Generalized Erlang and mortality levelling off distributions

We propose some simple generalizations of the Erlang distribution and study their properties and their connections to other existing well known distributions. In particular, we examine the mortality rate functions and their shapes which informs us how the population ages under these models. With suitable selections of parameter values, we can achieve all the common monotonic or non-monotonic hazard rate shapes that occur in reliability and survival analysis. Some of these functions have an S shape which is suitable for the lifetime random variables that exhibit a late life deceleration phenomenon.