8.8 Consider a cash management scheme in which a company needs to maintain the available cash to be no more than $K. Whenever the cash level reaches K, the company buys treasury bills and reduces the cash level to x. Whenever the cash level reaches 0, the company sells enough treasury bills to bring the cash level up to x, where 0 , x , K. Assume that in any given period the probability that the cash level increases by $1 is p, and the probability that it decreases by $1 is q 5 1 2 p. We define an intervention cycle as the period from the point when the cash level is x until the point when it is either 0 or K. Let T denote the time at which the available cash first reaches 0 or K, given that it starts at level x.

a. What is the expected value of T?

b. What is the mean number of visits to level m up to time T, where 0 , m , K?