(Solved):   a . 5. A reservoir (zk) with total capacity of a volume units of water has, during each da ...

 

a . 5. A reservoir (zk) with total capacity of a volume units of water has, during each day, either a net inflow of two units with probability p or a net outflow of one unit with probability q. If the reservoir is full or nearly full, any excess inflow is lost in an overflow. (a) Derive difference equations in detail for this model for uk , k = 1, 2, ... a, the probability that the reservoir will eventually become empty given that it initially contains k units (b) By applying the results in (a), deduce that ua = Ua-1 & ua = Ua-2 (c) For p=1/3, q=2/3, solve the difference equations with boundary conditions. (d) Show that the reservoir is certain to be empty at some time in the future.