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A study on pedestrian choice between stairway and escalator in the transfer station based on floor field cellular automata
Physica A 392 (2013) 5089–5100
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Physica A journal homepage: www.elsevier.com/locate/physa
A study on pedestrian choice between stairway and escalator in the transfer station based on floor field cellular automata Xiangfeng Ji a,∗ , Jian Zhang a , Bin Ran b,a a
School of Transportation, Southeast University, No. 2 Sipailou, Xuanwu District, Nanjing, Jiangsu Province, China
b
Department of Civil and Environmental Engineering, University of Wisconsin, Madison, United States
highlights • • • • •
Pedestrian choice between stairway and escalator is studied. Random utility theory is used and a logit-based model is proposed. Parameters like familiarity, walking disutility, and time pressure are proposed. Counting rule based on the Large Number Law is proposed. Sensitivity analysis on the three parameters is done to show the choice distributions.
article
info
Article history: Received 12 November 2012 Received in revised form 23 April 2013 Available online 21 June 2013 Keywords: Pedestrian choice Cellular automata Micro-simulation Parameter Sensitivity analysis Counting rule
abstract Stairway and escalator are the main transfer facilities in the station where pedestrians make choices between them. A good understanding of pedestrian choices is helpful to raise the efficiency of transfer stations and lower the probability of disasters, such as stamps caused by congestion. This paper studies the choice behavior of pedestrians using random utility theory and floor field cellular automata. Among the factors influencing pedestrian choices, there are non-quantitative ones and quantitative ones. Thus, a method combining qualitative description and quantitative description is adopted. Subsequently, a logit model is presented to mimic the choice behaviors of pedestrians. In this model, there are three new important parameters, including familiarity, walking disutility, and time pressure. By using micro-simulation, a sensitivity analysis for these parameters is conducted. Besides, a counting rule based on the Large Number Law is presented to count the real data in transfer stations in Shanghai. After comparing the sensitivity analysis results and measurement data, several reference values of the three important parameters are obtained in uncongested and congested situations respectively. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Walking is essential for pedestrian movement in densely populated cities in the world such as Hong Kong and Shanghai, especially in Mass Transit Railway (MTR) transfer stations. During the peak hours, there are massive pedestrian flows and vertical transfer facilities like stairways and escalators are bottlenecks in MTR, which are the focus of this paper. Paying no special attention to crowd of pedestrians there can lower the efficiencies and even cause serious disasters, such as stamps. Therefore, knowledge of pedestrian’s demand is valuable in the planning and design of the facilities. Study on pedestrian movement starts from the VAKH of USSR in 1937 and has a history of more than 60 years in Pretechenskii and Milinski (1969) [1]. Since then, a number of models have been proposed and lots of software packages, based on these models, have been developed and commercialized.
∗
Corresponding author. Tel.: +86 25 83795356; fax: +86 25 83795356. E-mail addresses: [email protected] (X. Ji), [email protected] (J. Zhang), [email protected], [email protected] (B. Ran).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.06.011
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Models of pedestrian simulation can be divided into two categories, the macroscopic ones and microscopic ones. In the 1970s, Henderson [2] presented the earliest macroscopic model for pedestrian flow, which analyzed pedestrian flow in analogy to fluid. Hughes (2002) [3] established a two-dimensional macroscopic model to simulate pedestrian flow and obtained the fundamental diagram. Besides, Yinhua Xia et al. (2009) [4] and Dogbe (2010) [5] also obtained the similar result. As for the microscopic models, there exist social force model, cellular automata model and some related improved models. The social force model, first proposed by Helbing and Mornar (1995) [6], took pedestrians as self-driving particles. Movements of pedestrians are decided by self-driving force, repulsive forces from other pedestrians and obstacles. Later, Helbing et al. (2001, 2005, 2007) [7–9] improved this model from different aspects. Blue et al. (1997) [10] first used the cellar automata model to mimic pedestrian movements and then Blue and Adler simulated unidirectional flow, bidirectional flow (2001) [11] and four-way flow (2000) [12]. Although the above-mentioned models can simulate pedestrian flow well, they ignore the analysis of pedestrian behavior more or less. However, recent studies on pedestrian route choice and collision avoidance fill this void. For example, Hoogendoorn and Bovy (2004) [13] studied pedestrian route choice based on the utility theory under uncertainty and solved this problem with dynamic programming. Antonini et al. (2006) [14] proposed the pedestrian discrete choice model, which included movement direction and movement speed. The final utility was decided by expected direction, expected speed, actual location, actual speed, and so on. The model is verified with measurement data. Guo and Huang (2011) [15] formulated the route choice behavior of pedestrians in evacuation in closed areas with internal obstacles. The route choice is determined by the potential of discrete space with effective factors, such as route distance, pedestrian congestion and route capacity. Pedestrian’s collision appears when two or more pedestrians move to the same location at the same time interval because of the irregular characteristics of pedestrian movement. Besides the former rule-based collision avoidance [10– 12], Karamouzas et al. (2009) [16] presented a new local method for collision avoidance based on collision prediction, which reproduced emergent behavior like lane formation observed in real crowds. Asano et al. (2009) [17] mimicked the collision avoidance based on pure strategy Nash equilibrium and solved it by the augment allocation algorithm. Asano et al. (2010) [18] integrated pedestrian route choice and collision avoidance based on their paper in 2009 [17]. Some studies have been made on the pedestrian choice between stairway and escalator. Cheung and Lam (1998) [19] investigated pedestrian behavior during a choice between stairway and escalator with data from six MTR stations in Hong Kong and calibrated the travel time on the vertical facilities. Some other researchers who are not from traffic engineering also did some related research [20–24]. Webb et al. (2011) [20] investigated whether individuals mimic the stair/escalator choices of preceding pedestrians. Webb et al. (2011) [21] investigated the poster/banner on the pedestrian choice between stairway and escalator. Eves et al. (2008) [22] modeled the effects of speed of leaving the station and stair width on the choice of the stairs or escalator. Eves et al. (2009) [23] reported pedestrian responsiveness to an intervention on the choice of stairways and escalators. Olander et al. (2011) [24] tested whether contextual factors may affect the stair/escalator choice, such as the impact of escalator availability. Based on the existing models, this paper proposes a logit-based model to mimic the pedestrian choice between stairway and escalator. Three important parameters are presented, which are familiarity, walking disutility, and time pressure and a sensitivity analysis on these parameters is done after a micro-simulation using floor field cellular automata. During the sensitivity analysis, one parameter is fixed according to the questionnaire stated below and the other two change to find out the pedestrian’s choice probability distribution for the escalator in congested and un-congested situation respectively. The simultaneous change of the three parameters for the choice probability distribution for the escalator will be one of our further study directions. Finally, the reference values of these three parameters are given according to the comparison between the simulation result and real data collected in a transfer station of Shanghai. Compared to the former models, the model of this paper is a new one shown as follows. Firstly, influencing factors on the pedestrian choice are summarized and classified into qualitative ones and quantitative ones. Our logit-based model is proposed according to the quantitative factors. Secondly, three parameters are presented and a sensitivity analysis on these parameters is done after micro-simulation. This is the first appearance in the analysis of pedestrian’s choice between stairway and escalator as far as we know. Finally, the reference values of these parameters are given according to the comparison between the simulation result and real data collected in a transfer station of Shanghai. The remainder is organized as follows. Section 2 analyzes the influencing factors on the pedestrian choice between stairway and escalator. The model is developed in Section 3 and in Section 4, simulation and the sensitivity analysis are done. Some conclusions are obtained in Section 5.
2. Influencing factors on the pedestrian choice The pedestrian choice between stairway and escalator is a complex process and can be influenced by lots of factors, such as consciousness, travel time, pleasantness or comfort, trip purpose and safety in Ref. [13], physical characteristics of the study area, age and gender in Ref. [25] and availability and occupancy in Ref. [24]. Besides, the speed of escalator and fear of using escalator also can be the influencing factors. However, the study object is the choice of coming pedestrians between stairway and escalator during peak hour. Therefore, influencing factors in this paper are divided into three categories according to the following assumptions and the proposed model in this paper.
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Assumption 1. The speed of escalator is a little faster than the pedestrian walking speed. So pedestrians who have reached the escalator have no influence on choices of the following ones. That is to say, pedestrians who have reached the escalator will be removed immediately during the simulation process. Assumption 2. Pedestrians with health problem or fear of escalator are not included in this paper. Assumption 3. Gender distinction is not included in this paper. Visibility and familiarity When coming pedestrians make choices between stairway and escalator, there is a choice blind zone for the ones who are not familiar with the study area herein because of the shelter of pedestrians in front of them. Therefore, pedestrians who are familiar with the area will make choices according to their habit. Yet, randomness can be found for ones who are not familiar with the area. Visibility and familiarity are reflected by the familiarity parameter later. Energy conservation, safety and comfort Generally speaking, pedestrians are used to doing things with low energy consumption, especially for ones who are older or with big luggage. Besides, the length and gradient of the stairway also can cause more energy consumption. There may be pedestrians walking in the opposite direction on the stairway. Therefore, pedestrians on stairway will take more energy for the safety and comfort as a result of counterflow. Energy conservation, safety and comfort are reflected by the walking disutility parameter later. Walking time and time pressure Without loss of generality, pedestrians are inclined to choose escalators to get to the destinations, which is very obvious during the off-peak hours. But during the peak hours, it is slightly different. Because of the congestion in front of escalators, more and more pedestrians will choose stairways to shorten the travel time. Walking time and time pressure are reflected by the time pressure parameter later. 3. Model development According to the above-mentioned influencing factors, there are quantitative ones and non-quantitative ones. In the proposed model, quantitative factors are quantified and non-quantitative ones are reflected by the three parameters, which are familiarity, walking disutility and time pressure. When pedestrians come close to the stairway or escalator, it is hard to tell the stairway from escalator for the ones who are not familiar with the area here because of the shelter as mentioned in Section 2. Therefore, in this paper, a parameter θi (0 < θi ≤ 1) called familiarity parameter is introduced. The larger the value of θi is, the more likely pedestrians make choices according to their familiarity, or habit. Based on the random utility theory, pedestrians make choices according to the perceived disutility. And the perceived ⌢t disutility U ij is the generalized cost pedestrian i spends for facility j at time t which is as follows. ⌢t
U ij = Uijt + δijt
(3.1) ⌢t
where Uijt is the expected value of U ij and δijt is random error.
In this paper, is made up of two components, which are the walking disutility τijt and the congestion disutility φijt . The walking disutility reflects the energy consumption of a pedestrian and the congestion disutility reflects the waiting time as mentioned in Section 2. The more the congestion is, the more the waiting time is. The value of time is reflected by the time pressure parameter stated below. Uijt
3.1. Walking disutility In general, handbags pedestrians take will not influence their choices. In this paper, age and big luggage are considered to calculate walking disutility and an assumption is made as follows to simplify the calculation. Assumption 4. Pedestrians with big luggage usually choose the escalator to lower the energy consumption. Therefore, pedestrians with big luggage are treated as older pedestrian in this paper. Disutility caused by age is calculated by the log-sigmoid transfer function in an artificial neutral network [26]. Typically, the log-sigmoid transfer function is as follows. y=
1 1 + exp(−λx)
where λ is the sensitivity coefficient.
(3.2)
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Fig. 3.1. Graph of the log-sigmoid transfer function.
In this paper, the lifespan of a pedestrian is assumed to be between 0 and 100 and then all the ages are centered at age 50. That the median of age is 50 is based on two considerations. The first one is the assumption that the lifespan is from 0 to 100. The second one is the mathematical property of the log-sigmoid transfer function (3.2) whose symmetric point is (0, 0.5). Therefore, if we shift the symmetric point, the symmetric point should be (50, 0.5) based on our lifespan assumption. If not, some disutility values appear incorrect after we test different combinations of symmetric point and sensitivity coefficient value of λ. From the left graph of Fig. 3.1, we can see that from λ = 1 to λ = 0.1, the slope of curves change. With λ being from 1 to 0.2 large values settle at 0 and 1, which is not a real situation for describing the disutility for the escalator considering the age. Therefore, the value of λ is assigned as 0.1. With λ being 0.1, the function is as follows. Although the value of λ is somewhat arbitrary, it serves well in the simulation. The best fit value for λ can be calibrated with large amount of measured values of pedestrians. y=
1 1 + exp − x−1050
(3.3)
where x is pedestrian’s age and the graph is shown in the right one of Fig. 3.1. It is certain that pedestrians have different walking disutility if they choose different facilities. So a parameter nα (1 ≤ n ≤ 10) called walking disutility parameter is introduced to amend the above-mentioned function. The walking disutility of pedestrian i choosing the stairway or escalator at time t is
τijt = nα
1
1 + exp −
xi −50 10
(3.4)
where τijt is the walking disutility of pedestrian i choosing the stairway or escalator at time t and xi is the age of pedestrian i. That n = 1 denotes the disutility if pedestrians choose the escalator and that n > 1 denotes the disutility if pedestrians choose the stairway. 3.2. Congestion disutility Congestion disutility mainly relates to the number of crowded pedestrians in front of the stairway or escalator and the time pressure of pedestrians. The disutility caused by the number of crowded pedestrians is also calculated by the logsigmoid transfer function [26] which is similar to the above one. During the simulation, the queued number of pedestrians in front of the stairway or escalator is assumed to be 10 respectively, which is a typical crowded number of pedestrians based on our field observation described in the second paragraph of Section 4.5. And then all the queued numbers are centered at 5. From the left graph in Fig. 3.2, similar to the above discussion, we can see that from λ = 1 to λ = 0.1, the slope of curves change. With λ being from 0.9 to 0.1, the disutility for the choice of escalator changes linearly, which is not the general situation. Therefore, the value of λ is assigned as 1. With λ being 1, the function is as follows. Although the value of λ is also somewhat arbitrary, it serves well in the simulation. The best fit value for λ can be calibrated with large amount measured values of pedestrians, too. y=
1 1 + exp(−(x − 5))
(3.5)
where x is the number of crowded pedestrians in front of the stairway or escalator. And the function graph is shown in the right one of Fig. 3.2, which is also similar to the above one.
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Fig. 3.2. Graph of the log-sigmoid transfer function.
It is also necessary to amend the above-mentioned formula for the perception error of pedestrians. Because of the perception error of pedestrian, they can hardly know the real number of crowded pedestrians, especially when there are a certain amount of pedestrians in front of the stairway or escalator. Therefore, we use the perceived number K˜ to calculate the choice probability distribution for the escalator instead of real number, and K˜ = K + γ , where K is the real number and γ is the estimation error. γ is prescribed to follow Gaussian distribution, or normal distribution whose mean value K /5 is and variance is 1. That is γ ∼ N (K /5, 1). A normal distribution denotes that the estimation deviation from the real number of pedestrians is not very large. For example, in the congestion situation, the probability that the estimation deviation is one is about 30% and the probability that the estimation deviation is two is only about 5%. Pedestrians sometimes feel time pressure, especially during peak hour. Then a parameter β, β ∈ (0, 1], called time pressure parameter is introduced in this paper. The bigger β is, the more pressure pedestrians feel. Pedestrians who have bigger β are eager to transfer and vice versa. So the congestion disutility φijt is calculated as follows.
φijt = βit
1 1 + exp(−(K˜ ijt − 5))
.
(3.6)
All in all, it is assumed that all the random errors are independent of each other and obey a Gumbel distribution with a mean value of zero. Based on the random utility theory, the probability Pijt of pedestrian i choosing facility j at time t is Pijt =
exp(−θi Uijt ) 2
(k ∈ j).
(3.7)
exp(−θi Uikt )
k=1
4. Simulation, analysis and discussion 4.1. Study object analysis In the transfer station, pedestrians come to the stairway or escalator through transfer corridor and then make choices according to their individual characteristics and their interaction with environment. In this paper, the study object is simplified as in Fig. 4.1. The generation zone is where pedestrians are generated during simulation as they come through transfer corridor and the choice zone is where pedestrians make choices. The study object is discretized into 9 × 15 cells and the cell size is 0.4 × 0.4 m2 . The generation zone and choice zone of pedestrians occupy 9 cells respectively. The stairway occupies 3 cells and the escalator occupies 2 cells. The simulation step is 0.3 s, so the pedestrian speed is 1.33 m/s. In each step, N (1 ≤ N ≤ 9) pedestrians are generated randomly or Nˆ (1 ≤ Nˆ ≤ 9) pedestrians are generated fixedly. The characteristics of pedestrians are generated randomly according to rules. When pedestrians come to the choice zone, they begin to make choices and if they reach the stairway or escalator, they will be removed in the next time interval. 4.2. Floor field model introduction 4.2.1. Model analysis We use the floor field cellular automata model to mimic the choice behaviors of pedestrians in front of the stairway and escalator. This model was firstly proposed by Burstedde et al. (2001) [27]. In the model the space is discretized into small
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Fig. 4.1. Simplified study object.
Fig. 4.2. Moving direction and transition probability.
cells which can either be empty or occupied by exactly one pedestrian. Each of these pedestrians can move to one of its unoccupied neighbor cells at each discrete time step t → t + 1 according to certain transition probabilities as shown in Fig. 4.2. The probabilities are given by the interaction with two floor fields. These two fields S and D determine the transition probabilities in such a way that a pedestrian is more likely to move in the direction of higher fields. The S represents the static floor field and the D represents the dynamic floor field. 4.2.2. The static floor field The static floor field S does not evolve with time and such a field can be used to specify regions of space which are more attractive, e.g. an exit. In this paper, the static floor field describes the shortest distance to an exit. And the exit is a cell where pedestrians here will be removed from the simulation. The method of construction of the static floor field adopts the method in Ref. [27]. Pedestrians can only leave the study area through these above-mentioned exit cells. The explicit values of S in this paper are calculated with a distance metric:
Sij = min
(iTs ,jTs )
2 2 2 2 (iTs − il ) + (jTS − jl ) − (iTs − i) + (jTS − j) . max (il ,jl )
This means that the strength of the static floor field depends on the shortest distance to an exit. max(il ,jl ) { (iTs − il )2 + (jTS − jl )2 }, where (il , jl ) runs over all cells, is the largest distance of any cell to the exit (iTs , jTS ). This is just a normalization so that the field values increase with decreasing distance (iTs − i)2 + (jTS − j)2 to an exit and is zero for the cell farthest away from the door. The shortest distance shown in Fig. 4.3 and the corresponding static floor field for the choice of stairway or escalator respectively are shown in Fig. 4.4. 4.2.3. The dynamic floor field The dynamic floor field D is a virtual trace left by the pedestrians and has its own dynamics through diffusion and decay. It can model the attractive interaction between pedestrians. At t = 0, the dynamic field of all the cells is zero. Whenever a pedestrian moves from one cell to one of his neighboring cells, the dynamic field of the starting place is increased by one.
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Fig. 4.3. Shortest distances for stairway or escalator, respectively.
Fig. 4.4. Static floor field for stairway or escalator, respectively.
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Therefore, the dynamic field has only non-negative integer values and can be compared to a bosonic field. The dynamic floor field is time dependent and it has diffusion and decay determined by two parameters ρ ∈ [0, 1] and δ ∈ [0, 1], which means the widening and dilution of the trace. In each time interval, each boson of the dynamic field decays with probability δ and diffused with probability ρ to one of its neighboring cells. Finally, this yields D = D(t , ρ, δ). 4.2.4. Update rule of floor field model The update rules of the floor field model are as follows. (1) The dynamic floor field is modified according to its diffusion and decay rules. (2) For each pedestrian, the transition probability for a move to an unoccupied neighbor cell is determined by the local dynamics and the two floor fields. The values of the field are weighted with two parameters. And this yields the following. pij = N exp(kD Dij ) exp(kS Sij )(1 − nij )ξij where nij is the occupation number andnij = 0, 1;
cell, e.g. walls ξij is the obstacle number and ξij = 01 forbidden else. −1 . N denotes normalization and N = (i,j) exp(kD Dij ) exp(kS Sij )(1 − nij )ξij
(3) Each pedestrian chooses a target cell based on the transition probabilities determined in the previous step. (4) Because of the update rule being just part of the update rule of the model in this paper, the collision avoidance rule will be stated later. (5) D is increased by all moving pedestrians. 4.3. Update rules of simulation Update rules for the discrete simulation are inevitable, and the total update rules in this paper are as follows. (1) Pedestrians are generated in the generation zone randomly or fixedly and the characteristics of pedestrians are generated according to random rules. (2) Pedestrians move forward one cell or stop in each step before they reach the choice zone. (3) Pedestrians make choices according to their own characteristics and the interaction environment when they reach the choice zone. After the choice is made, the corresponding static floor field is fixed. (4) Pedestrians move according to the rules in Section 4.2.4. (5) If there are queues in front of stairways or escalators, pedestrians enter them. (6) If pedestrians have not reached stairways or escalators, (3)–(5) recycle. And if they have reached the so called exit in this paper, they will be removed in the next time interval. (7) When two or more pedestrians move to the same cell in one time interval, one pedestrian is chosen to move according to the probability. 4.4. Simulation procedure The simulation procedure in this paper is as follows in Fig. 4.5. 4.5. Sensitivity analysis and discussion A sensitivity analysis of these three parameters, which are familiarity θ , walking disutility nα and time pressure β , is made here. Because α is dimensionless, the value of it is prescribed to be 1. And then 25 samples are chosen to do linear interpolation from the sample space {(θ , n, β)|0 < θ ≤ 1, 1 ≤ n ≤ 10, 0 < β ≤ 1}. During the interpolation, one parameter is fixed and others are sampled from the sample space to analyze their influence on the choice probability distribution for the escalator. The fixed values of the parameters are based on the following questionnaires and statistics. When we video the real pedestrian choice movement in a transfer station of Shanghai during peak hours, a questionnaire is made simultaneously. There are three questions in the questionnaire. The first one is whether you are familiar with the facility here. The second one is whether you are eager for your purpose. And the last one is how many times the energy consumption for taking stairway as much as that for taking escalator. The three questions are for the three parameters respectively. The questionnaire consists of 500 pieces. Through the statistics of the questionnaire described above, we find that 75% of pedestrians are familiar with the facilities herein and 80% feel time pressure during peak hours. And the energy consumption for taking the stairway is about 7–9 times compared to that for the escalator. Therefore, the value of θ is fixed with 0.75, n is 7 and β is 0.8. And the 25 samples are as follows and × denotes Cartesian product. (θ , n, β) ∈ {0.15, 0.35, 0.55, 0.75, 0.95} × {1, 3, 5, 7, 9} × {0.2, 0.4, 0.6, 0.8, 1}. In this paper, an analysis is made on the two special situations, which are un-congestion situation and congestion situation. In the non-congestion situation, there is no crowd in front of the stairway or escalator and pedestrians can leave through the stairway or escalator according to their choices. The number of pedestrians in front of the stairway or escalator is prescribed to be two pedestrians. That is to say, in the un-congestion situation, the congestion effect is excluded. However,
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Fig. 4.5. Simulation procedure.
in the congestion, the analysis is made when there is some number of pedestrians in front of the stairway or escalator. In this paper, the number is 8. Some other situations and some transition situation are not included in this paper. During the simulation, 11,000 time intervals are run with the first 1000 time intervals to initialize pedestrian movement, which is the usual situation in Blue and Adler (2001) [11]. Un-congested situation In the un-congested situation, the results are as follows in Figs. 4.6–4.8. In Fig. 4.6, the value of θ is fixed with 0.75 and the sample space of (n, β) is {1, 3, 5, 7, 9} × {0.2, 0.4, 0.6, 0.8, 1}. In Fig. 4.7, the value of n is fixed with 7 and the sample space of (θ , β) is {0.15, 0.35, 0.55, 0.75, 0.95} × {0.2, 0.4, 0.6, 0.8, 1}. In Fig. 4.8, the value of β is fixed with 0.8 and the sample space of (θ , n) is {0.15, 0.35, 0.55, 0.75, 0.95} × {1, 3, 5, 7, 9}. The different colors of the surfs reflect different choice probabilities.
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Fig. 4.6. Choice probability distribution for the escalator with θ being 0.75.
Fig. 4.7. Choice probability distribution for the escalator with n being 7.
Fig. 4.8. Choice probability distribution for the escalator with β being 0.8.
In Fig. 4.6, it can be found that pedestrians are inclined to choose to save energy. Maybe this is why there are always crowded pedestrians in front of escalators. But, when the stairway is less energy-consuming, pedestrians may change their choices. There is a wave around point (0.3, 3, 0.5), which can reflect pedestrian’s hesitation in making a choice, because of the low walking disutility of the stairway and low time pressure. In Fig. 4.7, it can be found that the pedestrians’ familiarity with the facilities influences their choices greatly, or pedestrians are used to their habits. But, when pedestrians feel more time pressure, they may change their choices to stairways. In Fig. 4.8, it can be found that when the energy consumption of taking escalators is somewhat equal to that of taking the stairway, whether pedestrians are familiar with the facilities or not does not influence their choices. Only when the difference of energy consumption is obvious, pedestrians are inclined to making choices according to the familiarity. Congested situation In the congested situation, the results are shown in Figs. 4.9–4.11 and the values of parameters are as stated in the uncongested situation. The probability here is also for the choice of escalators.
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Fig. 4.9. Choice probability distribution for the escalator with θ being 0.75.
Fig. 4.10. Choice probability distribution for the escalator with n being 7.
Fig. 4.11. Choice probability distribution for the escalator with β being 0.8.
In Fig. 4.9, it can be found that pedestrians also will choose escalators although there is congestion, or queues, especially when stairways are more energy-consuming. But the choice probability is surely lower than that without congestion. In Fig. 4.10, it can be found that when time pressure is small, pedestrians will make choices according to their habit. But when pedestrians feel more time pressure, they will tend to stairways. In Fig. 4.11, it can be found that when taking the stairway is not very energy-consuming, the pedestrians’ choice probability distribution is relatively stable. Besides, when pedestrians are not familiar with the facilities here, the choice probability for the escalator is low and is almost equal to that for the stairway. When a comparison is conducted between congested and un-congested situations, the effect of congestion is obvious. The pedestrians’ choice probability for the escalator is smaller in the congestion situation.
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5. Conclusion Using the videoed data described in the second paragraph of Section 4.5, we obtain the choice probability distribution for the escalator based on the following counting rule. Counting rule In Fig. 4.1, pedestrians make choices 9 cells ahead, which is 3.6 m in reality. Therefore, a square whose length is 3.6 m and width is equal to that of the escalator or stairway is fixed and pedestrian occupancy or density are measured. In different densities, the number of pedestrians choosing the escalator or stairway is counted. According to the Large Number Law, when the number becomes extremely large, the frequency is approximate to the probability. Through comparing the counted distributions and simulated ones, some reference values of the three parameters in congestion and un-congestion situations can be obtained. In the un-congestion situation, the value of familiarity parameter θ is between 0.7 and 0.9 and the value of walking disutility parameter n is between 7 and 9. However, the time pressure parameter β is more fluctuant, whose value is between 0.4 and 0.9. While, in the congestion situation, the value of familiarity parameter θ is between 0.6 and 0.8 and the value of walking disutility parameter n is between 5 and 7. And the time pressure parameter β is more fluctuant than that in the non-congestion situation, whose value is between 0.2 and 0.9. The reason why β is so fluctuant may be that pedestrians are hesitating between saving time and saving energy, especially when there are crowded pedestrians. The model presented in this paper can be used to guide the planning of the transfer facilities in a station, especially the stairway and escalator. Besides, it can also be used to evaluate the existing facilities. Models in this paper can also be used to simulate pedestrian’s choice between stairway and escalator in an actual transfer station. When doing so, the practitioners should know the geometry characteristics of the actual stairway and escalator. Besides, some isolation facilities should be known, too. As described in the second paragraph of Section 4.5, a questionnaire about the three important parameters should be done. Acknowledgments Special thanks go to the anonymous reviewers for their great suggestions. This study was supported by the National Basic Research Program of China (973 Program-No. 2012CB725400). References [1] V.M. Pretechenskii, A.I. Milinski, Planning for Foot Traffic Flow in Building, The National Bureau of Standards, Amerind Publishing Co., 1978. Translated from the Russian publication which appeared in 1969, STROIZDAT Publishers, Moscow, 1969. [2] L.F. Henderson, The statistics of crowd fluids, Nature (1971) 381–383. [3] R.L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B (2002) 507–535. [4] Y. Xia, S.C. Wong, C.W. Shu, Dynamic continuum pedestrian flow model with memory effect, Physical Review E (2009). [5] C. Dogbe, On the Cauchy problem for macroscopic model of pedestrian flows, Journal of Mathematical Analysis and Applications (2010) 77–85. [6] D. Helbing, P. Molnar, Social force model for pedestrian dynamics, Physical Review E (1995). [7] D. Helbing, P. Molnar, I.J. Farkas, K. Bolay, Self-organizing pedestrian movement, Environment and Planning B: Planning and Design (2001) 361–383. [8] D. Helbing, L. Buzna, A. Johansson, T. Werner, Self-organized pedestrian crowd dynamics: experiments, simulations, and design solutions, Transportation Science (2005). [9] D. Helbing, A. Johansson, Dynamics of crowd disasters: an empirical study, Physical Review E (2007). [10] V.J. Blue, M.J. Embrechts, J.L. Adler, Cellular automata modeling of pedestrian movements, in: IEEE International Conference on Systems, Man, and Cybernatics, Conference Proceedings, 1997, pp. 2320–2323. [11] V.J. Blue, J.L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways, Transportation Research Part B (2001) 293–312. [12] V.J. Blue, J.L. Adler, Modeling four-directional pedestrian flows, Transportation Research Record (2000). [13] S.P. Hoogendoorn, P.H.L. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transportation Research Part B (2004) 169–190. [14] G. Antonini, M. Bierlaire, M. Weber, Discrete choice models of pedestrian walking behavior, Transportation Research Part B (2006) 667–687. [15] R.Y. Guo, H.J. Huang, Route choice in pedestrian evacuation: formulated using a potential field, Journal of Statistical Mechanics (2011) P04018. [16] I. Karamouzas, P. Heil, P.V. Beek, M.H. Overmars, A Predictive Collision Avoidance Model for Pedestrian Simulation, in: Lecture Notes in Computer Science, vol. 5884/2009, 2009, pp. 41–52. [17] M. Asano, T. Iryo, M. Kuwahara, A pedestrian model considering anticipatory behaviour for capacity evaluation, Transportation and Traffic Theory (2009). [18] M. Asano, T. Iryo, M. Kuwahara, Microscopic pedestrian simulation model combined with a tactical model for route choice behaviour, Transportation Research Part C (2010) 842–855. [19] C.Y. Cheung, H.K.Lam. William, Pedestrian route choices between escalator and stairway in MTR station, Journal of Transportation Engineering-ASCE (1998) 277–285. [20] O.J. Webb, F.F. Eves, L. Smith, Investigating behavioral mimicry in the context of stair/escalator choice, British Journal of Health Psychology (2011) 373–385. [21] O.J. Webb, F.F. Eves, J. Kerr, A statistical summary of mall-based stair-climbing interventions, Journal of Physical Activity and Health 8 (2011) 558–565. [22] F.F. Eves, A.L. Lewis, C. Griffin, Modeling effects of stair width on rates of stair climbing in a train station, Preventive Medicine 47 (2008) 270–272. [23] F.F. Eves, E.K. Olander, G. Nicoll, A. Puig-ribera, C. Griffin, Increasing stair climbing in a train station: the effects of contextual variables and visibility, Journal of Environmental Psychology 29 (2) (2009) 300–303. [24] E.K. Olander, F.F. Eves, Elevator availability and its impact on stair use in a workplace, Journal of Environmental Psychology 31 (2) (2011) 200–206. [25] P.N. Seneviratne, J.F. Morrall, Analysis of factors affecting the choice of route of pedestrians, Transportation Planning and Technology 10 (2) (1985) 147–159. [26] Z. Ge, Z. Sun, Matlab Application, Publishing House of Electronics Industry, 2007 (in Chinese). [27] C. Burstedde, K. Klauck, A. Schadschneider, J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A 295 (2001) 507–525.
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Physica A journal homepage: www.elsevier.com/locate/physa
A study on pedestrian choice between stairway and escalator in the transfer station based on floor field cellular automata Xiangfeng Ji a,∗ , Jian Zhang a , Bin Ran b,a a
School of Transportation, Southeast University, No. 2 Sipailou, Xuanwu District, Nanjing, Jiangsu Province, China
b
Department of Civil and Environmental Engineering, University of Wisconsin, Madison, United States
highlights • • • • •
Pedestrian choice between stairway and escalator is studied. Random utility theory is used and a logit-based model is proposed. Parameters like familiarity, walking disutility, and time pressure are proposed. Counting rule based on the Large Number Law is proposed. Sensitivity analysis on the three parameters is done to show the choice distributions.
article
info
Article history: Received 12 November 2012 Received in revised form 23 April 2013 Available online 21 June 2013 Keywords: Pedestrian choice Cellular automata Micro-simulation Parameter Sensitivity analysis Counting rule
abstract Stairway and escalator are the main transfer facilities in the station where pedestrians make choices between them. A good understanding of pedestrian choices is helpful to raise the efficiency of transfer stations and lower the probability of disasters, such as stamps caused by congestion. This paper studies the choice behavior of pedestrians using random utility theory and floor field cellular automata. Among the factors influencing pedestrian choices, there are non-quantitative ones and quantitative ones. Thus, a method combining qualitative description and quantitative description is adopted. Subsequently, a logit model is presented to mimic the choice behaviors of pedestrians. In this model, there are three new important parameters, including familiarity, walking disutility, and time pressure. By using micro-simulation, a sensitivity analysis for these parameters is conducted. Besides, a counting rule based on the Large Number Law is presented to count the real data in transfer stations in Shanghai. After comparing the sensitivity analysis results and measurement data, several reference values of the three important parameters are obtained in uncongested and congested situations respectively. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Walking is essential for pedestrian movement in densely populated cities in the world such as Hong Kong and Shanghai, especially in Mass Transit Railway (MTR) transfer stations. During the peak hours, there are massive pedestrian flows and vertical transfer facilities like stairways and escalators are bottlenecks in MTR, which are the focus of this paper. Paying no special attention to crowd of pedestrians there can lower the efficiencies and even cause serious disasters, such as stamps. Therefore, knowledge of pedestrian’s demand is valuable in the planning and design of the facilities. Study on pedestrian movement starts from the VAKH of USSR in 1937 and has a history of more than 60 years in Pretechenskii and Milinski (1969) [1]. Since then, a number of models have been proposed and lots of software packages, based on these models, have been developed and commercialized.
∗
Corresponding author. Tel.: +86 25 83795356; fax: +86 25 83795356. E-mail addresses: [email protected] (X. Ji), [email protected] (J. Zhang), [email protected], [email protected] (B. Ran).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.06.011
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Models of pedestrian simulation can be divided into two categories, the macroscopic ones and microscopic ones. In the 1970s, Henderson [2] presented the earliest macroscopic model for pedestrian flow, which analyzed pedestrian flow in analogy to fluid. Hughes (2002) [3] established a two-dimensional macroscopic model to simulate pedestrian flow and obtained the fundamental diagram. Besides, Yinhua Xia et al. (2009) [4] and Dogbe (2010) [5] also obtained the similar result. As for the microscopic models, there exist social force model, cellular automata model and some related improved models. The social force model, first proposed by Helbing and Mornar (1995) [6], took pedestrians as self-driving particles. Movements of pedestrians are decided by self-driving force, repulsive forces from other pedestrians and obstacles. Later, Helbing et al. (2001, 2005, 2007) [7–9] improved this model from different aspects. Blue et al. (1997) [10] first used the cellar automata model to mimic pedestrian movements and then Blue and Adler simulated unidirectional flow, bidirectional flow (2001) [11] and four-way flow (2000) [12]. Although the above-mentioned models can simulate pedestrian flow well, they ignore the analysis of pedestrian behavior more or less. However, recent studies on pedestrian route choice and collision avoidance fill this void. For example, Hoogendoorn and Bovy (2004) [13] studied pedestrian route choice based on the utility theory under uncertainty and solved this problem with dynamic programming. Antonini et al. (2006) [14] proposed the pedestrian discrete choice model, which included movement direction and movement speed. The final utility was decided by expected direction, expected speed, actual location, actual speed, and so on. The model is verified with measurement data. Guo and Huang (2011) [15] formulated the route choice behavior of pedestrians in evacuation in closed areas with internal obstacles. The route choice is determined by the potential of discrete space with effective factors, such as route distance, pedestrian congestion and route capacity. Pedestrian’s collision appears when two or more pedestrians move to the same location at the same time interval because of the irregular characteristics of pedestrian movement. Besides the former rule-based collision avoidance [10– 12], Karamouzas et al. (2009) [16] presented a new local method for collision avoidance based on collision prediction, which reproduced emergent behavior like lane formation observed in real crowds. Asano et al. (2009) [17] mimicked the collision avoidance based on pure strategy Nash equilibrium and solved it by the augment allocation algorithm. Asano et al. (2010) [18] integrated pedestrian route choice and collision avoidance based on their paper in 2009 [17]. Some studies have been made on the pedestrian choice between stairway and escalator. Cheung and Lam (1998) [19] investigated pedestrian behavior during a choice between stairway and escalator with data from six MTR stations in Hong Kong and calibrated the travel time on the vertical facilities. Some other researchers who are not from traffic engineering also did some related research [20–24]. Webb et al. (2011) [20] investigated whether individuals mimic the stair/escalator choices of preceding pedestrians. Webb et al. (2011) [21] investigated the poster/banner on the pedestrian choice between stairway and escalator. Eves et al. (2008) [22] modeled the effects of speed of leaving the station and stair width on the choice of the stairs or escalator. Eves et al. (2009) [23] reported pedestrian responsiveness to an intervention on the choice of stairways and escalators. Olander et al. (2011) [24] tested whether contextual factors may affect the stair/escalator choice, such as the impact of escalator availability. Based on the existing models, this paper proposes a logit-based model to mimic the pedestrian choice between stairway and escalator. Three important parameters are presented, which are familiarity, walking disutility, and time pressure and a sensitivity analysis on these parameters is done after a micro-simulation using floor field cellular automata. During the sensitivity analysis, one parameter is fixed according to the questionnaire stated below and the other two change to find out the pedestrian’s choice probability distribution for the escalator in congested and un-congested situation respectively. The simultaneous change of the three parameters for the choice probability distribution for the escalator will be one of our further study directions. Finally, the reference values of these three parameters are given according to the comparison between the simulation result and real data collected in a transfer station of Shanghai. Compared to the former models, the model of this paper is a new one shown as follows. Firstly, influencing factors on the pedestrian choice are summarized and classified into qualitative ones and quantitative ones. Our logit-based model is proposed according to the quantitative factors. Secondly, three parameters are presented and a sensitivity analysis on these parameters is done after micro-simulation. This is the first appearance in the analysis of pedestrian’s choice between stairway and escalator as far as we know. Finally, the reference values of these parameters are given according to the comparison between the simulation result and real data collected in a transfer station of Shanghai. The remainder is organized as follows. Section 2 analyzes the influencing factors on the pedestrian choice between stairway and escalator. The model is developed in Section 3 and in Section 4, simulation and the sensitivity analysis are done. Some conclusions are obtained in Section 5.
2. Influencing factors on the pedestrian choice The pedestrian choice between stairway and escalator is a complex process and can be influenced by lots of factors, such as consciousness, travel time, pleasantness or comfort, trip purpose and safety in Ref. [13], physical characteristics of the study area, age and gender in Ref. [25] and availability and occupancy in Ref. [24]. Besides, the speed of escalator and fear of using escalator also can be the influencing factors. However, the study object is the choice of coming pedestrians between stairway and escalator during peak hour. Therefore, influencing factors in this paper are divided into three categories according to the following assumptions and the proposed model in this paper.
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Assumption 1. The speed of escalator is a little faster than the pedestrian walking speed. So pedestrians who have reached the escalator have no influence on choices of the following ones. That is to say, pedestrians who have reached the escalator will be removed immediately during the simulation process. Assumption 2. Pedestrians with health problem or fear of escalator are not included in this paper. Assumption 3. Gender distinction is not included in this paper. Visibility and familiarity When coming pedestrians make choices between stairway and escalator, there is a choice blind zone for the ones who are not familiar with the study area herein because of the shelter of pedestrians in front of them. Therefore, pedestrians who are familiar with the area will make choices according to their habit. Yet, randomness can be found for ones who are not familiar with the area. Visibility and familiarity are reflected by the familiarity parameter later. Energy conservation, safety and comfort Generally speaking, pedestrians are used to doing things with low energy consumption, especially for ones who are older or with big luggage. Besides, the length and gradient of the stairway also can cause more energy consumption. There may be pedestrians walking in the opposite direction on the stairway. Therefore, pedestrians on stairway will take more energy for the safety and comfort as a result of counterflow. Energy conservation, safety and comfort are reflected by the walking disutility parameter later. Walking time and time pressure Without loss of generality, pedestrians are inclined to choose escalators to get to the destinations, which is very obvious during the off-peak hours. But during the peak hours, it is slightly different. Because of the congestion in front of escalators, more and more pedestrians will choose stairways to shorten the travel time. Walking time and time pressure are reflected by the time pressure parameter later. 3. Model development According to the above-mentioned influencing factors, there are quantitative ones and non-quantitative ones. In the proposed model, quantitative factors are quantified and non-quantitative ones are reflected by the three parameters, which are familiarity, walking disutility and time pressure. When pedestrians come close to the stairway or escalator, it is hard to tell the stairway from escalator for the ones who are not familiar with the area here because of the shelter as mentioned in Section 2. Therefore, in this paper, a parameter θi (0 < θi ≤ 1) called familiarity parameter is introduced. The larger the value of θi is, the more likely pedestrians make choices according to their familiarity, or habit. Based on the random utility theory, pedestrians make choices according to the perceived disutility. And the perceived ⌢t disutility U ij is the generalized cost pedestrian i spends for facility j at time t which is as follows. ⌢t
U ij = Uijt + δijt
(3.1) ⌢t
where Uijt is the expected value of U ij and δijt is random error.
In this paper, is made up of two components, which are the walking disutility τijt and the congestion disutility φijt . The walking disutility reflects the energy consumption of a pedestrian and the congestion disutility reflects the waiting time as mentioned in Section 2. The more the congestion is, the more the waiting time is. The value of time is reflected by the time pressure parameter stated below. Uijt
3.1. Walking disutility In general, handbags pedestrians take will not influence their choices. In this paper, age and big luggage are considered to calculate walking disutility and an assumption is made as follows to simplify the calculation. Assumption 4. Pedestrians with big luggage usually choose the escalator to lower the energy consumption. Therefore, pedestrians with big luggage are treated as older pedestrian in this paper. Disutility caused by age is calculated by the log-sigmoid transfer function in an artificial neutral network [26]. Typically, the log-sigmoid transfer function is as follows. y=
1 1 + exp(−λx)
where λ is the sensitivity coefficient.
(3.2)
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Fig. 3.1. Graph of the log-sigmoid transfer function.
In this paper, the lifespan of a pedestrian is assumed to be between 0 and 100 and then all the ages are centered at age 50. That the median of age is 50 is based on two considerations. The first one is the assumption that the lifespan is from 0 to 100. The second one is the mathematical property of the log-sigmoid transfer function (3.2) whose symmetric point is (0, 0.5). Therefore, if we shift the symmetric point, the symmetric point should be (50, 0.5) based on our lifespan assumption. If not, some disutility values appear incorrect after we test different combinations of symmetric point and sensitivity coefficient value of λ. From the left graph of Fig. 3.1, we can see that from λ = 1 to λ = 0.1, the slope of curves change. With λ being from 1 to 0.2 large values settle at 0 and 1, which is not a real situation for describing the disutility for the escalator considering the age. Therefore, the value of λ is assigned as 0.1. With λ being 0.1, the function is as follows. Although the value of λ is somewhat arbitrary, it serves well in the simulation. The best fit value for λ can be calibrated with large amount of measured values of pedestrians. y=
1 1 + exp − x−1050
(3.3)
where x is pedestrian’s age and the graph is shown in the right one of Fig. 3.1. It is certain that pedestrians have different walking disutility if they choose different facilities. So a parameter nα (1 ≤ n ≤ 10) called walking disutility parameter is introduced to amend the above-mentioned function. The walking disutility of pedestrian i choosing the stairway or escalator at time t is
τijt = nα
1
1 + exp −
xi −50 10
(3.4)
where τijt is the walking disutility of pedestrian i choosing the stairway or escalator at time t and xi is the age of pedestrian i. That n = 1 denotes the disutility if pedestrians choose the escalator and that n > 1 denotes the disutility if pedestrians choose the stairway. 3.2. Congestion disutility Congestion disutility mainly relates to the number of crowded pedestrians in front of the stairway or escalator and the time pressure of pedestrians. The disutility caused by the number of crowded pedestrians is also calculated by the logsigmoid transfer function [26] which is similar to the above one. During the simulation, the queued number of pedestrians in front of the stairway or escalator is assumed to be 10 respectively, which is a typical crowded number of pedestrians based on our field observation described in the second paragraph of Section 4.5. And then all the queued numbers are centered at 5. From the left graph in Fig. 3.2, similar to the above discussion, we can see that from λ = 1 to λ = 0.1, the slope of curves change. With λ being from 0.9 to 0.1, the disutility for the choice of escalator changes linearly, which is not the general situation. Therefore, the value of λ is assigned as 1. With λ being 1, the function is as follows. Although the value of λ is also somewhat arbitrary, it serves well in the simulation. The best fit value for λ can be calibrated with large amount measured values of pedestrians, too. y=
1 1 + exp(−(x − 5))
(3.5)
where x is the number of crowded pedestrians in front of the stairway or escalator. And the function graph is shown in the right one of Fig. 3.2, which is also similar to the above one.
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Fig. 3.2. Graph of the log-sigmoid transfer function.
It is also necessary to amend the above-mentioned formula for the perception error of pedestrians. Because of the perception error of pedestrian, they can hardly know the real number of crowded pedestrians, especially when there are a certain amount of pedestrians in front of the stairway or escalator. Therefore, we use the perceived number K˜ to calculate the choice probability distribution for the escalator instead of real number, and K˜ = K + γ , where K is the real number and γ is the estimation error. γ is prescribed to follow Gaussian distribution, or normal distribution whose mean value K /5 is and variance is 1. That is γ ∼ N (K /5, 1). A normal distribution denotes that the estimation deviation from the real number of pedestrians is not very large. For example, in the congestion situation, the probability that the estimation deviation is one is about 30% and the probability that the estimation deviation is two is only about 5%. Pedestrians sometimes feel time pressure, especially during peak hour. Then a parameter β, β ∈ (0, 1], called time pressure parameter is introduced in this paper. The bigger β is, the more pressure pedestrians feel. Pedestrians who have bigger β are eager to transfer and vice versa. So the congestion disutility φijt is calculated as follows.
φijt = βit
1 1 + exp(−(K˜ ijt − 5))
.
(3.6)
All in all, it is assumed that all the random errors are independent of each other and obey a Gumbel distribution with a mean value of zero. Based on the random utility theory, the probability Pijt of pedestrian i choosing facility j at time t is Pijt =
exp(−θi Uijt ) 2
(k ∈ j).
(3.7)
exp(−θi Uikt )
k=1
4. Simulation, analysis and discussion 4.1. Study object analysis In the transfer station, pedestrians come to the stairway or escalator through transfer corridor and then make choices according to their individual characteristics and their interaction with environment. In this paper, the study object is simplified as in Fig. 4.1. The generation zone is where pedestrians are generated during simulation as they come through transfer corridor and the choice zone is where pedestrians make choices. The study object is discretized into 9 × 15 cells and the cell size is 0.4 × 0.4 m2 . The generation zone and choice zone of pedestrians occupy 9 cells respectively. The stairway occupies 3 cells and the escalator occupies 2 cells. The simulation step is 0.3 s, so the pedestrian speed is 1.33 m/s. In each step, N (1 ≤ N ≤ 9) pedestrians are generated randomly or Nˆ (1 ≤ Nˆ ≤ 9) pedestrians are generated fixedly. The characteristics of pedestrians are generated randomly according to rules. When pedestrians come to the choice zone, they begin to make choices and if they reach the stairway or escalator, they will be removed in the next time interval. 4.2. Floor field model introduction 4.2.1. Model analysis We use the floor field cellular automata model to mimic the choice behaviors of pedestrians in front of the stairway and escalator. This model was firstly proposed by Burstedde et al. (2001) [27]. In the model the space is discretized into small
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Fig. 4.1. Simplified study object.
Fig. 4.2. Moving direction and transition probability.
cells which can either be empty or occupied by exactly one pedestrian. Each of these pedestrians can move to one of its unoccupied neighbor cells at each discrete time step t → t + 1 according to certain transition probabilities as shown in Fig. 4.2. The probabilities are given by the interaction with two floor fields. These two fields S and D determine the transition probabilities in such a way that a pedestrian is more likely to move in the direction of higher fields. The S represents the static floor field and the D represents the dynamic floor field. 4.2.2. The static floor field The static floor field S does not evolve with time and such a field can be used to specify regions of space which are more attractive, e.g. an exit. In this paper, the static floor field describes the shortest distance to an exit. And the exit is a cell where pedestrians here will be removed from the simulation. The method of construction of the static floor field adopts the method in Ref. [27]. Pedestrians can only leave the study area through these above-mentioned exit cells. The explicit values of S in this paper are calculated with a distance metric:
Sij = min
(iTs ,jTs )
2 2 2 2 (iTs − il ) + (jTS − jl ) − (iTs − i) + (jTS − j) . max (il ,jl )
This means that the strength of the static floor field depends on the shortest distance to an exit. max(il ,jl ) { (iTs − il )2 + (jTS − jl )2 }, where (il , jl ) runs over all cells, is the largest distance of any cell to the exit (iTs , jTS ). This is just a normalization so that the field values increase with decreasing distance (iTs − i)2 + (jTS − j)2 to an exit and is zero for the cell farthest away from the door. The shortest distance shown in Fig. 4.3 and the corresponding static floor field for the choice of stairway or escalator respectively are shown in Fig. 4.4. 4.2.3. The dynamic floor field The dynamic floor field D is a virtual trace left by the pedestrians and has its own dynamics through diffusion and decay. It can model the attractive interaction between pedestrians. At t = 0, the dynamic field of all the cells is zero. Whenever a pedestrian moves from one cell to one of his neighboring cells, the dynamic field of the starting place is increased by one.
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Fig. 4.3. Shortest distances for stairway or escalator, respectively.
Fig. 4.4. Static floor field for stairway or escalator, respectively.
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Therefore, the dynamic field has only non-negative integer values and can be compared to a bosonic field. The dynamic floor field is time dependent and it has diffusion and decay determined by two parameters ρ ∈ [0, 1] and δ ∈ [0, 1], which means the widening and dilution of the trace. In each time interval, each boson of the dynamic field decays with probability δ and diffused with probability ρ to one of its neighboring cells. Finally, this yields D = D(t , ρ, δ). 4.2.4. Update rule of floor field model The update rules of the floor field model are as follows. (1) The dynamic floor field is modified according to its diffusion and decay rules. (2) For each pedestrian, the transition probability for a move to an unoccupied neighbor cell is determined by the local dynamics and the two floor fields. The values of the field are weighted with two parameters. And this yields the following. pij = N exp(kD Dij ) exp(kS Sij )(1 − nij )ξij where nij is the occupation number andnij = 0, 1;
cell, e.g. walls ξij is the obstacle number and ξij = 01 forbidden else. −1 . N denotes normalization and N = (i,j) exp(kD Dij ) exp(kS Sij )(1 − nij )ξij
(3) Each pedestrian chooses a target cell based on the transition probabilities determined in the previous step. (4) Because of the update rule being just part of the update rule of the model in this paper, the collision avoidance rule will be stated later. (5) D is increased by all moving pedestrians. 4.3. Update rules of simulation Update rules for the discrete simulation are inevitable, and the total update rules in this paper are as follows. (1) Pedestrians are generated in the generation zone randomly or fixedly and the characteristics of pedestrians are generated according to random rules. (2) Pedestrians move forward one cell or stop in each step before they reach the choice zone. (3) Pedestrians make choices according to their own characteristics and the interaction environment when they reach the choice zone. After the choice is made, the corresponding static floor field is fixed. (4) Pedestrians move according to the rules in Section 4.2.4. (5) If there are queues in front of stairways or escalators, pedestrians enter them. (6) If pedestrians have not reached stairways or escalators, (3)–(5) recycle. And if they have reached the so called exit in this paper, they will be removed in the next time interval. (7) When two or more pedestrians move to the same cell in one time interval, one pedestrian is chosen to move according to the probability. 4.4. Simulation procedure The simulation procedure in this paper is as follows in Fig. 4.5. 4.5. Sensitivity analysis and discussion A sensitivity analysis of these three parameters, which are familiarity θ , walking disutility nα and time pressure β , is made here. Because α is dimensionless, the value of it is prescribed to be 1. And then 25 samples are chosen to do linear interpolation from the sample space {(θ , n, β)|0 < θ ≤ 1, 1 ≤ n ≤ 10, 0 < β ≤ 1}. During the interpolation, one parameter is fixed and others are sampled from the sample space to analyze their influence on the choice probability distribution for the escalator. The fixed values of the parameters are based on the following questionnaires and statistics. When we video the real pedestrian choice movement in a transfer station of Shanghai during peak hours, a questionnaire is made simultaneously. There are three questions in the questionnaire. The first one is whether you are familiar with the facility here. The second one is whether you are eager for your purpose. And the last one is how many times the energy consumption for taking stairway as much as that for taking escalator. The three questions are for the three parameters respectively. The questionnaire consists of 500 pieces. Through the statistics of the questionnaire described above, we find that 75% of pedestrians are familiar with the facilities herein and 80% feel time pressure during peak hours. And the energy consumption for taking the stairway is about 7–9 times compared to that for the escalator. Therefore, the value of θ is fixed with 0.75, n is 7 and β is 0.8. And the 25 samples are as follows and × denotes Cartesian product. (θ , n, β) ∈ {0.15, 0.35, 0.55, 0.75, 0.95} × {1, 3, 5, 7, 9} × {0.2, 0.4, 0.6, 0.8, 1}. In this paper, an analysis is made on the two special situations, which are un-congestion situation and congestion situation. In the non-congestion situation, there is no crowd in front of the stairway or escalator and pedestrians can leave through the stairway or escalator according to their choices. The number of pedestrians in front of the stairway or escalator is prescribed to be two pedestrians. That is to say, in the un-congestion situation, the congestion effect is excluded. However,
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Fig. 4.5. Simulation procedure.
in the congestion, the analysis is made when there is some number of pedestrians in front of the stairway or escalator. In this paper, the number is 8. Some other situations and some transition situation are not included in this paper. During the simulation, 11,000 time intervals are run with the first 1000 time intervals to initialize pedestrian movement, which is the usual situation in Blue and Adler (2001) [11]. Un-congested situation In the un-congested situation, the results are as follows in Figs. 4.6–4.8. In Fig. 4.6, the value of θ is fixed with 0.75 and the sample space of (n, β) is {1, 3, 5, 7, 9} × {0.2, 0.4, 0.6, 0.8, 1}. In Fig. 4.7, the value of n is fixed with 7 and the sample space of (θ , β) is {0.15, 0.35, 0.55, 0.75, 0.95} × {0.2, 0.4, 0.6, 0.8, 1}. In Fig. 4.8, the value of β is fixed with 0.8 and the sample space of (θ , n) is {0.15, 0.35, 0.55, 0.75, 0.95} × {1, 3, 5, 7, 9}. The different colors of the surfs reflect different choice probabilities.
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Fig. 4.6. Choice probability distribution for the escalator with θ being 0.75.
Fig. 4.7. Choice probability distribution for the escalator with n being 7.
Fig. 4.8. Choice probability distribution for the escalator with β being 0.8.
In Fig. 4.6, it can be found that pedestrians are inclined to choose to save energy. Maybe this is why there are always crowded pedestrians in front of escalators. But, when the stairway is less energy-consuming, pedestrians may change their choices. There is a wave around point (0.3, 3, 0.5), which can reflect pedestrian’s hesitation in making a choice, because of the low walking disutility of the stairway and low time pressure. In Fig. 4.7, it can be found that the pedestrians’ familiarity with the facilities influences their choices greatly, or pedestrians are used to their habits. But, when pedestrians feel more time pressure, they may change their choices to stairways. In Fig. 4.8, it can be found that when the energy consumption of taking escalators is somewhat equal to that of taking the stairway, whether pedestrians are familiar with the facilities or not does not influence their choices. Only when the difference of energy consumption is obvious, pedestrians are inclined to making choices according to the familiarity. Congested situation In the congested situation, the results are shown in Figs. 4.9–4.11 and the values of parameters are as stated in the uncongested situation. The probability here is also for the choice of escalators.
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Fig. 4.9. Choice probability distribution for the escalator with θ being 0.75.
Fig. 4.10. Choice probability distribution for the escalator with n being 7.
Fig. 4.11. Choice probability distribution for the escalator with β being 0.8.
In Fig. 4.9, it can be found that pedestrians also will choose escalators although there is congestion, or queues, especially when stairways are more energy-consuming. But the choice probability is surely lower than that without congestion. In Fig. 4.10, it can be found that when time pressure is small, pedestrians will make choices according to their habit. But when pedestrians feel more time pressure, they will tend to stairways. In Fig. 4.11, it can be found that when taking the stairway is not very energy-consuming, the pedestrians’ choice probability distribution is relatively stable. Besides, when pedestrians are not familiar with the facilities here, the choice probability for the escalator is low and is almost equal to that for the stairway. When a comparison is conducted between congested and un-congested situations, the effect of congestion is obvious. The pedestrians’ choice probability for the escalator is smaller in the congestion situation.
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5. Conclusion Using the videoed data described in the second paragraph of Section 4.5, we obtain the choice probability distribution for the escalator based on the following counting rule. Counting rule In Fig. 4.1, pedestrians make choices 9 cells ahead, which is 3.6 m in reality. Therefore, a square whose length is 3.6 m and width is equal to that of the escalator or stairway is fixed and pedestrian occupancy or density are measured. In different densities, the number of pedestrians choosing the escalator or stairway is counted. According to the Large Number Law, when the number becomes extremely large, the frequency is approximate to the probability. Through comparing the counted distributions and simulated ones, some reference values of the three parameters in congestion and un-congestion situations can be obtained. In the un-congestion situation, the value of familiarity parameter θ is between 0.7 and 0.9 and the value of walking disutility parameter n is between 7 and 9. However, the time pressure parameter β is more fluctuant, whose value is between 0.4 and 0.9. While, in the congestion situation, the value of familiarity parameter θ is between 0.6 and 0.8 and the value of walking disutility parameter n is between 5 and 7. And the time pressure parameter β is more fluctuant than that in the non-congestion situation, whose value is between 0.2 and 0.9. The reason why β is so fluctuant may be that pedestrians are hesitating between saving time and saving energy, especially when there are crowded pedestrians. The model presented in this paper can be used to guide the planning of the transfer facilities in a station, especially the stairway and escalator. Besides, it can also be used to evaluate the existing facilities. Models in this paper can also be used to simulate pedestrian’s choice between stairway and escalator in an actual transfer station. 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