$9.$ Consider two nodes A and B on a shared link, and both nodes need to transmit frames to node C$,$ which is also on the same broadcast link. Assume that there are no other nodes on the shared link that currently want to transmit packets. Assume that we use the Slotted Aloha protocol. Suppose that a collision just occurred at time t $,$ when A and B transmitted in the same slot.

i$)$ In this part, assume that each node retransmits the collided frame with probability p in each successive slot after $\backslash ($ t $\backslash ).$ Express the probability of having a successful time slot in terms of p $.$ Find the optimum value of p $,$ which maximizes the probability of a successful slot, and the resulting maximum value for probability of a successful time slot.

ii$)$ Calculate the average number of time slots elapsed after time $\backslash ($ t $\backslash )$ until both nodes successfully transmit their respective frames.