Probabilistic Modeling2011 - With discrete systems...

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3/4/2011 1 PROBABILISTIC MODELING (AN INTRO TO MARKOV MODEL) With discrete systems RENTAL CAR COMPANY EXAMPLE ´ A rental car company with branches in Orlando and Tampa ´ A customer can rent a car in one city and return in either city ´ Problem: imbalance in available cars to rent ´ Given: historical data on percentages of cars rented and returned to these cities ´ Question: what is the long term behavior
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3/4/2011 2 HISTORICAL DATA (A CAR’S LIFE) Orlando Tampa Orlando 0.6 0.4 Tampa 0.3 0.7 Next city Present city 0.4 0.3 0.7 0.6 Orlando Tampa MATHEMATICAL MODEL (MARKOV CHAIN) ´ Discrete time sequence ´ At each time moment, it has the same finite number of possible outcomes (values, states) ´ The states do not overlap and cover all possible outcomes ´ So, in time, the sequence moves from one state to another, one for each time step ´ There is a probability associated with this transition
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3/4/2011 3 TRANSITION MATRIX ´ The sum of the probabilities for transitioning from present state to the next state is equal to
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