A traveling salesman visits the three cities of Atlanta,...

A traveling salesman visits the three cities of Atlanta, Boston, and Chicago. The matrix $\left(\begin{array}{rrr}0 & .5 & .5 \\ 1 & 0 & .5 \\ 0 & .5 & 0\end{array}\right)$ describes the transition probabilities of his trips. Describe his travels in words, and calculate how often he visits each city on average.

Key Concepts

Markov Chains

A Markov chain is a stochastic process that undergoes transitions from one state to another, where the probability distribution for the next state depends solely on the current state and not on the history of past states. In contexts like modeling a traveling salesman’s route among several cities, the chain encapsulates the idea that the next city visited depends only on the city currently being visited according to fixed probabilistic rules.

Transition Matrix

The transition matrix is a structured representation of the probabilities of moving between different states in a Markov chain. Each entry in this matrix indicates the probability of transitioning from one state to another. In many texts the matrix is arranged so that either each row or each column sums to 1, meaning that from any given state the total probability of moving to some state is 1. This matrix structure is crucial in describing dynamics such as the salesman’s movement across cities, where each column (or row, depending on the convention) shows the possible next destinations and their probabilities.

Stationary Distribution

The stationary distribution of a Markov chain is a probability vector that remains unchanged by further transitions; in other words, if the process starts in the stationary distribution, it will stay in that distribution over time. In practical terms, it represents the long?term average proportion of time (or visits) the process spends in each state. When applied to the traveling salesman problem, calculating the stationary distribution allows one to ascertain on average how often the salesman visits each city as time goes to infinity.

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