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的等待时间分布。一种变体对成功游程采用不同的枚举方案，另一种情形考虑“1”的概率呈几何变化的二元试验。我们在二元试验的$q$-序列中，研究了指定长度成功游程（非重叠、重叠、至少、恰好、$\ell$-重叠）第$r$次出现等待时间的精确分布。主要定理包括k阶$1$、$2$、$3$、$4$型$q$-负二项分布，以及$\ell$-重叠情形下的k阶$q$-负二项分布。本文考虑一系列独立但非同分布的二元0-1试验，其中“1”的概率遵循几何规律变化。通过枚举组合学方法，我们获得了上述分布的精确表达式。