A Unified Framework For Traffic Assignment Deriving Static And Quasi-Dynamic Models Consistent With General First Order Dynamic Traffic Assignment Models

INSTITUTE of TRANSPORT and
LOGISTICS STUDIES
The Australian Key Centre in
Transport and Logistics Management
The University of Sydney
Established under the Australian Research Council’s Key Centre Program.
WORKING PAPER
ITLS‐WP‐14‐08
A unified framework for traffic
assignment: deriving static and
quasi‐dynamic models consistent
with general first order dynamic
traffic assignment models
Michiel C.J. Bliemer, Mark P.H. Raadsen,
Luuk Brederode1,3
, Michael G.H. Bell and
Luc J.J. Wismans2,3
1
Department of Transport and Planning, Delft University
of Technology, the Netherlands
2
Centre for Transport Studies, University of Twente, the
Netherlands
3
DAT.Mobility, the Netherlands
April 2014
ISSN 1832‐570X

NUMBER: Working Paper ITLS-WP-14-08
TITLE: A unified framework for traffic assignment: deriving static
and quasi-dynamic models consistent with general first order
dynamic traffic assignment models
ABSTRACT: This paper presents a theoretical framework to derive static,
quasi-dynamic, and semi-dynamic traffic assignment models
from a general first order dynamic traffic assignment model. By
explicit derivation from a dynamic model, the resulting models
maintain maximum consistency with dynamic models. Further,
the derivations can be done with any fundamental diagram, any
turn flow restrictions, and deterministic or stochastic route
choice. We demonstrate the framework by deriving static (quasi-
dynamic) models that explicitly take queuing and spillback into
account. These models are generalisations of models previously
proposed in the literature. We further discuss all assumptions
usually implicitly made in the traditional static traffic assignment
model.
KEYWORDS: Traffic assignment, unified framework, dynamic and static,
first order, generalised link transmission model, residual
queues
AUTHORS: Bliemer, Raadsen, Brederode, Bell and Wismans
CONTACT: INSTITUTE OF TRANSPORT AND LOGISTICS STUDIES
(C13)
The Australian Key Centre in Transport and Logistics
Management
The University of Sydney NSW 2006 Australia
Telephone: +612 9114 1824
E-mail: business.itlsinfo@sydney.edu.au
Internet: http://sydney.edu.au/business/itls
DATE: April 2014

A unified framework for traffic assignment: deriving static and quasi‐dynamic models
consistent with general first order dynamic traffic assignment models
Bliemer, Raadsen, Brederode, Bell and Wismans
1
1. Introduction
Traffic assignment models are used all over the world in strategic (long term) transport planning
and project appraisal to forecast future traffic flows and travel times. Road authorities around
the world apply mainly static models on large scale road networks for this purpose, although
exceptions exist (e.g., the government in the Netherlands applies a quasi-dynamic traffic
assignment model in regional and national transport modelling studies, and the San Francisco
government in the US recently implemented a dynamic traffic assignment model for assessing
transport projects in the city). In the last decade, there has been an increased interest in dynamic
traffic assignment models as they have clear advantages with respect to realism. Nowadays,
such dynamic models have become feasible on relatively large scale transportation networks
due to significant increases in computation power and memory as well as the development of
more efficient algorithms for solving these models. As is well-known, traditional static models
suffer from several serious limitations, such as traffic flows exceeding link capacities and
queues not being explicitly described. As a result travel time delays appear in the wrong
locations. Despite this obvious lack of realism, such static models remain popular because they
are mathematically rigorous, typically guarantee the existence and uniqueness of an equilibrium
solution, can be efficiently solved using fast converging algorithms, are relatively easy to
understand and very robust in applications. Dynamic models are significantly more complex,
often do not guarantee existence or uniqueness of an equilibrium solution, require much longer
run times and are significantly more difficult to calibrate than static models, requiring a large
investment in time and money when moving towards dynamic models in practice.
While governments use predominantly static models for long term transport planning, dynamic
models are widely applied for traffic operations analysis and evaluation of traffic management
measures. In particular microscopic simulation models have become hugely popular. However,
almost without exception, governments struggle with the inconsistencies between the static
models used for transport planning and the dynamic models for traffic operations. If
infrastructure decisions are based on outcomes of a static model (as in most cases), it is
problematic if the predicted travel time savings cannot be reproduced in a dynamic model (and
in real life). For example, consider a simple corridor with two bottlenecks according to a
traditional static model. Expanding the capacity of the second bottleneck may result in
significant travel time benefits in the static model, while it may not result in any travel time
benefits in the dynamic model because traffic flow is reduced by the first bottleneck and the
second bottleneck poses no longer a problem in the dynamic case (and in reality). Such
inconsistencies exist because static and dynamic models are built on different underlying
principles. The theoretical framework and the models presented in this paper remove these
inconsistencies.
Besides inconsistencies being problematic in decision making, they also have impacts on model
calibration. The current state of practice is that static models, specifically their networks and
origin-destination (OD) matrix, are used as a basis for developing any dynamic model (also in
the case of a much smaller sub-network in micro-simulation). However, due to inconsistencies
between static and dynamic models, calibrated static models are a very undesirable starting
point for creating a dynamic model for two main reasons. Firstly, link capacities in static models
do not indicate maximum flow but rather influence the travel time via link performance
functions. In dynamic models the link capacities indicate physical maximum flow. A dynamic
model is very sensitive to these capacities and their values should be set with much more care.
Secondly, the OD matrix has been calibrated to empirical traffic flow data such as link counts.
Since traditional static models do not constrain traffic flows to link capacities, the OD matrix is
calibrated in such a way that it becomes not very useful for dynamic models that include such
capacity constraints. As a result, it becomes a time consuming, complex and often
underestimated task to re-calibrate the OD matrix for use in dynamic models. The class of static

2
models derived in this paper adopts the traffic flow principles from dynamic models, such that a
static OD matrix calibrated for this new class of static models is a much better starting point for
building dynamic models.
In the past decade, several extensions to static models have been proposed to decrease the
present huge gap between static and dynamic traffic assignment models. In particular, novel
static (often referred to as quasi-dynamic) models have been proposed as a replacement of
traditional unconstrained static models, such that traffic flows and travel times are more in line
with dynamic models. As such, these new models have the potential to take away most of the
inconsistencies between static and dynamic models. In this paper, instead of trying to extend
currently existing static models, we will show how to derive static (and quasi-dynamic) traffic
assignment models directly from any first order dynamic traffic assignment model. As such,
maximum consistency is guaranteed. In particular, we aim to maintain consistency in spatial
interactions described by the route choice model, the link model (described by a fundamental
diagram), and the node model (described by turn flow restrictions) by only making different
assumptions on how the time dimension is handled (i.e., aggregated) within the model. We will
show how to derive first order static models and quasi-dynamic models that realistically
describe queues and spillback, based on a generalised first order macroscopic dynamic model
with any fundamental diagram, any turn flow restrictions, and route choice behaviour. It should
be noted that also for microscopic and mesoscopic dynamic models such a derivation can be
made, as long as the underlying (first order) spatial interaction mechanisms can be translated
into a specific fundamental diagram, specific turn flow restrictions, and a specific route choice
model. For example, the mesoscopic models in AIMSUN and Dynameq consider a triangular
fundamental diagram (per lane) and simplified intersection rules that can be directly used to
derive a consistent static (or quasi-dynamic) deterministic or stochastic traffic assignment
model.
This paper presents several contributions. (i)We present a class of generalised first order
dynamic traffic assignment models by combining existing state-of-the-art first order node and
link models with a rigorous stochastic user equilibrium route choice problem formulation.
(ii)We derive static and quasi-dynamic model classes directly from the generalised dynamic
model. Although it seems natural to consider static models as a special case of dynamic models,
as far as we are aware we are the first to perform such a derivation. (iii)We explicitly state the
‘static’ assumptions (which we call temporal interaction assumptions) that have been made
implicitly in static models in the past (often without realising). In addition, we explicitly state
the traffic behavioural assumptions (which we call spatial interaction assumptions) that have
been made implicitly in static models (again, often without realising), which include
assumptions with respect to the fundamental diagram, turn flow restrictions, and route choice
behaviour. Understanding these assumptions provide important insights into the model
properties. (iv)By relaxing these spatial interaction assumptions and thereby adopting more
mechanisms from dynamic models, this leads to novel more realistic static models.
Furthermore, our framework also allows relaxing one or more temporal interaction assumptions,
leading to novel semi-dynamic models.(v)We demonstrate that static and quasi-dynamic models
earlier proposed in the literature can be seen as special cases by making different temporal and
spatial interaction assumptions. Our unified framework therefore describes the whole spectrum
of all first order traffic assignment models, ranging from static to dynamic, and enables one to
create time-aggregated models consistent with any first order dynamic model.
2. Static, quasi-dynamic and dynamic traffic assignment
models
Many traffic assignment models have been proposed in the literature, ranging from static
macroscopic to dynamic microscopic and anything in between. Several have been implemented
in commercially available software. All traffic assignment models consist of two main

3
components, namely (i) a route choice model that determines how the travel demand is
distributed over available routes in the network, and (ii) a network loading model that
propagates (simulates) the traffic on the network. The network loading model is often described
in terms of a link model (that describes the traffic flows along a link) and a node model (that
describes turn flow restrictions through each node). In this paper we focus on macroscopic
models that represent traffic as continuous flows instead of discrete individual vehicles, since
we aim to derive static and quasi-dynamic model classes from a dynamic model.
2.1 Static traffic assignment models
Static traffic assignment models are well established. They were mainly developed in the 1950s
with the work of Wardrop (1952) and Beckmann et al. (1956), initially for solving for a
deterministic user equilibrium solution and later extended to a stochastic user equilibrium
(Daganzo and Sheffi, 1977; Fisk, 1980). Route choice is the main part of these static models,
while the network loading component is not capacity constrained by a turn flow restrictions and
does not allow for residual traffic (e.g., queues). Travel times for traversing links are calculated
using link performance functions.
Static traffic assignment model theory has progressed slowly over time. There have been some
attempts to include capacity constraints into static models (e.g., Smith, 1987; Yang and Yagar,
1994, 1995; Larsson and Patriksson, 1995; Bell, 1995) or residual queues (Bifulco and Crisalli,
1998; Nesterov and De Palma, 1998, 2000, 2003; Lam and Zhang, 2000; Smith, 2013). None of
these static models take spillback into account.
Although some of the static models mentioned in the previous paragraph have been termed
quasi-dynamic, there is no explicit modelling of the time dimension in the route choice nor in
the network loading model and therefore they are essentially static models. Some authors use
the term quasi-dynamic to indicate models that produce results more similar to dynamic models
(such as residual queues and spillback). Operational quasi-dynamic algorithms have been
proposed in order to take queuing and spillback into account. Examples are QBLOK (4Cast,
2009) and a model in VISUM described in Bundschuh et al. (2006). We would like to point out
that these models are merely described as algorithms without explicitly specifying the
mathematical problem that is being solved. The lack of a rigorous problem formulation prohibits
analysis of the properties of these models. The first quasi-dynamic model that describes
spillback in a rigorous problem formulation has recently been proposed by Smith et al. (2013).
Since the term quasi-dynamic does not express any specific properties of the model besides
outputs that may be more in line with dynamic models, we refer to these quasi-dynamic models
as static models due to the absence of any time dynamics.
As far as we are aware, Although the above mentioned residual queuing models are an
improvement over the traditional static traffic assignment models, they predict queues inside the
bottleneck link and not upstream the bottleneck link because none of these models include a
proper (first order) node model. Bliemer et al. (2013) present a novel static model that for the
first time includes a proper node model in static traffic assignment, such that residual queues
appear upstream bottleneck links.
2.2 Semi-dynamic traffic assignment models
Models in which the time dimension explicitly enters the model in one way or the other we call
semi-dynamic. For example, there can be some dynamics in the network loading model, or there
can be dynamics across time periods in the route choice. The most important feature of semi-
dynamic models is the ability to transfer residual traffic and queues from one time period to the
other.
The oldest semi-dynamic traffic assignment model that we are aware of is SATURN (Van Vliet,
1982), which explicitly considers dynamics between time periods (by transferring queues) and

4
limited dynamics within the network loading. SATURN is again an operational model without a
rigorous problem formulation. More recently, Davidson et al. (2011) proposed another
operational model that they termed clocktime assignment, in which they propagate the travel
demand over paths using one minute time steps, and keep track of traffic that is still on the
network at the end of each ten minute time period (in the form of a ‘too late trip matrix’).
Another approach is taken by Brederode et al. (2010) and Bliemer et al. (2012) by combining a
static model with a dynamic model. The static model results in vertical queues, while the
dynamic model describes the queue dynamics by tracing backward and forward shockwaves
resulting in physical queues and spillback. While this approach has both static and dynamic
elements, we will not refer to this approach as semi-dynamic, but rather call it a hybrid model.
2.3 Dynamic traffic assignment models
Dynamic traffic assignment models have gained in popularity since the 1990s. Two different
approaches can be distinguished in the literature, which we loosely refer to as the (strategic)
transport planning approach and the (operational) traffic management approach, referring to the
two application areas mentioned in the introduction. In the transport planning approach, network
modellers have proposed direct extensions to the static model, maintaining the strong route
choice component but adopting a relatively weak network loading model that is not consistent
with traffic flow theory. In the traffic management approach, traffic flow theorists have
proposed more complex network loading models consistent with traffic flow theory, but these
were mostly limited to small networks or unable to find a user equilibrium solution. In the past
five years, the two approaches seem to have converged to models that describe both route
choice behaviour and traffic flow theory in a realistic, rigorous, and practically feasible way.
This is an important observation in itself.
In the transport planning approach, essentially a time index was added to all variables, and
traffic flow propagation was based on link travel times described by link performance functions.
Such models are often referred to as whole link models in which the travel time is a function of
the link flow or density. Examples are Janson (1991), Astarita (1996), Ran and Boyce (1996),
Wu et al. (1998), Chen and Hsueh (1998), Xu et al. (1999), Chabini (2001) and Bliemer and
Bovy (2003). Flow propagation using link performance functions is not consistent with traffic
flow theory (Bliemer, 2007), and due to the lack of a node model, link flows are not capacity
constrained and no physical queues are described.
In the traffic management approach, traffic flow theory based on kinematic waves introduced by
Lighthill and Whitham (1955) and Richards (1956) is adopted to conduct the network loading.
This theory describes traffic flow operations much more realistically, but typically requires
solving complex partial differential equations. First order vehicle dynamics can be described by
a given fundamental diagram for each link that describes the relationship between flow and
density. In these first order models, vehicles are assumed to be able to instantaneously change
speed, such that trajectories are piecewise linear. Second order models make the extension to
speed dynamics, leading to more realistic smooth trajectories, however they require solving
complex second order partial differential equations. Messmer and Papageorgiou (1990)
developed METANET, which is based on a simplification of these differential equations and
computes an approximate solution. Although second order models can describe some specific
phenomena such as capacity drop and hysteresis, first order models already capture most
important traffic dynamics, such as queue formation, spillback and shockwaves. Furthermore,
Daganzo (1995a) has argued that (approximations of) second order models can have certain
inherent limitations and can be problematic. Hence, in this paper we restrict ourselves to first
order models. Daganzo (1994, 1995b) proposed the cell-transmission model (CTM) in which
the road infrastructure is discretised into smaller cells as to provide an approximate solution to
the partial differential equations. Simple node models describing turn flow restrictions for
merges and diverges are included as part of the network loading. Newell (1993) describes a

5
simplified theory of kinematic waves assuming a triangular fundamental diagram, which is an
exact method that does not require solving any partial differential equations. Yperman (2007)
extended this link model to a network level and named it the link transmission model (LTM).
Yperman further extended Newell’s theory to a piecewise linear fundamental diagram. Gentile
(2010) proposed the generalised link transmission model (G-LTM) in which a fundamental
diagram of any concave shape can be taken into account. In many of these models derived from
traffic flow theory, route choice is described by splitting rates at nodes. This approach may be
suitable for short term forecasting, but for long term planning in which a user equilibrium
solution is required, splitting rates are not suitable because they are not consistent with the
notion of such an equilibrium. For all models derived from traffic flow theory holds that travel
times can be calculated as a post-processing step after the network loading from the cumulative
inflows and outflows assuming first-in-first-out (FIFO), instead of relying on link performance
functions. More recently, node models have been recognised to play an important role in traffic
assignment, and new macroscopic node models have been formulated and investigated
(Flötteröd and Rohde, 2011; Gibb, 2011; Tampère et al., 2011; Jin, 2012a; Jin, 2012b).
2.4 From dynamic to static
This paper derives static traffic assignment models from a generalised first order dynamic (G-
FOD) traffic assignment model under certain temporal interaction assumptions for handling the
time dimension. In this paper we will use the term ‘generalised’ to indicate that it describes a
class of models with any given concave fundamental diagram, any turn flow restrictions, and
stochastic or deterministic route choice behaviour. The general equilibrium route choice
problem will be formulated as a variational inequality problem. Since the (first order) G-LTM
link model in combination with a first order node model presents an exact, general, and
analytically rigorous G-FOD traffic assignment model, we can – without loss of generality – use
this network loading model as the basis for our derivations. By deriving models explicitly from
a generic first order dynamic model, the resulting classes of models inherit many properties
from dynamic models, including possible queue formation, spillback, and the transfer of
residual traffic to other time periods. The class of models derived from the G-FOD model in
which time dynamics do not play any role we refer to as generalised first order static (G-FOS)
models, while models in which time dynamics play some role, we refer to as generalised first
order semi-dynamic (G-FOSD) models. Quasi-dynamic models are classified as static models,
given that time dynamics play no role. Given certain spatial interaction assumptions, which
include route choice behaviour, a certain fundamental diagram (note that a link performance
function can also be rewritten as a fundamental diagram), and turn flow restrictions, a specific
model results. Figure 1 illustrates our proposed theoretical framework. Most traffic assignment
models proposed in the literature can be derived from the G-FOD model under certain temporal
and spatial interaction assumptions. More importantly, novel static (quasi-dynamic) and semi-
dynamic models can be derived with more realistic spatial interactions consistent with dynamic
models. We believe that this theoretical framework provides a significant contribution to the
literature.

6
Figure 1 - Theoretical framework for deriving static and semi-dynamic traffic assignment models.
3. Generalised first order dynamic (G-FOD) traffic
assignment model
In this section we formulate the generalised (single-class) macroscopic G-FOD traffic
assignment model that we will use as the basis for deriving static models. First, we will
introduce some notation regarding the input data describing travel demand and infrastructure
supply. Consider a general network ( , )
G N A
 where N denotes the set of nodes and A denotes
the set of directed links. Each link a A
 is characterised by a length a
L (km) and a
fundamental diagram (see also Section 3.2). Let O N
 and D N
 be the set of origins and
destinations, respectively. Let od
P be the set of paths for origin-destination (OD) pair ( , ),
o d
where o O
 and ,
d D
 and let P denote the set of all paths, which is the union of all sets .
od
P
A path p P
 is represented by a series of consecutive links. Each OD pair ( , )
o d has an
associated travel demand rate ( )
od
M t (veh/h) for each departure time instant [ , ],
t T T
 where
T and T indicate the start and end time instants of the travel demand period, respectively.
Links connecting an origin or destination to the physical road network are assumed to have
sufficient capacity to accommodate the travel demand. Traffic assignment deals with the
problem of assigning the travel demand to available routes given certain route choice behaviour.
Since it is practically infeasible to describe route choice in continuous time (although the
network loading problem can be solved in continuous time under certain assumptions, the travel
demand period [ , ]
T T is typically split into I route choice time periods, where each time period i
corresponds to time interval 1
[ , )
i i
t t  with a length of 1 ,
i i i
T t t

  1, , .
i I
  These time periods
do not necessarily have to be equal in length. All vehicles departing in a certain route choice
time period are assigned simultaneously to routes (i.e., route split proportions can be assumed
stationary over each time period). In dynamic traffic assignment models, these periods typically
G-FOS model class
Temporal
Interaction
Assumptions
(TIA)
Specific traffic assignment model
Forward kinematic waves
Backward kinematic waves
Residual traffic transfer
G-FOD model class
Spatial
Interaction
Assumptions
(SIA)
Route choice behaviour
Fundamental diagram
Turn flow restrictions
G-FOSD model class

7
range from one to 15 minutes1
, and in static models these periods range from one to several
hours. The average travel demand rate over route choice time period i, denoted by ( ),
od
M i is
equal to  
1
( ) 1/ ( ) .
i
i
t
od od
i t
M i T M t dt

  For notational convenience, we will omit the superscript
od from path variables (as a path implies the origin and destination), unless strictly necessary.
The G-FOD model consists of a route choice model, a network loading model, and a travel time
calculation component that feeds the travel time resulting from the network loading model back
into the route choice model. The network loading model consists of a general first order link
model and a general first order node model. Each component will be described separately in this
section.
3.1 Route choice
Let , 1, ,
[ ( )]
p p P i I
f i  

f  (veh/h) denote the departing flow rates on each path p for each time
period i. Let all travellers minimise their perceived route travel times. The stochastic route travel
times are given by the sum of the actual route travel time and an error term. In this paper we
assume that the error term is extreme value type I distributed with a variance of 2 2
/6 ,
 
leading to a logit-type route choice model with scale parameter .
 Let ( )
p
c i be the average
travel time in equilibrium on path p for vehicles departing during time period i. Then the
perceived route travel time can be written as 1
( ) log ( )
p p
c i f i

 (see e.g., Chen, 1999). Zhou et
al. (2012) have formulated a variational inequality (VI) problem for finding a dynamic
stochastic user equilibrium solution *
f based on the C-logit model. Adapting their formulation
and extending it to include multiple route choice periods (see e g,. Bliemer and Bovy, 2003), we
can formulate the following VI problem:
 
* * *
( , )
1
( ) log ( ) log ( ) ( ) 0, ,
od
od
p p p p p
o d i
p P
c i f i f i f i



 
      
 
 
   f (1)
where od
p
 is a path commonality factor (Cascetta et al., 1996) leading to the C-logit model
with 0
  or a path-size factor (Ben-Akiva and Bierlaire, 1999) or corrected path-size factor
(Bovy et al., 2008) leading to the PS-logit or PSC-logit models, respectively, with 0.
  The
resulting C-logit or path-size(-correction) logit models aim to take path overlap into account,
although we would like to point out that none of these factors can handle path overlap in route
choice perfectly (Bliemer and Bovy, 2008). Although VI problem (1) cannot describe other
route choice models (e.g., more sophisticated logit or probit models), we believe that this
formulation is sufficiently general to include the majority of the (large scale) equilibrium
models proposed in the literature and implemented in practice.
Further,  is the set of feasible path flow rates given by the following flow conservation and
non-negativity constraints that bound the flows from above and below:
( ) ( ), ( , ), ,
od
od
p
p P
f i M i o d i

  
 (2)
( ) 0, , .
p
f i p P i
    (3)
1
This time period should not be confused with the simulation time step size/period in dynamic network
loading models, which are often in the range of 0.1 to 2 seconds.

8
The average route travel times ( )
p
c i are computed as an implicit function of the route flows
after they have been loaded onto the network. This network loading model is described next.
3.2 Network loading
In the network loading model, path flows f are simulated on the network using flow
conservation and flow propagation equations. Let ( )
ap
u t (veh/h) and ( )
ap
v t (veh/h) denote the
path-specific link inflow rate into link a and the outflow rate out of link a at time instant t
following path p, respectively. Flow along a path is always conserved, as expressed by the
following equation:
( ), if link is the first link on path ,
( )
( ), if link follows link on path .
p
bp
ap
f i b p
u t
v t b a p

 

(4)
The path-specific inflow and outflow rates can be summed over all paths to obtain the total link
inflow and outflow rates,
( ) ( ), and ( ) ( ).
a ap a ap
p P p P
u t u t v t v t
 
 
  (5)
Constraints (4) are general flow conservation constraints that are present in all network loading
models. For the first link, the path-specific link inflow rate is equal to the average path flow rate
in the corresponding time period. For all consecutive links on the path, the path-specific link
inflow rates are equal to the path-specific outflow rates of the previous link on the path.
We further introduce the following definitions of cumulative flows. Let
0 0
( ) ( ) , and ( ) ( ) ,
t t
a a a a
U t u w dw V t v w dw
 
  (6)
be the total (cumulative) number of vehicles that have entered and exited link a up till time
instant t, respectively, and let the corresponding path-specific cumulative flows be defined as
0 0
( ) ( ) , and ( ) ( ) .
t t
ap ap ap ap
U t u w dw V t v w dw
 
  (7)
The outflow rates are the result of flow propagation constraints that involve a link and a node
model. It is mainly in the choice of the link and node models that (static and dynamic) models
proposed in the literature differ. In the next subsections we will describe the flow propagation
constraints for general first order macroscopic models.
3.2.1 First order link model
The link model describes the sending flow that would like to leave a link at a certain time
instant, as well as the receiving flow that could potentially flow into a link (i.e., can be
accommodated) at a certain time instant. In this section we take concepts of the discrete-time
LTM model (Yperman, 2007) and the continuous-time G-LTM model (Gentile, 2010) as the
basis for formulating our first order link model, which we believe is the most elegant, efficient,
exact and general formulation of a first order link model that currently exists. Due to space
limitations, we refer to Yperman (2007) and Gentile (2010) for more detailed descriptions of
concepts, definitions, and derivations. In this paper we aim to use notation that is common for
analytical dynamic traffic assignment models in the literature, which is different from the
notation often adopted in traffic flow theory used in Yperman (2007) and Gentile (2010).

9
Furthermore, we put the G-LTM model into a rigorous route based framework, while Yperman
(2007) ignored routes completely and Gentile (2010) adopted split proportions at nodes.
The G-LTM model provides an exact formulation of the flow propagation along a link
consistent with first order traffic flow theory and takes any concave fundamental diagram as
input. Let the fundamental diagram be described by function ( ),
q k
  which describes the
relationship between flow q and density k, see Figure 2. Any point on this curve represents a
specific traffic state on a cross-section of a link.
Figure 2: General concave fundamental diagram (figure adapted from Gentile, 2010).
Specific link characteristics that can be obtained from the fundamental diagram are the free-
flow (wave) speed (0)
a
 (km/h), the jam wave speed (0)
a
 (km/h), the physical capacity a
C
(veh/h), the jam density a
K (veh/km), and a critical density a
 (veh/km). For densities smaller
than the critical density, traffic is uncongested and vehicles are not queuing. This uncongested
part of the fundamental diagram is indicated by ( ).
I k
 Densities larger than the critical density
are associated with congested conditions and queues. The congested part of the fundamental
diagram is indicated by ( ).
II k
 Each stable traffic state propagates through the network with
the speed of the corresponding kinematic wave, which is given by the slope of the fundamental
diagram at that traffic state. In the uncongested part a forward kinematic wave belonging to
traffic flow u has a speed of ( ),
u
 note that this is usually not the same as the vehicle speed
related to this u. In the congested part a backward kinematic wave belonging to traffic flow v
has a (negative) speed of ( ),
v
 again observe that this kinematic wave speed always differs
from the vehicle speed related to the traffic flow v. In the special case of a triangular
fundamental diagram as proposed in Newell (1993) and used by Yperman (2007), all forward
waves in the uncongested part have free-flow speed (0),
a
 and all backward waves in the
congested part all have (negative) jam wave speed (0).
a

Using the original ideas of Newell (1993), it is possible to directly relate the traffic states at the
beginning and at the end of a link without the need to know exactly where vehicles are on a link
(and therefore there is no need to split the link into smaller cells like the cell transmission model
1
( )
I u


C
K
(0)
 (0)

flow q
(veh/h)
density k
(veh/km)
0
1
( ) ( )
( )
I I
u
k u u
u



 
   
 
 
1
( ) ( )
( )
II II
v
k v v K
v



 
    
 
 
1
( )
II v



( )
I k
 ( )
II k

u
v
( )
u
 ( )
v

uncongested congested

10
does). As such, all information that is needed is contained in the cumulative flows ( )
a
U t and
( ).
a
V t FIFO of link flows is an important property of this model that we will exploit later.
Let ( , )
a
S t t t
  (veh) be the sending flow of link a during simulation time period ( , ),
t t t
 
where the simulation time step t
 is usually somewhere between 0.1 and 10 seconds. The
sending flow is the number of vehicles that would like to exit the link during this time period
(the node model described later determines how many vehicles can actually flow out due to
possible receiving flow constraints). The sending flows can be computed using cumulative
inflows and outflows, and forward kinematic wave speeds:
( )
( , ) min min ( ) ( ) , ,
( )
I
a
a
a a a I a a
u t
a
L
S t t t U t t V t k u L C t
u


 
 
 
   
         
   
 
 
 
 
 
 
(8)
where ( )
I
a t
 is the set of all past realistic uncongested traffic inflow states up to time instant t,
andoperator ( ) max{0, }
x x

 is used in Figure 2 to define ( )
I
k u
 as the non-negative distance2
.
The minimum operators in Equation (8) are needed because of the Newell-Luke Minimum
Principle, which states that the minimum cumulative flow dominates the others. This equation
can be derived as a direct extension of the formulation in Yperman (2007) for piecewise linear
fundamental diagrams, or can be derived using the continuous-time formulation in Gentile
(2010). The outer minimum returns the first argument if the link is uncongested (i.e., no queue),
while the second argument is returned when it is in congestion.
Let the inner minimum be attained at inflow rate *
( ).
a
u t 3
The average sending flow rate can
simply be computed as ( , ) ( , )/
a a
s t t t S t t t t
    (veh/h). Letting 0
t
  yields the
following continuous-time formulation:
 
 
 
*
*
*
, if ( ) ( ) 0,
( )
( )
, otherwise,
( )
a
a a a I a a
a a
a
a
a
a a
L
C U t V t k u t L
u t
s t
L
u t
u t


  
  
    
 

  
 
 

 

  
  

(9)
where  
   
* *
/ ( ) ( ) ( ) 0
a a a a a I a a
U t L u t V t k u t L

     indicates that the sending flow at time
instant t did not encounter any congestion. In the same way, let ( , )
a
R t t t
  (veh) be the
receiving flow of link a during simulation time period ( , ).
t t t
  The receiving flow is the
number of vehicles that could enter the link during this time period (the node model described
later determines how many vehicles will actually flow in due to possible sending flow
constraints). The receiving flows can be computed using cumulative inflows and outflows, and
backward kinematic wave speeds:
2
While in dynamic network models the term cannot be negative due to concavity of the fundamental
diagram, when deriving static and quasi-dynamic network loading models in Sections 5 and 6 we will
require this property.
3
Note that it is possible that multiple forward shockwaves arrive at exactly the same time at the end of a
link. While the minimum in Equation (10) is unique, there may be multiple traffic states (inflow rates) for
which this minimum is attained. In that case, the Newell-Luke Minimum Principle prescribes that the
minimum inflow rate should be chosen, which due to concavity of the fundamental diagram corresponds
to the most recent traffic state in set ( ).
I
a t


11
( )
( , ) min min ( ) ( ) , ,
( )
II
a
a
a a a a a II a a
v t
a
L
R t t t V t t U t K L k v L C t
v


 
 
 
   
          
   
 
 
 
 
 
 
(10)
where ( )
II
a t
 is the set of all past realistic congested traffic outflow states up to time instant t,
and ( )
II
k v
 is the non-negative distance indicated in Figure 2. The outer minimum returns the
first argument if the link is in a spillback state, while the second argument is returned when
there is no spillback. Let the inner minimum be attained at outflow rate *
( ).
a
v t 4
The average
sending flow rate can be computed as ( , ) ( , )/
a a
r t t t R t t t t
    (veh/h). Letting 0
t
 
yields the following continuous-time formulation:
 
 
 
*
*
*
, if ( ) ( ) 0,
( )
( )
, otherwise,
( )
a
a a a a a II a a
a a
a
a
a
a a
L
C V t U t K L k v t L
v t
r t
L
v t
v t


  
  
     
 

  
 
 

 

  
  

(11)
where  
   
* *
/ ( ) ( ) ( ) 0
a a a a a a a II a a
V t L v t U t K L k v t L

      indicates that a queue has
reached the beginning of the link and is spilling back to upstream links.
3.2.2 First order node model
The node model determines how much of the sending flows can actually leave each link given
the receiving flow restrictions on consecutive links on the paths. In other words, the node model
describes the turn flow restrictions through each node, possibly leading to restrictions on link
outflows. It is one of the most important components of a network model, as all queues originate
at nodes and the node model therefore significantly influences delays and travel times. While
link models have received significant attention by researchers in the past, only in the last decade
node model have been recognised to be of significant importance in order to describe realistic
traffic flows, in particular in an urban setting.
The general first order node model discussed in this section is assumed to follow the
requirements provided by Tampère et al. (2011) for first order macroscopic node models. To
summarise, these requirements are (i) general applicability (i.e., can handle cross-nodes with
any number of incoming and outgoing links), (ii) maximizing flows, (iii) non-negativity, (iv)
conservation of vehicles, (v) satisfying demand and supply constraints, (vi) obeying
conservation of turning fractions (CTF), and (vii) satisfaction of the invariance principle
(Lebacque and Khoshyaran, 2005). First-in-first-out (FIFO) diverging rules (Daganzo, 1995b)
satisfy requirement (vi), while fair merging rules (Jin and Zhang, 2003) satisfy requirement
(vii).
Several node models have been proposed in the last decade (Jin and Zhang, 2003; Jin and
Zhang, 2004; Bliemer, 2007; Jin, 2012a; Jin, 2012b) but do not satisfy all of the above
requirements. Recently, some first order node models have been proposed that satisfy all the
4
Similar to Footnote 2, the Newell-Luke Minimum Principle dictates that in case multiple backward
kinematic waves reach the beginning of the link at the same time, the one corresponding to the minimum
outflow rate should be taken, which is the most recent one (given that the fundamental diagram is
concave).

12
conditions (Tampère et al., 2011; Flötteröd and Rohde, 2011; Gibb, 2011). In this paper we
consider any first order node model that satisfies the requirements in Tampère et al. (2011).
The sending flows ( )
a
s t describe the demand constraints and the receiving flows ( )
a
r t describe
the supply constraints in the node model. To be more precise, we need the sending flows in each
direction. Consider a certain node ,
n N
 and let in
n
A A
 and out
n
A A
 denote the sets of
incoming and outgoing links of node n, respectively. Let ( )
ab
s t (veh/h) be the directional
sending flow rate from link in
n
a A
 at time instant t that is destined for link out
.
n
b A
 We also
call ( , )
a b a turn (in the most general sense, not just left and right but also straight). Further, let
( )
ab t
 be the split proportion of the sending flow ( )
a
s t heading for link b, such that we can
write
( ) ( ) ( ).
ab ab a
s t t s t

 (12)
Since the FOD link model satisfies the link FIFO rule, we can determine the split proportions at
link exit from the flow proportions at link entrance. If ( )
a t

denotes the time of link entrance of
a vehicle that reaches the end of the link at time instant t, i.e.  
( ) ( ),
a a a
U t V t

 then the split
proportions can be computed as follows:
 
( ) ( ) ,
ab bp ap a
p P
t t
   

  (13)
where ap
 is a link-path incidence indicator that is equal to one if link a is on path p and zero
otherwise, and the path-specific flow proportion at any given time instant t is given by
( )
( ) .
( )
ap
ap
a
u t
t
u t
  (14)
The actual directional outflow rates are computed by multiplying the (desired) directional
sending flows with certain reduction factors ( )
a t
 for each link,
( ) ( ) ( ),
ab a ab
v t t s t

 (15)
Note that there is only a single reduction factor for all turns ( , )
a b due to the fact that the FIFO
property is assumed on each link, which essentially assumes a single queue for all turn
directions. Such a FIFO diverging rule is assumed in most (but not all) node models. If
( ) 1,
a t
  then all sending flow can exit the link, i.e. no turn flow restrictions are imposed.
However, if ( ) 1,
a t
  some of the sending flow cannot exit and will be added to a queue on
link a. These reduction factors are determined by turn flow restrictions described by a first order
node model for each node ,
n N
 which we implicitly indicate by function ( ):
n
 
  in
in out
( ) ( ( ), ( ), , ).
n
a n a b b n n
a A
t s t r t a A b A
   

 
      (16)
It should be noted that function ( )
n
  in Equation (16) can be fairly complex and often not
closed form, as the outflow rate for a specific link a typically depends on sending flows from all
incoming links and receiving flows from all outgoing links. We would like to refer to Tampère
et al. (2011) for a detailed description of these relationships.

13
The link outflow rate ( )
a
v t can now be computed as
out
( ) ( ),
n
a ab
b A
v t v t

  (17)
such that the path-specific flow rates, which feed back into the flow conservation constraints
stated in Equation (4), is given by
 
( ) ( ) ( ).
ap ap a a
v t t v t
 
 (18)
3.3 Travel time calculation
In G-FOD models, travel time calculations are done after flow propagation as a post-processing
step. Because of the FIFO link property, travel times can easily be derived from the (piecewise
linear) cumulative inflows and outflows, see e.g., Newell (1993) or Long et al. (2011). Because
of the FIFO property, the xth
vehicle enters link a at time instant 1
( )
a
U x

and exits the link at
time instant 1
( ),
a
V x

hence the link travel time for this vehicle is 1 1
( ) ( )
a a
V x U x
 
 (h). We are
interested in computing the average path travel time for vehicles departing during a certain time
period i. Since the FIFO property also holds on a path level, the path travel time for the xth
vehicle entering path p can be calculated as 1 1
( ) ( ),
p p
a p a p
V x U x
 
 where p
a and p
a are the first
and last link on path p, respectively. Alternatively, this path travel time can also be computed as
the sum of consecutive link travel times,  
1 1
( ) ( ) .
ap ap ap
a
V x U x
  

 Noting that there are
1
( ) ( ) ( )
p p
a p i a p i p i
U t U t f i T
   vehicles departing along path p during time period i, the average
path travel time is given by
 
1
( )
1 1
( )
1
( ) ( ) ( ) .
( )
a p i
p
p p
a p i
p
U t
p a p a p
p i U t
c i V x U x dx
f i T

 
 
 (19)
This integral, which represents the total travel time spent by all vehicles departing during time
period i, is graphically illustrated in Figure 4 for a simple example in which there is only one
path (so we have omitted the path subscript) consisting of four links with lanes and capacities
given in Figure 3. Three one-hour time periods (i.e., 1
i
T  for all i) of travel demand are
considered, with average path flow rates also given in Figure 3. Figure 4 shows possible
cumulative link inflows and outflows. Congestion appears on links 2 and 3, where we for
simplicity have assumed that the links are long enough to prevent spillback. The total travel
time spent by all vehicles in all time periods is indicated by the grey shaded area. For each time
period i, we are only interested in the area enclosed from the left and right by 1( )
U t and 4 ( ),
V t
and from the bottom and top by 1( )
i
U t and 1 1
( ),
i
U t  respectively. Dividing this area by the total
number of vehicles that entered yields the average path travel time. The average path travel
times are input into VI problem (1).
While the difference between 1
( )
a
V t

and 1
( ),
a
U t

which is the horizontal distance between the
cumulative flow curves, indicates the travel time for a vehicle entering the link at time a, the
vertical distance between ( )
a
U t and ( )
a
V t indicates the number of vehicles on the link at time
instant a (see also the dark black vertical lines in Figure 4). Knowing this number of vehicles
will become useful in the derivations that follow, hence we let ( )
a
Q t (veh) define the number of
vehicles on link a at time instant t as

14
( ) ( ) ( ).
a a a
Q t U t V t
  (20)
The number of vehicles on a link at the beginning of each time period, ( ),
a i
Q t we refer to as
residual traffic from earlier time periods.
Figure 3 - Example corridor network.
Figure 4 - Using cumulative link inflows and outflows to determine travel times.
4. Temporal interaction assumptions
Now that we have defined a general class of first order dynamic traffic assignment models, in
this section we formulate ‘static assumptions’ that eliminate the time dimension in one or more
parts of the model, and from which we can derive static and quasi-dynamic models. We will
refer to these assumptions as Temporal Interaction Assumptions (TIA), and are related to
forward and backward kinematic waves, and to how residual traffic is dealt with. Each of these
assumptions aims to remove or simplify any time dynamics within the model, in particular in
the network loading model.
4.1 Forward kinematic waves
In the G-FOD model, the forward kinematic waves are responsible for the propagation of the
vehicles in the network. In the dynamic model, the speeds of forward kinematic waves, ( ),
a u

are equal to the slopes of the fundamental diagram in the uncongested branch, ( ).
I k
 This
presents two complexities in dynamic models. First, we need to trace each forward kinematic
wave over time. Secondly, since the traffic conditions vary over time, the forward kinematic
wave speeds also vary over time. The second complexity can be overcome by assuming a finite
stationary forward kinematic wave speed over time, but this does not solve the first complexity.
However, assuming infinite forward kinematic wave speeds, i.e., ( ) ,
a u
   removes the need
to explicitly trace the kinematic waves by essentially propagating the vehicles instantaneously
through the network. This is the underlying assumption that is implicitly made in all static
(1) 3000
(2) 6000
(3) 1000
f
f
f



1 6000
C  2 6000
C  3 4000
C  4 2000
C 
veh
5000
10000
time
(1) 3000
f 
(2) 6000
f 
(3) 1000
f 
1( )
U t 1 2
( ) ( )
V t U t
 2 3
( ) ( )
V t U t
 3 4
( ) ( )
V t U t
 4 ( )
V t
1
t 2
t 3
t 4
t 5
t 6
t

15
models. Note that assuming ( ) 0
a u
  is not meaningful, as vehicles would not propagate
forward and stay at the origin.
It is important to realise that in large networks, not all vehicles may be able to reach their
destination within the considered time period (resulting in residual traffic). However, in static
models this is mostly ignored, such that for example a vehicle with a three hour travel time will
reach its destination during a two hour time period. Hence, the instantaneous forward
propagation assumption should preferably be restricted by the time period under consideration.
These possible assumptions are summarised in Table 1, where ‘S’ refers to an assumption that
removes the time dimension completely and hence results in a static model, while ‘D’ refers to
an assumption that is underlying a dynamic model.
Table 1 - Temporal interaction assumptions.
Assumptions on forward kinematic waves
S Instantaneous propagation of forward kinematic waves with infinite speeds;
D Time varying forward kinematic waves with speeds equal to the slope in the
uncongested branch of the fundamental diagram;
Assumptions on backward kinematic waves
S(v) No propagation of backward kinematic waves;
S(h) Instantaneous propagation of backward kinematic waves with infinite speeds and
finite link storage capacity;
D Time varying backward kinematic waves with speeds equal to the slope in the
congested branch of the fundamental diagram;
Assumptions on residual traffic transfer
S No transfer of residual traffic to next time periods;
D Transfer of residual traffic to next time periods;
4.2 Backward kinematic waves
In the G-FOD model, the backward kinematic waves are responsible for possible spillback of
queues to upstream links. Similar to forward kinematic waves, in the dynamic model the speeds
of backward kinematic waves, ( ),
a v
 are equal to the slopes of the fundamental diagram in the
uncongested branch, ( ).
II k
 Again, this requires tracing each backward kinematic wave over
time, while the wave speeds vary depending on traffic conditions. The simplest assumption to
remove any dynamics in backward kinematic waves is to set ( ) 0,
a v
  i.e. not having any
kinematic waves that move backward. This essentially means that queues form as (vertical)
point queues at the end of a link and never cause any spillback effects. The same model would
result if we would assume infinite link storage (denoted by the maximum number of vehicles
that could be present on a link), i.e. .
a a
K L   In case of such infinite storage, the backward
kinematic waves will never have an impact on upstream links and hence does not cause
spillback. More interesting is the assumption of infinite backward kinematic wave speeds,
( ) ,
a v
   combined with finite link storage. In this case, queues consume limited physical

16
storage space and spillback occurs if the finite storage space is exceeded. The upstream
influence of the backward kinematic waves will not propagate indefinitely but are automatically
restricted by the finite link storage capacities during each time period.
Table 1 summarises the above mentioned assumptions on the backward kinematic waves.
Assumption ‘S(v)’ results in a static model without spillback (e.g., vertical queues), while
assumption ‘S(h)’ yields a static model with horizontal queues and spillback. Assumption ‘D’ is
the underlying assumption in a dynamic model.
4.3 Residual traffic transfer
Residual traffic at the end of a time period results when vehicles are not able to reach their final
destination within the considered time period. These residual vehicles are either (i) in
congestion in a residual queue near a bottleneck, or (ii) are not in congestion but simply are not
able to reach their final destination because the travel time to reach the destination is longer than
the considered time period (see also Section 4.1). This residual traffic influences the traffic
flows and travel times in the next time period. If there is residual traffic on a certain link a at the
beginning of time period i, then ( ) 0.
a i
Q t  This dependency of traffic across time periods can
be eliminated by assuming that any residual traffic has exited the network at the beginning of
each time period i, i.e., ( ) 0
a i
Q t  for all links a.
Hence, we can consider two assumptions on residual traffic transfer as summarised in Table XT.
In case no residual traffic transfer occurs, the time periods become completely independent,
leading to a static model.
4.4 Combinations of temporal interaction assumptions
Different combinations of the TIAs in Table 1 result in different model classes. However, not
every combination is useful or even meaningful. If all assumptions are static (S), then a G-FOS
model results in which the time dimension is completely eliminated, both within and across time
periods. Depending on whether assumption ‘S(v)’ or ‘S(h)’ is adopted for backward kinematic
waves, a static model with vertical or horizontal queues will result, respectively. In case of
vertical queues, there will be no spillback, while assuming horizontal queues yields the
possibility of having spillback. We will further illustrate these classes of G-FOS models in
Section 6. If all assumptions are dynamic (D), then a G-FOD model results as described in
Section 3. If the TIAs are a mix of static (S) and dynamic (D), then a G-FOSD model results in
which the time dimension is only eliminated in certain parts of the model. In particular
assuming ‘D’ in residual traffic transfer leads to interesting semi-dynamic models in which
residual traffic is transferred between time periods (like in SATURN).
In this paper we focus on deriving G-FOS models and leave deriving the significantly more
complex semi-dynamic models for future research.
Regarding the assumptions on the forward and backward kinematic waves, they are summarised
graphically in Figure 5 for the different model types. The G-FOS assumes infinite forward wave
speeds and zero or infinite backward wave speeds, as indicated by the horizontal and vertical
lines in Figure 5(b). In the G-FOD model in Figure 5(d), these wave speeds are equal to the
slopes of the fundamental diagram at the corresponding densities. For the G-FOSD models, the
meaningful combinations of the slopes are indicated in Figure 5(c). Figure 5(a) illustrates the
case for the traditional static traffic assignment model and will be further explained in Section
6.4.

17
Figure 5 - Fundamental diagrams and assumptions on wave speeds.
We would like to point out that each of the above model classes are generic first order models in
the sense that they can take any fundamental diagram, any turn flow restrictions, and
deterministic or stochastic route choice into account. We refer to these choices as spatial
interaction assumptions, which will be discussed next.
5. Spatial interaction assumptions
While the TIAs describe how the time dimension is handled in the model, the spatial interaction
assumptions (SIA) describe how traffic flows spatially interact and directly impact the realism
of the model. These spatial interactions are a combination of assumptions on route choice
behaviour, the fundamental diagram, and turn flow restrictions (yielding flow reduction factors).
As pointed out in the introduction, maintaining consistency in spatial interactions is important
when moving between static and dynamic models. These spatial interactions have been
extensively analysed theoretically and have been estimated empirically. Specific choices for the
SIAs in a class of first order models lead to a specific traffic assignment model.
5.1 Route choice behaviour
Regarding route choice behaviour, often it is assumed that travellers have perfect knowledge
regarding travel times, such that 0
  and 0,
  leading to a deterministic model. In case
perception errors are introduced in which users base their route choice decisions on perceived
travel times, 0
  and a (logit-based) stochastic traffic assignment model results. Hence, we
can define the following assumptions. In case of such a stochastic model, path overlap is taken
into account if 0.
  This can be either done via a path-size factor (leading to a path-size logit
model) or a commonality factor (leading to the C-logit model), denoted by .
od
p
 If no path
overlap is taken into account, the model simplifies to the well-known conditional (also referred
to as multinomial) logit model. The different SIAs regarding route choice behaviour are
summarised in Table 2.
( )
I
q k
C
( )
I
q k
C
u
v
( )
u

( )
v

( )
II
q k
( )
I
q k
C
u
v
( )
u

( )
v

( )
II
q k
u ( )
u

k
q
0
q
0
k
q
0
k
( )
I
q k
C
u
v
( )
u

( )
v

( )
II
q k
q
0
k
(a) Traditional static (b) G-FOS
(c) G-FOSD (d) G-FOD

18
Table 2 - Spatial interaction assumptions.
Assumptions on route choice behaviour
D Deterministic route choice behaviour;
S(m) Stochastic route choice behaviour using the multinomial logit model, ignoring path
overlap;
S(c) Stochastic route choice behaviour using the C-logit model, taking overlap into
account using a commonality factor;
S(p) Stochastic route choice behaviour using the path size logit model, taking overlap into
account using a path size factor;
Assumptions on the fundamental diagram
L-X Linear uncongested branch, no congested branch;
L-H Linear uncongested branch, horizontal congested branch;
G-X General concave uncongested branch, no congested branch;
L-L Triangular fundamental diagram;
Q-Q Quadratic fundamental diagram;
Q-L Quadratic uncongested branch, linear congested branch;
G-G General concave uncongested branch, general concave congested branch;
Assumptions on the turn flow restrictions
U No turn flow restrictions specified, leading to unconstrained flows;
S Separable link exit capacity constraints;
G General non-separable turn flow restrictions;
5.2 Fundamental diagram
The shape of the fundamental diagram, which relates traffic flow and density, plays an
important role in traffic flow theory and different shapes lead to different spatial traffic patterns
in the model. The fundamental diagram can be directly estimated from cross-sectional traffic
counts and speeds, and consists in general of an increasing concave uncongested branch, and a
decreasing concave congested branch. The first fundamental diagram is described in
Greenshields (1935). He proposed a linear relationship between speed and density, which results
in a quadratic fundamental diagram. Such a symmetric fundamental diagram may describe
uncongested flows pretty accurately, but performs poorly for congested states. A popular choice
in traffic flow theory, mainly because of its simplicity, has been an asymmetric triangular
fundamental diagram. While a linear relationship in the congested branch seems realistic, a
linear relationship in the uncongested branch is less realistic, as this would mean that the free-
flow speed is equal to the speed at capacity. Bliemer et al. (2013) therefore propose a quadratic-
linear fundamental diagram as a good compromise, in which the uncongested branch of the

19
fundamental diagram is quadratic, while the congested branch is linear. While fundamental
diagrams have been used extensively in dynamic traffic assignment models, static traffic
assignment models have mainly relied on link performance functions (also called volume-delay
functions), which describe the relationship between link travel time and link flow (volume) or
between the speed and flow. The most well-known link performance function is the BPR link
performance function (Bureau of Public Road, 1964). Rewriting the travel time in terms of
speed and re-arranging terms, the BPR function can be written in form of a fundamental
diagram with the following (inverse) density-flow relationship:
 
,2
1
, ,1
( ) ( )
( ) 1 ,
(0)
a
a a
I a a a
a a
u i u i
u i
C




 
 
 
    
 
 
 
(21)
where ,1
a
 and ,2
a
 are link type specific parameters (with typical values for a motorway of
0.15 and 4, respectively). This function is plotted in Figure 5(a). Two things can be immediately
observed from this figure. First, the BPR function only contains the uncongested branch of the
fundamental diagram and ignores the congested branch. In fact, any link performance function
only describes the uncongested branch. Secondly, the uncongested branch is also increasing
beyond the capacity flow rate. Davidson (1966) proposed an alternative to the BPR function in
which the travel times goes to infinite as the flow approaches capacity, such that the
uncongested branch of the fundamental diagram has a horizontal asymptote at capacity.
However, this provides computational problems when flow reaches capacity, and while others
have discussed modifications to eliminate these problems (e.g., Daganzo, 1977; Taylor, 1984), it
is clear that link performance functions do not realistically describe the relationship between
traffic flow and density. In the G-FOS models derived in this paper we therefore propose to use
realistic fundamental diagrams as used in dynamic models, and derive travel times using
principles from traffic flow theory.
Table 2 summarises the most common assumptions on the fundamental diagram that can be
distinguished, where the first letter indicates the shape of the uncongested branch of the
fundamental diagram, while the second letter is associated with the congested branch. For
example ‘L-L’ indicates a triangular fundamental diagram in which both the congested and
uncongested branch are linear. When using a link performance function, one implicitly assumes
that no congested branch exists (denoted with ‘X’).
5.3 Turn flow restrictions
While link models and fundamental diagrams have been researched extensively for many
decades, node models seem to only have received significant interest in the last decade. Given
that queues and travel time delays are mainly determined by such node models, it is surprising
to see that essentially all static traffic assignment models lack a node model description. In the
absence of a node model, it is implicitly assumed that all flows are unconstrained and can enter
and exit all links, i.e. 1
a
  for all links a. Under this assumption, no residual queues form. In
case 1
a
  for a certain link a, turn flow restrictions are in place and a residual queue forms at
the end of link a. However, depending on the specific formulation of such turn flow restrictions,
this residual queue may be (incorrectly) inside the bottleneck link, or (correctly) upstream the
bottleneck. A common choice (see e.g., Bifulco and Crisalli, 1998; Smith et al., 2013) is to
constrain the outflow rate only by the link capacity. Although previous authors who used this
constraint made no reference to a node model, we see this constraint as primitive node model. In
this case, the turn flow restrictions are formulated as ( ) / ( ),
a a a
t C s t
  such that the outflow is
restricted by the physical link (exit) capacity. Note that these reduction factors are separable,
i.e., only depend on the flow on the link itself. However, link inflow may exceed capacity and
therefore the residual queue will be (incorrectly) in the bottleneck link.

20
First order node models describe reduction factors more realistically as in Equation (16), but
depend on all sending and receiving flows from all links connecting the node and are therefore
non-separable, such that also the link travel times are non-separable. To illustrate, consider a
node in which links a and b merge into link c. Then according to the often used (capacity
proportional) fair merging principle (see Jin and Zhang, 2003),
  
 
( ) min 1, ( )/ ( ) /( )
a c a a a b
t r t s t C C C
   and   
 
( ) min 1, ( )/ ( ) /( ) .
b c b b a b
t r t s t C C C
   In
case of a node in which link a diverges into links b and c, the reduction factors are often
supposed to follow the (demand proportional) FIFO diverging rule (see Daganzo, 1995b),
yielding ( ) min{1, ( )/ ( ), ( )/ ( )}.
a b ab c ac
t r t s t r t s t
  For cross nodes with multiple inlinks and
outlinks, such explicit analytical expressions may not exist and one has to rely on implicit
functions that need to be solved iteratively (Tampère et al., 2011). In such non-separable cases,
the variational inequality problem (1) cannot be simplified into a mathematical optimisation
problem (see e g., Dafermos, 1980).
Table 2 summarises the different types of choices for the turn flow restrictions, where ‘X’
represents the absence of any turn flow restrictions such that flows are unconstrained, ‘S’
represents separable exit capacity restrictions, and ‘G’ refers to general node models with non-
separable turn flow restrictions, such as the class of first order node models described in
Tampère et al. (2011).
6. Generalised First Order Static (G-FOS) traffic assignment
model
In this section we will mathematically derive the G-FOS models resulting from the static TIAs
stated in Section 4. Further, we show that many existing static models proposed in the literature
are a special case of the G-FOS model class under specific SIAs (see Section 5).
6.1 Route choice
Due to the static assumption on residual traffic in which results in an empty network at the
beginning of each time period i (i.e., ( ) 0
a i
Q t  for all links a), traffic flows and travel times are
independent across time periods. Hence, we can model each time period i separately. VI
problem (1) for finding a dynamic logit-based stochastic user equilibrium solution can therefore
be written as a series of independent VI problems (one for each time period i) for finding static
logit-based stochastic user equilibrium solutions ( ) [ ( )]
p p P
i f i 

f for each period i:
 
* * *
( , )
1
( ) log ( ) log ( ) ( ) 0, ( ) ( ),
od
od
p p p p p
o d p P
c i f i f i f i i i



 
      
 
 
  f (22)
where the set of feasible static path flow rates for time period i, ( ),
i
 is determined by the
following constraints:
( ) ( ), ( , ),
od
od
p
p P
f i M i o d

 
 (23)
( ) 0, .
p
f i p P
   (24)
It is clear that if inequality (22) holds for each time period i, then also inequality (1) holds as it
is simply the sum of inequalities (22). In other words, the dynamic problem (1) is split into
several simpler independent static sub-problems (22).

21
6.2 Network loading
In the network loading, we are looking for average link inflow and outflow rates as a result of
the static assumptions of (unrestricted) propagation of forward shockwaves with infinite speeds
and no propagation of backward shockwaves. The flow conservation constraints in Equation (4)
hold for each time instant t, such that we can simply write
( ), if link is the first link on path ,
( )
( ), if link follows link on path ,
p
bp
ap
f i b p
u i
v i b a p

 

(25)
where we provide variables that are stationary with an index i to represent all time instants in
period 1
[ , ).
i i
t t  These average path-specific link inflow and outflow rates, ( )
ap
u i and ( ),
ap
v i can
be used to obtain link inflow and outflow rates, ( ) ( ),
a ap
p
u i u i
  and ( ) ( ).
a ap
p
v i v i
 
Now we look in more detail at the flow propagation constraints. Because we are only interested
in finding an average inflow rate and an average outflow rate for each link, minimisation over
realised inflow and outflow states in Equations (8) and (10) is trivial as there is only a single
average inflow rate and a single average outflow rate. Define sending flows 1
( ) ( , )
a a i i
S i S t t 

and receiving flows 1
( ) ( , )
a a i i
R i R t t 
 for the whole of time period i. We will first derive a
model with vertical queues (and no spillback), before we are deriving a model with horizontal
queues (and spillback).
6.2.1 Horizontal queues and spillback
Letting ( )
a u
  (following the static assumption for forward shockwaves), ( )
a v
   and
(following the static assumption for backward kinematic waves ‘S(h)’), ( ) 0
a i
Q t  (no residual
traffic), and using ,
i
t T
  Equations (8) and (10) simplify to
 
 
 
 
 
1
1 ,
( ) min ( ) ( ) ( ) ,
min ( ) ( ) ( ),
min ( ) ( ) ,
( ) ,
a a i a i I a a a a i
a i a i a i a i
a i a i a i
a i
S i U t V t u i L C T
U t u i T V t C T
Q t u i T C T
u i T



   
  
 

(26)
and
   
 
 
 
 
 
1
1 ,
( ) min ( ) ( ) ,
min ( ) ( ) ( ) ,
min ( ) ( ) ,
min ( ) , .
a a i a i a a II a a a a a i
a a i a i a a a i
a i a a a a i
a i a a a i
R i V t U t K L v i K L C T
V t v i T U t K L C T
v i T Q t K L C T
v i T K L C T



     
   
  
 
(27)
Hence, the average sending and receiving flow rates for time period i are
( ) ( ),
a a
s i u i
 and (28)
 
( ) min ( ) / , .
a a a a i a
r i v i K L T C
  (29)

22
To understand the meaning of the receiving flow, we can also state that ( )
a a
r i C
 if
 
( ) .
a a i a a
C v i T K L
  In other words, the receiving flow is equal to capacity if the number of
vehicles that remain on the link is smaller than the maximum number of vehicles that fit on the
link. It should be noted that we can also replace a
K with the true queuing density available from
the fundamental diagram via  
1
, ( )
II a a
v i

 in order calculate the queue lengths and spillback
more accurately.
Using Equations (12) and (13), the directional sending flows ( )
ab
s i can hence be determined as
( )
( ) ( ) ( ).
( )
ap
ab a bp bp ap
p P p P
a
u i
s i s i u i
u i
 
 
 
  (30)
The node model in Equation (16) can now be written as
  in
in out
( ) ( ( ), ( ), , ).
n
a n a b b n n
a A
i s i r i a A b A
   

 
      (31)
The directional outflow rates can then be computed as ( ) ( ) ( )
ab a ab
v i i s i

 (see Equation (15)),
and hence the path-specific outflow rates can be written as ( ) ( ) ( ).
ap a ap
v i i u i

 Because of
Equation (25) we have that ( ) ( ) ( ) ( ),
bp ap a ap
u i v i i u i

  where b is the link downstream link a
on path p. Recursively applying Equation (25) yields
( ) ( ) ( ),
ap
ap ap p a
a
u i f i i

  

  (32)
where ap
 consists of the set of links from the origin of path p to (but not including) link a.
Hence, the path-specific inflow into link a can be determined by multiplying the route flow with
reduction factors (resulting from the node model) along the path. Combining Equations (30) and
(32) yields
( ) ( ) ( ).
ap
ab ap bp p a
p P a
s i f i i

   

 
   (33)
Further, the receiving flows ( )
b
r i depend on the link outflows ( )
b
v i (see Equation (29)), which
can be written as
( ) ( ) ( ),
b bp ap
p p
v i v i u i
 
  (34)
where a is the link upstream link b on path p. Hence, combining Equations (29), (32) and (34)
gives
( ) min ( ) ( ), .
bp
b b
b bp p b b
p P b
i
K L
r i f i i C
T 
  

 
 
 
 
 
 
 
  (35)
Noting that the sending flows ( ) [ ( )]
ab
i s i

s in Equation (33) can be expressed merely in terms
of ( ) [ ( )]
a
i i


α (and a given vector of path flows that are input to the network loading
problem), that the receiving flows ( ) [ ( )]
b
i r i

r in Equation (35) can also be expressed in terms
of ( )
i
α (given the network characteristics), and that the node model determines reduction

23
factors ( )
i
α as a function of ( )
i
s and ( ),
i
r the network loading problem can be written as a
fixed point problem. The result will be fixed point *
α and resulting link inflows
*
( ) ( ) ( )
ap
a ap p a
p a
u i f i i

 

   and link outflows *
( ) ( ) ( ).
a a a
v t i u i

 The fact that the network
loading model requires solving a fixed point problem is a direct consequence from the static
assumption of instantaneous propagation of forward and backward kinematic waves. In the
dynamic case, the network loading problem is typically solved by traffic simulation in which the
link and node models are solved consecutively over time. In the static model there are
instantaneous dependencies between the link and node models, which result in the fixed point
problem.
6.2.2 Vertical queues and no spillback
Letting ( )
a u
  (following the static assumption for forward shockwaves), ( ) 0
a v
 
(following the static assumption ‘S(v)’ for backward kinematic waves, where we note that the
speeds approach zero from below as the backward kinematic waves have negative speeds),
( ) 0
a i
Q t  (no residual traffic), and using ,
i
t T
  Equations (8) and (10) simplify to
 
 
 
 
 
1
1 ,
( ) min ( ) ( ) ( ) ,
min ( ) ( ) ( ),
min ( ) ( ) ,
( ) ,
a a i a i I a a a a i
a i a i a i a i
a i a i a i
a i
S i U t V t u i L C T
U t u i T V t C T
Q t u i T C T
u i T



   
  
 

(36)
and
   
 
 
 
1
1 ,
( ) min ( ) ( ) ,
min 0 ( ) ,
.
a a i a i a a II a a a a i
a i a i
a i
R i V t U t K L v i L C T
U t C T
C T



        
   

(37)
Hence, the average sending and receiving flow rates for time period i are simply ( ) ( )
a a
s i u i

and ( ) .
a a
r i C
 The resulting model is therefore only different in the receiving flows, which are
fixed to capacity, such that spillback never occurs. All equations from Section 6.2.1 are the
same, except that ( ) .
b b
r i C

6.3 Travel time calculation
Deriving travel times in a model in which the time dimension is eliminated is not trivial. Travel
times can no longer be directly derived from the cumulative flow curves as in Equation (19) for
dynamic models due to the fact that there is no explicit time dimension and flows are
instantaneously propagated. However, we can compute the route travel times as a summation of
uncongested (free-flow) travel times and delays in queues, i.e.,
ff queue
( ) ( ) ( ),
p p p
c i c i c i
  (38)
where ff
( )
p
c i is the travel time in uncongested states, and queue
( )
p
c i is the possible additional
delay spent in one or more queues.
The delay in queues can be determined by the integral defined in Equation (19), noting that in a
static assignment the cumulative inflow and outflow curves are ‘squeezed’ together, reflecting
instantaneous flow propagation. In order to calculate travel times for all departing vehicles, we

24
have to let all travel demand flow out of the network. Consider again the simple corridor
network in Figure 3, The grey shaded areas in Figure 6 indicate the total delay faced by
departing vehicles in each of the three time periods. Figure 6(a) indicates the cumulative inflows
and outflows in case of vertical queues, while Figure 6(b) show the case of horizontal queues.
These figures will be further explained below. We note that in our G-FOS models, the path-
specific inflow rate into the first link is ( ),
p
f i while the path-specific outflow rate out of the last
link is ( ) ( )
p a
a p
f i i


 according to Equation (32). Hence, we can write 1
( ) / ( )
p
a p p
U x x f i

 and
1
( ) / ( ) ( ),
p
a p p a
a p
V x x f i i



  such that the average delay for vehicles departing during time
period i on path p is
 
1
( )
queue 1 1
( )
( )
2 2
0
1
( ) ( ) ( )
( )
1
( ) 2 ( ) ( ) 2 ( )
1
1 .
2 ( )
a p i
p
p p
a p i
p
p i
U t
p a p a p
p i U t
f i T
p i p a p
a p x
i
a
a p
c i V x U x dx
f i T
x x
f i T f i i f i
T
i



 
 

 
 
 
 
 
 
 
 
 
 
 



(39)
The delay is therefore a direct function of the reduction factors. If all reduction factors are equal
to one, i.e. there are no bottlenecks or queues encountered on the path, then the delay is zero.
This delay function can be applied both for vertical and horizontal queues.
The free-flow travel times ff
( )
p
c i calculated differently for vertical and horizontal queues. First,
consider the case of vertical queues. The queues are assumed to be point queues without any
physical length, hence the uncongested travel times can simply be computed dividing the link
length by the link speed. The link speed is given by  
1
,
( )/ ( ) ,
a I a a
u i u i

 hence the uncongested
travel time is simply
 
1
,
ff
( )
( ) .
( )
a a I a
p
a p a
L u i
c i
u i



  (40)
In case of horizontal queues, the queues will have a certain physical length, such that vehicles
on congested links can only drive in free-flow on part of the link. To this end, we calculate the
queue length by dividing the vehicles in the queue by the jam density, i.e., 1
( ) / ,
a i a
Q t K
 where
   
1
( ) ( ) ( ) ( ) 1 ( ) .
a i a a i a a i
Q t u i v i T u i i T

     Instead of taking the jam density, it is also
possible to more accurately determine the queuing density from the fundamental diagram,
 
1
, ( ) .
II a a
v i

 by using the queuing density resulting from the congested branch of the
fundamental diagram, yielding a queue length of  
1
1 ,
( )/ ( ) .
a i II a a
Q t q v i

 This leads to a the
following uncongested route travel time,
 
 
 
1
,
ff
1
,
( )
( ) 1 ( )
( ) ,
( ) ( ) ( )
I a a
a a i
p a
a p II a a a a
u i
u i i T
c i L
i u i u i





  

 
 
 

 
 (41)
As far as we are aware, this travel time calculation includes for the first time the complete
fundamental diagram in a static model.

25
To illustrate Equation (39), first consider the example corridor in Figure 3 with vertical queues.
In the first time period, only link 4 is a bottleneck and a queue will form at the end of link 3. It
can be easily shown that 1 2 4
(1) (1) (1) 1,
  
   and 3 (1) 2000 / 3000 0.667.
   Hence, the
average delay according to Equation (39) is  
1
2
1/(1 1 0.667 1) 1 0.25
     h. This can be
confirmed by looking at the queues, namely there are 1000 vehicles waiting after one hour at the
end of link 3. The last vehicle will therefore have a delay of 1000/2000 0.5
 h, while the first
vehicle faces no delay, hence the average delay is 0.25 h. In the second period, all vehicles from
period 1 are assumed to have flown out and the travel demand of 6000 veh/h during one hour,
such that links 3 and 4 will both be bottlenecks and a queue will appear at the end of links 2 and
3. The network loading problem will yield that 1 4
(2) (2) 1,
 
  2 (2) 4000 / 6000 0.667,
  
and 3 (2) 2000 / 4000 0.5.
   Hence, the average delay will be  
 
1
2
1/(1 0.667 0.5 1 1 1
     h.
This can be confirmed by looking at the delays on links 2 and 3. There will be a queue of 2000
vehicles on link 2, which has an outflow rate of 4000 veh/h, hence the last vehicle will have a
delay of 2000/4000 0.5
 h. There will also be a queue of 2000 vehicles on link 3, which has
an outflow rate of 2000 veh/h, hence the last vehicle in this queue will see a delay of
2000/2000 1
 h. However, note that all travel demand has to pass this queue, also the vehicles
on link 2. If it takes 1 h to let all 4000 vehicles flow out of link 3 during one hour, it will take
(6000/ 4000) (2000/ 2000) 1.5
  h for all 6000 vehicles to pass link 3. Hence, the last vehicle
faces a delay of 0.5+1.5 = 2 h, such that the average delay is indeed 1 h. The scaling factor of
6000/4000 is often forgotten, but is necessary in order to compute travel times for all departing
vehicles. In time period 3 the travel demand is low, such that no queues appear.
Figure 6- Using cumulative link inflows and outflows to determine delays in static assignment with
vertical queues (a) or horizontal queues (b).
veh
5000
10000
time
1
t 2
t 3
t 4
t 5
t 6
t
(1) 3000
f 
(2) 6000
f 
(3) 1000
f 
1 1 2 2 3 3 4 4
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
U t V t U t V t U t V t U t V t
      
1( )
U t
1 2 2 3 3 4 4
( ) ( ) ( ) ( ) ( ) ( ) ( )
V t U t V t U t V t U t V t
     
1 1 2
( ) ( ) ( )
U t V t U t
 
2 3 3 4 4
( ) ( ) ( ) ( ) ( )
V t U t V t U t V t
   
(b)
veh
5000
10000
time
1
t 2
t 3
t 4
t 5
t 6
t
(1) 3000
f 
(2) 6000
f 
(3) 1000
f 
1 1 2 2 3 3 4 4
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
U t V t U t V t U t V t U t V t
      
1 1 2
( ) ( ) ( )
U t V t U t
 
2 3
( ) ( )
V t U t

3 4 4
( ) ( ) ( )
V t U t V t
 
1 1 2
2 3
( ) ( ) ( )
( ) ( )
U t V t U t
V t U t
 
 
3 4 4
( ) ( ) ( )
V t U t V t
 
(a)

26
Now let us consider the second time period of the same corridor but now with horizontal
queues. Assume that all links have a length of 3 km each, and for convenience we will use the
following jam densities as queue densities, 1 2 600,
K K
  3 400,
K  and 4 200.
K  It can
easily be shown that the solving the network loading problem yields
1 (2) 2000 / 6000 0.333,
   2 3 4
(2) (2) (2) 1.
  
   In other words, the queue on link 3 is
spilling back upstream and reaches link 1, such that all links have an outflow rate of 2000 veh/h.
In total there are 4000 vehicles in the queue, of which 400 3 1200
  vehicles on link 3,
600 3 1800
  on link 2, and the remaining 1000 on link 1. According to Equation (39) the
average delay is  
 
1
2
1/(1 0.333 1 1 1 1
     h, the same as in the case with vertical queues. Note
that in this simple corridor case, it does not matter for the delay whether we consider horizontal
or vertical queues, as all that matters is the outflow rate out of the last link. This does not mean
that it does not matter for path travel times whether queues or horizontal or vertical. On the
contrary, if there was another route passing through link 1 and not through the other links, the
path travel time of this route would be affected with an additional delay on link 1 in case of
horizontal queues, while in case of vertical queues this delay on link 1 does not exist.
It is important to note the main differences between the delays in Figure 4 and Figure 6. The
assumption of no residual traffic transfer leads to empty networks at the beginning of each time
period. Clearly, this can reduce the modelled total delay experienced by travellers significantly,
as for example can be seen when comparing the delays for time period 3. Without considering
residual traffic at the beginning of each time period, there are no delays predicted in the static
model, while there are significant delays predicted in the dynamic model. This illustrates that
the assumption of no residual traffic transfer is a very strong assumption that preferably should
be relaxed, leading to semi-dynamic models.
6.4 Specific first order static models
Most static (and quasi-dynamic) models proposed in the literature are a special case of the class
of G-FOS models that we derived. Different spatial interaction assumptions lead to different
models.
The model proposed by Smith (2013) can be seen as a special case of the G-FOS model with
horizontal queues, in which route choice is assumed deterministic, the uncongested branch of
the fundamental diagram is assumed concave, and turn restrictions are defined by simple link
exit capacities. Hence, the G-FOS model with horizontal queues as derived in this paper
therefore extends the model of Smith (2013) to any fundamental diagram, to any first order node
model that describes the turn restrictions, and to stochastic route choice.
Bliemer et al. (2013) proposed a model with vertical queues in which the turn restrictions are
describes by the node model of Tampère et al. (2011), while route choice based on the
multinomial logit model and a quadratic-linear fundamental diagram are assumed, and hence is
a special case of the G-FOS model with vertical queues. Bliemer et al. (2013) have further
shown that the fixed point problem for network loading as derived in this paper can be
efficiently solved for large scale networks.
The most well-known static traffic assignment model is the deterministic model (i.e., ,
  
0
  ) proposed by Beckmann et al. (1956) using link performance functions, hence it is
implicitly assumed there is no congested branch of the fundamental diagram. Further, no turn
flow restrictions are considered in this formulation, hence 1
a
  for all links, yielding
unrestricted flow propagation along each route. This strong assumption yields that Equation
(32) simplifies to ( ) ( ),
ap ap p
u i f i

 hence the fixed point problem in Section 6.2 simplifies to the
simple relationship ( ) ( ),
a ap p
p
u i f i

  which is the familiar constraint in the traditional static
traffic assignment model. Hence, also the traditional static traffic assignment model is a special

27
case of the G-FOS model. Due to the fact that only the uncongested branch of the fundamental
diagram is described and no turn flow restrictions are considered, this model is essentially only
applicable to uncongested networks, which questions its suitability for use in heavily congested
areas.
Extensions to the deterministic route choice model have been proposed by Fisk (1980) and Zhou
et al. (2012) by including perception error. Fisk used the multinomial logit model for route
choice while Zhou et al. adopts the C-logit route choice model. Both stochastic traffic
assignment models use the simple network loading relationship from Beckmann et al. (1956).
Bifulco and Crisalli (1998) were the first to formulate the static network loading problem as a
fixed point problem and use a formulation similar to Equation (32). However, they did not
include a proper node model but instead assumed that reduction factors are derived from link
exit capacities, such that ( ) / ( ).
a a a
i C u i
  This means that outflow is constrained to capacity,
leading to residual (vertical) queues typically inside the bottleneck link. Others have implicitly
or explicitly also made this assumption, i.e. Lam and Zhang (2000) and Smith et al. (2013).
There are other static models proposed in the literature that cannot be derived from the G-FOD
model. For example, Yang and Yagar (1994), Larsson and Patriksson (1995) and Bell (1995)
propose capacity constrained static traffic assignment models in which the link inflow may not
exceed the link capacity, but no explicit residual queues are formed. Instead, Lagrange
multipliers of the link capacity constraints are computed to represent link delays. These delays
re-route path flows in order to ensure that no link flows exceed link capacity (assuming there is
sufficient capacity in the network). It is not possible to formulate a fundamental diagram, turn
flow restrictions and specify route choice behaviour that results in these capacity constrained
static models, and hence these models are not first order static models.
7. Conclusions and discussion
In this paper we have presented a theoretical framework in which we can derive static (or quasi-
dynamic) and semi-dynamic traffic assignment models from a general first order dynamic traffic
assignment model using temporal interaction assumptions on forward and backward kinematic
waves and on the transfer of residual traffic. Further, the framework is very general in that it
allows to take into account any fundamental diagram (i.e., a link model), any turn flow
restrictions (i.e., a node model), and deterministic or stochastic route choice behaviour. This
theoretical framework is then used to derive novel static (quasi-dynamic) traffic assignment
models that take queuing and spillback into account. We believe this is the first time that static
models have actually been rigorously derived from a dynamic model. The advantage of such a
derivation is that it maintains maximum consistency with a dynamic model. The resulting static
models with vertical and horizontal queues are essentially general forms of the models
presented by Bliemer et al. (2013) and Smith et al. (2013), respectively.
One of the main contributions of this paper is that we have explicitly formulated all underlying
assumptions made in static (quasi-dynamic) traffic assignment models proposed in the literature.
One of the assumptions implicitly made in static models is the absence of residual traffic
transfer by assuming that the network is empty at the beginning of each time period under
consideration. While this may not be a large problem if the period under consideration in static
assignment is sufficiently large (i.e., covers the entire peak period from the moment of queue
build up till the moment that all queues have disappeared), it will lead to significant
underestimations of travel times in case of successive static traffic assignment on smaller time
period (i.e., hourly static traffic assignment while the peak period is several hours long). In the
latter case, transfer of residual traffic seems very important. Our framework allows for deriving
models that consider residual traffic transfer, leading to what we call semi-dynamic models.
These models are significantly more complex and will be mathematically derived in future
research.

28
As a special case, the traditional static traffic assignment model proposed by Beckmann et al.
(1956) can be derived by assuming (i) infinite forward kinematic wave speeds, (ii) no backward
kinematic wave speeds, (iii) no residual traffic transfer, (iv) a concave function for the
uncongested branch of the fundamental diagram while no function is specified for the congested
branch, (v) no turn flow restrictions, and (vi) deterministic route choice. Several of these
assumptions are very strong, and essentially limit the applicability of this model in practice. The
novel static models derived in this paper relax several of these strong assumptions, making the
forecasts more realistic and in line with dynamic models.
Our derivations are based on the general link transmission model as an exact and general
formulation of a first order dynamic traffic assignment model. A limitation in our derivations is
that it only holds for a single vehicle class and cannot handle multiclass traffic. We are not
aware of any other first order dynamic model that provides an explicit, exact, and analytical
expression of flow rates, which is needed for our derivations. This does not mean we cannot
formulate multiclass static models. In fact, this can be done in a relatively straightforward
fashion by performing a single class network loading using passenger car units, but in the
uncongested part of the travel time calculation use for each vehicle class a different function for
the uncongested branch of the fundamental diagram. Queuing delays may be assumed equal for
all vehicle classes. This would provide a pragmatic way of including multiple vehicle classes,
without explicitly deriving a multiclass static model from a multiclass dynamic model.
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