Consider a generalized selective repeat ARQ system that operates in the following way: as in conventional go back $*$n$*$ and selective repeat, let $*$RN$*$ be the smallest numbered packet not yet correctly received, and letN$\_$min$*$ be the transmitter$$s estimate of $*$RN$*.$ Let $*$ytop$*$ be the largest$-$numbered packet correctly received and accepted $($thus$,$ for go back $*$n$*,*$ytop$*$ would be $*$RN $-1*,$ whereas for conventional selective repeat, $*$ytop$*$ could be as large as $*$RN $+$ n $-1*).$ The rule at the transmitter is the same as for conventional go back $*$n$*$ or selective repeat:

The number $*$z$*$ of the packet transmitted must satisfy:

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SN$\_\{\backslash$text$\{$min$\}\}\backslash$leq z $\backslash$leq SN$\_\{\backslash$text$\{$min$\}\}+$ n $-1$

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The rule at the receiver, for some given positive number $*$k$*,$ is that a correctly received packet with number $*$z$*$ is accepted if:

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RN $\backslash$leq z $\backslash$leq ytop $+$ k

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$(2\mathrm{.}46)$

$($a$)$ Assume initially that $*$z$*($rather than $*$SN $=$ z $\backslash$mod m$*)$ is sent with each packet and that $*$RN$*($rather than $*$RN $\backslash$mod m$*)$ is sent with each packet in the reverse direction. Show that for each received packet at the receiver:

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z $\backslash$leq RN $+$ n $-1$

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z $\backslash$geq ytop $-$ n $+1$

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$(2\mathrm{.}47)$

$($b$)$ Now assume that $*$SN $=$ z $\backslash$mod m$*$ and $*$RN$*$ is sent mod $*$m$*.$ When $*$z$*$ is received, Eq$.(2\mathrm{.}47)$ is satisfied, but erroneous operation will result if $*$z $-$ m$*$ or $*$z $+$ m$*$ lie in the window specified by Eq$.(2\mathrm{.}46).$ How large need $*$m$*$ be to guarantee that $*$z $+$ m $>$ ytop $+$ k$*[$i$.$e$.,$ that $*$z $+$ m$*$ is outside the window of Eq$.(2\mathrm{.}46)]?$

$($c$)$ Is the value of $*$m$*$ determined in part $($b$)$ large enough to ensure that $*$z $-$ m $<$ RN$*?$

$($d$)$ How large need $*$m$*$ be $($as a function of $*$n$*$ and $*$k$*)$ to ensure correct operation?

$($e$)$ Interpret the special cases $*$k $=1*$ and $*$k $=$ n$*.$