Exact distributions of constrained (k, ℓ) strings of failures between subsequent successes

Consider a sequence of binary (success–failure) random variables (RVs) ordered on a line. The number of strings with a constrained number of consecutive failures between two subsequent successes is studied under an overlapping enumeration scheme. The respective waiting time is examined as well. The study is first developed on sequences of independent and identically distributed RVs. It is extended then on sequences of dependent, exchangeability and Markovian dependency is considered, and independent, not necessarily identically distributed, RVs. Exact probabilities and moments are obtained by means of combinatorial analysis and via recursive schemes. An explicit expression of the mean value of the number of strings for both independent and dependent sequences is derived. An application in system reliability is provided. Copyright Springer-Verlag 2013