Problem $2[20$ points$]$

A terminal $\backslash ($ A $\backslash )$ wants to transmit $\backslash ($ N $\backslash )$ frames of length $\backslash ($ P $\backslash )$ bits per frame, to a terminal $\backslash ($ B $\backslash )$ via a direct link. The link corrupts each frame independently with probability $\backslash ($ q $\backslash ).$ Terminal $\backslash ($ A $\backslash )$ can choose between two strategies for the transmission of the $\backslash ($ N $\backslash )$ frames: $(1)$ apply an error correcting code that will increase the length of each frame from $\backslash ($ P $\backslash )$ to $\backslash ($ P$+$C $\backslash ),$ where $\backslash ($ C$>0\backslash ),$ but guarantee that corrupted frames can be recovered at $\backslash ($ B $\backslash ),$ or $(2)$ keep the frame length equal to $\backslash ($ P $\backslash )$ and retransmit the corrupted frame.

$($a$)[5$ points$]$: Calculate the delay for successfully receiving the $\backslash ($ N $\backslash )$ frames at $\backslash ($ B $\backslash )$ using the first strategy. Ignore the propagation delay and any delay in transmitting ACKs $($instant feedback$).$

$($b$)[\backslash (\backslash$mathbf$\{5\}\backslash )$ points$]$: Calculate the average delay for successfully receiving the $\backslash ($ N $\backslash )$ frames at $\backslash ($ B $\backslash )$ using the second strategy. You can find this average by looking at the average number of attempts per packet for successful reception. Ignore the propagation delay and any delay in transmitting ACKs $($instant feedback$).$

$($c$)[5$ points$]$: For what probability value $\backslash ($ q $\backslash ),$ does strategy $(2)$ incur the same average delay as strategy $(1)?$

$($d$)[5$ points$]$: Sketch $\backslash ($ q $\backslash )$ as a function of $\backslash ($ C $\backslash )$ for a range of $\backslash ($ C $\backslash )$ from $0$ to $\backslash (\backslash$infty $\backslash )$ and label the areas of $\backslash ($ q $\backslash )$ for which strategy $(1)$ is preferred to strategy $(2).$ You do not need an exact plot, just an approximate sketch and some sample values and lines drawn between them $($e$.$g$.$ at $\backslash ($ C$=0,$ C$=1\backslash ),$ and $\backslash ($ q $\backslash$rightarrow $\backslash$infty$)\backslash ).$