(12pt, Slotted ALOHA efficiency) Suppose there are a number of nodes, all with many frames to send using the slotted ALOHA protocol. Using slotted ALOHA, each node transmits a frame in a time slot with probability p. A successful frame delivery happens in a time slot if and only if there is exactly one node transmitting a frame in the time slot. (a) In case there are 3 nodes, what is the probability that some node (any one out of three nodes) succeeds for the first time in the 2nd slot? (b) In case there are N nodes, what is the probability that there is a successful frame delivery in a time slot? (c) In case there are N nodes, the efficiency of slotted ALOHA is defined as the correct answer of Problem 9(b). Find the value of p that maximizes the efficiency. (d) Using the results found in Problem 9(c), find the efficiency of slotted ALOHA when N approaches infinity. [Hint: Use the definition of Euler's constant, e = limw- (1 +1/N)"]