Perceived Trip Time Reliability and Its Cost in a Rail Transit Network

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Article Perceived Trip Time Reliability and Its Cost in a Rail Transit Network

Jie Liu 1,2 , Paul Schonfeld 3, Jinqu Chen 1, Yong Yin 1,* and Qiyuan Peng 1

1 School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China; [email protected] (J.L.); [email protected] (J.C.); [email protected] (Q.P.) 2 Faculty of Transportation Engineering, Kunming University of Science and Technology, Kunming 650093, China 3 A. James Clark School of Engineering, University of Maryland, College Park, MD 20740, USA; [email protected] * Correspondence: [email protected]

Abstract: Time reliability in a Rail Transit Network (RTN) is usually measured according to clock- based trip time, while the travel conditions such as travel comfort and convenience cannot be reflected by clock-based trip time. Here, the crowding level of trains, seat availability, and transfer times are considered to compute passengers’ Perceived Trip Time (PTT). Compared with the average PTT, the extra PTT needed for arriving reliably, which equals the 95th percentile PTT minus the average PTT, is converted into the monetary cost for estimating Perceived Time Reliability Cost (PTRC). The ratio of extra PTT needed for arriving reliably to the average PTT referring to the buffer time index is proposed to measure Perceived Time Reliability (PTR). To overcome the difficulty of obtaining passengers’ PTT who travel among rail transit modes, a Monte Carlo simulation is applied to generated passengers’ PTT for computing PTR and PTRC. A case study of Chengdu’s RTN shows that the proposed metrics and method measure the PTR and PTRC in an RTN effectively. PTTR, PTRC, and influential factors have significant linear relations among them, and the obtained linear Citation: Liu, J.; Schonfeld, P.; Chen, J.; Yin, Y.; Peng, Q. Perceived Trip regression models among them can guide passengers to travel reliably. Time Reliability and Its Cost in a Rail Transit Network. Sustainability 2021, Keywords: perceived time; reliability; travel condition; Monte Carlo simulation; linear regression 13, 7504. https://doi.org/10.3390/ su13137504

Academic Editors: João Carlos de 1. Introduction Oliveira Matias and Paolo Renna As travel demand increases, passengers in large cities experience increasing congestion, crowding, and low time reliability. Time reliability not only reﬂects the service quality Received: 15 May 2021 of transportation networks, but also affects passengers’ route choices [1] and travel satisfac- Accepted: 22 June 2021 tion [2]. Passengers who require high time reliability, such as commuters, are willing to Published: 5 July 2021 pay for such reliability [3]. Both transportation managers and passengers consider time reliability to be very important. Publisher’s Note: MDPI stays neutral Many studies have explored time reliability for road networks [4,5], Rail Transit with regard to jurisdictional claims in Networks (RTNs) [6,7], and bus networks [8,9]. Although these studies have proposed published maps and institutional afﬁl- measures for evaluating the time reliability of transportation networks, travel conditions iations. such as the number of transfer times, crowding, and seat availability in vehicles that affect passengers’ perceptions have been relatively neglected [10]. The reason is that time reliability is measured based on clock-based trip time, which neglects travel conditions. Let us assume that passengers A and B travel between the same Origin to Destination Station (OD) Copyright: © 2021 by the authors. pair and spend the same time to arrive at the destination station. However, if passenger A Licensee MDPI, Basel, Switzerland. gets a seat in an uncrowded vehicle while passenger B stands in a crowded vehicle, their This article is an open access article travel conditions vary greatly despite having the same time reliability. Therefore, both distributed under the terms and clock-based trip time and the passengers’ travel conditions are incorporated here to mea- conditions of the Creative Commons sure the Perceived Time Reliability (PTR) and Perceived Time Reliability Cost (PTRC). The Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ extra PTT needed for arriving reliably equaling the difference between the 95th percentile 4.0/). PTT and average PTT, is converted into the monetary cost for estimating PTRC. Referring

Sustainability 2021, 13, 7504. https://doi.org/10.3390/su13137504 https://www.mdpi.com/journal/sustainability Sustainability 2021, 13, 7504 2 of 22

to the buffer time index, the ratio of extra PTT needed for arriving reliably to the average PTT is proposed to measure PTR. To reveal the relations among PTR and influential factors, as well as the relations among PTRC and influential factors, multiple linear regression models were developed and determined by applying the stepwise regression method [11]. The remainder of this paper is organized as follows: the literature review discusses the previous research on time reliability. In the methodology section, the probability distributions of trip time components are introduced, the metrics of PTR and PTRC are proposed, multiple linear regression models are developed among PTR and influential factors, as well as among PTRC and influential factors, after which the PTR and PTRC estimation procedure are introduced. The PTR and PTRC for Chengdu’s RTN are estimated in the case study. The results of applying the stepwise regression method for obtaining the linear regression models are analyzed. Finally, the conclusions of the study are summarized.

2. Literature Review The concept of time reliability was commonly associated with the variability of time [12]. Researchers studied time reliability from the perspectives of operators [13], travelers [14], and planners [15] in recent years. To measure the time reliability of transportation networks from the perspective of operations, some indicators such as punctuality of transportation schedule indicator, on-time classification, coefficient of headways’ variation, and weighted delay indicators [16,17] were proposed. Although those indicators could measure the stability of transportation schedules and differences in the passengers’ waiting times, they could not guide the travelers’ decisions directly, nor did they analyze the time reliability in terms of the travelers’ total trip time. It has become much easier to obtain travelers’ trip times from Automatic Fare Collec- tion (AFC) data, Global Positioning Systems (GPS) data, and Automatic Vehicle Location (AVL) data. Therefore, researchers analyzed the time reliability for transportation networks with these data [17]. The distribution of time in a road network was estimated according to GPS data [18]. The time reliability of Chengdu’s urban rail transit network was measured by using AFC data [7]. The AVL data were used to estimate path travel time variance according to travel time variances on paths’ segments [19]. The AFC data were effective for obtaining travelers’ trip time and analyzing time reliability for urban rail transit networks or bus networks. However, these data could not be used to obtain travelers’ trip time for different transportation modes. Thus, if a person took a train with one ticket and then took a bus with another ticket, the trip time could not be fully determined from AFC data since the identification numbers of the two tickets differed. It was a common phenomenon that passengers used different tickets to travel on China’s RTNs since different companies managed different modes of rail transit (such as urban rail transit, high-speed railway, and suburban railway) and issued different kinds of tickets. Various metrics were proposed to measure the time reliability of transportation networks, as shown in Table1. The buffer time index was widely used for measuring time reliability in transportation networks since it not only measured the time reliability but also guided passengers to allow additional time for reliably reaching their destinations. Thus, the time reliability of London’s Underground was evaluated with the buffer time index according to the passengers’ trip time obtained from AFC data [20]. The buffer time index for London bus routes was evaluated using AVL data [21]. The design criteria for the time reliability metric from the passengers’ perspective were proposed by [22], who noted that the buffer time index satisfied the criteria for measuring time reliability. Sustainability 2021, 13, 7504 3 of 22

Table 1. Time reliability indicators and description.

Indicators Description Coefficient of variation [23] The ratio of the standard deviation to the mean. The ratio of the difference between the 90th percentile trip time and 50th percentile trip time to Skewness of time [24] the difference between the 50th percentile trip time and 10th percentile trip time. 90th or 95th percentile trip time used as the reliable 90th or 95th percentile trip time [25] trip time The difference between the average trip time and Buffer time [26] 95th percentile trip time. The percentage of buffer time with respect to the Buffer time index [26] average trip time. The probability that a trip arrives within the trip On-time arrival [27] time budget. Time unreliability [25] The fraction of late arriving trips. The minimum trip time threshold that satisfies a Total time budget [27] certain reliability requirement given by decision-makers at a certain confidence level. The conditional expectation of trip times exceeding Mean-excess total time [28] the corresponding total trip time budget at a given confidence level.

The methods found for measuring time reliability may be categorized into three groups: analytical approaches, statistical approximation, and simulation approaches. The analytical approaches can be applied under different traffic conditions. However, their main drawback is the high complexity of their estimation process for time distributions at the network level [29]. The Normal, Lognormal, Gamma, Truncated Normal/Lognormal, and Weibull distributions have been applied to compute time reliability using statistical methods [23,30]. The mixture of distribution models was used to measure time reliability since the single distribution model could not well represent the time distribution. However, it was difficult to determine the mix of distribution models. For simulation approaches, the time reliability of a transportation network could be estimated in different scenarios (extreme weather, transportation accidents, and transportation control). The simulation approaches can also be applied to measure the time reliability when demand and capacity fluctuate [31,32]. A simulation approach may be applied very flexibly, but at high computation cost, to measure the time reliability of transportation networks. The literature review shows that researchers emphasized measuring time reliability in terms of variability of clock-based trip time. However, the passengers’ travel conditions are neglected when measuring the time reliability of transportation networks. Here, passengers’ Perceived Trip Time (PTT), which integrates travel conditions (i.e., the number of transfer times, crowding, and seat availability in the vehicle), and the clock-based trip time are computed and used for measuring the PTR and PTRC in an RTN. It is difficult to apply analytical approaches and statistical approximation to measure the PTR for the RTN because the passengers’ trip components are complex and the distributions of trip time components are different. In addition, passengers may use different tickets to travel on different rail transit modes, which complicates matching the ticket numbers to obtain the passengers’ trip times. Therefore, a Monte Carlo simulation is used here to generate passengers’ PTT for measuring PTR and PTRC in an RTN. The simulation process for a large network only lasts a few minutes due to advances in computing power and speed. Sustainability 2021, 13, 7504 4 of 22

Sustainability 2021, 13, x FOR PEER REVIEW3. Methodology 4 of 22

In this section, the computation of the passengers’ PTT and the probability distributions of clock-based trip time components are introduced. After that, the metrics for measuringthe method PTR for andestimating PTRC onPTR paths, and amongPTRC based OD pairs, on passenger on lines, trip and assignment an RTN are and proposed. Monte Then,Carlo thesimulation method foris proposed. estimating PTR and PTRC based on passenger trip assignment and Monte Carlo simulation is proposed. 3.1. Passenger’s Clock-Based Trip Time and PTT 3.1. Passenger’s Clock-Based Trip Time and PTT The clock-based trip time on a path between an OD pair includes access, egress and transferThe walking clock-based time, trip waiting timeon time a path at the between origin and an ODtransfer pair includesstations as access, well as egress in-vehicle and transfertime. The walking clock-based time, waiting trip time time components at the origin are and shown transfer in Figure stations 1. asThe well clock-based as in-vehicle trip time.time Theis different clock-based from tripthe timepassengers’ components PTT. areStudies shown show in Figurethat the1. Thecomponents clock-based of clock- trip timebased is differenttrip time from are theperceived passengers’ differently PTT. Studies by passengers, show that thee.g., components waiting time of clock-basedhas a much triphigher time perceived are perceived value differentlythan in-vehicle by passengers, time in uncrowded e.g., waiting vehicles. time The has number a much of higher trans- perceivedfer times, valuecrowding than in in-vehicle vehicles, time and inreduced uncrowded seat availability vehicles. The also number increase of the transfer passengers’ times, crowding in vehicles, and reduced seat availability also increase the passengers’ PTT [33]. PTT [33].

Transfer waking at station

Access Waiting In-vehicle Egress at station at station on links at station

Figure 1. The clock-based trip time components. Figure 1. The clock-based trip time components.

TheThe crowding crowding level level and and seat seat availability availability inin trains, trains, as as well well as as the the number number of of transfer transfer timestimes affectaffect passengers’ passengers’ PTT PTT [ 34[34].]. Therefore,Therefore, thethe weighted weighted in-vehicle in-vehicle time time is is computed computed accordingaccording toto crowdingcrowding level level and and seat seat availability availability in in trains. trains. TheThe transfer transfer penalty penalty time time is is computed according to transfer time and the number of transfer times. ,, odwhich is esti- computed according to transfer time and the number of transfer times. tn,k, which is estimatedmated with with Equation Equation (1), (1), is is the the PTT PTT for for passenger passenger n traveling traveling from from station stationo to to station stationd on pathon pathk: :

, =,, +,, + ∑ , ∙ + ∑ ∙, +, ∙() + ,, (1) od = walk∈+ wait + vehicle∈,· + · walk + wait ·( )β2 + walk tn,k tn,k,o tn,k,o ∑e∈Eod tn,e βe ∑s∈Sod β1 tn,s tn,s mk tn,k,d (1) k k,trans where ,, , ,, , and ,, are walking time at station , walking time at station , and waitingwalk walktime at stationwait , respectively, for passenger traveling from station to where tn,k,o , tn,k,d , and tn,k,o are walking time at station o, walking time at station d, and waitingstation time on at path station . o,, respectively,, ∈ for is passengerin-vehicle timen traveling on link from . station is theo to set station of linksd vehicle od od∈ onfrom path stationk. tn, e to, estation∈ Ek is in-vehicleon path . time, on and link ,e. E,k is the, set of are links walking from time station and waiting time, respectively,walk for passengerwait travelingod on path . is the set of trans- o to station d on path k. tn,s and tn,s , s ∈ Sk,trans are walking time, and waiting time, fer stations from station to station on path od. is the number of transfer times respectively, for passenger n traveling on path k. Sk,trans is the set of transfer stations from The time weights =1.1 and =0.5 are used here according to [35]. The values of station o to station d on path k. mk is the number of transfer times The time weights β1 = 1.1 and stay the same when computing the PTTs for different paths. The time weight and β2 = 0.5 are used here according to [35]. The values of β1 and β2 stay the same when , ∈ computingis related to the crowding PTTs for differentin trains paths.and seat The availability time weight onβ elinkis related to crowding. The load in trainsfactor, which is the ratio of passengers to seatsod on a vehicle, is used to reflect the train crowding. and seat availability on link e, e ∈ Ek . The load factor, which is the ratio of passengers to seatsThe values on a vehicle, of at is different used to reﬂect load factors the train when crowding. passengers The sit values or stand of β aree at shown different in loadTable factors2 according when to passengers [36]. sit or stand are shown in Table2 according to [36].

Table 2. Values of at different load factors when sitting or standing. Load Factor (%) Sitting Standing 0–75 0.86 ---- 75–100 0.95 ---- 100–125 1.05 1.62 125–150 1.16 1.79 150–175 1.27 1.99 175–200 1.40 2.20 >200 1.55 2.44

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Table 2. Values of βe at different load factors when sitting or standing.

Load Factor (%) Sitting Standing 0–75 0.86 —- 75–100 0.95 —- 100–125 1.05 1.62 125–150 1.16 1.79 150–175 1.27 1.99 175–200 1.40 2.20 >200 1.55 2.44

3.2. Probability Distributions of Clock-Based Trip Time Components The values of trip time vary since clock-based trip time components are random variables subject to probability distributions. The probability distributions of clock-based trip time components are obtained and used in Monte Carlo simulation [37] to generate the passengers’ PTT.

3.2.1. Probability Distributions of Walking Time The passengers’ walking time at stations is affected by many factors, such as the degree of congestion, accessibility of facilities, and the passengers’ age and genders. Here, only two essential factors, i.e., walking distance (considers walking distances on the passageways, upstairs, and downstairs) and walking speed are considered when determining the walking time distribution. The walking speed follows an N µ, σ2 distribution on passageways, upstairs, and downstairs. The values of µ and σ are shown in Table3 according to [38].

Table 3. Values of µ and σ for walking speed distribution N µ, σ2 [38].

Walking Place µ (m/s) σ passageways 1.39 0.463 upstairs 0.79 0.236 downstairs 0.81 0.174

The walking time is the sum of the ratios of walking distance to walking speeds, and walk walk thus the walking time distributions on passageways tk,pass, upstairs tk,up , and downstairs walk tk,down in path k are calculated following Equations (2)–(4):

walk lpass twalk ∼ (2) k,pass N(1.39, 0.4632)

walk lup twalk ∼ (3) k,up N(0.79, 0.2362) lwalk tdown ∼ down (4) k,walk N(0.81, 0.1742) walk walk walk where lpass , lup , and ldown are walking distances on passageways, upstairs, and downstairs, respectively.

3.2.2. Probability Distributions of Waiting Time The waiting time distributions on frequency-based lines and schedule-based lines vary greatly because passengers can use their tickets for any trains on frequency-based lines but only for a speciﬁc train on schedule-based lines. When passengers wait on a frequency-based line, they may have an extra waiting time if some trains have insufﬁcient capacity. When passengers wait on a schedule-based line, they can board their intended Sustainability 2021, 13, 7504 6 of 22

trains as long as they have tickets for those trains since the tickets are sold according to the trains’ capacities. (1) Probability Distribution of Waiting Time on Frequency-based Lines The passengers’ arrivals at stations on frequency-based lines such as metro lines are Poisson-distributed [38]. Passengers waiting at frequency-based lines have normal waiting time and extra waiting time. The normal waiting time means that passengers can board the ﬁrst arriving train. The normal waiting time is a uniformly distributed random variable ranging from 0 to the headways on lines [39] as shown in Equation (5):

wait to,normal ∼ U(0, ho); ∀o ∈ Sfrequency (5)

wait where to,normal and ho are the passengers’ normal waiting times at station o and the headway of a line that includes o. Sfrequency is the set of stations that are on frequency-based lines. Passengers may have extra waiting times due to the ﬁrst arriving train not having enough capacity for all waiting passengers. One successful boarding at station o requires m independent trials, each with a probability of success po [40]. In addition to the ﬁrst arriving train, the probability that passengers need to wait longer for arriving trains on the platform represented as P (X = m) is estimated with Equation (6).

m P(X = m) = (1 − po) ·po; m = 0, 1, 2, . . . , ∀o ∈ Sfrequency (6)

po is the probability that passengers board a train successfully at station o, o ∈ Sfrequency. po is related to the passenger ﬂow and remaining capacity of a link e in the direction of intended boarding trains and connected with o directly, which is estimated with Equation (7):

Cape − fe po = (7) Cape

where Cape and fe are the capacity and passenger ﬂow on link e, which is in the direction of intended boarding trains and connected with o directly. According to Equation (6), the number of subsequent trains that passengers must wait wait for is geometrically distributed [40]. Thus, the extra waiting time to,extra at station o that equals the headway ho multiplied by the number of subsequent passing trains m has the following distribution (Equation (8)):

wait m P to,extra = m·ho = (1 − po) ·po; m = 0, 1, 2, . . . ∀o ∈ Sfrequency (8)

(2) Probability Distribution of Waiting Time on Schedule-based Lines To avoid missing their intended boarding trains, passengers arrive at the stations in advance. Passengers waiting on schedule-based lines have no extra waiting time since the tickets are sold according to the trains’ capacities. The probability density function of waiting time on schedule-based operation lines (such as high-speed railway lines) follows the mixed distribution of beta and gamma distributions as Equation (9) according to [41]:

Γ(ω + θ) − f twait = x = ζ· ·xω−1·(1 − x)θ 1 + (1 − ζ); ∀o ∈ S (9) o Γ(ω)·Γ(θ) schedule

where to,wait are the waiting times at station o, o ∈ Sschedule. Sschedule is the station set on schedule-based lines. ζ and 1 − ζ represent the fractions of the beta distribution and gamma distribution, respectively. Γ(ω), Γ(θ) and Γ(ω + θ) are gamma distributions. The parameters in Equation (9) are related to the headways of waiting lines, which can be determined from Table4 according to [41]. Sustainability 2021, 13, 7504 7 of 22

Table 4. Values of parameters in Equation (9) for different headways.

Headways (min) ζ ω θ 5 0.43 0.41 2.85 10 0.52 0.36 3.39 15 0.58 0.32 3.98 20 0.64 0.27 4.57 30 0.90 0.24 6.52

3.2.3. Probability Distribution of In-Vehicle Time The trains’ running time on a link is normally distributed, and the expected running time is proportional to the standard deviation, according to [40]. Therefore, the in-vehicle vehicle times on a link te are normally distributed as Equation (10):

2 vehicle vehicle vehicle te ∼ N ue , ue ·α ; ∀e ∈ E (10)

where ue,vehicle is the expected in-vehicle time on link e, e ∈ E, which can be estimated with the mean of train running time on link e. α is 0.04 according to [40]. E is the set of links in an RTN.

3.3. PTR and PTRC Metrics 3.3.1. PTR and PTRC on a Path The extra PTT needed for arriving reliably equals the 95th percentile PTT minus the average PTT, computed with Equation (11). The ratio of the extra PTT needed for arriving reliably to average PTT is used for measuring PTR and is computed with Equation (12):

od od od tk,reliable = tk,95 − tk,∗; ∀o, d ∈ S (11)

tod rod = k,reliable ∀o d ∈ S k od ; , (12) tk,∗ od od od where tk,reliable, tk,95, and tk,∗ are the extra PTTs needed for arriving reliably, 95th percentile PTT, and average PTT, respectively, from station o to station d on path k, o, d ∈ S. S is the od set of stations in an RTN. rk is PTR from station o to station d on path k. The extra PTT needed for arriving reliably is converted to a monetary cost according to Equation (13) to estimate PTRC.

od od ck = tk,reliable·γ; ∀o, d ∈ S (13)

od where ck is the PTRC from station o to station d on path k. γ is the value of the time parameter that converts PBT into a monetary cost, which is related to the passengers’ income [34].

3.3.2. PTR and PTRC among OD Pairs The PTR and PTRC between an OD pair equal the average PTR and average PTRC for passengers who travel between that OD pair. PTR and PTRC from station o to station d (rod and cod) are estimated with Equations (14) and (15), respectively:

= ∑k m rod·uod rod = k=1 k k ; ∀o, d ∈ S (14) uod

= ∑k m cod·uod cod = k=1 k k ; ∀o, d ∈ S (15) uod Sustainability 2021, 13, 7504 8 of 22

where M is the total number of travel paths selected by passengers from station o to station od od d. uk and u are passenger trips from station o to station d on path k and from station o to station d, respectively. uod In Equations (15) and (16), k = λod, thus uod can be estimated by computing the uod k k od probability of path k being selected by passengers λk when they travel from station o to station d. The probabilities of paths selected by passengers can be determined after passenger trip assignments.

3.3.3. PTR and PTRC on Lines

The PTR rl computed with Equation (16) and PTRC cl computed with Equation (17) on a line equal the respective average PTR and average PTRC for passengers whose origin station or destination station is on that line.

od od do do ∑o∈s ∑d∈S r ·u +r ·u r = l ; ∀l ∈ L (16) l uod + udo ∑o∈sl ∑d∈S

od od do do ∑o∈s ∑d∈S c ·u +c ·u c = l ; ∀l ∈ L (17) l uod + udo ∑o∈sl ∑d∈S od do where L is the set of lines in an RTN. Sl is the station set on line l. u and u are passenger trips from station o to station d and from station d to station o, respectively.

3.3.4. PTR and PTRC in an RTN The average PTR for all passengers computed with Equation (18) is used to estimate PTR on the RTN: ∑ ∑ rod·uod R = o∈S d∈S (18) u where R is the PTR of the RTN and u is passenger trips on the RTN. The PTRC in an RTN C computed with Equation (19) is estimated with the average PTRC for all passengers: ∑ ∑ cod·uod C = o∈S d∈S (19) u

3.4. PTR and PTRC Estimation Based on Passenger Trip Assignment and Monte Carlo Simulation To measure PTR and PTRC in an RTN, passenger trips are assigned to the RTN to determine load factors (which are shown in Table2) and probabilities of paths being selected by passengers. Here, a length-based C-logit stochastic user equilibrium model is applied to assign passenger trips. The model is solved by the method of successive weighted averages [42]. Then, a Monte Carlo simulation [37] is applied to generate passengers’ PTT according to probability distributions of trip time components for computing PTR and PTRC on the RTN.

3.4.1. The Length-Based C-Logit Stochastic User Equilibrium Model The length-based C-logit stochastic user equilibrium model is applied to assign passenger trips since it effectively reﬂects the overlapping effect among paths and the passengers’ travel characteristics [43]. The length-based C-logit Stochastic User Equilibrium model is shown in Equations (20)–(27):

1 = Z fe = ( ) = k m od· ( od) + ( ) + k m od· od min Z f ∑o∈S ∑d∈S o6=d ∑k= uk In uk ∑e∈E te x dx ∑o∈S ∑d∈S o6=d ∑k= uk ωk (20) ϑ , 1 0 , 1 subject to:   Lod od i=m k,i ωk = e·In∑ = q q ∀o, d ∈ S (21) i 1 od od Lk Li Sustainability 2021, 13, 7504 9 of 22

od od od uk = u ·λk ; ∀o, d ∈ S (22) od od exp(−ϑ·(t ∗ + ω )) λod = k, k ; ∀o, d ∈ S (23) k k=M (− ·( od + od)) ∑k=1 exp ϑ tk,∗ ωk = k=M od· od ∀ ∈ fe ∑o∈S ∑d∈S,o6=d ∑k=1 uk δek ; e E (24)

Seate = cseat·ne; ∀e ∈ E (25)

Cape = ccap·ne∀e ∈ E (26)

fe loade = ; ∀e ∈ E (27) Seate In the objective function, ϑ is a non-negative parameter. The higher the value of ϑ, the higher the accuracy of the passengers’ perception of paths’ trip time and the closer the assignment to the equilibrium assignment. te(x) is the PTT on link e, which equals od the in-vehicle time multiplied by its time weight (i.e., βe in Table2). ωk , which is the common factor for path k, is computed with constraint (21). It measures the similarities od of path k with other paths in the same OD pair according to length of common links. Lk,i od is the sum of common links’ lengths on paths k and i from station o to station d. Lk and od Li are lengths of paths k and i, respectively, from station o to station d. e is a parameter in Equation (26). If e = 0, then the length-based C-logit route choice model collapses to a Multinomial Logit model. If e = 1, then the route choice probabilities in the limiting case of N coincident paths tend to 1/N of those computed with a Multinomial Logit model applied while considering the coincident paths as a single path [44]. e = 1 is used here od according to [45]. In constraint (22), uk equals the passenger trips traveling from station o to station d (i.e., uod) multiplied the probability of path k being selected by passengers (i.e., od λk which is computed with constraint (23)). fe is the passenger ﬂow on link e, which is od computed with constraint (24). δek is a binary variable, which is 0 if the path k from station o to station d contains link e and 1 otherwise. Constraints (25) and (26) compute the hourly number of seats on link e (i.e., Seate) and the capacity of link e (i.e., Cape), respectively; they equal the frequency of trains passing through link e (i.e., ne) multiplied by the seats per train (i.e., cseat) and the capacity per train (i.e., ccap). loade in constraint (27) is the load factor on link e, which measures crowding levels in vehicles on that link.

3.4.2. The Method of Successive Weighted Averages The Length-based C-logit Stochastic User Equilibrium model is solved with the method of successive weighted averages [42] to assign passenger trips on the RTN, as follows:

Step 1: Set the iteration’s number h = 1, the algorithm variable γ0 = 1, the algorithm parameter a ≥ 0, and the stop iteration criterion ϕ. The effective path sets among OD pairs are determined using Yen’s algorithm. The PTTs of paths in the effective travel path sets are computed without considering passenger flow. Step 2: Passenger trips are assigned to the RTN with Equations (23) and (24) according to the PTTs of effective travel paths to compute passenger flow on each link, which is h represented as fe , ∀e ∈ E. Step 3: The PTTs of effective travel paths among OD pairs are computed according to h the passenger flows on links fe , ∀e ∈ E. The passenger trips among OD pairs h are assigned to the RTN again. The passenger flow on each link, ze , ∀e ∈ E, is recomputed. a ha Step 4: Let γ = γ − + h , θ = and h = h + 1. Passenger flows on links are updated h h 1 h γh with Equation (28): h+1 h h h fe = fe + θh· ze − fe (28) Sustainability 2021, 13, 7504 10 of 22

q h+1 h 2 ( fe − fe ) h+1 Step 5: Convergence assessment. If h ≤ ϕ, then stop iterating and fe is the ∑e∈E fe passenger flow after passenger trip assignment; otherwise, go to step 3. The passenger flows on links and the probabilities of paths being selected by passengers are determined according to the above five steps. The load factor on a link is estimated as the ratio of hourly passenger flow on that link to the provided seats per hour on that link. Thus, the weight of in-vehicle time βe is determined from Table2.

3.4.3. The Method of Successive Weighted Averages The PTR and PTRC are estimated by applying a Monte Carlo simulation after passenger trips are assigned to the RTN. According to the probability distributions of clock-based trip time components introduced in Section 3.2, the PTTs of H = 1000 passengers from station o to station d on path k are generated by applying Monte Carlo simulation as follows: Step 1: Initialize. Initialize the iteration number h = 1 and set the maximum iteration step H = 1000; Step 2: Generate PTT on path k for passenger n, n = h during h iteration. Step 2.1: Generate walking times at stations for passenger n, n = h. walk walk walk The walking distances on passageways lpass , upstairs lup , and downstairs ldown at origin station o, destination station d and transfer stations on path k are determined and then passenger n’s walking times on passageways, upstairs, and downstairs at each station are generated with Equations (2)–(4) in Section 3.2.1. Thus, passenger n’s walking times at walk walk origin station o, destination station d, and transfer stations represented as tn,k,o , tn,k,d , and walk od tn,s , s ∈ Sk,trans, respectively, are the sum of walking time on passageways, upstairs, and od downstairs at station o, station d and transfer station s, s ∈ Sk,trans. Step 2.2: Generate waiting times at stations for passenger n, n = h. od Passenger n’s waiting times at origin station o and transfer stations s ∈ Sk,trans on wait wait od path k represented as tn,k,o and tn,s , s ∈ Sk,trans, respectively, are generated. To generate a waiting time at a station for passenger n, the operation type of the line to which that the waiting station belongs is determined. If the operation type is a frequency-based line, then the waiting time at the station is generated as follows: The normal waiting time at the station is generated according to Equation (5); The link that is in the direction of intended boarding trains and connected with the waiting station directly is determined. Then, the probability of successful boarding is estimated with Equation (7). After that, an extra waiting time at the waiting station is generated with Equation (8). The total generated waiting time is the sum of the generated normal waiting time and the generated extra waiting time. If a waiting station belongs to a schedule-based line, then passengers waiting at that station do not have extra waiting time and the waiting time at that station is generated with Equation (9). Step 2.3: Generate the transfer penalty time for passenger n, n = h. walk wait β2 The equation ∑ ∈ od β1· tn,s + tn,s ·(mk) is used to computed the generated s Sk,trans transfer penalty time according to the generated walking time and waiting time at transfer stations in steps 2.1 and 2.2. Step 2.4: Generate the weighted in-vehicle time for passenger n, n = h. od The link set Ek on path k from station o to station d is determined. The weight of od in-vehicle time for each link e, e ∈ Ek (i.e., βe) is determined after passenger trips are od vehicle assigned. Passenger n’s in-vehicle time on each link e, e ∈ Ek (i.e., tn,e ) is generated vehicle according to Equation (10). Finally, ∑ od tn,e ·βe computes the generated weighted e∈Ek in-vehicle time for passenger n on path k. Step 2.5: Generate a PTT for passenger n. Equation (1) computes the generated PTT for passenger n traveling on path k according to generated times in steps 2.1 to 2.4. Step 3: Stop assessment. Sustainability 2021, 13, 7504 11 of 22

Let h = h + 1. If h < H, return to step 2; otherwise, go to step 4. Step 4: Estimate PTR and the PTRC from station o to station d on path k. The passengers’ PTTs are generated according to steps 1 to 3. The 95th percentile PTT is the 950th PTT in the generated 1000 PTTs’ rank sorted from smallest to largest. The od extra PTT needed for arriving reliably, represented as tk,reliable, equals the 95th percentile PTT minus the average PTT. Thus, the PTR and the PTRC on path k are estimated with Equations (12) and (13), respectively. The steps for estimating PTR and PTRs from station o to station d on path k are shown Sustainability 2021, 13, x FOR PEER REVIEW 11 of 22 in Figure2. Thus, PTR and PTRC among OD pairs, on lines as well as on the RTN are estimated with Equations (14)–(19).

Step 1: Initialize Let iteration number ℎ=1 and set the maximum iteration step = 1000;

Step 2: Generate PTT on path for passenger , = ℎ during ℎ iteration

Step 2.1 Generate walking times at stations for passenger , = ℎ.

Step 2.2 Generate waiting times at stations for passenger , = ℎ.

Step 2.3 Generate the transfer penalty time for passenger , = ℎ.

Step 2.4 Generate the weighted in-vehicle time for passenger , = ℎ.

Step 2.5 Generate the PTT for passenger .

ℎ=ℎ+1 Step 3 Yes ℎ<

Step 4: Estimate PTR and the PTRC from station to station on path

FigureFigure 2. 2. TheThe steps steps for for generating generating PTTs PTTs from station o toto station stationd on on path pathk. .

4.4. Case Case Study Study PTRPTR and and PTRC PTRC on on Chengdu’s Chengdu’s RTN RTN are are measur measured,ed, and the regression relating PTRS andand PTRC PTRC to to influential inﬂuential factors factors is is analyzed in this section. Some Some measures for enhancing PTRPTR and and reducing reducing PTRC PTRC on on Chengdu’s Chengdu’s RTN RTN are proposed based on the results.

4.1.4.1. Chengdu’s Chengdu’s RTN RTN TheThe transfer transfer and and terminal terminal stations stations on lines, as well well as as the the modes modes of of lines lines in in Chengdu’s Chengdu’s RTNRTN are are shown in Figure 33a.a. There werewere sixsix metrometro lineslines (lines(lines 11 toto 6),6), includingincluding threethree suburbansuburban railway railway lines lines (lines (lines 7 7 to to 9) 9) and and three three high-speed high-speed rail lines (lines 10 to 12), 12), as as well well asas 174 174 stations stations on on Chengdu’s RTN RTN in May 2019. Some Some details details of of the the metro network are shownshown in Figure 3 3b.b. TheThe public public transportation transportation trips trips in in urban urban and and suburban suburban areas, areas, as wellas well as between Chengdu and other nearby cities are mainly served by metro lines and suburban as between Chengdu and other nearby cities are mainly served by metro lines and subur- railway lines as well as high-speed rail lines. ban railway lines as well as high-speed rail lines. Sustainability 2021, 13, 7504 12 of 22 Sustainability 2021, 13, x FOR PEER REVIEW 12 of 22

159 164

174 157

70 LINE 3 Terminal station or 161 71 69 68 67 LINE 1 72 transfer station LINE 2 36 37 38 39 73 67 1 74 40 75 Metro lines 41 76 2 77 36 42 78 Suburban railway lines 43 79 13 147 149 151 3 LINE 5 12 0 1 44 148 150 80 131 4 81 9 Intercity high-speed rail lines Line 4 146 117 3 45 46 132 (128-102) 5 82 128 123 122 120 118 116 115 113 47 112 6 133 102 83 124 121 119 117 114 11 125 48 7 84 134 126 145 49 1 109 107 105 103

(3-3) 8 Line 5 85 5 LINE 4 127 50 110 108 106 104 102 9 1 90 128 LINE 5 87 86 135 144 10 52 LINE 4 88 11 53 56 57 56 66 54 55 156 89 58 143 12 136 59 90 93 91 13 60 101 Line 1 94 142 141 140 139 138 137 61 95 92 14 152 15 62 63 64 65 66 (1-35) 35 96 153 LINE 6 97 16 98 154 LINE 2 Line 11 99 17 100 155 18 (172-56) 101 156 19 LINE 3 LINE 6 20 23 24 LINE 1 21 25 26 27 28 29 30 31 32 33 34 35 22 170

171 172 (a) (b)

Figure 3. (a) Chengdu’s RTN network; (b) Chengdu’s Metro network. Figure 3. (a) Chengdu’s RTN network; (b) Chengdu’s Metro network.

4.2. Chengdu’s RTNRTN OperationOperation DataData andand SurveyedSurveyed DataData The operationoperation type, headways, capacity perper hour, andand providedprovided seatsseats perper hourhour onon Chengdu’s railrail transittransit lineslines duringduring morningmorning peakpeak periodsperiods (from(from 7:307:30 a.m.a.m. toto 9:309:30 a.m.)a.m.) areare shownshown inin TableTable5 5.. Table 5. Attributes of Lines in Chengdu’s RTN. Table 5. Attributes of Lines in Chengdu’s RTN. Headway Seats Capacity Line Operation Type Headway Seats Capacity Line Operation(min) Type (per Hour) (Passenger Trips per Hour) (min) (per hour) (Passenger Trips per hour) 1 frequency-based 2.00 348 × 30 1460 × 30 2 frequency-based1 frequency-based 2.73 2.00 348 × 22 348 × 30 1460 1460× 22× 30 3 frequency-based2 frequency-based 3.00 2.73 348 × 20 348 × 22 1460 1460× 20× 22 4 frequency-based3 frequency-based 3.00 3.00 348 × 20 348 × 20 1460 1460× 20× 20 5 frequency-based4 frequency-based 4.00 3.00 348 × 15 348 × 20 1460 1460× 15× 20 6 frequency-based 6.00 348 × 10 1460 × 10 5 frequency-based 4.00 348 × 15 1460 × 15 7 schedule-based 10.00 250 × 6 680 × 6 8 schedule-based6 frequency-based 15.00 6.00 250 × 4 348 × 10 680 1460× 4× 10 9 schedule-based7 schedule-based 15.00 10.00 250 × 4 250 × 6 680 680× 4× 6 10 schedule-based8 schedule-based 15.00 15.00 610 × 4 250 × 4 1280 680× ×4 4 11 schedule-based 10.00 610 × 6 1280 × 6 9 schedule-based 15.00 250 × 4 680 × 4 12 schedule-based 10.00 610 × 6 1280 × 6 10 schedule-based 15.00 610 × 4 1280 × 4 Seats= seats per train × frequency of trains; Capacity = capacity of train × frequency of trains. 11 schedule-based 10.00 610 × 6 1280 × 6 12 schedule-based 10.00 610 × 6 1280 × 6 With the support of the National Key R & D Program of China, our team conducted surveysSeats= seats on per Chengdu’s train × frequency RTN. The of trains; average Capa walkingcity = capacity distances of train on × passways, frequency of upstairs, trains. and downstairs for investigated stations are estimated to be nearly 300 m, 15 m, and 15 m, With the support of the National Key R & D Program of China, our team conducted respectively. Those walking distances are used to generate walking time using Monto Carlo surveys on Chengdu’s RTN. The average walking distances on passways, upstairs, and simulation when passengers access and egress stations since determining the walking distancesdownstairs on for passways, investigated upstairs, stations and are downstairs estimated for to 174be nearly stations 300 in m, Chengdu’s 15 m, and RTN 15 m, is difﬁcult.respectively. The Those transfer walking walking distances distances are on used upstairs to generate and downstairs walking are time estimated using Monto to be 10Carlo m. simulation The transfer when walking passengers distances access (sum and of egress walking stations distances since on determining passways, the upstairs, walk- anding distances downstairs) on passways, at all transfer upstairs, stations, and train downstairs running for times 174on stations all links, in Chengdu’s as well asthe RTN OD is tripdifficult. distribution The transfer in Chengdu’s walking RTNdistances during on morningupstairs peakand downstairs periods, are are obtained estimated from to anbe operator10 m. The and transfer a survey walking [7]. The distances passenger (sum trips of among walking OD distances pairs are shownon passways, in Figure upstairs,4. The and downstairs) at all transfer stations, train running times on all links, as well as the OD Sustainability 2021, 13, x FOR PEER REVIEW 13 of 22

Sustainability 2021, 13, 7504 13 of 22 trip distribution in Chengdu’s RTN during morning peak periods, are obtained from an operator and a survey [7]. The passenger trips among OD pairs are shown in Figure 4. The arcs represent the number of passenger trips traveling among OD pairs during morning arcs represent the number of passenger trips traveling among OD pairs during morning peakpeak periods. periods. Darker Darker arcs arcs represent represent higher higher passenger passenger trips trips among among OD OD pairs. pairs. To To limit limit this this paper’spaper’s length, length, line line 1 1 isis takentaken asas an example to show show transfer transfer walking walking distances distances on on line line 1’s 1’stransfer transfer stations stations and and train train running running times times on line on line 1’s links, 1’s links, which which are listed are listed in Table in Table 6; Table6; Table7, respectively.7, respectively.

FigureFigure 4. 4.Passenger Passenger trips trips among among OD OD pairs pairs during during morning morning peak peak periods. periods.

Table 6. Transfer walking distances at line 1’s transfer stations. Table 6. Transfer walking distances at line 1’s transfer stations. Transfer Direc- Walking Dis- Transfer Direc- Walking Distance Transfer Station Transfer Walking Transfer Walking Transfer Station Directiontion Distancetance (m) (m) Directiontion Distance(m) (m) 3 Line 1 to line 5 178 Line 5 to line 1 155 3 Line 1 to line 5 178 Line 5 to line 1 155 33 Line 11 to to line line 7 7 237 237 Line Line 7 7 to to line line 1 207 207 66 Line 11 to to line line 4 4 252 252 Line Line 4 4 to to line line 1 155 155 77 Line 11 to to line line 2 2 215 215 Line Line 2 2 to to line line 1 200 200 1010 Line 11 to to line line 3 3 200 200 Line Line 3 3 to to line line 1 104 104 13 Line 1 to line 5 252 Line 5 to line 1 126 13 Line 1 to line 5 252 Line 5 to line 1 126

TableThe 7. passengerTrain running trips times among on line OD 1’s pairslinks. are assigned to the network using the Length- based C-logit Stochastic User Equilibrium Model. The PTR and PTRC among OD pairs can be estimatedLink with MonteRunning Carlo simulationTime (min) according Link to the above data.Running In estimating Time (min) the PTRC,(Station–Station) the value of time Upstream parameter Downstream equals 30% of(Station–Station) household income Upstream per hour Downstream according to economist1–2 Kenneth Gwilliam’s 1.87 recommendation1.88 [46 18–19]. The average household1.35 income1.33 is 134,187 ¥/per2–3 year which 2.08 is obtained2.07 from the “Chengdu 19–20 Statistical 1.38 Yearbook-2018” 1.37 [ 47]. The work3–4 time per week 1.47 is 46 h according1.47 to the data 20–21 released by the 1.57 National Bureau1.57 of Statistics.4–5 There are nearly 1.57 52 weeks1.58 per year. Therefore, 21–22 the work 1.95hours are 23921.92 h per year and5–6 thus the average 1.22 household 1.25 income per hour 22–23 is 56.10 ¥/hour. 1.50 The value1.50 of time parameter6–7γ, which equals 1.35 30% of household1.33 income 23–24 per hour, is 16.83 1.83 ¥/hour. 1.83 7–8 1.15 1.20 24–25 1.70 1.78 8–9 1.15 1.17 25–26 1.78 1.77 9–10 1.32 1.32 26–27 1.93 1.93 10–11 1.27 1.28 27–28 2.00 2.25 11–12 1.35 1.33 28–29 1.75 1.50 Sustainability 2021, 13, 7504 14 of 22 Sustainability 2021, 13, x FOR PEER REVIEW 14 of 22

Table 7. Train running times on line 1’s links. 12–13 1.40 1.45 29–30 1.77 1.77 Link Running13–14 Time (min) 1.63 1.63Link 30–31 Running 1.58 Time (min) 1.58 (Station–Station) Upstream14–15 Downstream 1.50 (Station–Station)1.50 31–32 Upstream 1.50 Downstream1.50 15–16 1.17 1.17 32–33 1.33 1.33 1–2 1.87 1.88 18–19 1.35 1.33 2–3 2.0816–17 2.07 1.20 1.20 19–20 33–34 1.38 1.87 1.371.87 3–4 1.4717–18 1.47 1.78 1.78 20–21 34–35 1.57 2.00 1.572.00 4–5 1.57 1.58 21–22 1.95 1.92 5–6 1.22The passenger 1.25 trips among OD 22–23 pairs are assigned to 1.50 the network using 1.50 the Length- 6–7 1.35 1.33 23–24 1.83 1.83 7–8 1.15based C-logit Stochastic 1.20 User Equilibrium 24–25 Model. The PTR 1.70 and PTRC among 1.78 OD pairs 8–9 1.15can be estimated 1.17with Monte Carlo simulation 25–26 according 1.78 to the above data. In 1.77 estimating 9–10 1.32the PTRC, the value 1.32 of time parameter 26–27 equals 30% of household 1.93 income per hour 1.93 accord- 10–11 1.27ing to economist 1.28Kenneth Gwilliam’s 27–28 recommendation 2.00[46]. The average household 2.25 in- 11–12 1.35come is 134,187 ¥/per 1.33 year which is 28–29 obtained from the 1.75“Chengdu Statistical 1.50 Yearbook- 12–13 1.402018” [47]. The work 1.45 time per week 29–30is 46 h according to the 1.77 data released by 1.77the National 13–14 1.63 1.63 30–31 1.58 1.58 14–15 1.50Bureau of Statistics. 1.50 There are nearly 31–32 52 weeks per year. 1.50 Therefore, the work 1.50 hours are 15–16 1.172392 h per year and 1.17 thus the average 32–33 household income 1.33 per hour is 56.10 1.33¥/hour. The 16–17 1.20value of time parameter 1.20 γ, which equals 33–34 30% of household 1.87 income per hour, 1.87 is 16.83 17–18 1.78¥/hour. 1.78 34–35 2.00 2.00

4.3. PTR and PTRC Estimation 4.3. PTR and PTRC Estimation 4.3.1. PTR and PTRC among OD Pairs 4.3.1. PTR and PTRC among OD Pairs The PTRs and PTRCs on two paths from station 155 to station 13, which are shown The PTRs and PTRCs on two paths from station 155 to station 13, which are shown in Figurein Figure5, are 5, estimated. are estimated. This isThis shown is shown as an exampleas an example of measuring of measuring PTR and PTR PTRC and on PTRC paths. on paths.

159 164

174 157

161 67 36 1 Line 4 117 3 128 (128-102) 102

90 56 66 156 101 Line 1 (1-35) 35 Line 11 (172-56)

Path 1 170 13 Path 2 171 155 172

Figure 5. Two paths from station 155 to station 13. Figure 5. Two paths from station 155 to station 13.

TheThe cumulativecumulative distributionsdistributions ofof generated 1000 passengers’ PTTs PTTs on on paths paths 1 1 and and 2 2using using Monte Monte Carlo Carlo simulation simulation are are shown shown in Fi ingure Figure 6a,b,6a,b, respectively. respectively. The ThePTRs PTRs on paths on paths1 and 12 andare estimated 2 are estimated to be 0.135 to be and 0.135 0.203, and respectively, 0.203, respectively, according according to the mean to the PTT mean and PTT95th and percentile 95th percentile PTT on paths PTT on1 and paths 2. The 1 and PTRCs 2. The on PTRCspaths 1 onand paths 2 are 1 1.345 and 2¥ areand 1.3451.827¥ ¥. andThis 1.827 result¥. illustrates This result that illustrates PTR on that path PTR 1 is on more path reliable 1 is more than reliable on path than 2 and on path passengers 2 and pay less for traveling reliably on path 1 than on path 2. Passengers need to allow 0.135 SustainabilitySustainability 20212021, ,1313, ,x 7504 FOR PEER REVIEW 1515 of of 22 22

Sustainability 2021, 13, x FOR PEER REVIEW 15 of 22

timespassengers and 0.203 pay times less forthetraveling average PTTs reliably on Paths on path 1 and 1 than 2, respectively, on path 2. Passengersto reach station need 13 to reliably.timesallow and 0.135 The 0.203 times probabilities times and the 0.203 averageof times paths thePTTs 1 averageand on 2 Paths being PTTs 1 selectedand on Paths 2, respectively, by 1 andpassengers 2, respectively, to reach are 0.417 station to reachand 13 0.583,reliably.station respectively. 13 The reliably. probabilities TheTherefore, probabilities of pathsthe PTR of1 pathsandand 2PTRC 1being and from 2 selected being station selected by 155passengers byto passengersstation are 13 0.417are are 0.175 0.417 and and0.583,and 1.626 0.583, respectively. ¥, respectively.respectively. Therefore, Therefore, the PTR the and PTR PTRC and PTRC from fromstation station 155 to 155 station to station 13 are 13 0.175 are and0.175 1.626 and ¥, 1.626 respectively. ¥, respectively. 1 1 95th PTT is 53.22 min 95th PTT is 47.19 min 1 1 95th PTT is 53.22 min 95th PTT is 47.19 min 0.8 0.8

0.8 0.8 0.6 0.6 Mean of PTT is Mean of PTT is 47.37 min 0.6 PBT is 5.85 min 0.6 39.24 min PBT is 7.95 min Mean of PTT is 0.4 Mean of PTT is 47.37 min 0.4 PBT is 5.85 min 39.24 min PBT is 7.95 min 0.4 Cumulative distribution Cumulative 0.4 0.2 0.2 Cumulative distribution Cumulative Cumulative distribution Cumulative 0.2 0.2 Cumulative distribution Cumulative 0 0 35 40 45 50 55 60 25 30 35 40 45 50 55 60 0 PTT/min 0 PTT/min 35 40 45 50 55 60 25 30 35 40 45 50 55 60 (b) PTT/min(a) PTT/min (a) (b) Figure 6. (a) Cumulative distribution of generated 1000 passengers’ PTTs on path 1; (b) Cumulative distribution of gener- atedFigure 1000 6. passengers’(a) Cumulative PTTs distributiondistribution on path 1. of of generated generated 1000 1000 passengers’ passengers’ PTTs PTTs on on path path 1; (1;b )(b Cumulative) Cumulative distribution distribution of generated of gener- 1000ated 1000 passengers’ passengers’ PTTs PTTs on path on 1.path 1. The PTRs among OD pairs are estimated according to the proposed metrics and method.The The PTRs contours among of OD PTRs pairs among are OD estimated pairs in according Chengdu’s to RTN the proposedduring morning metrics peak and hoursmethod. are The Theshown contours contours in Figure of of PTRs PTRs 7a. Figure among among 7a OD OD shows pairs pairs the in Chengdu’s PTRs among RTN OD during pairs aremorning between peak 0 tohours 0.3, whichare shown means inin FigurethatFigure passengers7 a.7a. Figure Figure 7shoulda 7a shows shows allow the the PTRs0 toPTRs 0.3 among timesamong OD the OD pairs average pairs are are betweenPTT between to reach 0 to 0 theirto0.3, 0.3, whichdestinations which means means reliably. that that passengers passengers Figure should7b showsshould allow the allow 0fractions to 0 0.3 to times0.3 of times OD the pairs averagethe average in different PTT toPTT reach PTRto reach their in- tervalstheirdestinations destinations during reliably. morning reliably. Figure peak Figure7 periods.b shows 7b shows It the shows fractions the that fractions ofonly OD 1.57% of pairs OD of inpairs OD different inpairs’ different PTR PTRs intervals PTRexceed in- 0.2,tervalsduring which during morning demonstrates morning peak periods. peakthat only periods. It a showsfew It OD shows that pairs onlythat have only 1.57% low 1.57% ofPTR. OD of pairs’OD pairs’ PTRs PTRs exceed exceed 0.2, 0.2,which which demonstrates demonstrates that that only only a few a few OD OD pairs pairs have have low low PTR. PTR.

60 55.05 60 50 55.05

50 40 36.77 40 30 36.77 Destination station number Destination

Fraction (%) 30 20 Destination station number Destination

Fraction (%) 20 10 5.43 1.18 1.41 10 0.15 0.01 0 5.43 [0, 0.05) [0.05, 0.1)[0.1, 0.15) [0.15, 0.2) [0.2,1.41 0.25) [0.25, 0.3) [0.3, 0.35) 1.18 0.15 0.01 Origin station number 0 PTTR intervals [0, 0.05) [0.05, 0.1)[0.1, 0.15) [0.15, 0.2) [0.2, 0.25) [0.25, 0.3) [0.3, 0.35) Origin station(a) number PTTR(b) intervals (a) (b) FigureFigure 7. 7. (a(a) )PTRs PTRs among among OD OD pairs; pairs; (b (b) )Fractions Fractions of of OD OD pairs pairs in in different different PTR PTR intervals. intervals. Figure 7. (a) PTRs among OD pairs; (b) Fractions of OD pairs in different PTR intervals. TheThe PTRCs PTRCs among among OD OD pairs pairs are are estimated estimated a accordingccording to to PTRs PTRs among among OD OD pairs. pairs. The The contourscontoursThe of ofPTRCs PTRCs PTRCs among among OD OD pairs pairspairs are andand estimated thethe fractions fractions according of of OD OD pairs to pairs PTRs in in different amongdifferent PTRCOD PTRC pairs. intervals inter- The valscontoursare shownare shown of inPTRCs Figure in Figureamong8a,b, respectively,8a,b,OD pairs respectively, and on the Chengdu’s frac on tionsChengdu’s of RTN OD duringpairsRTN induring morningdifferent morning peakPTRC hours. inter-peak hours.valsFigure are Figure8a shown shows 8a showsin the Figure PTRCs the PTRCs8a,b, among respectively, among OD pairs OD pairs areon betweenChengdu’s are between 0.72 RTN 0.72¥ to during ¥ 5 to¥. 5 It ¥. meansmorning It means that peakthat the thehours.extra extra costs Figure costs of arriving8aof showsarriving at the destinationat PTRCs destination among stations stations OD reliably pairs reliably are for between for a passenger a passenger 0.72 ¥ are to are between5 ¥. between It means 0.72 0.72 that¥ to ¥the5.98 to 5.98extra¥. The ¥. costs The stations ofstations arriving whose whose at numbers destination numbers exceed excstationseed 156 156 are reliably onare suburbanon for suburban a passenger and and high-speed arehigh-speed between railways. rail- 0.72 ways.¥Figure to 5.98 Figure8a ¥. shows The 8a stationsshows PTRCs PTRCs amongwhose among Metro,numbers suburbanMetro, exceed suburban 156 railway, are onrailway, and suburban high-speed and andhigh-speed railway high-speed arerailway high. railways. Figure 8a shows PTRCs among Metro, suburban railway, and high-speed railway Sustainability 2021, 13, x FOR PEER REVIEW 16 of 22

SustainabilitySustainability 20212021,, 1313,, x 7504 FOR PEER REVIEW 1616 of of 22 22

are high. Figure 8b shows the fractions of OD pairs in different PTRC intervals during areFiguremorning high.8b Figure showspeak periods. the8b fractionsshows It showsthe of fractions OD that pairs nearly of in OD different 80% pairs of OD PTRC in differentpairs’ intervals PTRCs PTRC during are intervals between morning during 1 peak¥ and morningperiods.3 ¥. Itpeak shows periods. that nearly It shows 80% that of ODnearly pairs’ 80% PTRCs of OD are pairs’ between PTRCs 1 ¥are and between 3 ¥. 1 ¥ and 3 ¥.

48.89 6050

48.89 5040

30.65 4030 30.65 Destination stationDestination number 3020 Fraction (%) Destination stationDestination number 9.63 2010 7.58

Fraction (%) 2.81 9.63 7.58 0.45 10 0 2.81[0,1) [1,2) [2,3) [3,4) [4,5) [5,6) 0.45 Origin station number 0 CADSR intervals [0,1) [1,2) [2,3) [3,4) [4,5) [5,6) Origin station(a) number CADSR intervals(b) (a) (b) FigureFigure 8.8.( (aa)) PTRCsPTRCs amongamong ODOD pairs;pairs; ( (bb)) Fractions Fractions of of OD OD pairs pairs in in different different PTRC PTRC intervals. intervals. Figure 8. (a) PTRCs among OD pairs; (b) Fractions of OD pairs in different PTRC intervals. 4.3.2.4.3.2. PTRsPTRs andand thethe PTRCsPTRCs onon LinesLines andand onon thethe RTNRTN 4.3.2. ThePTRsThe PTRs and theand and PTRCs PTRCs PTRCs on on onLines lines lines andduring during on themorning morningRTN peak peak hours hours are estimated are estimated with Equa- with EquationstionsThe (16) PTRs and (16) and(17). and PTRCs They (17). are Theyon shown lines are during shownin Figure morning in 9a,b, Figure respectively. peak9a,b, hours respectively. are A higher estimated A value higher with of the valueEqua- PTR tionsofindicates the (16) PTR and lower indicates (17). reliability. They lower are The shown reliability. PTR in on Figure line The 6 9a,b,is PTR the respectively. lowest on line and 6 is PTRC A the higher lowest on line value and8 is of the PTRC the highest PTR on indicateslinein Figure 8 is lower the 9a,b, highest reliability.respectively. in Figure The Therefore, PTR9a,b, on respectively. line with 6 is limited the lowest Therefore, operational and PTRC with resources on limited line and8 is operational the manpower, highest inresourcespriority Figure can9a,b, and be respectively. manpower, given to improving Therefore, priority the can with manage be givenlimitedment to operational improving of lines with resources the a managementhigh valueand manpower, of PTR of lines and prioritywithhigh aPTRC highcan beto value givenincrease of to PTR improvingpassengers’ and high the PTR PTRC manage and to decrease increasement of the passengers’lines passengers’ with a high PTR PTRC. value and decrease of PTR and the highpassengers’ PTRC to PTRC. increase passengers’ PTR and decrease the passengers’ PTRC. 0.15 2.1

0.15 2.1 1.8 0.12 1.8 0.12 1.5 0.09 1.5 PTTR 1.2 0.09 CADSR/¥ PTTR 1.2 0.06 CADSR/¥ 0.9 0.06 0.9 0.03 0.6 123456789101112 123456789101112 0.03 Line 0.6 Line 123456789101112 123456789101112 (a) Line(b) Line (a) (b) Figure 9. (a) PTRs on lines; (b) CADRs on lines. FigureFigure 9. 9. ((aa)) PTRs PTRs on on lines; lines; ( (bb)) CADRs CADRs on on lines. lines. The PTR and PTRC on Chengdu’s RTN during morning peak hours are estimated to be 0.14,TheThe PTRand PTR and2.07 and PTRC¥, PTRC respectively. on on Chengdu’s Chengdu’s Therefore, RTN RTN during duringon average, morning morning passengers peak peak hours hours should are are estimated estimatedallow 0.14 to toof bebethe 0.14, 0.14, mean and and PTT 2.07 2.07 (equivalent ¥,¥, respectively. respectively. to 2.07 Therefore, Therefore, ¥) to arrive on on at average, average,their destination passengers passengers stations should should reliably. allow allow The0.14 0.14 fareof of thetheper mean mean passenger PTT PTT (equivalent (equivalenttrip is 6.08 to to¥ 2.07in 2.07 Chengdu’s ¥)¥) to to arrive arrive RTN at at their their during destination destination morning stations stations peak hours. reliably. reliably. Therefore, The The fare fare a perperpassenger passenger passenger must trip trip spend is is 6.08 6.08 over ¥¥ inin 1/3 Chengdu’s Chengdu’s of the basic RTN RTN travel during during fare inmorning morning equivalent peak peak extra hours. hours. time Therefore, Therefore, to reach the a a passengerpassengerdestination must must station spend spend reliably. over over 1/3 1/3 of of the the basic basic travel travel fare fare in in equivalent equivalent extra extra time time to to reach reach the the destinationdestination station station reliably. reliably. 4.4. Regression Analysis Relating PTRs and PTRCs to Influential Factors 4.4. Regression Analysis Relating PTRs and PTRCs to Influential Factors 4.4.4.4.1. Regression Regression Analysis Models Relating Relating PTRs PTR and and PTRCs PTRC to Influentialto Influential Factors Factors 4.4.1. Regression Models Relating PTR and PTRC to Influential Factors 4.4.1. Regression Models Relating PTR and PTRC to Influential Factors The regression models relating PTR and PTRC to influential factors are proposed. In theory, more influential factors in a regression model increase its degree of fit and interpretability. However, no model can include all the influential factors due to possible Sustainability 2021, 13, 7504 17 of 22

multicollinearity. Besides, too many factors in a regression model will lead to model overfitting. The factors that are closely related to passengers’ PTT and can be quantified are considered. Six influential factors (i.e., explanatory variables), which are the distances of paths X1, walking distances on paths X2, the average trip times on paths X3, the average waiting times on paths X4, average loading factor on paths’ links X5, and ratios of walking distances on paths to distances of paths X6 are considered in developing multiple linear regression models:

Y1 = µ0 + µ1X1 + µ2X2 + µ3X3 + µ4X4 + µ5X5 + µ6X6 + ε1 (29)

Y2 = τ0 + τ1X1 + τ2X2 + τ3X3 + τ4X4 + τ5X5 + τ6X6 + ε2 (30)

where Y1 and Y2 are PTRs and PTRCs on paths, respectively. µ1 to µ6 and τ1 to τ6 are regression coefficients for each explanatory variable. µi, ∀i = 1, 2, ... , 6 means the change in the mean of PTR corresponding to a unit change in Xi when other variables Xj, j 6= i & ∀j = 1, 2, . . . , 6 are held constant. τi, ∀i = 1, 2, ... , 6 means the change in the mean of PTRC corresponding to a unit change in Xi when other variables Xj, j 6= i & ∀j = 1, 2, . . . , 6 are held constant. µ0 and τ0 are constant values. ε1 and ε2 represent models’ error, which are random variables and whose mean is 0. The influential factors in the multiple linear regression models may have correlations among them. In addition, not all influential factors significantly influence the PTRs or PTRCs. The stepwise regression method [11] is used here to reduce multicollinearity among influential factors. It gradually introduces influential factors that have a high impact on the PTRs or PTRCs into the regression models. During the process of introducing the factors into a regression model, the significance test is performed for the new model when a new factor is introduced into it.

4.4.2. Multiple Linear Regression Analysis for PTRs and Influential Factors The result of applying the stepwise regression method to determine the appropriate multiple linear regression model among PTRs and influential factors is shown in Figure 10. Figure 10a demonstrates that with additional stepwise regression iterations, the root mean square error (RMSE) decreases and the model fitting improves. The result of applying the stepwise regression method shows that all factors except X2 (walking distances on paths) are included in the model. The multiple linear regression model is shown in Equation (31). The significance value of the model (i.e., the value of p in Figure 10b) is 6 × 10−6, which is below 0.05 (0.05 is a commonly used value for judging whether the significance test is passed). This demonstrates that the linear relation between PTR and influential factors is significant. The coefficient of determination (i.e., R_square in Figure 10b) that quantifies the degree of linear correlation is 0.82433. It indicates that Equation (31) has a high goodness- of-fit.

Y1 = 0.11015 + 0.00048X1 − 0.08547X3 + 0.24889X4 + 0.0144X5 + 0.45231X6 (31)

4.4.3. Multiple Linear Regression Analysis for PTRC and Influential Factors Figure 11 shows the RMSE after introducing factors into the regression model and the result of applying the stepwise regression for determining the appropriate multiple linear regression model that relates PTRC to influential factors. Figure 11a demonstrates that applying the stepwise regression method can improve the fit of the model. The result of applying the stepwise regression method shows that X1 (distances of paths) is not introduced into the multiple linear regression model; 4 × 10−6 (i.e., the value of p in Figure 11b) is the significance value of the model, which is below 0.05 (0.05 is a commonly used value for judging whether the significance test is passed). Therefore, the linear relation between PTRC and influential factors is significant. R_square in Figure 11b is 0.9014. It is Sustainability 2021, 13, 7504 18 of 22

close to 1 and thus demonstrates that Equation (32) has a high goodness-of-ﬁt. The multiple Sustainability 2021, 13, x FOR PEER REVIEWlinear regression model between PTRC and inﬂuential factors is shown in Equation18 (32). of 22

Y2 = −0.26522 + 0.32674X2 + 0.30112X3 + 6.26087X4 + 0.04168X5 + 1.34128X6 (32)

0.04

0.03

RMSE 0.02

0.01 123456Steps (a)

Coeff. t_stat p_val

X1 0.00048 11.7486 0.0000 Intercept = 0.110148 X2 -0.00393 -1.9353 0.0532 RMSE = 0.0129846 X3 -0.08547 -27.1060 0.0000 R_square = 0.824329 Adj R_sq = 0.823445 X4 0.24889 42.4425 0.0000 F = 932.861 X5 0.01440 20.8709 0.0000 p = 0.000006 X6 0.45231 34.5493 0.0000

-0.1 0 0.1 0.2 0.3 0.4 0.5 Coefficients with Error Bars (b) Sustainability 2021, 13, x FOR PEER REVIEW 19 of 22 FigureFigure 10. 10. ((aa)) The The value value of of RMSE RMSE when when introducing introducing influential inﬂuential factorsfactors intointo thethe model;model; ((bb)) TheThe resultre- sultof applying of applying the stepwisethe stepwise regression regression method. method.

4.4.3.1.5 Multiple Linear Regression Analysis for PTRC and Influential Factors 1Figure 11 shows the RMSE after introducing factors into the regression model and theRMS E 0.5result of applying the stepwise regression for determining the appropriate multiple 0 linear regression123456 model that relates PTRC to influential factors. Figure 11a demonstrates that applying the stepwise regression(a) method can improve the fit of the model. The result of applying the stepwise regression method shows that (distances of paths) is not in- -6 troduced into the multiple linear regressionCoeff. t_stat model; p_val 4 × 10 (i.e., the value of in Figure 11b)X1 is the significance value of the model,-0.00073 which-0.7285 0.4665 is below 0.05 (0.05 is a commonly used Intercept = -0.265224 valueX2 for judging whether the significance 0.32674 6.6239test is0.0000 passed). Therefore, the linear relation RMSE = 0.317039

betweenX3 PTRC and influential factors is0.30112 significan 6.0691 0.0000t. R_squareR_square in = 0.90141Figure 11b is 0.9014. It is

closeX4 to 1 and thus demonstrates that Equati 6.26087 38.5496on (32) 0.0000 has a highAdj R_sq goodness-of-fit. = 0.900914 The multiple linear regression model between PTRC and influentialF = factors1817.63 is shown in Equation X5 0.04168 2.4784 0.0134 (32). p = 0.000004 X6 1.34128 4.0401 0.0001

0246 (312 = −0.26522 + 0.32674 + 0.30112 + 6.26087 + 0.04168 + 1.34128 Coefficients with Error Bars ) (b)

Figure 11. (a) TheThe valuevalue ofof RMSERMSE whenwhen introducingintroducing inﬂuentialinfluential factors factors into into the the model; model; ( b(b)) The The result result of applying the stepwise regression method. of applying the stepwise regression method.

4.4.4. PTR Enhancement and PTRC Reduction AnalysisAnalysis

The absolute coefficients’ coefficients’ values for X4 andand X6 are aremuch much largerlarger thanthan thethe coefficientcoefficient valuesvalues forfor otherother influentialinfluential factorsfactors inin EquationsEquations (31)(31) andand (32).(32). ThisThis meansmeans thatthat thethe averageaverage waitingwaiting timestimes onon pathspaths ((X4) andand thethe ratiosratios ofof walkingwalking distancesdistances onon pathspaths toto distancesdistances ofof pathspaths ((X6) have a much higher influenceinfluence on paths’ PTRsPTRs andand PTRCsPTRCs thanthan otherother influentialinfluential factors.factors. A higher higher value value of of Y1 means a lower PTR. In In order order to to decrease decrease the the Y1 and Y2 andand thus to enhance PTR as well as reduce PTRC on the network, the values of and should be decreased. The percent decreases in and are shown in Figure 12 when the values of and are decreased.

Percentage decrease in 1 Percentage decrease in 2

9.67%

7.45%

4.91% 3.84%

1.90%

0.41%

10% reduction in 4 10% reduction in 6 10% reduction in 6 and 4

Figure 12. The percent decreases in and when the values of and are decreased.

Figure 12 shows that reducing the values of and is effective in decreasing and . Therefore, two ways for improving the PTR and decreasing PTRC on the RTN are proposed: (1) decreasing the headway on lines to decrease the average waiting time on paths, and (2) providing convenient facilities and channels for passengers to enter, exit, and transfer at the stations, thereby decreasing the walking distances on paths.

Sustainability 2021, 13, x FOR PEER REVIEW 19 of 22

1.5

RMS E 0.5

0 123456 (a)

Coeff. t_stat p_val

X1 -0.00073 -0.7285 0.4665 Intercept = -0.265224 X2 0.32674 6.6239 0.0000 RMSE = 0.317039

X3 0.30112 6.0691 0.0000 R_square = 0.90141

X4 6.26087 38.5496 0.0000 Adj R_sq = 0.900914 F = 1817.63 X5 0.04168 2.4784 0.0134 p = 0.000004 X6 1.34128 4.0401 0.0001

0246 Coefficients with Error Bars (b) Figure 11. (a) The value of RMSE when introducing influential factors into the model; (b) The result of applying the stepwise regression method.

4.4.4. PTR Enhancement and PTRC Reduction Analysis

The absolute coefficients’ values for and are much larger than the coefficient Sustainability 2021, 13, 7504 values for other influential factors in Equations (31) and (32). This means that the average19 of 22 waiting times on paths () and the ratios of walking distances on paths to distances of paths () have a much higher influence on paths’ PTRs and PTRCs than other influential factors. A higher value of means a lower PTR. In order to decrease the and and thusthus toto enhanceenhance PTRPTR asas wellwell asas reducereduce PTRCPTRC onon thethe network,network, thethe valuesvalues ofof X4 andand X6 shouldshould bebe decreased.decreased. The The percent percent decreases decreases in inY1 and andY2 are are shown shown in Figure in Figure 12 when 12 when the valuesthe values of X of4 and Xand6 are decreased. are decreased.

Percentage decrease in 1 Percentage decrease in 2

9.67%

7.45%

4.91% 3.84%

1.90%

0.41%

10% reduction in 4 10% reduction in 6 10% reduction in 6 and 4

Figure 12. The percent decreases in and when the values of and are decreased. Figure 12. The percent decreases in Y1 and Y2 when the values of X4 and X6 are decreased. Figure 12 shows that reducing the values of and is effective in decreasing Figure 12 shows that reducing the values of X4 and X6 is effective in decreasing Y1 andand Y2.. Therefore,Therefore, twotwo waysways forfor improvingimproving thethe PTRPTR andand decreasingdecreasing PTRCPTRC onon thethe RTNRTN areare proposed:proposed: (1)(1) decreasingdecreasing thethe headwayheadway onon lineslines toto decreasedecrease thethe averageaverage waitingwaiting timetime onon paths,paths, andand (2)(2) providingproviding convenientconvenient facilitiesfacilities andand channelschannels forfor passengerspassengers toto enter,enter, exit,exit, andand transfertransfer atat thethe stations,stations, therebythereby decreasingdecreasing thethe walkingwalking distancesdistanceson on paths. paths.

5. Conclusions Time reliability is one of the most important factors affecting the passengers’ sat- isfaction with transportation services. The assessment of time reliability has attracted considerable attention. However, the existing literature on time reliability neglects the passengers’ travel conditions since the clock-based trip time, which does not contain travel conditions, is used for measuring time reliability. This study measures the PTR according to the passengers’ PTTs, which reflect passengers’ travel conditions (crowding and seat availability in the vehicle and the number of transfer times on the path). The metrics for estimating PTRs and the cost passengers pay for reaching destination reliably (PTRCs) are developed. Due to the difficulty of obtaining passengers’ trip time traveling among multiple modes of rail transit, a Monte Carlo simulation is used here for estimating PTTR and PTRC that does not rely on massive passenger travel data and data analysis. The multiple linear regression models relating PTR and PTRC to influential factors are developed and determined by applying the stepwise regression method. The following conclusions are drawn from applying proposed models and method to Chengdu’s RTN during morning peak hours: Only a few OD pairs on Chengdu’s RTN during morning peak hours have low PTR. Nearly 80% of OD pairs’ PTRCs are between 1 ¥ and 3 ¥. The PTR on line 6 is the lowest, and PTRC on line 8 is the highest. The PTR and PTRC on Chengdu’s RTN are estimated to be 0.14, and 2.07 ¥, respectively. A passenger must allow 0.14 of the average PTT and spend over 1/3 of travel fare in equivalent extra PTT to reach the destination station reliably. The linear relations among PTR, PTRC, and their influential factors are significant. The obtained multiple linear regression models have high goodness-of-fit. The average waiting times on paths and the ratios of walking distances on paths to distances of paths affect PTR and PTRC much more than other factors. Decreasing the headway on lines decreases the average waiting time on paths, and providing convenient facilities and channels for passengers to enter, exit, and transfer at the stations to decrease the walking distances on paths are effective in enhancing PTR and reducing the PTRC in an RTN. Sustainability 2021, 13, 7504 20 of 22

The proposed models and method can measure PTR and PTRC in an RTN network by integrating travel conditions and trip time. According to the evaluation results, the lines with low PTR and with high PTRC can be identified, which helps operators allocate operational resources and manpower to those lines with high priority to improve PTR and reduce PTRC. The influences of factors on PTR and PTRC are quantified according to obtained multiple linear regression models, which not only help guide operators in effectively improving the PTR and reducing the PTRC in an RTN, but also guides passengers to prepare extra PTT and select a path with high PTR. Although the metrics and method only apply to an RTN, they can potentially apply to different transportation modes, such as bus networks. Although crowding level of trains, seat availability, and transfer times are considered here to measure PTR, other factors that affect the time reliability, such as the stability of the transportation and the probability of the transportation disturbances should be considered in further studies. Many passengers require transfers among multiple public transportation lines, such as rail transit and bus. Therefore, the PTR and PTRC evaluation should be extended to public transportation networks that include rail transit and bus lines.

Author Contributions: Conceptualization, J.L. and P.S.; methodology, J.L. and Y.Y.; software, J.C.; validation, J.L., J.C. and Y.Y.; formal analysis, Y.Y.; investigation, J.L., J.C. and Y.Y.; resources, Q.P.; data curation, Y.Y.; writing—original draft preparation, J.L. and Y.Y.; writing—review and editing, P.S. and Y.Y.; visualization, J.C.; supervision, Q.P.; project administration, Q.P.; funding acquisition, Q.P. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Key R & D Program of China, grant number 2017YFB1200700, and the China Scholarship Council, grant number 201907000071. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors thank the Chengdu’s rail transit manager for providing rele- vant data. We also acknowledge the support of the National Key R & D Program of China (2017YFB1200700), The first author is supported by the China Scholarship Council (No.201907000071). Conflicts of Interest: The authors declare no conflict of interest.

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