Artistic Ending of the Chapter

Let US now apply Markov chains to investigate the trajectory of the system, the horsewoman, riding in the forest. The possible states that the system can occupy on the basis of Fig.2-59 are defined as follows Si-the passage between trees 1 and 2 through which the system can move, S2-the passage between trees 4 and 3, S3-the passage between trees 4 and 5, S4-the passage between trees 7 and 6, 

In the following we consider each case separately. In case a the steady state occupation probability is different for each state the corresponding state vector reads, S(11) = [0.300, 0.400, 0.200, 0.100, 0.000]. As seen, S2 is of the highest probability, i.e. S2(l 1) = 0.400. Case b differs from a only by the values of psi thus, causing vanishing of ss(n) to occur after 6 steps rather than after 1 step. From thereon, the values of S(n) are the same as in a. Case c causes a problem to the horsewoman because S(2) = [0.25, 0.25, 0.25, 0.25, 0.00] thus she has a problem what state to occupy because all states acquire an identical probability. Case d reveals periodic characteristics of the states with period of v = 2 as follows  

In other words, in each step two states, S2 and S4 or Si and S3, acquire a certain occupation probability. Case e demonstrates the following periodic results at steady state 

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