AbstractIt has been shown [2] that for any ergodic birth‐death process the p.d.f. ofTon, the passage time from the reflecting state 0 to any levelnis log‐concave and hence strongly unimodal. It is also known (cf [2]) that the p.d.f. ofTn, n+1orTn+1, nfor such a process is completely monotone and hence unimodal. It has been conjectured that the p.d.f. for the passage timeTmnbetweenanytwo states is unimodal. An analytical proof of the result is presented herein, based on underlying renewal structure and methods in the complex plane. It is further shown that the p.d.f. ofTmncan always be written as the convolution of two p.d.f.s, one completely monotone and the secondPFand hence log‐co