The authors have declared that no competing interests exist.
‡ These authors also contributed equally to this work.
Pedestrian dynamics models the walking movement of individuals in a crowd. It has recently been used in the analysis of procedures to reduce the risk of disease spread in airplanes, relying on the SPED model. This is a social force model inspired by molecular dynamics; pedestrians are treated as point particles, and their trajectories are determined in a simulation. A parameter sweep is performed to address uncertainties in human behavior, which requires a large number of simulations. The SPED model’s slow speed is a bottleneck to performing a large parameter sweep. This is a severe impediment to delivering real-time results, which are often required in the course of decision meetings, especially during emergencies. We propose a new model, called CALM, to remove this limitation. It is designed to simulate a crowd’s movement in constrained linear passageways, such as inside an aircraft. We show that CALM yields realistic results while improving performance by two orders of magnitude over the SPED model.
Pedestrian dynamics deals with modeling the movement of individuals, often as a part of a crowd. Its has been used in a wide variety of applications, from panic simulation and crowd behavior analysis [
The most popular models for the simulation of pedestrian dynamics are social force models [
The Self Propelled Entity Dynamics (SPED) model is one notable social dynamics model [
A critical challenge in the use of such models, especially during epidemics, lies in dealing with the intrinsic uncertainties in human behavior. The above application handles it by parameterizing the sources of uncertainty and then performing a sweep of the parameter space to generate all possible scenarios [
The number of scenarios that need to be generated is large, leading to a high computational effort. Chunduri et al. [
The primary contribution of this paper lies in proposing a fast pedestrian dynamics model—
Various types of methods have been used in pedestrian dynamics, such as cellular automaton [
Social force models treat each pedestrian as a point particle, analogous to an atom in molecular dynamics. Molecular dynamics uses models that capture the actual attractive and repulsive forces between atoms, which govern the movement of atoms. In social force models, conceptual forces are defined that perform a similar role, either increasing the speed or decreasing it. Each pedestrian wishes to reach a certain destination, which motivates a propulsive force
Both SPED and CALM use the model for propulsive force given in
Social dynamics models typically differ in their definition of the repulsive force. The basic idea is that the speed of a pedestrian should decrease on getting close to other persons or fixed objects on their path.
SPED computes repulsion for each pedestrian as follows. It considers the distance
A modified version of the Lennard-Jones potential from molecular dynamics is used to compute repulsion when
Here,
We observed by profiling the code that the Lennard-Jones potential is the primary computational bottleneck. In addition, the methodology to reduce the speed in the intermediate range is numerically awkward for the following reason. In order to verify convergence, a typical test would be to repeat the simulation with a smaller time step size and check if the results are similar. However, the method adopted in SPED makes the velocity a function of the time step size, which would not yield similar results when the time step size is changed. We designed the CALM model to overcome both limitations, and also added an additional behavioral feature that ensures that the simulation always progresses.
We note that actual human movement is not precisely defined by any particular potential. Rather, it varies so much that it is sufficient to capture its qualitative tendencies, and then use a parameter sweep to examine the range of movement patterns. We, therefore, define a simpler repulsive force by using a single curve that yields results qualitatively similar to SPED. It must satisfy the basic requirements that the speed should be the desired speed |
We accomplish this by using
Use of the above repulsive force is equivalent to solving Newton’s law of motion with the net force defined by
Comparing the above with the expression in
We accomplish this goal as follows. We calculate the variation of
In fitting the CALM model, we assume
We incorporate the same behavioral features as SPED in our simulations. In addition, we include the following. We have designed the CALM model for simulation of movement in narrow passageways, such as airplane aisles. In these situations, people cannot walk side-by-side. As a consequence, sometimes, a deadlock situation can happen when two people try to get right of way but neither can move forward. The SPED model will not progress in these situations.
We have designed a mechanism to resolve this. One of the passengers will be declared the winner, with a random component to the decision, and get the right of way. This reflects human behavior in practice, where one person would yield to another.
As we discussed in the introduction section, using pedestrian dynamics models for simulation of passengers movement in the airplanes is a critically important application for public health policy analysis. Namilae et al. [
We initialize the simulation by inputting the initial positions of passengers and physical obstacles from a file. We also assign each passenger a value of |
After the initialization, each step of the simulation computes the position of all passengers at several different points in time. This is accomplished by using an explicit Euler scheme to solve
Find the nearest passenger or physical obstacle to
Compute the repulsion
Compute the propulsion
Update the velocity and position of
Check for deadlock and update
During the deplaning procedure, each passenger will go through a few different states. The initial state of each passenger is going
In addition to
According to [
Parameter | Minimum Value | Maximum Value |
---|---|---|
|
1.1 m/s | 1.3 m/s |
0.2 | 0.6 | |
0.2 | 0.7 | |
0.5 m | 1.6 m | |
0.2 | 0.8 | |
0.2 m | 1.5 m |
The CALM model can also be used for boarding of airplanes with implementation details being similar to deplaning to a large extent. The main difference lies in the state diagram of the passengers, which is roughly in the reverse order of deplaning, as shown in
We have implemented boarding with three zones. Other boarding strategies can be implemented without changing the code, just by using suitable input files.
Boarding involves a few parameters, similar to deplaning. The parameter
Parameter | Minimum Value | Maximum Value |
---|---|---|
|
1.1 m/s | 1.3 m/s |
0.5 m | 1.6 m | |
0.2 | 0.8 | |
0.2 m | 1.5 m | |
0.2 | 0.6 |
We first show that the CALM models results are consistent with empirically observed metrics for disembarkation in airplanes. We then show that the CALM model yields substantial performance gains over the SPED model.
We run all our experiments on the Frontera supercomputer at the Texas Advanced Computing Center. This system consists of 8008 compute nodes with 56 cores per node for a total of 448448 cores and ranks the 5th fastest supercomputer in the world. Each node contains two Xeon Platinum 8280 28C processors running at 2.7GHz with 128 GB memory. Nodes are connected through Mellanox Infiniband HDR-100 network connected in a fat tree topology.
We use a scrambled Halton low dispcrepancy sequence for performing efficient parameter sweep [
Implementations of the CALM model for running parameter sweeps of deplaning and boarding simulations can be found at
We validate our results by examining disembarkation times on three different types of airplanes. A single simulation does not capture the variety of human movement patterns, and so we perform a parameter sweep with 1000 different combinations of parameter values, covering the range mentioned in
Airplane | Deplaning time | Empirically-observed deplaning time |
---|---|---|
[8.21, 16.43] (min) | [10.71, 12.13] (min) | |
[9.32, 15.17] (min) | [11.82, 13.4] (min) | |
[1.46, 4.06] (min) | [2.94, 3.33] (min) |
As the results of our experiments show, the empirically-observed deplaning time ranges are subsets of the ranges produced by the CALM model. We can draw two conclusions from these results. First, the results demonstrate that the CALM model generates results for all the expected scenarios. Second, the CALM model provides results that are a little outside the normal range. This was a deliberate design choice because our application goal is to generate rare scenarios that can capture extreme events [
In addition to validating the deplaning times, we selected several random simulations and checked the video output to examine if the behavior was realistic, as is commonly done in validation of pedestrian dynamics [
Moreover, it is common in this field use other features of crowds movements, such as the relationship between the density (number of passengers per unit area) and space-mean velocity, to validate models of pedestrian movements [
Analyzing these characteristics of the flow at bottlenecks in passengers’ paths is critical as these bottlenecks usually have a significant role in crowd movements [
We print the current position and velocity of all passengers every 100 iterations (which is equal to 0.5 seconds of real-world time) in deplaning simulations and then compute the density and space-mean velocity of passengers in the bottleneck for each of the sample iterations. Eqs
In these equations,
For each of the four airplanes that we consider in this paper, we select three deplaning simulations randomly from the fastest third, the slowest third and the moderate third of 1000 simulations in our parameters sweep. Then we compute the density and space-mean velocity for these simulations. For each value of density, different values of space-mean velocities may be observed. We compute the average of these space-mean velocities for each value of density in our plot of the fundamental diagrams.
We compare the performance of CALM and SPED using a parameter sweep of size 1000 for disembarkation process on an Airbus A320 with 144 seats. We present the runtime of the parameter sweeps and the average runtime of a single simulation that is computing by getting the average runtime of 1000 simulations of the parameter sweep in
SPED model | CALM model | Speedup | |
---|---|---|---|
Average runtime of a single simulation | 277.59 (s) | 4.7 (s) | 59.06 |
Runtime of the parameter sweep | 9209.4 (s) | 156.86 (s) | 58.70 |
There are two significant reasons for this considerable performance difference between these two models. First, we used a simpler force formulation to decrease its computational time in our model. In particular, much of the reduction in time was obtained by eliminating the Lennard-Jones potential and using a single simple formula to determine the impact of repulsion. This was the primary reason for decrease in simulation time, and led to around a factor 20 improvement in performance. Second, we removed passengers that had reached their destinations from the simulation. Both SPED and CALM models use algorithms of
Pedestrian dynamics is finding increased applications in diverse real-world problems. However, slow performance of existing models has been a bottleneck for policy analysis, especially in an emergency. In decision support meetings, a result is required in the order of a couple of minutes. The new CALM model delivers results in that time frame in contrast to the SPED model. Our validation simulations also show that the CALM model produces results that are consistent with empirical observations.
The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. URL:
The authors would like to acknowledge Robert Pahle for providing a webservice for producing the video outputs of the simulations. They would also like to thank Pierrot Derjany for helpful discussions about the SPED model.