An analytical model is formulated to support understanding and underpin experimental development of laser action in the promising diode end-pumped Alexandrite system. Closed form solutions are found for output power, threshold, and slope efficiency that for the first time incorporate the combined effects of laser ground state absorption and excited state absorption (laser ESA), along with pump excited state absorption (pump ESA), in the case of an end-pumping geometry. Comparison is made between model predictions and experimental results from a fiber-delivered diode end-pumped Alexandrite laser system, showing the impact of wavelength tuning, crystal temperature, laser output coupling, and intracavity loss. The model is broadly applicable to other quasi-three-level lasers with combined laser and pump ESA. A condition for bistable operation is also formulated.