Exercise
(Coffman, Kaduta, and Shepp[Notes 1]) A computing center keeps information on a tape in positions of unit length. During each time unit there is one request to occupy a unit of tape. When this arrives the first free unit is used. Also, during each second, each of the units that are occupied is vacated with probability [math]p[/math]. Simulate this process, starting with an empty tape. Estimate the expected number of sites occupied for a given value of [math]p[/math]. If [math]p[/math] is small, can you choose the tape long enough so that there is a small probability that a new job will have to be turned away (i.e., that all the sites are occupied)? Form a Markov chain with states the number of sites occupied. Modify the program FixedVector to compute the fixed vector. Use this to check your conjecture by simulation.
Notes