28. Each day, one of possible elements is requested, the ith one with probability Pi9 i...

28. Each day, one of ç possible elements is requested, the ith one with probability Pi9 i > 1, Óº¹ = 1· These elements are at all times arranged in an ordered list which 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 il 9 ...,/,, (which is a permutation of 1,2, ...,/?), let n ( i l 9 i n ) denote the limiting probability. In order for the state to be /j , . . . , / „ , it is necessary that the last request was for ix, the last çïç-æº
request was for /2 , the last çïç-æº or i2 request was for /3 , and so on.
Hence, it appears intuitive that Verify when ç = 3 that the above are indeed the limiting probabilities.