Estimating the average generation of a collection of cells is helpful in understanding complex cellular differentiation processes, identifying carcinogenic cellular activities, and quantifying the ageing of the immune system. Different techniques based on both direct observations and indirect inference have been proposed, with benefits and limitations varying in the two categories. In this thesis we enhance the mathematical results underpinning one of these inference methods, firstly proposed by Weber et al. in 2016 [116] and based on a DNA coded randomised algorithm. Assuming some sort of structure in the growth of a cell population, with the use of Branching Processes and Renewal Theory, we establish improved convergence properties of the proposed estimator to the average generation. Expanding and homeostatic populations are studied, allowing the method to be used for more complex patterns of population dynamics that includes the succession of these two phases. Furthermore, we establish the possibility of using the same method in a two-type branching process, obtaining a possible criterion to distinguish among some differentiation models in hemapotoiesis. A quality study of the model allows also us to establish values of the parameters which improve the performance of the estimator. Computer simulations, with parametrisations coming from the immunology field, are along the results with both a validation and exploratory purpose.