Cell transmission model of dynamic assignment for urban rail transit networks

For urban rail transit network, the space-time flow distribution can play an important role in evaluating and optimizing the space-time resource allocation. For obtaining the space-time flow distribution without the restriction of schedules, a dynamic assignment problem is proposed based on the concept of continuous transmission. To solve the dynamic assignment problem, the cell transmission model is built for urban rail transit networks. The priority principle, queuing process, capacity constraints and congestion effects are considered in the cell transmission mechanism. Then an efficient method is designed to solve the shortest path for an urban rail network, which decreases the computing cost for solving the cell transmission model. The instantaneous dynamic user optimal state can be reached with the method of successive average. Many evaluation indexes of passenger flow can be generated, to provide effective support for the optimization of train schedules and the capacity evaluation for urban rail transit network. Finally, the model and its potential application are demonstrated via two numerical experiments using a small-scale network and the Beijing Metro network.


Introduction
The development of urban rail transit systems (URTS) has been proposed as a coping strategy to relieve traffic congestion around the world. A URTS not only has the properties of large capacity with less land occupation, but also has the advantages of energy conservation, environmental protection, high safety and reliability. It is a sustainable transportation mode.
Urban traffic demand is known as the time-varying origin-destination (O-D) demand. For example, during the morning rush hour, passengers gather from places of residence to their work place; during the afternoon rush hour, passengers disperse from their work place to other places. The time-varying demand makes the URTS operation complex.
The core of the URTS operation is the train schedule. A high-quality schedule must fit the time-varying flow, i.e., it should have a high train frequency when the flows are large in some time intervals. Train schedule design of urban rail network is a rather complex problem with considering the time-varying demand. To simplify the schedule design problem, it is always divided into the schedule design for each single rail line [1][2]. Niu [46]. In particular, Carey et al. [46] indicated that the usual recommended method for preserving FIFO will ensure FIFO for each cell taken separately, but does not fully ensure FIFO in the transition between cells or for links or for routes, and the paper is concerned with how to implement FIFO in the CTM. However, the CTM is not used to solve the transit assignment problem. For transit assignment problem, there are two categories: one is frequency-based, and the other is schedule-based. The latter considers time-varying O-D demand. Tong and Wong [5] proposed a schedule-based, stochastic, dynamic transit assignment model, and a stochastic minimum path is generated by a specially developed branch and bound algorithm. Nuzzolo et al. [6] developed a dynamic process assignment model, both within-day and day-to-day, and tested it on a realistically sized network to verify its applicability for operations planning. Nguyen et al. [7] presented a new graph theoretic framework for the passenger assignment problem that encompassed the departure time and the route choice. The implicit FIFO access to transit lines was taken into account by the concept of available capacity. Poon et al. [8] proposed a predictive transit dynamic user equilibrium model, and the generalised cost function encompassed four components: in-vehicle time, waiting time, walking time, and a line change penalty. Passengers queued at platforms under the single channel first-in-first-out discipline. By using time-increment simulation, the passenger demand was loaded onto the network and the available capacity of each vehicle was updated dynamically. Hamdouch and Lawphongpanich [9] and Hamdouch et al. [10][11] proposed a user equilibrium transit assignment model that took into account transit schedules and individual vehicle capacities explicitly. When loading a vehicle, on-board passengers continuing to the next stop had priority and waiting passengers could be loaded on a FCFS or in a random manner. Sumalee et al. [12] proposed a stochastic dynamic transit assignment model with an explicit seat allocation process. Two priority rules were assumed in the seat allocation simulation: passengers arriving earlier at a stop can access the available seats prior to those arriving later; standing passengers already onboard can access the available seats prior to those just boarding at the stop/station. Zhang et al. [13] proposed a new multi-class user reliability-based dynamic transit assignment model, and the in-vehicle capacity constraint for random passenger demand was handled by an in-vehicle congestion parameter. Nuzzolo et al. [14] presented a schedule-based dynamic assignment model for transit networks, which took into account congestion through explicit vehicle capacity constraints, and solved the queue formation and dispersion through FCFS rules, the failure-to-board experience, as well as experienced LoS attributes.
In order to obtain the space-time flow distribution in the urban rail network, without the restriction of schedules, we proposed the concept of continuous transmission for DAURTN. To solve the problem of DAURTN based on the continuous transmission and maximal line capacity constraints, we design the cell transmission mechanism, and develop the CTM. In the construction of the cell transmission rules, the model considers the priority principle, the queuing process, the capacity constraint, and congestion effect. We design an efficient algorithm for solving the shortest path in the urban rail network, which decreases the computation cost of the algorithm for the CTM to be implemented on a large-scale network. Using the method of successive average (MSA), the instantaneous dynamic user optimal state can be reached. Many important indexes are generated by the CTM, which provides the effective support for the optimization of train schedules and the capacity evaluation for the urban rail transit network. We take a small-scale network and the Beijing Metro network as two numerical examples to show the model and its potential application. A qu : set of queuing arcs which point from the station node to the platform node; A ar : set of arriving arcs which point from the platform node to the station node; A tr : set of transmission arcs; a: an arc in A; x a (t k ): number of passengers in the arc a 2 A in t k ; x qu a ðt k Þ: number of passengers along arc a 2 A qu in t k ; x ar a ðt k Þ: number of passengers along arc a 2 A ar in t k ; x tr a ðt k Þ: number of passengers along arc a 2 A tr in t k ; " x qu a ðt k Þ: number of passengers departing from arc a 2 A qu in t k ; c (  Quantitative description of relevant concepts in the urban rail network Urban rail network An urban rail network comprises a number of lines, and a line comprises a number of stations and sections. In urban rail system, trains are usually planned individually for each line. We assume that trains do not run across lines, and passengers can move across lines by transfer stations. Given an urban rail network with n stations and m lines. The set of stations is denoted as S = {s 1 ,s 2 ,Á Á Á,s n }. Urban rail lines are almost linear, and some complex urban rail lines, for example, annular lines or Y-style lines, can be decomposed into the linear lines, so we represent urban rail lines as linear. Each line includes two directional lines according to two opposite directions of operation, and the set of directional lines can be denoted as O = {L 1U ,L 1D ,L 2U , L 2D ,Á Á Á,L mU ,L mD }, where L lU , L lD denotes two directional lines of line l. We denote the direction variable as d 2 {U,D}, and the directional line as L ld 2 O. For describing the queuing of passengers at the station, a station can be extended into many platforms for each directional line, and each platform only serves for a unique direction line, so a directional line can be described as a sequence of the platforms. We denote s i ld as the ith platform of the directional line L ld , and n(l) as the number of the stations serving by line l. The urban rail network with a single line is illustrated in Fig 1, where the dashed line represents an urban rail operating line and doesn't belong to the urban rail network. The hollow nodes represent the station nodes and the solid nodes represent the platform nodes. The upside part in The above described network structure does not include the annular lines or Y-style lines (a Y-style line is a structure where two separate lines merge into one at a station), but it is easy to transform them into linear lines. For example, if s 1 ld and s nðlÞ ld are viewed as the same platform, the above network can describe the annular lines. For simplicity, we do not give specialized descriptions for lines with those special structures. Moreover, the above network does not describe the train stopping process in order to decrease the scale of the urban rail network.

Constrained continuous transmission
To avoid the restrictions of the schedules, the concept of continuous transmission is introduced. Continuous transmission means that each rail line can serve passengers at any time and passenger's traveling is not restricted by the train's schedule times, just like the road transportation.  To account for the urban rail's capacity constraint, continuous transmission has a capacity constraint, which means that the passenger transmission intensity of directional line L ld at any time cannot exceed the transmission capacity C l /τ l , where C l is the capacity of each trains in line l, and τ l is the minimum headway of line l. We denote [T 1 ,T 2 ] as the operation period of the urban rail network. Divide [T 1 ,T 2 ] into N equal intervals by the interval ΔT, and denote t k , k = 1,2,Á Á Á,N, as the kth interval. For each time interval, its transmission capacity is ΔT C l /τ l , L ld 2 O.
The benefit of the introduction of continuous transmission is that the space-time flow distribution obtained by DAURTN is not restricted by urban rail schedule, and can be used to evaluate and optimize the space-time resource allocation, for example, the schedule, the rolling stock circulation, and so on.

Priority principle
In each time interval, passenger flow cannot exceed the transmission capacity. When the flow exceeds the transmission capacity, only part of passengers can be transmitted during the current time interval, and surplus passengers have to wait at the station. According to the travel behavior of urban rail transit, the passengers' choices for different O-D pairs with capacity constrains must obey the following priority principles: Space priority principle: the flows of upstream stations along the directional line have priority over those of downstream stations to occupy capacities.
This principle is from the papers of [9][10][11] and it means when loading a vehicle, on-board passengers continuing to the next stop had priority. Under the space priority principle, passengers at upstream stations will not reserve capacity for the waiting passengers at downstream stations.
First-come-first-serve (FCFS) principle: passengers arriving earlier have priority over those arriving later to obtain service at a station.
Under the first-come-first-serve principle, the limited transmission capacity will be provided for passengers in the order of the batches arriving at the stations, and for one batch of passengers with different destinations, the method of equal proportional competition is used to determine the flow of departing passengers [10].

Demands and costs
The origins and destinations of all O-D pairs belong to the station node set S. We denote RS as the set of O-D pairs, and q rs (t k ), (r,s) 2 RS, 1 k N as O-D demands at time interval t k . In this study, it is assumed that passengers follow the instantaneous dynamic route choice principle [22], i.e., passengers choose the minimal cost routes under the currently time interval. Passenger flow reaches the instantaneous dynamic user optimal state that for each O-D pair at each decision node at each time interval, the instantaneous travel costs for all routes that are being used equal the minimal instantaneous route travel time [22].
In urban rail network (V,A), for any interval t k , 1 k N, denote x a (t k ) as the flow on arc a 2 A in t k . When a 2 A qu ,A ar or A tr , use x qu a ðt k Þ; x ar a ðt k Þ; x tr a ðt k Þ to replace x a (t k ) respectively. Especially for a 2 A tr , x tr a ðt k Þ is equal to the cumulating flow of differences between inflow and outflow from time interval t 1 to t k , with the concept of the continuous transmission. This method of calculation is similar to that for road link flow in dynamic route choice models [22]. We denote " x qu a ðt k Þ as the flow departing and transmitted from arc a 2 A qu in t k . The cost of arc a is denoted as c(a), a 2 A. When a 2 A qu ,A ar or A tr , we use c qu (a),c ar (a) or c tr (a) to replace c(a) respectively.
Based on the continuous transmission and instantaneous dynamic route choice principle, we calculate the queuing time by the flow state in the current time interval, i.e., the total queuing flow x qu a ðt k Þ and the transmitted passenger flow " x qu a ðt k Þ at the platform of the queuing arc a in the current time interval. Thus, the number of time intervals queuing at platform is x qu a ðt k Þ=" x qu a ðt k Þ, and the queuing time is DTx qu a ðt k Þ=" x qu a ðt k Þ. However, " x qu a ðt k Þ may tend to or be equal to zero, which will result in too large congestion cost, so we assume that denominator has a lower limit, which is set to be λC l , where λ is a parameter. Therefore, the queuing time estimated by passengers is and then, the queuing cost can be expressed as where θ is the converted factor for transforming time to cost. For an arriving arc a 2 A ar ,c ar (a) is the cost of the average transfer walking time, and needed to be expressed as a multiple of ΔT. We denote s a as the arrival station of arc a. Assuming that the average transfer time at s a is a constant, denote it as const(s a ), then Denote the number of time intervals for transferring at station s a as k(s a ), then where dxe is the function of minimum integer no less than x. For a transmission arc a 2 A tr , the cost c tr (a) is the sum of section travel cost and congestion cost, namely where gðx tr a ðt k ÞÞ is the congestion cost. For a transmission arc a 2 A tr , the total flow on arc a at time interval t k is x tr a ðt k Þ, and the total capacity of arc a is denoted as C a , so the congestion cost gðx tr a ðt k ÞÞ of arc a can be expressed as follows: which is similar to the power form used in BPR functions and the congestion functions in the papers of [10] and [47], and where η and α are cost parameters and η,α > 0. The congestion cost function is increasing with travel time and passenger flow volume. With the concept of constrained continuous transmission, the transmission arc a can be regarded as a train with the length ðs i ld ; s iþ1 ld Þ, of which the capacity per length unit is C l /τ l , so the capacity of the transmission arc a 2 A tr is calculated as The congestion cost gðx tr a ðt k ÞÞ can be obtained by substituting Eq (7) into Eq (6): In Eq (8), the travel time tðs i ld ; s iþ1 ld Þ is fixed and determined, and only x tr a ðt k Þ is variable. Hence, when η,α > 0, the calculation of congestion is feasible and the congestion influences in the travel choice of passengers.

Cell transmission model for DAURTN Cell transmission mechanism
To solve the DAURTN based on the continuous transmission, we build the CTM for the urban rail network (V,A). For describing the CTM, the cell transmission network is constructed based on cell from the network as follows.
We define each section as a cell chain, and each station as a station cell. For any section ðs i ld ; s iþ1 ld Þ, as travel time tðs i ld ; s iþ1 ld Þ is fixed, we divide the section into several transmission cells by ΔT. The transmission cells of one section compose a cell chain, and passengers flows can be transmitted forward between the transmission cells. Note that travel time tðs i ld ; s iþ1 ld Þ may not be exactly divided by ΔT, so the time length of the tail cell can be equal to or exceed ΔT. The number of cells divided is n i ld ¼ btðs i ld ; s iþ1 ld Þ=DTc, where bxc is a function of the maximum integer no larger than x. Therefore, we denote Cell(L ld ,i,j) as the jth cell in the cell chain of section ðs i ld ; s iþ1 ld Þ. Denote m(L ld ,i) as the number of cells in the ith cell chain, and for simplifying the notation, denote Cell(L ld ,i,end) the last cell in the cell chain of section ðs i ld ; s iþ1 ld Þ. Denote y h (L ld ,i, j,t k ) as the flow of Cell(L ld ,i,j) traveling to destination s h in t k . We also denote Cell(s u ) as the station cell of s u , and y h (s u ,t k ) as the flow of Cell(s u ) traveling to destination s h in t k .
The transmission relationship between cells is illustrated in Fig 3, where a hollow node represents a station cell, a solid node represents a transmission cell, a hollow rectangle represents the corresponding cell chain of a section, and the arrows represent transmission directions. In Next, we analyze the transmission mechanism in 3 steps.
Step 1: The transmission processes Cell(L ld ,i,end) ¼) Cell(L ld ,i + 1,1), and CellðL ld ; i; endÞ¼) Cellð" s iþ1 ld Þ As the length of time in the tail cell Cell(L ld ,i,end) is equals to or greater than ΔT, only a certain proportion of flow can outflow, and the proportion is Then we can obtain the detained flow of the tail cell where ' ' denotes that the value of the right variable is assigned to the left variable. According to the space priority principle, the outflow f h (L ld ,i,end,t k ) of tail cell Cell(L ld ,i, end) has only two choices, i.e., Cell(L ld ,i + 1,1) or Cellð" s iþ1 ld Þ, and the transmission choice is determined by the instantaneous dynamic route choice principle [22], i.e., the shortest path from platform s iþ1 ld to destination station s h at the current time interval in network (V,A). If the shortest path passes through s iþ2 ld , then flow f h (L ld ,i,end,t k ) is transmitted into cell Cell(L ld ,i + 1,1); otherwise, it is transmitted to station cell Cellð" s iþ1 ld Þ. The above transmission choice is similar to the all-or-nothing assignment, i.e., the flows follow the shortest path.
We denote the set of destinations to which the shortest path from platform s iþ1 ld passes through s iþ2 ld as In order to realize the FCFS in transmission mechanism, we introduce a variable x h (s u ,t v ), 1 v N, which represents the flows arriving at station s u in t v and detained at the station in t k .
Step 2: The transmission process Cell(L ld ,i,j) ¼) Cell(L ld ,i,j + 1) After Step 1, in the tail cell of the cell chain, there may be some detained flows, and then the flow of the tail cell equals to the detained flows plus the flows from Cell(L ld ,i,end − 1), so For other cells in the chain, it is only need to move flows from the forward cell to the backward cell in the chain, that is, Step 3: The transmission process Cellðs u Þ¼)CellðL ld ; i; 1Þ; s i ld 2 Sðs u Þ According to the principle of the space priority, flows from Cell(L ld ,i − 1,end) are transmitted to Cell(L ld ,i,1) and occupy the capacity of Cell(L ld ,i,1) with priority. Thus, the surplus capacity of Cell(L ld ,i,1) in t k is The flows in t k , which are queuing at station s u and head to Cell(L ld ,i,1), have to compete for the surplus capacity with the FSFC principle.
In order to determine the queuing flow, passengers at station cell Cell(s u ) first determine which platform to queue. Similar to the method in Step 1, passengers determined the platform by the shortest path from station s u to destination station s h at the current time interval in network (V,A). If the shortest path passes through s i ld 2 Sðs u Þ, then flows traveling to destination station s h queue on platform s i ld . We denote the set of destinations to which the shortest path from station s u passes through s i ld 2 Sðs u Þ as Then ( After the above processes, the flow of each cell make a choice by the shortest paths and are all transmitted to the next cell. But the cost of each arc will be changed with the variable flow, so the method of successive average (MSA) is adopted to reach the instantaneous dynamic user optimal state in each time interval. The variables in the above model are updated in MSA.

An efficient method for solving the shortest path
In the CTM, it is needed to solve the shortest path in t k from s 2 S [ S O to s u 2 S in network (V,A). We design a fast method for solving the shortest path as follows.
If the shortest path from s 2 S [ S O to s u 2 S passes through several transfer stations, then the shortest path can be divided into three segments at most. The first segment of the shortest path is from origin s to the first transfer station s 0 F 2 S, and its length is denoted as pðs; s 0 F Þ. The last segment of the shortest path is from the last transfer station s 1 F 2 S to destination s u , and its length is pðs 1 F ; s u Þ. As long as we solve the length of the shortest path between any two transfer stations pðs 0 F ; s 1 F Þ, we can obtain the cost of the shortest paths in three cases as follows:

> < > :
If the shortest path from s 2 S [ S O to s u 2 S does not pass through any transfer station, then it will only use one line and can be solved easily.
The above analysis indicates that the solving method for the shortest path from s 2 S [ S O to s u 2 S can be decomposed into 3 steps.
Step 1: calculate the shortest path from s 2 S [ S O to s u 2 S in each network G(L lU ,L lD ), which composed of a pair of opposite directional lines L lU ,L lD shown in Fig 1. Step 2: calculate the shortest path between each two transfer stations in the network.
Step ; " s i lU ; i ¼ v; v þ 1; Á Á Á ; nðlÞg, which can form two generated sub-networks of G(L lU ,L lD ). Obviously, the shortest paths from other nodes to s u in the two sub-networks are equal to solving the shortest paths in G(L lU ,L lD ). In the former sub-network, there are three cases. Thus we can solve the shortest paths from s nðlÞÀ i lD and " s i lU to s u in the order of i = 2,3,Á Á Á,v. In the latter sub-network, the solving method is similar. Therefore, the amount of calculation for the shortest paths from s 2 S [ S O to s u 2 S in network G(L lU ,L lD ) is only O(n(l) 2 ), and the sum of calculation for the shortest paths of the whole m lines is Oð P m l¼1 nðlÞ 2 Þ.
The method for solving the shortest paths for an annular line only needs to make some supplements based on the above method for linear lines. For any destination s u , we can divide the annular line into a linear line by s u , and there are only two ways of dividing. Then we can adopt the above method to solve the shortest paths from s 2 S [ S O to s u for each way of dividing, and the shorter one between them is the shortest path. For a Y-style line, the similar method for solving the shortest paths is feasible.
We have obtained the length of shortest path between each two transfer stations in each line in step 1. In step 2, we use the Floyd-Warshall algorithm to solve the shortest path between any two transfer stations, and the computational complexity is O(|S F | 3 ).
In step 3, we can solve the shortest paths from s 2 S [ S O to s u 2 S by formula (23), and the computational complexity is OðjSj Á P m l¼1 nðlÞÞ. Therefore, the computational complexity of the shortest path solving algorithm in the urban rail transmission network is OðjSj Á P m l¼1 nðlÞ þ jS F j 3 Þ, which is less than Oðð P m l¼1 nðlÞÞ 3 Þ of the classical methods, i.e., Dijkstra algorithm and Floyd algorithm. The urban rail network in Beijing composes 15 operating lines and 231 stations until July 2014. There are only 40 transfer stations, which are far less than other stations. Thus, the solving algorithm of the shortest path is effective for the real urban rail network. We test our method and Floyd algorithm for solving the shortest paths of Beijing urban rail network. Using the Matlab(R2010b) to program, the shortest path problem is calculated for 100 times, and we record the CPU times for the two methods. The average CPU time of Floyd algorithm is 10.31s, while the average CPU time of our method is 1.06s, with the computer (IntelCore 2.90GHz, 8GB RAM).

Evaluation indexes of passenger flow
The CTM can generate many important evaluation indexes, including time-varying section flow, operating line circulation volume, the detained passenger volume on the platform, and the queuing length on the platform.
As the flow x tr a ðt k Þ for a 2 A tr is equal to the sum of the flows of all cells in this cell chain, we defined that the time-varying section flow means the outflow of the last cell in each time interval, which means passenger flow transmitted by the line section in each time interval and can reflect the capacity constraints in CTM. Thus, the section flow is It is obvious that the time-varying section flow varies with ΔT, i.e., if ΔT becomes longer, then the time-varying section flow for each time interval is larger.
We can obtain the circulation volume of each directional line, which means the sum of travel mileages of passengers on each directional line, that is, and its unit is person kilometer. The circulation volume of the whole network can be obtained by summing the circulation volumes of all directional lines.
In CTM, we can calculate the detained flow volume on the platform as It is known that the queuing length on the platform depends on the headway τ l , which means the larger the headway is, the longer the queuing length will be. The queuing length on the platform can be used to evaluate the service level of urban railway network. In the CTM, it is noted that the queuing length for each time interval is longest at the beginning of the current time interval, i.e., before flow transmission of each cell at each time interval, while it is shortest when at the ending of time interval, for the reason that some passengers queuing at the platform are transmitted at the ending of the current time interval.
To eliminate the above influence and obtain the reasonable queuing length to evaluate the service level, we define that the queuing length of each platform at time interval t k means the total queuing flow during the time interval [t k ΔT − τ l ,t k ΔT] with headway τ l , and it is calculated at the beginning of the above time interval. Thus, the queuing length includes the flows of the platform transmitted during [t k ΔT − τ l ,t k ΔT] and the detained flow at the time interval t k . The length of time interval is ΔT and ΔT < τ l , so the headway τ l may cover more than one time interval. It is known that the flow from Cell(s u ) to CellðL ld ; i; 1Þ; s i ld 2 Sðs u Þ at time interval t k is that The flow at the platform where r meets 0 τ l − rΔT < ΔT. As the detained flow on the platform in t k is Zðs i ld ; t k Þ, so the queuing length on the platform in t k is  Fig 5(A); the density distributions of O-D demands departing from area 1 are shown in Fig 5(B). The demands within each area and between areas 2 and 3 are all equal to 0. The detailed O-D demands and density distributions are listed in Table 1. It can be seen from the density distributions of travel demands that all the morning peaks of O-D demands are the period from 7:00 to 9:00, while all the evening peaks of O-D demands are the period from 18:00 to 20:00. The O-D demands between areas 2 and 1 are larger than those between areas 3 and 1.
With the computer (IntelCore 2.90GHz, 8GB RAM), we use Matlab (R2010b) to program and solve the model for this example, and it takes 19s CPU time to solve this model. Table 2, and the circulation volume of the network is 7.92 10 5 person kilometers.

Analysis of indexes. The circulation volumes of all lines are listed in
As some indexes are time-varying during an operation day, and the data are too large to be listed, we only calculate the statistical indexes about directional line L 1U ¼ fs 1 1U ; s 2 1U ; s 3 1U ; s 4 1U g to demonstrate and analyze the model.
The time-varying section flow can be obtained by the method of MSA. There are three curves to show the time-varying section flows of ðs 1 1U ; s 2 1U Þ; ðs 2 1U ; s 3 1U Þ; ðs 3 1U ; s 4 1U Þ in Fig 6. Note that L 1U is a directional line in which passengers mainly depart from the CBD, and its travel demand distributions are shown in Fig 5(B). The time-varying section flows are similar to Fig  5(B). During evening peak hours, the section flow of ðs 2 1U ; s 3 1U Þ reaches transmission capacity 156.25 from 18:06:15 to 20:18:45. The section flow of ðs 3 1U ; s 4 1U Þ is the lowest among three curves, because there is no demand starting from " s 3 1U along L 1U . There are three curves to show the detained flows on s 1 1U ; s 2 1U ; s 3 1U in Fig 7. The detained flow on s 2 1U begins from 18:00:00, reaches the maximal value 155.91 at 20:02:30, and disappears at 20:20:00. It is for the reason that the demands from the upstream station " s 1 1U ¼ s 4 occupies most capacity, which make the demands departing from " s 2 1U during evening peak hours not be met.
Three curves are illustrated in Fig 8 to show the queuing lengths on s 1 1U ; s 2 1U ; s 3 1U . On s 1 1U , all passengers can be transmitted in time due to the sufficient capacity, then the queuing length Cell transmission model of dynamic assignment for urban rail transit networks distribution is similar to the demand distribution. On s 2 1U , the detained flow begins to appear from 18:00:00 due to the insufficient capacity. As the travel demand intensity will not change from 18:00:00, the queuing length continues to increasing. On s 3 1U , the queuing flow does not appear at any time bacause there is no departure and transfer passenger flow.
Analysis of passenger flow characteristics. To demonstrate the reasonability for dynamic assignment with the CTM, we show the path choice, the space priority principle and the flow moving among platforms of one station.
(a) Path choice As passengers follows the instantaneous dynamic route choice principle, we now analyze the path choice for the passengers from platform s 2 1U to station s 9 . We adopt two paths, i.e., path 1: ðs 2 1U ; s 1 ; s 2 3U ; s 3 3U ; s 3 ; s 3 2U ; s 4 2U ; s 9 Þ and path 2: ðs 2 1U ; s 3 1U ; s 2 ; s 2 2U ; s 3 2U ; s 4 2U ; s 9 Þ. The flows and costs of these two paths are shown in Fig 9, we can see that the costs of path 1 for all time intervals are larger than those of path 2, so all passengers choose the path 2 to travel to s 9 . Now we consider the path choice for the passengers from platform s 2 2D to station s 4 . We also adopt two paths, i.e., path 1: To explain this phenomenon, we compare the curve of the transmitted flow on s 2 1U with that of the passing flow in Fig 12. As shown in Fig 12, the passing flow on s 2 1U decreases from    Cell transmission model of dynamic assignment for urban rail transit networks On s 3 1D ; s 3 3D 2 Sðs 1 Þ, the value are 0, which indicates that the non-detained flows at any time are not restricted by the capacity. On s 2 1U 2 Sðs 1 Þ, the value in the period from 17:58:45 to 20:03:45 is larger than 0, which indicates that the capacity in this period cannot satisfy the non-detained flow, and the detained flow appears. The value in the period from 20:05:00 to 20:20:00 is less than 0, which indicates that the capacity in this period exceeds the nondetained demand, and the detained flow is transmitted. Note that the values of s 2 1U before 17:58:45 and after 20:20:00 are both equal to 0, i.e., the detained flows in the two periods are both equal to 0.
In view of the figure area, the detained flow from 17:58:45 to 20:03:45 is much larger than the transmitted flow from 20:05:00 to 20:20:00 on the platform s 2 1U , but the detained flow disappear after 20:20:00. To explain this phenomenon, we can see that for s 2 3U 2 Sðs 1 Þ, the values in the period from 18:00:00 to 20:20:00 are less than 0, that is, the non-detained flow in this period is less than the transmitted flows. At other times, the non-detained demand is equal to the transmitted flows. It is obvious that some detained passengers on s 2 1U change their travel route, and move to s 2 3U . In view of the figure area, the sum of the two negative areas is equal to the positive area.
Sensitivity analysis. We set the parameter α = 1.0,3.0,4.0,4.5,5.0 respectively, and don't change other parameters. The detained flows of platform s 2 1U are calculated by the CTM for different values of parameter α, shown in Fig 14. From Fig 14, It can be seen that the detained flows of platform s 2 1U decrease with the increasing of the parameter α. From the demand distributions in Table 1, we know that the demands from Area 1 to Area 2 are larger than those from Area 1 to Area 3. It results in that the congestion in L 1U is more than that in L 3U . With the increase of parameter α, the cost differences between the paths traveling from Area 1 to For parameter η, we now do the sensitivity analysis. Set parameter η = 1.0,1.8,1.9,2.0 and don't change other parameters. We also calculate the detained flows of platform s 2 1U for different values of parameter η. Fig 15 shows the curves of the detained flows of platform s 2 1U and we can see that the detained flow of platform s 2 1U decreases with the increasing of the parameter α. The reason is similar to that of parameter α. It results in that the cost differences between the paths traveling from Area 1 to Area 2 and from Area 1 to Area 3 increase with the increase of parameter η. Thus, the increase of parameter η makes the decrease of detained flows of platform s 2 1U .  We set ΔT = 1 min, and the total number of time intervals is 1020. The transmission capacity of a line within ΔT is that DT C max l =t l ¼ 1 Â 1200 2:5 ¼ 480. The relative gap is 10 −3 , and the total CPU time is 5.16h.

Example 2: Beijing Metro network
The circulation volumes of all lines are illustrated in Table 3. The total circulation volume of the network is 8.69 × 10 7 person kilometers.
We take Line 5 as an example to analyze the space-time flow distribution. There are 23 stations in Line 5 illustrated in Fig 16, where solid dots represent transfer stations, while hollow dots represent the non-transfer stations. There are 7 transfer stations including Songjiazhuang, Ciqikou, Chongwenmen, Dongdan, Dongsi, Yonghegong and Huixinxijie Nankou in Line 5. The passenger flow of the morning peak from Songjiazhuang to Dongsi are more intensive than those of the evening peak from Dongsi to Songjiazhuang. As the morning peak is the time period when passengers need to get to their workplaces at specified times, while the evening peak is the time period of going off duty, so the travel time period chosen by the passengers is relatively wider. As shown in Fig 17 and Fig  18, the section flow at transfer stations will vary greatly, because the section flow can gather  from other lines, or disperse to other lines via the transfer stations. In addition, the density of the passenger flow space-time distribution in each section is less than 480, so it satisfies the capacity constraints. Fig 19 shows the variation of CPU time with the relative gap. We can see that the convergence speed is fast with a low relative gap, then as the relative gap gets lower, the CPU time becomes longer, and the convergence speed gets slower. When the relative gap reaches 10 −4 , the CPU time reaches 18.5h, for the reason that the algorithm of MSA takes a long time to reach a high relative gap.

Conclusions
In this paper, the concept of continuous transmission is introduced to model the dynamic assignment for the urban rail network without the restriction of train schedules. Based on the cell transmission mechanism, the proposed CTM considers the priority principle, queuing process, capacity constraints and congestion effect. Using the MSA, the instantaneous dynamic optimal state can be reached at each interval. A fast and effective method is designed for solving the shortest path for the urban rail network. This method decreases the computing cost for solving the CTM, and it is applied efficiently to the large-scale urban rail network.
The CTM can generate some important evaluation indexes, including the time-varying section flow, the circulation volume, the detained flow on the platform, and the queuing length on the platform. It provides effective supports for optimizing the space-time resource allocation for the urban rail network. Finally, the model and its potential application are  Cell transmission model of dynamic assignment for urban rail transit networks The proposed method is assumed that the passengers follow the instantaneous dynamic route choice principle, and as a topic of further interest, we can explore the CTM based on the continuous transmission with predictive/ideal dynamic user equilibrium. In the future studies, this method of DAURTN can be employed for optimizing the schedule, frequency and rolling stock, and evaluating the transportation capacity of network.
Supporting information S1