We study the distributions of waiting times in variations of the negative binomial distribution of order $k$. One variation apply different enumeration scheme on the runs of successes. Another case considers binary trials for which the probability of ones is geometrically varying. We investigate the exact distribution of the waiting time for the $r$-th occurrence of success run of a specified length (non-overlapping, overlapping, at least, exactly, $\ell$-overlapping) in a $q$-sequence of binary trials. The main theorems are Type $1$, $2$, $3$ and $4$ $q$-negative binomial distribution of order $k$ and $q$-negative binomial distribution of order $k$ in the $\ell$-overlapping case. In the present work, we consider a sequence of independent binary zero and one trials with not necessarily identical distribution with the probability of ones varying according to a geometric rule. Exact formulae for the distributions obtained by means of enumerative combinatorics.

翻译：我们研究了k阶负二项分布变体中等待时间的分布。一种变体对成功游程采用不同的枚举方案，另一种情况考虑成功概率呈几何变化的二元试验。我们探究了在二元试验的q序列中，第r次出现指定长度成功游程（非重叠、重叠、至少、恰好、ℓ-重叠）的等待时间的精确分布。主要定理包括ℓ-重叠情形下的类型1、2、3和4的k阶q-负二项分布及k阶q-负二项分布。本文考虑一系列独立的二元（0和1）试验，其分布不必相同，其中1的概率按几何规则变化。通过枚举组合学方法，得到了分布的精确公式。