A pure birth Markov chain is a continuous time Markov chain {Z(t):t>=0} with state space S[reverse not equivalent]{0,1,2,...} such that for each i>=0 the chain stays in state i for a random length of time that is exponentially distributed with mean and then jumps to (i+1). Suppose b([dot operator]) is a function from (0,[infinity])-->(0,[infinity]) that is nondecreasing and [short up arrow][infinity]. This paper addresses the two questions: (1) Given {[lambda]i}i>=0 what is the growth rate of Z(t)? (2) Given b([dot operator]) does there exist {[lambda]i} such that Z(t) grows at rate b(t)?
