DTA models are generally classified into two main categories. The first group consists of analytical models that describe the average behavior of traffic with macroscopic traffic flow variables such as inflow rates and travel times. The advantage of these models is their ability to specify properties such as existence and uniqueness of the solution, and a lower computational cost. However, a guarantee for such properties severely restricts the function that maps route flows on travel times, i.e. the DNL model. These restrictions prevent realistic representation of traffic dynamics, such as queue spillback.
A second group contains the simulation-based models, which keep track of individual vehicles, or vehicle packets, at each time step. Here, there is no restriction on the DNL component. Such models describe certain traffic phenomena more accurately, though at a higher computational cost. Properties such as existence and uniqueness of the solution are more difficult to derive. A more detailed overview of the different DTA models and their properties can be found in Peeta & Ziliaskopoulos (2001), Zhang & Nie (2005), Viti & Tampere (2010).
The difference in application of the DNL model as illustrated above explains why less realistic models are still used: there can be cases where a realistic representation of traffic dynamics is less of importance with respect to the ability to describe traffic behavior with analytical models (e.g. in economic studies for a large region). In this subsection we will discuss the following three queuing models that are mostly used:
• Point queue model • Spatial queue model
• First-order kinematic wave model
3.2.1 Point queue model
The vertical queue or point queue model (PQ) is the most basic model that still describes queuing behavior, though in an approximate way. In this model vehicles travel at free flow speed along a link, and form a queue at the end of a link if the outflow rate exceeds the capacity of that link. There is no restriction on the inflow of the link. A continuous mathematical form of this model can be found in Zhang & Nie (2005) and is given as below:
0 if =0 and ( ) ( ) otherwise in f in f q t C d q t C dt
L v
L v
λ λ − < = − −
(3.2)29 0 0 ( ) if =0 and ( ) ( ) otherwise in in out q t t q t t C t C
q
=
− λ − <
(3.3)where qin is the inflow rate at time t
out
q is the outflow rate at time t
λ
(t) is the queue length at time t C is the saturation rate of the linkLis the length of the link
f
v is the free flow speed of the link
3.2.2 Spatial queue model
A second model is the horizontal queue or spatial queue model (SQ). It uses a node model to describe the interaction between upstream and downstream links. Congestion propagation through links can thus be modeled. For every link the demand D and supply S is calculated as follows: ( ) if ( )=0 and ( ) ( ) otherwise in f in f q t t q t C D t C
L v
λL v
− − < =
(3.4)(
)
( ) min , max( jam , 0)
S t = C k ⋅L−x (3.5)
where kjam is the jam density, x is the number of vehicles on the link. The demand D and
supply S are used as input for the node model to determine the outflow of the upstream link. In this thesis we use typical freeway diverge, merge and connecting nodes (see Daganzo (1995a) and Tampère et al. (2011)).
3.2.3 First-order kinematic wave model
A third model is based on kinematic wave theory, which was introduced by Lighthill & Whitham (1955) and Richards (1956). This theory considers traffic as a continuous flow characterized by aggregate traffic variables: density, speed, and flow. These variables obey a traffic flow conservation law that is complemented by an empirical relationship between two of the variables, while the third follows by definition of the flow being the product of speed and density. The solution of this model consists of kinematic waves, shockwaves and expansion fans propagating along the links of the traffic network. For more explanation on kinematic wave theory, we refer to appendix A.
When a triangular fundamental diagram is used to describe the relationship between flow and density, simplified kinematic wave theory after Newell (1993) can be exploited to solve model by only considering cumulative flow at the up- and downstream ends of the links in the network. This allows for an efficient numerical solution algorithm for simplified kinematic wave theory in networks, called Link Transmission Model (LTM), which was developed by Yperman (2007) and is extensively used throughout this thesis. LTM is briefly introduced hereafter; for more details about this model, we refer to Yperman (2007).
In each time step, demand and supply at the up- and downstream link ends are determined and used as input for the node model to determine the outflow. The same node models as above are used. The demand or sending flow is defined as the maximum amount of vehicles that can leave the downstream end of the link if this link end were connected to a reservoir with infinite capacity. The only constraints are the link’s capacity and the available inflow to the link, which is propagated downstream using the minimal travel time of vehicles:
0 ( ) min ( , ) ( L, ), f L D t N x t t N x t C v =
+ ∆ − −
(3.6)The supply or receiving flow is defined as the maximum amount of vehicles that can enter the upstream end of the link if this link end were connected to a reservoir with infinite demand. The only constraints are the link’s capacity and the allowable outflow (due to the spillback of queues over the downstream node), which is propagated upstream using the minimal wave travel time along the link:
0 ( ) min ( L, L ) jam ( , ), S t N x t t k l N x t C w =
+ ∆ − + −
(3.7)Here x0 and xL are the link boundaries, the cumulative vehicle numbers are N(x,t) and w is characteristic the wave speed for congestion. For more information on these parameters and the derivation of equation (3.6) and (3.7), we refer to Yperman (2007).
3.2.4 Comparison of queuing models
The three models discussed above differ from each other only in the case of congestion: when no bottleneck is activated, the propagation along the links follows free flow speed. As soon as a bottleneck is activated, the models behave differently. In the PQ model, there is no restriction on the inflow of the link. Thus, if the inflow exceeds the capacity, a queue starts building up in the bottleneck link itself, without affecting upstream links, as is depicted in Figure 3.3 (a). Therefore, spillback of this queue to upstream links is not modeled. In the SQ model, link interactions can be accounted for. As soon as the number of vehicles on a link exceeds kjam.l vehicles on it, congestion spills back to the upstream link. The vehicles in a
queue are stacked together as densely as possible, as is depicted in Figure 3.3 (b). In LTM, the density of vehicles in a queue is dependent on the speed at which they are driving. The mapping between density and speed is determined by a fundamental diagram. Vehicles with a non-zero speed will queue at a lower density than kjam. Therefore it takes less time for a queue
to reach the end of a link compared to SQ models, where the queue always has a density of
kjam, regardless of the outflow. LTM will thus predict earlier spillback than with SQ models,
31
(a) (b) (c)
Figure 3.3: Queuing in (a) PQ, (b) SQ, and (c) LTM, when flow moves from left to right