Transportation Research Part Bakamatsu/Publications/PDF/2015-TRb-8… · tive departure curve fD iðtÞ : t 2 Tg at time t. Furthermore, the state equation can be represented using

Transportation Research Part B 81 (2015) 808–829

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Transportation Research Part B

journal homepage: www.elsevier .com/ locate / t rb

The corridor problem with discrete multiple bottlenecks q

http://dx.doi.org/10.1016/j.trb.2015.07.0150191-2615/� 2015 Elsevier Ltd. All rights reserved.

q Acknowledgements. The authors would like to thank Minoru Osawa for his research assistance on the calculations of the examples in ProposComments from Takamasa Iryo on earlier versions of this paper are greatly appreciated. The authors express their gratitude to three anonymous reftheir careful reading of the manuscript and useful suggestions. The second author was partially supported by JSPS KAKENHI Grant No. 26820207. Tauthor was supported by JSPS KAKENHI Grant No. 26330022.⇑ Corresponding author.

E-mail addresses: [email protected] (T. Akamatsu), [email protected] (K. Wada), [email protected] (S. H

Takashi Akamatsu a, Kentaro Wada b,⇑, Shunsuke Hayashi a

a Graduate School of Information Sciences, Tohoku University, Aramaki Aoba 6-3-09, Aoba-ku, Sendai, Miyagi 980-8579, Japanb Institute of Industrial Science, The University of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo 153-8505, Japan

a r t i c l e i n f o

Article history:Received 15 April 2015Received in revised form 21 July 2015Accepted 21 July 2015Available online 14 August 2015

Keywords:Corridor problemDeparture time choice equilibriumComplementarity problemExistence and uniqueness of equilibrium

a b s t r a c t

This paper presents a transparent approach to the analysis of dynamic user equilibriumand clarifies the properties of a departure-time choice equilibrium of a corridor problemwhere discrete multiple bottlenecks exist along a freeway. The basis of our approach isthe transformation of the formulation of equilibrium conditions in a conventional‘‘Eulerian coordinate system’’ into one in a ‘‘Lagrangian-like coordinate system.’’ This enablesus to evaluate dynamic travel times easily, and to achieve a deep understanding of themathematical structure of the problem, in particular, about the properties of the demandand supply (queuing) sub-models, relations with dynamic system optimal assignment,and differences between the morning and evening rush problems. Building on thesefoundations, we establish rigorous results on the existence and uniqueness of equilibria.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Traffic congestion during morning and evening peak periods causes serious economic losses in most large cities. TheVickrey (1969) bottleneck model was proposed to understand and characterize this problem. It describes commuters’departure-time choice equilibrium at a single bottleneck and has been extensively studied by many researchers (e.g.,Hendrickson and Kocur, 1981; Smith, 1984; Daganzo, 1985). The model has been applied to many problems because ofits simplicity and clarity. For example, it has been used to evaluate traffic policies (e.g., Arnott et al., 1993a; Daganzo andGarcia, 2000; Akamatsu et al., 2006).

If there is more than one bottleneck, there are two streams of research for analyzing user equilibria. The first uses astraightforward generalization of Vickrey’s model, whereas the second considers the dynamic user equilibrium (DUE) ona general network.

In the former stream of research, Kuwahara (1990) and Arnott et al. (1993b) analyzed an user equilibrium in two-tandembottleneck networks, and Lago and Daganzo (2007) studied a similar problem where there are spillover and merging effects.These generalizations can provide valuable insights into the distribution of congestion among commuters with different ori-gins. However, it is almost impossible to obtain an equilibrium in more general cases where a larger number of bottlenecksexist in a freeway corridor because this stream of studies has only used an analytical approach. Indeed, Arnott and DePalma

ition 4.5.erees forhe third

ayashi).

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T. Akamatsu et al. / Transportation Research Part B 81 (2015) 808–829 809

(2011) studied a variant of the general problem, the so-called corridor problem, but they could not provide a completeequilibrium solution.

The latter stream of research applies a more computational approach. Although various mathematical formulations ofDUE have been proposed (see, Szeto and Wong, 2011), they are considerably complex ways to analyze properties of an equi-librium because they must handle the complicated nested structure between link and path travel times in a network.Consequently, many issues remain regarding DUE properties, such as their existence, uniqueness, and stability (Iryo,2013). Moreover, there is no algorithm that ensures convergence to a DUE solution, although many algorithms have beenproposed. This is because most algorithms require monotonicity of the problem to guarantee convergence. However, thisproperty does not hold for DUE problems, as we shall see later.

This paper presents a transparent approach for analyzing DUE and clarifies the properties of a departure-time choiceequilibrium for a corridor problem where discrete multiple bottlenecks exist along a freeway. The basis of our approachis the transformation of the equilibrium conditions from a conventional ‘‘Eulerian coordinate system’’ into a‘‘Lagrangian-like coordinate system.’’ This is valuable for the morning rush (many-to-one travel demand) problem and thereverse problem in the evening (one-to-many travel demand problem). We can use this approach to easily evaluate the tra-vel times without the abovementioned complication. We can then achieve deep insights into the mathematical structure ofthe problem, in terns of the properties of the demand and supply (queuing) sub-models, the relationships with dynamic sys-tem optimal assignment, and the differences between the morning and evening rush problems. Building on these founda-tions, we establish rigorous results on the existence and uniqueness of equilibria.

The remainder of this paper is organized as follows. In Section 2, we describe the dynamic user equilibrium for the cor-ridor problem in an Eulerian coordinate system and discusses why it is difficult to analyze the problem in this form. Section 3contains our reformulation of the equilibrium conditions in a Lagrangian-like coordinate system. To develop an understand-ing of the mathematical properties of the problem, we present a preliminary analysis for the deterministic demand case inSection 4. In Section 5, we examine the existence and uniqueness of equilibria. Section 6 provides numerical examples of theequilibrium flow patterns. Finally, Section 7 concludes the paper.

2. The corridor problem

2.1. Assumptions

Consider a freeway corridor consisting of N on-ramps (origin nodes) and a single off-ramp (destination node). Thesenodes are numbered sequentially from destination node 0 to the most distant on-ramp node N, as shown in Fig. 1. Weassume that there is a single bottleneck i with a finite capacity li in a segment between each pair ði� 1; iÞ of adjacent ramps.Therefore, there are N bottlenecks in the network numbered sequentially ð1;2; . . . ;NÞ from downstream to upstream. At eachbottleneck, a queue can form when the inflow rate exceeds the capacity. The queue evolution and the associated queuingdelay are assumed to be represented by a point queue model (described in 2.2). The distance between each pair ði� 1; iÞof adjacent bottlenecks measured by the free flow travel time is given by ci.

From each on-ramp i ði ¼ 1; . . . ;NÞ;Q i commuters in residential location i ði ¼ 1; . . . ;NÞ enter the network and reach theirdestination during the morning rush-hour T 2 ½0; T�. We assume that the travel demands fQ ig are given constants. Eachcommuter chooses the departure time of her/his trip from home (on-ramp) to workplace (destination) so as to minimizeher/his disutility. This disutility includes travel time, queuing delay at bottlenecks, and a ‘‘schedule delay cost’’ that is asso-ciated with deviation from the wished arrival time to the destination. We assume that all commuters are homogeneous, suchthat they have the same desirable arrival time (work start time) tw at the workplace, the same value of time, and the samepenalty function for the schedule delay. However, for the arrival/departure-time choice principle, we employ a random util-ity model that describes the heterogeneity of users. The main problem considered in this paper is, under the assumptionsmentioned above, to characterize a dynamic user equilibrium distribution of arrivals at the destination where no commutercould reduce her/his disutility by changing arrival times.

2.2. Arrival/departure time choice equilibrium

Under the assumptions above, we formulate a model that describes the equilibrium in the corridor network. At equilib-rium, the following three conditions should hold.

2.2.1. Queuing conditions at each bottleneckWe describe the queuing congestion at each bottleneck in the network using a point queue model, where a queue is

assumed to form vertically at the entrance of each bottleneck. The model can be represented by the following three condi-tions. First, the state equation for the number of users queuing at each bottleneck is

EiðtÞ ¼ AiðtÞ � DiðtÞ; 8t 2 T ; 8i 2 N ; ð2:1aÞ

where AiðtÞ [DiðtÞ] denotes the cumulative number of users who arrive at [depart from] bottleneck i by time t; EiðtÞ is thenumber of users queuing (existing) in bottleneck i at time t, and N � f1; . . . ;Ng. Because a queue has no physical length

Fig. 1. The corridor network with N on-ramps and N bottlenecks.

810 T. Akamatsu et al. / Transportation Research Part B 81 (2015) 808–829

in the point queue model, EiðtÞ is the vertical distance between the cumulative arrival curve fAiðtÞ : t 2 T g and the cumula-tive departure curve fDiðtÞ : t 2 T g at time t. Furthermore, the state equation can be represented using the arrival flow ratekiðtÞ and the departure (exit) rate xiðtÞ at each bottleneck. That is,

1 Theproduce

eiðtÞ � _EiðtÞ ¼ kiðtÞ � xiðtÞ; ð2:1bÞ

where the ‘‘dot’’ denotes the derivative with respect to time t (e.g., _AiðtÞ � dAiðtÞ=dt), and kiðtÞ and xiðtÞ are defined as

kiðtÞ � _AiðtÞP 0 and xiðtÞ � _DiðtÞP 0, respectively.The second condition is the exit flow model, which is the most characteristic of the point queue model. The departure

flow rate from the bottleneck i at time t is

xiðtÞ ¼li if EiðtÞ > 0;min :½kiðtÞ;li� if EiðtÞ ¼ 0;

�8t 2 T ; 8i 2 N ; ð2:2Þ

where li denotes the capacity of bottleneck i, which is a given constant for each bottleneck.The final condition concerns the relationship between the state variables and the queuing delay (the time taken to pass

through the bottleneck). In the point queue model, the queuing delay diðtÞ at bottleneck i for a user arriving at the bottleneckat time t is the horizontal distance between the cumulative arrival curve fAiðtÞ : t 2 T g and the cumulative departure curvefDiðtÞ : t 2 T g. That is,

diðtÞ ¼ EiðtÞ=li 8t 2 T ; 8i 2 N : ð2:3aÞ

Because the number of users queuing in the bottleneck EiðtÞ satisfies (2.1) and (2.2), the queuing delay (2.3a) can also bewritten equivalently as

_diðtÞ ¼ðkiðtÞ=liÞ � 1 if diðtÞ > 0max :½0; ðkiðtÞ=liÞ � 1� if diðtÞ ¼ 0

�: ð2:3bÞ

This implies that the queuing delay in the point queue model should satisfy

_diðtÞ ¼ ðkiðtÞ=liÞ � 1 if diðtÞ > 0;_diðtÞP ðkiðtÞ=liÞ � 1 if diðtÞ ¼ 0;

(; () 0 6 diðtÞ ? _diðtÞ � ½ðkiðtÞ=liÞ � 1�P 0: ð2:3cÞ

In this paper, we employ the complementarity condition (2.3c)1 instead of (2.3b) as the queuing delay model. This is because ithas an advantage over (2.3b) in terms of the time discretization of the model, and its essential features are consistent with theoriginal point queue model. Note that a natural time-discretization scheme for (2.3c) is given by

0 6 diðt þ DtÞ ? ðdiðt þ DtÞ � diðtÞÞ � ððkiðt þ DtÞ=liÞ � 1ÞDt P 0; ð2:3dÞ() diðt þ DtÞ ¼max :½0; diðtÞ þ ððkiðt þ DtÞ=liÞ � 1ÞDt�; ð2:3eÞ

where kiðt þ DtÞ � ðAiðt þ DtÞ � AiðtÞÞ=Dt. It is clear that the discretization scheme above has the advantage that it alwayspredicts a non-negative queuing delay, whereas a naïve discretization scheme for (2.3b) could result in a negative queuingdelays (for the detailed discussion, see Akamatsu, 2001; Ban et al., 2012).

As a final comment on the point queue model, note that the point queue model (i.e., the conditions (2.1), (2.2) and (2.3))implies the First-In-First-Out (FIFO) and flow propagation conditions at each bottleneck. That is,

AiðtÞ ¼ Diðt þ diðtÞÞ 8t 2 T ; 8i 2 N ; ð2:4aÞ() kiðtÞ=xiðt þ diðtÞÞ ¼ 1þ _diðtÞP 0: ð2:4bÞ

It also follows that the queuing conditions (2.1) and (2.2) combined with the FIFO condition (2.4) imply the queuing delay(2.3).

model (2.3c) deviates from the model (2.3b) in the case that diðtÞ ¼ 0. However, the model (2.3c) guarantees a non-negativity of the queuing delay ands a physically meaningful queue dynamics (for the detailed discussions, see Ban et al., 2012).


2.2.2. Flow conservations in the networkThe dynamic traffic flows in the network should satisfy the following two conservation conditions. The first one is that the

origin–destination (OD) travel demands for each origin must be assigned to each time point in the interval T 2 ½0; T�. That is,the time-dependent OD demand should satisfy

2 Recsystem.structur

bQ iðTÞ ¼ Qi 8i 2 N ; ð2:5aÞ

where Q i is the number of users who wish to travel from origin i to the destination (i.e., the total OD demand of origin i to the

destination), which is a given constant for each OD pair, and bQ iðtÞ denotes the cumulative number of users who depart fromorigin node i and arrive at the bottleneck i by time t. This can be equivalently written as
Z T
0qiðtÞdt ¼ Q i; ð2:5bÞ

where qiðtÞ is the arrival rate of the users with origin i at bottleneck i at time t, that is, qiðtÞ ¼ dbQ iðtÞ=dt.The second condition is that the inflow and outflow at each bottleneck at each time point must be equal. That is,

Diþ1ðt � ciþ1Þ þ bQ iðtÞ ¼ AiðtÞ; bQ NðtÞ ¼ ANðtÞ 8t 2 T ; i ¼ 1; . . . ;N � 1; ð2:6aÞ() xiþ1ðt � ciþ1Þ þ qiðtÞ ¼ kiðtÞ; qNðtÞ ¼ kNðtÞ; ð2:6bÞ

where ci is the free-flow travel time from bottleneck i to bottleneck i� 1, which is a given constant.

2.2.3. Equilibrium conditions for arrival/departure-time choiceIn accordance with random utility theory, each user’s disutility is expressed as v iðtÞ þ �iðtÞ, where �iðtÞ denotes the disu-

tility representing the users’ idiosyncratic choices for arrival/departure times, with the distribution of �iðtÞ is assumed to becontinuous; and v iðtÞ denotes the generalized travel cost defined as the sum of the ‘‘schedule cost’’ and the travel time to thedestination. The schedule cost for a user is the cost (penalty) caused by the difference between the user’s desired arrival timetw and the actual arrival time t. The desired arrival time tw is assumed to be the same for all users. The schedule cost (in theunit of travel time) is represented by the function pðtÞ, where t is the destination arrival time. This function is common to allusers, and is assumed to be convex and continuous, following previous studies on the departure time choice equilibrium(e.g., Smith, 1984; Daganzo, 1985; Kuwahara, 1990).

At equilibrium, no one can improve her/his own disutility by changing her/his trip schedule unilaterally. Let piðtÞ be thetravel time from each bottleneck to the destination, for a user arriving at bottleneck i at time t. It follows from the definitionthat the user’s arrival time at the destination is t þ piðtÞ, and the schedule cost for the user is pðt þ piðtÞÞ. Therefore, the equi-librium condition can be expressed as

qiðtÞ ¼ Q ibPiðtÞ t 2 T ; 8i 2 N ; ð2:7aÞbPiðtÞ � Pr½v iðtÞ þ �iðtÞ 6 v iðt0Þ þ �iðt0Þ 8t0 – t 2 T �; ð2:7bÞ

v iðtÞ � pðt þ piðtÞÞ þ piðtÞ; ð2:7cÞ

where bPiðtÞ is the fraction of users arriving at bottleneck i at time t.Note that the equilibrium concept employed in this paper is not the ‘‘reactive user equilibrium’’ but the ‘‘predictive user

equilibrium.’’ That is, travel time piðtÞ is defined as the time that a user arriving at bottleneck i at time t actually experiencesduring the course of a trip to the destination, which can be defined by the following recursive equations (see Fig. 2):

piðtiÞ � pi�1ðti�1Þ ¼ diðtiÞ þ ci; p0ðtÞ ¼ 0 8t 2 T ; 8i 2 N ; ð2:8aÞti�1 � ti ¼ diðtiÞ þ ci 8i 2 N : ð2:8bÞ

2.3. Difficulties of analyzing the problem in an Eulerian coordinate system

The recursive Eq. (2.8a) for the travel time piðtÞ has a complicated nested structure with respect to time (i.e., (2.8) is asystem of recursive equations for piðtÞ with time delays that are state-dependent and time-varying). This causes extreme dif-ficulties in analyzing the properties of the equilibrium conditions. Indeed, even if we are just evaluating piðtÞ, we must eval-uate every arrival time ftj; j ¼ 1; . . . ; i� 1g at the downstream bottlenecks for a user arriving at bottleneck i at time t usingthe recursive Eq. (2.8b). This, in turn, requires to evaluate all the queuing delays fdjðtjÞ; j ¼ 1; . . . ; i� 1g, which aretime-varying and dependent on other state variables. Thus, it is almost impossible to derive theoretical properties of the equi-libria defined by travel time function fpiðtÞg with such troublesome issues.2

ently, Han et al. (2013) proved the existence of the simultaneous route-and-departure choice dynamic user equilibrium within an Eulerian coordinateHowever, their approach may not provide more insights on the equilibrium like those presented in Section 4 because of the complicated nestede between link and path travel times.

Fig. 2. Travel time piðtÞ from each bottleneck/origin to the destination.


One of the reasons for this difficulty is that the problem is formulated in an ‘‘Eulerian coordinate system.’’ The Euleriansystem is not suitable for dealing with the travel time that a user actually experiences during the course of a trip. To explic-itly represent such travel time we must trace the time–space path of each user in the network. To handle such travel time ina simpler and more transparent manner, it is rather natural to employ a ‘‘Lagrangian-like coordinate system,’’ in which wecan easily trace the time–space paths of users. Thus, in the subsequent section, we reformulate the equilibrium conditioninto a Lagrangian-like coordinate system.

3. Reformulation in a Lagrangian coordinate system

In this section, we describe the reformulation of the equilibrium conditions for the corridor problem into aLagrangian-like coordinate system. This formulation is better suited to both the morning-rush problem presented in the pre-vious section and the reverse problem (the evening-rush). After reformulating the morning-rush problem, we brieflydescribe the evening-rush problem by demonstrating their differences.

3.1. Many-to-one travel demand (morning-rush) problem

The equilibrium concept along with the FIFO discipline of the bottleneck model defined in Section 2.2 implies that userswho arrive at the destination at the same time have the same arrival time at any bottleneck on the way to their destination,and that the order of arrival at the destination must be kept at any bottleneck from origins (for the detailed discussions onthis property, see Kuwahara, 1990; Kuwahara and Akamatsu, 1993; Akamatsu and Kuwahara, 1999; Akamatsu, 2001). Usingthis property, we can define the equilibrium arrival time siðsÞ at bottleneck i for a user arriving at the destination at time s,and an object period S 2 ½S; S� for the destination arrival time s that is sufficiently long for all demands to be served in S.Letting wiðsÞ be the queuing delay at bottleneck i for this user, sðsÞ and wðsÞ should satisfy

siðsÞ ¼ si�1ðsÞ � ðwiðsÞ þ ciÞ 8s 2 S; 8i 2 N : ð3:1aÞ

It follows from this and the boundary condition s0ðsÞ ¼ s 8s 2 S (note that s0ðsÞ is the time of arrival at the destination) thatsiðsÞ can be written as an explicit function of s and wðsÞ, that is,

siðsÞ ¼ s�Xi

j¼1

ðwjðsÞ þ cjÞ 8s 2 S; 8i 2 N : ð3:1bÞ

Similarly, we define riðsÞ as the departure time from bottleneck i for a user arriving at the destination at time s, which obvi-ously satisfies (see Fig. 3)

riðsÞ ¼ siðsÞ þwiðsÞ ¼ si�1ðsÞ � ci: ð3:2Þ

In addition to fsðsÞ;wðsÞ : s 2 Sg and frðsÞ : s 2 Sg, which represent the time–space paths of users, it is convenient to definethe arrival flow rate yiðsÞ at bottleneck i for a user arriving at the destination at time s as

yiðsÞ � dAiðsiðsÞÞ=ds ¼ kiðsiðsÞÞ � DsiðsÞ; ð3:3Þ

where D denotes the derivative operation with respect to the destination-arrival time, s (e.g., DsiðsÞ � dsiðsÞ=ds).Based on these variables labeled by the destination-arrival-time s (instead of the time t in the Eulerian coordinate sys-

tem), we can reformulate the equilibrium using the following three conditions.

3.1.1. Queuing conditions at each bottleneckFor users arriving at the destination at time s, the queuing delay wiðsÞ at bottleneck i is

wiðsÞ ¼ diðsiðsÞÞ; 8s 2 S 8i 2 N ; ð3:4aÞ

which implies

DwiðsÞ ¼ _diðsiðsÞÞ � DsiðsÞ: ð3:4bÞ

Fig. 3. Arrival/departure time at bottlenecks for a user arriving at the destination at time s.


Substituting in the queuing delay (complementarity) condition (2.3c) into this, we have

DwiðsÞ ¼ ðyiðsÞ=liÞ � DsiðsÞ if wiðsÞ > 0DwiðsÞP ðyiðsÞ=liÞ � DsiðsÞ if wiðsÞ ¼ 0

�: ð3:5Þ

As shown in Section 2.2, the point queue model defined in (2.1)–(2.3) always satisfies the FIFO conditions (2.4) at eachbottleneck. Similarly, the queuing delay condition (3.5) combined with the definitional Eqs. (3.1) and (3.2) implies theFIFO condition. We can confirm this as follows. For users arriving at bottleneck i at time t ¼ siðsÞ, the FIFO condition canbe written as

AiðsiðsÞÞ ¼ DiðriðsÞÞ; 8s 2 S; 8i 2 N ; ð3:6aÞor yiðsÞ ¼ xiðriðsÞÞ � DriðsÞ: ð3:6bÞ

Recalling the definitions (3.1) and (3.2) of sðsÞ and rðsÞ, we have that DsðsÞ and DrðsÞ are

DriðsÞ ¼ DsiðsÞ þ DwiðsÞ;DsiðsÞ ¼ Dsi�1ðsÞ � DwiðsÞ; Ds0ðsÞ ¼ 1: ð3:7Þ

Substituting these into (3.6b), we have yiðsÞ ¼ xiðsiðsÞ þwiðsÞÞ � Dsi�1ðsÞ, which reduces to

yiðsÞ ¼ liDsi�1ðsÞ if wiðsÞ > 0yiðsÞ 6 liDsi�1ðsÞ if wiðsÞ ¼ 0

�ð3:8aÞ

oryiðsÞ ¼ liDriðsÞ if wiðsÞ > 0yiðsÞ 6 liDriðsÞ if wiðsÞ ¼ 0

�: ð3:8bÞ

It is clear that the queuing delay condition (3.5) combined with (3.7) implies the FIFO condition (3.8), and that the FIFO con-dition (3.8) combined with (3.7) implies the queuing delay condition (3.5). Thus, (3.5) and (3.8) are mutually interchangeablewhen formulating the equilibrium condition.

3.1.2. Flow conservations in the networkThe flow conservation (2.5) for each OD pair in Section 2.2 can be represented as

Q iðSÞ ¼ bQ iðsiðSÞÞ ¼ Qi 8i 2 N ; ð3:9aÞ

where Q iðsÞ is the cumulative number of users who depart from origin node i and arrive at the destination by time s, which

implies bQ iðsiðsÞÞ ¼ QiðsÞ. Letting qiðsÞ be the arrival flow rate at the destination defined by

qiðsÞ ¼ DbQ iðsiðsÞÞ ¼ qiðsiðsÞÞ � DsiðsÞ;

(3.9a) can be equivalently represented as

Q i ¼Z S

SqiðsÞds 8i 2 N : ð3:9bÞ

The flow conservation (2.6) for each bottleneck can also be rewritten in terms of the destination arrival time s. Substituting(3.1), (3.2), and the FIFO condition (3.6a) into the conservation Eq. (2.6a) at t ¼ siðsÞ yields

Aiþ1ðsiþ1ðsÞÞ þ Q iðsÞ ¼ AiðsiðsÞÞ; QNðsÞ ¼ ANðsNðsÞÞ 8s 2 S; i ¼ 1; . . . ;N � 1; ð3:10aÞ

which is equivalent to

qiðsÞ ¼ yiðsÞ � yiþ1ðsÞ; qNðsÞ ¼ yNðsÞ 8s 2 S; i ¼ 1; . . . ;N � 1: ð3:10bÞ

3.1.3. Equilibrium conditions for arrival/departure-time choiceFor a user with origin i who arrives at the destination at time s, the schedule cost is given by pðsÞ, and the travel time is

s� siðsÞ. Therefore, the equilibrium condition can be expressed as


qiðsÞ ¼ Q iPiðsÞ s 2 S; 8i 2 N ; ð3:11aÞPiðsÞ � Pr½v iðsÞ þ ~�iðsÞ 6 v iðs0Þ þ ~�iðs0Þ 8s0 – s 2 S�; ð3:11bÞv iðsÞ � pðsÞ þ ðs� siðsÞÞ; ð3:11cÞ

where ~�iðsÞ denotes the disutility representing the users’ idiosyncratic taste for arrival times, and PiðsÞ is the fraction of userswith origin i who arrive at the destination at time s. No assumption of a specific functional form of PiðsÞ is needed in our proofof uniqueness for the DUE. However, we use the logit choice function to derive a sufficient condition for the existence of theDUE and for the numerical examples in the later sections. That is,

PiðsÞ � expð�hv iðsÞÞ=Z S

Sexpð�hv iðsÞÞds; ð3:12Þ

where h is the inverse of the variance of the idiosyncratic tastes, which implies that the assumption of the ~�iðsÞ are i.i.d.Gumbel distributions. As h! þ1 (i.e., ~�iðsÞ ! 0), the logit type demand condition reduces to that of the homogeneous user(or deterministic demand) case,

pðsÞ þ ðs� siðsÞÞ ¼ qi if qiðsÞ > 0;pðsÞ þ ðs� siðsÞÞP qi if qiðsÞ ¼ 0;

�8s 2 S; 8i 2 N ; ð3:13Þ

where qi denotes the equilibrium disutility for a user with origin i.

3.2. One-to-many travel demand (evening-rush) problem

In the evening-rush problem, the corridor network defined in Section 2.1 is used by commuters in reverse. That is, thereare N off-ramps (destination nodes) and a single on-ramp (origin node). These nodes are numbered sequentially from originnode 0 to the most distant off-ramp node N. Q i commuters in residential location i depart from the origin during evening rushhour, and exit from the network at off-ramp i (i.e., one-to-many travel demand). Each commuter chooses the departure timeof her trip from the workplace (origin) to home (destination) so as to minimize her disutility. The only difference in theassumptions of two problems is that, in the evening-rush, commuters are assumed to experience a schedule delay cost asso-ciated with deviation from wished departure time from the origin à la de Palma and Lindsey (2002). Under these assumptions,the one-to-many travel demand (evening-rush) problem describes a DUE distribution of departures from the origin such thatno commuter could reduce her disutility by changing her departure time.

Now, let us formulate the equilibrium conditions for the one-to-many model in a Lagrangian-like coordinate system. In asimilar manner to the many-to-one model, we can define the equilibrium arrival time riðsÞ at destination i for a user depart-ing from the origin at time s. In this network topology, this is also equal to the departure time from bottleneck i for the user. Theperiod S � ½S; S� of the origin departure time s is sufficiently long so that all demands can be served in S. Letting wiðsÞ be thequeuing delay at bottleneck i for this user, we see that rðsÞ and wðsÞ should satisfy (see Fig. 4)

riðsÞ ¼ ri�1ðsÞ þ ðwiðsÞ þ ciÞ 8s 2 S; 8i 2 N : ð3:14aÞ

It follows from this and the boundary condition r0ðsÞ � s (i.e., r0ðsÞ is defined as the departure time from the origin) thatriðsÞ can be written as an explicit function of s and wðsÞ:

riðsÞ ¼ sþXi

j¼1

ðwjðsÞ þ cjÞ 8s 2 S; 8i 2 N : ð3:14bÞ

The arrival time siðsÞ at bottleneck i for a user departing from the origin at time s also satisfies

siðsÞ ¼ riðsÞ �wiðsÞ ¼ ri�1ðsÞ þ ci:

We also define yiðsÞ as the arrival flow rate at bottleneck i for a user departing from the origin at time s and qiðsÞ as the depar-ture flow rate from the origin at time s.

We can now reformulate the equilibrium for the one-to-many model using these variables labeled by theorigin-departure-time s. This formulation can be obtained by slightly modifying the equilibrium conditions for the

Fig. 4. Arrival/departure time at bottlenecks for a user departing from the origin at time s.


many-to-one model, although the definitions (i.e., reference time) of the variables of the two models are different.Specifically, the queuing condition has exactly the same form as (3.8b). Additionally, for the flow conservations in the net-work, the condition for each OD pair and the condition at each bottleneck have exactly the same forms as (3.9b) and (3.10b),respectively. For the departure-time choice, we can obtain the equilibrium condition for the one-to-many model by replacingthe travel time ðs� siðsÞÞ in (3.11) and (3.13) by ðriðsÞ � sÞ.

4. Preliminary analysis

This section presents a preliminary analysis for the deterministic demand (homogeneous users) case. Although it is aparticular case of the general demand condition, the analysis for this case gives several new insights for understandingthe mathematical structure of the problem and the properties of the basic components of the underlying problem(i.e., the queuing and the demand models).

4.1. Equivalent linear complementarity problems

In contrast to the previous section, we begin with analyzing the One-to-Many model (DUE-E), and show that the modelcan be cast into a standard linear complementarity problem. We then apply a similar analysis to the Many-to-One model(DUE-M), which reveals that DUE-M has a slightly more complicated structure than DUE-E.

The equilibrium problem [DUE-E] for the One-to-Many model with homogeneous users consists of five conditions (i.e.,

(3.14), (3.8b), (3.9b), (3.10b), and (3.13)) and associated unknowns fwðsÞ ¼ ½w1ðsÞ; . . . ;wNðsÞ�T ; sðsÞ ¼ ½s1ðsÞ; . . . ; sNðsÞ�T ;qðsÞ ¼ ½q1ðsÞ; . . . ; qNðsÞ�

T; yðsÞ ¼ ½y1ðsÞ; . . . ; yNðsÞ�

T : s 2 Sg and q ¼ ½q1; . . . ;qN�T . However, they are somewhat redundant.

Note that the flow conservation holds as an equality between yðsÞ and qðsÞ, i.e.,

NyðsÞ ¼ �qðsÞ:

Here, N denotes the reduced link-node incidence matrix for the corridor network, in which a row corresponding to the des-tination node is deleted, and is invertible. That is,

N ¼

�1 1. .

. . ..

�1 1�1

266664377775() N�1 ¼ �

1 � � � � � � 11 � � � 1

. .. ..

.

1

266664377775 ¼ �LT ;

where L is a lower triangular matrix with all non-zero entries equal to 1. This implies that N is an M-matrix, and hence, yðsÞcan be uniquely determined from qðsÞ:

yðsÞ ¼ �N�1qðsÞ ¼ LT qðsÞ: ð4:1Þ

Note that the non-negativity condition yðsÞP 0 is ‘‘automatically’’ satisfied if qðsÞP 0. Hence, by substituting (4.1) into(3.8b), we can eliminate yðsÞ from the problem [DUE-E]. Similarly, the travel time rðsÞ � rðsÞ � s for each OD pair and thequeuing delay wðsÞ at each bottleneck always satisfy

�NT rðsÞ ¼ wðsÞ þ c; ð4:2Þ

where rðsÞ ¼ ½r1ðsÞ; . . . ;rNðsÞ�T ;1 ¼ ½1; . . . ;1�T ; s ¼ s1, and c ¼ ½c1; . . . ; cN�T . Because �NT is an M-matrix, rðsÞ can be uniquelydetermined from wðsÞ:

rðsÞ ¼ �ðNTÞ�1ðwðsÞ þ cÞ ¼ LðwðsÞ þ cÞ; ð4:3Þ

where the non-negativity condition r P 0 is satisfied if wðsÞ þ c P 0. It follows that changes in the departure time from eachbottleneck DrðsÞ can be represented by DwðsÞ. That is,

DrðsÞ ¼ 1� ðNTÞ�1DwðsÞ ¼ 1þ LDwðsÞ; ð4:4Þ

where DrðsÞ ¼ ½Dr1ðsÞ; . . . ;DrNðsÞ�T and DwðsÞ ¼ ½Dw1ðsÞ; . . . ;DwNðsÞ�T .Thus, we can obtain a more concise expression for the problem [DUE-E] by eliminating these redundant variables

fyðsÞ;rðsÞg. This, together with the observation that [DUE-E] only consists of linear equations and linear complementarityconditions for the homogeneous users case, leads to the following proposition:

Proposition 4.1. The arrival/departure-time choice equilibrium problem [DUE-E] for the corridor network is equivalent to thefollowing linear complementarity problem (LCP): Find fqðsÞ;wðsÞ : s 2 Sg and q such that

0 6 qðsÞ ? pðsÞ1þ LðcþwðsÞÞ � q P 0; 8s 2 S; ð4:5aÞ0 6 wðsÞ ? CDrðsÞ � LT qðsÞP 0; 8s 2 S; ð4:5bÞ


0 6 q ?Z S

SqðsÞds� Q P 0; ð4:5cÞ

where DrðsÞ � 1þ LDwðsÞ;rðsÞ � LðwðsÞ þ cÞ þ s;C 2 RN�N is a diagonal matrix whose i-th diagonal element designates the

capacity li of bottleneck i, and Q ¼ ½Q 1; . . . ;Q N�T .

Proof. It is obvious that [DUE-E] implies the LCP above. Hence, it is sufficient for us to prove that the LCP implies [DUE-E].The proof is by contradiction. Suppose, on the contrary, that there is a flow pattern fqjðsÞg that satisfies the complementaritycondition (4.5) but does not satisfy the equality (3.9b) for some j. That is,

qj �Z S

SqjðsÞds� Q j

!¼ 0; qj P 0; and

Z S

SqjðsÞds� Q j > 0:

It then follows that qj ¼ 0 and qjðsÞ > 0 for this j and some s because Qj > 0. For these particular s and j with positive flow

qjðsÞ > 0, the equilibrium condition (4.5) reduces to pðsÞ þ ½LðwðsÞ þ cÞ�j ¼ 0. However, ½LðwðsÞ þ cÞ�j ¼P j

i¼1ðwiðsÞ þ ciÞ > 0because c > 0 and wðsÞP 0. This contradicts the assumption that pðsÞP 0. h

In what follows, we restrict ourselves to the cases in which the arrival time at the destination s is discretized. That is, theunderlying time period s is divided into a finite number of intervals, K, labeled s 2 S � f1;2; . . . ;Kg, which have a finite dura-tion ds. We use this restriction because rigorous discussion of existence and uniqueness issues for continuous time modelswould lead us into unnecessary mathematical complications, and such digression would obscure the essential structure ofthe model.

In accordance with the discretization of s, the time-dependent variables fqðsÞ : s 2 Sg; fwðsÞ : s 2 Sg, and fDwðsÞ : s 2 Sg,and the schedule delay function fpðsÞ : s 2 Sg are represented as finite dimensional column vectors:

q � ½qðsÞ�Ks¼1 2 RNK ; w � ½wðsÞ�Ks¼1 2 RNK ; Dw � ½DwðsÞ�Ks¼1 2 RNK ; and p � ½pðsÞ1�Ks¼1 2 RNK :

Here, DwðkÞ 2 RN should be defined as the backward-difference:

DwðsÞ � wðsÞ �wðs� 1Þ for s ¼ 1; . . . ;K; and Dwð0Þ � 0;

because the discretized formulation of the queuing condition (4.5b) should correspond to the discretization scheme (2.3d) ofthe point queue model in Section 2. Note that the variable Dw 2 RNK defined above can be represented by Dw � ½DK � I�w,where � denotes the Kronecker product, I is the N by N identity matrix, and DK is the K by K matrix representing the firstorder backward difference operator:

DK ¼

1�1 1

. .. . .

.

�1 1

266664377775:

In this discrete time setting, the Proposition 4.1 immediately implies the following result.

Proposition 4.2A (One-to-Many model). The DUE problem for the One-to-Many model in the discretized time period S can berepresented as a standard finite dimensional linear complementarity problem:

½DUE-E-LCP� Find X � ½q;w;q�T such that 0 6 X ? FðXÞ �MXþ b P 0;

where M 2 Rð2Kþ1ÞN�ð2Kþ1ÞN and b 2 Rð2Kþ1ÞN are defined as

ð4:6Þ

and IK � diag½1; . . . ;1� 2 RK�K ; 1K � ½1; . . . ;1�T 2 RK ; and Q � ½Q 1=ds; . . . ;Q N=ds�T :
We then proceed to the analysis of the Many-to-One model, in which the reduced incidence matrix N and its inverse are
given by

N ¼

1 �1. .

. . ..

1 �11

266664377775() N�1 ¼

1 � � � � � � 11 � � � 1

. .. ..

.

1

266664377775 ¼ LT :


In exactly the same manner as the One-to-Many model, we can eliminate yðsÞ from the problem [DUE-M] using

yðsÞ ¼ N�1qðsÞ ¼ LT qðsÞ:

Similarly, the travel time sðsÞ � s� sðsÞ for each OD pair and the queuing delay wðsÞ at each bottleneck always satisfy

NT sðsÞ ¼ wðsÞ þ c;

where sðsÞ ¼ ½s1ðsÞ; . . . ; sNðsÞ�T . Because NT is an M-matrix, sðsÞ can be uniquely determined from wðsÞ:

sðsÞ ¼ ðNTÞ�1ðwðsÞ þ cÞ ¼ LðwðsÞ þ cÞ;

where the non-negativity condition sðsÞP 0 is satisfied if wðsÞ þ c P 0. It follows that changes to the departure time fromeach bottleneck DrðsÞ can also be represented by DwðsÞ. That is,

DrðsÞ � DsðsÞ þ DwðsÞ ¼ 1� DsðsÞ þ DwðsÞ ¼ 1þ ½I� L�DwðsÞ;

where DsðsÞ ¼ ½Ds1ðsÞ; . . . ;DsNðsÞ�T . Thus, by eliminating these redundant variables fyðsÞ; sðsÞg, we can represent the equilib-rium conditions for the Many-to-One model as a linear complementarity problem that has exactly the same form as inProposition 4.1. Similarly, in the discrete time setting, we obtain the linear complementarity problem:

½DUE-MðBÞ-LCP� Find X � ½q;w;q�T such that 0 6 X ? FðXÞ �MXþ b P 0;

where M 2 Rð2Kþ1ÞN�ð2Kþ1ÞN and b 2 Rð2Kþ1ÞN are defined as ð4:7Þ

Although this LCP is a natural and concise expression of the DUE for the Many-to-One model, some caution is needed. Forthe queuing model to be self-consistent, the cumulative arrival curve at each bottleneck should not be backward-bending. Inour formulation, this condition is expressed as DsiðsÞP 0 8i 2 N ;8s 2 S, which trivially holds for the [DUE-E-LCP] of theOne-to-Many model. Indeed, this is true because the queuing condition in the LCP ensures that DrðsÞP C�1LT qðsÞP 0always hold, and Ds is given from the origin to destinations as

Dr0ðsÞ � Ds1ðsÞ � 1; DriðsÞ � Dsiþ1ðsÞ � 1þ ðDw1ðsÞ þ � � � þ DwiðsÞÞ for i ¼ 1; � � � ;N:

The queuing condition of [DUE-M(B)-LCP], however, does not ensure DsNðsÞP 0 for the most upstream bottleneck. This isbecause Ds is defined from the destination to origins in a ‘‘backward’’ manner (i.e., the direction opposite to the directionof travel). That is,

Dr1ðsÞ � Ds0ðsÞ ¼ 1; DriðsÞ � Dsi�1ðsÞ � 1� ðDw1ðsÞ þ � � � þ DwiðsÞÞ for i ¼ 2; . . . ;N:

Since the complementarity queuing condition of [DUE-M(B)-LCP] ensures DrðsÞP 0, the condition DsiðsÞP 0 holds fori ¼ 0;1; . . . ;N � 1, but the condition for DsNðsÞ is missing. Thus, for the complete description of the problem, we should adda ‘‘consistency condition’’ to [DUE-M(B)-LCP]:

DsNðsÞ ¼ 1� ðDw1ðsÞ þ � � � þ DwNðsÞÞ; and DsNðsÞP 0: ð4:8Þ

As an alternative to the ‘‘backward type’’ formulation in which information on DsNðsÞ is missing from the complementarityqueuing condition, we can also consider another (‘‘forward type’’) type of formulation. The new formulation first representsthe arrival time siðsÞ at each bottleneck from the most upstream bottleneck to the destination in a forward direction,

siðsÞ ¼ sNðsÞ þ ðwNðsÞ þ � � � þwiþ1ðsÞÞ for i ¼ 0;1; . . . ;N � 1:

This implies

DriðsÞ � Dsi�1ðsÞ ¼ DsNðsÞ þ ðDwNðsÞ þ � � � þ DwiðsÞÞ 8i 2 N :

Substituting this into the queuing condition (4.5b), we can derive a forward type representation of the queuing modelthat explicitly includes DsNðsÞ.

0 6 wðsÞ ? C½DsNðsÞ1þ LTDwðsÞ� � LT qðsÞP 0:

Finally, we must have s0ðsÞ ¼ s and sNðsÞP 0 for this formulation to be consistent with the definition of the destination arri-val time based variables. This reduces to

DsNðsÞ þ ðDwNðsÞ þ � � � þ Dw1ðsÞÞ ¼ 1; and DsNðsÞP 0: ð4:9Þ

The following proposition summarizes the above discussion:


Proposition 4.2B (Many-to-One model). The DUE problem for the Many-to-One model in the discretized time period S can berepresented as the following two types of complementarity problems:

ðaÞ Backward type : ½DUE-MðBÞ-LCP� and the consistency condition ð4:8Þ:ðbÞ Forwardtype : ½DUE-MðFÞ-MCP� and the consistency condition DsN P 0

½DUE-MðFÞ-MCP� Find X � ½q;w;q�T and DsN such that

0 6 X ? FðXÞ � MXþ b P 0; and DsN þ ðDwN þ � � � þ Dw1Þ ¼ 1

where M 2 Rð2Kþ1ÞN�ð2Kþ1ÞN and b 2 Rð2Kþ1ÞN are defined as

ð4:10Þ

and DsN � ½CDsNð1Þ1; . . . ;CDsNðKÞ1�T 2 RNK :

4.2. Connections with dynamic system optimal assignment

In the equivalent LCPs in Propositions 4.2A and 4.2B, the matrix M has a skew-symmetric structure. Based on this prop-erty, we can convert the DUE problem into a variational inequality problem with only cost variables ðwðsÞ;qÞ.

Proposition 4.3. The problem [DUE-M(B)-LCP] is equivalent to the following variational inequality problem (VIP):

½DUE-VIP� Find ðw�ðsÞ;q�Þ 2 X such thatXK

s¼1

ðwðsÞ �w�ðsÞÞ � Dr�ðsÞ � ðq� q�Þ � Q P 0 8ðwðsÞ;qÞ 2 X

where Dr�ðsÞ � 1þ ½I� L�Dw�ðsÞ;X � fðwðsÞ;qÞP 0 : pðsÞ1þ LðwðsÞ þ cÞ � q P 0 8s 2 Sg:

It is clear from the Proposition 4.3 that the above variational inequality problem can be associated with linear program-ming (LP) problems:

Corollary 4.3. If the value of fDr�ðsÞ : s 2 Sg at equilibrium is known in advance, the problem [DUE-VIP] reduces to the followingparametric linear programming problem [DUE-PLP-D]:

minðw;qÞP0

:FPLP�D �XK

s¼1

wðsÞ � CDr�ðsÞ � q � Q subject to pðsÞ1þ LðwðsÞ þ cÞ � q P 0 8s 2 S

and its dual problem [DUE-PLP-P] is given by

minqP0

:FPLP�P �XK

s¼1

ðpðsÞ1þ LcÞ � qðsÞ

subject to CDr�ðsÞ � LT qðsÞP 0 8s 2 S; andXK

s¼1

qðsÞ � Q ¼ 0:

A few remarks are in order. First, for the special case of N ¼ 1 (i.e., a network with a single bottleneck) for theMany-to-One model, the equivalent variational inequality problem in Proposition 4.3 always reduces to the LP inCorollary 4.3 without any assumptions on Dr�. Specifically, the vector Dr� for this case reduces to Dr1ðsÞ � ds0ðsÞ=ds, andDr1ðsÞ � 1 always holds because r1ðsÞ � s0ðsÞ � s, by definition. It follows that the equilibrium solution of the problem[DUE] for N ¼ 1 is obtained by simply solving the LP with unknowns fw1ðsÞg and q1. This result is consistent with the findingof Iryo and Yoshii (2007). Second, note that [DUE-PLP-P] can be interpreted as a particular type of dynamic system optimalassignment. The OD flow pattern fqðsÞg is controlled so as to minimize the total disutility of the commuters in the network,which consists of the total schedule delay pðsÞ1 � qðsÞ and the total free flow travel time Lc � qðsÞ. The constraints are the phys-ical conditions that dynamic flows in the corridor network should satisfy. The second constraint is the conservation of the ODdemand. The first constraint concerns the traffic capacity, and prevents the queues build up at each bottleneck. We can inter-pret this as the inflow rate LT qðsÞ ¼ yðsÞ is always less than CDr�ðsÞ, so the capacity is given by CDr�ðsÞ rather than the actualcapacity C1. Finally, the solution fqðsÞg of the problem [DUE-PLP-P] is not necessarily unique and can be a convex set, whichimplies that the equilibrium OD flow pattern for the DUE problem with the deterministic demand model (i.e., the solutions ofthe LCPs in Propositions 4.2A and 4.2B) is not necessarily unique, even if the equilibrium cost pattern fwðsÞg is unique.


4.3. Properties of the queuing models

For the One-to-Many model, the queuing sub-model is given by

0 6 wðsÞ ? CDrðsÞ � LT qðsÞP 0 8s 2 S;

where DrðsÞ � 1þ LDwðsÞ. This can be interpreted as an implicit functional relationship wðqÞ from q to w. More specifically,suppose that an OD flow pattern q 2 RNK is given. Then, the corresponding queuing delay pattern w 2 RNK is given as thesolution of the following LCP:

½PQ-E� 0 6 w ? ½DK � CL�wþ b P 0; where b � 1K � C1� ½IK � LT �q:

By inspecting this LCP, we see that the function wðqÞ has the following useful property.

Proposition 4.4A (One-to-Many queuing model). For a given OD flow vector q 2 RNK , the solution w 2 RNK (i.e., the queuingdelays) of the problems [PQ-E] is uniquely determined. Furthermore, the solution wðqÞ is Lipschitz continuous with respect tochanges in q.

Proof. It is well known in the theory of linear complementarity problems that if M is a P-matrix then the LCP0 6 X ? MXþ b P 0 has a unique solution, and the solution is Lipschitz continuous with respect to changes in b(Mangasarian and Shiau, 1987). Therefore, it suffices for us to prove that M � ½DK � CL� is a P-matrix. Note that M is a lowertriangular matrix, in which all diagonal entries flig are positive. This implies that all the eigenvalues of M and its principalsubmatrices are positive, and hence, M is a P-matrix. h

Similarly, we can analyze the properties of the queuing condition for the Many-to-One model, which has two types of for-mulation (i.e., backward and forward). For each formulation, DrðsÞ is defined as

Backward type : DrðsÞ � 1þ ½I� L�DwðsÞForward type : DrðsÞ � DsNðsÞ1þ LTDwðsÞ:

Thus, in the backward type formulation, the queuing delay pattern w for a given OD flow pattern q is the solution to thefollowing LCP:

½PQ-MðBÞ� 0 6 w ? ½DK � C½I� L��wþ b P 0; where b � 1K � C1� ½IK � LT �q:

For the forward type formulation, suppose that an OD flow pattern q and a parameter vector DsN for the consistency con-dition are given. Then, the corresponding queuing delay pattern w is the solution to the following LCP:

½PQ-MðFÞ� 0 6 w ? ½DK � CLT �wþ b P 0; where b � DsN � ½IK � LT �q:

From these LCP representations, we have the following proposition concerning the properties of the Many-to-One queuingmodel.

Proposition 4.4B (Many-to-One queuing model). For a given OD flow vector q 2 RNK and a parameter vector DsN 2 RK for theconsistency condition of the forward-type formulation, the solution w 2 RNK (i.e., the queuing delays) of the problem [PQ-M(F)] isuniquely determined, whereas the solution to the problem [PQ-M(B)] is not necessarily unique. Furthermore, the solution w 2 RNK

to the problem [PQ-M(F)] is Lipschitz continuous with respect to changes in q and DsN.

Proof. As in the proof of Proposition 4.4A, it suffices to show that the LCP mapping M for [PQ-M(F)] is a P-matrix, and that itis not for [PQ-M(B)]. For the problem [PQ-M(F)], M � ½DK � CLT � is a block lower triangular matrix in which each diagonalblock CLT is an upper triangular matrix with positive diagonal entries flig. This implies that all the eigenvalues of M andits principal submatrices are positive, and hence, that M is a P-matrix. For the problem [PQ-M(B)], M � ½DK � C½I� L�� is alower triangular matrix with diagonal entries equal to zero, which implies that all the eigenvalues of M are zero, and hence,that M is a P0-matrix but is not a P-matrix. h

A few remarks are in order here. First, Propositions 4.4A and 4.4B state some positive aspects (uniqueness and Lipschitzcontinuity) of the function wðqÞ, but they do not imply stronger properties such as monotonicity (that has been the abusedassumption in the dynamic traffic assignment literature). Indeed, the function wðqÞ derived from the queuing model is notgenerally monotone. Second, these propositions show that the Many-to-One model is not exactly the reverse of theOne-to-Many model. Indeed, the One-to-Many model always yields a unique cost pattern wðqÞ for a given flow pattern q,but the Many-to-One model cannot give such a tight relationship between q and w. Although the forward type formulation[PQ-M(F)] admits a unique solution, it requires information on not only the flow pattern q but also DsN for the mostupstream bottleneck.


4.4. Non-monotonicity of the LCP mapping

Propositions 4.4A and 4.4B reveal that the matrices DK � CL and DK � CLT for the queuing sub-model are P-matrices, so itis reasonable to expect that the matrices M for the overall equilibrium conditions [DUE-E-LCP] or [DUE-M-LCP] also havesuch a useful property. The following proposition, however, reveals that this conjecture is not true.

Proposition 4.5. Each mapping FðXÞ �MXþ b of the LCPs, [DUE-E-LCP] and [DUE-M-LCP], in Propositions 4.2A and 4.2B is notmonotone. Furthermore,

(a) the One-to-Many model: FðXÞ is not a P-function in general,(b) the Many-to-One model: FðXÞ is not a P0-function.

Proof. If FðXÞ is monotone, then it implies that FðXÞ is a P-function (but the converse is not true). Hence, it is sufficient toshow a counter example for the matrix M being P-matrix (or P0-matrix). For this, we show analytically that some of the prin-cipal minors of M can be negative even in a small sized problem of N ¼ K ¼ 3.

(a) the One-to-Many model: for a ¼ f5;7;12;14;16g;detðMðaÞÞ ¼ ð2l1 � l2Þl2 < 0 if l2 > 2l1

(b) the Many-to-One model:

ðBackward typeÞ for a ¼ f3;4;8;12;14;18;21g;detðMðaÞÞ ¼ �l22 < 0;

ðForward typeÞ for a ¼ f2;4;9;12;14;18;21g;detðMðaÞÞ ¼ �l22 < 0;

where a#S � f1; . . . ;Kg denotes an index set of the matrix M;a is the complement of a, and MðaÞ ¼ ½Mij� denotes aprincipal submatrix of M in which i 2 a and j 2 a. h

The above proposition implies that it is very difficult to prove the uniqueness of the equilibria of [DUE-E] or [DUE-M] forthe homogeneous user case. One reason for this difficulty seems to lie in the demand condition, in which q is not a smoothfunction of w but is just an upper hemi-continuous correspondence (recall that [DUE-PLP-P] in Corollary 4.3 does not nec-essarily have a unique solution). To obtain sharper results, we will examine the uniqueness of the equilibria for a generalizedcase of heterogeneous users, which we describe in the next section.

5. Existence and uniqueness

5.1. Existence

To establish the existence of equilibria for the problem [DUE], we characterize the equilibrium as a fixed point problemconsisting of the demand-side and supply-side conditions discussed in the previous sections. The demand side condition forthe problem [DUE] can be generally represented for both cases of the deterministic demand (or homogeneous users) and thestochastic demand (or heterogeneous users) as follows.

q ¼ DðwÞ; ð5:1Þ

where DðwÞ is the set of time-dependent OD flow patterns that can be generated from the departure time choice modelwhen the queuing delay pattern is w. Specifically, the demand function D : Y ! X is defined in (3.11) for the heterogeneoususers case, and is defined in the complementarity condition (3.13) for the homogeneous users case. X denotes the set of fea-sible OD flow patterns generated from a total OD demand Q , which is defined as X � fq :

PKs¼1qðsÞ ¼ Q ; and q P 0g. The set

X is non-empty, compact and convex for finite Q . Y is the set of queuing delay patterns w determined from the supply side(queuing) condition mentioned below.

The supply side condition and the equilibrium of the problem [DUE] should be described separately for the One-to-Manymodel and the Many-to-One model because the two models have subtle differences (as discussed in the previous section).

5.1.1. One-to-Many modelFor the One-to-Many model, the supply side condition can be generally expressed as

w ¼WðqÞ: ð5:2Þ

As shown in Proposition 4.4A, the function W : X ! Y is defined as the solution of the linear complementarity problems[PQ-E] for a given OD flow pattern q is given. It is a many-to-one mapping and is Lipschitz continuous with respect tochanges in q. This, combined with the boundedness and convexity of the set X implies that Y is a non-empty, compactset. Combining (5.1) and (5.2), the equilibrium flow pattern q� of [DUE-E] for the One-to-Many model can be defined asthe solution of the following fixed point problem:


q� ¼ DðWðq�ÞÞ 2 X ð5:3Þ

and the equilibrium queuing delay patterns w� are given by Wðq�Þ.By applying Kakutani’s fixed point theorem (Kakutani, 1941), we have the following existence result:

Theorem 5.1A. For the One-to-Many DUE model, the equilibrium ðq�;w�Þ defined in (5.3) exists for both cases of heterogeneoususers and homogeneous users.

Proof. The mapping DðwÞ is (a) upper hemi-continuous (for the proof, see for example, Daganzo, 1983; Mas-Colell et al.,1995) for the homogeneous users case, and (b) continuous for the heterogeneous users case. It sends every w into a closedand convex set in X. Because the mapping WðqÞ is Lipschitz continuous, the composition DðWðqÞÞ of DðwÞ and WðqÞ sharesthe properties DðwÞ for all q for which WðqÞ is Lipschitz continuous. Thus, (5.3) defines an upper hemi-continuous mappingswithin X that is non-empty, compact and convex set. Then, Kakutani’s fixed point theorem guarantees the existence of q�. h

5.1.2. Many-to-One modelFor the Many-to-One model, we can similarly define the supply side condition as

w ¼Wðq;DsNÞ; ð5:4Þ

where we employ the forward type formulation (or the LCP [PQ-M(F)]) discussed in 4.3, and DsN should satisfy (4.9), whichcan be written as

DsN ¼ LðwÞ � 1�XN

i¼1

Dwi: ð5:5Þ

Thus, the equilibrium for the Many-to-One model can be defined as the following fixed point problem with respect toq ¼ ½q;DsN�:

q� ¼ ½DðWðq�ÞÞ; LðWðq�ÞÞ� 2 bX ð5:6Þ

and the equilibrium queuing delay pattern w� is given by Wðq�Þ, where bX ¼ X � T , and T denotes the feasible set of DsN ,which is defined as T ¼ fDsN : 0 6 DsN 6 1bg for some finite constant b.

The assumption that the set T is bounded from above (i.e., DsN 6 1b) is always satisfied for the queuing model (and finiteQ ) in this paper.

Proposition 5.1. For a finite OD demand pattern Q and a positive capacity pattern l – 0, the upper bound of DsN is at most1þ

PNi¼1Pi

j¼NQj=li.

Proof. From the definition of the queuing model, we have

DsiðsÞ ¼ Dsi�1ðsÞ � DwiðsÞ and � DwiðsÞ ¼ wiðs� 1Þ �wiðsÞ 6 wiðs� 1Þ 6Xi

j¼N

Q j=li:

This implies DsiðsÞ 6 Dsi�1ðsÞ þPi

j¼NQ j=li for i ¼ N; . . . ;1. ‘‘Solving’’ the recursive inequalities in backward direction fromi ¼ N to 1, we obtain

DsNðsÞ 6 DsN�1ðsÞ þ Q N=lN 6 DsN�2ðsÞ þ Q N=lN þXN�1

j¼N

Q j lN�1 6 � � � 6 Ds0ðsÞ þXN

i¼1

Xi

j¼N

Qj

, ,li ¼ 1þ

XN

i¼1

Xi

j¼N

Qj

,li

6 b 8s 2 S: �

For the lower bound of the set T (or the consistency condition DsN P 0), we require the following assumption regarding theschedule delay function.

Assumption 5.1B. There exists an appropriately defined schedule delay function pðsÞ for which the equilibrium of (5.6)always satisfies DsNðsÞP 0 8s 2 S.

Although it is difficult to give a necessary and sufficient condition for this assumption to be true in the problem with gen-eral demand functions, we can provide a sufficient condition for some typical demand functions.

Proposition 5.2. Suppose that the schedule delay function pðsÞ satisfies the following condition

for ðaÞ deterministic demand function case : DpðsÞP �1 8s 2 S; ð5:7aÞfor ðbÞ logit demand function case : DpðsÞP �1þ ð1=hÞ 8s 2 S: ð5:7bÞ


Then, Assumption 5.1B always holds at equilibrium.

Proof. The logit demand model always yields an interior solution with respect to q, so the equilibrium condition for userswith origin N can be written as

ðs� sNðsÞÞ þ pðsÞ þ ð1=hÞ ln qNðsÞ � qN ¼ 0: 8s 2 S:

This implies that

DsNðsÞ ¼ 1þ DpðsÞ þ �ðsÞ 8s 2 S; ð5:8Þ

where �ðsÞ � 1h

lnqNðsÞ

qNðs� 1Þ

� �¼ 1

hln 1þ DqNðsÞ

qNðs� 1Þ

� � 1

hDqNðsÞ

qNðs� 1Þ ¼1h

qNðsÞqNðs� 1Þ � 1� �

> �1h:

Taking the limit as h! þ1 of (5.8), we can also obtain DsNðsÞ ¼ 1þ DpðsÞ for the deterministic demand case. Thus, the con-dition (5.7) implies DsNðsÞP 0. h

Based on the fixed point problem (5.6), we can obtain the following existence result for the Many-to-One model:

Theorem 5.1B. For the Many-to-One DUE model, if Assumption 5.1B holds, the equilibrium ðq�;w�Þ defined in (5.6) exists for bothcases of heterogeneous users and homogeneous users.

Proof. The proof is almost the same as that of the One-to-Many model. The difference is that WðqÞ is replaced with WðqÞ,and the mapping LðWðqÞÞ is introduced. The mapping WðqÞ is Lipschitz continuous with respect to q because DsN and q arelinear terms constituting the constant vector b of the LCP [PQ-M(F)] (see Propositon4.4B). Using this, and because LðwÞ is alinear mapping of w, we find that the composition mapping LðWðqÞÞ is Lipschitz continuous with respect to q. It also follows

that ðDðWðqÞÞ; LðWðqÞÞÞ is a mapping from bX to bX itself, because Proposition 5.1 and Assumption 5.1B imply that LðWðqÞÞsends every q into the closed and convex set T. Thus, (5.6) defines an upper hemi-continuous mapping within bX that isnon-empty, compact and convex set. Then, Kakutani’s fixed point theorem guarantees the existence of q�. h

The following remarks conclude this subsection. First, condition (5.7a) in the Many-to-One model is consistent with thesufficient condition for the existence of equilibria in Vickrey’s bottleneck model (Smith, 1984). This condition limits thedecreasing rate of schedule cost function and prevents the cumulative arrival curve at the most upstream bottleneck frombeing backward-bending. Second, for the One-to-Many model, we do not need a similar condition on the schedule cost toguarantee the existence of equilibria. The reason for this difference in the two models can be clearly understood from struc-tures of queuing models discussed in Section 4.3. It is also consistent with the finding of de Palma and Lindsey (2002) for asingle bottleneck.

5.2. Uniqueness

As shown in the previous section, it is difficult to prove the uniqueness of the solution of [DUE] for the homogeneoususers case because the equivalent LCPs do not have monotonicity or P-property. In this subsection, we examine the unique-ness of the equilibria for the heterogeneous users case. Specifically, we assume that the demand condition is given by (3.11),which is based on the random utility theory. Note that we do not impose any particular type of functional form (such as logitor probit) for the choice function in the analysis below.

For the convenience of exposition, we slightly change the notation in this subsection: qi 2 RK and wi 2 RK ði ¼ 1; . . . ;NÞare column vectors with elements qiðsÞ and wiðsÞ ðs ¼ 1; . . . ;KÞ, respectively; q 2 RNK and w 2 RNK denote column vectorswith block elements qi and wi ði ¼ 1; . . . ;NÞ, respectively.

5.2.1. One-to-Many modelThe demand-side condition for the heterogeneous users defined in (3.11) is represented as qi ¼ Q iPiðviðwÞÞ, where viðwÞ

is a vector (for i th origin) with elements v iðsÞ �Pi

j¼1ðwjðsÞ þ cjÞ þ pðsÞ ðs ¼ 1; . . . ;KÞ;Pi : RNK ! RK is a column vector

½Pið1Þ; . . . ; PiðKÞ�T of the choice probability function PiðsÞ : RNK ! R, which is defined in (3.11b). This can be written as the fol-lowing complementarity condition:

0 6 q ? q� QPðwÞP 0; ð5:9Þ

where P : RNK ! RNK is a column vector ½P1; . . . ;PN �T , and Q � diag½Q1IK ; . . . ;Q NIK � 2 RNK�NK . The supply-side (queuing) con-dition for the One-to-Many model can be written as

0 6 w ? ½CL � DK �wþ C1� 1K � ½LT � IK �q P 0: ð5:10Þ


Thus, we can put the demand–supply equilibrium condition in a single nonlinear complementarity problem (NCP).

Proposition 5.3A (One-to-Many model). The problem [DUE-E] for the One-to-Many model with heterogeneous users isequivalent to the following NCP, and has a solution.

[DUE-E-NCP]: Find X � ½q;w�T 2 R2NK such that

ð5:11Þ

We prove the uniqueness of the solution of [DUE-E-NCP] by using the following lemma, which is based on thePoincaré-Hopf’s index theorem and the theory of complementarity problems.

Lemma 5.1. (Mas-Colell, 1979; Kolstad and Mathiesen, 1987; Simsek et al., 2007): For a nonlinear complementarity problem:

Find x such that 0 6 x ? FðxÞP 0; ð5:12Þ

where F : RNþ ! RN is continuously differentiable, define N by N ¼ f1; . . . ;Ng;BðxÞ ¼ fi 2 N jxi > 0g; JBðF;xÞ to the principal sub

matrix of the Jacobian matrix of F corresponding to the indices of BðxÞ, and detðJBðF;xÞÞ its determinant. If BðxÞ ¼£, then definedetðJBðF;xÞÞ ¼ 1. Suppose a solution to the problem (5.12) exists, and at each solution x�i to the NCP that strict complementaritycondition holds (i.e., x�i ¼ 0 implies Fiðx�Þ > 0). Then (1) if we have detðJBðF;xÞÞ > 0 for all solutions to the NCP then there is pre-cisely one solution; and conversely. (2) If there is only one solution to the NCP then detðJBðF;xÞÞP 0.

In order to apply this lemma to [DUE-E-NCP], we first need the Jacobian matrix of GðXÞ. A straightforward calculationleads to the following lemma:

Lemma 5.2A. The Jacobian matrix $GðXÞ of GðXÞ in [DUE-E-NCP] is given by

ð5:13Þ

where H � diag½Q 1H1; . . . ;Q NHN �;Hi � ½hiðs; tÞ� is the Jacobian matrix of the choice probability function PiðviðwÞÞ where the ðs; tÞelement is hiðs; tÞ � @PiðsÞ=@v iðtÞ; ðs ¼ 1; . . . ;K; t ¼ 1; . . . ;KÞ.

In examining the properties of $GðXÞ, the following lemmas are useful.

Lemma 5.3(1). The Jacobi matrix HiðviðwÞÞ is negative semi-definite 8vi 2 RK and is negative definite on the tangent space TX offPi : 1T Pi ¼ 1;Pi P 0g.

Proof. See for example, Sandholm (2010, Chap.6, pp.213). h

Lemma 5.3(2). The matrices L and DK are positive definite, and their eigenvalues are all equal to þ1.

Proof. xT Lx ¼PN

i¼1xiPN

j¼1xj ¼ ð1=2ÞPN

j¼1xj

� �2þ ð1=2Þ

PNj¼1x2

j > 0 8x 2 RN – 0

xTDK x ¼PK

k¼1ðxk � xk�1Þxk ¼ ð1=2ÞPK

k¼1ðxk � xk�1Þ2 > 0 8x 2 RK – 0.

Because both of L and DK are lower triangular matrices with all the diagonal entries being 1, their eigenvalues also allequal to 1. h

Now, we can examine the positivity of the determinant of $GðXÞ.

Lemma 5.4A. detð$GðXÞÞ ¼ detðA�HÞ; where A � ðLTÞ�1C� DK .

Proof. detð$GðXÞÞ ¼ detðCL � DK � ðLT � IKÞHðL � IKÞÞ

¼ detðLT � IKÞ � detð½LT � IK ��1½CL � DK �½L � IK ��1 �HÞ � detðL � IKÞ

¼ detðLT � IKÞ � detððLTÞ�1C� DK �HÞ � detðL � IKÞ:

It follows from Lemma 5.3(2) that detðLT � IKÞ ¼ detðL � IKÞ ¼ þ1, and hence,

detð$GðXÞÞ ¼ detððLTÞ�1C� DK �HÞ ¼ detðA�HÞ: �


Lemma 5.5A. The determinant of $GðXÞ is positive for all X P 0.

Proof. Since the matrix A�H is a block upper triangular matrix, in which each diagonal block is liDK � Q iHi, the determi-

nant of A�H is given byQN

i¼1 detðliDK � Q iHiÞ. Note here that each of the matrix liDK � Q iHi 8i 2 N is positive definitebecause DK is positive definite (Lemma 5.3(2)), and Hi 8i 2 N are negative semi-definite (Lemma 5.3(1)), which implies thatkkðliDK � Q iHiÞ > 0 8k; i , where kkðMÞ denotes k th eigenvalue of a matrix M. Thus,

$GðXÞ ¼YN

i¼1

detðliDK � Q iHiÞ ¼YNi¼1

YK

k¼1

kkðliDK � Q iHiÞ > 0: � ð5:14Þ

By extending the above analysis to the case of Jacobian matrices at equilibrium points, we obtain the uniqueness resultfor the One-to-Many model.

Theorem 5.2A. For the One-to-Many DUE model with heterogeneous users, the equilibrium defined in (5.11) is unique.

Proof. See Appendix A. h

5.2.2. Many-to-One modelThe demand condition of the Many-to-One model can be represented in the same manner with that of the One-to-Many

model, but the supply-side condition of the Many-to-One model is slightly different (see the discussions in Section 4), and itis given by

0 6 w ? ½CLT � DK �wþ C1� DsN � ½LT � IK �q P 0; ð5:15aÞDsN þ ðDwN þ � � � þ Dw1Þ ¼ 1K and DsN P 0: ð5:15bÞ

Thus, if we impose Assumption 5.1B so that the consistency condition DsN P 0 is ‘‘automatically’’ satisfied at equilibrium,the problem [DUE-M] for the Many-to-One model with heterogeneous users is equivalent to the following mixed comple-mentarity problem (MCP):

[DUE-M-MCP]: Find X � ½q;w�T 2 R2NK and DsN such that

0 6 X ? GSðq;wÞ � GDðwÞ þ GsðDsNÞP 0; ð5:16aÞDsN þ ðDwN þ � � � þ Dw1Þ ¼ 1K ; ð5:16bÞ

To prove the uniqueness by using Lemma 5.1, it is convenient to convert the problem into a standard NCP. For this pur-pose, we convert the consistency condition (5.15b) into the following pair of complementarity conditions:

0 6 x ? 1K � ðzþ DK w1 þ � � � þ DK wNÞP 0 and 0 6 z ? gðx; zÞP 0; ð5:17Þ

where z � DsN;gðx; zÞ ¼ ½gðxð1Þ; zð1ÞÞ; . . . ; gðxðKÞ; zðKÞÞ�T , and g : R2þ ! R is a continuous and smooth function of x and z that

satisfies

@g=@x > 0 8x P 0; @g=@z > 0 8z P 0; argx:fgðx; zÞ ¼ 0g > 0; 8z P 0; argz:fgðx; zÞ ¼ 0g > 0;8x P 0:

A simple example of such a function is gðx; zÞ � f 1ðzÞ � f 2ðxÞ, where f 1ðzÞ � lnðzþ 1Þ; f 2ðxÞ � ð1=xÞ. These properties of gimply that ðx; zÞ > 0 as well as the equality condition ðDK w1 þ � � � þ DK wNÞ þ z ¼ 1K are always maintained in the comple-mentarity system (5.17). Thus, we have the following equivalent NCP for the Many-to-One model.

Proposition 5.3B (Many-to-One model). If Assumption 5.1B holds, the problem [DUE-M] for the Many-to-One model withheterogeneous users is equivalent to the following standard NCP, and has a solution.

[DUE-M-NCP]: Find X � ½q;w�T 2 R2NK and Y � ½x; z�T 2 R2K such that

0 6XY

� �? GðX;YÞ �

GSðq;wÞ � GDðwÞ þ G12YG21Xþ G2ðYÞ

� �P 0 ð5:18Þ


Note that if Assumption 5.1B does not hold, there is a possibility that the solution to the complementarity condition (5.17)cannot be satisfied, which implies that the solution to [DUE-M-NCP] does not exist even if the solution to [DUE-M-MCP]exists.

The positivity of the determinant of the Jacobian of the problem [DUE-M-NCP] can be examined by the similar manner tothat in the One-to-Many model.

Lemma 5.2B. The Jacobian matrix $GðX;YÞ of GðX;YÞ in [DUE-M-NCP] is given by

Lemma 5.4B. detð$GðX;YÞÞ ¼ detðA�HÞ ¼ detðB11Þ � detðB22Þ

Lemma 5.5B. The determinant of $GðX;YÞ is positive for all ðX;YÞP 0 if an arbitrary pair of element ðqNðsÞ;wNðsÞÞ of ðqN;wNÞ isfixed (or deleted from the unknown variables).

Proof. Because detð$GðX;YÞÞ ¼ detðA�HÞ ¼ detðB11Þ � detðB22Þ, we show that detðB11Þ > 0 and detðB22Þ > 0 in turn. Thematrix B11 is a block upper triangular matrix, in which each diagonal block is a positive definite matrix liþ1DK � Q iHi.Hence, detðB11Þ reduces to the products of eigenvalues kkðliþ1DK � Q iHiÞ > 0 8k; i:

detðB11Þ ¼YN�1

i¼1

detðliþ1DK � Q iHiÞ ¼YN�1

i¼1

YK

k¼1

kkðliþ1DK � Q iHiÞ > 0:

As for the matrix B22, a simple calculation yields

detðB22Þ ¼ gx � detð�Q NHNÞ;

where gx �Q@gðx; zÞ=@xjx¼xiðsÞ;z¼ziðsÞ > 0 8ðx; zÞP 0. Since the demand side Jacobian HN is negative definite on the tangent

space TX (see Lemma 5.3(2)), deleting a single row and column of HN guarantees that detðB22Þ ¼ gx � detð�Q NHNÞ > 0. h

Comparing these lemmas and those for the One-to-Many model, we see first that the main difference is concerning theequilibrium conditions for the most upstream bottleneck. Specifically, the submatrix B22 of the Jacobian $GðX;YÞ stems fromthe equilibrium condition for the most upstream (N th) bottleneck/Origin, which corresponds to the last (N th) diagonalblock, lNDK � QNHN , of the matrix A�H in the One-to-Many model. Second, the determinant of the matrix B22 for theMany-to-One model reduces to the determinant of the demand side matrix �Q NHN only, while the determinant oflNDK � Q NHN for the One-to-Many model is decided from two terms stemming from the supply-side condition ðlNDKÞand the demand-side condition (�QNHN). This reflects the fact that the equilibrium condition of the Many-to-One modelis lacking in the queuing condition for the most upstream bottleneck. This fact together with the positive semi-definitenessof �HN (i.e., its rank is K � 1) also leads to the indeterminacy of the ðqN;wNÞ, and the requirement of deleting a single pair ofelements ðqNðsÞ;wNðsÞÞ from ðqN ;wNÞ in Lemma 5.5B; in the One-to-Many model, the possible indeterminacy from thesemi-definiteness of HN is recovered by the positive definiteness of the supply-side term lNDK . Finally, the possible indeter-minacy of the ðqN;wNÞ in the Many-to-One model may not be a serious problem in practice because it can be easily avoidedby taking the assignment time period S long enough, and setting, say, qNð1Þ ¼ wNð1Þ ¼ 0; furthermore, it cannot occur ifthere is at least a single time period s at which the most upstream bottleneck is not active (i.e., wNðsÞ ¼ 0) at equilibrium,which may be often the case.

Now, we are in a position to state the uniqueness theorem for the Many-to-One model.

Theorem 5.2B. For the Many-to-One DUE model with heterogeneous users, suppose that Assumption 5.1B holds. Then theequilibrium ðq�;w�Þ defined in (5.18) is unique, if an arbitrary pair of element ðqNðsÞ;wNðsÞÞ of ðqN ;wNÞ is fixed (or deleted fromthe unknown variables).

Fig. 5. An equilibrium flow pattern for the Many-to-One model with the desired arrival time tw ¼ 40 (first row: aggregate cumulative curves; second-fourthrows: disaggregate cumulative curves for each origin).


Proof. The difference between the Jacobians of the One-to-Many model and the Many-to-One model is just theblock-submatrix B22 discussed above. Since this submatrix cannot be affected by the operations deleting the rows and col-umns of the Jacobian matrix corresponding to BðxÞ ¼ fi 2 N jxi > 0g in Lemma 5.1 (recall that ðx; zÞ cannot be zero from thedefinition of the function g), the proof is almost the same as that of Theorem 5.1A, and omitted here. h

6. Numerical examples

This section presents numerical examples of an equilibrium flow pattern for each model by solving the problems[DUE-E-NCP] and [DUE-M-MCP]. For the computation of the equilibrium, we use the ReSNA (Regularized SmoothingNewton Algorithm)3 that was originally developed for solving the second-order cone complementarity problems(SOCCP) (Hayashi et al., 2005). Because the SOCCP involves the NCP as a subclass, ReSNA is also applicable to NCP in a directmanner. In Hayashi (2015), the global convergence of the algorithm is proved under the (Cartesian) P0 assumption, which isa natural extension of P0 assumption for NCP or MCP (a weaker condition than monotonicity).4 Although our problemsmay not be P0, our experiments have successfully obtained equilibrium solutions in most cases when the logit parameter his not so high.

We considered a corridor network with the number of bottlenecks N ¼ 3 and employed a piecewise linear schedule delayfunction with early [late] penalty parameter 0:5 and late [early] penalty parameter 2:0 for the Many-to-One model[One-to-Many model]. We used the logit choice function (i.e., the discrete time version of (3.12)) with the parameterh ¼ 60. The free flow travel times, capacities of bottlenecks, and total demands were give by c ¼ 0;l ¼ ½30;20;10�T , and

Q ¼ ½100;200;300�T , respectively. The number of assignment intervals was 60, which was sufficiently long for all thedemands to be served in assignment period S.

The results for these numerical examples are shown in Figs. 5 and 6. These figures illustrate the aggregate (first rows) anddisaggregate cumulative curves (second-fourth rows). Each disaggregate curve represents the origin-specific (ordestination-specific) cumulative number of users at each bottleneck. The aggregate curve at each bottleneck is the sum ofthe disaggregate curves at the same bottleneck. The red and blue lines represent the cumulative arrival and departure curves,

3 Website of ReSNA: http://www.plan.civil.tohoku.ac.jp/opt/hayashi/ReSNA/.4 In Hayashi et al. (2005), the quadratic convergence of the algorithm is also proved under monotonicity assumption.

http://www.plan.civil.tohoku.ac.jp/opt/hayashi/ReSNA/

Fig. 6. An equilibrium flow pattern for the One-to-Many model with the desired departure time tw ¼ 20 (first row: aggregate cumulative curves; second-fourth rows: disaggregate cumulative curves for each destination).


respectively. The vertical lines in the figures for bottleneck 1 represent the desired arrival [departure] time at the destination[origin], which is common to all users in the Many-to-One [One-to-Many] model.

From these figures, it is clear that equilibrium flow patterns for the two models have different characteristics even thoughthe situations are only the reverse of each other. Specifically, for the Many-to-One model, the disaggregate arrival and depar-ture patterns among different origins are very different, but the patterns for the One-to-Many model are not significantlydifferent for different destinations. This seems to be related to the difference between two models discussed inSection 4.3. Future work should investigate this in detail, along with other characteristics of the equilibrium flow patterns.

7. Concluding remarks

This paper considered a departure-time choice equilibrium for a corridor problem where discrete multiple bottlenecksexist along a freeway. We first transformed the equilibrium conditions for Many-to-One and One-to-Many models fromthe conventional Eulerian coordinate system to a Lagrangian-like coordinate system. This approach allowed us to achieve adeep understanding of the mathematical structure of the problem. Specifically, we analyzed the equilibrium for the homo-geneous users case and revealed the following properties: (1) the supply (queuing) sub-model of the Many-to-One modelhas a slightly more complicated structure than that of the One-to-Many model, although the demand sub-models of bothmodels have same structure; (2) the mappings of the equivalent linear complementarity problems of both models are notmonotone in general, which implies that the uniqueness of the equilibria is not necessarily guaranteed for the homogeneoususers case. We then examined the existence and uniqueness of equilibria for heterogeneous users case and obtained the fol-lowing results: (3) equilibria exist in both models for the case of homogeneous users and heterogeneous users case, but theMany-to-One model requires an assumption on the schedule penalty function; (4) the equilibria in both of the models for thecase of the heterogeneous users are unique. Finally, we gave numerical examples of the equilibrium flow pattern of eachmodel.

Although we focused on the theoretical properties of equilibria for the corridor problem due to space limitations, a sys-tematic numerical experiment is required to characterize the equilibrium flow and cost patterns. The algorithm used inSection 6 does not give guaranteed convergence, but our uniqueness result supports the case for proceeding with such anumerical experiment because we should, at most, find one equilibrium solution. Such an experiment is currently being con-ducted by the authors, and the outcome will be reported in the near future.


Appendix A. Proof of Theorem 5.2A

We prove the uniqueness of the solution of [DUE-E-NCP] by using Lemma 5.1. Since the equilibrium q� of [DUE-E-NCP] isalways an interior solution, it is sufficient for us to consider the set of time periods during which equilibrium queuing delayw� is positive. For this, we first define a subset Si of the index set S � f1; . . . ;Kg by SiðwiÞ ¼ fs 2 S jwiðsÞ > 0gði ¼ 1; . . . ;NÞ,and the associated matrix IðSiðwiÞ;SÞ 2 RKi�K that is a submatrix of IK consisting of the set of rows with indices in SiðwiÞ,where Ki � jSij. Let BðwÞ be the union of fSiðwiÞ : i ¼ 1; . . . ;Ng;BðwÞ ¼

SNi¼1SiðwiÞ, its associated matrix BðwÞ be

BðwÞ � diag½IðS1ðw1Þ;SÞ; . . . ; IðSNðwNÞ;SÞ� 2 Rm�n; where m �XN

i¼1

Ki; n � NK;

and JBðGðq�;w�ÞÞ be the principal submatrix of the Jacobian of Gðq�;w�Þ corresponding to the indices of BðwÞ. We also use thefollowing abbreviation of the notation for the simplicity of exposition:

Iði; �Þ � IðSiðwiÞ;SÞ; Ið�; iÞ � Iði; �Þ ¼ IðS;SiðwiÞÞ;Iði; jÞ � IðSiðwiÞ;SÞ IðS;SjðwjÞÞ � Iði; �Þ Ið�; jÞ:

Under this setting, the Jacobian matrix JBðGðq�;w�ÞÞ can be written as

JBðGðq�;w�ÞÞ �IK

B

� �$Gðq�;w�Þ

IK

B

� �T

where A22 ¼ BðCL � DKÞBT ;A21A12 ¼ BðLT � IKÞHðL � IKÞBT . Hence, detðJBÞ is given by

detðJBÞ ¼ detðA11ÞdetðA22 � A21A�111 A12Þ ¼ detðA22 � A21A12Þ;

¼ detðBðCL � DKÞBT � BðLT � IKÞHðL � IKÞBTÞ;� detðJðSÞB � JðDÞB Þ;

where JðSÞB � BðCL � DKÞBT , and JðDÞB � BðLT � IKÞHðL � IKÞBT . In order to simplify the demand-side matrix JðDÞB , we first repre-

sent the principal submatrix of the Jacobian Hðw�Þ corresponding to SiðwiÞ as Hi � Iði; �ÞHiIð�; iÞ ði ¼ 1; . . . ;NÞ. Then the

matrix JðDÞB can be written as

JðDÞB � BðLT � IKÞHðL � IKÞBT ¼ LT SHST L ¼ LT HL;

where LT is a block upper triangular matrix in which ði; jÞ block is Iði; jÞ and S � diag½Ið1; �Þ; . . . ; IðN; �Þ�. We used the identityIði; iÞIði; �Þ ¼ Iði; �Þ; Iði; iÞIði; jÞ ¼ Iði; jÞ, and Iði; jÞIðj; �Þ ¼ Iði; �Þ if Si #Sj. Note here that the inverse of the block upper triangular

matrix LT can be given as an block upper triangular (and double diagonal) matrix in which each diagonal block is Iði; iÞ andði; jÞ block is �Iði; jÞ if j ¼ iþ 1, zero otherwise, because Iði; j� 1ÞIðj� 1; jÞ ¼ Iði; jÞIðj; jÞ ði < jÞ always hold. By using this, we

can diagonalized the matrix JðDÞB as follows:

ðLTÞ�1

JðDÞB L�1 ¼ SHST ¼ H;

which implies that detðJðSÞB � JðDÞB Þ is simplified as

detðJðSÞB � JðDÞB Þ ¼ detðLT ½ðLTÞ�1ðJðDÞB � JðSÞB ÞL�1�LÞ;

¼ detðLTÞdetððLTÞ�1

JðSÞB L�1 � HÞdetðLÞ;

¼ detððLTÞ�1

JðSÞB L�1 � HÞ:

Here the last equality follows from the fact detðLÞ ¼ detðLTÞ ¼ þ1 because LT is a block upper triangular matrix with all diag-

onal entries equal to þ1. Corresponding to this diagonalization, we needs to calculate ðLTÞ�1

JðSÞB L�1 to evaluate detðJðSÞB � JðDÞB Þ.By defining Dði; jÞ � Iði; �ÞDK Ið�; jÞ, we see that the supply-side matrix JðSÞB � BðCL � DKÞBT reduces to a block lower triangular

matrix whose ði; jÞ block is liDði; jÞ. Then, a straightforward calculation reveals that ðLTÞ�1

JðSÞB L�1 � H is also a block lower tri-

angular matrix in which each diagonal block is liDði; iÞ. Therefore, each diagonal block of ðLTÞ�1

JðSÞB L�1 � H is

liDði; iÞ � QiHi ¼ Iði; �ÞðliDK � Q iHiÞIð�; iÞ, and hence, detðJðSÞB � JðDÞB Þ is given by the products of detðIði; �ÞðliDK �HiÞIð�; iÞÞ.Note here that each of the matrix Iði; �ÞðliDK � Q iHiÞIð�; iÞ is positive definite for any choice of the set Si because DK is positivedefinite (Lemma 5.3(2)), and Hi ði ¼ 1; . . . ;NÞ are negative semi-definite (Lemma 5.3(1)), and these properties are inherited to


their principal submatrices. This implies that all the eigenvalues of Iði; �ÞðliDK �HiÞIð�; iÞ are positive. Thus, we can concludethat for BðwÞ corresponding to all solutions of [DUE-E-NCP],

detðJBðGðq�;w�ÞÞÞ ¼ detðJðSÞB � JðDÞB Þ ¼YN

i¼1

detðIði; �ÞðliDK � Q iHiÞIð�; iÞÞ > 0;

and Lemma 5.1 guarantees that the solution is unique. h

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Transportation Research Part Bakamatsu/Publications/PDF/2015-TRb-8… · tive departure curve fD iðtÞ : t 2 Tg at time t. Furthermore, the state equation can be represented using