In reliability analysis, accelerated life testing is the most common way to assess a product's life. Under such test settings, products are tested at higher-than-usual levels of stress to induce early failure. The goal of accelerated life analysis is to utilize the test data to extrapolate a product's life distribution and its associated parameters at a normal stress level. The current study introduces the geometric process model for the analysis of accelerated life testing with an exponential life distribution under constant stress. The geometric process describes a simple monotone process and has been applied to a variety of situations such as the maintenance problems in engineering. By assuming that the lifetime under increasing stress levels forms a geometric process, we derive the maximum likelihood estimators of the exponential parameter under three different censoring schemes: no censoring, Type I and Type II censoring. For each censoring type, we also derive confidence intervals for the parameters using both asymptotic distribution and the parametric bootstrap method. The performance of the estimators is evaluated by a simulation study with different pre-fixed parameters. The study also considers whether the assumption of geometric process is satisfied under a wide-acknowledged log-linear relationship between life and stress. Some simulated numerical examples are presented to illustrate how to pre-design the stress levels so that the geometric process model could be applied. We also use these numerical examples to compare the performance of the proposed geometric model and the traditional failure time regression model. The results of simulation studies, as well as the advantage of the proposed model are summarized in the conclusion part, and the prospect of future works is briefly discussed.