Finding Probability Mass Function (PMF) Given a Geometrically Distributed Random Variable and a Negative Binomial Random Variable? I am a non-student working through the first edition of Yates and Goodman's text, Probability and Stochastic Processes. On page 115, question 3.6.9 goes like this:

Each millisecond at a telephone switch, a call independently arrives with probability $p$. Each call is either a data call $(d)$ with probability $q$ or a voice call $(v)$. Each data is a fax call with probability $r$. Let $N$ equal the number of milliseconds required to observe the first $100$ fax calls. Let $T$ equal the number of milliseconds you observe the switch waiting for the first fax call. Find the marginal PMF $P_T(t)$ and the conditional PMF $P_{N|T}(n,t)$. Lastly, find the conditional PMF $P_{T|N}(t,n).$

I figured out the marginal PMFs, $P_N(n)$ and $P_T(t)$:
$\quad \quad \quad \quad \quad P_N(n) =\begin{cases} \binom{n-1}{99}pqr^{100}(1-pqr)^{n-100},\quad n=100, 101,... \\ 0,\quad \text{otherwise} \end{cases}$
$\quad \quad \quad \quad \quad \space P_T(t) =\begin{cases} pqr(1-pqr)^{t-1},\quad t=1,2... \\ 0,\quad \text{otherwise} \end{cases}$
How would I go about computing the joint PMF $P_{N,T}(n,t)$? 
My understanding is that the joint PMF is the intersection of the probabilities of both random variables $N$ and $T$. Thus, my intuition tells me that the joint PMF will have a domain of $n,t \geq 100$. 
 A: Hint #1 You should start from first principles rather than trying at guessing among standard distributions: consider the joint distribution of the 100 $T_i$'s, which are (i) independent and (ii) geometric $\mathcal{G}(pqr)$. In other words, the $T_i$'s are iid:
$$T_i\stackrel{\text{i.i.d.}}{\sim} \mathcal{G}(\rho)\qquad i=1,\ldots,100$$
with $\rho=pqr$. You can then deduce from this joint distribution on $(T_1,\ldots,T_{100})$, the distribution of $T_1$ conditional on 
$$N=∑^{100}_{i=1}T_i=n$$
Hint #2 The joint pmf of $(T_1,\ldots,T_{100})$ is provided by the product of their respective pmfs:
$$\begin{align*}
\mathbb{P}((T_1,\ldots,T_{100})&=(t_1,\ldots,t_{100}))=\prod_{i=1}^{100}\rho(1-\rho)^{t_i-1}\\&=\rho^{100}(1-\rho)^{-100}(1-\rho)^{ \sum_{i=1}^{100} t_i}\end{align*}$$
since they are independent
Hint #3 If $\sum_{i=1}^{100} t_i=n$, can you plug $n$ in the above joint pmf and deduce the distribution of $(T_1,\ldots,T_{100})$ given $N=n$? What are the consequences of the pmf being constant in $(t_1,\ldots,t_{100})$ given $n$?
