Random Access Channel

Three users are competing for access to the random access channel (RACH) of a cellular communication system. A simple ALOHA protocol is used to control access to this channel, i.e., in a given time slot a user attempts to transmit with some probability $p$ and the base station informs mobiles upon successful reception via a message on the access grant channel (AGCH).

1. Assume first that a successful random access can occur if and only if exactly one transmission attempt is made. If three users are executing the above protocol to access the RACH, what is the probability of a success?
2. Assuming no additional users attempt to access the same RACH, what is the expected number of time slots until all three users have successfully transmitted their random access message?
3. How would you select the transmit probability $p$ in light of your results?
4. A more realistic model for the RACH is as follows. A single transmission is successful with probability $0.9$ (transmission errors), while one message is successfully received when two messages are transmitted with probability $0.5$. When all three users transmit in the same time slot, one of the three messages will be received correctly by the base station with probability $0.3$. The non-zero probabilities in the latter two cases are due to a phenomenon called capture.

   If $p=1$, what is the expected number of time slots until all users have successfully accessed the RACH?

5. For a general $p$, what is the expected number of time slots until all three users have successfully accessed the RACH?
6. How would you select $p$ for this channel?