From whom are looking an initial approach to know the best time to bill your customers for some subscription services, this paper can be a good start.
In my current company, this is a very challenging problem.
Abstract: A system and method for recurring billing of periodic subscriptions are disclosed. The system attempts to maximize a metric like long term customer retention while tailoring the subscription billing to the customer, using machine learning. The system is initially trained with a set of training data — a large corpus of records of subscription billings — including successes, billing failures, and customer cancellations. Any available metadata about the users or the type of subscription is also attached and may be used as features for the machine learning model. Such metadata may include, for example, customers’ age, gender, demographics, interests, and online behavioral profile/history, as well as metadata to identify the type of service being billed, such as music subscriptions, delivery subscriptions or other types of subscriptions, or the payment instrument. The system is used to predict the subscription model for a given user with relevant user-related constraints, while optimizing acceptability to that user.