A computer network consists of several stations connected by various media $($usually cables$)$ There are certain instances when no message is being transmitted. At such "suitable instances", each station will send a message with probability $p,$ independently of other stations. However if two or more stations send messages, a collusion will corrupt messages and they will be discarded. These messages will be retransmitted until they reach their destination. Suppose network consists of $N$ stations.

a$)$ What is the probability that at a "suitable instance" a message is initiated by one of the stations and will go through without a collision?

b$)$ Show that, to maximize the probability of a message going through with no collisions, mean value of initiated messages should be exactly one message.