Question: 1. Let {X,;t 2 0} be a pure birth process with rate Ax > 0, for x E 8 = {0,1,2,. . . }. (a)

1. Let {X,;t 2 0} be a pure birth process with rate Ax > 0, for x E 8 = {0,1,2,. . . }. (a) Write the backward equations (KBE) and use it to solve for Pm(t). (b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed. (c) Suppose Ax = A is constant for all x E S. Prove by induction that Px_k'x(t) = (At)ke'M / k! for k = 0,. . .,x and x = 0,1,2,.... (d) Suppose A = 3 arrivals per minute. Find the probability that, given there were 5 arrivals in 7 minutes, the rst arrival was in the rst minute. (e) Suppose A = 3 and that there have been 100 arrivals so far. What is the probability that in the next 4 minutes, there will be at most 1 arrival

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