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