Influence of driver behavior in the emergence of traffic gridlocks

For each vehicle, the acceleration $a(t_{i})$ is set by one of three acceleration modes, which we call GO, CAR IN FRONT and STOP. These modes are a way to mimic how the driver controls the vehicle in an urban traffic scenario. The motivation behind the acceleration modes is the flexibility they provide in order to construct a modular decision tree that implements the driving behaviors we are interested in, namely the careful and the aggressive.

The acceleration in each mode is set as follows:

• •

  GO: The vehicle moves with constant acceleration $a_{\text{go}}$ until it reaches a maximum speed $v_{\text{max}}$. It is given as

  $\displaystyle a(t_{i})$

  $\displaystyle=$

  $\displaystyle\left\{\begin{array}[]{ll}0,&v(t_{i})\geq v_{\text{max}}\\ a_{\text{go}},&v(t_{i})<v_{\text{max}}.\\ \end{array}\right.$

  (6)

• •

  CAR IN FRONT: The vehicle moves at a speed such that it keeps a safe distance to the leading vehicle[8]. The safe distance is the necessary space such that the time elapsed from bumper to bumper as measured at a fixed location is $\delta t_{\text{s}}$. The acceleration is

  $$a(t_{i})=\frac{d(t_{i})-d_{\text{s}}(t_{i})}{\delta t\,\delta t_{\text{s}}},$$

  (7)

  where $d(t_{i})$ is the distance to the leading vehicle (bumper to bumper) at a time $t_{i}$ (see Fig. 1) and $d_{\text{s}}(t_{i})=v(t_{i})\,\delta t_{\text{s}}$ is the safe separation between the cars such that at the current speed $v(t_{i})$ the vehicle would take a time $\delta t_{\text{s}}$ to traverse. In this mode, the cars keep the same time separation, which will be a larger (shorter) length when going at a faster (slower) speed. In practice, when taking one iteration step for the velocity, Eq. (2) reduces to $v(t_{i+1})=d(t_{i})/\delta t_{\text{s}}$. In other words, the speed is set to the value that is needed to keep a time $\delta t_{\text{s}}$ to the leading vehicle.

• •

  STOP: A car will stop when the traffic light is red. The acceleration in this case will be

  $$a(t_{i})=-\frac{v(t_{i})^{2}}{2d_{\text{STP}}(t_{i})},$$

  (8)

  where $d_{\text{STP}}(t_{i})$ is the distance to the point where the car has to be at rest (see Fig. 1). By taking this approach, the car breaks smoothly, reaching zero speed as $d_{\text{STP}}(t_{i})$ approaches zero. In this mode we set $v(t_{i})=0$ whenever $d_{\text{STP}}(t_{i})<d_{\text{min}}$ or $d(t_{i})<d_{\text{min}}$. The first condition ensures that a car stops before entering an intersection and the second one avoids collisions. The fixed quantity $d_{\text{min}}$ is the minimum separation distance between cars.

We will show in Sec. 3 that the acceleration modes reproduce the fundamental diagram for freeway traffic shown in Refs. \refcitehelbing2001traffic,daganzo2005variational,geroliminis2008existence.

The decision as to which acceleration mode is used at a given time is determined by the values of the distances mentioned above, i.e. $d(t_{i})$, $d_{\text{TL}}(t_{i})$ and $d_{\text{STP}}(t_{i})$. We assume that the process of deciding which acceleration mode will be used starts by determining what is closer: either the leading vehicle or the traffic light at the nearest intersection. If the leading vehicle is closer, the driver will just follow it at a safe distance, regardless of the traffic light. On the contrary, if the traffic light is closer, then the driver will disregard the leading vehicle and will maneuver according to the color of the signal. In the first case we say that the driver is car-guided, and in the second case he is TL-guided (Traffic Light). This decision is made at the beginning of each iteration by comparing the distances $d(t_{i})$ and $d_{\text{TL}}(t_{i})$ (see Fig. 1)

Once this step is done the next ones will depend on the color of the traffic light. We consider red, yellow and green phases with duration $T_{r}$, $T_{y}$ and $T_{g}$, respectively. We use the traffic light color as the starting point of the decision trees that will implement the two types of driving behavior, i.e. careful and aggressive.

We focus our attention on the traffic flux and speed attained when individual drivers behave in one of two ways, which we call the aggressive and the careful types, following Ref. \refciteehlert2001microscopic. The distinction between them occurs when the driver has to make a decision at the same time that he is approaching the intersection of two perpendicular streets. We assume that the traffic flow at the intersection is controlled by a traffic light and that all streets have a single lane with a fixed sense of motion. When the vehicle density is high, there is a chance that some drivers will block the intersection, either moving slowly or standing still completely. If the potentially blocking driver, has enough green light time to clear the intersection, then he just goes through it at a low speed. The problem arises when the traffic light changes from green to yellow and red while the car is still traversing the intersection. At this point the vehicles waiting on the perpendicular lane have now the green light, but they will not be able to move because the intersection is blocked, thus increasing congestion.

We define the careful driving behavior or careful type as that in which the driver will never block the intersection. The careful driver achieves this by waiting to have enough distance headway to the leading vehicle so that she can effectively clear the intersection. The aggressive type is the opposite. This driver will follow the leading vehicle at a safe distance but he will not have any consideration about obstructing an intersection whatsoever.

From now on, for clarity, we will drop the time dependence from the distances considered.

In order to test our driving algorithm, we present first some results for a simplified setting. Using the city layout, let’s consider the situation in which cars travel along a unique street only, and all traffic lights are set to green during the whole simulation. This setting mimics the conditions of a continuous road of length $L$ with periodic boundary conditions. We set $L=20(B_{\text{w}}+S_{\text{w}})$ (see Fig. 1), corresponding to a total length $L=2$ km, according to the values listed in Table 3.1.

We randomly distribute a number of vehicles $N$ with zero initial velocity along the street. This defines the density $\rho=N/L$. The cars start moving according to the rules outlined in Sec. 2. We follow their motion for a three-hour simulated period, during which we compute the instantaneous average speed $\bar{v}(t_{k})$ over all vehicles at intervals of 1 s according to

where $v_{j}(t_{k})$ is the speed of the $j$-th vehicle at a time $t_{k}$. Now we take the last 300 s of the simulation and compute the time averaged speed $V=\sum_{k}\bar{v}(t_{k})/300$. These time averages have a standard deviation that is less than 4% the value of $V$. Knowing $\rho$ and $V$ the traffic flow is given as $Q=\rho V$. By taking the last 300 s of a three-hour simulation we make sure that the transient part of the dynamics is over and we are left with the steady state.

The results are shown in Fig. 4. We plot the traffic flow $Q$ (panel A) and the average velocity $V$ (panel B) versus the vehicle density $\rho$. We consider the two extreme cases: one where all drivers are aggressive and the other where all drivers are careful. Our results for the aggressive type are in agreement with the fundamental diagrams in Refs. \refcitechowdhury1999self,chowdhury2000statistical,treiber2000congested,helbing2001traffic,mhirech2011traffic,qi2016impact. The traffic flow $Q$ exhibits a linear growth at low densities, reaching a maximum when the safe separation is $d_{\text{s}}=v_{\text{max}}\,\delta t_{\text{s}}$. This corresponds to a critical density $\rho_{\text{cr}}=1/(\ell+d_{\text{s}})\approx 11$ vehicles/km. From this point the flow decreases linearly until the density is $\rho_{\text{max}}=1/(\ell+d_{\text{min}})\approx 143$ vehicles/km. When density reaches $\rho_{\text{max}}$ all vehicles are at their minimum separation $d_{\text{min}}$, thus they cannot be any closer and $Q$ goes to zero.

In the case where all drivers are careful (0% aggressive), the traffic flow exhibits a different behavior. It starts increasing linearly at low densities, as in the previous case, but it drops to a constant value over a wide range of densities. This is due to the fact that careful drivers try to stop at every intersection if they don’t have enough space to position themselves without blocking the intersection after crossing it. This behavior is desired at higher densities, thus it is not optimal for lower ones. Since we are interested in urban traffic and congestion at high densities we do not perform further optimization for the careful type at lower densities.

The fact that the aggressive type behavior reproduces the fundamental diagram for freeway traffic validates our ad-hoc driving algorithm formulation, which is qualitatively similar to those presented in Refs. \refcitehelbing1998generalized,treiber2000congested. Other feature we found in our simulations is the existence of stop-and-go waves, which are a well-known property of freeway traffic. [17]

We consider a square city of size $10\times 10$ blocks, each block being a $B_{\text{w}}+S_{\text{w}}$ square per side. The total road length is $L=200(B_{\text{w}}+S_{\text{w}})$, which amounts to 20 km. The density $\rho$ and average velocity $V$ are defined in the same way as in section 3.1.

Initially a number of vehicles $N$ are randomly distributed on all streets. All initial velocities are zero. We consider three driving behavior configurations for the $N$ drivers: all aggressive, half aggressive-half careful and all careful. We will name these configurations by the percentage of aggressive drivers they have, i.e. 100-, 50- and 0%-aggressive, respectively. For each configuration we consider that there is a probability that a driver will take a turn at the next intersection. Ideally, the amount of turns a driver will take depends solely on the departure and destination points within the city. In reality it is influenced by other factors such as traffic incidents along the way, a preferred route or the driver’s empirical knowledge of the state of the city traffic. To cover a wide range of turning probabilities we set up a series of runs, each with a fixed probability to take a turn. For instance, a run with 0% turning probability means that all drivers will just go straight ahead, the amount of cars in each lane will remain constant throughout the simulated time. The probabilities considered here are 0, 10, 25, 50 and 75%. For the type of city we are analyzing, i.e. streets consisting of a single lane with alternating and unique senses, a trip from one location to another can be accomplished by taking turns in 10 to 25% of the traversed intersections.

Traffic lights are set up in two different ways: synchronized (sync-) and random (rand-) modes. In sync-mode all traffic lights change at the same time. Initially, the lights are in green phase for both senses along the $x$-direction, consequently they are in red phase for both senses along the $y$-direction. The duration of the three phases is the same for all traffic lights (see Table 3.1). In rand-mode each traffic light is out of phase by a random fraction of the cycle. The extent of the cycle is still the same for all traffic lights but now the shifting sequence has been randomly ordered.

Figure 5 shows traffic flow and average velocity for urban traffic using the three driving behavior configurations, the five turning probabilities and traffic lights in sync-mode. Figure 6 shows similar results but setting traffic lights to rand-mode. The effect of turning is evident for sync-mode traffic lights. In rand-mode, turning probabilities make no significant differences for both the flow and the average velocity. Our results are similar to those reported in Ref. \refcitemhirech2011traffic for urban traffic with signalized intersections.

The cases with 100% aggressive drivers give higher fluxes at lower densities and traffic lights in sync-mode. This is no surprise, since sync-mode enables half of the city grid to momentarily behave as free roads during the green phase. However, the high flux is cut down abruptly to zero when a critical density is reached. This sharp change from a finite value to zero flow signals a total gridlock, meaning that all vehicles are standing still without moving. This constitutes a first order phase transition [6, 1, 18]. The cause of this lies in the distinctive feature of the aggressive type: the driver can block the intersection box. At low densities such behavior is of little or no consequence. But as soon as the streets become crowded and the drivers that are waiting to clear the intersection are actually blocking the street perpendicular to their motion, the traffic collapses and nobody can move.

Gridlocks emerge in sync- and rand-modes and for all turning probabilities whenever we have 100% and 50% aggressive drivers, as can be seen in panels A, B, D and E in Figs. 5 and 6. The only cases in which gridlock is avoided is with 0% aggressive drivers. The main feature of the careful driver is that she will never block the intersection. She will wait until there is enough space ahead to actually traverse and clear the intersection. In cases of high vehicle densities, this behavior keeps the traffic flowing, as can be seen in panels C and F in Figs. 5 and 6.

There is always traffic flow in the 0% aggressive configuration. It goes to zero only at high densities, when vehicles approach their allowed minimum separation. We do not find any traffic jam when all drivers are careful. This poses the question: how many aggressive drivers are needed in order to produce a gridlock? We explored this situation performing runs with 1, 3, 6, 12 and 25% aggressive drivers, the traffic flow for these cases (25% not shown) are presented in Fig. 7. We can see that traffic jams arise with as little as 1-3% aggressive drivers. In the 1% case, gridlocks are more frequent with increasing turning probability. In the 3% case, only the 10% probability turning curve is free from gridlocks until $\rho\approx 60$ vehicles/km, for higher turning probabilities gridlocks are present.

The effect of a higher turning probability is translated into a decrease in flow for the sync-mode traffic lights setting (Fig. 5). In rand-mode it has no noticeable effect on the flow (panels A and D in Fig. 6). It makes the gridlock onset appear at lower densities [1] (panels B and E) and correlates with a smaller flow for 0% aggressive drivers (panels C and F). This can be understood noticing that an aggressive driver that takes a turn into a congested lane, can end up blocking the intersection, instead of just going ahead had it not turned.

We chose to simulate a three-hour period of congested traffic, which is enough to collapse the flow even with a small number of aggressive drivers. The time it takes for a gridlock to appear depends on the vehicle density. The higher the density the sooner it emerges. We can say that a traffic jam could be avoided if higher densities dissolve quickly, instead of lasting for as long as three hours. Consequently, the system would have a higher tolerance to aggressive drivers, requiring larger amounts of them for a gridlock to happen. We therefore take the three-hour period as a worst case scenario.