  This report aims to characterise certain sojourn time distributions that
naturally arise from semi-Markov models. To this end, it describes a family of
discrete distributions that extend the geometric distribution for both finite
and infinite time. We show formulae for the moment generating functions and the
mean and variance, and give specific examples. We consider specific
parametrised subfamilies; the linear factor model and simple polynomial factor
models. We numerically simulate drawing from these distributions and solving
for the Maximum Likelihood Estimators (MLEs) for the parameters of each
subfamily, including for very small sample sizes. The report then describes the
determination of the bias and variance of the MLEs, and shows how they relate
to the Fisher information, where they exhibit appropriate concentration effects
as the sample size increases. Finally, the report addresses an application of
these methods to experimental data, which shows a strong fit with the simple
polynomial factor model.
