$3.$ TCP Dynamics and Buffer Sizing. Consider two hosts $\backslash (\backslash$mathbf$\{$S$\}\backslash )$ and $\backslash (\backslash$mathbf$\{$D$\}\backslash )$ connected through a router $\backslash (\backslash$mathbf$\{$R$\}\backslash )$ as depicted in the figure below. The capacity and delay of the links connecting $\backslash (\backslash$mathbf$\{$S$\}\backslash )$ to $\backslash (\backslash$mathbf$\{$R$\},\backslash$mathbf$\{$R$\}\backslash )$ to $\backslash (\backslash$mathbf$\{$D$\}\backslash ),$ and $\backslash (\backslash$mathbf$\{$D$\}\backslash )$ back to $\backslash (\backslash$mathbf$\{$S$\}\backslash )$ are shown in the figure. The source node $\backslash (\backslash$mathbf$\{$S$\}\backslash )$ starts a TCP connection with destination $\backslash (\backslash$mathbf$\{$D$\}\backslash ).$

We make the following assumptions:

$-$ The initial congestion window size is $1.$

$-$ We ignore the $3-$way handshake.

$-$ There is no slow$-$start, and the TCP connection is in additive$-$increase, multiplicative$-$decrease mode from the beginning.

$-$ The link connecting the router $\backslash (\backslash$mathbf$\{$R$\}\backslash )$ to $\backslash (\backslash$mathbf$\{$D$\}\backslash )$ has a buffer of size $5$ packets.

$-$ The TCP connection from $\backslash (\backslash$mathbf$\{$S$\}\backslash )$ to $\backslash (\backslash$mathbf$\{$D$\}\backslash )$ is used to deliver a very large number of packets $($i$.$e$.$ flow will not end for the cases we consider here$).$

$3$a$)$ What is the maximum congestion window size for the TCP flow? What is the maximum rate that this TCP connection can achieve $($in packets per second$)?$