Waiting Time Distributions for Pattern Occurrence in a Constrained Sequence

A binary sequence of zeros and ones is called a (d,k)-sequence if it does not contain runs of zeros of length either less than d or greater than k, where d and k are arbitrary, but fixed, non-negative integers and d < k. Such sequences find an abundance of applications in communications, in particular for magnetic and optical recording. Occasionally, one requires that (d,k)-sequences do not contain a specific pattern w. Therefore, distribution results concerning pattern occurrence in (d,k)-sequences are of interest. In this paper we study the distribution of the waiting time until the r th occurrence of a pattern w in a random (d,k)-sequence generated by a Markov source. Numerical examples are also provided.