Capacity of subway station is an important factor to ensure the safety and improve the transportation efficiency. In this paper, based on the M/G/C/C statedependent queuing model, a probabilistic selection optimization model is proposed to assess the capacity of the station. The goal of the model is to maximize the output rate of the station, and the decision variables of the model are the selection results of the passengers. Finally, this paper takes a subway station of Shanghai Metro as a case study and calculates the optimal selection probability. The proposed model could be used to analyze the average waiting time, congestion probability, and other evaluation indexes; at the same time, it verifies the validity and practicability of the model.
Subway station plays an important role in gathering and switching passengers. Especially during peak hours, passengers may occupy the station in high densities, which has become normal phenomenon recently. As we known, high density may cause unpredictable accident, such as stampede. Therefore, how to define and calculate the capacity of subway station (CSS) has attracted more and more scholars’ attention.
Over the past few decades, there have been many literatures related to CSS. The research can be classified into two categories according to the method: simulation and mathematic model. Simulation is used to describe passenger’s route choice behavior under different scenarios. For example, Kaakai et al. [
Mathematic model is usually designed from macroscopic aspect. Most of the existing literatures consider inner structure of subway station as a crucial factor to the capacity. For example, Xu [
Thus, in order to analyze the capacity of subway station, the relationship between passenger route choice and inner structure of subway station was firstly determined. This paper presents a novel definition of CSS and a queuing network based optimization model was proposed. Finally, a case study using real data from Shanghai Metro is developed to verify the validity and feasibility of this method.
The rest of this paper is organized as follows. Section
Currently, there is no unified definition of CSS. Different organizations have different interpretations. For example, CSS defined by Transportation Research Board (TRB) of United States is “the maximum number of passengers passing or occupying the equipment under normal circumstances” [
As we know, CSS is related to three subprocesses: inbound process, outbound process, and alighting/boarding process (as shown in Figure
Inbound and outbound process at the subway station.
The number of passengers that can be served by subway station contains two categories: inbound passengers and outbound passengers. Therefore, the number of passengers passing through the subway station in a unit time
When travelling in the station, passengers will be affected by the layout of facilities, passenger flow organization, station capacity, and other factors. Therefore, it is necessary to clarify the various constraints. First of all, the number of passengers passing per unit time cannot exceed the station’s demand.
Therefore, this paper defined CSS as the maximum inbound passengers that can be served by one subway station in a given time period without any unpredicted incident such as train delay and fire.
To facilitate problem formulation, we make the following assumptions:
Alightingboarding process in the station obeys “firstalightingthenboarding” principle.
There are inbound passengers at stairs, escalators, and other facilities as an M/G/C/C state
There is no passenger control strategy in the station.
All passengers could board the train which is arriving at the platform.
The order of inbound process must be followed.
According to the definition of CSS in the previous section, it can be seen that CSS is related to inner structure and arrangement of the facilities. Therefore, subway station is modeled as a queuing network system that contains
The queuing network structure of passenger inbound process.
In addition, according to different levels in the subway station, this queuing network model can be divided into several submodels. For example, as shown in Figure
In this paper, a traditional single queuing model is extended to a network queuing model. Firstly, topology of the subway station is modeled. Then, connection structure among the fare gates, escalators, stairs, corridors, platforms, and other facilities is established.
The process between entering the fare gate and boarding the train is called delivery process. Passengers in this process will go through several levels of selection process. For example, passengers can choose stairs, escalators, or elevator after entering the station. After they arrive at the platform, they could choose waiting area 1 or 2 and so on.
According to assumption 5, passengers will not return to the station hall from platform. The sum of output selection probabilities of each facility is 1; for example, the sum of the probabilities of selecting facilities B1, B2, B3, and B4 from A1 in level A is 1. This can be formulated as follows:
For any facility in the network structure model, during the process of walking or queuing, the speed of the passenger is affected by the gathering density. Therefore, this paper uses the statedependent M/G/C/C queuing model to describe the queuing process of the nodes of the facilities. The probability of having
In crowded conditions, the relationship between passenger speed and density is closely related to the factors such as travel destination, personal attributes, and travel time. In general, the greater the passenger flow density, the smaller the speed of movement. When the density reaches a certain limit, the passenger flow will not be able to move. This paper has analyzed the video data of the Shanghai subway station; it is found that pedestrian moving speeddensity feature is similar to the classical BPR model in road traffic. Therefore, this paper uses the exponential speeddensity function to describe the data:
In this formula,
For any facility in the network structure diagram, when its service number exceeds the system service capacity, congestion will occur. And the probability of the occurrence can be calculated by using the following formula:
Considering the service strength of each facility in the network, the number of passengers expected to be served at any node must be within a reasonable range:
In this model, formula (
The proposed case study is a typical station (QiBao station) in Shanghai subway system located in line 9. Figure
The arrival rate of passengers entering QiBao station in peak hours.
Facility number  The number of people entering the station  Arrival rate  

Fare gate for entrance 1  G101  992  1.3 persons/s 
G102  820  
G103  982  
G104  921  
G105  991  


Fare gate for entrance 2  G201  753  0.91 persons/s 
G202  474  
G203  692  
G204  681  
G205  691  


Fare gate for entrance 3  G301  873  1.00 person/s 
G302  621  
G303  691  
G304  721  
G305  711 
The layout of the station hall in QiBao station.
The layout of the platform in QiBao station.
Figure
Geometric dimensions of the virtual corridors.
Virtual corridor  Length (m)  Width (m) 

Corridor 1  10  1 
Corridor 2  8  1 
Corridor 3  12  1 
Corridor 4  12  1 
Corridor 5  8  1 
Corridor 6  8  1 
Corridor 7  5  1 
Corridor 8  5  1 
Corridor 9  5  1 
Corridor 10  5  1 
Corridor 11  12  1 
Corridor 12  12  1 
Corridor 13  9  1 
Corridor 14  9  1 
Corridor 15  14  1 
Corridor 16  14  1 
Corridor 17  6  1 
Corridor 18  6  1 
Queuing network of QiBao station.
According to Table
Optimal selection probability.
Selection probability  Value 

A1 
0.231 
A1 
0.535 
A1 
0.234 
A2 
0.212 
A2 
0.637 
A2 
0.151 
A3 
0.231 
A3 
0.548 
A3 
0.221 
B1 
0.261 
B1 
0.101 
B1 
0.182 
B1 
0.11 
B1 
0.09 
B1 
0.256 
B2 
0.212 
B2 
0.15 
B2 
0.13 
B2 
0.15 
B2 
0.147 
B2 
0.211 
B3 
0.198 
B3 
0.158 
B3 
0.143 
B3 
0.149 
B3 
0.154 
B3 
0.198 
It can be seen from Table
Calculation of facilities and equipment.
Facility  Average waiting time  Congestion probability 

Fare gate for entry 1  2.066  0.195 
Fare gate for entry 2  1.508  0.184 
Fare gate for entry 3  1.908  0.167 
Escalator  7.188  0.553 
Stairs  3.098  0.350 
Elevator  5.331  0.401 
Corridor 1  1.501  0.095 
Corridor 2  1.503  0.090 
Corridor 3  1.511  0.085 
Corridor 4  1.608  0.075 
Corridor 5  1.548  0.070 
Corridor 6  1.118  0.077 
Corridor 7  1.981  0.074 
Corridor 8  2.011  0.073 
Corridor 9  1.999  0.078 
Corridor 10  2.113  0.090 
Corridor 11  1.430  0.091 
Corridor 12  1.406  0.099 
Corridor 13  1.876  0.089 
Corridor 14  1.770  0.088 
Corridor 15  1.734  0.081 
Corridor 16  1.777  0.099 
Corridor 17  1.234  0.093 
Corridor 18  1.654  0.086 
Based on the analysis of the characteristics of passenger receiving service from equipment in subway station, this paper constructs a network queuing model based on M/G/C/C. The model takes the maximum output rate of the station as the objective function, and the decision variable is the probability of selection of the service flow line by passengers. At the same time, it takes a case analysis of the Shanghai subway station as the background and calculates the optimal output selection probability of each station. Evaluation criteria such as average waiting time and congestion probability are analyzed, and the validity and practicability of the model are verified.
The study found that stairs and escalators are still the “bottlenecks” which determine the carrying capacity of subway stations. Adjusting the protection of flow line of the escalators, the stairs can be a good way to reduce the probability of equipment congestion and can reasonably cope with the impact of the tide of high passenger flow on the station. The proposed network model can provide an effective means for quantitative analysis of station carrying capacity. But if it is applied to the actual operation and management, it is necessary to study its matching with the line transport capacity.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is jointly supported by the National Key Research and Development Plan of China (Grant no. 2017YFC0804903), Scientific Research Foundation for Doctors in Shanghai University of Engineering Science (201611), Young Teachers Training Funding of Universities in Shanghai (ZZGCD15114), and National Natural Science Foundation of China (Grant no. 71601110).