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Project supported jointly by the National Natural Science Foundation of China (Grant No. 61233001) and the Fundamental Research Funds for Central Universities of China (Grant No. 2013JBZ007).

Analysis of dynamic features of pedestrian flows is one of the most exciting topics in pedestrian dynamics. This paper focuses on the effect of homogeneity and heterogeneity in three parameters of the social force model, namely desired velocity, reaction time, and body size, on the moving dynamics of bidirectional pedestrian flows in the corridors. The speed and its deviation in free flows are investigated. Simulation results show that the homogeneous higher desired speed which is less than a critical threshold, shorter reaction time or smaller body size results in higher speed of flows. The free dynamics is more sensitive to the heterogeneity in desired speed than that in reaction time or in body size. In particular, an inner lane formation is observed in normal lanes. Furthermore, the breakdown probability and the start time of breakdown are focused on. This study reveals that the sizes of homogeneous desired speed, reaction time or body size play more important roles in affecting the breakdown than the heterogeneities in these three parameters do.

The movement of pedestrian flows is an extremely complex procedure. Since the early 1970s, the study on pedestrian dynamics has attracted the attention of researchers. The modeling methods have been emerging in order to study pedestrian behaviors in normal conditions during evacuation.^{[1–5]}

Generally, pedestrian models can be divided into three categories: macroscopic, mesoscopic, and microscopic models. Some good overviews on pedestrian models are presented in Refs. [6] and [7]. Pedestrian models play an important role in understanding the dynamic features of pedestrian flows which are characterized by different self-organization behaviors. Zipper effect,^{[8]} lane formation in the bidirectional flow,^{[9]} strips in the crossing flow,^{[10]} and other self-organization phenomena^{[11]} have already been investigated using various pedestrian models or walker experiments. Moreover, pedestrian models are also used to assess designs and operations of building constructions such as stations, airports, and stadiums.^{[12]} The social force model^{[13]} is considered as the reference model in this paper. One reason is that the social force model as a microscopic model can qualitatively represent self-organization behaviors such as arching and lane formation.^{[13]} Another reason is that the social force model considers both physical factors and motivational factors, which makes the simulated pedestrians more realistic.^{[14]}

In recent studies, the bidirectional flow as one of the typical pedestrian flows has attracted more and more attention of scientists because of its various dynamic movement features. Pedestrians in the bidirectional flow are spontaneously organized in lanes with a uniform walking direction if the density of pedestrians is high enough.^{[15]} According to Ref. [16], there are three phases in the bidirectional flow: the freely moving phase, the coexisting phase, and the uniformly jamming phase. Additionally, phase transition from coexisting phase to congestion occurs with an increase in entrance density in the bidirectional flow.^{[17]} Congestion starts when pedestrians cannot maintain speeds in their desired directions, which may result in a temporary or total standstill which implies breakdown.^{[18]} However, we are still not clear whether breakdown will occur after congestion starts, or when this breakdown occurs. Besides, the bidirectional flow usually contains pedestrians in heterogeneity.^{[19]} Some people may walk faster because of their preference or due to some urgent matters, others may walk slower because of gender, age, disability, and some people may have better abilities to regulate their velocities according to the real-time situation of the pedestrian flow. The movement state, however, is more sensitive to which types of pedestrian flows is also unclear.

On the basis of previous studies, the dynamic phenomena of both free flows and congestion flows with homogeneity or heterogeneity are mainly focused on in this paper. The mean speed of the free flow and the fluctuations in speed are investigated in order to reflect the effects of homogeneity and heterogeneity in different parameters of the social force model on free flow dynamics. By comparing the values of the mean speeds and their standard deviations, we can obtain that the free flow dynamics is more sensitive to which parameter of the social force model with homogeneity or heterogeneity. Besides, some observed walking behaviors in this paper further contribute to better understanding different free flows. Moreover, the performances of congestion flows reflected by the breakdown probability and the start time of breakdown are studied in this paper. By comparing these values of probabilities and the corresponding start time, we can have a more in-depth understanding of the impacts of different pedestrians' factors on breakdown. The study on the breakdown phenomenon of pedestrian flows with homogeneity or heterogeneity in this paper can provide some theoretical supports for corridor designers to assess the layout of the corridor under an expected traffic.

This paper continues with the description of the social force model in Section 2. Section 3 presents the setup of the simulation scenarios, where sixteen scenarios are set. The detailed analysis of the walking speed of the bidirectional flow is carried out in Section 4. Section 5 presents simulation results of the breakdown probability and also the start time of breakdown. After analyzing the simulation results, the key discoveries are reviewed and the future work is looked into in Section 6.

In the social force model of Helbing *et al.*,^{[13]} pedestrians are driven by the desired force, *i* and *j*, *f*_{i j}; and the interaction force between pedestrian *i* and walls *w*, *f*_{iw}. The corresponding motion equation for each pedestrian *i* is:

*m*

_{i}denotes the mass of pedestrian

*i*, and

*v*

_{i}(

*t*) denotes the actual walking velocity at time instant

*t*.

The desired force

*τ*

_{i}is the adaptation time to accelerate the current velocity to the desired velocity.

The interaction force *f*_{ij} defines the pedestrian’s psychological tendency to steer away from others and the physical force that occurs when the distance between two pedestrian centers *d*_{ij} is less than the sum of the radii of these two pedestrians *r*_{i j} = *r*_{i} + *r*_{j}:

Here, *A*_{i} is the interaction strength and *B*_{i} is the range of the repulsive interactions. *j* to *i*, and *r*_{i} denotes the position of pedestrian *i*. cos(*φ*_{i j}) = −*n*_{i j}·*e*_{i}, and *e*_{i} = *υ*_{i}/||*υ*_{i}||. 0 ≤ *λ*_{i} ≤ 1, which introduces an anisotropic effect of pedestrians’ vision field on the motion according to Helbing *et al.*^{[15]} Note that the anisotropic effect means pedestrians in front of the current pedestrian may have larger impacts on him or her than people behind with changing the parameter *λ*_{i}. We assume that *λ*_{i} = 0.3 for all pedestrians in this paper. For more details about this vision field, we refer the readers to Refs. [4] and [15]. *k* denotes the body compression coefficient, and *κ* denotes the coefficient of sliding friction. *g*(*x*) is zero if pedestrians do not touch each other (*d*_{i j} > *r*_{i j}), otherwise it is equal to the argument *x*.

The interaction force between pedestrian *i* and walls *w*, *f*_{iw}, is handled analogously:

The parameters of the original social force model are specified in Table

In the theater, subway station, airport or other public places, there always exist corridors having bidirectional flows. It is a very meaningful job to understand flow efficiency that may be affected by different types of flows in order to avoid breakdown to occur as much as possible. This paper assumes that the bidirectional pedestrian flow moves in a corridor whose size is 10 m × 4 m as shown in Fig.

In order to study the dynamic features in the homogeneous and heterogeneous bidirectional flows, Table ^{[18]} It should be noted that scenario 0 is set as the reference scenario. Only one parameter in scenarios 1–15 replaces the corresponding parameter in the reference scenario, and others remain the same. We assume that pedestrians’ reaction time *τ* uniformly distributes between 0.20 s and 0.50 s with its deviation *σ*_{τ} in scenarios 6 and 7, and radius *r* uniformly distributes between 0.200 m and 0.300 m with its deviation *σ*_{r} in scenarios 11 and 12. Pedestrians’ desired velocity, however, satisfies the normal distribution with the standard deviation *σ*_{v} in scenarios 1 and 2. Scenario 5 refers to pedestrians on both sides either having faster desired speeds 1.50 m/s or slower desired speeds 1.20 m/s, and the number of pedestrians with faster desired speeds is the same with that with slower ones. Analogously, we set the scenarios 10 and 15.

The performance of free flows is characterized by the mean speed of the flow and the fluctuation in speed. The performance of congestion flows, however, is indicated by the breakdown probability and the start time of breakdown.

In the free bidirectional flow, pedestrians can walk freely and lanes start to form. In this section, the detailed comparisons and interpretations of pedestrians’ movement in different scenarios of Table

The walking speed and flow rate are two important parameters to reflect the free flow dynamics to some degree. Generally, the flow rate is not only related to the walking speed but also to the density of pedestrians. After the repeated simulation runs, we find that the flow rates in different scenarios of Table

Figure *T*. Δ*T* is set to 0.1 s during the simulation which considers many factors such as avoiding overlapping. Note that we choose the time period from 50 s to 150 s to eliminate the effect of initial conditions, and the inflow is set to 1 p/s from both sides. Table

It can be clearly observed that there is a positive correlation between the value of desired speed less than a critical threshold and the mean speed by comparing the data of scenarios 0, 3, and 4 in Table

In order to have a deeper understanding of the effect of the difference of desired speeds in scenario 5, we set the following symbols which can help to better analyze the motion features of the bidirectional flow: in Fig. *v*_{min}0 = 1 m/s to enlarge the differences between pedestrians. The width of the corridor remains the same which is 4 m and the inflow is set to 1 p/s.

From the movement of pedestrians within the yellow oval in Fig. *t* = 140 s, but overtaking behavior occurs after a few seconds. In a lane, there always exist pedestrians with faster desired speeds and also with slower ones, but we can find an interesting phenomenon that pedestrians with the same desired velocities tend to follow each other to develop a queue in a small scale. This lane formation is inside a lane. It can be explained as: Assume a moving state that there is no local lane formation inside a typical lane of the bidirectional flow, then pedestrians with faster desired speeds or with slower ones may walk together in a queue, and have the same desired walking direction to leave the corridor. According to Eqs. (*f*_{i j} which can result in the current moving state to remain stable for a while to realize the local equilibrium. The local inner lanes usually form after a series of unstable moving states. Note that the lanes are always broken and re-formed after a period of time especially on the boundaries because of the entrance of new pedestrians.

Figure

Figure

The moving bidirectional flow could transit from free flow to congestion flow under some circumstances, and breakdown may occur after congestion starts. Literature showed that different pedestrians have different features such as habits and physical states, and these different features may result in breakdown. In this section, we will focus on the breakdown issue of the bidirectional flow for the scenarios in Table

Corresponding to Ref. [18] where the size of the corridor is the same with the one in our paper, total breakdown is defined when at least 60 pedestrians walk very slowly during five consecutive seconds. Here, we assume that this slow velocity has a maximum value of 0.2 m/s.

During our simulation the inflow value is constant even though the breakdown occurs, and each simulation lasts 1500 s. The statistics results, namely the breakdown probability and the start time of breakdown, are obtained after 100 simulation runs. It should be noted that 1500 s for each simulation is long enough for the study of breakdown because the length of the corridor is 10 m, and a pedestrian only needs about 10 s to pass the corridor in the free flow.

Figure

Figure *et al.* pointed that if the desired speed of pedestrians is lower than 1.5 m/s, the efficiency of leaving will increase with an increase in desired speed; and if the desired speed is higher than 1.5 m/s, the efficiency of leaving will decrease with increasing the desired speed because of pushing which can result in additional friction effects.^{[13]} Accordingly, it is not the arbitrarily large desired speed can reduce the breakdown probability. We can further conclude that if smaller breakdown probability is expected to be obtained in the bidirectional flow, this pedestrian flow should better contain more pedestrians with relatively high desired speed less than the threshold value, and pedestrians’ desired speeds should better have smaller heterogeneities.

Figure

Figure

The start time of breakdown with the change of inflow for the scenarios in Table

Figures

By comparing Figs.

Comparing Figs.

Through comparing Figs.

By analyzing Figs.

In sum, from the simulation results it can be drawn that the sizes of the three parameters in the social force model, namely, the desired speed, reaction time, and body size, not only play an important role in breakdown probability of the bidirectional flow but also have a significant influence on the start time of breakdown. The heterogeneity in desired speed has a much larger impact on breakdown than that in reaction time or in body size does, and the effect of the heterogeneity in body size on breakdown is a little larger than that in reaction time.

Based on the social force model, we have presented and analyzed the dynamic features of the bidirectional pedestrian flow for sixteen different scenarios in this paper. The speeds in the free flows are focused on for the homogeneous and heterogeneous studies of the effects of the parameters in the social force model on the flow dynamics. The simulation results show that the homogeneous higher desired speed less than a critical threshold, shorter reaction time or smaller body size can yield higher free speed of the bidirectional flow. In particular, the large heterogeneity in desired speed results in the free speed to become slow and together with frequent fluctuations in the instant, while the heterogeneity in reaction time or in body size does not play an important role in free flows. Moreover, the breakdown probability and the start time of breakdown for different scenarios are considered and analyzed. We can draw our conclusion that heterogeneity in desired speed which can not only result in breakdown probability to become larger but also affect breakdown to occur earlier has larger impacts on breakdown than that in reaction time or body size. Besides, the homogeneous higher desired speed less than a critical threshold, short reaction time or small body size can delay the start time of breakdown.

Our study provides the effect of different types of pedestrian flows on the flow efficiency, which can not only enlighten us to understand more about the dynamics of bidirectional flows but also provide some theoretical supports to assess the layout of corridor in the facility design under the expected traffic to avoid jams or even breakdown. Moreover, the best design of the corridor can be implemented with a pre-determined probability of breakdown for different types of pedestrians. Our future work will in particular focus on the lane formation of the bidirectional flow such as the time evolution of the number of lanes and the width of lanes.

**Reference**