Post on 03-Feb-2020
Stationary Flow Approximation In Floor-FieldModels Using Markov-Chain Theory
Pavel Hrabak and Frantisek Gaspar
Czech Technical University in Prague
MATTS 2018Thursday October 18, 2018
P. Hrabak and F. Gaspar (CTU) date 1 / 25
What is it about?
P. Hrabak and F. Gaspar (CTU) date 2 / 25
Motivation
Outline
1 Motivation
2 Cellular model as Markov Chain
3 Our contribution
4 How was it done?
5 Conclusions
6 Bonus – Heterogeneity
P. Hrabak and F. Gaspar (CTU) date 3 / 25
Motivation
Cellular evacuation model
Lattice consisting of cells 40× 40 cm.How to model exit of the width 75 cm?Two cells representing 80 cm are “too much”.
P. Hrabak and F. Gaspar (CTU) date 4 / 25
Motivation
Not the width, but maximal flow
In Seyfried et al. 2009 linear dependence J = 1.9 · width suggested.Same bottleneck width, different flow value.
Key feature is the “maximal flow through the bottleneck”Measured experimentally or estimated by some linear model
J = 1.9 · width + β2 · property2 + β3 · property3 + . . .
How to adjust the cellular model to produce wanted bottleneck flow?
P. Hrabak and F. Gaspar (CTU) date 5 / 25
Motivation
Locally dependent friction
Locally dependent “friction parameter”.
Wanted equation saying:“For this flow value choose this value of the friction parameter”
Finding implicit equations of the form
J = J(parameters)
How?
P. Hrabak and F. Gaspar (CTU) date 6 / 25
Cellular model as Markov Chain
Outline
1 Motivation
2 Cellular model as Markov Chain
3 Our contribution
4 How was it done?
5 Conclusions
6 Bonus – Heterogeneity
P. Hrabak and F. Gaspar (CTU) date 7 / 25
Cellular model as Markov Chain
Floor-field modelsFloor-field conception
Static floor-field
Sb = dist(b,bexit)
Probabilistic choice of next cell
Pr(bact → b) ∝ exp{−kS · Sb}
Conflicts and frictionConflict: k agents to one cell.
No one wins with prob. φ(ζ, k).
Cellular lattice
exit
b0
b1
b4
b3
b2
S0
S1
S4
S3
S2
0
P. Hrabak and F. Gaspar (CTU) date 8 / 25
Cellular model as Markov Chain
Flow approximation by Ezaki et. al 2012T. Ezaki, D. Yanagisawa, and K. Nishinari, 2012
They investigated 25× 25 cells lattice regarding the stationary flowMaximal bottleneck flow approximated by means of reduced state space
Markov chain with 24 states having stationary distribution π, thus
J ≈∑
s
jsπs
P. Hrabak and F. Gaspar (CTU) date 9 / 25
Cellular model as Markov Chain
Assumptions of the approximation
kS ≈ +∞, thus almost deterministic motionSurrounding cells always occupied in congestionResult: “it works well”
1
1Taken from T. Ezaki, D. Yanagisawa, and K. Nishinari, 2012P. Hrabak and F. Gaspar (CTU) date 10 / 25
Our contribution
Outline
1 Motivation
2 Cellular model as Markov Chain
3 Our contribution
4 How was it done?
5 Conclusions
6 Bonus – Heterogeneity
P. Hrabak and F. Gaspar (CTU) date 11 / 25
Our contribution
Questions to be answered by our research
Does this work for general kS ∈ (0,+∞)?
Does this work for multiple-cell wide bottlenecks?
Is the assumption of always occupied surrounding cells valid?
P. Hrabak and F. Gaspar (CTU) date 12 / 25
Our contribution
General kS
P. Hrabak and F. Gaspar (CTU) date 13 / 25
Our contribution
Multiple-cell exits
P. Hrabak and F. Gaspar (CTU) date 14 / 25
Our contribution
Assumption of always occupied surrounding cells
Simulation of the 25× 25 cells latticeTheoretical result with r0 ∈ [0,1], probability of the nearest cell beingempty.
P. Hrabak and F. Gaspar (CTU) date 15 / 25
Our contribution
Flow depending on friction – main result
P. Hrabak and F. Gaspar (CTU) date 16 / 25
How was it done?
Outline
1 Motivation
2 Cellular model as Markov Chain
3 Our contribution
4 How was it done?
5 Conclusions
6 Bonus – Heterogeneity
P. Hrabak and F. Gaspar (CTU) date 17 / 25
How was it done?
Building the transition matrix P algorithmically
P. Hrabak and F. Gaspar (CTU) date 18 / 25
How was it done?
Stationary flow calculation
Stationary distribution satisfying
π = π · P
Due to numerical issues
π = p(0) · limn→+∞
Pn ≈ p(0) · Pn0
The flow calculation
J ≈∑
s
js · πs , js = E(outflowing agents | s)
P. Hrabak and F. Gaspar (CTU) date 19 / 25
Conclusions
Outline
1 Motivation
2 Cellular model as Markov Chain
3 Our contribution
4 How was it done?
5 Conclusions
6 Bonus – Heterogeneity
P. Hrabak and F. Gaspar (CTU) date 20 / 25
Conclusions
Results
Algorithm for building the transition matrix (MATLAB)Symbolic matricesFlow depending on frinction ζ
Valid rather for lower values of ζ
P. Hrabak and F. Gaspar (CTU) date 21 / 25
Bonus – Heterogeneity
Outline
1 Motivation
2 Cellular model as Markov Chain
3 Our contribution
4 How was it done?
5 Conclusions
6 Bonus – Heterogeneity
P. Hrabak and F. Gaspar (CTU) date 22 / 25
Bonus – Heterogeneity
Heterogeneity
H ≥ 1 types of agentsVarious attraction strength to the exit
kS = (kS;1, . . . , kS;H)
Various friction abilityζ = (ζ1, . . . , ζH)
Generalized friction functionφ(ζ, k)
AggressivenessA = (A1, . . . ,AH)
Pr(j ∈ {1, . . . , k}wins) ∝ Aij
Input distributionβ = (β1, . . . , βH)
P. Hrabak and F. Gaspar (CTU) date 23 / 25
Bonus – Heterogeneity
Heterogeneity
Problem: how to determine the distribution of nearest cell occupation
r = (r1, . . . , rH) .
Substituted from the simulation =⇒ approximation works well
P. Hrabak and F. Gaspar (CTU) date 24 / 25
Bonus – Heterogeneity
Thank you for your attention!
P. Hrabak and F. Gaspar (CTU) date 25 / 25