This paper contains a study of the following problem: Each of two events recurs with definitely known period and duration, while the starting time of each event is unknown. It is desired that, before the elapse of a certain time, the events occur simultaneously and that this "overlap" be of at least a given minimum duration. The probability of this satisfactory coincidence is first evaluated, and it is found that the solution, while mathematically adequate, is of no value for practical application. This circumstance arises from the possibility that, with certain rational ratios of the periods, the events may "lock in step". Accordingly, an attempt is made to smooth the probability function with respect to small variations in the ratio of the periods. Due to difficulties in manipulating the number-theoretic expressions involved, this smoothing is carried through only by the use of certain approximations. Moreover, because of these same difficulties, an averaged value of the probability itself is not obtained, but, in its stead, there is derived a formula for that fraction of randomly related repeated trials in which the original probability will be less than one-half. Thus, the original problem is not completely solved. The results obtained, however, do allow one to compare the relative advantages of different situations and to make a rough estimate of the likelihood of success. Generally speaking, the analysis is applicable whenever the ratio of "on time" to "off time" is small for each event.
"Probability of Coincidence for Two Periodically Recurring Events." Ann. Math. Statist. 19 (1) 16 - 29, March, 1948. https://doi.org/10.1214/aoms/1177730286