2011 Elsevier Ltd. All rights reserved.

k hasditiontraveical w969),

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0191-2615/$ - see front matter 2011 Elsevier Ltd. All rights reserved.

Corresponding author. Tel.: +1 951 234 2190.E-mail addresses: richard.arnott@ucr.edu (R. Arnott), edepalma@math.ucr.edu (E. DePalma).

1 Tel.: +1 951 827 1581.2 The authors are indebted to Caltrans and the USDOT for research award UCTC FY 09-10, awarded under Caltrans for transfer to USDOT. Arnott would like to

thank seminar participants at U.B.C., Dong Hua University, Kyoto University, the University of Tokyo, and the Free University of Amsterdam, as well as sessionparticipants at the 2004 and 2008 AEA Meetings for helpful remarks. Yasushi Atsumi, Malachy Carey, Marvin Kraus, Stephen Ross, and above all Ian Sue Wing,provided especially useful comments, either at presentations or on early versions of the paper. He would like to thank Danqing Hu, Junling Wang, XiaomengYang and Li Zeng for excellent research assistance.

Transportation Research Part B 45 (2011) 743768

Contents lists available at ScienceDirect

Transportation Research Part B

journal homepage: www.elsevier .com/ locate/ t rbdoi:10.1016/j.trb.2011.01.004by visualizing a departure rate surface over a metropolitan area. What does it look like at a point in time, and how does itchange over the rush hour? Similarly, what do the ow, density, and velocity surfaces look like, and how do they evolve?

This paper takes a modest step forward in examining the spatial equilibrium dynamics of rush-hour congestion. It laysout perhaps the simplest possible model with continuous time and space that can address the issue. The metropolitan areais modeled as a single trafc corridor of uniform width joining the suburbs to the central business district (CBD), a point inspace; the population entering each point along the corridor over the rush hour is taken as given; except for their locations,vehicles are identical, contributing equally to congestion, having a common work start-time, and having a common trip pricefunction that is linear in travel time and schedule delay; congestion takes the form of classic ow congestion; and there is1. Introduction

In recent years,2 considerable worcentral feature is the trip-timing conwhere trip price includes the cost ofand the toll, if applicable. The theoretbehind a single bottleneck (Vickrey, 1taining a bottleneck.

While insightful, the work does nbeen done examining the equilibrium dynamics of rush-hour trafc congestion. The, that no vehicle can experience a lower trip price by departing at a different time,l time, the cost of traveling at an inconvenient time (termed schedule delay cost),ork on the topic has been in the context of Vickreys model of a deterministic queuewith some papers treating extensions to very simple networks, with each link con-

vide much insight into the spatial dynamics of rush-hour trafc congestion. StartThe corridor problem: Preliminary results on the no-toll equilibrium

Richard Arnott a,1, Elijah DePalma b,aDepartment of Economics, University of California, Riverside, Riverside, CA 92506, United StatesbDepartment of Statistics, University of California, Riverside, CA 92506, United States

a r t i c l e i n f o

Article history:Received 13 January 2010Received in revised form 16 January 2011Accepted 16 January 2011

Keywords:Morning commuteCongestionCorridorEquilibrium

a b s t r a c t

Consider a trafc corridor that connects a continuum of residential locations to a point cen-tral business district, and that is subject to ow congestion. The population density func-tion along the corridor is exogenous, and except for location vehicles are identical. Allvehicles travel along the corridor from home to work in the morning rush hour, and havethe same work start-time but may arrive early. The two components of costs are traveltime costs and schedule delay (time early) costs. Determining equilibrium and optimumtrafc ow patterns for this continuous model, and possible extensions, is termed TheCorridor Problem. Equilibria must satisfy the trip-timing condition, that at each locationno vehicle can experience a lower trip price by departing at a different time. This paperinvestigates the no-toll equilibrium of the basic Corridor Problem.

but be

744 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768There are good reasons to believe that the Corridor Problem is important. On the practical side, solving the problemwouldprovide a point of entry to understanding the spatial dynamics of rush-hour trafc congestion, which is surely important inthe enlightened design of road and mass transit networks. On the theoretical side, the problem has posed a stumbling blockto the development of three lines of theoretical literature on the economics of trafc congestion. During the 1970s severalpapers were written on the economics of trafc congestion in the context of the monocentric city and related models (Solowand Vickrey, 1971; Solow, 1972; Kanemoto, 1976; Arnott, 1979), assuming that trafc ow is constant over the day. Theirfocus was on second-best issues, in particular on how the underpricing of urban auto congestion distorts land use and affectscapacity investment rules. Are the insights from that literature substantially modied when account is taken of the ebb andow of trafc? At around the same time, Beckmann and Puu (e.g., Beckmann and Puu, 1985) started work on two-dimen-sional, steady-state continuous ow models of trafc congestion. Solving the Corridor Problem might provide insight intohow to extend their work to non-stationary trafc ow. Later, Arnott et al. (1994) attempted to generalize the bottleneckmodel to a trafc corridor, modeled as a series of bottlenecks with entry points between them. Because of the models lin-earity, the solution degenerated into the treatment of multiple cases, the number rising geometrically with the number ofbottlenecks. Thus, despite its elegant simplicity in other contexts, the bottleneck model does not appear well suited to exam-ining the spatial dynamics of trafc congestion.

There is some prior work on the equilibrium spatial dynamics of urban trafc congestion. In the context of the monocen-tric model, Yinger (1993) assumed that vehicles at the urban boundary are the rst to depart and depart together, and arefollowed by successive cohorts from increasingly more central locations, and solved for the implied spatial dynamics of con-gestion over the rush hour. Ross and Yinger (2000) proved that the departure pattern assumed in Yinger (1993) does notsatisfy the trip-timing condition, and that no other simple departure pattern does either. In earlier work, Arnott (2004) con-jectured an equilibrium departure pattern but, since he was unable to prove his conjecture, investigated a discretized variantof the problem, with buses and bus stops, termed the bus-corridor problem. Congestion takes the form of bus speed vary-ing inversely with the number of passengers. The numerical examples of the bus-corridor problem presented there are con-sistent with the form of departure set conjectured for the corridor problem proper, but do not prove the conjecture since thediscretization alters the problem. Tian et al. (2007) derive the equilibrium properties of a variant of the bus-corridor problemin which congestion takes the form of crowding costs that increase in the number of passengers, provide some solution algo-rithms, and present numerical examples. The numerical examples of this variant of the bus-corridor problem are also con-sistent with the form of the departure set conjectured for the Corridor Problem proper in Arnott (2004), but again do notprove the conjecture because the problem is somewhat different.

Section 2 presents the basic model and states the problem. Section 3 derives some implications of the trip-timing condi-tion. Section 4 states the heuristic reasoning underlying an initial proposed solution. Section 5 undertakes the mathematicalanalysis of the initial proposed solution, in the process demonstrates that the initial proposed solution is not consistent inone respect with the trip-timing condition, and modies the proposed solution. Section 6 develops an algorithm to solvenumerically for a departure pattern consistent with the modied proposed solution. The algebraic calculations used to derivethe results in Section 6 are placed in the Appendix at the end of the paper. Section 7 presents the results of the numericalalgorithm and a graph of an observed commuting pattern on a Los Angeles freeway. Section 8 concludes.

2. Model description

Consider a trafc corridor of constant width that connects a continuum of residential locations, the suburbs, to a pointcentral business district (CBD) that lies at the eastern end of the corridor, as shown in Fig. 1. Location is indexed by x, thedistance from the outer boundary of residential settlement towards the CBD, that is located at x. N(x)dx denotes the exoge-neous number of vehicles departing between x and x + dx over the rush hour. It is assumed that N(x) is strictly positive forx 2 0; x.

A glossary listing all the notation used in the paper is located at the end of the paper, following the references.

2.1. Trip cost

Each morning all vehicles travel from their departure location to the CBD and have the common desired arrival time, t.Late arrivals are not permitted, and in the absence of a toll the common travel cost function is

C atravel time btime early;where a is the value or cost of travel time, and b is the value or cost of time early. It is assumed that a > b, which is supportedby empirical evidence (Small, 1982). Let T(x, t) denote the travel time of a vehicle that departs from x at time t. Then t + T(x, t)is the vehicles arrival time, so that t t Tx; t is its time early, so that the trip cost may be written ascause of the problems difculty feel justied in reporting what progress we have made.no-toll. The paper poses the simple question: What pattern(s) of departures satisfy the trip-timing condition? We term thecorresponding problem and extensions, including determination of socially optimal allocations, The Corridor Problem.

Unless some insight has eluded us, answering this question in the context of even so basic a model is surprisingly difcult(but if it were not difcult, it would likely have been solved). We have not yet succeeded in obtaining a complete solution,

Cx; t aTx; t bt t Tx; t: 1

2.2. Continuity equation

Classical ow congestion is assumed, which combines the equation of continuity with an assumed relationship between

with V0 < 0. It is typically assumed that: (i) V steadily decreases with q, so that ow, F = qV, is a smooth convex function of q;

minus the outow. Letting n(x, t)P 0 denote the entry rate onto the road, the equation of continuity is

2.3. Tr

Thset of

Thdepar

It sequilied, w

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 745(ii) ow is zero with zero density and also with jam density. The equation of continuity is simply a statement of conservationof mass for a uid, that the change in the number of vehicles on a section of road of innitesimal length equals the inowvelocity and density. Recall that we have assumed the road to be of constant width. Accordingly, at location x and time t, letq(x, t) denote the density of vehicles per unit length, and let v(x, t) denote velocity. The relationship between velocity anddensity is written as

vx; t Vqx; t;There are no-tolls, so that, at each time and location, trip price equals trip cost.Fig. 1. Trafc corridor, space-time diagram.@q@t

@@x

qV nx; t:

ip-Timing condition (TT)

ere are two equilibrium conditions. The rst is that all vehicles commute. If we let D denote the departure set, i.e., the(x, t) points for which departures occur in equilibrium, then the condition that all vehicles commute can be written asZx;t2D

nx; tdt Nx 8x 2 0; x: 2

e second equilibrium condition is the trip-timing condition (TT), that no vehicle can experience a lower trip price byting at a different time. Letting p(x) denote the equilibrium trip price at location x, the TT condition can be written as

Cx; t px 8x; t 2 D Equality Component of the TT;Cx; tP px 8x; t R D Inequality Component of the TT: 3

tates that at no location can the trip price be reduced by traveling outside the departure set at that location. A no-tollbrium is a departure pattern, n(x, t)P 0, and a trip price function, p(x), such that the equilibrium conditions are satis-ith T(x, t) obtained from the solution to the continuity equation.

3. Imp

3.1. Re

Fro

depar

sible?be ide

traveltravel

increawhich

3.2. Co

b b

746 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768bTax 0 8 x; a 2 intA;while differentiation of (6) with respect to a yields

bTxax; a V 0qt1 bTaV2

7bTa ba 8x; a 2 intAvx; a T x; a Vqx; a T x; aNote that we use a subscript to indicate a partial derivative, where appropriate. Differentiation of (5) with respect to a and

then x yieldsbTxx; a 1 1 : 6It will prove convenient at this point to make the transformation of variables

ax; t t Tx; twhere a(x, t) is the arrival time at the CBD of a vehicle that departs location x at time t. If bT x; a is the travel time of a vehiclethat arrives at the CBD at time a, then the inverse transformation is

tx; a a bT x; awhich relates departure time to arrival time. The trip-timing condition, expressed in terms of arrival time, is

px abT x; a bt a 8x; a 2A; 5where A, the arrival set, is the set of all (x,a) for which the arrival rate is positive. The advantage of working in terms ofarrival time is that bT x; a tracks the cohort of vehicles that arrives at time a. Since a vehicle with arrival time a passes loca-tion x at time a bT x; a,

bT x; a bT x dx; a dxvx; a bT x; a

and sonstant departure rate within interior of departure setsection formalizes this intuitive argument.ses to be the same requires that the travel time between stops 1 and 2 be the same for the rst and second buses,requires that at bus stop 1 the same number of passengers board the rst and second buses. The argument in the nexttime increase for those boarding at stop 2 equals the travel time increase between stops 2 and 3,. For these travel timetime on the rst bus by the same amount for passengers boarding at stop 1 as for those boarding at the stop 2. Thetime increase for those boarding at stop 1 equals the travel time increase between stops 1 and 2, 2 and 3,. The travelrepresented in the earlier cohort are then slowed down by the same amount. This result can be illustrated by considering thebus-corridor discretization of the same problem, for which the speed of a bus is related to its number of passengers.

Suppose that the rst bus to depart picks up passengers at stops 1 and 2, and that the second bus to depart picks up pas-sengers at stops 1, 2, and 3. The trip-timing equilibrium condition requires that travel time on the second bus be higher thanOneway this result can be achieved, and we shall show is the only one, is for the departure function for a later cohort tontical to that for an earlier cohort, except for the addition of vehicles closer to the CBD, since cars entering at all locationsThe same is true for any time and location, (x0, t0) such that (x0, t0) and (x0, t0 + Dt) are in the departure set. How is this pos-ture set, then the difference between their arrival times is

Da aa bDt: 4Hence, over the departure set at each location, travel time increases linearly in the departure time at the rate bab. In partic-ular, if two vehicles leave the same location, x, one at time t, the other at time t +D t, and if both (x, t) and (x, t +Dt) are in them (1) and (3),

Tx; t px bt t

a b 8x; t 2 Dlications of the trip-timing condition

lation between arrival and departure times

From equality of mixed partial derivatives, it follows that the right-hand side of (7) equals zero, and since V0, 1 bTa and Vare all strictly nonzero, it follows that

qtx; a bT x; a 0 8x; a 2 intA;or

qtx; t 0 8x; t 2 intD: 8(8) states that trafc density is constant at a particular location over the interior of the departure set at that location. Sinceqt=0, the continuity equation reduces to

@

@xqVq nx; t:

Differentiating this equation with respect to t yields

@n@t

@@tqVx

@

@xqVt

@

@x0 0:

Thus, at each location, the departure rate is constant over the interior of the departure set:

nx; t nx 8x; t 2 intD: 9

4. Proposed departure set

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 747Fig. 2. Horn-shaped proposed departure set. The upper boundary is a vehicle trajectory corresponding to the nal cohort of vehicles to arrive at the CBD. Thedashed line indicates a sample vehicle trajectory, with decreasing velocity within the departure set (Region I) and increasing velocity outside the departureset (Region II). Note that the slope of the dashed line equals the slope of the upper boundary, up to the point of leaving the departure set.Consider two vehicle trajectory segments, running from some x0 to x00, both of which are in the interior of the departureset, an earlier one and a later one. For each x, trafc density is the same for both trajectories (from (8)), as is the departurerate (from (9)). Furthermore, at all locations between x0 and x

0 0the travel time of the later trajectory exceeds the travel time of

the earlier trajectory by the same amount. This requires that travel time between x00 and the CBD be higher for the later tra-jectory, that in turn requires that more vehicles enter the road between x00 and the CBD for the later trajectory. One way thiscan be achieved is for the rst departure time at each location to be later for more central locations. Put alternatively, latertrajectories pick up vehicles at increasingly central locations.

Fig. 2 displays a departure set consistent with this reasoning. Time is renormalized so that t = 0 corresponds to the start ofthe morning commute (at x = 0.) The departure set is connected. The lower boundary gives the time of the rst departure ateach location, and the upper boundary the time of the last departure at each location. A sample trajectory is shown as thedashed line. The rst trajectory contains vehicles from only the most distant location. Succeeding trajectories contain vehi-cles from locations successively closer to the CBD, as well as from all more distant locations. The last trajectory, which arrivesat the CBD exactly at the desired arrival time, t, contains vehicles from all locations. We refer to the departure set as Region I,and the region below the departure set as Region II.

Since the pattern of density by location in the interior of the departure set does not change over time, at any location thenumber of vehicles entering at more distant locations must equal the ow at that location. At more central locations there-

constant. Now suppose that the departure set over an innitesimally small location interval, [x,x + dx], is disconnected.

departs x inside the departure set at an earlier time, a vehicle that departs x outside the departure set at a later time will

depart the same location but at a later time incur greater travel time cost.

748 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768Property 4: The departure set is connected and does not contain holes.Property 4 easily follows from Properties 1 and 3, and the requirement that the population density be nonzero at all loca-tions up to the edge of the metropolitan area. Since the population density is nonzero at all locations the departure set isnonempty at all locations. From Property 3 the departure set at a given location is a connected set. From Property 1 theupper boundary of the departure set is a vehicle trajectory, which is a connected set. Thus, the departure set is the unionof connected sets, each of which has nonempty intersection with a connected set, and is therefore connected.

4.2. Proposed departure set

In his earlier work on the Corridor Problem, Arnott (2004) had established Properties 1 and 2, and conjectured Properties3 and 4. Furthermore, he was able to solve numerically for equilibrium of the (discretized) bus-Corridor Problem on theassumption that, at each bus stop, the same number of passengers board each bus picking up passengers at that stop, exceptperhaps the rst. Accordingly, Arnott conjectured that the departure set takes the form shown in Fig. 2, with the start of theincur less travel time cost up to the point when it either enters the departure set (i.e., joins a cohort of vehicles that departwithin the departure set) or reaches the CBD. This is inconsistent with the TT condition, that requires that vehicles thatThen a vehicle departing from x outside of the departure set will experience a lower trafc density, and hence a highervelocity, over [x,x + dx], than a vehicle departing within the departure set at the same location. Therefore, to a vehicle thatWe have proved that, at a particular location in the interior of the departure set, density (8) and the entry rate (9) must bewere different on the upper boundary, then the velocity as a function of location for the last cohort would not be v(x),which can be shown to imply violation of the trip-timing condition.fore, within the departure set the ow rate must be higher, which is inconsistent with hypercongestion. Thus, along a vehicletrajectory velocity decreases from x = 0 to the lower boundary of the departure set, and then speeds up from the lowerboundary of the departure set to the CBD as trafc thins out since no vehicles are entering the road. Thus, a vehicle trajectoryis convex in the interior of Region I and concave in the interior of Region II.

The above line of reasoning leaves open the properties of the boundary of the departure set. The rest of this section willsketch a more formal derivation of the properties of the departure set.

4.1. General properties of the departure set

we shall argue that the departure set has the following properties:

Property 1. The upper boundary of the departure set is a vehicle trajectory.

Property 2. At any location, the departure rate on the upper boundary of the departure set is the same as in the interior of thedeparture set.

Property 3. At any location, the departure set at that location is a connected set.

Property 4. The departure set is connected and does not contain holes.

Property 1: Upper boundary of the departure set is a vehicle trajectory.If the upper boundary of the departure set is a vehicle trajectory, the trajectory must arrive at the CBD exactly at t. Sup-pose not, and that the trajectory arrives at the CBD at t0 < t. Then at any location there is a departure time for which avehicle can depart, experience no trafc congestion, and arrive at the CBD between t0 and t, experience less travel timeand arrive less early than all other vehicles departing from that location, which is inconsistent with the trip-timing con-dition.Suppose that the nal cohort of vehicles to arrive at the CBD does not contain vehicles that depart from x = 0, and that thelatest departure from x = 0 in the departure interval is at t0. Then a vehicle departing x = 0 slightly after t0 can travel at free-ow travel speed until it meets the cohort of vehicles that departs from x = 0 at t0, hence experiencing a lower trip costthan the vehicle that departs x = 0 at t0.With some modication, the same line of reasoning can be applied to establish that the nal cohort must contain vehiclesfrom every location.Property 2: At any location, the departure rate on the upper boundary of the departure set is the same as in the interior.(8) indicates that, at each location, density and therefore velocity are constant in the interior of the departure set. Hence,in the interior of the departure set velocity can be written as v(x) and density can be written as q(x). If the departure rate

Property 3: Over any innitesimally small location interval, [x,x + dx], the departure set is a connected set

at eacthat aan arbitrary distribution of population along the corridor. For obvious reasons, Arnott termed the proposed departure set, a

Astures.

equilibrium solution to exist with an arbitrary distribution of population along the corridor. Accordingly, we shall addressthe qu

Secied d

dt dx dq

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 7491 qV0 nx :

In particular, the ow, F = qV, satises

qV0dq nxdx;and, since initially F = 0,

Fx qVx Z x0nx0dx0: 10

Thus, within the departure set, the ow at a particular location equals the sum of the departure rates at upstream loca-tions. The ow function can be solved for, given the departure rate function, and vice versa.

At any location, the population equals the departure rate there times the width of the departure set there. Knowledge ofany two allows for the solution of the third.Region I (the departure set), and Region II. Section 5.4 lays out the procedure we employ for describing trajectories andpoints on the lower boundary of the departure set. Section 5.5 derives one of three governing equations, and applies it todemonstrate that the proposed departure set is inconsistent with equilibrium. Section 5.6 presents the modied departureset, which modies the proposed departure set to be consistent with the governing equation. Section 5.7 states the threegoverning equations and discusses how they work together, and Section 5.8 derives the three governing equations.

The central insight on which the analytical results and the numerical analysis build is that once the lower boundary of thedeparture set, as well as the ow rate at every point along the boundary, are solved for, the rest of the solution followsstraightforwardly. Thus, the focus is on solving for the lower boundary of the departure set consistent with the three gov-erning equations.

5.1. Continuity equation: method of characteristics

The continuity equations in Regions I and II are rst-order, quasi-linear partial differential equations for the density, q, asa function of (x, t) within each region. We may solve each equation by the method of characteristics, that converts the PDEinto a system of ODEs, whose solution yields characteristic curves in the xt plane, with q determined in each region as afunction along these curves (Evans, 2002, Rhee et al., 1986). In the following two sections we apply this method of charac-teristics to Regions I and II.

5.2. Region I

We showed earlier that the TT condition implies that, at each location, the departure rate is constant over the interior ofthe departure set and along its upper boundary, and shall assume furthermore that the departure rate is the same along thelower boundary as well. The continuity equation in Region I is therefore

@q@t

qV0 @q@x

nx

where 0 indicates a derivative with respect to q, i.e., qV0 ddq qV. The characteristic curves for this PDE satisfytion 5 derives the full set of equations that together determine an equilibrium departure rate distribution for the mod-eparture set. Section 5.1 through Section 5.3 analyze the continuity equation within each of two regions, the interior of5. Mathematical analysis: analytic resultsestion: What distributions of population along the corridor are consistent with the modied departure set?modied departure set. Like the proposed departure set, the modied departure set does not provide enough freedom for anwe shall see, Arnotts proposed departure set was incorrect, as there must be a zone bordering the CBD with no depar-Refer to a departure set that modies the proposed departure set, taking this feature of the solution into account, thehorn-shaped departure set.h location, velocity is constant over that locations departure time interval), and in Region II accelerates. He conjectureddjustment of the lower and upper boundaries provides enough freedom for the trip-timing condition to be satised forrush hour endogenous. We refer to this as the proposed departure set. Region I is the departure set, and Region II is the regionin the 0;0 ! x;t rectangle below the departure set. The upper boundary of the departure set is the vehicle trajectoryarriving at the CBD at t, and includes departures from all locations. The departure set is connected; a vehicle trajectory(shown in Fig. 2 as a dashed line) starts at x = 0, in Region I is parallel to the upper boundary of the departure set (since,

Since population is nonzero at all locations, (10) implies that ow is a strictly increasing function of x on the interior of thedeparture set, reaching at most capacity ow at the entry point closest to the CBD. Hence, we conclude that an equilibriumsolution to the Corridor Problem does not permit hypercongestion. Furthermore, this implies a one-to-one relationship onthe interior of the departure set between ow, velocity and density. Newell (1988) proved that a trafc corridor with a sin-gle-entry point does not admit hypercongestion. The result here is more general since entries are possible at every pointalong the corridor. Recall that we have assumed a road of constant width; hypercongestion could presumably occur in amore general model, e.g., if road capacity were to shrink as the CBD is approached.

5.3. Region II

The continuity equation in Region II is the well-known homogeneous equation

@q@t

qV0 @q@x

0:

(qV)0 is the rate at which ow changes with density, and its reciprocal is the rate at which density changes with ow. Asdiscussed in Newell (1993), the characteristic curves in Region II are straight lines which are iso-density curves with slopedtdx 1qV0. These characteristic lines emanate from the lower boundary of the departure set, and completely determine thedensity eld in Region II (see Fig. 3). Since ow is non-decreasing on the lower boundary of the departure set, the slopesof the characteristic lines are non-decreasing, and therefore the characteristic lines are non-intersecting. This excludes

part ohort oat the

0

750 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768Fig. 3. Characteristic lines in Region II, which are iso-density curves, along with a sample vehicle trajectory.at time u is

TIu Z bu0

1vx0dx

0

where v(x) is the velocity in the departure set at location x. Thus, the (x, t) coordinates along the lower boundary curve areparametrized as (b(u),u + TI(u)).

5.5. Arrival ow rate

An important consequence of the TT condition is that it enables us to determine the ow rate at the CBD in terms of theow rate at the lower boundary of the departure set. Let F(b(u)) denote the ow within the departure set at location x = b(u),f the overall solution. We parametrize the lower boundary curve of the departure set as follows (refer to Fig. 4). A co-f vehicles departs x = 0 at time u, reaches the lower boundary curve of the departure set at location x = b(u), and arrivesCBD at time a(u). From (4), is au aabu xV . The travel time to the lower boundary curve for a vehicle departing x = 0the possibility of shocks occurring in Region II, and implies that density is continuous in Region II. Note that the initial vehicletrajectory, with V = V0, coincides with a characteristic line. Also note that the slope of a characteristic line at a point on thelower boundary is greater than the slope of a trajectory curve at that point, since 1qV0 1VqV 0 > 1V.

5.4. Parametrization of lower boundary curve

We rst normalize time so that the rush hour starts at t = 0; t is then the length of the rush hour, which is determined as

Since da a , we may determine the arrival ow rate at the CBD as

dA du a b

He

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 7515.6. Modication of proposed departure set

The ow of a cohort of vehicles decreases from the lower boundary of the departure set to the CBD by a factor of aba . Also,

ow mtwo psistenCBD, wwhatCBD. A

Sinimum

5.7. Th

Thdeparsuch tthis relem w

Cocurveacteriat thethreelowinFlow at x; a da

Fbuada

a

Fbua

nce, ow for a cohort of vehicles strictly decreases from the lower boundary to the CBD by the multiplicative factor aba .0 0 0

du abso Fbu R bu0 nx0dx0. If we follow the vehicle trajectory that departs x = 0 at time u, leaves the departure set at locationb(u), and arrives at the CBD at time au aabu xV0, then we may write the cumulative number of arrivals at the CBD by timea, A(a) as

Aa Z ua Z bu0

nxdxdu0 Z ua

Fbu0du0: 11

Fig. 4. Trajectory departing x = 0 reaches the lower boundary at location x = b(u). Velocity function within the departure set at location x is v(x) = v(b(u)).ust be continuous from the lower boundary of the departure set to the CBD. With the proposed departure set, theseroperties cannot be simultaneously satised for the last cohort of vehicles. Thus, the proposed departure set is incon-t with equilibrium. If the departure set is modied so that there is zero population density over an interval before theith the interval being determined so as to satisfy the rst condition, then both conditions can be satised. Thus, in

follows we consider a modied departure set, which is still horn-shaped, but has zero population density near themodied departure set is shown in Fig. 5.

ce ow is an increasing function along the lower boundary curve and since hypercongestion does not occur, the max-ow must occur at the tip of the horn, and this maximum ow must be less than or equal to capacity ow.

ree governing equations: summary

e TT condition implies that, at each location, the departure rate is constant over the interior and upper boundary of theture set. We have made an assumption concerning the departure rate along the lower boundary of the departure sethat, at each location, the departure rate is constant over the entire departure set, including the lower boundary. Usingsult, we shall derive three equations, (12)(14), that must be satised by an equilibrium solution to the Corridor Prob-ith the modied departure set.nsider a vehicle trajectory departing x = 0 at time u0 and arriving at the CBD at time a(u) (Fig. 6). The characteristicthrough this arrival point originates from the lower boundary at location b(uf), where uf < u0 (see Fig. 6). Since the char-stic curve is a straight line of constant ow, we may equate the ow at the lower boundary for the cohort uf to the owCBD for the cohort u0. Hence, for each departure time u0 there will correspond a unique value of uf. We now state thegoverning equations that must be satised for each pair of u0 and uf values, and we derive these equations in the fol-g sections.

Sincan elsolutiand thx = 0,that thtime a

5.8. D

5.8.1.Re1

ddqqVjb

Fig. 6

752 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768Fig. 5. The ow for a cohort must decrease from the tip of the departure set to the CBD by the multiplicative factor aba .u0 a bax buf ddq qVjbuf

uf Z buf 0

1vx0 dx

0 xV0

" #12

Fbuf a ba Fbu0 13Z u0uf

Fbu0du0 q qVqV0

buf x buf : 14

ce we have established a one-to-one correspondence within the departure set between density, ow and velocity, weiminate all the terms involving density, q, and velocity, V, and rewrite all three equations in terms of only ow, F. Aon to these three equations consists of the function b(u) describing the x-coordinates of the lower boundary curve,e ow function along the lower boundary curve, F(b(u)). Since the outer boundary of residential settlement is atthe function b(u) must satisfy b(0) = 0, and since initially there is no trafc, the ow function must satisfy F(0) = 0. Notee initial width of the departure set (i.e., the duration of time over which individuals depart at x = 0) and the work start-t the CBD, t, will be determined as part of the solution and are not given a priori.

erivation of the three governing equations

First governing equationferring back to Fig. 6, since the TT condition implies that a aabu0 xV0, and since the slope of the characteristic line isuf , by calculating the slope directly we have

. Arrival of the vehicle trajectory departing x = 0 at time u0 intersects the characteristic line that originates from (b(uf),uf + TI(uf)), where uf < u0.

aabu0 xV0 uf TIuf

x buf

1ddq qVjbuf

:

We may solve for u0 in terms of uf to obtain the rst governing equation

u0 a bax buf ddq qVjbuf

uf Z buf 0

1vx0dx

0 xV0

" #:

5.8.2. Second governing equationFlow on the lower boundary decreases along a trajectory to the CBD by the multiplicative factor aba , i.e., if F(b(u)) is the

ow at the lower boundary of the cohort that departed at time u, and au aabu xV0 is the arrival time at the CBD of thiscohort, then the ow at x; au equals aba Fbu. Referring to Fig. 6, since characteristic lines in Region II are iso-ow lines,this allows us to express the ow within the departure set at location b(uf) in terms of the ow within the departure set atlocation b(u0), yielding the second governing equation

Fbuf a ba Fbu0:

Since hypercongestion does not occur within the departure set, there is a one-to-one relation between ow, density andvelocity. We may therefore also use this second governing equation to determine the density (or velocity) at b(uf) in terms ofthe density (or velocity) at b(u0).

He

0 0 0 0 I 0

Ththat d

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 753Fig. 7. Trajectory in Region II parametrized as ~xu;~tu, where uf 6 u 6 u0.Au 0

Fbu du :

Denote the cumulative ow as a function of the space-time coordinate in Region II as bAx; t, to distinguish it from thecumulative ow along a trajectory in Region II, A(u). Following Newell (1993), if we move along a characteristic line in RegionII from (x, t) to (x + dx, t + dt), then the cumulative ow along this line satisese cumulative ow, or cumulative number of arrivals, is constant along a trajectory in Region II. By (11), for the cohorteparts x = 0 at time u the cumulative ow along the cohorts trajectory in Region II isZ u

0 0~xuf ;~tuf x; aa bu0 xV0:nce,

~xu ;~tu bu ;u T u 5.8.3. Third governing equationConsider a trajectory that departs x = 0 at time u0, and parameterize the (x, t) coordinates of the portion of this trajectory

in Region II as ~xu;~tu, uf 6 u 6 u0, where ~xu;~tu is the point on the trajectory in Region II that intersects the charac-teristic line emanating from the lower boundary of the departure set at the point (b(u),u + TI(u)) (see Fig. 7).

ow.Eac

suppo

the upmidwtrunca

givenbution

We

6.1. Gr

Gre

754 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768eenshields velocitydensity relation

enshields linear velocitydensity relation iserning equations with this Relation and scale parameters implemented. We then give a broad overview of our numericalstrategy before presenting the details. All algebraic calculations used to derive the results in this section have been placedin the Appendix at the end of the paper.ow value less than or equal to capacity ow at the tip of the horn and a given a, there is a unique population distri-solving the Corridor Problem with the modied departure set.begin by introducing Greenshields Relation, choosing appropriate scale parameters, and then restating the three gov-that is identical to the original departure set over the regions not truncated. The ow at the tip of this truncated departure setis less than capacity ow. Based on the results of this section, we can further conclude (with Greenshields relation) that, for a

bper boundary of the departure set. This vehicle trajectory intersects the lower boundary of the departure set at someay point, and traverses Region II to reach the CBD. If we now let this vehicle trajectory be the upper boundary of a new,ted departure set, removing all other trajectories that depart after it, then we obtain a truncated departure set solutionture set, since ow is non-decreasing). Consider any vehicle trajectory that departs x = 0 before the last departure, i.e., belowh one of these solutions admits a continuous family of truncated solutions that do not reach capacity ow. To see this,se that we have a departure set solution that reaches capacity ow (that must necessarily occur at the tip of the depar-dbAdx

q qVqV0 15a

dbAdt

qqV0 qV ; 15b

where q and V are the constant density and constant velocity along the characteristic line. If we integrate (15a) along thecharacteristic line from (b(u),u + TI(u)) to ~xu;~tu, we obtain

bA~xu;~tu bAbu; u TIu q qVqV0

bu~xu bu; uf 6 u 6 u0:

Since along the trajectory ~xu;~tu the cumulative ow is the constant value Au0 R u00 Fbu0du0, and since on the

lower boundary at b(u) the cumulative ow is Au R u0 Fbu0du0, we may rewrite this expression asZ u0u

Fbu0du0 q qVqV0

bu~xu bu; uf 6 u 6 u0: 16

In particular, when u = uf, we obtain the third governing equationZ u0uf

Fbu0du0 q qVqV0

buf x buf :

This equation relates the integral of the ow along the lower boundary from b(uf) to b(u0), to the distance from the CBD tothe lower boundary at b(uf), and the density and velocity at b(uf).

6. Numerical analysis (Greenshields)

Ideally, given an arbitrary population density we would like to construct a solution to the Corridor Problem and, if multi-ple solutions exist, characterize all possible solutions. However, it is not clear if a solution will exist for an arbitrary popu-lation density, e.g., we have already shown that the population must be zero some nite distance before the CBD. In the lastsection we showed how a solution with the modied departure set can be determined by solving for a lower boundary curveof the departure set and a ow along this lower boundary curve. We also showed how a specic solution uniquely deter-mines a population density. Thus, we seek to characterize all possible solutions consisting of lower boundary curves andows along these lower boundary curves, from which we could extract all possible population densities that admit solutionsto the Corridor Problem with the modied departure set. We have determined that any solution must satisfy the three gov-erning equations. What is not clear, however, is whether or not the three governing equations admit a solution, and, if so,whether or not they admit a unique solution.

In this section we consider a specic velocitydensity relation (Greenshields), and provide a numerically constructiveproof that, for a given ratio of parameters, ba, the three governing equations admit a unique solution for a modied depar-ture set and a ow distribution on that departure set that reaches capacity ow (note that x and V0 are scale parametersthat will only determine the scaling along the distance and time axes). As discussed, this solution will uniquely determinea population density. Thus, in this section we will conclude that, using Greenshields relation and given ba, there is a uniquepopulation density that admits a solution to the Corridor Problem with the modied departure set that reaches capacity

that V0 = 1 and 12 6 V 6 1. We also choose the units of population so that the jam density, qJ = 4, that results in the ow, Fvaryincost toNeweis useing eq

must simultaneously solve (17) subject to the constraints b(0) = 0 and F(0) = 0. We will rst seek a departure set solutionthat resoluti

(17

0 to thary cutions,

of lineinitial

are unand acapac

A cthis se

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 755iquely determined. Thus, this procedure provides a numerically constructive proof that, given Greenshields relationratio of parameters ba, there is a unique solution to the Corridor Problem with the modied departure set that reachesity ow.omplication arises when we attempt to construct the nal segment of the lower boundary curve, since the uf values ingment do not have a corresponding u0 value with which they can be paired, and thus the rst and third governingcurve and ow values on the rst discretized subinterval. Hence, by making valid linear approximations we numerically con-struct a solution to the three governing equations, and at each step of our numerical procedure the solution values we obtainar equations yields the lower boundary curve and ow values for the next discretized subinterval. Futhermore, theseed values for this numerical procedure are uniquely determined by linearly approximating the lower boundarynumerical procedure takes the lower boundary curve and ow values at one discretized subinterval, inputs them into thediscretized versions of the rst and third governing equations for the next discretized subinterval, yielding a pair of linearequations for the lower boundary curve and ow values for the next discretized subinterval. The unique solution to this paire capacity ow value of 1. If our discretization is ne enough, then we can linearly approximate both the lower bound-rve and the ow over each discretized subinterval, that enables us to discretize the rst and third governing equa-(17a) and (17c), i.e., to restate them in a form on each discretized subinterval that does not involve integrals. Our0 f

governing Eq. (17b) to exactly determine the ow values at each of the discretization points, with ow values ranging fromaches capacity ow. As mentioned earlier, the existence of such a solution will imply a continuous family of truncatedons that do not reach capacity ow.) involve a natural pairing of u values, u and u . We utilize this pairing to discretize the problem, using the secondFbuf a ba Fbu0 17bZ u0uf

Fbu0du0 1wbuf 2

wbuf 1 buf : 17c

6.3. Overview of numerical solution

To determine any solutions to the Corridor Problem with the modied departure set under Greenshields relation, weg from 0 to a capacity ow value of 1. The only relevant parameter, then, is ba < 1, that is the ratio of the unit time earlythe unit travel time cost. Finally, let w denote the slope of the ow vs. density curve, w = 2V 1 where 0 6 w 6 1.

ll (1993) refers to w as the wave velocity, and although it is not necessary to introduce this additional function, itful in simplifying the algebraic manipulations that follow. Using these units and notation we restate the three govern-uations in a form that will be useful for our numerical procedure:

u0 a ba1 buf wbuf uf

Z buf 0

21 1 Fx0p dx0 1

" #17aV V0 1 qqJ

!

q qJ 1VV0

;

where V0 is the free-ow velocity and qJ is the jam density. We write the ow in terms of velocity as

F qV qJV0

VV0 V;

that achieves its maximum value, capacity ow, at V V02 . Since we have shown that hypercongestion does not occur,V02 6 V 6 V0, and we may write velocity in terms of ow as

V V02

11 4F

qJV0

s !:

Also,

qV0 ddq

qV 2V V0:

6.2. Scale parameters and notation

The natural units of distance and time are x and xV0, respectively. Thus, we choose our units such that x 1 andxV0 1, so

equations will no longer be applicable. We alleviate this problem by guessing a value of the lower boundary curve for thesubsequent discretized subinterval, using our guess to numerically construct a vehicle trajectory that should theoreticallyintersect the lower boundary curve exactly at the point that we guessed, and choosing repeated guesses until we nd thatthe vehicle trajectory does intersect our point at some desired level of tolerance. This procedure is re-iterated until the nalsegment has been constructed. The following sections provide the details of the numerical procedure, with all algebraic cal-culations relegated to the appendix.

6.4. Iterated sequence of discretized ow values

Over the departure set, we seek a solution such that the ow increases from 0 to a capacity ow value of 1. We choose aninitial ow value, 0 < F0 6 1, determine the point on the lower boundary curve of the departure set where this ow value isattained, track the trajectory curve through this point until it reaches the CBD, and then backtrack along the characteristicline intersecting this point until reaching the lower boundary curve. This has already been graphically illustrated in Fig. 6. By(17b), the ow at this new iterated value is F1 F0aba . Continuing this iterative procedure, as we approach the residentialboundary (x = 0), the nth iterated ow value is Fn F0 aba

n, that approaches 0.

We graphically illustrate this iterative procedure in Fig. 8, with an initial ow value of F0 = 1, capacity ow. Note that(17b) allows us to develop a sequence of ow iterates, Fi F(b(ui)), without knowing the corresponding ui and bi b(ui)values.

As illustrated in Fig. 8, if we begin with F0 = 1 and iterate this procedure N times, then we partition the lower boundarycurve into N + 1 segments, where the rst segment begins with a ow value of 1 and ends with a ow value of aba . We furthersubdivide the ow values on this rst segment into k subdivisions, equally spaced between the values aba and 1. We applythe iterative procedure to each of the k ow values within this segment, generating k ow values within each of the subse-quent segments. Hence, if we consider the rst N segments constructed, each having k subdivisions, then we obtain a se-quence with a total of Nf = Nk ow values. We illustrate this idea in Fig. 9, where we have reordered the sequence of owvalues such that F1 is the smallest ow value in the sequence, increasing to the capacity ow value, FNf FNk 1. The gen-eration of this sequence of ow iterates derives from (17b), and does not require knowledge of the corresponding ui or b(ui)values. We state a closed form formula for all ow values in this sequence, given values of N and k:

Sin

and w

756 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768Fig. 8. Iterated ow values. Iterative procedure tracks the intersection at the CBD of a trajectory curve with a characteristic line.ce V 12 11 F

p and w

1 F

p, we may use this sequence to obtain corresponding sequences of vi v(b(ui))

i w(b(ui)) values.FiNjk a ba j1 a b

a ik

1 a ba

; 81 6 j 6 N;1 6 i 6 k 18

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 7576.5. Discretization of the rst and third governing equations

Fig. 9. Sequence of ow values from F1 to FNf FNk 1. For all i, 1 6 i 6 (N 1)k, the ith ow value satises Fi Fikaba .ThCorres1 6 i 6

If wof u:

Asmay bhave berning

and th

Alttemptknowndiscrebe sole (x, t) coordinates of the lower boundary curve are parametrized in terms of u as bu;u TIu bu;uR bu0

dx0vx0 .

ponding to our sequence of ow values, we denote the sequence of coordinates of the lower boundary curve as (bi, ti),Nf. Similarly, we denote the iterated sequence of cumulative ow values as Ai Aui

R ui0 Fbu0du0.

e choose k large enough, then over each subinterval (ui1, ui), we may approximate F(b(u)) and b(u) as linear functions

Fbu Fi1 Fi Fi1ui ui1 u ui1; u 2 ui1;ui

bu bi1 bi bi1ui ui1 u ui1; u 2 ui1; ui:19

shown in the Appendix, we may use these linear approximations to determine expressions for the ti and Ai values, thate substituted into (17a) and (17c) to yield discretized versions of these two equations. The algebraic manipulationseen placed in the Appendix, and we present here only the nal results, namely the discretized version of the rst gov-equation,

uik a ba

1 biwi

ti1 ui ui1 4bi bi1Fi Fi1 log1wi1wi1

wi wi1

1

; 1 6 i 6 Nf k; 20

e discretized version of the third governing equation,

uik uik1 2Fik Fik11wi2

wi1 bi Aik1 Ai

" #; 8i;1 6 i 6 Nf k: 21

hough the three governing equations, (12)(14), are valid for any monotonic velocitydensity relationship, if we at-to discretize the rst and third governing equations then we will generally obtain two nonlinear equations in two un-s (e.g., using Underwoods relation). However, if we use Greenshields velocitydensity relationship then thetized versions of the rst and third governing equations are a pair of linear equations in two unknowns, that mayved exactly.

758 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768To initiate the above iterative procedure, we must provide values for b1, t1, u1, . . ., u1+k and A1, . . ., A1+k. If we choose N largeenough, then F1, . . ., F1+k will be close to 0, and over the interval (0,u1+k) we can approximate F(b(u)) as a linear function of u,Fbu F1u1 u. Using this linear approximation we then apply the discretized versions of the rst and third governing equa-tions, that enables us to solve for u1 and b1, and, hence, determine all necessary initializing seed values. The algebraic detailsare provided in the Appendix, with the initial seed values given in (A-8).

6.8. Final segment

After implementing the above iterative procedure, we will have determined all Fi, wi, ui and Ai values for i = 1, . . ., Nf, andwill have determined all bi, ti values for i = 1, . . ., Nf k. The only remaining values to determine arebNfk1; tNfk1; . . . ; bNf ; tNf , corresponding to the (x, t) coordinates of the lower boundary curve in the nal segment. Fur-thermore, since we have determined uNf (the departure time at x = 0 of the nal cohort of vehicles that arrives at the CBDexactly at time t), from (4) we can calculate t as t aabuNf 1.

In constructing the nal segment, the rst governing equation will no longer be applicable, since the characteristic linesemanating from the nal segment will intersect the CBD at a point greater than t, that does not contain the intersection ofany vehicle trajectories. Recall that in deriving the third governing equation, we rst determined the change in the cumu-lative ow along a characteristic line from the lower boundary of the departure set up to a vehicle trajectory (16). We sim-plied this equation by only considering the point where the vehicle trajectory intersects the CBD, i.e., by setting u = uf sothat ~xu x in (16), yielding the third governing equation. To determine the nal segment of the lower boundary curvewe use (16) in its more general form.

To construct this nal segment of k pieces, given a known value of bi on one piece we guess a value for bi+1 on the nextpiece. We then further subdivide this single piece intom subdivisions. The characteristic lines emanating from the endpointsof these m subdivisions partition the vehicle trajectory curve through bi+1 into m pieces. We assume that, over each of the mpartitions, the vehicle trajectory curve can be approximated by a linear function, that will be valid if the subdivisions arechosen ne enough, i.e., if m is chosen large enough. We then use this linear approximation for the vehicle trajectory curveon each subdivision to explicitly calculate the vehicle trajectory curve. If our initial guess for bi+1 was correct, then the vehicletrajectory curve we calculate should intersect the lower boundary curve at bi+1. We try different values of bi+1 until obtainingone such that the vehicle trajectory curve we calculate based on the bi+1 value intersects the lower boundary curve at bi+1with sufciently small error. Using this bi+1 value we calculate ti+1 based on our linear approximation for b(u) on thei + 1th piece. This procedure is iterated until all bi, ti values have been determined.

The algebraic details of this procedure are provided in the Appendix, along with a graph illustrating this procedure(Fig. 14).

7. Numerical results (Greenshields)

7.1. Departure set solutions

We implement the numerical procedure for the ratio of parameter values ba 0:2;0:4;0:6 and 0.8. A graph of each depar-ture set, along with the vehicle trajectory corresponding to the nal cohort of vehicles, is displayed in Fig. 10. We have plot-ted all graphs with the same axis, to discern the behavior of the solution with respect to the ratio of parameter values. As theunit time early cost, b, approaches the unit travel time cost, a, the width and length of the departure set decreases, approach-ing free-ow condition of zero trafc departing x = 0 at time u = 0 and arriving at the CBD at time t 1. In each of thesegraphs the ow reaches the capacity ow value of 1 at the tip of the departure set. At this point the slope of the characteristiccurves will be innite, and since the slope of the lower boundary curve must be greater than the slope of the characteristiccurves, it too must be innite. We observe this behavior in our numerical solutions, as the tip of the departure set becomesvertical.6.6. Iterative procedure

Using (18) we may determine all Fi andwi values, for 1 6 i 6 Nf. Suppose, that for a given value of i, i < Nf k, we know thevalues b1, . . ., bi1, t1, . . ., ti1, u1, . . ., ui+k1 and A1, . . ., Ai+k1. Then the discretized versions of the rst and third governingequations, (21) and (20), are a pair of linear equations in the unknown quantities bi and ui+k, in terms of known quantities.We may solve these equations to obtain the values of bi and ui+k, and then use these values in (A-3) and (A-1) to determineAi+k and ti. This procedure may be iterated until i = Nf k. The algebraic details are provided in the Appendix, where we de-rive (A-4) and (A-5), which completely summarize the core of our iterative procedure. Specically, given the initial seed val-ues b1, t1, u1, . . ., u1+k and A1, . . ., A1+k, we iteratively use (A-4) and (A-5) to determine b1; . . . ; bNfk, t1; . . . ; tNfk, u1; . . . ;uNf andA1; . . . ;ANf . At the conclusion of this procedure, the only undetermined quantities will be the (bi, ti) values in the last segment,from i = Nf k + 1 to i = Nf.

6.7. Initializing seed values

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 759isco

so

7.

thmwth

inRe

FiggrCBTo understand why the departure set solutions areba, the more rapidly must congestion increase to sarresponds to capacity ow, the higher is ba the shoThe following table lists the numerical values (tolutions.

Numerical results with x 1, V0 = 1 and qJ = 4Width of departure set at x = 0 (time units)Tip of departure set (distance, time) unitst (time units)Total population (population units)

2. Corresponding population densities

We previously showed how the TT condition imple upper boundary of the departure set the departuined the ow, we may numerically differentiate (1e may determine the population density at each loce departure set at that location.In Fig. 11 we have graphed the population densittegral of the population density is the total populatgion II for the nal cohort of vehicles. Note that if w

. 10. Numerically constructed departure set solutions for varioaphed with a solid line. The upper boundary curve of the departuD at t), is graphed with a dashed line.so sensitive to thetisfy the trip-timingrter must be the rutwo decimal plac

ba Values

0.2

0.89(0.88, 1.93)2.110.43

ies that, at each locare rate, n(x, t), is co0) to determine theation by multiplyin

ies corresponding tion, which also eque considered a tru

us ratios of parameter vare set, that correspondsvalue of ba, recall frocondition. Since t

sh and the smalleres) of several imp

0.4

0.28(0.72, 1.10)1.470.13

tion, within the innstant (9). Hence,constant departurg the departure rat

o the sample deparals the cumulativencated departure s

lues, 0 < ba < 1. The lowto the trajectory of the m (4) that Da aabhe maximum levelmust be the popuortant features of o

0.6

0.10(0.52, 0.67)1.240.03

terior of the departonce we have nume rate at each locate at that location b

ture set solutionsow value along thet solution that did

er boundary curve of thnal cohort of vehicles (Dt. The higherof congestionlation.ur numerical

0.8

0.02(0.27, 0.31)1.100.005

ure set and onerically deter-ion. From (2),y the width of

in Fig. 10. Thee trajectory innot reach the

e departure set isthat arrives at the

760 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768capacless o

7.3. In

Topose t

Suppo

Thus,

Suppoof the

Fig. 11differenthat locity ow value of 1, then the corresponding population densities would have the same general shape but would includeverall population and would reach zero earlier.

terpretation of results

gain a more intuitive feeling for our results, we transform our results using more realistic values of x, V0 and qJ. Sup-he residential settlement is x 10 mi long. To satisfy x 1 we must choose distance units so that1 distance unit 10 mi:se free-ow velocity is V0 = 50 mi/h. To satisfy V0 = 1 we must choose time units so that

50 mi=h 10 mi15 h

1 distance unit15 h

1:

1 time unit 15h 12 min :

se the jam density for a single trafc lane is 1 vehicle16 ft . If the road has a constant width of four lanes, then the jam densityroad is 4 vehicles16 ft . To satisfy qJ = 4 we must choose population units so that

4 vehicles16 ft

1320vehiclesmi

13;200 vehicles10 mi

4 3300 vehicles1 distance unit

4:

. Population densities corresponding to the departure set solutions in Fig. 10. We calculated the departure rate at each location by numericallytiating the ow values with respect to location. The population density is obtained by multiplying the departure rate by the departure set width atation.

1 population unit 3300 vehicles:

su unit f e city w 000 veh:2;0:4; depart tions a onding n

de s. 10 an pt that e the a s of dis ean e follow sts the n alues of feature din

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 7617.4. Comparison with empirical dataA mtion apliedbe quipose,dominpolycefth, tworkeval timthe diilar litical ev

Tosmoot

3 WeSystemNumerical results with x 10 mi, V0 = 50 mi/h and qJ 4 vehicles16 ftWidth of departure set at x = 0 (min) 10.7 3.4 1.2 0.2Tip of departure set (miles, min) (8.8, 23.2) (7.2, 13.2) (5.2, 8.0) (2.7, 3.7)t (min) 25.3 17.6 14.9 13.2Total population per four-lane road (vehicles) 1419 429 99 16.5odel is just a model. There are several good reasore considerably more complex than our model sugfor only a family of population distributions alonte different. Second, in US metropolitan areas, an iand our analysis does not apply to non-work tripsant as an employment center, and employmentntric. Fourth, employers respond to increased trafhe bottleneck model implies that, in early morningrs with higher ab ratios commute earlier since theye. Since those with higher ab ratios tend to have hi

stribution of income along the corridor. Nevertheleany of caveats can be given about the monocentricidence.compare the predicted equilibrium departure pattehed ow rates along interstate 60W leading into do

are very thankful to Jifei Ban for collecting the raw data and pr(PeMS, http://pems.dot.ca.gov/), and the local linear regressiona Values

0.2 0.4 0.6 0.8bns to believe that tgests. First, the eqg the trafc corridncreasingly large pr. Third, in almost alocations are becomc congestion by orush-hour travel fare relatively moregher income, the spss, a model is not vecity model, and yet

rn with an actual cwntown Los Angele

oducing the graphs. Datasmoothing and the graphhe spatial dynamicuilibrium trafc dyor, and the actualoportion of trips dll metropolitan areing increasingly dffering less concenrom a common oriaverse to congestiatial dynamics of try useful unless it hmany of its predict

ommuting patterns from 4 am to 10

was collected from thes were produced usings of rush-hour tranamics we characpopulation distrio not have a commas, the CBD is beecentralized, distrated work startgin to a commonon than an inconvrafc ow shouldas predictive conions are supporte

, Fig. 12 presents3

am on Tuesday, M

Caltrans Performancethe R-packages loct athe previous table, but using the current system of units.

d population along the axes in those gures. Th ing table li umerical v the same s presente

nsities have the same graphs as shown in Fig d 11, exce we must us bove unit tance, tim

For each of the four ratios of parameters ba 0 0:6;0:8, the ure set solu nd corresp populatio

ch roads in the city, in which case a population or the entir ould be 99, icles.To put this number into perspective, let us suppose that the city is 4 miles wide and that 5% of the city area is used forroads from the suburbs to the CBD. Since the typical lane width is 11 feet, there would be [(5280)(5)(0.05)] [(4)(11)] = 30Thus,

Fig. 12. Flow rates along freeway 60-westbound extending a distance of 15 miles from downtown Los Angeles, collected from 4 am to 10 am on a Tuesdaymorning. Flow rates are obtained from vehicle counts per 5 min interval, collected at 23 irregularly spaced stations. Linear local regression is used to smooththe raw ow values over time and space.fc conges-terized ap-bution mayuting pur-

coming lesspersed, andtimes. Anddestination,enient arri-depend ontent. A sim-d by empir-

a surface ofay 4, 2010.

Measurementnd lattice.

horn to the tip. The actual surface looks quite different.

762 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768Fig. 13 displays the data in a form that provides a less discriminating test. According to the theory, the ow-weightedmean time at a location should increase with proximity to the CBD, and indeed it does. Even though the theory passes thisweak test, it is clear that much more research will be needed before we can say that we understand the spatial dynamics ofrush-hour trafc congestion reasonably well.

8. Concluding comments

8.1. Directions for future researchData are from 23 trafc stations irregularly spaced along a 15-mile segment of freeway. Each trafc station records the numberof vehicle counts per 5-min interval. Linear local regression is used to smooth the trafc ow rates over time and space. Accord-ing to the theory, the trafc ow surface should have a horn-shaped base, with the wide end of the horn extending along thetime axis and with the tip on the far upper corner (at least at 9 am), and its height should be increasing from the wide end of the

Fig. 13. Flow-weighted mean times at each location from the raw data collected for Fig. 12. The ow-weighted mean time steadily increases as we near theCBD, which is in agreement with our analytic solution.This paper takes a preliminary step in developing a theory of the spatial dynamics of metropolitan trafc congestion.Even the preliminary step is really only half a step. When we started this research, we hoped to provide a complete solu-

tion to the Corridor Problem. Instead, we have succeeded in providing a solution for only a restricted family of populationdistributions along the trafc corridor. What remains to be done to obtain a complete solution? Frankly, we dont know. Ini-tially we conjectured that there are mass points along the lower boundary of the departure set, but these cannot be accom-modated with LWR trafc congestion. Then we conjectured that there is queuing at entry points, but in a decentralizedenvironment queuing occurs only when there is a capacity constraint on entry, which is not a feature of the model. If notmass points or queuing, what? To deal with this puzzle, we have adopted the strategy of solving a sequence of simpler mod-els culminating in the Corridor Problem. Building on Newell (1988), we have solved for the equilibrium and optimum for asingle-entry corridor (Arnott and DePalma, 2011) for a particular form of the congestion technology The next step is to solvefor the equilibrium and optimum for a two-entry corridor (the origin and further along the corridor), the step after that for ann-entry corridor, and the step after that a continuum-entry corridor as a limit. The Corridor Problem is intriguing partly be-cause it is so difcult. In the real world, the spatial dynamics of trafc congestion are well behaved, being much the samefrom day to day, strongly suggesting that an equilibrium exists. Why then is it so difcult to characterize?

In searching the literature, we uncovered no empirical papers on the spatial dynamics of rush-hour trafc congestion atthe level of a metropolitan area. There is a branch of literature in trafc ow analysis that looks at the dynamics of trafccongestion along individual freeways, with a view towards understanding how trafc jams form and dissipate, and how theyaffect ow, but all the papers in this literature take as given the pattern of entries onto the road. There is also an emergingliterature (Geroliminis and Daganzo, 2007) that documents the existence of a stable macroscopic diagram for rush-hour traf-c congestion in the center of metropolitan areas, but it does not examine the relationship between rush-hour trafc dynam-ics at different locations. Most large metropolitan areas, at least in Europe and North America, have undertaken at least onelarge travel diary survey. Using these data, it should not be difcult to document the empirical regularities of rush-hour traf-c dynamics.

Toso farditiondynampatterporati

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 763More generally, household and rm location decisions, household travel time decisions, and rm work-start-time deci-sions are all interdependent. Addressing these interdependencies requires a full-blown general (economic) equilibriummod-el, in which prices adjust to clear land, labor, and other markets at all locations and times. Current computable generalequilibrium models of metropolitan transportation and land use (Anas and Liu, 2007) are conceptually sophisticated anddescriptively rich. They have not been applied to the study of intra-day trafc dynamics, but could be extended to do so.

8.2. Conclusion

Determining equilibrium trafc ow over the course of the day for an entire metropolitan area is an important unsolvedproblem in urban transport economic theory and in transportation science. The problem appears to be an important one.Traditional network equilibrium analysis, as well as the theory of land use, employs models that abstract from intra-day var-iation in trafc. Since trafc congestion is an inherently nonconvex phenomenon, one wonders how much bias and loss theaggregation implicit in this abstraction entails. This paper investigated a simple model of the spatial dynamics of morningrush-hour congestion in the absence of tolls. There is a single corridor connecting a continuum of residential locations to thecentral business district. There is an exogenous density of identical commuters along the corridor, all of whommake a morn-ing trip to the CBD and have a common desired arrival time. The constant-width road is subject to classical ow congestion.A vehicles trip price is linear in travel time and early arrival time (late arrival is not permitted). Equilibrium satises the trip-timing condition that no vehicle can lower its trip price by altering its departure time. What is the equilibrium pattern ofdepartures? We termed this problem and related extensions (such as the social optimum, and the equilibrium with heter-ogeneous commuters and price-sensitive demand), The Corridor Problem.

Even though the no-toll equilibrium Corridor Problem assumes away many features of a much more complex reality, itappears very difcult to solve. We have not yet succeeded in obtaining a complete solution to the problem, and this paperpresented preliminary results.

Consider two locations x0 and x00, with x0 further from the CBD than x00. Vehicles from both locations travel in one cohort (agroup of cars arriving at the CBD at the same time) arriving at the CBD at an earlier time a0 and another at a later time a00.Since the vehicles departing x0 and x00 experience the same decrease in schedule delay when traveling in cohort a00 rather thanin cohort a0, to satisfy the trip-timing equilibrium condition they must experience the same increase in travel time. Buildingon this observation applied to all locations and cohorts, we proved that successive cohorts are identical except for the addi-tion of vehicles at more central locations. The rst cohort contains only vehicles from the locations most distant from theCBD, the second cohort is identical (in entry rate, ow, and density at the most distant locations) except that it adds vehiclesat the next most distant locations, and so on, until the last cohort contains vehicles from all locations.

Based on these results, Arnott had conjectured that therewould exist an equilibriumwith this qualitative departure patternfor an arbitrary distribution of population along the road. However, in constructing numerical solutions of equilibria with thisqualitative departure pattern, DePalma discovered that such an equilibrium exists (and when it is exists, is unique) only for atwo-parameter family of population distributions along the road. Thus, either an equilibrium does not exist for other popula-tion distributions, or our specication of the problem is incomplete, or our solutionmethod somehowoverconstrains the prob-lem. We have been unable to uncover the root of the difculty, which is why we are unable to provide a complete solution.

Once a full solution to the no-toll equilibrium of the Corridor Problem is obtained, there are many interesting applicationsand extensions to be investigated. But a full solution to the basic problemmust come rst. The previous subsection proposeda research strategy with this goal.

Appendix A

A.1. Discretization of the rst governing equation

The (x, t) coordinates of the lower boundary curve are parametrized in terms of u as bu; u TIu bu;u R bu0 dx0vx0 . Corresponding to our sequence of ow values, the sequence of coordinates of the lower boundary curve

are (bi, ti), 1 6 i 6 Nf. The time coordinates, ti, can be expressed as

ti ui Z bi0

21 1 Fx0p dx0 ui1

Z bi10

21 1 Fx0p dx0 ui ui1

Z bibi1

21 1 Fx0p dx0

ti1 ui ui1 Z bibi1

21 1 Fx0p dx0:achieve analytical tractability, theory employs toy models. These models elucidate basic principles but can take us onlyin understanding empirical phenomena. Bridging the gap are theory-based programming models. The trip-timing con-is a variational inequality. Given the current state of the art, it should be possible to model the equilibrium spatialics of rush-hour trafc congestion as the solution to variational inequality problems, taking as given the actual spatialns of residential locations and of employment locations and their work start-times. The major difculty will be incor-ng household activity scheduling and non-commuting travel.

bi bi1

We no

Z ui 2 dbdu Z ui 2 bibi1uiui1

A.2. D

Let A Au R ui Fbu0du0. From the third governing equation, (17c), for all i, 1 6 i 6 N k,

ui i

Sub

uik uik1 1wi2

We rearrange this equation to solve for ui+k, yielding our discretized version of the third governing equation, presentedearlie

764 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768As with the discretized version of the rst governing equation, this equation is a linear equation in the unknown quan-tities ui+k and bi, and if we had used another velocitydensity relation, then the discretized version of the third governingequation would have been a nonlinear equation in these two unknown quantities.r in the paper in (21):

uik uik1 2Fik Fik11wi2

wi1 bi Aik1 Ai

" #; 8i; 1 6 i 6 Nf k:Aik1 2 Fik Fik1 Ai wi 1 bi:Aik Z uik0

Fbu0du0 Aik1 Z uikuik1

Fbu0du0 Aik1 uik uik12 Fik Fik1 : A-3

stituting this expression for Ai+k into (A-2) yieldsWe use this linear approximation to directly approximate Ai+k asFbu Fi1 Fi Fi1ui ui1 u ui1; u 2 ui1; ui:If we choose k large enough, then over each subinterval (ui1,ui), we may approximate F(b(u)) as a linear function of u, as in(19):i i 0 f

Z uikFbu0du0 Aik Ai 1wi

2

w1 bi: A-2iscretization of the third governing equationNote that this equation is a linear equation in the unknown quantities ui+k and bi. If, instead of using Greenshields rela-tion, we had used another velocitydensity relation, then the discretized version of the rst governing equation would havebeen a nonlinear equation in these two unknown quantities.uik a ba

1 biwi

ti1 ui ui1 4bi bi1Fi Fi1 log1wi1wi1

wi wi1

1

; 1 6 i 6 Nf k:Substituting this approximation for ti into the rst governing equation, (17a), yields a discretized version of the rst gov-erning equation, presented earlier in the paper in (20): ti1 ui ui1 4bi bi1Fi Fi1 log1wi1wi1

wi wi1

: A-1ti ti1 ui ui1 ui1 1

1 Fbu

p du ti1 ui ui1 ui1 1

1 Fi1 FiFi1uiui1 u ui1

q du

w use these linear approximations to approximate ti:bu bi1 ui ui1 u ui1; u 2 ui1;ui:Fbu Fi1 Fi Fi1ui ui1 u ui1; u 2 ui1;uiIf we choose k large enough, then over each subinterval (ui1, ui), we may approximate F(b(u)) and b(u) as linear functions ofu, as in (19):

titiesi = Nf

Q Q

1 1

u F

Basedas

In par

Rep

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 765u1F12

aa b 1

1w1w1

1 b1: A-7

(A-6) and (A-7) are a pair of linear equations that may be solved to obtain u1 and b1. Since ui u1F1 Fi for i = 1, . . ., 1 + k, wemay determine u2, . . ., u1+k. Since our linear approximation implies that Ai F1u1

u2i2 for i = 1, . . ., 1 + k, we may determine A1, . . .,

A1+k. Finally, using (A-1) we may determine t1. To summarize,u1a

a b2

1 1w1w1

4F1

log1w1

2w1 1 1w1 b1: A-6

on our linear approximation for F(b(u)) over the interval (0,u1+k), we may calculate the cumulative ow on this interval

Au Z u0

Fbu0du0 F1u1

u2

2u 2 0; u1k:

ticular, for i = 1, . . ., 1 + k, Ai u12F1 F2i . From the third governing equation, (A-2),

A1k A1 1w12

w11 b1:

lacing Ai+k withu12F1

F21k and A1 withu1F12 , and recalling that

F1kF1

aab, we obtain 2" # 2If we replace u1+k with 1F1 F1k and recall that1kF1

aab, then after some algebraic simplication we obtain " #

u1k a ba

1 b1w1

u1 4b1F1 log1w1

2

w1 1

1

:(A-4) and (A-5) completely summarize the core of our iterative procedure. Specically, given the initial seed values b1, t1,u1, . . ., u1+k and A1, . . ., A1+k, we iteratively use (A-4) and (A-5) to determine b1; . . . ; bNfk, t1; . . . ; tNfk, u1; . . . ;uNf and A1; . . . ;ANf .At the conclusion of this procedure, the only undetermined quantities will be the (bi, ti) values in the last segment, fromi = Nf k + 1 to i = Nf.

A.4. Initializing seed values

To initiate the above iterative procedure, we must provide values for b1, t1, u1, . . ., u1+k and A1, . . ., A1+k. If we choose N largeenough, then F1, . . ., F1+k will be close to 0, and over the interval (0,u1+k) we can approximate F(b(u)) as a linear function of u.We apply the discretized versions of the rst and third governing equations, that enable us to solve for u1 and b1, and, hence,determine all necessary initializing seed values. Specically, suppose that over the interval (0,u1+k) we approximateFbu F1u u. Therefore, for i = 1, . . ., k + 1, ui u1F Fi. From the discretized version of the rst governing equation, (20),bi 1 3Q2Q4 ti ti1 ui ui1 bi bi1Quik Q1 biQ2 Aik Ai 1wi

2

wi1 bi

A-5The updated values may be calculated as follows:Q4 a ba

1wi

Q

Q3 a ba

1wi

ti1 ui ui1 Qbi1 1Q2 21wi

wiFik Fik1 A- 4 ik ik1 i

2Q1 uik1 2

F F1wi2

w Aik1 Ai

" #

Q 4

Fi Fi1 log1wi1wi1 wi wi1bi and ui+k, and then use these values in (A-3) and (A-1) to determine Ai+k and ti. This procedure may be iterated untilk. Specically, at each step calculate the quantities Q, Q1, . . ., Q4 in terms of the known quantities: A.3. Iterative procedure

We may solve the discretized versions of the rst and third governing equations, (20) and (21), for the unknown quan-

b1 1w11w1 F12

2w1 log 1w12 1w21 F12

u1 21w121 b1

F1w1 aab 2

1

t1 u1 4b1F1 log1w1

2

w1 1

A- 8

ui u1F1 Fi; i 2; . . . ;1 k

Ai ui2F1 F2i ; i 1; . . . ;1 k:

A.5. Final segment

After implementing the above iterative procedure, we will have determined all Fi, wi, ui and Ai values for i = 1, . . ., Nf, andwill have determined all bi, ti values for i = 1, . . ., Nf k. The only remaining values to determine are bNfk1; tNfk1; . . . ;bNf ; tNf , corresponding to the (x, t) coordinates of the lower boundary curve in the nal segment. Furthermore, since wehave determined uNf (the departure time at x = 0 of the nal cohort of vehicles that arrives at the CBD exactly at time t), from(4) we can calculate t as t aabuNf 1.

Suppose that (bi, ti) is known for some Nf k 6 i < Nf, and we wish to determine (bi+1, ti+1). Subdivide the ow from itsvalue Fi at (bi, ti) to its value Fi+1 at (bi+1, ti+1) into m equal values, and let (Fi)j denote the ow value at the jth subdivision,

so Fq

Co i+1 i+1

Fig. 14(bi)0 = bthen ~xm

766 R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768. For i = Nf k + 1 to i = Nf 1, proceed as follows. Guess a value of bi+1, and subdivide the lower boundary curve from bi to bi+1 into m segments,i,. . ., (bi)j, . . ., (bi)m = bi+1. Iteratively construct the vehicle trajectory through (bi+1, ti+1) from the point ~x0;~t0 to ~xm;~tm. If bi+1 was chosen correctly, bi1.nates in Region II as ~x;~t. The characteristic line emanating from ((bi)j, (ti)j) intersects this vehicle trajectory at the point~xj;~tj, j = 0, . . ., m. Beginning with j = 0, proceed as follows. Approximate the trajectory curve through the point ~xj;~tj as astraight line with slope dtdx 1Vij, whose equation is given by

x ~xj Vijt ~tj:j m

tij ti jm ui1 ui 4bi1biFi1Fi log1wij1wi

fwij wig

h i:

A-9

nsider the vehicle trajectory that passes through the lower boundary at (b , t ), and denote its space-time coordi-v ij 12 1 1 Fij, for j = 0, . . ., m. If we use the linear approximations for F(b(u)) and b(u), (19), and the resulting timecoordinate along the lower boundary curve, (A-1), then we may approximate the space-time coordinates along the lowerboundary curve corresponding to the sequence with j = 0, . . ., m as ((bi)j,(ti)j), where

bi bi j bi1 biij Fi jm Fi1 Fi, j = 0, . . ., m. Based on these values we may calculate wij 1 Fij andq

F: ow, density times velocityn(x,t):D: setp(x): eqa: arrivbT x; a:Region

R. Arnott, E. DePalma / Transportation Research Part B 45 (2011) 743768 767Region II: (x,t) points below the departure setb(u): x-coordinate of the lower boundary as a function of uTI(u): travel time to the lower boundary as a function of uI: (x,t) points within the departure setA: set of (x,a) points for which arrival rate is positiveu: departure time of a vehicle departing location x = 0entry rate, or departure rate, onto the road at (x,t)of (x,t) points at which departures occuruilibrium trip price of a departure at location xal time at the CBDtravel time to the CBD of a departure from location x and arriving at time aNow consider the characteristic line emanating from ((bi)j+1, (ti)j+1), that has slope dtdx 1wij1, and whose equation is givenby

x bij1 wij1t tij1:The (x, t) intersection of these two lines is taken as the (j + 1)st point along the vehicle trajectory, ~xj1;~tj1, i.e.,

~tj1 wij1tij1 bij1 ~xj Vij~tj

wij1 Vij~xj1 wij1~tj1 tij1 bij1:

If m is chosen large enough, and we iterate this procedure from j = 0, . . ., m 1, then the nal point on our trajectory,~xm;~tm, should coincide with the point on the lower boundary curve, (bi+1, ti+1). The above procedure depends upon our ini-tial choice for bi+1, and the magnitude of our error in choosing bi+1 can be measured by the difference between ~xm and bi+1.Thus, to determine bi+1, we construct a function that calculates this error for various values of bi+1, and choose the value ofbi+1 that minimizes this error. Once we have determined bi+1 we update ti+1, and then iteratively repeat this procedure untilobtaining bNf ; tNf . A graph illustrating these concepts is provided in Fig. 14.

References

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Glossary

CBD: Central Business DistrictTT: Trip-Timing conditionx: distancet: timex: location of CBDt: work start-time at the CBDN(x): population densityC: total travel costa: unit cost of travel timeb: unit cost of time early arrivalT(x,t): travel time from (x,t) to the CBDq(x,t): density (vehicles/length) at (x,t)v(x,t): velocity at (x,t)V(q): velocity as a function of density

A(u): cumulative ow along the lower boundarybAx; t: cumulative ow in Region IIu0,uf: pair of departure times from x = 0, as in Fig.6~x;~t: space-time coordinates of a vehicle trajectory in Region IIV0: maximum, free-ow velocity (Greenshields)qJ: jam density at which velocity is zero(Greenshields)w: wave-velocity or slope of the ow-density curve, ddq qVi, j: dummy indicesN: number of segment divisions of the lower boundary curvek: number of subdivisions within each segmentNf = Nk: total number of points along the lower boundary curveui,bi,Fi,vi,wi,Ai,ti: function values along the lower boundary curveQ,Q1, . . . ,Q4: quantities calculated in the iterative procedurem: number of subintervals into which we divide each subdivision when