Consider nodes A$,$ B and C on a shared $10$Mbps wireless channel operating CSMA$/$CA$,$ as

shown in the figure below. Suppose node A has one $128-$byte frame to send to node B$,$ and

node C hasone$250-$byte frame to send to node B$.$ The propagation delay between A and B is

$5$ bit time; the propagation delay between C and B is $10$ bit time. Suppose DIFS time is $50$

microseconds and SIFS time is $10$ microseconds. The system starts at time $0.$ Both node A

and node C begin transmitting its respective frame to B after one DIFS time, and then

collision occurs at B$.$ The CSMA$/$CA protocol at A and C follows the conventions that: $($i$)$ after

the $($collided$)$ frame transmission, A $($or C$)$ will wait for an amount of time equaling

propagation delay$+$SIFS time$+$ACK transmission time$+$ propagation delay for ACK; $($ii$)$ if no

ACK is received by then, A $($or C$)$ will attempt to retransmit after sensing the channel idle for

DIFS time and then waiting for $512(2$k$-1)$ bit time, where K$=1$; A $($or C$)$ will repeat steps $($i$)($ii$)$

with K being the number of its retransmissions, until ACK for its frame transmission is

successfully received. At B$,$ when it has successfully received a frame, it will send a $100-$bit

ACK after SIFS time. One frame$$s transmission is successful if B has received the frame and

sent its ACK out without collision $($i$.$e$.,$ without receiving signal of the other frame

transmission$).$

$(1)$ What is the start time $($in microseconds$)$ of the successful retransmission from A$,$ after

which A successfully transmits its entire frame and receives ACK without collision? What is

the start time $($in microseconds$)$ of the successful retransmission from C$,$ after which C

successfully transmits its entire frame and receives ACK without collision?

$(2)$ Let T be the time when both A and Chavecompleted the successful transmission of their

respective frame and received their respective acknowledgement. Calculate the efficiency of

the CSMA$/$CA channel during the time interval $[0,$ T$]$