Modeling the number of insureds' cars using queuing theory

In this paper, we propose to model the number of insured cars per household. We use queuing theory to construct a new model that needs 4 different parameters: one that describes the rate of addition of new cars on the insurance contract, a second one that models the rate of removal of insured vehicles, a third parameter that models the cancellation rate of the insurance policy, and finally a parameter that describes the rate of renewal. Statistical inference techniques allow us to estimate each parameter of the model, even in the case where there is censorship of data. We also propose to generalize this new queuing process by adding some explanatory variables into each parameter of the model. This allows us to determine which policyholder's profiles are more likely to add or remove vehicles from their insurance policy, to cancel their contract or to renew annually. The estimated parameters help us to analyze the insurance portfolio in detail because the queuing theory model allows us to compute various kinds of useful statistics for insurers, such as the expected number of cars insured or the customer lifetime value that calculates the discounted future profits of an insured. Using car insurance data, a numerical illustration based on a portfolio from a Canadian insurance company is included to support this discussion.