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Math Help - Markov Model

  1. #1
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    Markov Model

    I was given a scenario and have to represent the facts using the Markov model of X(underscore n)=A^n x(underscore 0)

    The facts are: there are 3 modes of transportation- train, bus or car. If a train is taken, there is a 0.2 probability that there will be a signal failure. If this happens the following day, a person will take the bus, otherwise he will take a train again. If a bus is taken, there is a 0.3 probability that it is late. If this happens, the person will take the car, otherwise he will take the bus again. If a car is taken, there is a 0.4 probability the roads will be congested. If this happens the following day, the person will take the train, otherwise he will take the car again.
    The first day of travelling, the person goes by train.

    How would I represent this situation? Would I use a Stochastic matrix somehow?
    Also, how would I go about finding the outcome of the 5th day, and the long state vector?

    Thank you
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  2. #2
    MHF Contributor
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    This should probably be in the linear and abstract algebra forum.

    I am not 100% on my transition matrix but here is what I obtained.

    \begin{bmatrix}<br />
 & train & bus & car\\ <br />
train & .8 & 0 & .4\\ <br />
bus & .2 & .7 & 0\\<br />
car & 0 & .3 & .6<br />
\end{bmatrix} and \mathbf{x}_0=\begin{bmatrix}<br />
1\\ <br />
0\\ <br />
0 <br />
\end{bmatrix}

    X is your eigenvector matrix and D is your diagonal matrix
    A^5=XD^5X^{-1}\mathbf{x}_0
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