People arrive at a queue according to the following scheme: During each minute of time either $0$ or $1$ person arrives. The probability that $1$ person arrives is $p$ and that no person arrives is $q = 1 - p$. Let $C_r$ be the number of customers arriving in the first $r$ minutes. Consider a Bernoulli trials process with a success if a person arrives in a unit time and failure if no person arrives in a unit time. Let $T_r$ be the number of failures before the $rth$ success.
Is $C_r$ just a regular binomial distribution?
$$p_{C_r}(k)=\binom{r}{k}p^k(1-p)^{r-k}$$
So $E(C_r)=rp$ and $Var(C_r) = rp(1-p)$
Is $T_r$ a modification of a negative binomial distribution where $x$ is the number of failures rather than the total number of trials?
$$p_{T_r}(x)=\binom{x+r-1}{r-1}p^r(1-p)^{x}$$
The answer is indeed yes, to both questions.
So now you may find $\mathsf E(T_r)$ and $\mathsf{Var}(T_r)$.