4. a) Consider a one server Markov queue where customers arrive in a Poisson process, rate )i, and, on seeing i. customers in the system, join the queue with probability r Service times are independent. exponentially distributed with parameter ,ii. and independent of the arrival process. Let the Markov chain {X(t) : f. 2 {1} denote the number of customers in the system. (i) Carefully stating any results to which you appealI determine the generator matrix Q. (ii) Determine the stationary distribution of the number of customers in the system and write down its expected value. b) In a storage facility, the removal of items is due either to demandI or deterioration. The demand for items is a Poisson process, rate A and the lifetime of items are independent, exponentially distributed with parameter u, and independent of the demand process. On each occasion the stock drops to zero. N new items are immediately placed in the storage facility. Let the Markov chain {X(f.) : t 2 {l} with state space {1, . . . .N} denote the number of items in the storage facility. (i) Determine the stationary distribution of the Markov chain. (ii) Determine the average number of items in the storage facility. (iii) Determine the expected time for the facility to first empty following replenishment