Starting-wave and optimal density in a queue

Publication Type:

Conference Paper


PED - Pedestrian and Evacuation Dynamics (2012)


Starting-wave; optimal density; queuing


Various kinds of self-driven many-particles (SDP) systems, such as pedestrian dynamics, vehicular traffic and traffic phenomena in biology have attracted a great deal of attention in a wide range of fields during the last few decades. Most of these complex systems are interesting not only from the point of view of natural sciences for fundamental understanding of how nature works but also from the points of view of applied sciences and engineering for the potential practical use of the results of the investigations. Especially, the interdisciplinary investigations for the dynamics of jamming phenomena in SDP systems, so-called Jamology, have been progressed by developing sophisticated mathematical model considered as a system of interacting particles driven far-from equilibrium. These contributions to analyze the mechanism of jamming formation tell that one of the most important factors to cause the jamming phenomena is the sensitivity, which indicates the time delay of reaction of pedestrians or drivers to the stimulus. As an example, if the reactions of drivers are extremely sensitive, they can avoid the traffic jam by adjusting their behavior immediately to their front car’s movement. The reaction time of pedestrians is similarly important toward smooth movement of crowd. Moreover, we would like to point out that the wave of successive reaction in a queue, so-called starting-wave, plays a significant role for the waiting time of queuing system of pedestrians and vehicles, since quick-start in walking accomplishes the more smooth movement of crowds. In this contribution, first of all we investigate the relation between the propagation speed of pedestrians’ reaction and their density by using mathematical model based on the stochastic cellular automata. Then, the optimal density to minimize the travel time of last pedestrians in a queue to reach the head position of the initial queue is investigated under taking into account the propagation time of starting-wave. Finally we verify these results obtained from mathematical model by performing the experiments of pedestrians.
Our mathematical model is built on the stochastic cellular automata, which recently prevails to model the stochastic transport in complex systems. Let us imagine that the passage is partitioned into L identical cells that each cell can accommodate at most one particle (pedestrian) at a time. Note that, in the following, we refer to “particle” as a representation of a pedestrian in a model and “pedestrian” as a person itself. The length of each cell corresponds to 0.5 meters by considering the reasonable volume exclusion effect of pedestrians. Moreover, a total number of N indicates the particles which are placed at equal distance H cell. The update rules of our cellular automaton model are as follows: first of all, only the particle at front of a queue moves forward. Then the following particle only the second particle in the queue can move forward. After the second particle moves forward, the next particle can start to move in sequence. These rules of pedestrians’ walking are applied in parallel to all particles. Note that, in our model, in order to investigate the propagation speed of successive reactions, all of the following particles can not move forward before the starting-wave reaches to them. Therefore, unlike the usual stochastic cellular automaton models such as ASEP, ZRP, in our model, only if the next cell is empty
and predecessor had already moved, following particles can move forward with probability p(h) which depends on their headway distance h. This hopping probability of particles p(h), which indicates the velocity of particles, is given in analogy with the idea of Optimal Velocity (OV) function, which is often introduced into the mathematical model for vehicular traffic as a desired velocity of drivers depending on headway distance. This function is motivated by the common expectation that drivers have their desired velocity to do the comfortable driving. We have measured the propagation speed of starting-wave which is derived from the length of initial queue divided by the elapsed time under each given initial density which is decided by the initial value of particles on the cells, that is, N/L. The results obtained from mathematical model show the power law in the relation between propagation speed of starting-wave and the initial density of pedestrians.
If there is an optimal density to minimize the delay to get out from a queue or crowd, the control of density is possible to reduce the waste of waiting time. Moreover, if pedestrians stand in a line with large headway (low density), the starting-wave propagates fast under the long queue. Whereas, if they stand in a line with small headway (high density), the starting-wave propagates slowly under the short queue. Which situation decreases the waste of waiting time? The optimal initial distribution for a queue is investigated here by considering the fundamental relation characterized by the power law. The results obtained from mean field theory based on the mathematical model show that the optimal density does exist at a density, which depends on their walking velocity.
We have found that these results, that is, fundamental relation characterized by the power law and the existence of optimal density, are verified by performing the experiments of pedestrians.
In this contribution, we have investigated the propagation speed of pedestrians’ reaction in relaxation process of a queue so-called starting-wave. The faster the starting-wave propagates, the more smoothly the crowd moves. We have revealed the existence of optimal density, where the travel time of last pedestrians to reach the start line for the initial queue is minimized by both analytical calculations and experiments. This optimal density inevitably plays a significant role to design not only the initial queue of pedestrians but also the traffic problems.