Joint distributions associated with countably many runs in sooner waiting time problems

We consider the runs enumerations in the waiting time problem related to countably many runs. Two types of random variables, i.e., the waiting time until the first occurrence of a run among the infinite number of runs and the numbers of occurrences of runs of several lengths observed at that time are of main interest. The corresponding joint distributions are investigated in terms of the probability generating functions. We provide the effective computational tools for the evaluation of the probability mass functions by making use of the potential partition polynomials. Finally, applications with numerical results are given in order to demonstrate feasibility of our theoretical results.