Understanding the value of acquiring or retaining subscribers is crucial for subscription-based businesses. While customer lifetime value (LTV) is commonly used to do so, we demonstrate that LTV likely over-states the true value of acquisition or retention. We establish a methodology to estimate the monetary value of acquired or retained subscribers based on estimating both on and off-service LTV. To overcome the lack of data on off-service households, we use an approach based on Markov chains that recovers off-service LTV from minimal data on non-subscriber transitions. Furthermore, we demonstrate how the methodology can be used to (i) forecast aggregate subscriber numbers that respect both aggregate market constraints and account-level dynamics, (ii) estimate the impact of price changes on revenue and subscription growth and (iii) provide optimal policies, such as price discounting, that maximize expected lifetime revenue.

The majority of cancer-related fatalities are due to metastatic disease. Chemotherapeutic agents are administered along with radiation in chemoradiotherapy (CRT) to control the primary tumor and systemic disease such as metastasis. This work introduces a mathematical model of CRT treatment scheduling to obtain optimal drug and radiation protocols with the objective of minimizing metastatic cancer cell populations at multiple potential sites while maintaining a desired level of control on the primary tumor. Dynamic programming framework is used to determine the optimal radiotherapy fractionation regimen and the drug administration schedule. We design efficient DP data structures and use structural properties of the optimal solution to reduce the complexity of the resulting DP algorithm. We derive closed-form expressions for optimal chemotherapy schedules in special cases. The results suggest that if there is only an additive and spatial cooperation between the chemotherapeutic drug and radiation with no interaction between them, then radiation and drug administration schedules can be decoupled. In that case, regardless of the chemo- and radio sensitivity parameters, the optimal radiotherapy schedule follows a hypofractionated scheme. However, the structure of the optimal chemotherapy schedule depends on model parameters such as chemotherapy-induced cell kill at primary and metastatic sites, as well as the ability of primary tumor cells to initiate successful metastasis at different body sites. In contrast, an interactive cooperation between the drug and radiation leads to optimal split-course concurrent CRT regimens. Additionally, under dynamic radio sensitivity parameters due to the reoxygenation effect during therapy, we observe that it is optimal to immediately start the chemotherapy and administer a few large radiation fractions at the beginning of the therapy, while scheduling smaller fractions in later sessions. We quantify the trade-off between the new and traditional objectives of minimizing the metastatic population size and maximizing the primary tumor control probability, respectively, for a cervical cancer case. The trade-off information indicates the potential for significant reduction in the metastatic population with minimal loss in the primary tumor control.