Pedestrian Walking Behavior Revealed through a Random Walk Model

This paper applies method of continuous-time random walks for pedestrian flow simulation. In the model, pedestrians can walk forward or backward and turn left or right if there is no block. Velocities of pedestrian flow moving forward or diffusing are dominated by coefficients. The waiting time preceding each jump is assumed to follow an exponential distribution. To solve the model, a second-order two-dimensional partial differential equation, a high-order compact scheme with the alternating direction implicit method, is employed. In the numerical experiments, the walking domain of the first one is two-dimensional with two entrances and one exit, and that of the second one is two-dimensional with one entrance and one exit. The flows in both scenarios are one way. Numerical results show that the model can be used for pedestrian flow simulation.


Introduction
In recent years, modeling pedestrian flow has attracted considerable attention, partly because the model serves as basis for efficient crowd evacuation management and pedestrian facility operations.However, the research is still in its infancy owing to the complexity of human being's behaviors.
Most of the existing models for pedestrian flow are of microscopic nature, describing in detail the interactions among pedestrians, and between pedestrians and obstacles.Those models include, among others, cellular automata models 1-7 , lattice gas models 8-12 , the social force models 13 , the centrifugal force models 14 , and the floor field models 15, 16 .In cellular automata models, the walking space is two-dimensional and divided into cells.Each cell can either be empty, be occupied by exactly one pedestrian, or contain an obstacle.Cellular automata models are widely used for capturing pedestrian walking behaviors, such as bi-direction movement 1, 3, 4 , pedestrian counter flow with different walk velocities as the probability of N jumps occurring during the time interval 0, t .Consequently, the probability that there is no jump occurring during the time interval 0, t is Applying the Laplace transform to both sides of 2.1 leads to Note that Ψ 1, t represents the probability that the first jump occurs at time t and there are no further jumps after it until t.We thus have where Ψ 0, t and ψ t are simplified as Ψ and ψ.Similarly, we obtain The Laplace form of 2.4 is given by

2.5
Let P x, y, t be the probability of observing the walker at site x, y at time t, and let p N x, y denote the probability that the walker is at the position x, y after N jumps.We then have The Laplace transformation of 2.6 yields p N x, y ψ N s .

2.7
In the domain of Ω with the length of the lattice edge being h as shown in Figure 1 , the law of total probability implies that  where r 1 to r 4 represents the probability for the walker to proceed forward, turn left, walk back, and turn right, respectively.The probabilities are assumed to be known and 4 i 1 r i 1. Applying Taylor's expansion to RHS of 2.8 at the point x, y leads to

2.9
where O h 2 denotes higher-order terms that are omitted hereinafter.We thus obtain the following: ∂ 2 ∂y 2 p N−1 x, y .

2.10
Define f x r 1 − r 3 /h, f y r 2 − r 4 /h, where f x is the direction force along the x-axis and f y is the force along the y-axis.Consequently, 2.10 can be recast as where Applying the Laplace transformation to 2.11 , we have where

2.13
The specific form of 2.13 depends on the choice of the waiting time distribution ψ τ .Several distributions are plausible based on the nature of pedestrian walking behaviors.Here we assume that ψ τ follows the exponential distribution, that is, where λ is the mean of waiting time.The probability density function of waiting time in the Laplace space can be written as Assume the domain Ω is divided evenly into spaced cells of length Δx along x-axis and length Δy along y-axis, and δ x P ij , δ y P ij , δ 2 x P ij , and δ 2 y P ij represent the approximations to the first and second derivatives of P with respect to x or y at node x i , y j .Based on the standard central finite difference method, 3.1 can be discretized as follows: In the above, the truncation error, that is, τ ij , is where δ x , δ y , δ 2 x , and δ 2 y are the first-and second-order central difference operators.
Differentiating 3.1 with respect to x or y once and twice, respectively yields, approximations of higher-order derivatives as follows:

3.6
The difference between LHS of 3.5 and L x A y L y A x P ij is expressed as  and the inputs are where d 0, 8 ∪ 12, 20 .We further assume r 1 to r 4 to be 0.70, 0.15, 0.0, and 0.15, respectively, and λ 0.045 and h 0.05.
Figure 3 is snapshots of numerical solutions of the pedestrian flow at times t 1, 2, . . ., 9, respectively, to show the movement pattern of the pedestrians.The density increases steadily with the increase of entering flow and reaches its maximum at time t 5.The density centered at either group is becoming less while the density between these two groups is becoming larger as the pedestrians are walking forward.For each group of pedestrians, density at the center is always larger than those in the surroundings.The reason is that the people around the block are much easier to disperse than those in the middle.As the time is close to t 9, the density is approaching zero and only some late-entering or slowwalking people remain in the platform.From the snapshots, we can observe the phenomena of dispersion and advection of the pedestrian flow.
To further illustrate the model, additional experiments were conducted with the whole left boundary, that is, x 0, as the entrance.In addition, the inflow is supposed to be steady with P 0, y, t 1.0, where y ∈ 0, 20 and t ∈ 0, 30 . Figure 4 plots the density along x at t 1, 2, . . ., 8, respectively.It can be observed that there is a sharp decrease in each curve, indicating that only a few pedestrians are fast walkers.The results are consistent with the phenomenon we may observe in reality that no matter how crowded a platform is, the density close to the exit is always less than the jam density.To reveal the impact of direction choice behavior on the flow patterns, Figure 5 plots the average density of the platform along time under various scenarios where r 1 0.80, 0.60, 0.40, 0.33, respectively, and probabilities of left and right turns are r 2 r 4 1 − r 1 /2 and walking backward is not allowed.It is shown that the time the flow becomes steady is significantly dependent on r 1 .The larger r 1 is, the faster the flow reaches to a steady state.The numerical results coincide with actual pedestrian moving behavior.Moreover, a smaller value of r 1 leads to a lower density in the steady state.
Figure 6 illustrates the average density across the time under different entering flow intensity at PE 1.0, 0.8, 0.6, 0.4, 0.2, respectively.It shows the steady density is only slightly lower than the input flow intensity.In addition, it is observed that no matter what intensity the inflow is, the time the flow reaches to its steady state remains almost the same.

Conclusion
This paper is an application of continuous-time random walks approach to pedestrian flow simulation.The model is capable of describing macroscopic phenomena such as forward moving and dispersion of pedestrian flow.In addition, by varying coefficients of the model, some microscopic phenomena such as route/direction choice behaviors can be replicated.To solve the model, a high-order compact scheme with the alternating direction implicit method is applied.Numerical results validated both the model and the numerical method.The model formulation in the paper only accounts for the distribution of the waiting time.In our future study, the probability distribution of jump length will be considered to further enhance the validity of the model.To make the model more practically applicable, we also plan to incorporate bi-direction flow and more realistic boundary conditions, such as input and output between platform and trains in platform.

Figure 2 :
Figure 2: Schematic illustration of the one-way pedestrian flow in a platform.The length of the platform is 30 and the width is 20.The left dashed lines represent two entrances and the right one is exit.Solid lines are walls.

Figure 3 :
Figure 3: Density of pedestrian flow at different times.