- 3 3 4 5 8 20 Marks Items Arrive In A Queue During Time Intervals 0 1 1 2 Let An Denote The Number Of 1 (125.71 KiB) Viewed 38 times
= = 3. (3 + 4 + 5 + 8 20 marks) Items arrive in a queue during time intervals (0,1), (1,2), Let An denote the number of arrivals in the interval (n - 1,n), and assume that (An: n > 1) are mutually independent and identically distributed with pmf a; P(An j) = 1, j = 0,1,2,3. The arriving items queue in a buffer with capacity 4: if arrivals exceed the buffer capacity, then the excess is lost. A single server dispatches one item per unit time (if any are waiting), with these dispatches occurring at times n = 1,2, ... (by which is meant that the dispatch at time n occurs after the arrivals An, and prior to the next arrivals An+1). Let X, denote the buffer level at time n, immediately before any dispatch. We know that (Xn) is a Markov chain, with state space S = {0,1,...,4}. Denote by P [Pijli; the transition probability matrix for (Xn). Suppose that the queue is empty initially, i.e. Xo = 0. (a) Give the one-step transition probability matrix for (Xn). (b) Find the mean occupation times (moj(10) : j ES) for (Xn). (c) Find the expected time until the buffer reaches its capacity for the first time. (d) Let Y, denote the number of items that go to waste from arrivals during the interval (n – 1,n). Show that E(Yn+1|X, = 0) = 1pO3 + Hint: First evaluate E(Yn+1|X, j), for each j e S, and then apply the Law of Total Expectation: E(Yn+1|X, = 0) = Ejes E(Yn+1|X, = j, X, = 0)P(Xn = j|X, = 0).] = = + = +