Consider the below chain network, where a node can $($i$)$ remain

silent $($ii$)$ send a message $($iii$)$ successfully receive a message only if

there is no collision in a time slot $($the entire time in this problem is

divided into time slots$).$ All "ACKs" are link layer acknowledgements.

The data packet and the ACK both consume one entire time slot. The circular region around the node shows the transmission range of that node. Assume that a global controller schedules transmissions carefully in a given time slot, so that there is never a collision at any of the nodes. Ignore any specific IEEE standard or CSMA etc. Instead, assume that transmissions occur when directly allowed by the controller through some out$-$of$-$band operation.

Suppose A sends message to C$,$ and also D sends a message to B $($no

ACKs$).$ What is the combined maximum rate $($messages$/$slot$)$

considering interference range equal to that of transmission range?

If ACKs are required, and the interference range is twice that of

transmission range, what is the combined maximum rate for the

previous case?

A sends message to D$,$ and C sends another message to D$.$ Assuming the presence of ACK and interference range equal to twice transmission range, what is the combined maximum rate

$($messages$/$slot$)?$