43. Each day, one of n possible elements is requested, the ith one with probabilityI mage. These elements are at all times arranged in an ordered list that is revised as follows: The element selected is moved to the front of the list with the relative positions of all the other elements remaining unchanged. Define the state at any time to be the list ordering at that time and note that there are n! possible states.

(a) Argue that the preceding is a Markov chain.

(b) For any state Image (which is a permutation of Image

), let Image) denote the limiting probability. In order for the state to be Image, it is necessary for the last request to be for , the last non- request for Image, the last non- or Image request for Image, and so on. Hence, it appears intuitive that Verify when Image that the preceding are indeed the limiting probabilities.