Let X\documentclass[12pt]{minimal}
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\begin{document}
$$_1$$
\end{document}, X\documentclass[12pt]{minimal}
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\begin{document}
$$_2$$
\end{document}, ... be a sequence of independent and identically distributed random variables, which take values in a countable set S = {0, 1, 2, ...}. By a pattern we mean a finite sequence of elements in S. For every i = 0, 1, 2, ..., we denote by P\documentclass[12pt]{minimal}
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$$_i$$
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$$_{i,1}$$
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$$_{i,2}$$
\end{document}... a\documentclass[12pt]{minimal}
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$$_{i,k_i }$$
\end{document}" the pattern of some length k\documentclass[12pt]{minimal}
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$$_i$$
\end{document}, and E\documentclass[12pt]{minimal}
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$$_i$$
\end{document} denotes the event that the pattern P\documentclass[12pt]{minimal}
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$$_i$$
\end{document} occurs in the sequence X\documentclass[12pt]{minimal}
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$$_1$$
\end{document}, X\documentclass[12pt]{minimal}
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$$_2$$
\end{document}, .... In this paper, we have derived the generalized probability generating functions of the distributions of the waiting times until the r-th occurrence among the events \documentclass[12pt]{minimal}
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$$\{ E_i \} _{i = 0}^\infty$$
\end{document}. We also have derived the probability generating functions of the distributions of the number of occurrences of sub-patterns of length l(l < k) until the fiurrence of the pattern of length k in the higher order Markov chain.
