Probabilistic Analysis in real time network : Error in data and feedback channel. Two users communication

Question

I'm studying probabilistic analysis in real time network. We have learned how many attempts in average are required to transmit a packet when there is no error on the feedback channel.

With no error in the feedback

$p$ Probability packet not send

$1-p$ Probability packet send

$i$ transmission attempts

$Pr(n=i) = p^{i-1} (1-p)$

$E[n] = \sum_{i=1}^{\infty}i p^{i-1} (1-p)$

With some rules of derivation and geometric series :

$E[n] = \frac{1}{1-p}$

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I need to extend the analysis for the case of errors in the feedback channel.

How to find how many attempts in average are required to transmit a packet when there is error in the data channel and the feedback channel?

I have tried something :

$p$ Probability packet not send

$1-p$ Probability packet send

$i$ transmission attempts

$q$ Probability ACK not send

$1-q$ Probability ACK send

$Pr(n=1) = (1-p)(1-q)$

$Pr(n=2) = p(1-p)(1-q) + q(1-p)^2(1-q)$

$Pr(n=2) = (1-p)(1-q) (p+ q(1-p))^{2-1}$

$Pr(n=i) = (1-p)(1-q)(p+q(1-p))^{i-1}$

I'm a little confused. The number of possibilities increase very fast. For 2 attempts : The data fails one time and succeeds the second time with no error in the feedback or the data succeeds in both case but the feedback fails the first time.

But when I arrive at 20 attempts the number of possibility reach $2^{i-1} = 2^{19}$. Which looks too big for me.

How can I check if I'm on the right track?