Estimating Life Cycle Sales of Technology Products with Frequent Repeat Purchases: A Fractional Calculus-Based Approach.

The article focuses on the development and evaluation of the generalized diffusion model with repeat purchases (GDMR), a novel sales growth model designed to predict product sales trajectories when repeat purchases or subscription renewals constitute a significant portion of sales, particularly for technology products. The GDMR extends the classic Bass diffusion model by incorporating fractional calculus through a noninteger-order integral equation, introducing a coefficient of repeat purchases to capture varying frequencies of repeat buying behavior. Empirical analyses using multiple technology product sales data sets demonstrate that the GDMR outperforms benchmark repeat-purchase models, generic time series models, and machine learning approaches in both fitting historical sales and forecasting future sales, even when only aggregate sales data are available. Additionally, the model can incorporate marketing mix variables, enhancing its practical applicability for firms' strategic planning in production, inventory, and marketing. The study also contributes to the interpretation of fractional calculus by providing a business-contextualized understanding of fractional integrals related to adoption and repeat purchase dynamics.