Distributions associated with (k1,k2) events on semi-Markov binary trials

We derive neat expressions for the probability generating functions of relevant waiting times associated with (k1,k2)(k1,k2) events on semi-Markov binary trials. These lead to evaluation of relevant probabilities associated with numbers of occurrence of such events on a string of a fixed length. Our methodology is general enough and provides a template for treating more general events than those of type (k1,k2)(k1,k2). Also, the same template is extendable to semi-Markov trials with more than two outcomes.