Moments of first passage times in general birth-death processes

We consider ordinary and conditional first passage times in a general birth-death process. Under existence conditions, we derive closed-form expressions for the kth order moment of the defined random variables, k >= 1. We also give an explicit condition for a birth-death process to be ergodic degree 3. Based on the obtained results, we analyze some applications for Markovian queueing systems. In particular, we compute for some non-standard Markovian queues, the moments of the busy period duration, the busy cycle duration, and the state-dependent waiting time in queue.