Population-based cancer screening is often asked but hardly addressed by a question: “How many rounds of screening are required before identifying a cancer of interest staying in the pre-clinical detectable phase (PCDP)?” and also a similar one related to the number of screens required for stopping screening for the low risk group. It can be answered by using longitudinal follow-up data on repeated rounds of screen, namely periodic screen, but such kind of data are rather complicated and fraught with intractable statistical properties including correlated multistate outcomes, unobserved and incomplete (censoring or truncation) information, and imperfect measurements. We therefore developed a negative-binomial-family-based discrete-time stochastic process, taking sensitivity and specificity into account, to accommodate these thorny issues. The estimation of parameters was implemented with Bayesian Markov Chain Monte Carlo method. We demonstrated how to apply this proposed negative-binomial-family-based model to the empirical data similar to the Finnish breast cancer screening program.