In a particular computer network, the host computer broadcasts a packet of data to $N$ receivers. The

host computer then waits to receive an acknowledgment message from each of the $N$ receivers before

proceeding to broadcast the next packet. If the host does not receive all acknowledgments within a cer$-$

tain time period, it will rebroadcast the packet. The host computer is then in "retransmission mode".

It will continue retransmitting the packet until, for one of the transmissions, all $N$ acknowledgments

are received. Then it will proceed to broadcast the next packet.

Let $p$ be the probability that a given receiver successfully receives a packet from the transmitter

and successfully acknowledges that packet reception. Assume that the events of packet transmis$-$

sion$/$acknowledgment success are independent between receivers or for separate transmission at$-$

tempts.

$($a$)$ For the protocol described above, where we require that all $N$ of the acknowledgments be received

in response to a given transmission attempt, find $($for arbitrary $m)$ the probability that a packet is

successfully broadcast in $m$ or fewer attempts.

$($b$)$ Suppose that we change the protocol such that the transmitter only needs to receive acknowledg$-$

ments from those receivers for which it has not had a positive acknowledgment for this packet on

a prior transmission. $($For example, suppose there are three receivers: $A,B,$ and C $.$ If receiver $A$

gets the packet on the first transmission and sucessfully acknowledges such, but $B$ and $C$ do not,

the transmitter will retransmit. On this second transmission, the transmitter only looks for successful

acknowledgments from $B$ and $C$ to declare successful broadcast of this packet.$)$ Find $($for arbitrary

$m)$ the probability that a packet is successfully broadcast in $m$ or fewer attempts.