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.