Consider a wired communication network of strategic importance. As shown in the figure below, the network consists of 3 stations: A, B, and C, and 3 links: L1, L2, and L3. Messages can be communicated from station B to station C, using either link L2 or link L3. Messages from station A can be communicated to station C by using either of the combinations, {1, L2) or {L1, L3). Assume that the reliability (defined as 1-probability of failure) of links L1, L2 and L3 is 70%, 80%, and 60% respectively. Assume that the probability of failure of each link is independent of that of any other link. L2 L2 L1 L1 L3 L3 L4 Parts a, b and c Part d Only for this part of the problem, assume that a message dispatcher is not aware of whether or not any link in the network has failed. For a single attempt at dispatching a message from station A to station C, assume that a message dispatcher randomly chooses between links L2 and L3 with probability 50% each. what is the probability that the message actually gets delivered to station C in that single attempt? a. b. For this and all subsequent parts of this problem, assume that the message dispatcher is well aware of the failure status of each link, i.e., the dispatcher is knows whether or not each link in the network has failed. So if only one of L2 and L3 fails then the dispatcher is able to select the other link for dispatching a message. What is the probability that a message from station A actually gets delivered to station C? By how much does the knowledge of link failure statuses increase the probability of the message actually getting delivered? (i.e., how much is this answer higher than that of part a?) The network operator wants to reconfigure the network so as to maximize the probability of a message actually getting delivered from station A to station C. The same three links L1, L2 and L3 with reliabilities of 70%, 80% and 60% respectively need to be used in the reconfigured network and there should be exactly one link from station A to station B and exactly two links between station B and station C. But we are free to decide which of the three links should be from station A to station B. Find the best network configuration that will maximize the probability of a message from station A to station C being delivered. As before assume that a message dispatcher is well aware of the failure status of each link before dispatching the message. C. d. An additional direct link L4 is now added between stations A and C. But L4 has only 40% reliability and the probabilities of failure of each of the four links L1, L2, L3 and L4 are independent of each other. Assuming again that a message dispatcher is well aware of the status of each link, what is the probability that a message actually gets delivered from station A to station C
