On the unimodality of passage time densities in birth‐death processes
- 1 March 1981
- journal article
- Published by Wiley in Statistica Neerlandica
- Vol. 35 (1) , 49-55
- https://doi.org/10.1111/j.1467-9574.1981.tb00710.x
Abstract
It has been shown [2] that for any ergodic birth‐death process the p.d.f. of Ton, 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 Tn, n+1 or Tn+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 Tmn 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 Tmn can always be written as the convolution of two p.d.f.s, one completely monotone and the second PF and hence log‐concave.Keywords
This publication has 3 references indexed in Scilit:
- Markov Chain Models — Rarity and ExponentialityPublished by Springer Nature ,1979
- Exponential spectra as a tool for the study of server-systems with several classes of customersJournal of Applied Probability, 1978
- Log-concavity and log-convexity in passage time densities of diffusion and birth-death processesJournal of Applied Probability, 1971