# A Stairs Evacuation Model Considering the Pedestrian Merging Flows

Authors: Xia Zhong Zheng; Dan Tian; Ming Zhang; Chaoran Hu; Liyang Tong
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

Pedestrian merging flows are common in a stairs evacuation process, which involves complex interactions among pedestrians that substantially restrict the efficiency of the stairs evacuation process. Analyzing the pedestrian merging flows process and improving the efficiency of stairs evacuation are urgent and essential tasks. A novel simplified stairs evacuation model for simulating and analyzing the stairs evacuation process, which considers the impact of merging flows, is proposed in this process. The dynamic pedestrian output rate of a floor platform is calculated by the number of pedestrians on the floor platform. The merging ratio determined by the design size of stairs is adopted to determine the ratio between the stairs pedestrian flow and the floor pedestrian flow in the pedestrian output rate of the floor platform. To evaluate the stairs evacuation process is divided into three stages based on the pedestrian merging flows process, and the evacuation time at each stage is computed by the dynamic pedestrian output rate of the floor platform. The stairs evacuation capacity is calculated by the evacuation time and the number of pedestrians. A case study of a six-floor building evacuation is investigated, and the reliability and feasibility of the proposed model is verified. By establishing different merging ratios, the optimal merging ratio is obtained by comparing the evacuation capacities of different merging ratios, which provides a reference of stairs design for designers.


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## Body

## 1. Introduction

In the context of a building evacuation, stairs can be regarded as an emergency area, and the evacuation performance of the stairs is related to the safety of pedestrians [1, 2]. Due to the high density of pedestrians and the particular characteristics of staircases as slopes, congestion is more likely to occur in the staircase of a building, and casualties and property damage may occur during an emergency [3]. For example, 11 students were injured and three students were nearly killed during a stampede at Xian Yang Second Primary School in China on November 18, 2006. These frequent incidents motivate us to explore the process of stairs evacuation to decrease the number of stairs evacuation incidents and provide scientific and reliable references for emergency evacuation and building design.As the main egress component of buildings, the process of stairs evacuation has prompted a considerable number of researchers to study this topic in the past decade [4]. Different evacuation models, which range from simple hand calculations to more complex computation, have been developed. In a previous study, the evacuation process can be expressed by evacuation simulation models, such as BGRAF, EXODUS, and SIMULEX [5]. These three models provide a simulation platform for evacuation, expressing the movement characteristics of people without resistance, but it is difficult to reflect the behavioral changes of people during the movement. Cellular automata model [6, 7] divides the evacuation area evenly into several cells, each of which has a state of pedestrian occupancy or unoccupied. The evacuated pedestrians can be simulated by the defined behavior parameters for evacuation process. Lattice gases model [8] enhances the way of regional division, using nonuniform division of evacuation space. The grid around each pedestrian is dynamic, closely related to the actual movement of the pedestrian, more in line with the evacuation of high-density people. Fluid-dynamic models [9] regard the internal evacuation network movement of the building as fluid movement, and simulates the movement process of the crowd to establish a macroscopic fluid dynamics model of pedestrian movement. The macroscopic model can effectively reflect evacuation time and the evacuation process, but lack the microscopic factors such as the psychological effects of pedestrians and behavioral changes during evacuation. Social force models [10, 11] and agent-based model [12] give behavioral information of evacuated pedestrians, corrects behavior parameters such as evacuation speed, and optimizes the simulation model. The simulation process is close to objective reality. In evacuation networks, the congestion process is an important indicator of the efficiency in evacuation process and often occurs at key evacuation nodes. Queuing theory [13] has advantages in analyzing local congestion processes, and can effectively analyze the distribution of pedestrian flow and bottleneck identification during evacuation. The evacuation process is divided into more categories according to different understandings of scholars, and can be analyzed from multiple angles such as macroscopic and microscopic. The previous research interpretation and application of the results highly depend on the scholar’s understanding of the methods employed for the simulation of pedestrian movement, and they may fail to highlight the essential reason causing the congestion in the stairs evacuation [14]. Different from the traditional evacuation analysis situation, the stair evacuation process focuses on the evacuation bottleneck generated by the local evacuation feature, and analyzes the causes of congestion. The microscopic evacuation process is subject to more influencing factors and has a significant impact on evacuation results. Thus, the factors that influence stairs evacuation is important.Three main factors that influence stairs evacuation are generally presented by researchers during the analysis of evacuation data [15], namely, (1) walking speeds [16], (2) pedestrian flows [17], and (3) pedestrian densities [18]. In particular, these factors are not the essential reason that influences a stairs evacuation; the essential reason is the pedestrian merging flows in the stairs [3]. The merging of two pedestrian flows that approach a stair landing from the floor and that approach from the stair is referred to as pedestrian merging flows on stairs [20]. Due to pedestrian flow from upper stairs and floors, the walking speeds of pedestrians decrease, and the flows and pedestrian densities increase [21]. Therefore, pedestrian flow merging should be considered in the study of the stairs evacuation process.Merging flows involve complex interactions among pedestrians, which will restrict the efficiency of the evacuation process. Different experimental studies have been conducted to illuminate the influence of merging flows in a stairs evacuation [22–24]. Pedestrian merging behavior was described by experiments, and the relationship between the width of a channel and the pedestrian density can be obtained [25]. Researchers determined that the merging ratio can impact flows and stairs evacuation times during stairs evacuation [26]. In a previous study, the merging ratio was defined as approximately 50 : 50. However, the merging ratio is related to the pedestrian density [14]. Thus, Sano T introduced a mathematical model to calculate the evacuation times on stairs, which takes into account the impact of merging flows [14]. Different merging ratios were designed to analyze the influence of pedestrian merging flows on evacuation times, and the merging behavior was explored by the merging ratio [27]. In practice, the merging ratio represents the movement rule of stair flows and floor flows, which only denotes the pedestrian movement status in pedestrian merging; however, pedestrian merging flows are regarded as a part of a stairs evacuation and cannot be adopted to analyze the entire process of stairs evacuation [28]. The current evacuation model provides a series approach to analyze the entire stairs evacuation process [29–31]. This model does not fully express the dynamic change process of the floor pedestrian flow and stairs pedestrian flow in a stairs evacuation. An essential assumption of this model is that the pedestrian flow of stairs is in the steady-state, in which the staircase is filled with pedestrians and has the same density for each floor at the start; this assumption disregards the initial/final stage of a stairs evacuation [32]. Therefore, the development of a stairs evacuation model that fully considers pedestrian merging flows and appropriately describes the entire process of stairs evacuation has theoretical significance and practical value.Pedestrian merging flows is an importance reason that influences a stairs evacuation; it directly causes the congestion phenomenon in a stairs evacuation. However, the relationship between the pedestrian merging flows and the evacuation time by the traditional evacuation model in the entire process of stairs evacuation is not easily expressed. To explore the relationship between pedestrian merging flows and stairs evacuation times and obtain the optimal merging ratio to guide the design of stairs, we propose a stairs evacuation model that considers pedestrian merging flows to describe the process of stairs evacuation in this study. The output rate function of stairs can be determined by queuing theory, and the merging ratio is determined by the width of stairs and the floor channel. The entire process of stairs evacuation is divided into three parts based on the pedestrian merging flows. We can obtain the dynamic variation process of the pedestrian flow rate of stairs and floors, and the evacuation time is computed to measure the stairs evacuation capacity. The different merging ratios are simulated to explore the relationship between pedestrian merging flows and stairs evacuation times, and the optimal merging ratio can be obtained by analyzing the stairs evacuation capacity.The remainder of the paper is organized as follows: The output rate function of stairs and merging ratio are determined in Section2. In Section 3, the stairs evacuation model is constructed, and the stairs evacuation capacity is calculated. An illustrative example of the model’s utility and applicability is shown, and the optimal merging ratio is obtained in Section 4. Section 5 provides a detailed discussion of the advantages of the proposed methods. The conclusions and future research directions are provided in Section 6.


## 2. Analysis of Stairs Pedestrian Merging

The main factors that influence a stairs evacuation include the pedestrian output rate of a floor platform and merging ratio. This section introduces a simplified mathematical model for the representation and calculation of the output rate of a floor platform and merging ratio on staircases in case of simultaneous total evacuation of a multistory building.


### 2.1. Preliminaries

A set of assumptions need to be introduced to explain the field of applicability of the model before the model is established. Many scholars adopt these assumptions to analyze the pedestrian evacuation.Assumption 1.
The pedestrian merging flows in the stairs are continuous without any delay in the process of merging on stairs. In general, the evacuation is a continuous process without any restriction; thus, the merging is not interrupted in the evacuation process.Assumption 2.
The width of the stairs is steady. According to the principle of the stairs design, the width of the stairs is uniformly valued. According to the design criterion of a staircase, the width of the stairs is set to a constant value.Assumption 3.
The inclination of the stairs should not exceed 45°. In general, the inclination is defined as 30°, and the number of stair steps is set to 18°.Assumption 4.
No pedestrians come from the ground floor. In the process of evacuation, the pedestrian movement object is a structured exit. Thus, no pedestrian come from the ground floor.Assumption 5.
The pedestrian output rate is determined by the width of the stairs and the floor channel. The floor pedestrian flow and stairs pedestrian flow represent a uniform distribution in the floor platform after pedestrian merging.According to the previously mentioned assumption and the pedestrian merging process, the stairs pedestrian merging process is described in Figure1.Figure 1
Structure and structure name of the stairs.


### 2.2. Calculation of Pedestrian Output Rate of Floor Platform

The pedestrian output rate of the floor platform is the pedestrian output number of floor platforms during the unit time in the merging flow process. When the floor channel pedestrian flow merging and staircase pedestrian flow merging occur on the stairs, the pedestrian merging process can be regarded as the queuing system, and the floor platform is the server. The pedestrian merging process satisfies the following rules:(1)
Every pedestrian possesses independent movement characteristics;
(2)
The stairs pedestrian merging obeys a Poisson distribution;
(3)
The density of pedestrians can influence the merging process;
(4)
The stairs evacuation adopts the FCFS (First Come First Served) involved in the scheduling theory.Based on the previously mentioned principle, the M/G/c/c proposed in the queuing theory is developed to calculate the pedestrian output rate of the floor platform [33, 25]. Thus, we assume that the pedestrian passing rate of the floor channel and the pedestrian passing rate of stairs in unit time have fixed values and are defined as qifloor and qistair, respectively. The pedestrian output rate of the floor platform is related to the number of pedestrians in the floor platform. According to the third rule, when the pedestrian flow in the floor platform is n, the probability can be represented as(1)PnN=n=qifloor+qistairET1nn!fnfn−1⋯f2f1P0,whereN, which is the free variable, represents the number of pedestrians in the floor platform; n is the actual pedestrian flow in the floor platform; ET1 is the expected service time per pedestrian in the floor platform; and fn is the pedestrian service rate when the number of pedestrians in the floor platform equal n, which is described as(2)fn=vnv1,wherev1 represents the average movement speed when one pedestrian is on the stairs; vn is the average movement speed of n pedestrians on the stairs based on the relationship between the density of pedestrians and the velocity. Considering the emergency situation, pedestrians will increase the speed of movement due to the need to escape, so the speed-density function needs to be corrected. This paper uses the method proposed by Ding et al. [34] correct the pedestrians movement speed during emergency evacuation. The calculation method is as follows: vn is denoted as vn=ϑv; ϑ is a correction coefficient related to the pedestrian density with the value ϑ=1.49−0.36D; D is the pedestrian density in the stairs; v is the normal movement velocity of pedestrians; and P0 is the probability when no pedestrians exist on the floor platform, which is expressed as(3)P0−1n=0=1+∑j=lcqifloor+qistairET1jj!fjfj−1⋯f2f1,c represents the capacity of the floor platform, and the value of c is determined by the size of the floor platform, that is, c=Dwl, w is the width of floor platform, and l is the length of floor platform. According to the probability of the number of pedestrian flows, the pedestrian output rate of the floor platform is denoted as(4)Ei=qifloor+qistair1−PnN=n,whereEi is the pedestrian output rate of the floor platform. When the number of pedestrian flows in the floor platform attains the maximum, the pedestrian output rate of floor platform is Eic, and Pc is the probability that the number of floor platforms attain the maximum capacity.


### 2.3. Definition Pedestrian Flow Rate

The merging ratio represents the relationship between the floor channel pedestrian flow and the stairs pedestrian flow based on the design principle of a structure, and the width of the floor channel and stairs determine the number of pedestrian flows. Thus, the pedestrian passing rate of the floor channel and stairs can be influenced by the width of the floor channel and stairs. Zheng [35] proposed the floor merging ratio by the merging number of stairs pedestrians and floor channel pedestrians in unit time. In this paper, the merging ratio is calculated by the design pedestrian pass rate for stairs and floors, which can be defined as(5)k=LfloorLfloor+Lstair,1−k=LstairLfloor+Lstair,wherek represents the merging ratio, Lfloor is the design width of the floor channel, and Lstair is the design width of the stairs.


## 2.1. Preliminaries

A set of assumptions need to be introduced to explain the field of applicability of the model before the model is established. Many scholars adopt these assumptions to analyze the pedestrian evacuation.Assumption 1.
The pedestrian merging flows in the stairs are continuous without any delay in the process of merging on stairs. In general, the evacuation is a continuous process without any restriction; thus, the merging is not interrupted in the evacuation process.Assumption 2.
The width of the stairs is steady. According to the principle of the stairs design, the width of the stairs is uniformly valued. According to the design criterion of a staircase, the width of the stairs is set to a constant value.Assumption 3.
The inclination of the stairs should not exceed 45°. In general, the inclination is defined as 30°, and the number of stair steps is set to 18°.Assumption 4.
No pedestrians come from the ground floor. In the process of evacuation, the pedestrian movement object is a structured exit. Thus, no pedestrian come from the ground floor.Assumption 5.
The pedestrian output rate is determined by the width of the stairs and the floor channel. The floor pedestrian flow and stairs pedestrian flow represent a uniform distribution in the floor platform after pedestrian merging.According to the previously mentioned assumption and the pedestrian merging process, the stairs pedestrian merging process is described in Figure1.Figure 1
Structure and structure name of the stairs.


## 2.2. Calculation of Pedestrian Output Rate of Floor Platform

The pedestrian output rate of the floor platform is the pedestrian output number of floor platforms during the unit time in the merging flow process. When the floor channel pedestrian flow merging and staircase pedestrian flow merging occur on the stairs, the pedestrian merging process can be regarded as the queuing system, and the floor platform is the server. The pedestrian merging process satisfies the following rules:(1)
Every pedestrian possesses independent movement characteristics;
(2)
The stairs pedestrian merging obeys a Poisson distribution;
(3)
The density of pedestrians can influence the merging process;
(4)
The stairs evacuation adopts the FCFS (First Come First Served) involved in the scheduling theory.Based on the previously mentioned principle, the M/G/c/c proposed in the queuing theory is developed to calculate the pedestrian output rate of the floor platform [33, 25]. Thus, we assume that the pedestrian passing rate of the floor channel and the pedestrian passing rate of stairs in unit time have fixed values and are defined as qifloor and qistair, respectively. The pedestrian output rate of the floor platform is related to the number of pedestrians in the floor platform. According to the third rule, when the pedestrian flow in the floor platform is n, the probability can be represented as(1)PnN=n=qifloor+qistairET1nn!fnfn−1⋯f2f1P0,whereN, which is the free variable, represents the number of pedestrians in the floor platform; n is the actual pedestrian flow in the floor platform; ET1 is the expected service time per pedestrian in the floor platform; and fn is the pedestrian service rate when the number of pedestrians in the floor platform equal n, which is described as(2)fn=vnv1,wherev1 represents the average movement speed when one pedestrian is on the stairs; vn is the average movement speed of n pedestrians on the stairs based on the relationship between the density of pedestrians and the velocity. Considering the emergency situation, pedestrians will increase the speed of movement due to the need to escape, so the speed-density function needs to be corrected. This paper uses the method proposed by Ding et al. [34] correct the pedestrians movement speed during emergency evacuation. The calculation method is as follows: vn is denoted as vn=ϑv; ϑ is a correction coefficient related to the pedestrian density with the value ϑ=1.49−0.36D; D is the pedestrian density in the stairs; v is the normal movement velocity of pedestrians; and P0 is the probability when no pedestrians exist on the floor platform, which is expressed as(3)P0−1n=0=1+∑j=lcqifloor+qistairET1jj!fjfj−1⋯f2f1,c represents the capacity of the floor platform, and the value of c is determined by the size of the floor platform, that is, c=Dwl, w is the width of floor platform, and l is the length of floor platform. According to the probability of the number of pedestrian flows, the pedestrian output rate of the floor platform is denoted as(4)Ei=qifloor+qistair1−PnN=n,whereEi is the pedestrian output rate of the floor platform. When the number of pedestrian flows in the floor platform attains the maximum, the pedestrian output rate of floor platform is Eic, and Pc is the probability that the number of floor platforms attain the maximum capacity.


## 2.3. Definition Pedestrian Flow Rate

The merging ratio represents the relationship between the floor channel pedestrian flow and the stairs pedestrian flow based on the design principle of a structure, and the width of the floor channel and stairs determine the number of pedestrian flows. Thus, the pedestrian passing rate of the floor channel and stairs can be influenced by the width of the floor channel and stairs. Zheng [35] proposed the floor merging ratio by the merging number of stairs pedestrians and floor channel pedestrians in unit time. In this paper, the merging ratio is calculated by the design pedestrian pass rate for stairs and floors, which can be defined as(5)k=LfloorLfloor+Lstair,1−k=LstairLfloor+Lstair,wherek represents the merging ratio, Lfloor is the design width of the floor channel, and Lstair is the design width of the stairs.


## 3. Stairs Evacuation Model

This section introduces a new simplified mathematical model that represents the process of stairs evacuation. Considering the merging process of stairs as a critical point, the process of stairs evacuation can be divided into stages: the stage before pedestrian merging flows, the pedestrian merging flows stage and the stage after pedestrian merging flows.


### 3.1. Calculation of Evacuation Time before Merging Flow

Before pedestrian merging flows, the floor pedestrian flow will enter the floor platform, while some pedestrians may deviate from a structure. Thus, the duration time of the stage before pedestrian merging flows is defined ast′; the number of pedestrian flow in the stage before merging flows is low; the density of pedestrians is low; and pedestrians move at the normal speed. The length of stairs is assumed to be l, and the duration time of the stage before pedestrian merging flows can be defined as(6)t′=lσv,whereσ is the adjustment coefficient of the movement velocity in the process of downward stairs. In the stage before pedestrian merging flows, some pedestrians near the exit of a structure will leave the structure. Similarly, some pedestrians will transfer to the next floor from this floor. We assume that the pedestrians on each floor have the same velocity, and the pedestrian passing rate for all floors before merging flows occur in the stairs is constant. Thus, the output number of floor pedestrians before merging flows occur can be expressed as(7)Si′=qifloort′,whereSi′ is the output number of floor pedestrians before merging flows. Based on the feature of stairs evacuation, the pedestrian output number of structures is equal to the output number of the first floor.


### 3.2. Calculation of Evacuation Time during Merging Flow

Assume that the pedestrian flow of each floor isGi and that the pedestrian number of floors is m. The pedestrian output rate has attained the design requirement of stairs when the process of merging flows starts. However, the capacity of the floor platform does not exceed the maximum value. Thus, the number of pedestrians in the floor platform will attain the maximum value with the development of the merging flows process; the process of merging flows tends to attain stabilization; and the output rate of the floor platform is equal to the sum of the passing rate of the floor channel and stairs. The passing rate of the floor channel can be calculated as follows:(8)qi′floort=qifloorUit−C<0,kEicUit−C=0.Similarly, the passing rate of stairs can be described as(9)qi′stairt=qistairUit−C<0,1−kEicUit−C=0,whereqi′floort and qi′stairt represent the pedestrian passing rate of the floor channel and stairs, respectively, which is a function related to time; t is the duration of merging; C is the maximum capacity of the floor platform; and Uit−C is the judgement function of the pedestrian number of floor platforms. Because the pedestrian number of floor platforms is less than the maximum value of floor platforms, Uit−C≤0, where Uit is the function of the pedestrian number of floor platforms, which is related to the evacuation time:(10)Uit=qi′floort+qi′stairt−∫0tQitdt+rUit−Δt−C<0,CUit−Δt−C=0Δ
t is the time interval; r represents the existing pedestrian number of floor platforms before merging flows occur; and Qit is the floor platform output rate at the t-th time. According to Equation (10), Qit can be described as(11)Qit=EiUit−ΔtUit−Δt−C<0,EicUit−Δt−C=0,whereEiUit−Δt represents the pedestrian output rate in the floor platform when the pedestrian number of floor platforms is Uit−Δt, which can be defined as follows:(12)EiUit−Δt=qi′floort−Δt+qi′stairt−Δt1−PUit−ΔtN=Uit−Δt.In the process of stairs evacuation, the difference in the stairs design causes a difference in the evacuation environment in the stairs and a different evacuation principle. Once the evacuation of floor pedestrians is complete, that is, once the pedestrian passing rate of the floor channel is 0, the merging flows process ends. Based on the evacuation situation of each floor, three evacuation scenarios are possible:Scenario 1.
Wheni=1, according to the pedestrian output number of floor channels and merging ratio and combining the feature of stairs design, the pedestrian output number of floor channels M1 in the merging process can be expressed as follows:(13)M1t=q1′floort.Scenario 2.
When1<i<n, according to the merging feature of stairs, the pedestrian output rate in the i-th floor may be influenced by the i−1-th floor. When the pedestrian output rate for the i−1-th floor is less than the theoretical pedestrian output rate in the i-th floor Qi, and the practical pedestrian output rate for the floor platform on the i-th floor is equal to the pedestrian number of stairs on the i−1-th floor, conversely, the practical pedestrian output rate of the floor platform on the i-th floor Qi′ can be described as follows:(14)Qi′t=qi−1′stairqi−1′stairt<Qit,Qitqi−1′stairt≥Qit.
The floor pedestrian flow determines the duration of pedestrian merging flows. When the evacuation of pedestrian flows on the floor is completed, the process of merging ends. Comparing the pedestrian passing rate of the floor channel and floor platform, the pedestrian passing rate of the floor channelsMi on the i-th floor can be obtained as follows:(15)Mit=minqi′floor,kQi′t.Scenario 3.
Wheni=n, the output number of floor platform pedestrians is equal to the output number of floors, and the output number of floor channel pedestrians is influenced by the output number of stair on the n−1-th floor as the merging flows process develops. Thus, the pedestrian passing rate of the floor channel Mnt on the n-th floor can be denoted as follows:(16)Mnt=qn′floortqn′floort≤qn−1′stairt,qn−1′stairtqn′floort>qn−1′stairt.
Some pedestrians have left their initial floor before pedestrian merging flows. The number of each floor can be calculated by the output number of floor pedestrians.(17)SQimerging=Gi−Si′.
The stairs pedestrian flow and floor pedestrian flow comprise the main factor of stairs pedestrian merging flows. According to the definition of stairs pedestrian merging and the output rate of floor pedestrian flow, the merging duration of each floorTimerging can be calculated as follows:(18)SQimerging=∫0TimergingMitdt.
We assume that the duration of the pedestrian merging flows process is determined byTimerging, and the maximum value of Timerging is the duration of the stairs pedestrian merging flows process Tihui. Tihui can be described as follows:(19)t″=maxi=1mTimerging.


### 3.3. Calculation of Evacuation Time after Merging Flow

When the merging flows process ends, all pedestrians in a structure are on the stairs and leave the structure by the exit. The stage of stairs evacuation is referred to as the stage after merging. To calculate the evacuation time after merging flows, the remainder pedestrian flow after merging flows is the main factor in calculating the evacuation time after merging. Assume that the remainder pedestrian flow isSQhou, which can be calculated as(20)SQremain=∑i=1mGi−S1′−∫0t″Q1tdt.∑
i
=
1
m
G
i represents the initial pedestrian flow in the structure, S1′ is the output number of pedestrians before merging flows, and ∫0t″Q1tdt is the output number of pedestrian flows in the merging flows process.The remaining pedestrians in the stairs can transfer to the exit of the structure, and the duration after merging flowst″′ can be calculated by the remaining pedestrians and the pedestrian output rate of the structure exit, which can be denoted as follows:(21)SQremain=∫0t″′Q1tdt.


### 3.4. Determination of Stairs Evacuation Capacity

The capacity of stairs evacuation is the efficiency value of stairs evacuation, which represents the average evacuation number in unit time. The more efficient the evacuation capacity is, the shorter the amount of evacuation time, and vice versa. Thus, the evacuation time can be obtained to calculate the capacity of the stairs, the total evacuation time of the stairs is obtained by computing the duration for each evacuation stage, which is described as follows:(22)T=t′+t″+t″′.Combining the pedestrian number of a structure and the total evacuation time, the capacity of the stairs can be expressed as follows:(23)S=∑i=1mGiT.To clearly express the stairs evacuation capacity, the stairs evacuation process is simulated, and the simulation flow chart of the evacuation is established to describe the stairs merging flows process (Figure2).Figure 2
Flow chart of stairs evacuation process.


## 3.1. Calculation of Evacuation Time before Merging Flow

Before pedestrian merging flows, the floor pedestrian flow will enter the floor platform, while some pedestrians may deviate from a structure. Thus, the duration time of the stage before pedestrian merging flows is defined ast′; the number of pedestrian flow in the stage before merging flows is low; the density of pedestrians is low; and pedestrians move at the normal speed. The length of stairs is assumed to be l, and the duration time of the stage before pedestrian merging flows can be defined as(6)t′=lσv,whereσ is the adjustment coefficient of the movement velocity in the process of downward stairs. In the stage before pedestrian merging flows, some pedestrians near the exit of a structure will leave the structure. Similarly, some pedestrians will transfer to the next floor from this floor. We assume that the pedestrians on each floor have the same velocity, and the pedestrian passing rate for all floors before merging flows occur in the stairs is constant. Thus, the output number of floor pedestrians before merging flows occur can be expressed as(7)Si′=qifloort′,whereSi′ is the output number of floor pedestrians before merging flows. Based on the feature of stairs evacuation, the pedestrian output number of structures is equal to the output number of the first floor.


## 3.2. Calculation of Evacuation Time during Merging Flow

Assume that the pedestrian flow of each floor isGi and that the pedestrian number of floors is m. The pedestrian output rate has attained the design requirement of stairs when the process of merging flows starts. However, the capacity of the floor platform does not exceed the maximum value. Thus, the number of pedestrians in the floor platform will attain the maximum value with the development of the merging flows process; the process of merging flows tends to attain stabilization; and the output rate of the floor platform is equal to the sum of the passing rate of the floor channel and stairs. The passing rate of the floor channel can be calculated as follows:(8)qi′floort=qifloorUit−C<0,kEicUit−C=0.Similarly, the passing rate of stairs can be described as(9)qi′stairt=qistairUit−C<0,1−kEicUit−C=0,whereqi′floort and qi′stairt represent the pedestrian passing rate of the floor channel and stairs, respectively, which is a function related to time; t is the duration of merging; C is the maximum capacity of the floor platform; and Uit−C is the judgement function of the pedestrian number of floor platforms. Because the pedestrian number of floor platforms is less than the maximum value of floor platforms, Uit−C≤0, where Uit is the function of the pedestrian number of floor platforms, which is related to the evacuation time:(10)Uit=qi′floort+qi′stairt−∫0tQitdt+rUit−Δt−C<0,CUit−Δt−C=0Δ
t is the time interval; r represents the existing pedestrian number of floor platforms before merging flows occur; and Qit is the floor platform output rate at the t-th time. According to Equation (10), Qit can be described as(11)Qit=EiUit−ΔtUit−Δt−C<0,EicUit−Δt−C=0,whereEiUit−Δt represents the pedestrian output rate in the floor platform when the pedestrian number of floor platforms is Uit−Δt, which can be defined as follows:(12)EiUit−Δt=qi′floort−Δt+qi′stairt−Δt1−PUit−ΔtN=Uit−Δt.In the process of stairs evacuation, the difference in the stairs design causes a difference in the evacuation environment in the stairs and a different evacuation principle. Once the evacuation of floor pedestrians is complete, that is, once the pedestrian passing rate of the floor channel is 0, the merging flows process ends. Based on the evacuation situation of each floor, three evacuation scenarios are possible:Scenario 1.
Wheni=1, according to the pedestrian output number of floor channels and merging ratio and combining the feature of stairs design, the pedestrian output number of floor channels M1 in the merging process can be expressed as follows:(13)M1t=q1′floort.Scenario 2.
When1<i<n, according to the merging feature of stairs, the pedestrian output rate in the i-th floor may be influenced by the i−1-th floor. When the pedestrian output rate for the i−1-th floor is less than the theoretical pedestrian output rate in the i-th floor Qi, and the practical pedestrian output rate for the floor platform on the i-th floor is equal to the pedestrian number of stairs on the i−1-th floor, conversely, the practical pedestrian output rate of the floor platform on the i-th floor Qi′ can be described as follows:(14)Qi′t=qi−1′stairqi−1′stairt<Qit,Qitqi−1′stairt≥Qit.
The floor pedestrian flow determines the duration of pedestrian merging flows. When the evacuation of pedestrian flows on the floor is completed, the process of merging ends. Comparing the pedestrian passing rate of the floor channel and floor platform, the pedestrian passing rate of the floor channelsMi on the i-th floor can be obtained as follows:(15)Mit=minqi′floor,kQi′t.Scenario 3.
Wheni=n, the output number of floor platform pedestrians is equal to the output number of floors, and the output number of floor channel pedestrians is influenced by the output number of stair on the n−1-th floor as the merging flows process develops. Thus, the pedestrian passing rate of the floor channel Mnt on the n-th floor can be denoted as follows:(16)Mnt=qn′floortqn′floort≤qn−1′stairt,qn−1′stairtqn′floort>qn−1′stairt.
Some pedestrians have left their initial floor before pedestrian merging flows. The number of each floor can be calculated by the output number of floor pedestrians.(17)SQimerging=Gi−Si′.
The stairs pedestrian flow and floor pedestrian flow comprise the main factor of stairs pedestrian merging flows. According to the definition of stairs pedestrian merging and the output rate of floor pedestrian flow, the merging duration of each floorTimerging can be calculated as follows:(18)SQimerging=∫0TimergingMitdt.
We assume that the duration of the pedestrian merging flows process is determined byTimerging, and the maximum value of Timerging is the duration of the stairs pedestrian merging flows process Tihui. Tihui can be described as follows:(19)t″=maxi=1mTimerging.


## 3.3. Calculation of Evacuation Time after Merging Flow

When the merging flows process ends, all pedestrians in a structure are on the stairs and leave the structure by the exit. The stage of stairs evacuation is referred to as the stage after merging. To calculate the evacuation time after merging flows, the remainder pedestrian flow after merging flows is the main factor in calculating the evacuation time after merging. Assume that the remainder pedestrian flow isSQhou, which can be calculated as(20)SQremain=∑i=1mGi−S1′−∫0t″Q1tdt.∑
i
=
1
m
G
i represents the initial pedestrian flow in the structure, S1′ is the output number of pedestrians before merging flows, and ∫0t″Q1tdt is the output number of pedestrian flows in the merging flows process.The remaining pedestrians in the stairs can transfer to the exit of the structure, and the duration after merging flowst″′ can be calculated by the remaining pedestrians and the pedestrian output rate of the structure exit, which can be denoted as follows:(21)SQremain=∫0t″′Q1tdt.


## 3.4. Determination of Stairs Evacuation Capacity

The capacity of stairs evacuation is the efficiency value of stairs evacuation, which represents the average evacuation number in unit time. The more efficient the evacuation capacity is, the shorter the amount of evacuation time, and vice versa. Thus, the evacuation time can be obtained to calculate the capacity of the stairs, the total evacuation time of the stairs is obtained by computing the duration for each evacuation stage, which is described as follows:(22)T=t′+t″+t″′.Combining the pedestrian number of a structure and the total evacuation time, the capacity of the stairs can be expressed as follows:(23)S=∑i=1mGiT.To clearly express the stairs evacuation capacity, the stairs evacuation process is simulated, and the simulation flow chart of the evacuation is established to describe the stairs merging flows process (Figure2).Figure 2
Flow chart of stairs evacuation process.


## 4. Case Study

To present the applicability of the merging flows model, an exemplary case study of a teaching building is presented here. The number of floors is 6, the width of stairs is 1.5 m, the length of the floor platform is 3.3 m, the width of the floor platform is 1.8 m, and the length of stairs is 4 m. The building is designed for primary teaching on all floors. For simplicity, the number of pedestrians on a floor is assumed to be 100. The case study applies assumptions based on the Chinese building code, but any other relevant reference may have been employed for their definitions.


### 4.1. Simulation of Stairs Evacuation Process

To qualitatively evaluate the results obtained with the pedestrian merging flows model for evacuation on stairs, the process of stairs evacuation is simulated. The results are shown in Figures3 and 4; Figure 3 shows the pedestrian output rate of the floor platform for different floors; and Figure 4 shows the pedestrian number of floor platforms for different floors. The duration time of the stage before pedestrian merging flows is approximately 14 s; the duration after pedestrian merging flows is approximately 100 s; and the duration of the pedestrian merging flows process is approximately 327 s (Figure 3). The pedestrian output rate of the floor platform decreases with the development of the pedestrian merging flows process, and correspondingly, the pedestrian number of floor platforms increases (Figure 4). As the pedestrian number on the floor platform increases, the degree congestion of the floor platform increases, and the pedestrian output rate of the floor platform decreases. Thus, the simulation results satisfy the practical evacuation principle.Figure 3
Flow output rate of floor platform.Figure 4
Pedestrian flow floor platform.


### 4.2. Analysis of Stairs Evacuation Capacity

To analyze the stairs evacuation capacity and obtain the optimal stairs merging ratio, different merging ratios are simulated. According to the requirement of building design, the merging ratios are defined as follows: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. Simulating the merging process of different merging ratios, the evacuation time can be described as shown in Figure5.Figure 5
Evacuation time of stairs for the different merging ratios.Figure5 shows the relationship between the evacuation time and the merging ratio, and the relationship between the floor and the stairs can be explored by Equation (5). When the merging ratio is less than 0.5, which represents that the stairs pedestrian flow is prior in the process of stairs evacuation. When the merging ratio is 0.4, the stairs evacuation time is the shortest; when the merging ratio is 0.9, the stairs evacuation time is the longest.To explore the reason that influences the stairs evacuation time, the merging flows time of each floor, the passing rate of the floor channel, the passing rate of the stairs, and the output rate of the floor platform are calculated, as shown in Figure6.Figure 6
Influence of merging ratio for the merging flows time, the passing rate of floor channel, the passing rate of stairs, and the output rate of floor platform for different floors.
(a)
(b)
(c)
(d)Figure6(a) is presented to show the influence of the merging flows time and the merging ratio, which have the shortest evacuation time and the shortest merging flows time. Similarly, the merging ratio, which has the longest evacuation time, possesses the longest merging flows time. The merging flows duration time indicates the congestion degree of stairs. When the merging flows duration time is shorter, the degree of congestion is lower. For any merging ratio, the longer merging flows duration concentrate on the third floor, fourth floor, and fifth floor. Thus, the merging ratio can reduce the merging duration and congestion degree but cannot change the distribution of congestion scale (Figure 6(a)). Figure 6(b) shows the change rule for the pedestrian passing rate of the floor channel. The pedestrian passing rate of the floor channel is related to the stairs evacuation time, and the pedestrian passing rate of floor channel pedestrians increases as the stairs evacuation time decreases. However, the pedestrian passing rate of floor channel pedestrians is influenced by the width of the floor channel and the pass rate of the floor platform. Because the merging ratio is greater than 0.5, the width of the floor channel is greater than the width of the stairs. Although the evacuation time is longer, some floor channels have a high pedestrian output rate. Because the width of floor channel enables more floor pedestrian flow to pass the floor channel, the limitation of the pedestrian output rate of the floor platform causes a decrease in the pedestrian passing rate of the floor channel as the evacuation process develops. Figure 6(c) shows the variation rule of the pedestrian passing rate of stairs and indicates the relationship among the merging ratio, floor, and pedestrian passing rate of stairs. Intuitively, the pedestrian passing rate of stairs for each floor is higher in the hierarchy, and the pedestrian passing rates of stairs on the first and sixth floors are lower. Different merging ratios have different pedestrian passing rates of stairs. When the merging ratio equals 0.4, the pedestrian passing rate of stairs is greatest on each floor, which denotes that the relationship between the stairs evacuation time and the pedestrian passing rate of stairs is positively related. Figure 6(d) illustrates the pedestrian output rate of floor platforms on each floor for different merging ratios. Due to the influence of the number of floor platforms and the limitation of the pedestrian passing rate of stairs, the principle of the pedestrian output rate of the floor platform is similar to the pedestrian passing rate of stairs. According to Equation (23), the evacuation capacity of stairs can be described as in Figure 7Figure7 shows the evacuation capacity of stairs for different merging ratios. The evacuation capacity of stairs is greatest when the merging ratio is 0.4; when the merging ratio is 0.9, the evacuation capacity of stairs is the lowest. When the merging ratio exceeds 0.5, the floor pedestrian flow is slower than the evacuation process, but the evacuation capacity of stairs decreases as the merging ratio increases. When the merging ratio is 0.5, the pedestrian output rate of the floor is equal to the pedestrian passing rate of stairs, and the evacuation capacity of stairs is not optimal. When the merging ratio is 0.4, the evacuation capacity of stairs is optimal. Thus, when the stairs pedestrian flow is preferentially evacuated, the evacuation capacity improves. However, it is not true that the evacuation capacity of stairs becomes more powerful as the priority degree of the stairs pedestrian flow increases. When the merging ratio is 0.1, the evacuation capacity of the stairs is low.Figure 7
Evacuation capacity of stairs for different merging ratios.


## 4.1. Simulation of Stairs Evacuation Process

To qualitatively evaluate the results obtained with the pedestrian merging flows model for evacuation on stairs, the process of stairs evacuation is simulated. The results are shown in Figures3 and 4; Figure 3 shows the pedestrian output rate of the floor platform for different floors; and Figure 4 shows the pedestrian number of floor platforms for different floors. The duration time of the stage before pedestrian merging flows is approximately 14 s; the duration after pedestrian merging flows is approximately 100 s; and the duration of the pedestrian merging flows process is approximately 327 s (Figure 3). The pedestrian output rate of the floor platform decreases with the development of the pedestrian merging flows process, and correspondingly, the pedestrian number of floor platforms increases (Figure 4). As the pedestrian number on the floor platform increases, the degree congestion of the floor platform increases, and the pedestrian output rate of the floor platform decreases. Thus, the simulation results satisfy the practical evacuation principle.Figure 3
Flow output rate of floor platform.Figure 4
Pedestrian flow floor platform.


## 4.2. Analysis of Stairs Evacuation Capacity

To analyze the stairs evacuation capacity and obtain the optimal stairs merging ratio, different merging ratios are simulated. According to the requirement of building design, the merging ratios are defined as follows: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. Simulating the merging process of different merging ratios, the evacuation time can be described as shown in Figure5.Figure 5
Evacuation time of stairs for the different merging ratios.Figure5 shows the relationship between the evacuation time and the merging ratio, and the relationship between the floor and the stairs can be explored by Equation (5). When the merging ratio is less than 0.5, which represents that the stairs pedestrian flow is prior in the process of stairs evacuation. When the merging ratio is 0.4, the stairs evacuation time is the shortest; when the merging ratio is 0.9, the stairs evacuation time is the longest.To explore the reason that influences the stairs evacuation time, the merging flows time of each floor, the passing rate of the floor channel, the passing rate of the stairs, and the output rate of the floor platform are calculated, as shown in Figure6.Figure 6
Influence of merging ratio for the merging flows time, the passing rate of floor channel, the passing rate of stairs, and the output rate of floor platform for different floors.
(a)
(b)
(c)
(d)Figure6(a) is presented to show the influence of the merging flows time and the merging ratio, which have the shortest evacuation time and the shortest merging flows time. Similarly, the merging ratio, which has the longest evacuation time, possesses the longest merging flows time. The merging flows duration time indicates the congestion degree of stairs. When the merging flows duration time is shorter, the degree of congestion is lower. For any merging ratio, the longer merging flows duration concentrate on the third floor, fourth floor, and fifth floor. Thus, the merging ratio can reduce the merging duration and congestion degree but cannot change the distribution of congestion scale (Figure 6(a)). Figure 6(b) shows the change rule for the pedestrian passing rate of the floor channel. The pedestrian passing rate of the floor channel is related to the stairs evacuation time, and the pedestrian passing rate of floor channel pedestrians increases as the stairs evacuation time decreases. However, the pedestrian passing rate of floor channel pedestrians is influenced by the width of the floor channel and the pass rate of the floor platform. Because the merging ratio is greater than 0.5, the width of the floor channel is greater than the width of the stairs. Although the evacuation time is longer, some floor channels have a high pedestrian output rate. Because the width of floor channel enables more floor pedestrian flow to pass the floor channel, the limitation of the pedestrian output rate of the floor platform causes a decrease in the pedestrian passing rate of the floor channel as the evacuation process develops. Figure 6(c) shows the variation rule of the pedestrian passing rate of stairs and indicates the relationship among the merging ratio, floor, and pedestrian passing rate of stairs. Intuitively, the pedestrian passing rate of stairs for each floor is higher in the hierarchy, and the pedestrian passing rates of stairs on the first and sixth floors are lower. Different merging ratios have different pedestrian passing rates of stairs. When the merging ratio equals 0.4, the pedestrian passing rate of stairs is greatest on each floor, which denotes that the relationship between the stairs evacuation time and the pedestrian passing rate of stairs is positively related. Figure 6(d) illustrates the pedestrian output rate of floor platforms on each floor for different merging ratios. Due to the influence of the number of floor platforms and the limitation of the pedestrian passing rate of stairs, the principle of the pedestrian output rate of the floor platform is similar to the pedestrian passing rate of stairs. According to Equation (23), the evacuation capacity of stairs can be described as in Figure 7Figure7 shows the evacuation capacity of stairs for different merging ratios. The evacuation capacity of stairs is greatest when the merging ratio is 0.4; when the merging ratio is 0.9, the evacuation capacity of stairs is the lowest. When the merging ratio exceeds 0.5, the floor pedestrian flow is slower than the evacuation process, but the evacuation capacity of stairs decreases as the merging ratio increases. When the merging ratio is 0.5, the pedestrian output rate of the floor is equal to the pedestrian passing rate of stairs, and the evacuation capacity of stairs is not optimal. When the merging ratio is 0.4, the evacuation capacity of stairs is optimal. Thus, when the stairs pedestrian flow is preferentially evacuated, the evacuation capacity improves. However, it is not true that the evacuation capacity of stairs becomes more powerful as the priority degree of the stairs pedestrian flow increases. When the merging ratio is 0.1, the evacuation capacity of the stairs is low.Figure 7
Evacuation capacity of stairs for different merging ratios.


## 5. Discussion

For the majority of stair configurations and population types, evacuation models tend to produce approximately a 1 : 1 merging ratio when representing evacuation on stairs. The results derive from some of the evacuation experiments: the merging ratio is related to the configurations of stairs and doors. This paper introduces and analyzes the study of merging flows in buildings. Different merging ratios are adopted to simulate different stairs evacuation situations. The scope of this research is not limited to analyzing the stairs evacuation process by simulation. Conversely, the paper provides and exemplifies a methodological contribution. For this reason, a pedestrian merging model, which represents the impact of different merging ratios on evacuation, is significant.A visual case is adopted to illustrate the evacuation capacity of stairs for different merging ratios. By analyzing the simulation results, we can determine that the merging ratio influences the stairs evacuation time. In this paper, the merging ratio is defined by altering the width of the floor channel and stairs. Thus, the pedestrian passing rate of the floor channel and stairs can constantly change as the merging ratio changes. When the merging ratio is equal to 0.4, the stairs evacuation capacity is optimal. In general, 0.5 is regarded as the optimal merging ratio [2], but the pedestrian output rate of the floor platform is calculated by the pedestrian number of floor platforms. Appropriately increasing the width of the stairs or decreases the width of the floor channel can improve the pedestrian output rate of the floor platform. The pedestrian output cardinal of the floor platform increases by increasing the width of the stairs. Decreasing the width of the floor channel can limit the floor pedestrian flow and cause the number of floor platforms to decrease and the pedestrian output rate to increase. When the merging ratio increases, the stairs evacuation capacity decreases. However, the width of the stairs or the width of the floor channel cannot infinitely decrease. When the merging ratio is less than 0.4, the stairs evacuation capacity gradually decreases.In the novel evacuation model, the pedestrian merging flows process is adopted to analyze how the merging process influences the stairs evacuation capacity. Although the total evacuation time of pedestrians is not directly affected by pedestrian merging flows on stairs, the calculation of the evacuation times for each individual floor depends on the pedestrian merging flows; thus, many researchers focused on the pedestrian merging flows in a stairs evacuation [22]. In general, experimental research was the most common method for exploring pedestrian merging behavior, and the merging flows area was defined to analyze the human behavior in a real staircase [36]. These experiments provided sufficient data of pedestrian merging flows to research merging behavior, but could not explain and analyze why pedestrian merging flows affect stairs evacuation. However, the study offered specific suggestions to decrease the influence of pedestrian merging flows. The simulation method was adopted to express the pedestrian merging flows process during stairs evacuation. Unlike the classical evacuation model [37, 38], the proposed evacuation model considered the passing rates of stairs and floors and employed the output rate to calculate the stairs evacuation time.The merging ratio was usually adopted in any merging flows process. Some novel simplified mathematical models for the calculation of stairs evacuation times take into account the impact of the merging ratio. The impact of the merging ratio on pedestrian flows and evacuation times was calculated for each floor in congested situations [13, 25]. Important conditions of these models was that the steady state conditions of pedestrian flows were considered, and the staircase was filled with pedestrians at the same density as the constant conditions at the start [32, 39]. In this paper, the entire stairs evacuation process was simulated to analyze the impact of pedestrian merging. The merging flows process was regarded as the criterion to divide the stairs evacuation process into three steps. Unlike this model, the proposed model expressed a dynamic evacuation process, in which the pedestrian passing rate of stairs and floors was variable over time. Based on the queuing behavior of the floor platform and the merging behavior, the pedestrian output rate of the floor platform was changed as the merging process proceeded. The number of pedestrian flows can influence the output rate, and the output rate decreases as the number of pedestrian flow increase [38, 39]. Thus, a probabilistic approach was adopted to calculate the pedestrian output rate of the floor platform, which expresses the variable process of the pedestrian output rate of the floor platform in the stairs evacuation. With a change in the pedestrian output rate of the floor platform, the pedestrian output rate of stairs and floors changed, and the stairs evacuation process attained a dynamic balance by the interactions among the floor platforms, stairs and floor channels. The model provided a straightforward function to express the stairs evacuation capacity by a quantificational approach.


## 6. Conclusions

This paper introduces a novel stairs evacuation model for the calculation of evacuation times considering the impact of merging ratios in stairs. The main contributions of the paper are summarized as follows:(1)
The M/G/c/c proposed in the queuing theory was adopted to calculate the pedestrian output rate of the floor platform, which was applied to obtain the stairs evacuation time. The stairs evacuation process was divided into three steps according to the pedestrian merging. The merging flows process was a dynamic feedback mechanism, and the interaction among the stairs, floor channels and floor platforms caused the evacuation process to attain a balanced state. The stairs evacuation capacity was computed by the evacuation time and the number of pedestrians on each floor.
(2)
A case was introduced to verify the reliability and feasibility of the model and analyze the evacuation capacity of different merging ratios. The simulation results have been compared with existing evacuation data obtained from the evacuation drill in the building introduced in the case study. The results show that the model results are similar to the practical evacuation results. This model identifies the impact of pedestrian merging flows on evacuation times for each floor in a simpler and explicit way. The optimal merging ratio is computed by simulating different merging flows scenarios.Other evacuation behaviors can be incorporated in the model to express the stairs pedestrian merging flows process, such as adopting an automatic evacuation decision mechanism that describes the pedestrian behaviors in the pedestrian merging flows process. Many factors influence the pedestrian merging flows process, and the interactions of factors determined the complex of pedestrian merging flows. Thus, a novel approach that considers the interactions of factors will be explored to describe the pedestrian merging flows process.



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*Source: 7615479-2019-12-10.xml*