Problem 24

N transmitters share the same communication channel. On the hour, each transmitter will use the channel with probability p independent of the actions of the other transmitters. The duration of each transmission is exactly one hour. Each hour one of three situations can occur:

1.

None of the transmitters uses the channel. The channel remains unused until the next hour.

2.

Exactly one of the transmitters uses the channel and no errors occur in the transmission.

3.

More than one transmitter uses the channel. The channel can be used by only one transmitter without incurring errors. In this situation, the simultaneous transmissions interfere with each other and all transmissions are garbled during the next hour.

1.

What is the probability of each of these situations occuring?

2.

What value of p maximizes p_x, the probability of an error-free transmission? What value does p_x approach as N becomes large?

3.

Using the value of p found above, what is the probability that n hours pass between times the channel is used successfully?

4.

Let N=2, what is the average amount of time the channel is used successfully during any given hour?