  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.
