論文 / 著書情報
Article / Book Information
Title

Fundamental diagram of urban rail transit considering train‒passenger
interaction

Authors

Toru Seo, Kentaro Wada, Daisuke Fukuda

CitationTransportation

Pub. date2022, 4Creative Commons

Information is in the article.

Powered by T2R2 (Science Tokyo Research Repository)

Transportation
https://doi.org/10.1007/s11116-022-10281-0

Fundamental diagram of urban rail transit considering
train–passenger interaction
Toru Seo1· Kentaro Wada2 · Daisuke Fukuda3

Accepted: 18 March 2022
© The Author(s) 2022

Abstract
Urban rail transit often operates with high service frequencies to serve heavy passenger
demand during rush hours. Such operations can be delayed by two types of congestion:
train congestion and passenger congestion, both of which interact with each other. Thisdelay is problematic for many transit systems, since it can be amplified due to the interaction. However, there are no tractable models describing them; and it makes difficult toanalyze management strategies of congested transit systems in general and tractable ways.
To fill this gap, this article proposes simple yet physical and dynamic model of urban railtransit. First, a fundamental diagram of transit system (i.e., theoretical relation among
train-flow, train-density, and passenger-flow) is analytically derived considering the aforementioned physical interaction. Then, a macroscopic model of transit system for dynamictransit assignment is developed based on the fundamental diagram. Finally, accuracy of themacroscopic model is investigated by comparing to microscopic simulation. The proposedmodels would be useful for mathematical analysis on management strategies of urban railtransit systems, such as optimal dynamic pricing for travel demand management.
Keywords Public transport · Rush hour · Fundamental diagram · Macroscopic fundamental
diagram · Dynamic transit assignment

Introduction
Urban rail transit systems such as metro is handling significant transportation needs of metropolitan areas (Vuchic 2005). Its most notable usage is the morning commute, in whichheavy passenger demand is concentrated in a short time period. It is known that such transit systems often suffer from delays caused by congestion, even if no serious incidents or
accidents occur (Kato et al. 2012; Tirachini et al. 2013; Kariyazaki et al. 2015). Therefore,
This paper is an enhanced version of Seo et al. (2017a, 2017b).
* Toru Seo
seo.t.aa@m.titech.ac.jp
1

Tokyo Institute of Technology, 2‑12‑1‑M1‑13 O‑okayama, Meguro, Tokyo 152‑8552, Japan

2

University of Tsukuba, 3F1211, 1‑1‑1 Tennodai, Tsukuba, Ibaraki 305‑8573, Japan

3

The University of Tokyo, 7‑3‑1 Hongo, Bunkyo‑ku, Tokyo 113‑8656, Japan

13

Vol.:(0123456789)

Transportation

appropriate management of transit systems is required; especially, travel demand management for mass transit systems has been gaining attention recently (Halvorsen et al. 2019;
Huan et al. 2021).
One of the approaches to find management strategies of transit systems is theoreticalanalysis with simplifications, such as use of certain static models with constant travel time(de Cea and Fernández 1993; Tabuchi 1993; Kraus and Yoshida 2002; Tian et al. 2007;
Gonzales and Daganzo 2012; Trozzi et al. 2013; de Palma et al. 2015a, b). In this approach,general policy implications can be obtained thanks to the simplicity and tractability of theanalysis. However, they may not be sufficient to investigate dynamic (i.e., time-varying)
operation and demand management strategies.
In congested transit systems, dynamical interaction among trains and passengers plays
essential roles to determine the system’s operational behavior, and the travel time can be
dynamically and significantly changed due to this interaction. For instances, there are two
types of congestion in transit systems:• train-congestion: congestion involving consecutive trains using the same tracks,• passenger-congestion: congestion of passengers who are boarding to a train, namely,
bottleneck congestion at the doors of a train while it is stopped at a station (Lam et al.
1998; Wada et al. 2012; Kariyazaki et al. 2015),1

and these two types of congestion interact with each other and cause delay (Newell andPotts 1964; Kusakabe et al. 2010; Wada et al. 2012; Kato et al. 2012; Tirachini et al. 2013;
Kariyazaki et al. 2015; Cuniasse et al. 2015). The “knock-on delay” is a well-known phenomenon describing “a train’s delay which is caused by other trains in front of it” (Careyand Kwieciński 1994). Some of the knock-on delays can be considered as caused by the
dynamic train–passenger interaction (Kariyazaki et al. 2015). Specifically, trains aredelayed due to the large number of boarding and alighting passengers at stations, and suchdelay propagates to the subsequent trains—this can be a train equivalent of the “bus bunching” (Newell and Potts 1964; Daganzo 2009). For example, assume that passenger-congestion happened temporally due to high demand. It would extend the dwelling time of a train
at a station. Then, this extended dwell time would interrupt the operation of subsequent
trains, and cause train-congestion on the track. It would deteriorate the passenger throughput, and thus the passenger-congestion at stations would intensify. This kind of dynamical
phenomena cannot be captured by static models.
A consequence of such dynamical passenger–train interaction can be found in macroscopic states of transit systems. Figure 1 shows observed 3-dimensional relations amongstates of transit systems, that is, train-flow (train/h), train-density (train/km), and passenger-flow (passenger/h). The visualization is based on the concepts of the fundamental diagram (Greenshields 1935) of urban traffic. Although Fig. 1a and 1b show data from completely different transit systems with completely different scales (e.g., the number of cars
per one train was 10 for Tokyo and 1–6 for Boston), they have similar patterns. First, as thepassenger-flow increases, the train-density increases; this could be a result of transit operators responding to increased passenger-demand. Second, as the passenger-flow increases,
the average service speed of trains and train-flow decreases; this could be a result of theaforementioned congestion due to the interaction among trains and passengers. It would be
1

Note that passenger-congestion differs from in-vehicle passenger-crowding (Kumagai et al. 2020), which
results in discomfort due to standing and crowding, but is not necessarily cause any delay directly.

13

Transportation

(a)

(b)

Fig. 1  Observed 3-dimensional relations among train-flow, train-density, and passenger-flow. The x-coordinates represent the train-flow (i.e., the number of trains per a kilometer), the y-coordinates represent thetrain-flow (i.e., the number of passing trains per an hour), the slope from the origin to an arbitrary point
indicates the train-speed (i.e., the average speed of trains) of that point, and the color of each point represents the passenger-flow (i.e., average demand of passengers per an hour) of that point. “pax” in the figure
is an abbreviation for “passenger”. Data sources: Fukuda et al. (2019), Zhang and Wada (2019), Tokyu
Railways, and Massachusetts Bay Transportation Authority

preferable if we have a theoretical model of this phenomena, because it would be useful toobtain general principles on transit operations; however, to our knowledge, such a model
does not exist in the literature.2
This study derives a theoretical relation among the state variables of transit systems
similar to Fig. 1 based on the microscopic operation principles. It is modeled as a fundamental diagram (FD), which is a well-known concept in the vehicular traffic flow theory.
The original FD describes relation between vehicular flow and density, and it can be used
to describe dynamic evolution of traffic by combining with other principles in a tractablemanner (Lighthill and Whitham 1955; Richards 1956; Mahmassani et al. 1984; Geroliminis and Daganzo 2007). In fact, several recent studies have employed FDs of transit systems to describe train-congestion by modeling the relation between train-flow and traindensity (Cuniasse et al. 2015; Corman et al. 2019; de Rivera and Dick 2021). The novel
feature of this study is the incorporation of passengers in an analytical way. Furthermore,this study develops a dynamic transit assignment method based on the proposed FD.
This study proposes tractable models of the dynamics of urban rail transit considering
the physical interaction between train-congestion and passenger-congestion. In “Microscopic Model of Rail Transit System” section, a microscopic model of a rail transit system
is introduced based on a passenger boarding model and a train cruising model. In “Fundamental Diagram of Rail Transit System” section, the operation performance of the microscopic model is analyzed. Specifically, a mathematically tractable relation among trainflow, train-density, and passenger-flow is derived—namely, a fundamental diagram (FD).
The model can be also viewed as a variation of 3-dimensional macroscopic fundamental

2
Several detailed operation models have been proposed to capture the detailed mechanism of the dynamics
of interaction (see Vuchic 2005; Koutsopoulos and Wang 2007; Parbo et al. 2016; Cats et al. 2016; Li et al.
2017; Alonso et al. 2017; Cunha et al. 2021, and references therein), and these have been used to develop
efficient operation schemes. However, these models are often based on microscopic simulation, and thus
their purposes tend to be case-specific optimization and evaluation. It would be difficult to use them toderive the relation depicted in Fig. 1 or to obtain general or strategic policy implications for management
strategies, as they are essentially complex and intractable.

13

Transportation

diagram (Mahmassani et al. 1984; Geroliminis and Daganzo 2007) with an analytical derivation. This is the key contribution of this study. In “Dynamic Model Based on Fundamental Diagram” section, a macroscopic loading model of a transit system is developed basedon the proposed FD. The model describes the aggregated behavior of trains and passengers
in a urban-scale spatial domain based on the FD. In Section “Validation of the Macroscopic Model”, the approximation accuracy and other properties of the proposed macroscopic model are investigated through a comparison with microscopic simulation. Section“Conclusion” concludes this article. Note that empirical validation of the proposed model
based on actual data is out of scope of this study. Such validation is now being conductedby some of the authors and preliminary results that support the model have been obtained(Fukuda et al. 2019; Zhang and Wada 2019).

Microscopic Model of Rail Transit System
This section introduces a microscopic model of rail transit system, from which we derive
the FD in Section “Fundamental Diagram of Rail Transit System”. It consists of two
microscopic operation principles, namely, a passenger boarding model which describes thetrain’s dwell behavior at a station for passenger boarding and a train cruising model which
describes the cruising behavior on the railroad. This microscopic model has been proposedby Wada et al. (2012) to analyze train bunching.

Rail Transit Operation Principles
Consider a railway system on a single line track, where trains and stations are indexed bym and i, respectively. We assume that all trains stop at every station. Let tm,i be the arrival
time of train m at station i. Then, a dynamical system that represents each train motion is
given bytm,i+1 = tm,i + bm,i + cm,i

∀m, i

(1)

where bm,i is the passenger boarding time of train m at station i, and cm,i is the trip time oftrain m between stations i and i + 1, which are determined by the two operational submodels (see Figure 2).
The passenger boarding time is modeled using a queuing model. That is, the flow-rateof passenger boarding is assumed to be constant, 𝜇p; and there is a buffer time (e.g., timerequired for door opening/closing), gb, for the dwell time. Then, the dwell time of a train at
a station, bm,i , is represented asbm,i =

qp,i hm,i𝜇p+ gb,i ,(2)

where qp,i is the (possibly time-dependent) passenger demand flow rate at station i,
hm,i ≡ tm,i − tm−1,i is the time-headway, and thus qp,i hm,i is the number of waiting passengers
at the station.3 This can be considered as a special case of Lam et al. (1998), which have3

In reality, there are passengers alighting a train, in addition to ones boarding. By carefully distinguishing the two types of passengers and replacing the terminology in the main text, the discussions in the main
text are valid and the final results are not altered. For example, “the number of boarding passengers” canbe replaced with “the sum of the number of boarding passengers and the number of alighting passengers”.

13Transportation

Fig. 2  Illustration of the microscopic model of rail transit system

empirically analyzed the boarding time in Hong Kong metro in crowded situations. All
passengers waiting for a train at a station are assumed to board the first train arrived.
The cruising behavior of a train is modeled using the Newell’s simplified car-following
model (Newell 2002).4 In this model, a train travels by maintaining the minimum safety
clearance. Specifically, xm,i (t), position of a train m between stations i and i + 1 at time t, isdescribed as{}xm,i (t) = min xm,i (t − 𝜏) + v𝜏, xm−1,i (t − 𝜏) − 𝛿 ,(3)where m − 1 indicates the preceding train of train m, 𝜏 is the physical minimum time-headway, v is the desired cruising speed that is determined by a (fixed-block or moving-block)signal control system, and 𝛿 is the minimum spacing.
We call the traffic is in free-flowing regime if the train travels between stations at thefree-flow speed vf (the maximum speed of the track), i.e., the train motion is represented by
the first term with v = vf in the minimum operation of Eq. (3). We call the traffic is in congested regime if the train is required to decrease its speed to maintain both the safety headway 𝜏 and distance 𝛿. In this case, the second term of Eq. (3) is active. The speed profile inthis regime may differ in different train operators (i.e., signal control systems) and drivers.
We employ one of the simplest approximations of this speed profile, that is, the train travels between stations at a constant speed while maintaining the minimum safety clearance.

Footnote 3 (continued)
However, it will complicate the discussions; therefore, we ignore passengers alighting a train.
For the boarding time function (2), more complicated functional forms such as a nonlinear function inwhich the boarding time is significantly extended in overcrowded situations may be considerable insteadof the linear function. In fact, Seo et al. (2019) have derived another FD based on such a nonlinear boarding time function, but its functional form is extremely complicated. In order to simplify the discussion, we
employ the linear function in this paper.
4
Newell’s simplified car-following model is a special case of the well-known road traffic flow model, theLighthill–Whitham–Richards (LWR) model (Lighthill and Whitham 1955; Richards 1956; Newell 1993).
Although the LWR model is known as a “macroscopic” model based on continuum fluid approximation,
Newell (2002) showed that it is equivalent to a microscopic car-following model proposed by his paper.

13

Transportation

Validity of Assumptions
In this model, the dwell time of a train is determined by the number of boarding passengers,
not by a pre-determined timetable. Although this seems like inconsistency between the proposed model and actual schedule-based train operations, this can be considered as a reasonableapproximation of average operation pattern of schedule-based train operations. The reasonsare as follows. First, in a congested urban areas, it is common that passenger boarding time is
not negligible and occasionally delay transit operation, as reviewed in Section “Introduction”.
Therefore, in order to maintain a scheduled operation based on a timetable, this timetable has
to be determined considering the passenger demand (e.g., Niu and Zhou 2013). Consequently,
the dwelling time in such timetable can be considered as similar to the proposed passenger
boarding model (2) where qp,i hm,i is interpreted as an average number of waiting passenger
and gb is interpreted as a buffer time to deal with fluctuation of the demand.
Second, the passenger boarding model with a constant capacity is consistent with the modelling of ordinary pedestrian flows for a fixed-width bottleneck (Lam et al. 1998; Hoogendoorn and Daamen 2005). Meanwhile, there is no stock capacity for passengers in the presented model; in other words, a train can transport infinite number of passengers. This is alimitation of the current model; however, unless the passenger demand is excessive level (e.g.,where not all of the waiting passenger can board an arriving train), this limitation will not be
problematic.
We have assumed that all passengers waiting for a train at a station board the first train
arrived. However, in very crowded situations in reality, not all passengers can board the firsttrain and they will form a queue in a platform. This phenomenon is not directly captured by
the proposed model, but it can be captured by extending it. Specifically, a new model that
describes queuing at the platform can be developed and attached to the proposed model, sothat the outflow from the platform queuing model becomes the passenger boarding flow qp,i inthe proposed model. This extension is left for the future work.
The train cruising model (3) can be considered as a lower order but a reasonable approximation of railway operation in a macroscopic scale. First, the model’s fundamental operating principle is consistent with practical controls and existing studies (Carey and Kwieciński
1994; Higgins and Kozan 1998; Huisman et al. 2005): each train has to maintain a headway
and a spacing that are greater than the given minimum ones. Second, the car-following modelis almost equivalent to the “moving block signal control” (Dicembre and Ricci 2011). In fact,de Rivera and Dick (2021) and Corman et al. (2019) have employed car-following approximations of railways with moving block systems to analyze their performance. Additionally,
the “fixed block signal control” can be considered as an inefficient moving block signal andapproximated as a car-following model in the macroscopic scale (Li et al. 2005; Kariyazaki
et al. 2015). The other assumption in the cruising model is the constant speed assumption
in the congested regime. This approximates a train operation with low acceleration rates for
ensuring the comfort of passengers and less energy loss. In addition, this assumption can
describe the congested situation where the train stops between stations, on average, as we willshow in the numerical experiment in Section “Validation of the Macroscopic Model”.

13

Transportation

Fundamental Diagram of Rail Transit System
In this section, we derive an FD of a rail transit system described by the microscopic model
formulated in Section “Microscopic Model of Rail Transit System”. The FD is defined as
the relation among train-flow, train-density, and passenger-flow.

Steady State of Rail Transit System
We consider the steady state of the proposed microscopic model. The steady state isdefined as a traffic state that does not change over time and/or space. In the other words,
in the steady states, traffic state variables (flow, density, and average speed) in a certain
time–space region are invariant with respect to temporal and/or spatial change of the position of the time–space region. There may exist a special functional that relates traffic statevariables in steady state traffic; if such a function exists, it is often called as an FD of thetraffic (Daganzo 1997).
To derive the traffic state in the steady state, let us consider a homogeneous rail transit
system in which the stations and passenger demands are homogeneously distributed overthe line, i.e., li = l , qp,i = qp and other parameters ( gb , vf , 𝜏 , and 𝛿) are the same for all stations and trains. Then, the steady state is defined as a state that, for a given steady passenger demand qp, the time-headway between successive trains, H, is time-independent. Notethat qp < 𝜇p must be satisfied; otherwise, passenger boarding will never end.
Transit systems under different steady states are illustrated as time–space diagramsin Fig. 3. In each sub-figure, train m arrives at and departs from station i, then travels tostation i + 1 at cruising speed v, and finally arrives at station i + 1. The main differences
between each sub-figure are density of trains and, consequently, traffic regime. In Fig. 3a,the density is small so that the speed v is equal to the free-flow speed vf and hf is greaterthan zero; therefore the state is classified into the free-flowing regime. In Fig. 3b, the density is medium so that the speed is equal to vf and hf is equal to zero; therefore, the stateis classified into the critical regime. In Fig. 3c, the density is large so that the speed is lessthan vf ; therefore, the state is classified into the congested regime.

Fundamental Diagram
In general, the followings are considered as the traffic state variables of a rail transit system:
•
•
•
•
•
•

train-flow q,
train-density k,
train-mean-speed v̄ ,passenger-flow qp,
passenger-density kp,
passenger-mean-speed v̄ p.

Among these, there are three independent variables: for example, the combination of q, k,
and qp. This is because of the identities q = k̄v and qp = kp v̄ p, and v̄ = v̄ p.5

5Note that the mean speed v̄ differs from the cruising speed v; the former takes the dwelling time at a station and cruising between stations into account, whereas the latter only considers the cruising time.

13

Transportation

(a)

(b)

(c)
Fig. 3  Time–space diagrams of rail transit system under steady states13

Transportation
Fig. 4  Numerical example of
the FD

Now suppose that the relation among the independent variables of the traffic state under
every steady state is expressed using a function Q as
(4)

q = Q(k, qp ).

The function Q is regarded as an FD of the rail transit system. In fact, by assuming thatthe rail transit operation principle follows Eqs. (2) and (3), the FD function is analyticallyderived aswith⎧ lk − qp ∕𝜇p
if k < k∗ (qp ),⎪ g + l∕v ,bf
Q(k, qp ) = ⎨l𝛿⎪−(k − k∗ (qp )) + q∗ (qp ), if k ≥ k∗ (qp ),⎩ (l − 𝛿)gb + 𝜏lq∗ (qp ) =k∗ (qp ) = −1 − qp ∕𝜇p
gb + 𝛿∕vf + 𝜏(l − 𝛿)∕vf − 𝜏(gb + 𝛿∕vf + 𝜏)𝜇p l

qp +

(5)

(6)

,

gb + l∕vf(gb + 𝛿∕vf + 𝜏)l,(7)

where q∗ (qp ) and k∗ (qp ) represent train-flow and train-density, respectively, at a criticalstate with passenger-flow qp. For the derivation, see Appendix A. Although the FD equations (5)–(7) look complicated, they represent a simple relation: a piecewise linear (i.e.,triangular) relation between q and k under fixed qp. See Fig. 4 for a numerical example of
the FD which we will explain later.

Discussions
The FD is interpreted as a function that describes transit operation performance (train-flowq, headway H = 1∕q, and mean-speed v̄ = q∕k ) under a given train supply (train-density k)
and passenger demand (passenger-flow qp) for the given technical parameters of the transit
system (𝜇p , gb , vf , 𝜏, 𝛿, l ). Therefore, it can be considered as a similar concept to the macroscopic fundamental diagram (MFD) (Geroliminis and Daganzo 2007; Daganzo 2007),
which describes a road network throughput under a given number of vehicles and technicalparameters of the road network.

13

Transportation
Table 1  Parameters of thenumerical example

Parameter

Value

u

70 km/h

𝜏
𝛿
𝜇p
gb
l

1/70 h
1 km
36000 pax/h
10/3600 h
3 km

Numerical Example
First of all, for ease of understanding, we show a numerical example of the FD in Fig. 4.
The parameter values are presented in Table 1. In the figure, the horizontal axis represents
train-density k, the vertical axis represents train-flow q, and the plot color represents passenger-flow qp. The slope of the straight line from a traffic state to the origin represents themean speed v̄ of the state.
For example, the figure can be read as follows. Suppose that the passenger demand per
station is qp = 16000 (pax/h). If the number of trains in the transit system is given by thetrain-density k = 0.3 (train/km), then the resulting train traffic has a train-flow of q ≃ 15
(train/h) and a mean speed of v̄ ≃ 50 (km/h). This is the traffic state in the free-flowing
regime. There is a congested state corresponding to a free-flowing state: for the aforementioned state with (q, k, v̄ ) ≃ (15 veh/h, 0.3 veh/km, 50 km/h), the corresponding congested
state is (15 veh/h, 0.55 veh/km, 27 km/h). The critical state under qp = 16000 (pax/h) is(22 veh/h, 0.42 veh/km, 52 km/h). Notice that this state has the fastest mean speed under
the given passenger demand. The triangular q–k relation mentioned before is clearly shownin the figure; the “left edge” of the triangle corresponds to the free-flowing regime, the“top vertex” corresponds to the critical regime, and the “right edge” corresponds to thecongested regime.
By comparing the theoretical FD (Fig. 4) with the actual data (Fig. 1), some similarities
can be found. The two features found in the actual data (as the passenger-flow increases,
the train-density increases; and as the passenger-flow increases, the train-speed and trainflow decrease) can be interpreted that the actual data are from a part of free-flow regimeof the theoretical FD. Furthermore, in the high train-density regime in the Tokyo data
(Fig. 1a), we observed a slight drop of passenger-flow; this might be a congested regime of
the theoretical FD. From these results, we can say that the theoretical model explains theactual data to some extent.

Detailed Features of Fundamental Diagram
The FD has the following theoretical features which are analytically derived from Eq. (5).
They can easily be found in the numerical example in Fig. 4.
As mentioned, the traffic state of a transit system is categorized into three regimes (freeflowing, critical, and congested), as in the standard traffic flow theory. Therefore, there isa critical train-density k∗ (qp ) for any given qp. Train traffic is in the free-flowing regime if
k < k∗ (qp ), in the critical regime if k = k∗ (qp ), or in the congested regime otherwise. The

13

Transportation

congested regime can be considered as inefficient compared with the free-flowing regime,
because the congested regime takes more time to transport the same volume of passengers.
The critical regime is the most efficient in the sense that its travel time (i.e., 1∕̄v, 1∕̄vp)
and in-vehicle crowding (i.e., number of passengers per train, qp ∕q) are minimum undera given passenger demand. However, the critical regime requires more trains (i.e., higher
train-density) than the free-flowing regime; therefore, it may not be the most efficient if theoperation cost is taken into account.
Even in the critical regime, the mean speed v̄ is inversely proportional to passenger
demand qp. This means that travel time increases as passenger demand increases. In addition, the size of the feasible area of (q, k) narrows as qp increases. Thus, the operational
flexibility of the transit system declines as the passenger demand increases.
Flow and density of trains in the critical regime satisfy the following relations:q∗ (qp ) =1lk∗ (qp ) −.
(l − 𝛿)∕vf − 𝜏(l − 𝛿)∕vf − 𝜏(8)(Here, we have assumed (l − 𝛿)∕vf − 𝜏 ≠ 0.) Therefore, the critical regime is represented
as a straight line whose slope (l∕[(l − 𝛿)∕vf − 𝜏]) is either positive or negative in the q–kplane. This implies a qualitative difference between transit systems. Specifically, if theslope is positive, a transit operation with constant train-density would transition from thefree-flowing regime to the congested regime as passenger demand increases (Fig. 4). On
the contrary, if the slope is negative, such an operation would transition from free-flowing
to congested as passenger demand decreases. This seems paradoxical, but it is actually reasonable because the operational efficiency can be degraded if the number of trains is excessive compared to passenger demand.
The FD describes an transit system’s performance under a steady state operation as
mentioned. Under the presence of well-designed adaptive control strategies, such as schedule-based and headway-based control (Daganzo 2009; Wada et al. 2012), the steady state is
likely to be realized. This is because the aim of such adaptive control is usually to eliminate
bunching—in other words, such control makes the operation steady. Therefore, it can be
expected that the FD could be useful to describe average performance of actual transit system, which is usually not steady due to heterogeneity among passenger demand and train
supply. This issue is numerically validated in Section “Results”.
Last but not least, it is worth mentioning that all parameters in the proposed model havean explicit physical meaning. Therefore, the parameter calibration required to approximate
an actual transit system is relatively easy.

Relation to the Macroscopic Fundamental Diagram
The proposed FD resembles the MFD (Geroliminis and Daganzo 2007; Daganzo 2007)
and its extensions (e.g., Geroliminis et al. 2014; Chiabaut 2015) as mentioned. They
are similar in the following sense. First, they both consider dynamic traffic. Second,
they both describe the relations among macroscopic traffic state variables in which thetraffic is not necessarily steady or homogeneous at the local scale (i.e., they use areawide aggregations based on Edie’s definition; see Appendix B). Third, they both have
unimodal relations, meaning that there are free-flowing and congested regimes, where
the former has higher performance than the latter; in addition, there is a critical regime
where the throughput is maximized. Therefore, it is expected that existing approaches
for MFD applications, such as modeling, control and the optimization of transport13

Transportation

Fig. 5  Railway system as an input–output systemsystems (e.g., Daganzo 2007; Geroliminis and Levinson 2009; Geroliminis et al. 2013;
Fosgerau 2015), are also suitable for the proposed transit FD.
However, there are substantial differences between the proposed FD and the existing
MFD-like concepts. In comparison with the original MFD (Geroliminis and Daganzo
2007; Daganzo 2007) and its railway variant (Cuniasse et al. 2015), the proposed FD
has an additional dimension, that is, passenger-flow. In comparison with the threedimensional MFD of Geroliminis et al. (2014), which describes the relations among
total traffic flow, car density, and bus density in a multi-modal traffic network, theproposed FD explicitly models the physical interaction among the three variables. In
comparison with the passenger MFD of Chiabaut (2015), which describes the relation
between passenger flow and passenger density when passengers can choose to travel by
car or bus, in the proposed FD, passenger demand can degrade the performance (i.e.,
speed) of the vehicles because of the inclusion of the boarding time.

Dynamic Model Based on Fundamental Diagram
Recall that the proposed FD describes the relationship among traffic variables under
the steady state. It means that the behavior of a dynamical system in which demandand supply change over time is not described solely by the FD. This feature is the sameas in the road traffic FD and MFDs. In this section, we formulate a model of urban
rail transit operation where the demand (i.e., passenger-flow) and supply (i.e., traindensity) change dynamically. In this proposed model, individual train and passenger
trajectories are not explicitly described; therefore, the model is called macroscopic.
The proposed model is based on an exit-flow model (Merchant and Nemhauser
1978; Carey and McCartney 2004) in which the proposed FD is employed as the exitflow function. Specifically, the transit system is considered as an input–output system,
as illustrated in Fig. 5. The exit-flow modeling approach is often employed for areawide traffic approximations and analysis using MFDs, such as optimal control to avoid
congestion (Daganzo 2007) and analyses of user equilibrium and social optimum inmorning commute problems (Geroliminis and Levinson 2009). The advantage of thisapproach is that it would be possible to conduct mathematically tractable analysis ofdynamic, large-scale, and complex transportation systems, where the detailed traffic dynamics are difficult to model in a tractable manner—this is the case for transit
operations.

13

Transportation

Formulation
Let a(t) be the inflow of trains to the transit system, ap (t) be the inflow of passengers, d(t)be the outflow of trains from the transit system, and dp (t) be the outflow of passengers, ontime t respectively. We set the initial time to be 0. Let A(t), Ap (t), D(t), and Dp (t) be thetcumulative values of a(t), ap (t), d(t), and dp (t), respectively (e.g., A(t) = ∫0 a(s)ds). Let T(t)be the travel time of a train that entered the system at time t, and let its initial value T(0) be
given by the free-flow travel time under q = a(0) and qp = ap (0). To simplify the formulation, the trip length of the passengers is assumed to be equal to that of the trains.6 Thismeans that T(⋅) is the travel time of both the trains and the passengers. These functions areinterpreted as follows:
•
•
•
•
•

a(t): trains’ departure rate from their origin station at time t.
ap (t): passengers’ arrival rate at the platform of their origin station at time t.
d(t): trains’ arrival rate at their final destination station at time t.
dp (t): passengers’ arrival rate at their destination station at time t.
T(t): travel time of a train and passengers from origin (departs at time t) to destination.
Note that the arrival time at the destination is t + T(t).

Therefore, in reality, a(⋅) and ap (⋅) will be determined by the transit operation plan andpassenger departure time choice, respectively. Then, d(⋅), dp (⋅), and T(⋅) are endogenously
determined through the operational dynamics.
In accordance with exit-flow modeling, the train traffic is modeled as follows. First, theexit flow d(t) is assumed to bed(t) = Q(k(t), ap (t))(9)

where the FD function Q(⋅) is considered to be an exit-flow function.7 This means thatthe dynamics of the transit system are modeled by taking the conservation of trains into
account as follows:

L

dk(t)
= a(t) − Q(k(t), ap (t)),
dt

(10)

where L represents the length of the transit route. This exit-flow model has been employed
in several studies to represent the macroscopic behavior of a transportation system (e.g.,
Merchant and Nemhauser 1978; Carey and McCartney 2004; Daganzo 2007). Note that the
average train-density k(t) is defined ask(t) =

A(t) − D(t)
,
L

(11)

6
This assumption is reasonable if the average trip length is shared by trains and passengers. If they are different, a modification such as Tp (t) = T(t)∕𝜆, where 𝜆 is the ratio of average trip length of the passengers tothat of the trains, would be possible.
7
If np is considered as the sum of the number of passengers who are boarding and alighting (as mentioned
in note 3), we can simply define d(t) to be equal to Q(k(t), ap (t) + dp (t)). Such a model is also computableusing a similar procedure.

13

Transportation

which is consistent with Eq. (10). Based on above functions and equations, d(t) and D(t)are sequentially computed—in other words, the train traffic is computed using the initialand boundary conditions and the exit-flow model based on the FD.
The passenger traffic is derived as follows. By the definition of the travel time of trains,

A(t) = D(t + T(t))(12)

holds. As A(t) and D(t) have already been obtained, the travel time T(t) such that Eq. (12)holds is computed. Then, Dp (t) and dp (t) are computed from the definition of the travel
time of passengers, which is also T(t):Ap (t) = Dp (t + T(t)).

(13)

Discussion
The proposed macroscopic model computes train out-flow d(t) and passenger out-flow dp (t)based on the FD function Q(⋅), the initial and boundary conditions a(t), ap (t), and T(0).
The advantages of the proposed model are twofold. First, it is a dynamic model, meaningthat it can describe dynamic changing (i.e., time-varying) passenger demand, travel time,fee, and so on. This is a significant advantage compared with static models, in which thesevariables are static and constant. Second, it is analytically tractable, so that optimal control
strategy may be derived as an analytical solution. This is a significant advantage comparedwith microscopic simulation models, which are usually not analytically tractable and computationally costly. Thus, the proposed model might be suitable for some specific purposes,
such as to derive the optimal solution for dynamic pricing under time-varying passenger
demand.
It is reasonable to expect that the proposed model can accurately approximate the macroscopic behavior of a transit operation with high-frequency operation (i.e., small timeheadway) under moderate changes in demand and/or supply, although the FD is derived
based on the steady state assumption. This is because exit-flow models are reasonably
approximate a dynamical system’s behavior when the changes in inflow are moderate compared with the relaxation time of the system. In the next section, the quantitative accuracyof the model is validated through numerical experiments.

Validation of the Macroscopic ModelIn this section, we validate the quantitative accuracy of the macroscopic model by comparing its results with that of the microscopic model (i.e., Eqs. (2) and (3)).

Simulation Setting
The parameter values of the transit operation are listed in Table 1 for both the microscopic
and macroscopic models. The railroad is considered to be a one-way corridor. The stations
are equally spaced at intervals of l, and there are a total of 10 stations. Trains enter the railroad with flow a(t); in the microscopic model, a discrete train enters the railroad from the
upstream boundary station if ⌊A(t)⌋ (i.e., integer part of A(t)) is incremented. In the microscopic model, trains leave the railroad from the downstream boundary station without any

13Transportation

restrictions, other than the passenger boarding and minimum headway clearance. Passengers arrive at each station with flow ap (t).
The functions a(t) and ap (t) are exogenously determined to mimic morning rush hours,with each having a peak at t = 2. The flow before the peak time increases monotonically,
whereas the flow after the peak time decreases monotonically—in other words, the socalled S-shaped A(t) and Ap (t) (c.f., Fig. 7) are considered. The parameters of these functions are the minimum train supply amin , the maximum train supply amax, the minimum, and the maximum passenger demand amax
. The functional forms
passenger demand aminpp
are described in Appendix C. The simulation duration is set to 4 h for the baseline scenario
in Section “Baseline scenario” and to 8 h for the sensitivity analysis in Section “Sensitivity
analysis of the demand/supply conditions” (the reason will be explained later).
The microscopic model without any control is asymptotically unstable, as proven by
Wada et al. (2012); this means that time-varying demand and supply always cause train
bunching, making the experiment unrealistic and useless. Therefore, the headway-based
control scheme proposed by Wada et al. (2012) is implemented in the microscopic model
to prevent bunching and stabilize the operation. This scheme has two control measures:
holding (i.e., extending the dwell time) and an increase of free-flow speed, similar to
Daganzo (2009). The former is activated by a train if its following train is delayed, and isrepresented as an increase in gb in the microscopic model. The latter is activated by a train
if it is delayed, and is represented as an increase in vf up to a maximum allowable speed
vmax. In this experiment, vmax is set to 80 km/h and vf is 70 km/h. This control scheme canbe considered realistic and reasonable, as similar operations are executed in practice. See
Appendix D for further details of the control scheme. Note that the boundary conditions
are the trajectory of the first train x0 (t) ∀t , the initial position of all the trains xm (0) ∀m (thisis converted to the departure time of all the trains from the most upstream station xm (tm0 ) ∀m
where tm0 denotes the departure time of train m), and the passenger demand to each station
qp. Also note that the microscopic model can be considered as an operation under planned
frequency (1/a(t)) with a buffer time-based stabilization or regularization scheme (Appendix D).

Results
First, to examine how well the proposed model reproduces the behavior of the transit system under time-varying conditions, the results for the baseline scenario are presented in
Section “Baseline scenario”. Then, a sensitivity analysis of the demand/supply conditions
is conduced and applicable ranges of the proposed model are investigated in Section “Sensitivity analysis of the demand/supply conditions”.

Baseline scenario
The baseline scenario with parameter values amin = 10 (train/h), amax = 15 (train/h),amin= 0.1𝜇p (pax/h), and amax
= 0.5𝜇p (pax/h) is investigated first. This situation can beppconsidered as ordinary crowded. A solution of the microscopic model is shown in Fig. 6
as a time–space diagram. The colored curves represent the trajectories of each train traveling in the upward direction while stopping at every station. Around the peak time period(t = 2), train congestion occurs; namely, some of the trains stop occasionally between stations in order to maintain the safety interval. The congestion is caused by heavy passenger
demand; therefore, the situation during rush hour is reproduced.

13

Transportation

Fig. 6  Result of the microscopic model in the baseline scenario

(a)

(b)

Fig. 7  Result of the macroscopic model in the baseline scenario

Fig. 8  Comparison between themacroscopic and microscopic
models in the baseline scenario

The result given by the macroscopic model is shown in Fig. 7 as cumulative plots.
Fig. 7a shows the cumulative curves for the trains, where the blue curve represents the
inflow A and the red curve represents the outflow D. Fig. 7b shows those of passengers
in the same manner. Congestion and delay are observed around the peak period (it ismore remarkable in the passenger traffic). For example, during the peak time period,dp (t) is less than ap (t) and ap (t� ), where t′ is time such that t = t� + T(t� ). This means thatthe throughput of the transit system is reduced by the heavy passenger demand. Consequently, T(t) is greater during peak hours than in off-peak periods such as T(0), meaningthat delays occur due to the congestion.

13

Transportation

Fig. 9  Comparison between the microscopic and macroscopic models under different demand/supply conditions

The macroscopic and microscopic models are compared in terms of the cumulative
number of trains in Fig. 8. In the figure, the solid curves denote the macroscopic modeland the dots denote the microscopic model. The macroscopic model reproduced the microscopic model’s result accurately; the relative difference of total travel time was roughly 4%.
For example, the congestion and delay during the peak time period are captured very well.
However, there is a slight bias: the macroscopic model gives a slightly shorter travel time.
This is mainly due to the large-scale unsteady state (i.e., train bunching) generated in the
microscopic model; the delay caused by such large-scale bunching cannot be recovered by
the microscopic model under the implemented headway-based control scheme (for details,
see Appendix D). It means that if the control is schedule-based, the bias could be reduced.

Sensitivity analysis of the demand/supply conditions
The accuracy of the macroscopic model regarding the dynamic patterns of demand/supply
is now examined. This is worth investigating it quantitatively, because it is qualitatively
clear that the exit-flow model is valid if the speed of demand/supply changes is “sufficiently” small as discussed in Section “Discussion”. Specifically, the sensitivity of the peak
and train supply amax is evaluated by assigning various values topassenger demand amaxpthese parameters. The simulation duration is set to 8 h to take the residual delay after t = 4(h) in some scenarios into account. The other parameters are the same as in the baseline
scenario.
The results are summarized in Fig. 9 and Table 2. It shows the relative difference in total
travel time (TTT) of trains between the microscopic and macroscopic models for various. The minimum train supply andpeak train supply amax and peak passenger demand amaxp=
0.1𝜇p = 6000 (pax/h). The relapassenger demand are set as amin = 10 (train/h) and aminptive difference can be considered as an error index of the macroscopic model. The negativevalues indicate that TTT of the macroscopic model is smaller.
According to the results in Fig. 9 and Table 2, the accuracy of the macroscopic model
is high when the maximum passenger demand is not extremely large. This is an expected
result, as the speed of demand change is slow in these cases. TTT given by the macroscopic model is almost always less than that of the microscopic model; this might be due to13Transportation
Table 2  Comparison of total travel time (TTT) of the microscopic and macroscopic models under different
demand/supply conditions(a) Max train supply 10.0 train/h
Max pax demand (pax/h)

6000

9000

12000

15000

18000

21000

24000

27000

30000

TTT of macro model (train h)

40.9

41.9

42.9

43.8

44.8

45.8

46.7

47.7

48.7

TTT of micro model (train h)

40.9

41.7

42.7

43.7

44.8

46.0

48.2

53.4

56.4

6000

9000

12000

15000

18000

21000

24000

27000

30000

(b) Max train supply 12.5train/h
Max pax demand (pax/h)
TTT of macro model (train h)

43.3

44.2

45.2

46.1

47.1

48.1

49.0

50.0

51.0

TTT of micro model (train h)

43.7

44.9

45.7

46.8

48.6

50.1

51.2

57.9

91.5

(c) Max train supply 15.0 train/h
Max pax demand (pax/h)

6000

9000

12000

15000

18000

21000

24000

27000

30000

TTT of macro model (train h)

45.5

46.5

47.5

48.4

49.4

50.4

51.3

52.3

53.2

TTT of micro model (train h)

46.7

47.3

48.4

49.7

51.3

52.3

55.1

65.5

78.1

9000

12000

15000

18000

21000

24000

27000

30000

(d) Max train supply 17.5 train/h
Max pax demand (pax/h)

6000

TTT of macro model (train h)

47.8

48.8

49.8

50.7

51.7

52.6

53.6

54.6

55.5

TTT of micro model (train h)

49.4

50.3

51.8

52.3

53.8

58.6

66.0

70.6

88.3

9000

12000

15000

18000

21000

24000

27000

30000

(e) Max train supply 20.0 train/h
Max pax demand (pax/h)

6000

TTT of macro model (train h)

50.1

51.1

52.0

53.0

54.0

54.9

55.9

56.9

58.3

TTT of micro model (train h)

52.0

53.6

54.1

55.6

56.9

58.2

66.1

67.4

119.9

the aforementioned inconsistency between the steady state assumption of the macroscopic
model and headway-based control of the microscopic model.
The relative error increases suddenly when the demand exceeds a certain value, around20000–22000. This sudden change is a result of extraordinary large-scale train bunchingin the microscopic model. This bunching often occurs in cases with excessive passenger> 𝜇p ∕2. These excessive passenger demand can be considered as
demand, such as amaxpextraordinary and unrealistically high. For example, a train need to stop at a station for
more than 10 min to process the passenger demand with 20000 pax/h. Additionally, thedwell time becomes longer than the cruising time between adjacent stations in this situation; in the real world, this does not occur even in rush hours .
As for the sensitivity of the train supply a(⋅), there is a weak tendency for faster variations in supply to cause larger errors. This is also an expected result. In any case, the error
is small.
From these results, we conclude that the proposed model accurately reproduced themicroscopic model’s results accurately under ordinary passenger demand. On the other
hand, the proposed model could not reproduce extraordinary and unrealistic situations withexcessive train bunching. This might be acceptable for representing transit systems during
normal rush hours.

13Transportation

Conclusion
The main contribution of this paper is that it analytically derived a closed-form expressionof FD of rail transit systems based on microscopic operation principles. The FD determines operation performance of rail transit systems (i.e., flow, headway, and mean speed)
based on supply of trains and passenger demand. Furthermore, this paper proposed an efficient, macroscopic dynamic assignment method based on the FD. The results of numericalexperiments showed that the proposed method accurately reproduced results of a microscopic simulation model in macroscopic scales under realistic situations.
Specifically, the following three models of an urban rail transit system have been analyzed in this paper:• Microscopic model: A model describing the trajectories of individual trains and passengers based on Newell’s car-following model and passenger boarding model. This isrepresented in Eqs. (2) and (3), and is solved using simulations.
• Fundamental diagram: An exact relationship among train-flow, train-density, andpassenger-flow in the microscopic model under a steady state. This is represented in
Eqs. (4)–(7). It is a closed-form equation.
• Macroscopic model: A model describing train and passenger traffic using an exit-flow
model whose exit-flow function is the FD. This is represented in Eqs. (9), (11), (12),and (13), and is solved using simple simulations.

The FD and macroscopic model are the original contributions of this study, whereas the
microscopic model was proposed by Wada et al. (2012).
The FD itself implies several insights on transit system, such as relation between mean
speed of the system and passenger demand. In addition, according to the results of thenumerical experiment, the macroscopic model can reproduce the behavior of the microscopic model accurately, except for cases with unrealistically excessive demands. The proposed model is able to analyze dynamical changes in operation and demand (which is notpossible for existing static models) while keeping the analytical tractability high (which isusually not possible for existing detailed simulation). Thus, we can expect that the model
can contribute for obtaining general policy implications on management strategies of railtransit systems, such as pricing and control for morning commute problems.
Following future works are considerable. First, rigorous empirical validation on
the existence of the FD is required. In fact, several preliminarily results on it have beenreported (Fukuda et al. 2019; Zhang and Wada 2019) as shown in Fig. 1. Second, as anapplication of the FD, analysis of operation and demand management for transit systems
is important. For example, the morning commute problem (Zhang et al. 2021) has been
analyzed, and its departure time choice equilibrium and an optimal dynamic timetable havebeen derived. Third, queuing phenomena on platforms are important in highly crowded
situations as explained in Section “Validity of Assumptions”. This could be done by adding
a platform queueing model to the outer layer of the exit flow model. Forth, extensions tomore complicated and realistic railway operation, such as skip-stop operations and variable
stopping patterns are considerable. Such heterogeneous operation schemes could be modeled by the following approach. When derive the FD equation, we need to consider train
operation in a specific spatial-temporal region which is the minimum unit component ofsteady operation; in the current paper, we consider two trains and two stations as in Fig. 2.
If the operation is heterogeneous, we can consider a larger spatial-temporal region to obtain13

Transportation

the minimum unit component of the operation. After defining such spatial-temporal region,
we may able to derive an FD equation by employing a similar approach to this paper. Control theoretical approaches (Van Breusegem et al. 1991; Farhi 2019; Schanzenbacher et al.
2021) may be also useful to extend the proposed model to consider such heterogeneity as
well as bottlenecks and railway networks.

Appendix A: Derivation of FD
This appendix describes derivation of the FD expressed in Eqs. (5)–(7). Consider a loopedrail transit system8 under steady state operation. Let L be the length of the railroad, S be thenumber of the stations, M be the number of trains, H be the time-headway of the operation,
tb be the dwelling time of a train at a station, tc be the cruising time of a train between adjacent stations, and qp be the passenger demand flow rate per station. Note that the distance
between adjacent stations l is L/S and the number of passengers boarding a train at eachstation is qp H .
The time-headway of the operation is derived as follows. The round trip time of a train
in the looped railroad is S(tb + tc ), and M trains pass the station during that time. Then, theidentities NH = S(tb + tc ) andH=

gb + t c
M∕S − qp ∕𝜇p(A.1)

hold. Moreover, by the definition of headway and Newell’s car-following rule, the timeheadway H must satisfy

𝛿 + v𝜏
+ hf .
v(A.2)

gb + 𝛿∕v + 𝜏+ hf .
1 − qp ∕𝜇p(A.3)

H = tb +This reduces to

H=

The q–k relation in a free-flowing regime is derived as follows. As the train-flow is 1/H andtrain-density is M/L by definition, Eq. (A.1) is transformed toq(k) =

kl − qp ∕𝜇p
gb + l∕vf

(A.4)

.

The train-flow and train-density under a critical state, (q∗ , k∗ ), are derived as follows. By
substituting v = vf and hf = 0 into Eq. (A.3) and using the identity q = k̄v, we obtainq∗ =1 − qp ∕𝜇p
gb + 𝛿∕vf + 𝜏

,

(A.5)

8
Many other assumptions are acceptable on the shape of the track as long as we can define the steady state
operation shown in Fig. 3.

13

Transportationk ∗ = k0 +(1 − qp ∕𝜇p )(gb + l∕vf )(gb + 𝛿∕vf + 𝜏)l

(A.6)

,

where k0 is the minimum train-density where the train-flow is zero, namely, k0 = qp ∕(𝜇p l).
The q–k relation in a congested regime is derived as follows. First, the k–v relation in acongested regime is easily derived from the q–v relation (A.3) with hf = 0 and the identityq = k̄v:

k(v) = k0 +(1 − qp ∕𝜇p )(gb + l∕v)
(gb + 𝛿∕v + 𝜏)l

(A.7)

.

Now, consider dq∕dk , which is identical to (dq∕dv) ⋅ (dv∕dk). This is derived as

dq
l𝛿
,
=
dk
(𝛿 − l)gb − 𝜏l(A.8)

which is constant and negative; therefore, the q–k relation is linear in a congested regime.
Then, recalling that the linear q–k curve passes the point (q∗ , k∗ ) with a slope of dq∕dk , theq–k relation in a congested regime is derived asq(k) =l𝛿
k + q0(𝛿 − l)gb − 𝜏l

(A.9)

dq ∗
⋅k .
dk

(A.10)

with

q0 = q∗ −Eqs. (5)–(7) are constructed based on Eqs. (A.4), (A.5), (A.6), (A.9), and (A.10).

Appendix B: Consistency of the FD and Edie’s generalized definitionof traffic state
It is noteworthy that Eqs. (4) and (5) are consistent with Edie’s generalized definition (Edie
1963) of traffic states; because from this consistency we can confirm that the FD is consistent with the fundamental definition of traffic. For steady-state transit operation, Edie’s traffic state is derived as

q=

k=

v̄ =

1
,
H

qp H∕𝜇p + gb + l∕v

(B.1)

,

(B.2)

l
.
qp H∕𝜇p + gb + l∕v

(B.3)

lH

These relations are derived by applying Edie’s definition to the “minimum componentof the time–space diagram” of the steady state, which is a parallelogram-shaped area in
Fig. 3 whose vertexes are time–space points of (i) train m departs from station i, (ii) train m

13

Transportation

arrives at station i + 1, (iii) train m − 1 arrives at station i + 1, and (iv) train m − 1 departs
from station i. One can easily confirm that Eqs. (B.1)–(B.3) satisfy the FD equation. In fact,
the FD equation is also derived from Eqs. (B.1)–(B.3) and the constraint (A.3) induced byNewell’s car-following model.

Appendix C: S‑shaped supply and demand functions
The train supply and passenger demand in the experiments are given by the following
functions:⎧ amin + (amax − amin ) t , if t < 2,2⎪
a(t) = ⎨ amin + (amax − amin ) 4−t , if 2 ≤ t < 4,2
⎪ amin ,if 4 ≤ t,⎩(C.1)⎧ amin + (amax − amin ) t , if t < 2,
p
p
2
⎪ p
max − amin ) 4−t , if 2 ≤ t < 4,+(aap (t) = ⎨ aminppp2
⎪ amin ,if 4 ≤ t.
⎩ p

(C.2)

Both functions have a minimum value at t = 0 and t ≥ 4 and a minimum value at t = 2, andchange linearly in between.

Appendix D: Adaptive control scheme in the microscopic modelThis appendix briefly explains the adaptive control scheme for preventing train bunching, proposed by Wada et al. (2012). This scheme consists of two control measures: holding at a station and increasing the maximum speed during cruising.
First, the scheme modifies the buffer time for dwelling (originally defined as gb in Eq. (2))
of train m at station i towithgb ∶= max{0, gb − Em (i)}

(D.1)

(
)
Em (i) = (1 − 𝛼)𝜀m (i) + 𝛼𝜇p 𝜀m (i) − 𝜀m−1 (i) ,(D.2)

where 𝜀m (i) ≡ tm (i) − Tm,i represents the delay, tm (i) represents the time at which train marrives at station i, Tm,i represents the scheduled time (i.e., without delay) at which train mshould arrive at station i, and 𝛼 ∈ [0, 1] is a weighting parameter. This scheme representsa typical holding control strategy, similar to the bunching prevention method of Daganzo(2009), which extends the dwelling time of a vehicle if the headway to the preceding vehicle is too small and vice versa.
Second, the scheme modifies the free-flow cruising speed vf such that the interstation travel
time is reduced by}{min l∕vf − l∕vmax , max{0, Em (i) − gb } .
(D.3)

13

Transportation

This means that, in the event of a delay, the train tries to catch up by increasing its cruising
speed up to the maximum allowable speed vmax (which implies that the free-flow speed vf isa “buffered” maximum speed).
Meanwhile, the proposed train operation model in this study does not have a schedule—it
is a frequency-based operation. Therefore, in this study, the scheduled headway in the scheme(Tm,i − Tm−1,i) is approximated by the planned frequency (1∕a(tm (i))). Thus, we set 𝛼 = 1 andsubstitute Em (i) with()
𝜇p tm (i) − tm−1 (i) − 1∕a(tm (i)) .
(D.4)
The stationary state of the operational dynamics under the original scheme is basically
identical to the steady state defined in Section 3.1. There may be small difference in thecongested regime because of the operation scheme; however, this will not be problematicsince heavily congested regime will not occur. In the case of 𝛼 < 1, the scheme makes thetrain operation asymptotically stable, meaning that the operation schedule is robust to small
disturbances. In the case of 𝛼 = 1, the scheme prevents the propagation and amplificationof delay, but does not recover the original schedule (the small ‘shift’ found in Fig. 8 is due
to 𝛼 = 1). Note that these control measures do not interrupt passenger boarding or violate
the safety clearance between trains, meaning that most of the fundamental assumptions of
the proposed FD are satisfied.
Author Contributions TS: Conceptualization, Methodology, Validation, Formal analysis, Investigation,
Resources, Writing - Original Draft, Writing - Review & Editing, Visualization, Funding acquisition. KW: Conceptualization, Methodology, Formal analysis, Investigation, Writing - Review & Editing, Funding acquisition.
DF: Conceptualization, Methodology, Investigation, Writing - Review & Editing, Funding acquisition.
Funding This work was supported by Japan Society for the Promotion of Science under KAKENHI Grantin-Aid for Scientific Research (B) #17H03320 and #20H02268.
Availability of data and material The data used to produce Fig. 1 are partially available at Fukuda et al.
(2019), Zhang and Wada (2019), and Massachusetts Bay Transportation Authority.
Code availability The code of the simulation in Section “Validation of the Macroscopic Model” is available
upon request.

Declarations
Conflicts of interest The authors declare no conflicts of interest associated with this manuscript.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as longas you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to thematerial. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directlyfrom the copyright holder. To view a copy of this licence, visithttp://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/.

13

Transportation

References
Alonso, B., Munoz, J.C., Ibeas, A., Moura, J.L.: A congested and dwell time dependent transit corridor
assignment model. J. Adv. Transport. (2017)
Carey, M., Kwieciński, A.: Stochastic approximation to the effects of headways on knock-on delays of
trains. Transport. Res. Part B 28(4), 251–267 (1994)
Carey, M., McCartney, M.: An exit-flow model used in dynamic traffic assignment. Comput. Oper. Res.
31(10), 1583–1602 (2004)
Cats, O., West, J., Eliasson, J.: A dynamic stochastic model for evaluating congestion and crowding effects
in transit systems. Transport. Res. Part B 89, 43–57 (2016)
Chiabaut, N.: Evaluation of a multimodal urban arterial: The passenger macroscopic fundamental diagram.
Transport. Res. Part B 81, 410–420 (2015)
Corman, F., Henken, J., Keyvan-Ekbatani, M.: Macroscopic fundamental diagrams for train operations—are
we there yet? In: 2019 6th International Conference on Models and Technologies for Intelligent Transportation Systems, pp. 1–8 (2019)
Cunha, J., Reis, V., Teixeira, P.: Development of an agent-based model for railway infrastructure projectappraisal. Transportation (2021)
Cuniasse, P.-A., Buisson, C., Rodriguez, J., Teboul, E., de Almeida, D.: Analyzing railroad congestion in adense urban network through the use of a road traffic network fundamental diagram concept. Public
Transport. 7(3), 355–367 (2015)
Daganzo, C.F.: Fundamentals of Transportation and Traffic Operations. Pergamon, Oxford (1997)
Daganzo, C.F.: Urban gridlock: macroscopic modeling and mitigation approaches. Transp. Res. Part B
41(1), 49–62 (2007)
Daganzo, C.F.: A headway-based approach to eliminate bus bunching: systematic analysis and comparisons.
Transp. Res. Part B 43(10), 913–921 (2009)de Cea, J., Fernández, E.: Transit assignment for congested public transport systems: an equilibrium model.
Transp. Sci. 27(2), 133–147 (1993)
de Palma, A., Kilani, M., Proost, S.: Discomfort in mass transit and its implication for scheduling and pricing. Transp. Res. Part B 71, 1–18 (2015)
de Palma, A., Lindsey, R., Monchambert, G.: The economics of crowding in public transport. Working
Paper (hal-01203310) (2015b)
de Rivera, A.D., Dick, C.T.: Illustrating the implications of moving blocks on railway traffic flow behavior
with fundamental diagrams. Transp. Res. Part C 123, 102982 (2021)
Dicembre, A., Ricci, S.: Railway traffic on high density urban corridors: capacity, signalling and timetable.
J. Rail Transp. Plan. Manag. 1(2), 59–68 (2011)
Edie, L.C.: Discussion of traffic stream measurements and definitions. In: Almond, J. (ed.) Proceedings ofthe 2nd International Symposium on the Theory of Traffic Flow, pp. 139–154 (1963)
Farhi, N.: Physical models and control of the train dynamics in a metro line without junction. IEEE Trans.
Control Syst. Technol. 27(4), 1829–1837 (2019)
Fosgerau, M.: Congestion in the bathtub. Econ. Transp. 4, 241–255 (2015)
Fukuda, D., Imaoka, M., Seo, T.: Empirical investigation of fundamental diagram for urban rail transit using
Tokyo’s commuter rail data. In: TRANSITDATA2019: 5th International Workshop and Symposium(2019)
Geroliminis, N., Daganzo, C.F.: Macroscopic modeling of traffic in cities. In: Transportation Research
Board 86th Annual Meeting (2007)
Geroliminis, N., Haddad, J., Ramezani, M.: Optimal perimeter control for two urban regions with macroscopic fundamental diagrams: A model predictive approach. IEEE Trans. Intell. Transp. Syst. 14(1),348–359 (2013)
Geroliminis, N., Levinson, D.M.: Cordon pricing consistent with the physics of overcrowding. In: Lam,
W.H.K., Wong, S.C., Lo, H.K. (eds.) Transportation and Traffic Theory 2009. Springer, pp. 219–240(2009)
Geroliminis, N., Zheng, N., Ampountolas, K.: A three-dimensional macroscopic fundamental diagram formixed bi-modal urban networks. Transport. Res. Part C 42, 168–181 (2014)
Gonzales, E.J., Daganzo, C.F.: Morning commute with competing modes and distributed demand: user
equilibrium, system optimum, and pricing. Transp. Res. Part B 46(10), 1519–1534 (2012)
Greenshields, B.D.: A study of traffic capacity. Highway Res Board Proc. 14, 448–477 (1935)
Halvorsen, A., Koutsopoulos, H.N., Ma, Z., Zhao, J.: Demand management of congested public transportsystems: a conceptual framework and application using smart card data. Transportation 47(5), 2337–
2365 (2019)
Higgins, A., Kozan, E.: Modeling train delays in urban networks. Transp. Sci. 32(4), 346–357 (1998)

13

Transportation
Hoogendoorn, S.P., Daamen, W.: Pedestrian behavior at bottlenecks. Transp. Sci. 39(2), 147–159 (2005)
Huan, N., Hess, S., Yao, E.: Understanding the effects of travel demand management on metro commuters’ behavioural loyalty: a hybrid choice modelling approach. Transportation (2021)
Huisman, D., Kroon, L.G., Lentink, R.M., Vromans, M.J.C.M.: Operations research in passenger railway
transportation. Stat. Neerl. 59(4), 467–497 (2005)
Kariyazaki, K., Hibino, N., Morichi, S.: Simulation analysis of train operation to recover knock-on delay
under high-frequency intervals. Case Stud. Transp. Policy 3(1), 92–98 (2015)
Kato, H., Kaneko, Y., Soyama, Y.: Departure-time choices of urban rail passengers facing unreliable service: Evidence from Tokyo. In: Proceedings of the International Conference on Advanced Systems
for Public Transport 2012 (2012)
Koutsopoulos, H., Wang, Z.: Simulation of urban rail operations: application framework. Transp. Res.
Rec. 2006, 84–91 (2007)
Kraus, M., Yoshida, Y.: The commuter’s time-of-use decision and optimal pricing and service in urban
mass transit. J. Urban Econ. 51(1), 170–195 (2002)
Kumagai, J., Wakamatsu, M., Managi, S.: Do commuters adapt to in-vehicle crowding on trains? Transportation (2020)
Kusakabe, T., Iryo, T., Asakura, Y.: Estimation method for railway passengers’ train choice behavior
with smart card transaction data. Transportation 37(5), 731–749 (2010)
Lam, W.H.K., Cheung, C.Y., Poon, Y.F.: A study of train dwelling time at the Hong Kong mass transit
railway system. J. Adv. Transp. 32(3), 285–295 (1998)
Li, K.P., Gao, Z.Y., Ning, B.: Cellular automaton model for railway traffic. J. Comput. Phys. 209(1),179–192 (2005)
Li, S., Dessouky, M.M., Yang, L., Gao, Z.: Joint optimal train regulation and passenger flow control
strategy for high-frequency metro lines. Transp. Res. Part B 99, 113–137 (2017)
Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. a theory of traffic flow on long crowded roads.
Proc. R. Soc. Lond. A 229(1178), 317–345 (1955)
Mahmassani, H.S., Williams, J.C., Herman, R.: Investigation of network-level traffic flow relationships:
some simulation results. Transp. Res. Rec. 971, 121–130 (1984)
Merchant, D.K., Nemhauser, G.L.: A model and an algorithm for the dynamic traffic assignment problems. Transp. Sci. 12(3), 183–199 (1978)
Newell, G.F.: A simplified theory of kinematic waves in highway traffic. Transp. Res. Part B 27(4),281–313 (Part I, II, and III) (1993)
Newell, G.F.: A simplified car-following theory: a lower order model. Transp. Res. Part B 36(3), 195–
205 (2002)
Newell, G.F., Potts, R.B.: Maintaining a bus schedule. In: Proceedings of the 2nd Australian Road
Research Board. Vol. 2 (1964)
Niu, H., Zhou, X.: Optimizing urban rail timetable under time-dependent demand and oversaturated conditions. Transp. Res. Part C 36, 212–230 (2013)
Parbo, J., Nielsen, O.A., Prato, C.G.: Passenger perspectives in railway timetabling: a literature review.
Transp. Rev. 36(4), 500–526 (2016)
Richards, P.I.: Shock waves on the highway. Oper. Res. 4(1), 42–51 (1956)
Schanzenbacher, F., Farhi, N., Leurent, F., Gabriel, G.: Feedback control for metro lines with a junction.
IEEE Trans. Intell. Transp. Syst. 22(5), 2741–2750 (2021)
Seo, T., Wada, K., Fukuda, D.: Fundamental diagram of urban rail transit considering train–passenger
interaction. arXiv preprint arXiv:​ 1708.​02147​(updat​ed in 2021) (2017a)
Seo, T., Wada, K., Fukuda, D.: A macroscopic and dynamic model of urban rail transit with delay andcongestion. In: Transportation Research Board 96th Annual Meeting (2017b)
Seo, T., Wada, K., Fukuda, D.: Transit fundamental diagram and its application to analysis of high frequency transitsystems. Proceedings of the Infrastructure Planning 60, (in Japanese) (2019)
Tabuchi, T.: Bottleneck congestion and modal split. J. Urban Econ. 34(3), 414–431 (1993)
Tian, Q., Huang, H.-J., Yang, H.: Equilibrium properties of the morning peak-period commuting in a
many-to-one mass transit system. Transp. Res. Part B 41(6), 616–631 (2007)
Tirachini, A., Hensher, D.A., Rose, J.M.: Crowding in public transport systems: effects on users, operation and implications for the estimation of demand. Transp. Res. Part A 53, 36–52 (2013)
Trozzi, V., Gentile, G., Bell, M.G.H., Kaparias, I.: Dynamic user equilibrium in public transport networks with passenger congestion and hyperpaths. Transp. Res. Part B 57, 266–285 (2013)
Van Breusegem, V., Campion, G., Bastin, G.: Traffic modeling and state feedback control for metro
lines. IEEE Trans. Autom. Control 36(7), 770–784 (1991)
Vuchic, V.R.: Urban Transit: Operations, Planning, and Economics. Wiley, New York (2005)

13

Transportation
Wada, K., Kil, S., Akamatsu, T., Osawa, M.: A control strategy to prevent delay propagation in high-frequency railway systems. Journal of Japan Society of Civil Engineers, Ser. D3 (Infrastructure Planningand Management) 68(5), I_1025–I_1034 (2012), (in Japanese; extended abstract in English was presented at the 1st European Symposium on Quantitative Methods in Transportation Systems and available at https://​www.​resea​rchga​te.​net/​publi​cation/​28182​3577)
Zhang, J., Wada, K.: Fundamental diagram of urban rail transit: An empirical investigation by Boston’s
subway data. In: hEART 2019: 8th Symposium of the European Association for Research in Transportation (2019)
Zhang, J., Wada, K., Oguchi, T.: Morning commute in congested urban rail transit system: A macroscopic
model for equilibrium distribution of passenger arrivals. arXiv preprint arXiv:​2102.​13454 (2021)
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps andinstitutional affiliations.

Toru Seo is an Associate Professor at Tokyo Institute of Technology. He had earned his doctoral degree in2015 at Tokyo Institute of Technology. His primary research interests are transportation science, traffic flow
theory, transportation data analysis, and transportation system modeling and management.
Kentaro Wada is an Associate Professor at the Faculty of Engineering, Information and Systems, Universityof Tsukuba. He is also a Research Fellow at the Institute of Industrial Science, the University of Tokyo. His
expertise includes dynamic traffic assignment, traffic flow theory, and transportation system design/optimization. He had his Ph.D. in Information Sciences from Tohoku University in 2013.
Daisuke Fukuda is a professor in transportation studies at the University of Tokyo in the department of civilengineering. He has his Ph.D. from the University of Tokyo in 2004 and was an associate professor from2005 to 2020. He conducts research in travel behavior analysis and transportation economics.

13