Let {Xt ; t 0} be a pure birth process with rate x > 0, for x S = {0, 1, 2, . . . }.

(a) Write the backward equations (KBE) and use it to solve for Px,x(t).

(b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed.

(c) Suppose x = is constant for all x S. Prove by induction that Pxk,x (t) = (t) k e t/k! for k = 0, . . . , x and x = 0, 1, 2, . . . .

(d) Suppose = 3 arrivals per minute. Find the probability that, given there were 5 arrivals in 7 minutes, the first arrival was in the first minute.

(e) Suppose = 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.