On the unimodality of passage time densities in birth-death processes

It has been shown [2] that for any ergodic birth-death process the p.d.f. of T_on , the passage time from the reflecting state 0 to any level n is log-concave and hence strongly unimodal. It is also known (cf [2]) that the p.d.f. of T_{n, n+1} or T_{n+1, n} for such a process is completely monotone and hence unimodal. It has been conjectured that the p.d.f. for the passage time T_mn between any two 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. of T_mn can always be written as the convolution of two p.d.f.s, one completely monotone and the second PF and hence log-concave.