Several renewal processes related to the Type I and Type II counter models are defined and studied. The distribution and characteristic functions for the secondary (or output) process of the Type I counter model are obtained explicitly. Both the non-stationary and stationary probabilities of the state of the counter, (locked or unlocked), are derived. Integral equations determining the distribution and characteristic functions for the secondary process of the Type II counter model are obtained. Also it is shown that a more general model proposed by Albert and Nelson  may be solved explicitly in terms of a corresponding Type II counter model. An example of this general model is given. Related with each model is a discrete renewal process which is also studied.
"On Renewal Processes Related to Type I and Type II Counter Models." Ann. Math. Statist. 29 (3) 737 - 754, September, 1958. https://doi.org/10.1214/aoms/1177706533