General

From our general framework, we have the following requirements to a traffic simulation:

• Vehicles need to be able to follow plans. This implies that the simulation needs to be dynamic (i.e. time-dependent), and that some notion of individual vehicles needs to be present in the simulation.

• The simulation needs to be reasonably fast. A computational speed of at least 100 times faster than real time (i.e. simulating 24 hours of traffic in 0.24 hours of computing time) is desirable in order to obtain bearable waiting times for the feedback/learning. This computing speed can be achieved by selecting small scenarios, by using simple models, or by parallel computing. This text concentrates on the last two aspects.

The important numbers characterizing a road from the perspective of transportation planning are:

• Free speed. This is the speed that vehicles drive on a link when no other constraints are present.

• Flow capacity. This is the maximum number of vehicles per time unit that can move over a link when no other constraints are present. In city traffic, the flow capacity is often determined by a traffic light at the end.

• Storage constraint. This is the maximum number of vehicles that can be on a link under jammed conditions.

The first two numbers are also used in all traditional transportation planning software (based on static assignment, see Chap. 28) and are therefore typically available with standard data files for transportation planning. The third number is necessary when a link is full and no more vehicles can enter, causing spillback. Without the storage constraint, flow demand above the flow capacity would allow an unlimited number of vehicles on the link, which is clearly not realistic.

The queue model bases its dynamics on free speed, flow capacity, and storage constraint only. Typical input data are, for each link $a$, the attributes free flow velocity $v_{0,a}$, length $L_a$, capacity $C_a$ and number of lanes $n_{lanes,a}$. Free flow travel time is calculated by $T_{0,a}={L_a / v_{0,a}}$. The storage constraint of a link is calculated as $N_{sites,a} = {L_a \cdot n_{lanes,a}}/ \ell$, where $\ell$ is the space a single vehicle in the average occupies in a jam, which is the inverse of the jam density. One can use $\ell = 7.5~\hbox{m}$, as for the CA technique.

The arguably simplest intersection logic (47) is that all links are processed in arbitrary but fixed sequence, and a vehicle is moved to the next link if (1) it has arrived at the end of the link, (2) it can be moved according to capacity, and (3) there is space on the destination link (see Algorithm A in Fig. 18.1). More formally, the following happens:

• Free speed: A vehicle that enters link $a$ at time $t_0$ cannot leave the link before time $t_0 + T_{0,a}$, where $T_{0,a}$ is the free speed link travel time as explained above.

• Flow capacity: The condition ``vehicle can be moved according to capacity'' is determined as
  
  $$N < int(C_a) ~~\hbox{or}~~\Big(N = int(C_a) ~~\hbox{and}~~ rnd < fr(C_a) \Big)$$
  
  
  where $int(C_a)$ is the integer part of the capacity of the link (in vehicles per time step), $fr(C_a)$ is the fractional part of the capacity of the link, and $N$ is the number of the vehicles which already left the same link in the same time step. $rnd$ is a random number such that $0\leq rnd < 1$. What it is meant by this formula is that the vehicles can leave the link if leaving capacity of the link has not been exceeded yet in this time step. If the capacity per time step is non-integer, then we move the last vehicle with a probability which is equal to the non-integer part of the capacity per time step.

• ``Space on destination link'': If the destination link is full, the vehicle will not move across the intersection.