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Problem A
Unloading Commuter Trains
Main details
• Each Train Car Has Only Two Exits, Each
Train Car Has A Center Aisle And There
Are Two Seats On One Side And Three
Seats On The Other For Each Row Of
Seats
• Exit The Car, And Then Make Their Way
To A Stairway, The Stairway Could
Accommodate Two Columns Of People
Main details
• Assume There Are N Cars To A Train, Each Car
Has Length D. The Length Of The Platform Is P,
And The Number Of Stairs In Each Staircase Is
Q.
• Final goal: Specifically Optimize (Minimize)
The Time Traveled To Reach Street Level To
Exit A Station. Estimate The Amount Of Time
For A Passenger To Reach The Street Level Of
The Station To Exit The Complex .
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SIMPLY OPTIMIZATION
min f(x)
s.t
the object function
the constrints
Solution:
according to the knowledge of differential and integral calculus
f
=0 (i=1,2, --- n)
 xi
Make some equotions and find the solutions. The point we get is
called the stationary point , generally It is the optimal solution.
Simulation graphics
N CARS
P
Q
D
three problems to focus on
1. find out the variables of problem
ex: the number and the place of the
stairways
2
stairway
1
2. Sseveral assumptions for the
practical problems
(1)the macro model
divide the people to several blocks
instead of study the individuals
ONE UNIT BLOCK
TOTAL D BLOCKS
Ignore the influence of each other in one block
Main parameter:average velocity,rate of flow
(2)the microcosmic model
ex: (1)normal walking speed t~（0.9，1.5）
(2)
5
d
D
set k rows each car
so the tatol number of the pessengers is 5*k
per car
(3)how much people can stand in one
square metre ,there are too many parameters to
consider like frictional force and even
psychological factors
3. set the goal
min f(X)= f1 ( x)  f 2 ( x) 
·········
The first part is to get out of the car
detail: the width of door and the location of the
passenger
if the platform is big enough for all the passengers
in the car then all of them will be waiting on the
platform for getting on the stairway ········to the
second part
The second part is to get on the stairway
platform
stairway
The third part is to get out of the stair that is
also the time it takes to go through the stairway
Reference model
• (1) Social Force model
One of the most challenging tasks in computer
vision is analysis of human activity in crowded
scenes. In addition, research in sociology and
behavioral sciences provide mathematical
models of pedestrian behavior patterns such
as Social Force Model. In this paper, we
introduce a computer vision method based on
particle advection to detect and localize
abnormal crowd behavior using the Social
Force model.
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Social Force model
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• In this model, the individual
dynamics of pedestrians is modeled
as:
Here t is just like the resistance
13
Considering the effect of panic
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M/M/c model
In queueing theory, a discipline within the
mathematical theory of probability, the M/M/c
queue (or Erlang–C model[1]:495) is a multiserver queueing model.[2] In Kendall's notation
it describes a system where arrivals form a
single queue and are governed by a Poisson
process, there are c servers and job service
times are exponentially distributed It is a
generalisation of the M/M/1 queue which
considers only a single server. The model with
infinitely many servers is the M/M/∞ queue.
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Thank you!
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