aspects: propagation mechanism of the S0ðtÞ¼ÀkIðtÞSðtÞð1Þ complex network and influential factors of the congestion 0 propagation, are combined by li [24] who proposed a I ðtÞ¼kIðtÞSðtÞÀrIðtÞð2Þ coordination game model for the traffic congestion propa- R0ðtÞ¼rIðtÞð3Þ gation and explored the influential factors (e.g., road 123 132 Urban Rail Transit (2018) 4(3):130–140

In the classical SIR model it is assumed that S, I and 3 Congestion Propagation Model R are the fraction of the population. The parameters k and r reflect some external forces that influence the course of Based on the SIR epidemic model, stations that are easily the disease outbreak, for example, changes in the envi- affected by the congestion are categorized under suscepti- ronment (seasonality) or changes in contact rates. Investi- ble stations. Stations that were previously susceptible sta- gations of this model and extensions have been developed tions and were affected by the congestion are categorized by many researchers, including Dietz and Heesterbeek under congested stations. The recovered stations are those [29], Anderson and May [30], Bailey [31], Brauer [32], which have recovered from the congestion. Theoretically, Diekmann [33], Keeling and Rohani [34] and Thieme [28]. for the statement simplicity, stations after recovery from This modeling metaphor can be gainfully employed to congestion obtain permanent immunity from congestion. examine the congestion propagation. Figure 1 shows a Table 1 lists the definition of indices, sets and parame- simple propagation congestion process among an urban rail ters used in the mathematical formulations of congestion transit network based on three categories of SIR epidemic propagation model. model. The first station on the left-hand side represents These variables SðtÞ, IðtÞ and RðtÞ represent three dif- susceptible station (susceptible individual), that is, station ferent types of stations at a particular time, respectively. which is usually adjacent to the initial congested stations is SðtÞ represents the number of susceptible stations, IðtÞ easily affected and delayed by congestion. When train runs represents the number of congested stations, and RðtÞ to the next station, susceptible stations are affected by represents the number of recovered stations. oversaturated station (infectious individual) and become Assume that the congestion propagation rate is k and oversaturated themselves (they become infected). Also recovery rate is r. During the time period [t, t ? Dt], the over time, a proportion of stations will become recovered congested station increment is kIðtÞSðtÞDt and the recov- stations (recovered individuals). In the SIR model, the ered station increment is rIðtÞDt. The state transition infectious population will recover and develop immunity to equations can be formulated as follows, and Fig. 2 shows infection. the congestion propagation. This paper aims to fill the gap of previous research by Sðt þ DtÞÀSðtÞ¼ÀkIðtÞSðtÞDt ð4Þ providing a research on congestion propagation based on Iðt þ DtÞÀIðtÞ¼kIðtÞSðtÞDt À rIðtÞDt ð5Þ SIR epidemic model. We propose a congestion propagation model and use the increment of the number of con- Rðt þ DtÞÀRðtÞ¼rIðtÞDt ð6Þ gested stations to evaluate the congestion incidence of the where Sðt0Þ¼S0 [ 0; Iðt0Þ¼I0 [ 0; Rðt0Þ¼R0 ¼ 0. network. As a result, the model generates the proper The increment of the number of congested stations is measurements for different oversaturated conditions. applied to evaluate the congestion propagation. It can be Furthermore, in order to simulate the congestion propa- defined as follows: gation process, we propose a separate method to calculate DI t Dt N k rI t the value of congestion propagation rate. The outcome of ð þ Þ¼ p À ð Þð7Þ this model produces a quantitative analysis of the effi- where p represents the departure station of the metro line. ciency of measurements taken to relieve the oversaturated These formulas describe the increment or decrement of conditions the number of stations among three different categories over time, which can be used to simulate the congestion propagation process.

Epidemic Propagation 4 The Method for Calculating Propagation Rate

Susceptible Transmission Infectious Recovery Recovered Individual rate(%) Individual rate(%) Individual In SIR model, the transmission rate of an infectious disease is the rate of infection given contact. It is the epidemiological analogue of a rate constant in chemical reactions. The direct Congestion Propagation measurement of the transmission rate is essentially impos- sible for most infections. But if we wish to predict the Susceptible Propagation Oversaturated Recovery Recovered Station rate(%) Station rate(%) Station changes caused by public health programs, we need to know the transmission rate [30]. The transmission rate of many acute infectious diseases varies signiﬁcantly in time and Fig. 1 Comparison between epidemic propagation and congestion frequently exhibits signiﬁcant seasonal dependence [35–37]: propagation 123 Urban Rail Transit (2018) 4(3):130–140 133

Table 1 Notation of congestion Symbol Deﬁnition propagation model t A time step Dt The headway between two subways S Set of stations in the network k Congestion propagation rate r The proportion of stations recovered from oversaturated conditions

Ni The number of stations connected to station i [ S IðtÞ The number of congested stations at t SðtÞ The number of susceptible stations at t RðtÞ The number of stations which have recovered from the congestion at t TNðtÞ The total number of stations in the network

Fig. 2 Congestion propagation between t and t ? Dt

Some epidemic cases peak in winter, while other cases peak with passenger transfers and parameter B associated with in spring or in summer. The transmission rate of many acute time. infectious diseases varies significantly in time, but the underlying mechanisms are usually uncertain. They may 4.1 Formula Construction include seasonal changes in the environment, contact rate, immune system response. The transmission rate has been We use the total passenger flow in one station and the thought difficult to measure directly. transfer passenger flow to represent the five influential Similarly, in urban rail transit, propagation rate repre- factors: passenger flow characteristic, average passenger sents the congestion probability between two adjacent transfer time for transfer convenience, congestion occur- stations when one of them encounters oversaturated con- ring time t, the station capacity Fj at station j and the dition. There are various influential factors of the propa- average train headway. The passenger arrival is set gation rate. In order to compute the propagation rate, these homogeneous for the station so that the application of the factors are classified into six classes: passenger flow average train headway to represent the passenger waiting characteristic, train departure interval, passenger transfer time is feasible (e.g., 2 min). convenience, the time of congestion occurring, the initial Table 2 lists the general indices, sets and parameters congested station and station capacity [38]. We divided used in the mathematical formulation for calculating these parameters into two classes: parameter A associated propagation rate.

123 134 Urban Rail Transit (2018) 4(3):130–140

P P Table 2 Notations for congestion propagation rate model Nj Nj i¼1 Flowi;jx1 þ Flowj À i¼1 Flowi;j x2 Symbol Definition A ¼ ð13Þ Flowj U Set of transfer routes between two stations Transferi;jx1 þ Travelj;k þ Wj LL Set of lines (such as line 4) B ¼ ð14Þ Tll r Indexes of routes in U i Indexes of stations in S j Indexes of stations in S 4.2 Parameter Calibration k Indexes of stations in S There are three methods for the weight analysis of the Flowj Total passenger flow in station j [ S parameter calibration: analytic hierarchy process, expert Fj The maximum passenger flow in station j [ S scoring method and variation coefficient method. However, Flowi,j Transfer passenger flow from station i [ S to j [ S in the previous research, there are few literatures on the Transferi,j Average transfer walking time from station i [ S to j [ S definition of the interrelation among parameters. Wj Average waiting time for train arrival in station j [ S To elaborate the relationship among different parame- Travel j,k Average travel time from station j [ S to k [ S ters, we deduct the gray system model to calculate the Tll Average total running time for line ll [ LL values of a and b. A Parameter associated with passenger transfers The gray system model aims to calculate the degree of B Parameter associated with time association between the behavior factor (i.e., congestion a Calibration value of A propagation) and the relevant factors (i.e., passenger flow b Calibration value of B and passenger behavior). If the developing trend between the behavior and the relevant factors is consistent, the degree of gray incidence would be large. If the trend is less well defined, the degree of gray incidence would be small The quantitative model was constructed as follows: [39]. The gray system model is considered to be an analysis P P Nj Nj of the geometric proximity among different factor i¼1 Flowi;jx1 þ Flowj À i¼1 Flowi;j x2 k ¼ a sequences and the behavior sequence. The proximity is Flowj described by the degree of gray incidence, which is Transfer x þ Travel þ W þ b i;j 1 j;k j ð8Þ regarded as a measure of the similarities of data that can be T ll arranged in sequential order. In this model, data are col- s.t. lected for behavior sequence X0 and relevant factor sequence X over the same time period. 1 passengers coming from other metro lines i x ¼ 1 0 passenger coming directly to station without transfer

ð9Þ Table 3 Notation for parameter calibration 1 passengers coming directly to station without transfer x ¼ Symbol Deﬁnition 2 0 passenger coming from other metro lines X Behavior sequence ð10Þ 0 Xi Relevant factor sequence of parameter i [ W x1 þ x2 ¼ 1 ð11Þ xiðjÞ The elements of Xi cðX ; X Þ The gray correlation degree of X and X Flowj PFj ð12Þ p q p q Nj D j The absolute deviation of x j and x j Flowi;j 0ið Þ 0ð Þ ið Þ where i¼1 represents passenger transfer rate, and ÀÁP Flowj Dmin The bipolar minimum deviation of x0ðjÞ and xiðjÞ Nj FlowjÀ Flowi;j i¼1 represents the rate without transfer. Dmax The bipolar maximum deviation of x0ðjÞ and xiðjÞ Flowj k An arbitrary given number k 2ð0; 1Þ Travelj;k þ Wj is for the time spent between two adjacent W Set of uncalibrated parameters stations in the same metro line. If passengers come from V Set of input data another metro line, we add Transferi;jx1, namely walking i Index of uncalibrated parameters W time in the transfer channel to formula (8). j Index of input data V We deﬁne passenger-transfer-associated parameter A and time-associated parameter B as follows: p Index of uncalibrated parameters W q Index of uncalibrated parameters W

123 Urban Rail Transit (2018) 4(3):130–140 135

The notation for parameter calibration is shown in Dmin ¼ min min D0iðjÞð25Þ Table 3. i j Dmax ¼ max max D0iðjÞð26Þ Definition 1 According to the notation, we can define a i j and b as follows: Then, D0iðjÞ represents absolute deviation of x0ðjÞ and xiðjÞ, X X a ¼ cð 0; AÞð15Þ and Dmin and Dmax represent bipolar minimum deviation and bipolar maximum deviation, respectively(refer to b ¼ cðX0; XBÞð16Þ [39]). Definition 2 Assume that the behavior data sequence is defined as follows: Theorem 1 For an arbitrary given number k 2ð0; 1Þ,we T define: X0 ¼ðx0ð1Þ; x0ð2Þ; ...; x0ðnÞÞ ð17Þ Dmin þ kDmax cðX0ðjÞ; XiðjÞÞ ¼ ð27Þ D0iðjÞþkDmax Xn Besides, X0 is equal to k when a and b are equal to 1 1 c ¼ cðX ; X Þ¼ cðx ðjÞ; x ðjÞÞ ð28Þ (refer to [39]). 0 i n 0 i And the correlation factor data sequence is assumed to j¼1 be: T Xi ¼ðxið1Þ; xið2Þ; ...; xiðnÞÞ Then, the value c ¼ cðX0; XiÞ can be calculated by using . . formula (28). . . ð18Þ T Xm ¼ðxmð1Þ; xmð2Þ; ...; xmðnÞÞ : 5 Case Study where xiðjÞ [ 0; i ¼ 0; 1; 2...m; j ¼ 1; 2...n:. 1 Xn In this section, we focus on how to use the congestion Suppose that cðX ; X Þ¼ cðx ðjÞ; x ðjÞÞ ð19Þ 0 i n 0 i propagation model and the propagation rate to conduct the j¼1 quantitative analysis of the above proposed model. Such It satisfies the following four properties: analysis can bring about clearity of the operation condition of the urban railway network. The congestion propagation ðÞi Normality: 0\cðX0; XiÞ1; when X0 ¼ Xi ð20Þ rate model can be used to generate measures to solve the , cðX0; XiÞ¼1

ðÞii Integrity: cðXp; XqÞ 6¼ cðXq; XpÞwhen p 6¼ q; ð21Þ 26 Line 13 where Xp; Xq 2 X ¼fxkjk ¼ 0; 1; 2 ÁÁÁm; m 2g: 22 25 ðÞiii Even symmetry: cðXp; XqÞ¼cðXq; XpÞ 22 ð Þ 21 , x ¼fXp; Xqg 24 Line 2 1 Line 4 XIZHIMEN STATION ðÞiv Proximity: the smaller jx0ðjÞÀxiðjÞj; ð23Þ 19 20 2 13 the bigger cðx0ðjÞ; xiðjÞÞ:

Then, cðX0; XiÞ is called the gray correlation degree of Line 6 16 17 3 12 18 X0 and Xi ,cðx0ðjÞ; xiðjÞÞ represents the correlation coeffi- X X j cient of 0 and i at the th point, and the four properties 4 11 (i), (ii), (iii) and (iv) are called four axioms of the gray 15 Line 9 10 correlation(refer to [39]). Line 1 14 27 28 5 9 23 Definition 3 Assume that Xi ¼ðxið1Þ; xið2Þ; ...; xiðnÞÞ which is defined by the above Definition 1, where i ¼ 0; 1; 2...m. We define: 687

Fig. 3 A part of Beijing metro network D0iðjÞ¼jx0ðjÞÀxiðjÞj ð24Þ

123 136 Urban Rail Transit (2018) 4(3):130–140 oversaturated condition, improve congestion and ﬁnally According to Deﬁnition 2, when the value of parameters enable us to analyze the incidents for future reference. a and b is equal to 1, the behavior data sequence X0 is equal The case study takes the oversaturated condition to the value of k. occurred at Beijing Xizhimen Station which is a big Equation (8) can be written as: interchange station of line 2, line 4 and line 13. The rail P P Nj Flow x Flow Nj Flow x transit network is shown in Fig. 3. The calculation results i¼1 i;j 1 þ j À i¼1 i;j Þ 2 k ¼ by using the model is to verify the effectiveness of the SIR Flowj epidemic model. Transferi;jx1 þ Travelj;k þ Wj þ ð29Þ Tll 5.1 Data Preparation By using the data shown in Tables 4 and 5, the relevant factor sequences can be calculated as follows:We assume The data collected on September 8, 2014, at Xizhimen k = 0.5; by using formulae (23–26), the calibration degree Station are applied to calibrate a and b. The data measured can be calculated as follows: and collected include the passenger number, average passenger walking speed, the railway departure interval, the apeak ¼ cpeakðX0; XAÞ¼0:54: running time between two stations and the total running anormal ¼ cnormalðX0; XAÞ¼0:57: time from the origin to the destination. These data and various comparisons with other research are shown in bpeak ¼ cpeakðX0; XBÞ¼0:46: Tables 4, 5 and 6. bnormal ¼ cnormalðX0; XBÞ¼0:43: The transfer rate from line 4 to line 2 differs greatly compared to the data obtained by Li [40], while the other transfer rates are similar. The collected data of the project 5.3 Propagation Simulation evaluation report published by China Metro Engineering Consulting Company [41] are more accurate than the data In the literature, the methods of calculating recovery rate r obtained from Li. As a result, the collected data are are limited. In order to simulate the propagation rate, we selected as the basis to calibrate the parameters. Table 5 assume that the congestion recovery rate is 0.10, by displays the value of parameters associated with time from referring to Ref. [22]. From the network graph, some three lines in Xizhimen Station. parameters are given, NXizhimen =5,I0 =1,r= 0.10. In the In Table 6 the transfer walking speed in Beijing metro aims to calculate the propagation rate, we refer to formula line 5 and Haidian Huangzhuang Station which is an (8); the correlation degrees a and b are given by Sect. 5.2 interchange station of line 10 and line 4 are also list for a (apeak = 0.54, anormal = 0.57, bpeak = 0.46,bnormal = 0.43). comparison with the data collected at Xizhimen Station. Considering the oversaturated situation, the parameters in The calculation resulst for calibration model is shown in peak hours are chosen for the following calculation so that Table 7. Eq. (8) can be written as: P P Nj Nj i¼1 Flowi;jx1 þðFlowj À i¼1 Flowi;jÞx2 5.2 Calibration Value Calculation kpeak ¼ 0:54 Â Flowj Transferi;jx1 þ Travelj;k þ Wj In this section, we use MATLAB to calculate the value of a þ 0:46 Â Tll and b. ð30Þ

Table 4 Value of the passenger transfer rate Metro lines Peak-hour transfer rate (%) Normal-hour transfer rate (%) Transfer rate (%) in 2015 (by Li [40])

Line 4 to line 2 13 9 75 Line 13 to line 2 36 27 27 Line 2 to line 4 27 28 51 Line 13 to line 4 31 2 8 Line 2 to line 13 40 39 43 Line 4 to line 13 6 5 7

123 Urban Rail Transit (2018) 4(3):130–140 137

Table 5 Value of the time-associated parameters Metro lines Walking time in Walking time in Waiting time in Waiting time in Traveling time between Total running peak hours (min) normal hours (min) peak hours (min) normal hours (min) two stations (min) time (min)

Line 4 to line 2 3.73 2.65 2.00 4.50 2.00 39.00 Line 13 to line 2 6.63 3.60 2.00 4.50 2.00 39.00 Line 2 to line 4 3.83 2.72 2.50 4.00 2.00 79.00 Line 13 to line 4 5.27 3.01 2.50 4.00 2.00 79.00 Line 2 to line 13 6.78 3.59 2.67 5.50 3.00 50.00 Line 4 to line 13 8.12 5.10 2.67 5.50 3.00 50.00

Table 6 Passenger walking speed in transfer channel in peak hours Metro lines 4–2 13–2 2–4 2–13 2–13 4–13 Beijing metro line 5 Haidian Huangzhuang Station

Walking speed (m/s) 0.817 0.735 0.795 0.726 0.604 0.647 0.620 [42] 0.740 [43]

Table 7 The input data for calibration value calculation Metro lines Peak-hour Peak-hour Peak-hour Normal-hour Normal-hour Normal-hour feature data correlation factor correlation factor feature data correlation factor correlation factor sequence X0 data sequence XA data sequence XB sequence X0 data sequence XA data sequence XB

Line 4 to line 2 3.73 0.13 0.19 0.32 0.09 0.23 Line 13 to line 2 6.63 0.36 0.28 0.52 0.27 0.26 Line 2 to line 4 3.83 0.27 0.10 0.39 0.28 0.11 Line 13 to line 4 5.27 0.31 0.12 0.13 0.02 0.12 Line 2 to line 13 6.78 0.4 0.25 0.63 0.39 0.22 Line 4 to line 13 8.12 0.06 0.27 0.32 0.05 0.28

Table 8 Congestion propagation rate kpeak at different lines Using the data shown in Tables 4 and 5, the value of

Metro lines Propagation rate (%) propagation rate kpeak can be calculated, which is shown in Table 8. Line 4 to line 2 16.7 The peak-hour propagation rate kpeak is selected, and in Line 13 to line 2 31.3 this case the propagation rate of line 4 is used in simulation Line 2 to line 4 16.2 kline 4 = 28.7% by considering the oversaturated conditions. Line 13 to ling 4 20.9 The input of these values into the above formula (8) sim- Line 2 to line 13 31.8 ulates the congestion propagation process as follows: Line 4 to line 13 14.4 The process of congestion propagation in Fig. 4 illus- Line 2 32.3 trates that the increment of congested stations increases Line 4 28.7 rapidly within the ﬁrst 5 min and that there is a decline at Line 13 20.1 time step 8 and a slight decline between time step 8 and 25.

We adjust the parameter Ni in order to compare the result with the actual circumstances, and the simulation is shown in Fig. 5a. Figure 4 illustrates the alternatives of measures taken by operators which include traveling past stations 1

123 138 Urban Rail Transit (2018) 4(3):130–140

It is shown that when combined with the above analysis of the propagation rate, the propagation of congested station will expand when the propagation rate increases. Thus, it is more efﬁcient to reduce the number of adjacent sta-

tions (reduce Ni) by traveling past some of the stations without stopping. Furthermore, there are other measures that can improve the oversaturated condition by reducing the propagation rate k, which includes restricting passenger ﬂow at the station entrance, adjusting the train interval, improving transfer convenience and facilitating access to stations.

6 Comparison

Liu [40] introduced the time algorithm to calculate the Fig. 4 Process of congestion propagation congestion propagation rate. t À t k ¼ 1 2 ð31Þ and 3 without stopping, and the adjacent stations of the T initial congested station (N1) reduce time step from 5 to 3; as a result, the number of congested stations declines to 6 In this equation, T represents the train operating time in t within 25 min, and the increment begins to stabilize the network, 1 represents the time point when an over- between time step 10 and 25. The adjustment of its value saturated condition begins at a station (e.g., Xizhimen (e.g., 5, 4, 3, 2) provides a better illustration of the impact Station), and t2 represents the time point when an oversaturated condition begins at the connected station (e.g., of parameter Ni, and the simulation is shown in Fig. 5a. Figure 5b shows the evolutions of propagation rate Chegongzhuang Station). equal to 0.23, 0.20, 0.16 and 0.10, respectively. A con- It is challenging to apply this formulation to practical clusion can be drawn from this figure; the total number of use as the value of parameters t1 and t2 is difficult to congested stations will decline if the propagation rate is measure. Hence, the result may lack accuracy. Moreover, reduced. However, by comparing (b) with (a), it is clear the data of parameters cannot be collected before the that the reduction in adjacent stations will significantly occurrence of oversaturated conditions. Consequently, this reduce the increase in the total number of congested sta- method is not available and lacks the ability to prevent tions and the corresponding improvement is more efficient. oversaturated conditions. By comparison, the method proposed in Sect. 4 can be used in a variety of applications and the data of parameters

Fig. 5 Congestion propagation simulation under different values of parameters 123 Urban Rail Transit (2018) 4(3):130–140 139 can be collected more easily. For example, the method can urban transit. Thus, the models proposed in this paper be used in generating effective measures for the oversatu- optimize the recovery measures and more importantly rated condition. Firstly, the parameter data and possible simulate the current circumstances and forecast the prop- measures are the input. Secondly, the operators can cal- agation trends in urban transit systems. culate the propagation rate and the cumulative number of congested stations. Finally, the operators can select the Acknowledgements Funding was provided by the Fundamental most effective measure, by referring to the value of con- Research Funds for the Central Universities (2017JBM029). gestion propagation rate and the simulated number of Open Access This article is distributed under the terms of the congested stations. Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a 7 Conclusion link to the Creative Commons license, and indicate if changes were made. This paper presents a model based on SIR epidemic model that comprises a congestion propagation model and a propagation rate calculation method to utilize the propa- References gation theories of the oversaturated conditions in rail 1. Ortigosa J, Menendez M (2014) Analysis of quasi-grid urban transit. The application of the congestion propagation structures. Cities 36:18–27 model aims to simulate the propagation process of the 2. Lu K, Han B, Zhou X (2018) Smart urban transit systems: from whole network under oversaturated conditions. In this integrated framework to interdisciplinary perspective. Urban Rail paper, a new approach consisting of a separate method that Transit 4:49 3. Li SB, Wu JJ, Gao ZY, Lin Y,d Fu BB (2011) Bi-dynamic calculates the congestion propagation rate is studied and analysis of traffic congestion and propagation based on complex presented. The congestion propagation rate model is used network. Acta Physica Sinica 60(5) to generate measures to solve the oversaturated condition, 4. Khattak A, Yangsheng J, Lu H et al (2017) Width design of urban improve congestion and finally enable us to analyze the rail transit station walkway: a novel simulation-based optimization approach. Urban Rail Transit 3:112 incidents for future reference. The two models establish a 5. Ji Y, Geroliminis N (2012) Modeling congestion propagation in solid foundation for the study of large-scale network con- urban transportation networks. Presented at the 12th Swiss gestion propagation. The influential factors of the propa- Transport Research Conference, Locarno, Switzerland gation rate can be classified into six classes: passenger flow 6. Roberg-Orenstein P, Abbess CR, Wright C (2007) Traffic jam simulation. J Maps 2007:107–121 characteristic, train departure interval, passenger transfer 7. Scott D, Novak D, Aultman-Hall L, Guo F (2006) Network convenience, the time of congestion occurring, the initial robustness index: a new method for identifying critical links and congested station and station capacity. Nevertheless, there evaluating the performance of transportation networks. J Transp are some other influential factors for these models. Hence, Geogr 14(3):215–227 8. D’Acierno L, Botte M, Placido A et al (2017) Methodology for further research on other influential factors is suggested to determining dwell times consistent with passenger flows in the extend the models. The congestion propagation model case of metro services. Urban Rail Transit. 3:73 provides an extensive analysis of the propagation process 9. Zhu S, Levinson DM, Liu H, Harder K (2010) The traffic and related to congestion. Moreover, the model depicts the behavioral effects of the I-35W Mississippi river bridge collapse. Transp Res A Policy Pract 44(10):771–784 forecasting and trends of congestion so that traffic con- 10. Sundara P, Puan OC, Hainin MR (2013) Determining the impact trollers obtain a quantitative observation of the process. of darkness on highway traffic shockwave propagation. Am J The congestion propagation rate model distinguishes the Appl Sci 10(9):1000–1008 efficiency of the alternatives related to congestion 11. Zhang XY (2014) Traffic congestion dissipation model based on traffic wave theory under typical traffic incidents. Beijing Jiao- improvement measures. As a result, traffic controllers tong University, Beijing evaluate the outcome and subsequently select the optimal 12. Zang JR, Song GH, Wan T, Gao Y, Fei WP, Yu L (2017) A measure to resolve the oversaturated conditions. The temporal and spatial model of congestion propagation and dis- expansion of urban transit networks magnifies the propa- sipation on expressway caused by traffic incidents. J Transp Syst Eng Inf Technol 17:179–185 gation of congestion and oversaturated conditions. The 13. Meng L, Zhou X (2014) Simultaneous train rerouting and goal of the comprehensive quantitative model is to inte- rescheduling on an n-track network: a model reformulation with grate congestion propagation analysis with efficiency dif- network-based cumulative flow variables. Transp Res B ferentiation of propagation rate to promote effective 67(3):208–234 14. Wang P, Goverde RMP (2017) Multi-train trajectory optimiza- congestion recovery measures. The paper hopes to initial- tion for energy efficiency and delay recovery on single-track ize a new approach in the quantitative model of the con- railway lines. Transp Res B Methodol 105(11):340–361 gestion propagation through the utilization of the two models and integrate the two results for the development of 123 140 Urban Rail Transit (2018) 4(3):130–140

15. Carey M, Kwiecin´ski A (1994) Stochastic approximation to the 29. Dietz K, Heesterbeek JAP (2000) Bernoulli was ahead of modern effects of headways on knock-on delays of trains. Transp Res B epidemiology. Nature 408(6812):513–514 Methodol 28(4):251–267 30. Anderson RM, May RM (1991) Infective diseases of humans: 16. Louwerse I, Huisman D (2014) Adjusting a railway timetable in dynamics and control. Oxford University Press, Oxford case of partial or complete blockades. Eur J Oper Res 31. Bailey NTJ (1957) The mathematical theory of epidemics. 235(3):583–593 Charles Griffin, London 17. Cadarso L, MarıńA´ ngel (2012) Integration of timetable planning 32. Brauer F, van den Driessche P, Wu J (eds) (2008) Mathematical and rolling stock in rapid transit networks. Ann Oper Res epidemiology. Springer, Berlin 199(1):113–135 33. Diekmann O, Heesterbeek H, Britton T (2013) Mathematical 18. Goverde RMP (2010) A delay propagation algorithm for large- tools for understanding infectious disease dynamics. Princeton scale railway traffic networks. Transp Res C Emerg Technol University Press, Princeton 18(3):269–287 34. Keeling M, Rohani P (2007) Modeling infectious diseases in 19. Bu¨ker T, Seybold B (2012) Stochastic modelling of delay prop- humans and animals. Princeton University Press, Princeton agation in large networks. J Rail Transp Plan Manag 35. Becker NG, Britton T (1999) Statistical studies of infectious 2(1–2):34–50 disease incidence. J R Stat Soc Ser B Stat Methodol 61:287–307 20. Zhou YF, Zhou LSH, Yue YX (2011) Synchronized and coor- 36. Ponciano JM, Capistrań MA (2011) First principles modeling of dinated train connecting optimization for transfer stations of nonlinear incidence rates in seasonal epidemics. PLoS Comput, urban rail networks. J China Railw Soc 33:9–16 Biol, p 7 21. Turns Stephen R (2012) An introduction to combustion concepts 37. Dowell SF (2001) Seasonal variation in host susceptibility and and applications, 2nd edn. McGraw Hill Education, New York cycles of certain infectious diseases. Emerg Infect Dis 7:369–374 22. Duan LW, Wen C, Peng QY (2012) Transmission mechanism of 38. Cats O, West J, Eliasson J et al (2016) Dynamic stochastic model sudden large passenger flow in urban rail transit network. J Railw for evaluating congestion and crowding effects in transit systems. Transp Econ 34:79–84 J Transp Res Methodol 89:43–57 23. Li P (2012) Research on passenger flow distribution character- 39. Deng JL (2002) The basis of grey theory. The Press of Huazhong istics and congestion transmission for urban rail transit operation University of Science and Technology, Wuhan network. Beijing Jiaotong University Master Degree Thesis 40. Li Z, Liu Y, Wang J (2015) Modeling and simulating traffic 24. Li Y, Liu Y, Zou K (2016) Research on the critical value of traffic congestion propagation in connected vehicles driven by temporal congestion propagation based on coordination game. J Procedia and spatial preference. J Wirel Netw 22(4):1121–1131 Eng 137:754–761 41. Yu HX (2008) The research of transfer passenger organization in 25. Liu LF (2013) Crowding propagation and organization coordi- the Xizhimen Station. Beijing Jiaotong University Master Degree nation of urban rail network under larger passenger flow. China’s Thesis Southwest Jiaotong University Master Degree Thesis 42. Xu QW (2008) Analysis and modeling of individual passenger 26. Wu L (2013) Analysis of unexpected passenger demand and behavior in urban railway transit hub. Beijing Jiaotong University resulting congestion in urban rail transit. China’s Southwest Master Degree Thesis Jiaotong University Master Degree Thesis 43. Du P, Liu CH, Liu ZHL (2009) Walking time modeling on 27. Kermack W, McKendrick A (1927) A contribution to the math- transfer pedestrians in subway passages. J Transp Syst Eng Inf ematical theory of epidemics. Proc R Soc Lond A 115:700–721 Technol 4:103–109 28. Thieme HR (2003) Mathematics in population biology. Princeton University Press, Princeton

123