Consider the following Ethernet model. Transmission attempts are made at random times with an average spacing of λ slot times; specifically, the interval between consecutive attempts is an exponential random variable x = −λ log u, where u is chosen randomly in the interval 0 ≤ u ≤ 1.

An attempt at time t results in a collision if there is another attempt in the range from t − 1 to t + 1, where t is measured in units of the 51.

2-µs slot time; otherwise the attempt succeeds.

(a) Write a program to simulate, for a given value of λ, the average number of slot times needed before a successful transmission, called the contention interval. Find the minimum value of the contention interval. Note that you will have to find one attempt past the one that succeeds in order to determine if there was a collision. Ignore retransmissions, which probably do not fit the random model above.

(b) The Ethernet alternates between contention intervals and successful transmissions. Suppose the average successful transmission lasts 8 slot times (512 bytes). Using your minimum length of the contention interval from above, what fraction of the theoretical 10-Mbps bandwidth is available for transmissions?