Monitoring heterogeneous serially correlated usage behavior in subscription-based services

Effective monitoring of usage behavior necessitates applying accurate stochastic models to represent customer heterogeneous time-dependent behavior. In this research, it is assumed that the sequence of customer visits over a subscription period occurs based on the Poisson process, while usage at each purchase occasion follows an autoregressive Bernoulli model of first order. The autocorrelated observations are derived from a two-state Markov chain model. Generalized linear models are employed to describe heterogeneous behavior across customers. In order to monitor the number of visits as well as the fraction of visits eventuated in a purchase, control statistics are defined on the basis of generalized likelihood ratio (GLR) test. Furthermore, in the case of the marginal logistic model for dependent observations, a chi-square test statistic based on the asymptotic multivariate normal distribution of quasi-likelihood estimates is employed. Performances of the monitoring schemes are compared with an illustrative case provided by simulation. Results indicate that the adjusted Shewhart c chart resembles the deviance residual control chart for monitoring the frequency of customer visit. On the other hand, the GLR statistic based on the conditional logistic regression is more powerful in detecting unnatural usage behavior when compared with the chi-square control statistic based on the marginal logistic model.