21. 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 the other stations. However, if two or more stations send messages, a collision will corrupt the messages and they will be discarded. These messages will be retransmitted until they reach their destination. Suppose that the 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, exactly one message, on average, should be initiated at each “suitable instance.”

(c) Find the limit of the maximum probability obtained in

(b) as the number of stations of the network grows to ∞.