intra-period (within-day) dynamic models for continuous services

Networks and Spatial Economics, 3: (2003) 271–296©C 2003 Kluwer Academic Publishers, Manufactured in the Netherlands.

Intra-Period (Within-Day) Dynamic Modelsfor Continuous Services

ENNIO CASCETTADepartment of Transportation Engineering, University of Naples “Federico II”, Via Claudio 21,80125 Napoli, Italyemail: [email protected]

PIERLUIGI COPPOLADepartment of Civil Engineering, University of Rome “Tor Vergata”, Via del Politecnico 1, 00133 Roma, Italyemail: [email protected]

Abstract

To simulate important aspect of some transportation systems (e.g. demand peaks, temporary capacity variations,temporary over-saturation of supply elements, and formation and dispersion of queues) a new class of models,referred to in the literature as Dynamic Traffic Assignment (DTA) models, have been recently developed. AlthoughDynamic Traffic Assignment to networks is a relatively new research subject, a great number of models have beenproposed in the last two decades. These can be divided in two main classes according to the typology of servicethey aim at simulating. These are continuous services, considering transportation services available at any timeand accessible from several points, such as the services offered by individual road modes (car, bicycle etc.), andscheduled services simulating services available only at certain times and that can be accessed only at certainlocations (terminals, stations, airports etc.). In this paper the focus is on continuous services. Models proposedin the literature are reviewed and classified according to basic assumptions on the flow structure, i.e. whether acontinuous or a discrete approach is followed, and on the representation of time (discrete vs. continuous). A generalmodeling framework consisting of supply, demand, and demand-supply interaction models, and including mostof the existing specifications is presented both for the discrete time-discrete flow and continuous time continuousflow cases.

Keywords: continuous services, discrete time-discrete flow, continuous time-continuous flow

1. Introduction

Traditional mathematical models for the simulation of transportation system are basedon the assumptions of intra-period stationariety. This is equivalent to assuming that allsignificant variables are constant, at least on average, over successive sub-intervals of areference period long enough to allow the system to reach a condition of stationariety. Thisassumption, although acceptable for many applications, does not allow the satisfactorysimulation of some transportation systems such as heavily congested urban road networks.In fact, some important phenomena cannot be reproduced by traditional intra-period staticmodels, including demand peaks, temporary capacity variations, temporary over-saturationof supply elements, and formation and dispersion of queues.

272 CASCETTA AND COPPOLA

To simulate these aspects, different intra-periodal or within-day dynamic models, typi-cally referred to in the literature as Dynamic Traffic Assignment (DTA) models have beendeveloped. These have different formulations and levels of complexity depending on thetype of supply system involved. Transportation services and representative supply modelscan be divided in two main classes: continuous and scheduled. The first case considersservices available at any time and accessible from several points, such as the services of-fered by individual road modes (car, bicycle etc.). On the other hand scheduled servicesare available only at certain times and can be accessed only at certain locations (terminals,stations, airports etc.). In the rest of this paper we focus on continuos services (e.g. roadnetworks) models only; examples of dynamic traffic assignment models for discrete ser-vices can be found in the papers by Hickman and Bernstein (1997) and by Nuzzolo et al.(1999).

The general framework for intra-period dynamic models for continuos services dependson the basic assumptions on the flow structure, i.e. whether a continuous or a discreteapproach is followed, and on the representation of time, i.e. continuous-time and discrete-time models. In principle, all the combinations are possible but typically flow-discretizationimplies the time-discretization and vice versa.

In continuous flow-continuous time, also referred to in the literature as analytical models(Astarita, 1996; Carey, 1992; Chabini and He, 1997; Friesz et al., 1989, 1993; Ran et al.,1993; Wu et al., 1995; Xu et al., 1999), users are modeled as “particles” of a mono-dimensional, partly compressible fluid, moving at different rates through the system. Onthe other hand, discrete flow-discrete time, also known as simulation model (Ben Akivaet al., 1997; Cascetta and Cantarella, 1991; Cantarella et al., 1999; Jayakrishnan et al.,1994; Yang and Koutsopoulos, 1996) assume that users are discrete units; these can bepackets, e.g. groups of vehicles sharing the same trip, or individual vehicles. Flows are heredefined as the number of user units moving in a time interval.

The paper is organized as follows. In Section 2 an overall formulation for dynamicsupply models is formalized respectively for the continuous flow-continuous time caseand for the discrete flow-discrete time one. In Section 3 within-day dynamic demandmodels are reviewed. Finally, in Section 4 a general within-day dynamic formulationof demand-supply interaction model is formally stated for congested and uncongestedsystems.

Note that throughout the remainder of this paper, in order to simplify notation and analysis,a single user class and fixed origin-destination demand flows will be assumed.

2. Within-day dynamic supply models

Within-day dynamic supply models, like static models, express flows and performancesof the system (flows on individual links, travel times, generalized costs, etc.) as functionsof the path flows and the characteristics of the physical system. Although the componentsof a dynamic supply model are the same (see figure 2.1), the within-day dynamics impactthe relationships between link and path flows and costs which are no longer linear as instatic models.

INTRA-PERIOD (WITHIN-DAY) DYNAMIC MODELS FOR CONTINUOUS SERVICES 273

2.1. Continuous flow-continuous time supply models

Continuous flow-continuous time models were the first to be investigated by the scientificcommunity (Merchant and Nemhauser, 1978). These models can be further classified by therepresentation of space. Space discrete (link-based) models are closer to static models: thebasic variables influencing link performances such as densities and speeds are defined withrespect to links. Other continuous flow model (i.e. continuous space models) are based on adirect application of differential equations systems derived from continuous space models oftraffic flows for each link (Lighthill and Whitham, 1955; Payne, 1971; Papageorgiou et al.,1989) together with the equations assuring flow conservation at each node. The solution ofthese models, at least in theory, allows the definition of variables such as flow, speed, anddensity at each point s and at each instant τ . The solution of such models however requiresa discretization in space, �s, hence from the solution point-of-view they can be consideredsimilar to discrete space model through a duly definition of link length (i.e. �s = La). Thefollowing will focus mainly on link-based models.

2.1.1. Variables and consistency conditions. Variables of continuous flow link-based dy-namic supply models can be classified into three groups: topological, flow/occupancy,time/cost variables.

Topological variables. The topological features of a journey are modeled through a graphmodel. For these systems all concepts related to graph models for continuous transportationservices such as centroids, paths, incidence matrices etc. (see, for instance, Cascetta, 2001)extend directly to within-day dynamic networks. Let:

a: be the index of a link of length La ;k: be the index of a path, made up of a sequence of links ak

1, ak2, . . . , ak

nk where aki is the i th

link of path k, nk the number of link of path k;ak

i+1: be the link following aki on path k;

aki−1: be the link preceding ak

i on path k.

Flow and occupancy variables. For analytical convenience it will be assumed that all theflow variables are continuous and continuously differentiable functions of time τ ≥ 0. Let

dod (τ ): be the origin-destination demand flow at time τ , i.e. the flow rate of users leavingzone (node) o at time τ to zone (node) d;

d(τ ): be the vector of O-D flows at time τ ;hk(τ ): be the path flow of users who start their journey at time τ and follow path k;hod (τ ): be the path flow vector with components given by the flows hk(τ ): relative to each

path connecting od pair, k ∈ Kod ;h(τ ): be the total vector of path flows for all the O-D pairs at time τ . Under the assumptions

made, users are no longer discrete elements but fluid particles leaving on path k at a timedensity given by hk(τ ), the time-space trajectories of the particles leaving at each pointin time can be traced along the links making up the path. The number of users leavingon path k in the infinitesimal interval (τ, τ + dτ ) is equal to hk(τ )dτ ;

f kas(τ ): be the user flow following path k and crossing section s of link a at time τ . Unlikethe static case, it is not possible to define a generic link flow since flow crossing at the


SUPPLY MODEL

PATH USERCOSTS AND

PERFORMANCES

SERVICE ANDCONNECTION

CHARACTERISTICS

GRAPH

PATHPERFORMANCE

MODEL

PATHS (LINKSEQUENCES)

LINKPERFORMANCES

LINK FLOWSLINK

PERFORMANCEMODEL

NETWORK FLOWPROPAGATION

MODEL

PATHFLOWS

IMPACTFUNCTIONS

EXTERNALIMPACTS

Congested System

Within-day Dynamic System

Figure 1.1. Schematic representation of within-day dynamic supply models (Cascetta, 2001).

time τ the different sections of a link usually is not constant over the link. Among thesections of a link, entrance (s = 0) and exit (s = La) are particularly relevant;

uka(τ ) = f k

a,0(τ ): be the flow traveling on path k and entering link a at time τ (in-flow),uk

a(τ ) ≥ 0;wk

a(τ ) = f ka,La(τ ): be the flow traveling on path k leaving link a at time τ (out-flow),

wka(τ ) ≥ 0;

fa,s(τ ), ua(τ ), wa(τ ): be the total flow crossing section s, entering and leaving link a,at time τ , respectively, and relate to the path-specific variables through the followingrelationships:

fa,s(τ ) =∑

k

f ka,s(τ ) (2.1a)

ua(τ ) =∑

k

uka(τ ) (2.1b)

wa(τ ) =∑

k

wka(τ ) (2.1c)


Ua(τ ), Wa(τ ): be the cumulative in-flow and out-flow on link a at time τ ; respectively; theyexpress the total number of users who entered and left the link up to a given point in timeτ . Cumulative flows relate to flow rates through the following equations:

Ua(τ ) =∫ τ

0ua(t) dt

Wa(τ ) =∫ τ

0wa(t) dt

xa(τ ): be the number of users on link a at time τ or link occupancy;ka(τ ) = xa(τ )/La : be the users density on link a at time τ .

Temporal profiles of flow variables must satisfy conservation equations since flows cannotbe created or dispersed at any point of the network except centroid nodes. If no flows aregenerated and/or absorbed at a node i (i.e. the node i is not a centroid node) flow conservationconditions require that in-flows, out-flows and path flows satisfy the following equations:

uka1

k(τ ) = hk(τ ) (2.2a)

ukak

i+1(τ ) = wk

aki(τ ) (2.2b)

and summing over all paths, Eq. (2.2b) yield:∑a∈F S(i)

ua(τ ) =∑

a∈BS(i)

wa(τ ) (2.2c)

Equation (2.2c) constrains the total out-flow of the links belonging to its backward starBS(i) to equal the total in-flow on all the links belonging to its forward star FS(i) at anytime τ for a node i that is not a centroid. Equations (2.2) can be extended easily for centroidnodes distinguishing between paths ending and/or starting in node i .

Link density at time τ can be expressed as a function of (non-negative and integrable) in-flow and out-flow temporal profiles. The differential equation expressing flow conservationon a link is:

dxa(τ )

dτ= ua(τ ) − wa(τ ) (2.3)

which, once integrated, leads to the following result:

xa(τ ) = La · ka(τ ) =∫ τ

0ua(t) dt −

∫ τ

0wa(t) dt = Ua(τ ) − Wa(τ ) (2.4)

Equation (2.4) expresses the relationship among density and cumulative in-flow and out-flow on a link. The number of users on link a at time τ is the difference between thecumulated in-flow and out-flow at that time.

Travel time and cost variables. In within-day dynamic supply models the travel time hasa dual role. It is a performance variable included as an attribute of the generalized perceivedcost, as in within-day static models. On the other hand, it ensures the internal consistency ofthe relationships between some variables of the model. For this reason, travel time is denotedwith a specific variable different from the other performance variables. Moreover, travel


times of links and paths may assume different values for different time instants; this maydepend on different transportation supply and/or congestion conditions. For this reason, anumber of new variables related to travel time must be introduced. It will be assumed thattravel times are continuous and continuously differentiable functions of the absolute timeτ . Let

t fa (τ ): be the forward travel time, i.e. the time to cross link a for a flow particle entering the

link at time τ ;tba (τ ): be the backward travel time, i.e. the time to cross the link a for a flow particle leaving

the link at time τ ;t La (τ ): be the leaving-time function, representing the leaving time of a particle entering link

a at time τ ;t−1a (τ ): be the inverse travel time function, representing the entrance time of a particle

leaving link a at time τ ;eca(τ ): be the generalized extra cost for crossing link a entering at time τ . The generalized

extra cost expresses the perceived disutility of link a with the exception of the traveltime. It includes other performance variables, e.g. time variable tolls, homogenized indisutility terms.

Temporal consistency of the model requires that the different travel times satisfy the fol-lowing relationships:

t fa (τ ) = tb

a

(τ + t f

a (τ ))

(2.5a)

tba (τ ) = t f

a

(τ − tb

a (τ ))

(2.5b)

t−1a (τ ) = τ − tb

a (τ ) (2.5c)

t La (τ ) = τ + t f

a (τ ) (2.5d)

Travel times themselves must be consistent. In fact, under the assumption of partly com-pressible mono-dimensional fluid, travel time functions must be modeled such that a fluidparticle entering at time τ ′′ on link a can never reach, or overtake, another particle thatentered the same link at an earlier time τ ′ < τ ′′. If this were the case, it would imply thefluid left between τ ′ and τ ′′ be compressed to a zero space (infinite density) or, in thecase of overtaking, the mono-dimensionality assumption of the fluid (no turbulence alonga link) would be violated. This condition is usually referred to in the literature as strongFirst-In-First-Out (FIFO) rule, and can be stated formally as:

τ ′ + t fa (τ ′) < τ ′′ + t f

a (τ ′′) ∀τ ′ < τ ′′ (2.6a)

similarly for the backward travel time:

τ ′ − tba (τ ′) < τ ′′ − tb

a (τ ′′) ∀τ ′ < τ ′′ (2.6b)

A weak FIFO rule is obtained when the strict inequality is substituted by weak inequalitywithin the above conditions. For sake of brevity this topic will not be discussed in the fol-lowing. Relationships (2.6a) and (2.6b) imply that τ + t f

a (τ ) and τ − tba (τ ) are well-defined

strictly increasing functions of τ , i.e. a single value of exit or entrance time correspondto each value of τ . On the other hand, as said above, without the FIFO rule, two particles


could cross the same section at the same time. Thus, a single value of absolute time maycorrespond to different values of speed and acceleration in the same point in space and theinverse of time function t−1

a would be ill-defined. It can be shown easily that a sufficientcondition for a FIFO discipline (2.6a) is the following:

dt fa (τ )

dτ> −1 ∀τ (2.7a)

Similarly in terms of backward travel time it can be shown that a sufficient condition for aFIFO discipline (Eq. (2.6b)) is:

dtba(τ )

dτ< +1 ∀τ (2.7b)

The physical interpretation of Eqs. (2.7) is that to avoid FIFO rule violation, the travel timecannot decrease more rapidly than the absolute time.

Several equivalent conditions have been proposed to impose a FIFO discipline (see, forexample, Friesz et al., 1993; Astarita, 1996; Chabini and Kachani, 2000). One of the mostintuitive states that a FIFO discipline exists if and only if the total number of vehiclesentering a generic link a by time τ equals the total number of vehicles exiting after a timeinterval equal to the forward travel time of link a at any time τ :

Ua(τ ) = Wa(τ + t f

a (τ )) ∀τ (2.8a)

similarly in terms of backward travel time:

Ua(τ − tb

a (τ )) = Wa(τ ) ∀τ (2.8b)

By differentiating the above relationships it follows that:

ua(τ ) = wa(τ + t f

a (τ )) ·

(1 + dt f

a (t)

dτ(τ )

)(2.9a)

ua(τ − tb

a (τ )) ·

(1 − dtb

a (t)

dτ(τ )

)= wa(τ ) (2.9b)

Equations (2.9) express the relationship between link in-flows and out-flows in a dynamiccontext. From Eq. (2.9a) the in-flow on a link at time τ is equal to out-flow at the corre-sponding exit time (i.e. the absolute time after the link forward travel time) multiplied bya factor larger than one (i.e. the out-flow is less than the in-flow) if the flow on the link isslowing down (dt/dτ > 0) and vice versa. Similar considerations can be derived analyzingEq. (2.9b): the out-flow on a link at time τ is equal to the in-flow at the correspondingentrance time multiplied by a factor larger than one (i.e. the out-flow is bigger than thein-flow) if the flow on the link is accelerating (dt/dτ < 0) and vice versa. Note that inthe static case since link travel times are constant over time, in-flows always equal out-flows. Moreover, since a positive in-flow on a link, ua(τ ) > 0, implies a positive out-flow,wa(τ ) > 0, conditions (2.7) for FIFO discipline can be derived again from Eqs. (2.9).

Once that the main variables and their consistency relationships have been introduced,the other components of the supply model, namely link performance functions, path per-formance functions and network flow propagation model can be analyzed.


2.1.2. Link performance and travel time functions. Fundamental to dynamic supplymodels are the link travel time functions expressing travel time as a function of link flowsfor congested networks. Most models proposed in the literature adopt functions assumingexplicitly, i.e. travel time functions (Friesz et al., 1993; Fernandez and de Cea, 1994), orimplicitly, i.e. exit functions (Wie et al., 1990), the travel time on a link depending on thenumber of users traveling on the link. Implicit exit time functions express directly the out-flow of a given link as a function of the link occupancy wa(τ ) = wa(xa(τ )). These functions,however, lead to a number of theoretical inconsistencies (see, for example, Astarita, 1996)and will not be considered in the following.

Travel time functions express the travel time ta(τ ) of a particle arriving at the beginningof the link a at time τ as a function of the relevant traffic condition variables. Most modelsproposed adopt “separable” travel time functions, i.e. functions expressing the travel timet fa (τ ) in terms of the instantaneous occupancy on the same link xa(τ ):

t fa (τ ) = ta(xa(τ )) (2.10)

Several functional forms have been proposed for Eq. (2.10), not all of which however leadto results consistent with the FIFO rule. One of the proposed functions is the linear traveltime function (Friesz et al., 1993):

t fa (xa(τ )) = t0

a + 1

Qa· (xa(τ ))

where Qa is, as usual, the capacity of link a. It can be shown that a linear travel time functionimposes the FIFO discipline and the consistency of the model. Furthermore the out-flowwa never exceeds the capacity of link a.

2.1.3. Path performance and travel time functions. Specific time variables can also beassociated with paths. Let

T fak

i(τ ) [or equivalently T f

ak (τ )]: be the forward travel time to link ai [a] along path k; i.e.

the time needed to reach the beginning of link ai [a] following path k and leaving at timeτ from the beginning of link ak

1 ;T b

aki(τ ) [or equivalently T b

ak (τ )]: be the backward travel time necessary to reach the beginning

of link ai [a] following path k and arriving at time τ .

Temporal consistency of the model requires that the different travel times satisfy the fol-lowing relationships:

T fak

i(τ ) = T b

aki

(τ + T f

aki(τ )

)(2.11a)

T bak

i(τ ) = T f

aki

(τ − T b

aki(τ )

)(2.11b)

Moreover, let:

TT fk (τ ): be the forward total travel time of path k, i.e. the time needed to traverse path kstarting at time τ ;

TT bk (τ ): be the backward total travel time on path k, i.e. the time needed to traverse path karriving at time τ ;


ECk(τ ): be the path k extra-generalized cost starting at time τ ;gk(τ ): be the total generalized cost along path k leaving at time τ .

Equation (2.11a) can be applied recursively from the first link ak1 to the generic link ak

i ofpath k. This results in a “nested” sum of link travel times:

T fak

i(τ ) = t f

ak1(τ ) + t f

ak2

(τ + t f

ak1(τ )

)+ t f

ak3

(τ + t f

ak1(τ ) + t f

ak2

(τ + t f

ak1(τ )

))+ · · · + t f

aki−1

(τ + t f

ak1(τ ) + · · · + t f

aki−2

(τ + t f

ak1(τ ) + · · ·

))(2.12a)

Similarly Eq. (2.11b) can be applied from link aki to the first link ak

1 of path k:

T bak

i(τ ) = tb

aki−1

(τ ) + tbak

i−2

(τ − tb

aki−1

(τ ))

+ tbak

i−3

(τ − tb

aki−1

(τ ))

− tbak

i−2

(τ − tb

aki−3

(τ ))

+ · · · + tbak

1

(τ − tb

aki−1

(τ ) − · · · − tbak

2

(τ − tb

aki−1

(τ )

− · · · − tbak

3

(τ − tb

aki−1

(τ ) − · · ·)))

(2.12b)

Previous equations can be easily extended to express total path travel time, path extra-cost(assuming link-wise additive attributes) and generalized cost as functions of link traveltimes:

TT fk (τ ) = t f

ak1(τ ) + t f

ak2

(τ + t f

ak1(τ )

)+ · · · + t f

aknk

(τ + · · ·) = T fak

nk

(τ ) + t fak

nk

(τ + · · ·)(2.13a)

ECk(τ ) = ecak1(τ ) + ecak

2

(τ + t f

ak1(τ )

)+ · · · + ec f

aknk

(τ + · · ·) (2.13b)

gk(τ ) = βt TT fk (τ ) + ECk(τ ) (2.13c)

The relationships between the vectors of forward path travel time functions TT f (τ ),with one component for each path in the network and the vectors of forward link traveltime functions t(τ ),with one component for each link in the network, can be expressedsymbolically as:

TT f (τ ) = �(t(τ ′), τ ′ > τ ) (2.14)

Equations (2.14) are the within-day dynamic equivalent of the link-wise cost compositionexpressed by supply model for static networks. In the static case the order in which linkperformance attributes or costs are summed to obtain path costs is irrelevant. This is no longertrue for within-day dynamic supply models in which link times and costs have to be summedup in their topological order along path k to satisfy the temporal succession of crossed links.

2.1.4. Dynamic Network Flow Propagation models. The Dynamic Network Flow Propa-gation (or DNL, Dynamic Network Loading) model simulates how time-varying continuousinput path flows propagate through the network inducing time-varying in-flows, out-flowsand link occupancies.

The simplest case is that of a single-link network. The link flow propagation modelcan be expressed formally by combining the different consistency equations introduced inthe previous section and the travel time function. In fact, the whole model expressing the


continuous link flow dynamics is specified as a function of a single input variable, usually in-flow, since the four variables defining the dynamics of the link, namely ua(τ ), wa(τ ), xa(τ )and t f

a (τ ), are connected by three equations:

dxa(τ )

dτ= ua(τ ) − wa(τ )

t fa (τ ) = ta(xa(τ ))

and, under the FIFO rule condition, such as:

ua(τ ) = wa(τ + t f

a (τ )) ·

(1 + dt f

a (τ )

dτ

)

The DNL model can be extended to general networks if the FIFO condition is satisfiedby link and path travel times. In this case the conservation and link dynamics equationshave to be stated with respect to specific path values:

ukak

1(τ ) = hk(τ ) ∀k (2.15a)

ukak

i(τ ) = wak

i−1(τ ) ∀k; ∀i = 1 . . . nk−1 (2.15b)

dxka (τ )

dτ= uk

a(τ ) − wka(τ ) ∀k; ∀a (2.15c)

xa(τ ) =∑

k

δak xka (τ ) ∀a (2.15d)

t fa (τ ) = ta (xa(τ )) ∀a (2.15e)

wka

(τ + t f

a (τ )) = uk

a(τ )(1 + dt f

a (τ )dτ

) ∀a (2.15f)

in addition to the boundary conditions (e.g. u(0) = w(0) = x(0) = 0).The above equations give an implicit representation of the DNL model; they can also be

reformulated in such a way to bear a closer resemblance to their static counterpart, whichcan be shown to be a particular case. To see this, consider the generic path k up to link a assingle link (see figure 2.1). At the generic time instant τ , the in-flow on this “link” is givenby hk(τ − T b

ak (τ )) where T bak (τ ) is the travel times needed to traverse it, while the out-flows

can be computed by applying Eq. (2.9b). By summing over all the paths containing link a,we obtain the in-flow on the link a at time τ, ua(τ ):

ua(τ ) =∑

k

δak · hk(τ − T b

ak (τ )) ·

(1 − dT b

ak (τ )

dτ

)(2.16)

being δak equal to 1 if link a belongs to path k, 0 otherwise (i.e. the generic element of thelink-path incidence matrix).

Note that in the static case in-flows are constantly equal to out-flows, path flows areconstantly equal to hk and both link and path travel time are constant over time:

ua(τ ) = wa(τ ) = ua = wa = fa

hk(τ ) = hk


Figure 2.2. Representation of the Dynamic Network Flow Propagation model.

t fa (τ ) = tb

a (τ ) = ta

dt fa (τ )

dτ= dtb

a (τ )

dτ= 0

Moreover, the fundamental diagram relations yields:

fa = xa

La· va = xa

ta

where va is the link speed and La is the link length. Then the system of Eqs. (2.15) becomesthe linear system:

f a =∑

k

δakhk = � · h

expressing the network flow propagation model for static networks.On the other hand if path travel times are constant over time (

dT ba,k (τ )dτ

= 0), e.g. the networkis uncongested and there are no changes in supply, the in-flow profile is the summation ofthe path-flow profiles for the paths including the link, shifted by the time needed to reachthe link:

ua(τ ) =∑

k


a,k(τ ))

(2.17)

2.1.5. Formalization of the whole supply model. The equations introduced in the previoussections express the dependence of in-flows, out-flows occupancies, link and path traveltimes and costs on path flows leaving in previous time instants. The relevant equations


defining the overall supply model for congested networks respecting FIFO rule, can beexpressed symbolically as:

f = �[t(τ ), h(τ )] (2.18a)

t(τ ) = t( f (τ ′), τ ′ ≤ τ ) (2.18b)

TT f (τ ) = �(t(τ ′), τ ′ > τ ) (2.18c)

where:

t(τ ): is the vector of link travel times at time τ ;TT f (τ ): is the vector of forward path travel times at time τ ;f (τ ): denotes the vector of relevant flow or occupancy input variables for travel time

functions at time τ ;h(τ ): is the path flow vector at time τ ;�: expresses symbolically the relationship between link and path travel times, see

Eqs. (2.12);�: expresses symbolically the Dynamic Network Loading model, see Eqs. (2.15);

Note that Eqs. (2.18) reflect the fact that for congested networks the time to cross each linkat a time τ depends on the flow on all the links of the network in the previous time τ ′, as itdepends on the travel time to reach the link along the generic path k and, thus, depends onthe travel time on links preceding a along each path k.

The solution of the dynamic supply model described is based on time discretization of thedifferential equation defining it. Given the large number of differential equations involved,the sequence in which they are processed is also relevant. Note that this formulation of thesupply model assumes that the relevant congestion variables influencing travel times are link-related occupancies at the time of arrival of a given particle at each link. This assumption,however convenient from the computational point of view and “closer” to the static model,is appropriate only for deterministic queuing links and for very short running links.

2.2. Discrete flow-discrete time supply models

Discrete flow-discrete time can be based on two different approaches according to the wayin which space is treated. Mesoscopic models simulate the network performances at anaggregated level; as in the link-based continuous flow-continuous time models, aggregatedvariables of capacity, flows and occupancy are used. The traffic, however, is representeddiscretely by tracing the trips of single packets; each packet is characterized by a departuretime and by a path up to the destination. It is usually assumed that the packets are con-centrated at a point (concentrated or piled packets); this assumption is more realistic thesmaller the size of the packets. Mesoscopic models can be applied to networks of generalform and extended to simulate queue-formation and spill-backs with reasonable computingtimes. On the other hand, they do not allow a detailed simulation of traffic phenomena(overtaking, lane-changing, etc.). Microscopic models explicitly simulate the time-spacetrajectory of each individual vehicle (speed, acceleration, etc.) and its interactions withnearby vehicles (overtaking, lane-changing, etc.) given the departure time from the origin


and the path followed to the destination, as well as some individual characteristics (such asdesired speed, driving style, etc.). These models (which in traffic theory literature are oftenreferred to as micro-simulation models) can be very accurate. However, they do not allowthe explicit formulation of the whole assignment model or the analysis of its theoreticalproperties. Moreover, it is very difficult to calibrate all the parameters of the model andconsiderable computational resources are required. In the following we will refer only tomesoscopic models.

In this class of models users are considered as discrete units; they can be vehicles or groupsof vehicles moving over the network and experiencing the same trip. In the following,discrete units will be referred to as packets including the special case of single-vehiclepackets. Moreover the reference period is subdivided into intervals [ j] (in the following,intervals are assumed to be of equal duration DT for simplicity sake). These models oftenassume that relevant flow variables are averaged over time intervals. They also assume thatusers begin their trips at a characteristic time instant, τ j , of an interval [ j]. This may be thebeginning or mid-point of the interval. In principle the duration of departure intervals candiffer from the duration of averaging intervals. For example, some models use very shortdeparture intervals while averaging the variables over longer intervals. In the following,to simplify notation, only the single-interval case will be considered, the generalizationto multiple intervals is rather straightforward. Furthermore it will be assumed that therepresentative instant of each interval is its final point, i.e. τ j = [ j] · DT .

A general framework for discrete flow-discrete time models is more difficult to formalizethan for continuous models, since there are several possibilities to discretize the relevantvariables. The framework proposed in the following reflects a general formulation.

2.2.1. Variables and consistency conditions. Like continuous models, variables and their“structural” relationships must first be defined.

Time variables. The discretization of time requires the introduction of other time variablesin addition to the generic absolute time τ . Let

τ ( j): be the generic instant of time interval [ j], τ ( j) ∈ ([ j − 1] · DT, [ j] · DT);τ j : be the characteristic instant of time interval [ j], here assumed to be its end-point,

τ j = [ j] · DT .

Topological variables. Topological variables are the same as in the continuous-flowcontinuos-time case and will not be restated.

Flow and occupancy variables. The flow variables have the same definitions as in thecontinuous case, but in discrete flow models they represent “counts”, i.e. number of users ina generic interval [ j], rather than flows, i.e. temporal densities. In the following, however,they will be referred to indifferently as units (in a time interval) or flows to simplify thenotation and the extension of continuous flow results. Let

dod [ j]: be the number of users moving between the pair od leaving in the representativeinstant of interval [ j];

hk[ j]: be the number of users starting their trip along path k, k ∈ Kod , in (the representativeinstant of interval [ j]); hk[ j] can be seen as the dimension of the packet k j ;


f ka,s[ j], uk

a[ j], wka[ j]: be respectively the number of users moving on path k and crossing

section s of link a, the number of users on path k entering and leaving link aduringinterval [ j];

fa,s[ j], ua[ j], wa[ j]: be respectively the total, summed over all path, number of userscrossing section s of link a, the total number of users entering and leaving link a duringinterval [ j].

Equations (2.1) expressing the total flows as sum of path flows and (2.2) expressing flowconservation at nodes hold also in the discrete case.

Flow variables can be defined also with respect to any sub-interval of interval j e.g. theinterval [τ j−1, τ ( j)] up to time τ ( j) in this case they will be denoted as f k

as[τ ( j)] and soon. Let

xa(τ j ), xa(τ ( j)): be the number of users on the link (i.e. the link-occupancy) respectivelyin time instants τ j and τ ( j);

x̂a[ j]: be the average occupancy on link a during interval [ j]. It obviously results that:

x̂a[ j] = 1

DT

∫ [ j]DT

[ j−1]DTxa(τ ( j)) dτ ( j)

Ua[τ j ]Ua[τ ( j)], Wa[τ j ], Wa[τ ( j)]: be respectively the cumulated in-flows and out-flowsof link a up to the representative instant of interval [ j] and to a generic time instantwithin that interval respectively. Cumulated in-flows and out-flows are related to intervalspecific values as:

Ua(τ j ) =∑j ′< j

ua[ j] (2.19a)

Wa(τ j ) =∑j ′< j

wa[ j] (2.19b)

In-flows and out-flows are also related to link occupancy through link conservation equationsanalogous to Eqs. (2.3) and (2.4):

xa(τ j ) − xa(τ j−1) = ua[ j] − wa[ j] (2.20a)

xa(τ j ) = Ua(τ j ) − Wa(τ j ) (2.20b)

Travel time and cost variables. In general, link and path travel times are continuousvariables related to generic time instant τ as in the continuous case; in the discrete case,however, not all instants τ are meaningful since not all correspond to the arrival (or departure)of a packet. In the following time and cost variables will be introduced with respect to ageneric instant τ . Let

t fa (τ ), tb

a (τ ): be the forward and backward travel time on link a for a packet respectivelyentering or leaving the link at time τ . Forward and backward link travel times are relatedthrough mutual consistency equations identical to Eqs. (2.5) which will not be restated.


Since in discrete flow models users are identifiable units (packets k j ), it is possible to definetemporal variables associated to the specific packet. Let:

τ ua [k j ], τw

a [k j ]: be respectively the entrance and exit times on link a of packet k j . Consistencywith travel times requires that:

τwa [k j ] = τ u

a [k j ] + t fa

(τ u

a [k j ])

(2.21a)

τ ua [k j ] = τw

a [k j ] − tba

(τw

a [k j ])

(2.21b)

The FIFO discipline also applies to discrete models if it is assumed that packets cannotovertake each other, or if no explicit overtaking mechanism is introduced. The formalrepresentation of the FIFO rule is identical to that for continuos flow models. Alternativeconditions for the FIFO rule, analogous to those introduced for continuous models can bestated. It should be observed, however, that, for discrete models, this condition is easier toensure since a packet is identified by the very nature of the model rather than implicitlythrough the trajectory crossing a given point at a given time.

As for the continuous case, the general discrete dynamic supply model can be formalizedthrough link and path performance functions and the network flow propagation model.

2.2.2. Link performance and travel time functions. The dependence of link travel time onlink “flow” variables for congested networks can be expressed through a number of models.It is possible to specify separable and non-separable cost functions, the latter possiblyallowing for spill-back effects from downstream links. The simpler separable travel timefunctions are similar to the functions adopted for running and queuing links described inSection 2.1.2.

Forward travel time on running link a can be expressed as a linear function of arrivaltime, thus varying for different time instants τ ( j) within interval j :

t fa (τ ( j)) = t0

a + 1

Qa· xa(τ ( j)) (2.22)

or alternatively through the average speed computed as a function of link density as in thefundamental diagram of traffic flow:

t fa (τ ( j)) = La

va(xa(τ ( j))/La)(2.23)

Given the discrete nature of the models, several assumptions can be made on the compu-tation of travel times for packets entering the link in a given interval. It is possible to assumethat the travel times are equal for all packets entering the link in a given interval. In thiscase the occupancy variable in Eqs. (2.22) and (2.23) correspond to a representative timeof interval j , typically its start-point, τ j−1. This implies that travel times are constant forall users entering the link during the interval. Alternatively, travel times can be computedas functions of the average link occupancy during the previous interval, x̂a[ j − 1] or thesame interval, x̂a[ j]; in the latter case, however, link travel time for users entering the linkduring the interval depends on users entering the link later in the same interval. This maycause inconsistencies and counter-intuitive results and should be avoided.


2.2.3. Path performance and travel time functions. The concepts of forward and back-ward travel time needed to reach link ak

i along path k leaving or arriving in a given instantcan be immediately extended to discrete supply models. These variables will be denotedby T f

aki(τ j ) and T b

aki(τ ( j)) respectively to stress the fact that departures can occur only at the

representative time of each interval, τ j , while arrivals can be at any time during the intervalτ ( j). Equations (2.11) expressing the relationships between forward and backward traveltimes apply also to the discrete case. Similarly the forward (backward) total travel time onpath k for a given departure (arrival) time can be defined also for the discrete case, denotingthe variables with TT f

k (τ j ) and TTbk (τ ( j)) respectively.

Similarly the relationship between link and path travel times is analogous to Eqs. (2.12)and, when applied recursively, leads to a “nested” structure corresponding to Eqs. (2.13):

T fak

i(τ j ) = t f

ak1(τ j ) + t f

ak2

(τ j + t f

ak1(τ j )

)+ t f

ak3

(τ j + t f

ak1(τ j ) + t f

ak2

(τ j + t f

ak1(τ j )

))+ · · · + t f

aki−1

(τ j + t f

ak1(τ j ) + · · · + t f

aki−2

(τ j + t f

ak1(τ j ) + · · ·

))(2.24)

In the discrete flow case, however, Eq. (2.24) can be expressed more straightforwardly byusing the arrival times of the generic packet to link ak

i , τuak

i[k j ], as:

T fak

i(τ j ) = t f

ak1(τ j ) + t f

ak2

(τ u

ak2[k j ]

)+ t f

ak3

(τ u

ak3[k j ]

)+ · · · + t f

aki−1

(τ u

aki−1

[k j ])

(2.25)

The same construct applies to total path travel time TT fk (τ j ), and to other path-additive

attributes ECk(τ j ) and finally to total path cost gk(τ j ):

TT fk (τ j ) = t f

ak1(τ j ) + t f

ak2

(τ u

ak2(τ j [k j ]

)+ t f

ak3

(τ u

ak3[k j ]

)+ · · · + t f

aknk

(τ u

aknk

[k j ])

= T fak

nk

(τ j ) + t fak

nk

(τ u

aknk

[k j ])

(2.26a)

ECk(τ j ) = ec fak

1(τ j ) + ec f

ak2

(τ u

ak2[k j ]

)+ ec f

ak3

(τ u

ak3[k j ]

)+ · · · + ec f

aknk

(τ u

aknk

[k j ])(2.26b)

gk(τ j ) = βt TT fk (τ j ) + ECk(τ j ) (2.26c)

Formally the relationship between the vector of total path travel time, TT f (τ j ), for agiven departure time τ j and travel times on the links making up each path, can be expressedsymbolically as:

TT f (τ j ) = �(t(τ ′), τ ′ ≥ τ j ) (2.27)

Equation (2.27) is the equivalent of Eq. (2.14) in the continuos flow-continuous time case.

2.2.4. Dynamic Network Flow Propagation models. Unlike the continuous-flow case, theDynamic Network Flow Propagation model for discrete flows can be formulated explicitlysince packets can be identified while moving over the network. In this case the in-flow onlink a in the interval [ j] can be expressed as:

ua[ j] =∑

k

∑l≤ j

δak [l, j] · hk[l] (2.28a)


where the δak (l, j) are zero/one variables analogous to the elements of the static link-pathincidence matrix; they are equal to one if the packet kl (of intensity hk[l]) enters link aduring interval j , 0 otherwise:

δak[l, j] ={

1 if τ ua [kl] ∈ ([ j − 1]DT, [ j]DT )

0 otherwise

Obviously the δak[l, j] are all equal to zero if link a does not belong to path k.Equation (2.28a) can also be formulated using a matrix notation as:

ua[ j] =∑l≤ j

�[l, j] · h[l] (2.28b)

which is close to the static counterpart f = �h.Similar equations can be stated for the out-flow, wa[ j], from the generic link a at time

interval j :

wa[ j] =∑

k

∑l≤ j

δ′ak[l, j] · hk[l] (2.29a)

where the δ′ak[l, j] is equal to one if the packet kl (of intensity hk[l]) leaves link a during

interval j , 0 otherwise:

δ′ak[l, j] =

{1 if τw

a [kl] ∈ ([ j − 1]DT, [ j]DT )

0 otherwise

and in matrix terms:

wa[ j] =∑l≤ j

�′[l, j] · h[l] (2.29b)

Note that the elements of dynamic incidence matrices depend on link travel times and, forcongested networks, on link flows and occupancies. In this respect they should be denotedas:

δak[l, j] = δak[l, j](t(τ ′); τ ′ ∈ (τl , τ j ))

The overall DNL model relating link flows and occupancies to path flows can be expressedcombining the previous equations:

xa(τ j ) − xa(τ j−1) = ua[ j] − wa[ j] (2.30a)

ua[ j] =∑l≤ j

�[l, j] · h[l] (2.30b)

wa[ j] =∑l≤ j

�′[l, j] · h[l] (2.30c)

τ uak [kl] = τl + T f

aki(τl) (2.30d)

τwak [kl] = τ u

ak [kl] + t fak

(τ u

ak [kl])

(2.30e)

t fa (τ ( j)) = La

Va(xa(τ j−1)/La)(2.30f)


The above set of equations has been specified under the assumption that link travel timefunctions depend on link occupancy at the beginning of each interval; the model can beexpressed in a similar form with reference to a generic time instant τ ( j).

2.2.5. Formalization of the whole supply model. Equations (2.30) can be expressed sym-bolically as non-linear vector functions relating link flows (in-flows and out-flows) andoccupancies for an interval j , to the vector of path flows leaving in intervals from l to j andthe link travel times successive to τl and previous to the end of interval j, τ j :

f [ j] = �(h[l], t(τ ′); l ≤ j, τ ′ ∈ [τl , τ j ]) (2.31a)

Expression (2.31a) can be further combined with the equation relating link travel times tolink occupancies for congested dynamic network loading models:

f [ j] = �(h[l], t(x(τ ′)); l ≤ j, τ ′ ∈ [τl , τ j ]) (2.31b)

The global supply model is completed by the symbolic relationships relating path traveltimes to link travel times:

TT f (τl) = �(t(x(τ ′)); τ ′ ≥ τl) (2.32)

3. Dynamic demand models for continuous service systems

Demand models express the relationship between path flows and path costs. In a within-daydynamic context, flow rates may vary within the simulation period, therefore, unless fixedflow rates are assumed, there is the need to simulate an additional travel choice dimension,that is the departure time from origin. In fact, the “minimal” demand model, that is themodel simulating the travel choice to be included in all assignment models, relates to pathand departure time choice.

The flow of users following a path k connecting the O-D pair od and starting at timeτ, hk(τ ) can be represented with elastic demand profile models, simulating in addition topath, departure time choice given the desired arrival time at destination, τd , or the desireddeparture time from the origin τo. These kind of models are described in this section. Thecontinuous time-continuous flow case will be discussed first. Let:

dod (τd ): be the flow of trips between the pair od with desired arrival time τd ;pod ,k (τ/τd ): be the choice probability of departing at time τ following path k, given the

O-D pair od and the desired arrival time τd ;Vk(τ/τd ): be the systematic utility of path k and departure time τ , given the desired arrival

time τd ;Vod (τ/τd ): be the vector of systematic utilities relative to all the paths connecting the pair

od, k ∈ Kod , for a given departure time τ and desired arrival time τd .

The demand conservation condition over the whole reference interval [O, T ] can beformally expressed as:

hk(τ ) =∫ T

0dod (τd )pod,k(τ/τd ) dτd


The first models to derive the within-day departure rates were Multinomial Logit (Abkowitz,1980; Small, 1982). Path flows are here considered to be the outcome of a two-stage decisionprocess: the choice of departing at a given time τ and the choice of using path k conditionalto the choice of departing at time τ :

pod,k(τ/τd ) = pod (τ/τd ) · pod [k/τ, τd ] (3.1)

Choice probabilities of departure time τ and path k are usually expressed with random utilitymodels as a function of the systematic utilities of available path-departure time alternatives:

pod,k(τ/τd ) = pod,k(Vod (τ ′/τd ), ∀τ ′) (3.2)

Such probabilities are obtained from a mixed continuous (departure time)/discrete (path)alternatives multinomial Logit or Nested Logit. The specification of a simultaneous Multi-nomial Logit model for Eq. (3.2) is:

pod,k(τ/τd ) = exp(Vk(τ/τd ))∑k ′∈Kod

∫ T0 exp(Vk ′ (τ ′/τd )) dτ ′

(3.3)

Examples of Nested Logit Specification are given by Vythoulkas (1990) and Ben Akivaet al. (1986).

Systematic utility functions proposed for the simulation of the combined path-departuretime choice include, in addition to path attributes, the schedule delay, i.e. the penalty forarriving early or late with respect to the desired arrival time. In case of desired arrival timeτd , it results:

Vk(τ/τd ) = βt TT fk (τ ) + ECk(τ ) + βeEAPk(τ, τd , TTk(τ ))

+ βlLAPk(τ, τd , TTk(τ )) (3.4a)

where:

EAPk(τ, τd , TT fk (τ )): is the penalty related to early arrival with respect to τd departing in τ

and following path k. This penalty is usually considered only if the early arrival is abovea minimum threshold �e:

EAPk(τ, τd,TT f

k (τ )) = τd − �e − (

τ + TT fk (τ )

)if τd − �e − (

τ + TT fk (τ )

)> 0

= 0 otherwise

LAPk (τ, τd,, TT fk (τ )): is the penalty related to a delay with respect to τd departing in τ and

following path k. This penalty is usually considered only if the delay is above a minimumthreshold �l :

LAPk[τ, τd , TT f

k (τ )] = τ + TT f

k (τ ) − τd − �l if τ + TT fk (τ ) − τd − �l > 0

= 0 otherwise

In case of desired departure time from the origin τo, the expression of the systematic utilityis still a function of path travel time and schedule delay, but in this case the schedule delay


does not depend on the path travel time TT fk (τ ):

Vk(τ/τo) = βt TT fk (τ ) + ECk(τ ) + βeEDP(τ, τo) + βlLDP(τ, τo) (3.4b)

where:

EDP(τ, τo): is the penalty related to early departure with respect to τo departing in τ , usuallyconsidered only if the early departure is above a minimum threshold �e:

EDP(τ, τo) = τo − �e − τ if τo − �e − τ > 0

= 0 otherwise

LDP(τ, τo): is the penalty related to a delay with respect to τo departing in τ , usuallyconsidered only if the delay is above a minimum threshold �l :

LDP(τ, τo) = τ − τo − �l if τ − τo − �l > 0

= 0 otherwise

All the coefficients β in Eqs. (3.4) are negative. Furthermore, the schedule delay penaltiesshould have coefficients βe and βl with absolute values greater than the travel time coefficient(|βe| > |βt |, |βl | > |βt |) in order to avoid unrealistic user behavior, e.g. large probabilitiesfor alternatives with very high early/late arrival penalties but with smaller travel times.Empirical results for work related trips show that the disutility of late arrivals is larger thanthat for early arrivals (|βe| < |βl |).

The extension of dynamic demand models to the discrete case is rather straightfor-ward. The only difference is that alternative departure times are the discrete intervals..[ j − 1], [ j], [ j + 1], or representative instants . . . τ j−1, τ j , τ j+1 . . .. Simultaneous depar-ture time and path choice probabilities are thus expressed as pod,k[τ j/τd ]. A multinomialLogit specification can be (Cascetta et al., 1982):

pod,k[τ j/τd ] = exp(Vk[τ j/τd ])∑τ j ′

∑k ′∈Kod

exp(Vk ′ [τ j ′/τd ])

Alternatively a model specification introducing a correlation structure among adjacent de-parture intervals, e.g. with a Probit model (Mahamassani and Liu, 1998), can be adopted.The previous results for choice models and systematic utility specifications apply also tothe discrete departure time case. Discrete departure time models can be adopted for thecontinuous flows. In fact, some specifications of continuous departure time choice modelassume that travelers do not choose among an infinite number of departure instant, but ratheramong a finite number of times intervals (e.g. 5 minutes long), and that actual departuretimes are uniformly distributed within the chosen interval. In this case the probability ofleaving at time τ ( j),within interval j , and following path k computed with a MultinomialLogit model would be:

pod,k(τ ( j)/τd ) = 1

DT

exp(Vk[ j/τd ])∑j ′∑

k ′∈Kodexp(Vk ′ [ j ′/τd ])


4. Dynamic demand-supply interaction models

Demand-supply interaction models for within-day dynamic continuous services systemsare conceptually analogous to the equivalent static systems. In the following sections someformal results will be given, making reference to the continuous and discrete cases, for un-congested network as well as for congested network. The latter will be approached throughequilibrium models; dynamic process models that require further demand models simulat-ing learning, or utility updating, and choice updating mechanisms will not be treated inthe following (Cantarella and Cascetta, 1995). Dynamic Traffic Assignment (DTA) modelsare rather complex and few operational formulations have been developed. Furthermore,there are few theoretical results to date for the analysis of the properties of the resultingflow configurations (see for instance Kaufman et al., 1998b) compared to those obtainedfor static models. For sake of simplicity, in the following only (within-day dynamic) de-mand models with desired departure time τo will be illustrated. The extension to the caseof desired arrival time is rather straightforward.

4.1. Uncongested network dynamic assignment models

In case of uncongested network, link travel times do not depend on link occupancies. Thusthe dynamic traffic assignment model in the continuous flow-continuous time case, can bespecified as:

t f (τ) = t0(τ ) (4.1a)

TT f (τ ) = �(t0(τ )) (4.1b)

Vod (τ/τo) = βt TT(τ ) + EC(τ ) + βeEDP(τ, τo) + βlLDP(τ, τo) (4.1c)

h(τ ) =∑τo

P(Vod (τ/τo)) · d(τo) (4.1d)

f (τ ) = �(h(τ ), t0(τ )) (4.1e)

Equations (4.1c) and (4.1d) represent the within-day dynamic demand models. On the otherhand, Eqs. (4.1a), (4.1b) and (4.1e) make up the supply model representing respectively thelink performance model, the path performance model and the dynamic network flow prop-agation model. The uncongested dynamic assignment model (UND) can be deterministic(DUND) or stochastic (SUND) depending on the path choice model used in Eq. (4.1d).

The Dynamic Network Loading model (DNL) has been formulated symbolically in termsof a characteristic link flow vector, f , since, if FIFO rule holds, the different formulationsin terms of in-flow, out-flow or link occupancy are equivalent. For instance, Eq. (4.1e) canbe stated in terms of in-flows as (see Section 2.1.4):

ua(τ ) =∑

k


a,k(τ ))

where, since the network is uncongested, the backward travel times T ba,k are independent of

flows for each link a, but in general depend on the specific time τ :

T ba,k(τ ) = T 0

a,k(t0(τ )) (4.1f)


From Eqs. (4.1) it results that in principle both demand and link travel times vary with τ .However, given the absence of congestion, Eqs. (4.1) can be solved sequentially to obtainpath performances and link flows. In uncongested networks it is usually assumed that linktravel times are constant with respect to τ , i.e. t0

a (τ ) = t0a . Thus the system of Eqs. (4.1)

becomes:

t f (τ ) = t0 (4.2a)

TT f (τ ) = �(t0) (4.2b)

Vod (τ/τo) = βt TT(τ ) + EC(τ ) + βeEDP(τ, τo) + βlLDP(τ, τo) (4.2c)

h(τ ) =∑τo

P(Vod (τ/τo)) · d(τo) (4.2d)

f (τ ) = �(h(τ ), t0) (4.2e)

Here the only elements varying within-day are the demand flows inducing time-varyingpath and link flows. In particular Eq. (4.2b) becomes:

TTk(τ ) =∑

k

δak · t0a ∀τ

or

TT(τ ) = �T · t0 ∀τ

In the discrete-flow case, the uncongested network assignment models can be formallyspecified as:

t f (τ j ) = t0j (4.3a)

TT f (τ j ) = �(t0j

)(4.3b)

Vod (τ j/τo) = βt TT f (τ j ) + EC(τ j ) + βeEDP(τ j , τo) + βlLDP(τ j , τo) (4.3c)

h(τ j ) =∑τo

P(Vod (τ j/τo)) · d(τo) (4.3d)

f [ j] = �(h(τ j ), t0

j ′ ; j ′ < j)

(4.3e)

Note that in the above equations time dependency can be expressed equivalently as therepresentative time instant of interval j ,τ j , or simply as [ j]. Equations (4.3c) and (4.3d)represent the within-day dynamic demand models while Eqs. (4.3a), (4.3b) and (4.3e)represent respectively the link performance model, the path performance model and thedynamic network flow propagation model, components of the overall supply model. TheDNL can also be stated as:

f [ j] =∑l≤ j

�[l, j] · h[l]

where, if link travel times are constant for all time intervals of the simulation period, i.e.tf a(τ ) = t0

a , the matrix � does not depend on the starting interval l, but only on the differencebetween j and l.


4.2. Dynamic user equilibrium assignment models

For congested networks, dynamic equilibrium assignment can be specified through fixed-point models (Cantarella, 1997; Kaufman et al., 1998a) by combining supply and demandmodels both for fixed and elastic demand profiles. For within-day dynamic systems thedependency of travel times on link flows (occupancies) introduces two feedback cycles:in addition to the path cost and flow cycle (typical of within-day static user-equilibriummodels), in the dynamic case link flows depend on travel times.

In the continuous-flow case, user-equilibrium models can be formally stated as a fixed-point problem in travel times, costs and flows derived from the following system of non-linear equations:

t f (τ ) = t f ( f (τ )) (4.4a)

TT f (τ ) = �(t f (τ ′); τ ′ ≤ τ ) (4.4b)

Vod (τ/τo) = βt TT f (τ ) + EC(τ ) + βeEDP(τ, τo) + βlLDP(τ, τo) (4.4c)

h(τ ) =∑τo

P(Vod (τ/τo)) · d(τ ) (4.4d)

f (τ ) = �(h(τ ), t f (τ ); τ ′ ≤ τ ) (4.4e)

Equation (4.4e) expresses the dependency of link flow vector at time τ , f (τ ), on the pathflow vectors h and on link travel time vectors t in all previous time instants τ ′ < τ . Thiscan be more explicitly stated, for instance, in terms of in-flows on the generic link a, as (seeSection 2.1.4):

ua(τ ) =∑

k


ak (τ )) ·

(1 − dT b

ak (τ )

dτ

)

where the dependency of link flow at τ on the travel times of all links of the network andin all previous instants τ ′ is embedded in the backward travel time, given by the recursiveEq. (2.12b) here restated for readers convenience:

T bak

i(τ ) = tb

aki−1

(τ ) + tbak

i−2

(τ − tb

aki−1

(τ ))

+ tbak

i−3

(τ − tb

aki−1

(τ ))

− tbak

i−2

(τ − tb

aki−3

(τ ))

+ · · · + tbak

1

(τ − tb

aki−1

(τ ) − · · · − tbak

2

(τ − tb

aki−1

(τ )

− · · · − tbak

3

(τ − tb

aki−1

(τ ) − · · ·)))

A formal fixed-point specification in link flows of dynamic user-equilibrium continuos-flowmodels is the following:

f ∗(τ ) = �

( ∑τo

P(βt�(t( f ∗(τ ′)) + EC(t( f ∗(τ ′)) + βeEDP(τ, τo)

+ βlLDP(τ, τo)) · dτo , t f ( f ∗(τ ′)); τ ′ < τ

)

Dynamic user equilibrium models can be deterministic or stochastic depending on the path-departure time choice model. Existence and uniqueness conditions for continuous-flowdynamic user equilibrium models are currently being studied.


In the discrete-flow case, the models can be formally formulated as follows:

t f (τ j ) = t f (f (τ j )) (4.5a)

TT f (τ j ) = �[t f (τ j ′ )]; j ′ = 1 . . . j] (4.5b)

Vod (τ j/τo) = βt TT f (τ j ) + EC(τ j ) + βeEDP(τ j , τo) + βlLDP(τ j , τo) (4.5c)

h(τ j ) =∑τo

P(Vod (τ j/τo)) · d(τo) (4.5d)

f (τ j ′ ) = �(h(τ j ′ ), t f (τ j ′ ); j ′ = 1 . . . j) (4.5e)

Equation (4.5e) is the analogous of (4.1e) for the uncongested network case. It can be statedmore explicitly as:

f [ j] =∑l≤ j

�[l, j] · h[l] (4.5f)

The difference with respect to the uncongested network is that, in this case, � is a functionof links travel times t in all the previous intervals up to interval j :

�[l, j] = �[l, j](t f (τi ); i = l . . . j) (4.5g)

A formal fixed-point specification of a dynamic user equilibrium models is the following.

f ∗[τ j ] =∑τo

∑l=1... j

�[l, j](t f ( f ∗[i]; i = l . . . j)).

P[βt�(t f ( f ∗[i]; i = l . . . j)) + EC(t f ( f ∗[i]; i = l . . . j))

+ βeEDP(τl , τo) + βlLDP(τl , τo)] · dτo

Existence and uniqueness conditions for the fixed-point formulation have not been stated;however, in this case it is more difficult to arrive at general conditions, if possible at all,given the discreteness of time and packets.

5. Conclusions

In this paper, Intra-period Dynamic Traffic Assignment (DTA) models for continuous ser-vices systems (e.g. road networks) are reviewed and a general modeling framework con-sisting of supply, demand, and demand-supply interaction models, is presented.

The overall formulation depends on the basic assumptions on the flow structure, i.e.whether a continuous or a discrete approach is followed, and on the representation of time,i.e. whether a continuous or a discrete time representation is adopted. In principle, all thecombinations are possible but typically flow-discretization implies the time-discretizationand vice versa.

Continuous flow-continuous time models represent the “natural” extensions of staticsupply, demand and assignment models and allow the study of mathematical properties thatare useful in understanding the logic of the model and in designing solution algorithms.At least in theory, these models allow the definition of variables such as flow, speed, anddensity at each point s and at each instant τ . The solution of such models, however, requiresa discretization in space, �s, hence from the solution point-of-view they can be considered


similar to discrete space model through a duly definition of link length. Moreover, discreteflow-discrete time models are more consistent with reality and with the computationalframework.

Either approach requires within-day dynamic demand and supply models. However,while on the demand side, within-day dynamics can be captured by adding a new choicedimension to traditional demand models (i.e. departure time choice), within-day dynamicsupply models require a substantial reformulation of within-day static supply models. Withrespect to continuous models, it is more difficult to formalize a discrete flow-discrete timemodels in a general way, since there are several possibilities to discretize the relevantvariables. The general framework, proposed in the paper only for mesoscopic models,reflects a general formulation.

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