Number of runs of ones of length exceeding a threshold in a modified binary sequence with locks

Let us consider a sequence of binary (zero-one) trials. A counter registers ones but once an one is registered the mechanism is locked for a fixed number of trials following each registration. On such a sequence of a fixed length we define a random variable enumerating runs of ones of length exceeding a threshold. Exact recursive expressions are obtained for the probability mass function, generating functions and moments, as well as an exact closed formula for the mean of this random variable. A simulation algorithm is also provided for approximating values of this random variable on such a modified binary sequence. Statistical inference problems associated with the probability of ones are examined by numerical techniques and simulation. An application connected to the chance that a stochastic process remains or not in statistical control is discussed. Illustrative numerical examples exemplify further the theoretical results.