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A within-day dynamic traffic assignment model for urban road networks

Giuseppe Bellei, Guido Gentile and Natale Papola

Universit degli Studi di Roma La Sapienza

ABSTRACT

In this paper a new formulation of within-day dynamic traffic assignment is presented, where dynamic

user equilibrium is expressed as a fixed point problem in terms of arc flow temporal profiles. Specifically, it

is shown that, by extending to the dynamic case the concept of Network Loading Map, is no more needed to

introduce a Dynamic Network Loading in order to ensure the temporal consistency of the supply model. On

this basis it is possible to devise efficient assignment algorithms, whose complexity is equal to the one

resulting in the static case multiplied by the number of time intervals in which the period of analysis is

divided. With specific reference to a Logit path choice model, an implicit path enumeration network loading

procedure is obtained as an extension of Dials algorithm; then, the fixed point problem is solved through the

Bathers method.

Keywords: Dynamic User Equilibrium, Fixed Point Problem, Temporal Profiles, Network Loading Map,

Implicit Path Formulation, Dials Algorithm

1 INTRODUCTIONThe quantitative analysis of road network traffic is usually performed through static assignment models,

yielding the transport demand-supply equilibrium under the assumption of within-day stationariety. This

implies that the relevant variables of the system are assumed to be constant within the reference period. The

static assignment models satisfactorily reproduce congestion effects on traffic flow and cost patterns; on the

other hand they are not able to reproduce some important dynamic phenomena, such as the formation and

dispersion of vehicle queues due to the temporary over-saturation of road intersections. The Within-Day

Dynamic Traffic Assignment (WDDTA) models are conceived to overcome this limit.

Many different approaches have been proposed in the literature. Here we recall: the approaches based on

simulation (DYNASMART, Jayakrishnan, Mahmassani and Hu, 1994; CONTRAM, Taylor, 1990); the

empirical solving methods devised in analogy with static assignment (Hamerslag, 1988; Bellei and Bielli,

1990); the formulations based on the optimum control theory, referring to traffic assignment alone (Ran,

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Boyce and LeBlanc, 1993), or combined with route guidance (Papageorgiu, 1990); the formulations based on

nonlinear programming (Carey and Subrahmanian, 2000); the approaches based on the discretization in time

and space (Daganzo, 1994, 1995); the approaches based on time-space hyper-networks (Drissi Kaitouni and

Hameda Benchekroun, 1992); finally, the approaches based on variational inequalities (Addison and

Heidecker 1993, Friesz et al. , 1993; Smith, 1993). Its also worth mentioning the so called doubly-dynamic

assignment models, based either on stochastic processes (Cascetta and Cantarella, 1991), or on variational

inequalities (Friesz, Bernstein and Stough, 1996).

Though a wide body of literature has been produced on this subject (only some works are quoted here),

the variety of theoretical approaches and mathematical formulations gives evidence that, contrariwise to the

static assignment, a generally accepted framework has not been attained jet. Moreover, the existing solving

methods applicable to real-size networks either are heuristics, or require a thick time discretization which

needs considerable computing times on standard hardware (Wisten and Smith, 1997).

In this paper, with reference to congested road urban networks and to any probabilistic path choice model,

a new formulation of WDDTA is presented and a solution algorithm is devised for the Logit case. The

approach adopted consists in regarding WDDTA as a dynamic user equilibrium, which is consistently

formalized through a fixed point problem. As a consequence, some results concerning the static assignment

models and algorithms are extended to the dynamic case.

In the static case, the user equilibrium can be formalized either by combining the demand and the supply

models (left side of figure 1), or by combining the Network Loading Map (NLM) and the arc performance

function (right side of figure 1). The two approaches are equivalent from an analytical point of view, but the

second one is more convenient as it allows developing implicit path enumeration algorithms, which are

generally more efficient.

[Figure 1]

In the dynamic case, on the contrary, approaches based on the definition of a dynamic NLM yielding the

arc flow temporal profiles for given arc performance temporal profiles are not yet available. In this paper, in

order to bridge the gap, we realize an extension to the dynamic case of the NLM, when formalizing, and of

the Dials algorithm, when solving the problem. Here below the relevant aspects of these extensions are

examined.

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actually achieve in this paper only with reference to the Logit path choice model.

To focus our attention on the innovations here introduced and to simplify notation, a single user class and

a fixed origin-destination demand will be considered, regarding only path choice as elastic. In particular, the

demand is assumed to be rigid with respect to the departure time choice. On the supply side, the arc

performance model is based on a macroscopic, space-discrete flow model, which is non-separable with

respect to time (performances at a given instant depend on the previous history of flows) but is separable

with respect to space (the spill back phenomenon is not represented), where capacity constraints are

introduced, thus explicitly representing vehicle queues that carry-over the delay into subsequent instants. The

extension to more general contexts will be the subject of future research.

2 THE MATHEMATICAL MODELIn this section the components of the system described in figure 1 are formally introduced. As the analysis

is carried out within a dynamic context, the model variables are temporal profiles, here represented as

piecewise C1

functions of the time variable .

Users trips on the road network are modelled through a strongly connected oriented graph G = (N, A),

whereNis the set of the nodes andA is the set of the arcs. Each arc a is identified by its initial node TL(a),

referred to as tail, and by its completing node HD(a), referred to as head; that is: a = (TL(a),HD(a)). The

origins and the destinations of user trips constitute a subset Cof nodes, referred to as centroids.

In the following it is assumed that, when travelling from nodex to destination d, users consider a specific

subsetKxd

of paths fromx to d, referred to as efficient paths. The definition of path efficiency utilized in this

paper is based on the notion ofnode topological order(Nguyen, Pallottino and Inaudi, 1996). With reference

to a given destination d, we assume that the distance on graph G from the generic node x to dis monotone

non-decreasing with the node topological orderTOxd. Distances are measured on the basis of a specified arc

constant, independent of congestion and time, such as the length, the zero-flow cost or the cost obtained in

some static assignment. The topological order of the nodes can be then determined through a label setting

shortest path algorithm. We denote N(k)d the k-th node in increasing topological order, with 1 k |N|,

where, in general, |X| is the cardinality of a discrete setX. An arc is efficient if its tail has a higher topological

order than its head, while a path is efficient if all its arcs are efficient. The sets of the efficient arcs exiting

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pairs;

Tkxd

() time when users following path kKxdand leaving nodex at time reach destination d;

Tkxd

a() time when users following path kKxd

and leaving nodex at time enter arc aAkxd

;

Tkxd

a-1

() time when users following path kKxdand entering arc aAkxd

at time leave nodex;

T temporal profile of the (AK 1) sub-path exit time vector relative to the OD pairs, where

AK= oC, dC, kKod |Akod

|.

The supply model is based on a space-discrete macroscopic flow model, where users are represented as

particles of a mono-dimensional partly compressible fluid (Cascetta, 2001), and the performance temporal

profiles are assumed to be continuous. Moreover, capacity constraints are introduced, thus explicitly

representing vehicle queues. In this context the First In First Out (FIFO) rule holds true, although the

resulting flow temporal profiles are inherently discontinuous because whenever the queue vanishes on an arc,

the outflow reduces instantaneously from the capacity to the inflow. Then, with reference to the exit time

temporal profiles, by assuming the mild hypothesis that all queues vanish within time , for given < ,

we have:

ta() < ta() , Tk

xd

a() < Tkxd

a(). (1)

On the basis of (1), at each , where the derivative of the profile exists, it is:

dta()/d> 0 , dTkxd

b()/d> 0. (2)

The monotonicity expressed by (1) ensures that the temporal profiles of the exit times are invertible. Its

worth noting that (1) and (2) also hold with reference to the inverse temporal profiles of the exit times; that

is, forta()-1

and Tkxd

a()-1

.

Having introduced the main variables of the mathematical model, in the following subsections we focus

on the relations among them.

2.1Arc performance modelCongestion phenomena are formally represented assuming that the temporal profiles of the arc exit times

and of the arc costs depend on the temporal profiles of the arc inflows, that is:

t= t(f), (3)

c = c(f). (4)

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The performance functionals (3) and (4) are now specified with reference to a particular link-node model,

where the generic arc a represents a road link of length La consisting ofna parallel lanes that terminate at a

regulated intersection. Specifically, we assume that the generic nodex represents a traffic light with a fixed

cycle time CTx , where each entering arc has a dedicated phase in order to avoid explicit representation of

turning movements. At the final section of the link we then have a bottleneck, since the capacity is there

reduced from naQ to Qa = naQ(1-ra), which is however assumed to be strictly positive and bounded,

where Q is the vehicle capacity of a standard lane, is the vehicle occupancy coefficient and ra is the

effective red share. When the inflowfa() exceeds the capacity Qa an over-saturated queue occurs. This is to

be added to the under-saturatedqueue, proper to an intermittent service, that reaches its maximum at the end

of the red light and then disappears at some instant within the end of the green light. The arc is then modeled

by considering two serial phases: a running phase, which simulates the link travel time including the over-

saturationdelay due to the bottleneck, and a waiting phase, which simulates the under-saturation delay due

to the traffic light.

Dealing with urban networks, where the speed limit is usually low, we can assume, with reference to the

running phase, that the vehicles travel at the maximum permitted arc speed Va , until they have to queue.

From that point to the bottleneck, coherently with the well-known theory of kinematics waves (Daganzo,

1997), the vehicles proceed at the over-saturated queue speed, as depicted in figure 2.

[Figure 2]

Because the details of the actual temporal profile of the delay due to the traffic light is useless in the

context of traffic assignment, we can assume, with reference to the waiting phase, that a vehicle reaching the

intersection at time

experiences the average under-saturation delay a() corresponding to the flow

completing the running phase at the same time eR a(); that is:

a() = 0.5ra2CTHD(a) /(1-(1-ra)eR a()/Qa). (5)

Based on the facts that: a) in case there is no over-saturatedqueue, the flow completing the running phase

at time , when a vehicle entered at reaches the intersection, is equal to the inflow at time , and b)

because of vehicle conservation and FIFO rule, the vehicles entered between two given instants must exit

between the corresponding exit times, the temporal profile of the arc exit time resulting from the above link-

node model may be expressed as:

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( )( )

( ) ( )( )

2

1,

2 1 1

a HD a k kaa EQa BQa

a a a a

r CTLt

V r f Q

= + +

, (6)

( )( )

( )1

d ,

2 kBQa

a HD ak k kaa BQa a BQa EQa

a a

r CTLt f

V Q

= + + +

, (7)

where it is: ta(BQ ak) = BQ a

k+La /Va +0.5raCTHD(a) , while BQ a

k+La /Va and EQ a

k+La /Va are, respectively,

the instants when the over-saturatedqueue appears and vanishes for the k-th time on arc a. These instants are

implicitly expressed through the following relations:

( ) ( ) ( )

( ) ( )

: ,2

: d

k

EQa

kBQa

a HD ak k k k aBQa a BQa a a BQa BQa

a

k k k

EQa a a EQa BQa

r CTLf Q t

V

f Q

+ > = + +

= . (8)

In subsection 5.1 a numerical method to calculate recursively the arc exit times at a sequence of instants will

be presented.

Due to the approximation introduced in (6) when considering the average delay caused by the traffic light,

the temporal profile of the arc exit time does not satisfy: a) the FIFO rule, if the outflow decreases too

quickly; b) the continuity hypothesis, in correspondence of discontinuities of the inflow. To overcome these

drawbacks, in the context of the proposed numerical method we adopt a slightly variation of such

approximation, satisfying, under mild assumptions, the FIFO rule and the continuity hypothesis, which

would be anyhow verified in a more detailed representation.

Note that, in the above link-node model, a capacity constraint is defined only connected with the

bottleneck, while the flow crossing any other section of the arc, and in particular the entering flow, can take

any non-negative value. Moreover, the spill back phenomenon is not represented, implicitly assuming that

the portion of queue exceeding the length of the link is vertical.

The arc cost for users entering at time is simply assumed to be:

ca() = (ta() -) +ma(), (9)

where ma is the temporal profile of the monetary cost, while is the Value of Time.

2.2Path performance model

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For each path kKxd and arc aAkxd

, the travel time of the sub-path between node x and node TL(a) at

time is the sum of the travel times of its arcs bAkxd

a , each of them referred to the time Tkxd

b() when users

leavingx at reach TL(b); that is:

Tkxd

a() = +bAkxda [tb(Tkxd

b()) -Tkxd

b()]. (10)

Equation (10), applied recursively, yields the sub-path exit time temporal profiles as a function of the arc exit

time temporal profiles. This is formally expressed through the following functional:

T= T(t). (11)

Assuming additive costs, the generalized cost of path kKxd at time is the sum of the costs of its arcs

aAkxd

, each of them referred to the time Tkxd

a() when users leaving nodex at reach node TL(a); that is:

Ckxd

() = aAkxdca(Tkxd

a()). (12)

Those (12), which refer to the OD pairs, are formally expressed through the following functional:

C= C(c, T). (13)

2.3Path choice modelIn modelling travel demand we follow the behavioural approach based on random utility theory, where it

is assumed that users are rational decision-makers who, when making their travel choice: a) consider a

positive finite number of mutually exclusive travel alternatives constituting their choice set; b) associate

with each travel alternative of their choice set a perceived utility, not known with certainty and thus regarded

by the analyst as a random variable; c) select a maximum utility travel alternative. More specifically, we

assume that the path choice model isprobabilistic (Cantarella, 1997).

All users travelling from origin o to destination d, are assumed to be identical to each other with respect

to any characteristic influencing the travel behaviour. In particular, they share the same choice set of travel

alternatives, constituted by the efficient path set Kod

. With reference to the choice model we use the

following notation:

Pkod

() probability that users traveling from origin o to destination dand departing at time choose path

kKod ;

P temporal profile of the (K 1)path choice probability vector;

Wod

() expected maximum perceived utility for users traveling from origin o to destination dand departing

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at time , referred to as OD satisfaction.

Assuming the mean value of the perceived disutility of the generic travel alternative to be given, as usual,

by the generalized cost of the corresponding path, the choice probability of path kK od and the OD

satisfaction of OD pair (o, d), at time , are formally expressed, respectively, as:

Pkod

() = Pkod

(Cjod

():jKod), (14)

Wod() = W od(Cjod

():jKod). (15)

In compact form, we then have:

P= P(C). (16)

The flow on path kKod at time is given by the corresponding choice probability multiplied by the

demand flow, both referred to that same instant; that is:

Fkod

() =Pkod

()dod(). (17)

In compact form, we then have:

F= diag( d)P, (18)

where is the (KD)path-OD pair incidence matrix.

2.4Network flow propagation modelIn the classical network flow propagation model based on explicit path formulation, the inflow of arc a at

time is given by a weighted sum of the flows on all the paths kKaod

= {kKod: aAkod

} including that arc,

each of them referred to the departure time Tkod

a-1() of those users who, following that path, reach node

TL(a) at time ; that is:

fa() = oC, dC, kKaodFkod(Tkoda-1())(dTkoda-1()/d). (19)

Note that the weight dTkod

a-1()/d is specific to the dynamic case. It stems from the fact that travel times

vary over time, so that those users who follow a given sub-path, leave its initial node at a certain rate and, in

general, arrive to its final node at a different rate, which is higher than the previous one, if the travel time

along the sub-path is decreasing, and lower, otherwise (for details, see for example Cascetta, 2001).

Equations (19) are formally expressed by the following functional:

f= (F, T). (20)

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Note that, because the graph is strongly connected, the link capacities are strictly positive and the demand

is bounded, then there exists a time , such that: < < , the demand is satisfied and the resulting arc

inflow temporal profiles are null outside the period of analysis [0, ].

2.5Implicit path formulationWe now introduce a new implicit path formulation addressing both network flow propagation and path

choice. This is founded on the concepts of arc conditional probability and node satisfaction; the

corresponding notation is introduced below:

pad() probability of using arc a, conditional on being at node TL(a) at time , when traveling to

destination d;

p temporal profile of the (|A||C| 1) arc conditional probability vector;

wxd() expected value of the maximum perceived utility at time relative to the paths Kxd, connecting

nodex to destination d;

w temporal profile of the (|N||C| 1) node satisfaction vector.

Based on the above definition, the choice probability of the generic path kKod at time is equal to the

product of the conditional probabilities of its arcs aAkod

, each of them referred to the time Tkod

a() when

users leaving o at reach TL(a); that is:

Pkod

() = aAkodpad(Tk

oda()). (21)

The main idea underlying the implicit path formulation is to decompose the path choice in a sequence of

arc choices taken at each node among the arcs of its efficient forward star, while travelling towards the

destination. While, in general, pad() depends on the sub-path utilized to reach node TL(a), this

decomposition is possible when the arc conditional probabilities at each node are equal for all users directed

to the same destination. This property of the path choice model can be formally expressed as:

( )

( ) ( )( )

( ) ( )( )( )( )

( ) ( )( )( ) ( )( )

( )( )

1

1

TL a d oda ha

d TL a d d odb hb

TL a d oTL aodj k h

j K k K d

a TL a d oTL aodj k h

j Kb FSE TL a k Kb FSE TL a

P P T

pP P T

= =

, (22)

where hK oTL(a) is the generic sub-path utilized to reach TL(a) at time , having left origin o at time

ThoTL(a)

()-1

, KaTL(a)d

is the set of sub-paths from TL(a) to d including arc a, and Khaod

= {kKod : aAkod

,

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hKoTL(a) ,AhoTL(a)

Akod

} is the set of paths from o to dconstituted by sub-path h from o to TL(a) and by

any sub-pathjKaTL(a)d

from TL(a) to d(see figure 3).

[Figure 3]

In the following we assume that (22) holds true. In the context of the arc choice taken at a given node x by

users travelling to destination d: a) the choice set is constituted by the subsets of paths beginning with a same

arc bFSE(x)d; b) the systematic disutility associated to each alternative of the choice set can be expressed as

the difference between the arc cost and the node satisfaction of its head, synthesizing the costs of the

efficient sub-paths connecting that node to the destination; c) the choice probabilities and the satisfaction are,

respectively, the arc conditional probabilities and the node satisfaction. We then have:

pad() = pa

d(cb() - wHD(b)

d(tb()): bFSE(x)

d) , aFSE(x)d, (23)

wxd() = wx

d(cb() - wHD(b)

d(tb()): bFSE(x)

d). (24)

In general the form of functions padand wx

din (23) and (24) is different from that of Pk

odand W

odin (14) and

(15), respectively, depending on the correlation among the random residuals associated to the paths

connecting the node to the destination. Indeed, in section 4 it will be proved, with reference to the

Multinomial Logit path choice model, that (22) holds true, while the form of probabilities and satisfactions

relative to path and arc choices coincide.

Equations (23) are expressed in compact form by the functional:

p = p(w, c, t), (25)

while the recursive relations (24) are formally expressed by the functional:

w= w(c, t). (26)

Referring to users traveling to destination d, the flowfad

() entering arc a is given by the arc conditional

probabilitypad() multiplied by the flow entering node TL(a) at time ; the latter is given, in turn, by the sum

of the outflows of the arcs entering TL(a), and of the demand flow from TL(a) to d, which is null when

TL(a)C; that is:

fad() =pa

d()[dTL(a)d() + bBSE(TL(a))dfb

d(tb

-1())(dtb-1()/d)], (27)

where dtb-1

()/d has the same meaning as dTkod

a-1

()/din (19).

Recalling the definition of arc conditional probability, equation (27) implicitly expresses the conservation

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at node TL(a) of the flows directed towards destination d. The arc inflowfa() is simply given by the sum of

the flowsfad() relative to all the destinations:

fa() = dCfad(). (28)

On the basis of (27) and (28) the temporal profile of the arc inflow vector is formally expressed by the

following functional:

f= (p, t; d). (29)

3 FORMULATION OF THE WITHIN-DAY DYNAMIC TRAFFIC ASSIGNMENTIn the previous section a mathematical model reproducing the within-day dynamic of road traffic has been

formalized, both with an explicit and an implicit path formulation. On this basis the demand-supply approach

depicted in figure 1 turns into the scheme on the left side of figure 4, while the NLM-based approach with

implicit path formulation is represented by the scheme on the right side.

[Figure 4]

3.1Dynamic network loadingThe DNL is a sub-problem of WDDTA. It consists in the seeking, for given path flows, of an arc flow

pattern consistent with the sub-path travel times through the arc performance model. The importance of the

DNL stems from the fact that its solution is a necessary step to guarantee the internal temporal consistency of

the supply model.

By combining (3) and (11) with (20) it yields the explicit path formulation of the DNL:

f= (F, T(t(f))), (30)

while by combining (3) with (29) it yields implicit path formulation of the DNL:

f= (p, t(f); d). (31)

The solutions, if any, to the fixed point problems (30) and (31) are formally expressed, in that order, by

the following maps:

fDNL(F), (32)

fDNL(p ; d). (33)

3.2Network loading map

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In analogy with the static case, the NLM is a functional relation which allows determining, for given arc

performances, an arc flow pattern consistent with the demand flows through the path choice model.

By combining (11), (13), (16) and (18) with (20) it yields the explicit path formulation of the NLM:

f= (diag( d)P(C(c, T(t))), T(t)), (34)

while by combining (26) and (25) with (29) it yields the implicit path formulation of the NLM:

f= (p(w(c, t), c, t), t; d). (35)

3.3Fixed point formulations of the within-day dynamic traffic assignmentOn the basis of the above results, WDDTA can be formalized as a fixed-point problem in terms of arc

flow temporal profiles based on the DNL, or on the NLM, and, in both cases, with explicit or implicit path

formulation.

An explicit path formulation based on the DNL, similar to that in Cascetta (2001), can be obtained by

combining (3), (4), (11), (13), (16) and (18) with (32):

fDNL(diag(d)P(C(c(f), T(t(f))))). (36)

The corresponding implicit path formulation is obtained by combining (3), (4), (26) and (25) with (33):

fDNL(p(w(c(f), t(f)), c(f), t(f)); d). (37)

An explicit path formulation based on the NLM can be obtained by combining (3) and (4) with (34):

f= (diag( d)P(C(c(f), T(t(f)))), T(t(f))), (38)

while using (35) instead of (34) the corresponding implicit path formulation is obtained:

f= (p(w(c(f), t(f)), c(f), t(f)), t(f); d). (39)

We now prove that the formulations based on the DNL and those based on the NLM are equivalent.

Proposition 1. Any solutionf* to problem (38) solves problem (36) and vice versa.

Proof. Letf* be a flow pattern, and let: t* = t(f*), c*= c(f*), T* = T(t*), C* = C(c*, T*),P* = P(C*),

F* = diag(d)P*.

By contradiction, assume thatf* solves problem (38) but does not solve problem (36). Becausef* solves

problem (38) we have:f* = (F*,T*), while becausef* does not solve problem (36) it is: f*DNL(F*).

The latter expresses thatf* cannot be a solution to the DNL problem (30), implying thatf* (F()*,T*),

which contradicts the former.

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Again by contradiction, assume now that f * solves problem (36), but does not solve problem (38).

Becausef* does not solve problem (38) we have: f* (F*,T*), while becausef* solves problem (36) it

is: f *DNL(F*). The latter expresses that f * is a solution to the DNL problem (30), implying that

f* = (F*,T*), which contradicts the former.

By a similar argument it can easily be established that any solution to problem (39) solves problem (37)

and vice versa.

4 THE CASE OF MULTINOMIAL LOGITThe explicit and implicit formulations of the path choice model presented, respectively, in sub-section 2.3

and 2.5 are now specified with reference to Multinomial Logit. In this case, for each OD pair (o, d), the

choice probability of path kKodat time , formally expressed by (14), is given by:

( )

( )

( )

exp

expod

od

k

od

k od

j

j K

C

PC

=

, (40)

while the satisfaction, formally expressed by (15), is given by:

( )( )

ln expod

od

kod

k K

CW

=

. (41)

Proposition 2. With reference to the Multinomial Logit path choice model, (22) holds true, while the

form of functions pad

and wxd

in (23) and (24) coincide with that of Pkod

and Wod

in (14) and (15),

respectively.

Proof. Consider a destination dand a given time .

With reference to the generic arc a, based on (40), it is:

( ) ( )( )

( ) ( )( )

( ) ( )( )

1exp

expTL a d TL a d a a

TL a d

TL a d

jTL a d

j TL a d j K j K j

j K

CP

C

=

. (42)

As all the paths inKhaodinclude sub-path hKoTL(a) and coincide with the paths in Ka

TL(a)dbut forh, on the

basis of (12) and (40), it is:

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( )

( ) ( ) ( )( )

( )

ln exp ,

0, otherwise

d

d

a aHD a

d

x a FSE x

c w tx d

w

+

=

. (48)

The latter coincides with the Logit specification of the satisfaction associated with the arc choice at the

generic nodex, formally expressed by (24).

By definition, the denominator on the left hand side of (22) is equal to 1. Then, we have:

pad() = jKaTL(a)dPj

TL(a)d(). (49)

Based on (42), and utilizing also (45) and (47), (49) becomes:

( )

( ) ( ) ( )( ) ( ) ( )exp

d d

a aHD a TL ad

a

c w t w

p

+

=

(50)

while, utilizing (46), it can be alternatively expressed as:

( )

( ) ( ) ( )( )

( ) ( ) ( )( )

( )( )

exp

expd

d

a aHD a

d

a d

b bHD b

b FSE TL a

c w t

pc w t

+ =

+

. (51)

The latter coincides with the Logit specification of the probability associated with the choice of the generic

arc a, formally expressed by (23).

It is worth noting that proposition 2 may be proved for any distribution of independent random residuals

which is stable with respect to the maximum operator.

We now show that, as expected, using the arc conditional probabilities (50), in (21) yields the path

probability (40). With reference to path kKod, at time we have:

( ) ( )( ) ( ) ( )( )( ) ( ) ( )( )1

expod

k

od od d od d od

k a k a a k a k aHD a TL a

a A

P c T w t T w T

= +

. (52)

Let aAkod

and bAkod

be two consecutive arcs of path k. Clearly, it is: ta(Tkod

a()) = Tkod

b() and

HD(a) = TL(b). Then, within the summation in (52), each second addendum is the opposite of the third

addendum of the successive term. Thus (52) becomes:

( ) ( )( ) ( )( ) ( )1expod

k

od od d od d

k a k a d k o

a A

P c T w T w

= + . (53)

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Based on (12) and (44), (53) coincides with (40).

5 NUMERICAL METHOD

In this section, the main blocks of the implicit path algorithm solving the NLM-based fixed point

formulation depicted on the right hand side of figure 4 are presented.

In order to implement the proposed mathematical model, the period of analysis [0, ] is divided intoI

time intervals identified by the sequence of instants (0

, , i, ,

I), with

0= 0 and

I= ; moreover

we set: I+1

= . In the following we assume to approximate the generic temporal profile xj through either a

piece-wise constant or a piece-wise linear function defined by the values taken at such instants, so that for

the two cases, in that order, we have:

xj(0) =xj

0, xj() =xj

i, (i-1, i], i = 1, ,I, (54)

xj(0) =xj

0, xj() =x

j

i-1+ (-

i-1) (xj

i-xj

i-1)/(

i-

i-1), (i-1, i], i = 1, ,I. (55)

Thenxj can be numerically represented through a (1 I+1) row vectorxj = (xj0, , xj

i, , xj

I), while the

temporal profile x of the generic (J 1) vector can be represented through a (J I+1) matrix

x = (x1, , xj, , xJ)T.

Specifically, the temporal profiles of the flows are assumed piece-wise constant, while those of the other

variables are assumed piece-wise linear. Clearly, this assumption introduces an approximation with respect

to the mathematical model; however, it is of much help when devising the numerical method presented in

this section.

The state of the network at time 0

is assumed to be known; here, without loss of generality, an initially

unloaded network is considered.

5.1Arc performancesWith reference to a generic arc a, we now show how to calculate the numerical representation of the arc

exit time temporal profile ta resulting from the model presented in subsection 2.1, based on the hypothesis

that the arc inflow is constant during each time interval. To this end it is convenient to handle the running

phase and the waiting phase separately.

Thus, let tRaibe the instant when a vehicle entering at time

icompletes the running phase. Because the

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temporal profile of the arc inflow complies with (54), based on (8) the over-saturated queue can only arise at

some instant tR ai. Then, referring to the generic interval (tR a

i-1, tR a

i], there are two cases, depending on

whether, at the end of the interval, the over-saturated queue is present or not. In the first case there is a queue

all through the interval and the flow completing the running phase equals the capacity Qa. Because of vehicle

conservation and FIFO rule, we have:

Qa(tR ai-tR a

i-1) =fa

i(i-i-1), (56)

from which a recursive expression fortR aiis obtained:

tR ai= tR a

i-1+(

i-

i-1) fa

i/Qa . (57)

In the second case we simply have:

tR ai=

i+La /Va . (58)

As in the first case (58) yields a value oftR ailess than that given by (57), and vice versa in the second case, it

is in general:

tR ai= max{tR a

i-1 +(i-

i-1) fai/Qa ,

i+La /Va}. (59)

Note that in the second case the queue may vanish at any instant within the interval (tR ai-1

, tRai], which can

be easily determined (Bellei and Bielli, 1996). Instead, assumption (55) referred to the temporal profile of the

running time implies that the queue vanishes at some instant tR ai, thus introducing an approximation with

respect to the actual temporal profiles resulting from an arc inflow temporal profile complying with (54). In

figure 5 are depicted both the actual temporal profiles (thick lines) and those resulting from the

approximation introduced (dashed lines) with reference to the flow completing the running phase eR a() (a),

the queue length qR a() (b), the running time tRa()-(c), for a given arc inflow temporal profilefa() (d).

[Figure 5]

The approximate temporal profile eR a() is constant during each interval (tR ai-1

, tRai], where it takes value

eR ai, which is less than or equal to the capacity Qa. Specifically, because of vehicle conservation and FIFO

rule, we have:

eR ai(tR a

i-tR a

i-1) =fa

i(i-i-1). (60)

Clearly, the flow beginning the waiting phase coincides with the flow completing the running phase.

Then, the actual temporal profile of the waiting time is the continuous, saw-toothed shaped function,

depicted by a thin line in figure 6, corresponding to the delay due to a traffic light working in under-saturated

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conditions. Such a detailed representation is definitely useless, given that an appropriate duration of time

intervals in the context of WDDTA ranges between 5 and 15 minutes, which is about 10 times longer than

common cycle times.

The average delay ai suffered by vehicles entered during the time interval (i-1, i] and completing the

running phase during the interval (tR ai-1

, tRai] is given by (5) calculated in eRa

i. Then, on the basis of (60), we

have:

ai= 0.5ra

2CTHD(a) /[1-(1-ra) fai/Qa(

i-

i-1)/(tR a

i-tR a

i-1)]. (61)

The temporal profile of the average delay, depicted by a constant thick line in figure 6, would constitute a

suitable approximation of the waiting time temporal profile, except that, at each time tR ai, it does not satisfy

the continuity hypothesis and, when eRai< eRa

i-1, the FIFO rule.

To overcome these drawbacks, we propose, as an effective approximation, the continuous piece-wise

linear function obtained by connecting the points (tR ai, 0.5(a

i+a

i+1)), depicted by a dashed line in figure 6.

It can be easily proved that, if (i -

i-1) CTHD(a) /8 for each i = 1, ,I, the resulting arc exit time temporal

profile satisfies the FIFO rule; such a condition is very likely to be verified in practice.

[Figure 6]

The above assumptions and results lead to the following procedure for calculating the arc exit times as a

function of the arc inflows.

function t(f)

foreachaAta

0 =La /Va

fori = 1 toI

tai= max{ta

i-1+(

i-

i-1) fa

i/Qa ,

i+La /Va}nexti

ta0

= ta0

+0.25ra2

CTHD(a)(1 + 1/[1 -(1-ra) fa1

/Qa1

/(ta1

-ta0

)])fori = 1 toI-1

tai= ta

i+0.25ra

2CTHD(a)(1/[1 -(1-ra) fai/Qa(

i-

i-1) /(ta

i-ta

i-1)] +

+1/[1 -(1-ra) fai+1

/Qa(i+1

-i) /(ta

i+1-ta

i)])

nexti

taI

= taI

+0.25ra2CTHD(a)(1/[1 -(1-ra) fa

I/Qa(

I-

I-1) /(ta

I-ta

I-1)] + 1)

nextaend function

Then, Based on (9), the arc costs can be easily computed as a function of the arc exit times through the

following procedure.

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function(t)

foreachaAfori = 0 toI

cai= (ta

i-i) +mai

nextinexta

end function

5.2 Multinomial Logit implicit path choiceThe implicit path formulation presented in subsection 2.5 is here implemented with reference to the

Multinomial Logit path choice model discussed in section 4. Coherently with this approach, one destination

at the time is considered.

Since users choose only efficient paths, the system of node satisfactions defined by equations (24) is of

the triangular type and can be solved by progressing in increasing topological order, starting from the

destination. Following this approach, the node satisfactions are computed below, based on (48), as a function

of the arc performances.

function w(c, t)

w = 0

foreachdCfork= 2 to |N|

x =N(k)d

foreachaFSE(x)d

j = 1fori = 0 toI

dountiljta

i

j =j +1

loop

( ) ( )( ) ( )

1

1 1

1

exp

d j d j

HD a HD ad ji i ja aHD a j j

d i d ix x

w wc w t

w w

+ +

= +

nexti

nextafori = 0 toI

wxdi

= ln(wxdi

)

nextiwx

d I+1= wx

d I

nextxnextd

end function

Then, based on (50), the arc conditional probabilities can be computed as a function of the node

satisfactions and of the arc performances through the following procedure.

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function p(w, c, t)

p = 0

foreachdCforeachaA

ifONTL(a)d

> ONHD(a)dthen

j = 1fori = 0 toI

dountiljta

i

j =j +1

loop

( ) ( )( ) ( )

( )

1

1 1

1

exp

d j d j

HD a HD ad j d ii i ja aHD a TL aj j

d ia

w wc w t w

p

+ +

=

nexti

end if

nextanextd

end function

In the above two procedures, the do loop cycle is needed to determine j such that j-1< ta

i

j, so as to

calculate wHD(a)d(ta

i) consistently with (55), while the assumption that wHD(a)

d() = wHD(a)

d Ifor every >

I, implied

by the instruction wxd I+1

= wxd I

, is supported by the hypothesis that all queues vanish within time < .

5.3Network flow propagationThe implicit path formulation presented in subsection 2.5 is here implemented with reference to the

network flow propagation model expressed by the system of arc inflows defined by equations (27). Again,

one destination at the time is considered.

Since users choose only efficient paths, this system is of the triangular type and can be solved by

processing the nodes in decreasing topological order and determining, for each node, the arc inflow temporal

profiles of its efficient forward star. This way, in fact, when an arc is considered, the inflow temporal profile

of each arc belonging to the efficient backward star of its tail has already been determined.

Indeed, instead of dealing with backward inflows at the moment of determining the inflow temporal

profile of the current arc, it is more convenient to propagate forward in time and space the inflow of the

current arc towards the arcs belonging to the efficient forward star of its head, as this operation does not

involve the inverse temporal profiles of the arc exit time. This way, when an arc is considered, its inflow

temporal profile has already been determined. Specifically, the inflow forward propagation consists of the

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three steps below.

a) Determining the actual arc outflow temporal profile ad, corresponding to an arc inflow temporal

profile complying with (54) and an arc exit time temporal profile complying with (55). Based on vehicle

conservation and FIFO rule, ad turns out to be constant during each interval (tai-1, tai], where it takes value:

ad i

=fad i

(i -i-1) /(tai-ta

i-1), (62)

thus not complying with assumption (54).

b) Transforming ad

into an equivalent profile ead

constant during each time interval (i-1

, i], preserving

vehicle conservation. This can be obtained by applying, for each interval ( tai-1

, tai], the procedure described

below and depicted in figure 7, which calculates the corresponding contribution ofad

to ead.

[Figure 7]

functionspread

1) ifjta

ithen

ead j

= ead j

+ ad i

(tai-ta

i-1) /(

j-j-1)else

2) ead j

= ead j

+ad i

(j -tai-1) /(

j-j-1)

j =j +1

3) dountiljta

i

ead j

= ead j

+ad i

j =j +1

loop

4) ead j

= ead j

+ ad i

(tai-

j-1) /(j-

j-1)

end if

end function

If condition 1 is met, we have (tai-1

, tai](j-1, j]; then the contribution of a

d ibelongs entirely to ea

d j.

Otherwise, only the outflow exiting during the interval (tai-1,

j] is assigned to ead j

(step 2). Then, until the

current interval (j-1

, j] belongs to (ta

i-1, ta

i],j is iteratively incremented and exactly the value a

d iis summed

up to ead j (step 3). At the end of this cycle, the outflow exiting during the interval (j-1, tai] is assigned to ead j

(step 4), wherej can be equal toI+1.

It should be noted that, by means of this transformation, the first vehicles constituting the actual outflow ad i

are slowed down, while the last are accelerated; however this effect is not relevant in practice.

c) Splitting ead

among the arcs belonging to FSE(HD(a))d. Based on assumption (55) referred to the arc

conditional probabilities, the contribution of the outflow of arc a during the generic time interval (i-1

, i] to

the inflow of arc bFSE(HD(a))d throughout the same interval is ead i

0,5 (pbd i

+pbd i-1).

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Following this approach, the arc inflows are computed below, as a function of the arc conditional

probabilities and of the arc exit times.

function(p, t ; d)f= 0

for eachdCfd

= 0 , ed

= 0

foreach oCfor eachaFSE(o)d

fori = 1 toI

fad i

= dodi0.5 (pa

d i-1+pa

d i)

nextinexta

nextofork= |N| to 2 step -1

x =N(k)d

for eachaFSE(x)dj = 1

dountiljta

0

j =j +1

loop

fori = 1 toI

ad i

=fad i(i -i-1) /(ta

i-ta

i-1)

callspread

nexti

foreachbFSE(HD(a))dfori = 1 toI

fbd i

=fbd i

+ead i

0.5

(pb

d i-1

+pbd i

)nextinextb

nextanextkf= f+f

d

nextdend function

6 SOLUTION ALGORITHMThe implicit path formulations of WDDTA (37) and (39), though equivalent in terms of solutions, differ

noticeably in terms of fixed-point operator: the first one involves solving the DNL problem, while the second

one requires just the evaluation of the NLM. In this section, Bathers method (Bottom and Chabini, 2001) is

applied to both formulations, and, in the next section, the resulting algorithms are compared referring to the

Multinomial Logit path choice model.

Bathers method is an accelerated averaging algorithm solving general fixed-point problems, which can

be seen as a relatively minor modification of the Method of Successive Averages (MSA) (Sheffi and Powell,

1982). In the MSA, the successive deign point(the point at which the fixed point operator is evaluated) is

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taken to be the current estimate of the fixed point. Bathers method, instead, is intended to exploit the

potential advantages of using one method to select the design points, and a different method to estimate the

fixed point. This approach, in fact, allows a more aggressive exploration of the feasible space and a more

effective employment of the information generated during the search in order to estimate the fixed point.

Specifically, at each iteration, the current design point f* is obtained by adding to the average of previous

design points favg

an appropriate fraction of the average of previous deviations fdev

, where the current

deviation is defined to be the difference between the fixed point operator and the current design point at

which it is evaluated.

At the cost of retaining two auxiliary vectors, Bathers method outperforms the MSA, exhibiting a

convergence rates four or more times faster.

Making reference to the numerical method presented in the previous section, the procedure implementing

this fixed point algorithm for both implicit path formulations of WDDTA is outlined below.

functionWDDTA0) k= 01) f= 0 , f* = f , f

avg= 0 , f

dev= 0

2) do until (||f-f*|| < ork> kmax)andk> 03) k= k+1

4) f= f*5) t = t(f)

6) c = (t)

7) w = w(c, t)

8) p = p(w, c, t)

9) NLM) f= (p, t ; d) | DNL) f= DNL(p ; d)

10) fdev

= (1 -1/k) fdev +1/k(f-f*)11) f

avg= (1 -1/k) favg +1/kf*

12) f* = favg

-k1 / 3

fdev13) loop

14) f= favg

end function

After the initialization of the arc inflows (step 1), the fixed point operator is iteratively evaluated (steps 5-8)

until a stop criterion is met (step 2). Specifically, at each iteration k, on the basis of the current arc inflows,

the arc performances are determined (steps 5-6). Then, the node satisfactions are calculated (step 7) in order

to compute the arc conditional probabilities (step 8), on which basis the demand is loaded on the network

(step 9). At last, the auxiliary flows are updated through an averaging process (step 10-11) and the successive

design point is determined (step 12).

The two algorithms solving the fixed-point problems (37) and (39) are the same, but for step 9, where a

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DNL problem is to be solved instead of performing a straight network flow propagation based on the current

travel times. Specifically, the determination of a point of the map DNL(p ; d) can be attained by solving the

fixed point formulation (31) of the DNL problem, again through bathers method, by means of the following

procedure.

functionDNL(p ; d)

h = 0

y * = f , yavg

= 0 , ydev

= 0

do until (||f-y *|| hmax)andh > 0h = h +1

f= y *

t = t(f)

f= (p, t ; d)

ydev = (1 -1/k) ydev +1/k(f-y *)

yavg

= (1 -1/k) yavg

+1/ky *y * = yavg -h1 / 3ydevloop

f= yavg

end function

Note that the above procedure ensures that, at each iteration k, the capacity constraints are satisfied, as the

arc flows resulting from the DLN are reciprocally consistent with travel times. On the contrary, when

applying the NLM the capacity constraints are satisfied only at the equilibrium.

The proposed algorithms are not proven to converge to a fixed point. They should be then used as

heuristics which have been found to give good results in practice.

7 NUMERICAL APPLICATIONThe network of Sioux Falls, consisting of 76 directed arcs and 24 centroids, has been considered for a

numerical application of the proposed model. To this end, the known daily demand has been distributed

consistently with an arbitrary temporal profile simulating a morning peak.

Since the formulations (37) and (39) are equivalent, if the solution were unique, the corresponding

algorithms would yield the same arc flow pattern. Indeed, the numerical results show that both algorithms

converge to the same solution, as qualitatively shown in figure 8.

[Figure 8]

On this basis, the comparisons between the two approaches is to be made in terms of efficiency of the

corresponding algorithms. In this regard, note that the computational burden to calculate the arc conditional

probabilities through the composite function p(w(c, t), c, t) is about the same as that required to evaluate

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many times less than that of the bi-level formulation, as one half the number of iterations needed for the

DNL to converge.

The extension of this approach to more general cases (e.g., elastic demand with departure time choice,

multimodal network, spillback phenomenon, extra-urban networks) will be the subject of future research.

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REFERENCES

Addison J. D. , Heydecker B. G. (1993) A mathematical model for dynamic traffic assignment, in

Transportation and Traffic Theory, ed. C. F. Daganzo, Elsevier, Amsterdam, 171-184.

Bellei G., Bielli M. (1990) Dynamic assignment for trip planning systems evaluation, in Atti del

Convegno Nazionale ANIPLA: LAutomazione nei Sistemi di Trasporto, 520-535.

Bellei G. , Bielli M. (1996) Dynamic traffic assignment in congested networks, inAdvanced Methods in

Transportation Analysis, ed.s L. Bianco, P. Toth, Sprinter-Verlag, Heidelberg, 263-298.

Bellei G., G. Gentile, N. Papola 2002 Network Pricing Optimization in Multi-user and Multimodal

Context with Elastic Demand, Transportation Research B, 36/9 pp 779-798.

Bottom J. , Chabini I. (2001) Accelerated averaging methods for fixed point problems in transportation

analysis and planning, inPreprints of Tristan IV, Azores, Portugal, 69-79.

Cantarella G. E. (1997) A general fixed-point approach to multimode multi-user equilibrium assignment

with elastic demand, Transportation Science 31, 107-128.

Carey M. , Subrahmanian E. (2000) An approach to modelling time-varying flows on congested

networks, Transportation Research B 34, 157-183.

Cascetta E. , Cantarella G. E. (1991) A day-to-day and within-day dynamic stochastic assignment

model, Transportation Research A 25, 277-291.

Cascetta E. (2001) Transportation systems engineering: theory and methods, Kluwer Academic

Publisher, 384-386..

Daganzo C. F. (1994) The cell transmission model: a dynamic representation of highway traffic

consistent with hydrodynamic theory, Transportation Research B 28, 269-287.

Daganzo C. F. (1995) The cell transmission model Part II: Network traffic, Transportation Research B 29, 79-93.

Daganzo C. F. (1997)Fundamentals of Transportation and Traffic Operations, Pergamon, 97-112.

Dial R. B. (1971) A probabilistic multipath traffic assignment algorithm which obviates path

enumeration, Transportation Research 5, 83-111.

Drissi Kaitouni O. , Hameda Benchekroun A. (1992) A dynamic traffic assignment model and solution

algorithm, Transportation Science 26, 119-128.

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Figure 1

supplymodel

demandmodel

networkloadingmap

arcperf.

function

network flow

propagation model

arc performance

model

path performance

model

path

performances

arc

performances

path

flows

arc

flows

demand

flows

path choice

model

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32

Figure 2

arctg Va

fa()

naQ

Qa

density

flow

B

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33

Figure 3

oC dC

xTL(a)N

aA

hKoTL(a)

jKaTL(a)d

time

Tkod

a()-1 tb()

cb() -wHD(b)d(tb())

bFSEd(x)

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35

Figure 5

arctg Va

Qa

La

i

space

i-1

qRai

tRai-1

tRai

EQ a

fai

fai-1

fai+1

fai+2

time

La /Va

Qa

qRai+1

flow

flow

time

i-2 i+1

i+2

tRai-2

tRai+1

tRai+2

(a) eRa()

(b) qRa()

(c) tRa()-

(d)fa()

time

time

time

eRai-1

eRai+2

i

BQ a

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37

Figure 7

qai-1

time

i

i-1

tai-1

La

ad i

contribution ofad i to ea

d

space tai

qai

fad i

La

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Figure 8

6.00 AM

10.00 AM 11.00 AM9.00 AM

7.00 AM 8.00 AM

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39

Figure 9

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

n of evaluations of function (p, t ; d)

error (us/h)

WDDTA with NLM WDDTA with DNL

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