Kurtis Cahill James Badal

 Introduction  Model a Maze as a Markov Chain  Assumptions  First Approach and Example  Second Approach and Example  Experiment  Results  Conclusion

 Problem: To find an efficient approach of solving the rate of visitation of a cell inside a large maze  Application: To find the best possible place to intercept information

 Allows Stochastic principles to be applied to the problem  Each maze cell will be model as a state in Markov Chain  The Markov Chain will be one recurrent class

 To reduce the complexity of the problem and simulation, certain assumptions will be applied: 1.

Unbiased transition to adjacent cells 2.

Random walk can’t be stationary 3.

No isolated cells inside the maze

 

r i –

Steady-state rate of the Chain

i

th state of the Markov

p ji –

Probability of moving from state next step

j

to state

i

on the

The transition matrix for the random walk on this maze

System of Steady State Rate Equations

Row Reduced System of Steady State Rate Equations



r i –

Steady-state rate of the Chain

i

th state of the Markov  

p –

Proportionality constant

n i –

Number of connections to the

i

th cell

Solution to System of Steady State Rate Equations

 Random Walker starts at a certain maze location and walks 10 8 steps  At each step the random walker increments the visit count of the most recently visited cell  The mean and standard deviation are measured at the end of the experiment  The measured result is compared to the calculated result

Random Walk result of a 2x2 Maze

Random Walk result of a 5x5 Maze

Random Walk result of a 10x10 Maze

Random Walk result of a 20x20 Maze

Random Walk result of a 40x40 Maze

 Modeled the maze as a Markov Chain  Applied Stochastic principles to the maze  First Approach is n 3 complexity  Second Approach is n complexity  Tested the calculated result with the measured result