Fire Exits

Deddy Koesrindartoto

Iowa State University

 

Matt Golder

New York University

 

 

Objective

 

20 people in a room.  Fire breaks out.  Model it!

 

Questions One Might Ask:

  1. Where is the optimal place to locate an exit?
  2. How does the number of exits affect survivability?
  3. How does the structure of the room affect survivability?
  4. How does the speed with which the fire spreads affect survivability?
  5. How does the location of the fire affect survivability?

These are just some of the questions one might be interested in modeling.

 

Basic Model

 

Setup:

  • Room is an N x K grid
  • People are arranged randomly to the squares of the grid.
  • There can be only one person in any single square of the grid
  • Fire starts in a randomly chosen square of the grid
  • A single exit is placed somewhere on the perimeter of the grid.

 

Allowable Moves:

1.      Fire

    • Moves in four directions (up, down, left, right) by one square each time it gets to move.  This is to capture the intuition that fires can spread in multiple directions at once.
    • Thus, if the fire is located centrally in the room at some coordinates (n,k) at time t, then at time t+1 it will be located at (n,k), (n-1,k), (n+1,k), (n, k+1) and (n,k-1).
    • The fire cannot spread beyond the room

2.      People

    • People can move in any one of four directions (up, down, left, right) so long as the square they are moving to is empty.
    • They employ the following decision rule

1.      Move to empty square as close to exit as possible

2.      If there are two empty squares equally close, then choose randomly

3.      If the square closest to exit is occupied, move to the empty square that is furthest from the fire

4.      If there are two square equally distant from the fire, then choose randomly

5.      If there are no empty squares that satisfy these criteria, then do not move

    • The situation that we tried to capture is the one in which the people know where the exit and the fire is.  We also wanted to capture the intuition that if people could not reach the exit, they would move away from the fire in an attempt to prolong their survival (perhaps rescue would come).

 

Sequence of Moves:

  • People get to move first.  They move in a particular sequence.  The person closest to the door moves first, then the second closest, then the third closest etc..  If there are two people equally close then the person to move is chosen randomly.  Each agent gets a chance to move.  This is repeated x number of times.  In other words, each agents gets the opportunity to move x times.
  • Fire gets to move after the people have moved x times.  One can think of x as determining the speed with which the fire spreads relative to the speed with which the people can move.  This is a parameter that we vary in our simulations.
  • This sequencing is then repeated until there are no people in the room or the fire has occupied the whole room.

 

Death and Survival Rules:

  • If the fire moves to a square occupied by a person, that person dies.
  • If a person moves to the exit, that person survives.

    • Research Design

     

    Question:

    We are interested in issues of institutional design.  We address two questions.

     

    1. Where should we place the exit to maximize survivability?  Specifically, does placing the exit in the corner or in the middle of one of the walls increase the survival rate?
    2. How does the size of the room relative to the number of people in the room affect the survival rate?

     

    Hypotheses:

     

    1. Survival rate is higher when the exit is located in the middle of the wall instead of the corner
    2. The larger the room relative to the number of people, the greater the survival rate.

     

    Simulations:

    • Set room size
    • Set number of people
    • Set speed with which fire spreads.  One can think of the speed with which the fire spreads as being a function of the sprinkler system i.e. fire spreads more slowly with a good sprinkler system.
    • Simulate 20 times
    • Repeat with different room size
    • Calculate the average percentage of people that survive in the various scenarios

     

    Results

     

    We show our results below in Figure 1 and Table 1.  We simulated the situation where there were 20 people in a room of various sizes (100x100, 50x50, 25x25) in which the exit was sometimes placed in the corner and sometimes placed in the middle of a wall.

     

    Figure 1 provides strong evidence to confirm our hypotheses.  Placing the exit in the middle of the wall and having a larger room size relative to the number of people both save lives.  These results are confirmed in Table 1.  On average, 85% of the people die in the small room where the exit is in the corner compared to only 43% of the people in the small room with the exit in the middle of the wall.  Having the exit in the middle is clearly superior on safety grounds.  In fact, the results indicate that placing the exit in the middle of the wall of a small room is much better for survivability (only 8.48 people die on average) than having a large room in which the exit is in the corner (11 people die on average).

     

     

     

    Table 1: Mean Number of Deaths as Room Size and Exit Location Varies

    (20 People Alive at the Beginning)

     

    Room Size

    Position of Exit

    Corner

    Middle

    100x100

    11

    (6.48)

    4.90

    (3.42)

    50x50

    14.81

    (6.69)

    7.29

    (3.94)

    25x25

    16.95

    (6.36)

    8.48

    (4.76)

    Standard deviations in parentheses

     

     

    Conclusion

     

    The policy implications of our analysis are quite clear.  Irrespective of the size of the room, fire exits should be placed in the middle of the wall rather than a corner.  Irrespective of where the fire exit is placed, a larger room size relative to the number of people occupying it is preferable to save lives.  The best design would be to have a large room with the fire exit in the middle of the wall.

     

    There are obviously several ways in which this analysis could be improved.  For example, one might want to know whether adding more exits increases the number of people who survive.  One would think that an additional exit would lead to more people getting out alive, but this may not be the case.  Even if it is, one would like to know whether adding an additional exit increases the survival rate in a linear or non-linear manner.  This type of modeling would also enable us to answer questions such as: how many exits are required to maintain a given survival rate as the size of the room varies. 

     

    Clearly, computational models such as this have a role to play in designing rooms to take account of possible fires.  However, there are several ways in which this analysis could be applied to social science research with a few modifications.  For example, one might apply this type of model to a banking collapse in which investors rush to get their money out before the bank runs out of money.  Alternatively, one can think of situations involving civil wars or genocide where a particular population is trying to flee militias or government forces and where a neighboring country is offering safe haven (if only they can get there).