$5)[20$ points$]$

A$.$ Consider two nodes, A and B$,$ that use the slotted ALOHA protocol to contend for a channel. Suppose node A has more data to transmit than node B$,$ and node A$\text{'}$s retransmission probability $\backslash ($ p$\_\{\backslash$mathrm$\{$A$\}\}\backslash )$ is greater than node B $\text{'}$s retransmission probability $\backslash ($ p$\_\{\backslash$mathrm$\{$B$\}\}\backslash )$

i$.$ Provide a formula for node A$\text{'}$s average throughput. What is the total efficiency of the protocol with these two nodes? $(5$ points$)$

ii$.$ If $\backslash ($ p$\_\{\backslash$mathrm$\{$A$\}\}=2$ p$\_\{\backslash$mathrm$\{$B$\}\}\backslash )$ is node A$\text{'}$s average throughput twice as large as that of node B $?$ Why or why not? If not, how can you choose $\backslash ($ p$\_\{\backslash$mathrm$\{$A$\}\}\backslash )$ and $\backslash ($ p$\_\{\backslash$mathrm$\{$B$\}\}\backslash )$ to make that happen? $(5$ points$)$

B$.$ Suppose four active nodes, nodes A$,$ B$,$ C and D$,$ are competing for access to a channel using slotted ALOHA. Assume each node has an infinite number of packets to send. Each node attempts to transmit in each slot with probability p $.$ The first slot is numbered slot $1,$ the second slot is numbered slot $2,$ and so on$.$

i$.$ What is the probability that node A succeeds for the first time in slot $5?(5$ points$)$

ii$.$ What is the probability that the first success occurs in slot $3?(5$ points$)$