Exercise 6. Type A, B, and C items are placed in a common buffer, each type arriving as part of an independent Poisson process with average arrival rates, respectively, of a, b, and c items per minute. For the first four parts, assume the buffer is discharged immediately whenever it contains a total of ten items. (1) What is the probability that, of the first ten items to arrive at the buffer, only the first and one other are of type A? (2) What is the probability that any particular discharge of the buffer contains five times as many type A items as type B items? (3) Determine the PDF, expectation, and variance of the total time between consecutive dis- charges of the buffer. (4) Determine the probability that during a particular five minute interval there are exactly two arrivals of each type. For the next two parts, assume the buffer is discharged immediately whenever it contains a total of three type A items. (5) Determine the PDF, expectation, and variance of the total time between consecutive dis- charges of the buffer. (6) For an observer arriving at a random time, long after the process began, obtain the PDF's of: (a) U, the time until the arrival of the next item at the buffer input. (b) V, the time until the next discharge of the buffer