J. Cent. South Univ. (2013) 20: 837-844 DOI: 10.1007/s1177101315552
Probabilistic model for remain passenger queues at subway station platform XU Xinyue(许心越) 1,2 , LIU Jun(刘军) 2 , LI Haiying(李海鹰) 1 , ZHOU Yanfang(周艳芳) 2 1. State Key Lab of Rail Traffic Control & Safety, Beijing Jiaotong University, Beijing 100044, China; 2. School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China © Central South University Press and SpringerVerlag Berlin Heidelberg 2013 Abstract: The remain passenger problem at subway station platform was defined initially, and the period variation of remain passenger queues at platform was investigated through arriving and boarding analyses. Taking remain passenger queues at platform as dynamic stochastic process, a new probabilistic queuing method was developed based on probabilistic theory and discrete time Markov chain theory. This model can calculate remain passenger queues while considering different directions. Considering the stable or variable train arriving period and different platform crossing types, a series of model deformation research was carried out. The probabilistic approach allows to capture the cyclic behavior of queues, measures the uncertainty of a queue state prediction by computing the evolution of its probability in time, and gives any temporal distribution of the arrivals. Compared with the actual data, the deviation of experimental results is less than 20%, which shows the efficiency of probabilistic approach clearly. Key words: subway; platform; remain passenger; queuing theory; probabilistic theory; Markov chain
1 Introduction Congestion and pollution problems in China’s large cities made the construction of the subway system necessary. Many subway stations experience very high levels of pedestrian density especially at platform. Indeed, several statistical analyses of accident data were performed in China, Europe and United States of America, such as those in Refs. [1–4], and showed that a lot of injuries occur during the travelers’ boarding and alighting and that these injuries are closely linked to the design of station platform [5–6]. Moreover, transit platform have critical passenger holding capacities, which if exceeded, could result in passengers being pushed onto tracks. Because of the importance, lots of papers researched the platform problem. A number of studies focused on alighting and boarding flows of passengers. The relationships between the dwelling time of trains and the crowding situations at Light Rail Transit (LRT) stations in Hong Kong are firstly determined, and regression models are established for the dwelling delays of train by WILLIAM et al [7]. Research on alighting and boarding times at Dutch railway stations by DAAMEN and HOOGENDOORN [8] was done focusing on the dwell time of trains. Factors assessed included passengers’ distribution on the platform, alighting and boarding times, station type, type of train
service, vehicle characteristics and period of day. Measurements of boarding and alighting times for different train types were researched by HEINZ and ANDERSON [9]. The time characteristics of boarding passengers were analyzed and the piecewise mathematical model for average boarding time based on the field data of passenger boarding time was presented by CAO and YUAN [10]. Moreover, inflow of passengers was also studied. Arrival was considered as the continuous and steady progress and can be assumed to follow a Poisson distribution by ÖZGÜR and MIRAC [11] and RODRIGO [12]. But little interest has been given to the change law of the passenger at platform. Remain passenger problem at subway station platform was defined initially in this work, and a new probabilistic approach was presented based on probabilistic theory and discrete time Markov chain theory. The probabilistic approach could give a theoretically quantitative prediction for remain passenger queue length in the end of each cycle time, which can be used for designing the solution for passenger flow organization.
2 Problem defining and notations 2.1 Basic concepts and definitions By observing the travelers’ distribution on a platform, passengers could be classified as inflow, outflow, alighting passenger and boarding passenger, as
Foundation item: Project(2011BAG01B01) supported by the Major State Basic Research and Development Program of China; Project(RCS2012ZZ002) supported by the State Key Lab of Rail Traffic Control and Safety, China Received date: 2012–03–01; Accepted date: 2012–05–23 Corresponding author: XU Xinyue, PhD; Tel: +86–15011516578; Email:
[email protected]
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shown in Fig. 1. The train runs by specially cycle, but passengers can only alight and board during dwell time, that is: TDk = tk¢ - t k
(1)
where k represents cycle number, TDk is dwell time duration, tk ¢ is departure time at kth cycle and tk is arrival time at kth cycle.
Fig. 1 Passenger flow between platform and train
The cycle starts from the departure time of a train and ends to the departure time of the successive train in the same direction. Figure 2 shows the description of the cycle. Obviously, Tk = tk¢ +1 - tk¢ = TSk + TDk , TSk = tk +1 - tk ¢
(2)
where Tk is cycle time of kth cycle; TSk is separation time duration. Fig. 3 Relationship between arrivals, boarding passengers, queue length and total delay in oversaturated case and with undersaturated conditions and positive initial queue length with uniform arrivals and departures: (a) Cumulative arrivals and number of boarding passengers in oversaturated case; (b) Cumulative arrivals and number of boarding passengers in undersaturated case with non zero initial queue
Fig. 2 Definition of cycle and phase
The phenomenon, which the passenger arriving into platform will not be able to depart (board) within the same cycle and should wait for the next, can be defined as remain passenger problem. Obviously, remain passenger problem in subway has directions according to train operation direction and the similar process in different directions. Considering a singletrack remain passenger problem of one platform, if the platform is oversaturated (Fig. 3(a)), i.e. there are more passenger arrivals than maximum boarding capacity (depending largely on train idle capacity and boarding rate), the train arriving in the cycle will not be able to depart within the same cycle (remain queue). A remain queue may be observed also if an undersaturated period follows an oversaturated one and
a nonzero initial queue is assumed at the start of the cycle (Fig. 3(b)). Figure 3 shows the dynamic behavior of queues when constant arrival and boarding rate are assumed. If one assumes perfect uniform arrivals and boarding within one cycle, respectively, at rates a and b, the expectation value of the queue length in a cycle will increase linearly with a during the separation phase and if a 0 . Equations (6) and (8) can be rewritten by b max
pij = pij (1) =
å pij (t, b1 )P(b1 )
(11)
b 1 = 0 Q max
P (Q0 = j , k ) =
å P (Q0 = i, k - 1) × pij
(12)
i = 0
Figure 5 shows an example of queue transition
Fig. 6 Evolution of queue length probabilities for x=0.975
4.2 Variable cycle The train runs in fixed headway in normal situation,
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but the headway varies when an emergency occurs. We presume that the dwell time is fixed and the separation time is changed. In fact, the dwell time can’t be changed given specially passengers, but separation time can be changed by train speed. From previous studies [10, 12, 14], we consider the deviating time distribution from the fixed headway as a normal distribution, as shown by
(THk - TH ) ~ N (0, e )
(13)
where T H represents the average headway time, e represents the deviating. As shown in previous sections, apart from these exceptions, it is a generally accepted hypothesis to consider the arrivals at an isolated platform within a cycle following a Poisson process [10–11, 14]. In practice, the demand is subdivided into periods of stationary conditions, in which the average arrivals do not change significantly from each other. According to the definition of Poisson distribution, this average value represents the only parameter, which defines its shape. To sum up, Eqs. (3)–(5) can be rewritten by b max
pij ( k ) =
å pak ( j + bk - i) × pb (bk )
(14)
b k = 0
ìb max -i ï å pak ( ak £ n), "i £ bk , ak Î [0, a max ] pi 0 = í b k =0 ï î 0, Otherwise
piQ max
Fig. 7 Transition matrix for variable cycle at different cycles
(15)
ì b max ï å pak (ak ³ Qmax - i + bk }, "Qmax - i + bk £ a max = íb k =0 ï î 0, Otherwise (16)
where j≠0, 0≤j+bk–i≤Qmax, bk≤Amax, pak(x) represents the probabilistic of arrivals, and pb(x) represents the probabilistic of departures. Figure 7 shows an example of different queue transition probabilities for variable cycle in a saturated case (l=2.4, x = 1.1). Obviously, different transition probabilities are achieved for different cycles. If the queue is zero, there is over 99% chance that it will remain non zero at the following cycle, which verified that remain passenger queue inevitable happened when arrivals is larger than departures. 4.3 Different crossing types of platform There are two crossing types of platforms in subway station, namely side and island. Given one platform for different types, the method of calculating remain passenger is different. The flag u represents the up direction of trains and the flag d represents the down direction of trains.
1) Side platform. For the type, the trains of double directions arrive at different platforms and the train at one platform has single direction, so remain passenger queues can be directly calculated by the probabilistic singletrack queuing model. 2) Island platform. There are two remain passenger queues on the platform, which share the same space of platform. Some specific behaviors weren’t taken into account such as a passenger who decides to board again into the public transport because of alighting by mistake. The platform can be assumed by the up and down separately, independently of each other, as shown in Fig. 8.
Fig. 8 Schematic diagram of platform used separately
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So, the total remain passenger queues in any time t at platform can be calculated by d
u
Qmax
E[Q0 (t )] =
å
Q max
d
j × P (Q0 = j , k ) +
j =0
å j × P(Q0 =
u
j , k )
j = 0
(17) where kd
k d +1
ku
k u +1
å Tkdd £ t
