Two nodes, A and B, are each attempting to transmit one small frame to each other over a shared ethernet medium. Assume the signal propagation delay is zero, and a small frame takes 50 s to transmit, in the absence of collisions. But when A and B both first attempt to transmit, a collision occurs, both A and B detect the collision, and thus both A and B must attempt to retransmit. 

(a) Ethernet uses an exponential-backoff scheme upon collision detection, but let's first analyze a simpler scheme: each node picks, independently and at random, to use either a 5 s delay or a 10 s delay, waits that amount of time, then attempts to retransmit. What is the chance they will have a second collision during this retransmission?

(b) What is the chance they will have three consecutive retransmission collisions (i.e. the initial collision, plus three additional collisions due to bad luck for the retransmissions)?

(c) After the initial collision, approximately how long, on average, will one of the nodes need to wait before it successfully begins to retransmit without causing a collision? Hint: calculate how many re-collisions there will be, on average. You can solve this numerically in a spreadsheet, by just enumerating a few of the most common scenarios, or you can derive an exact formula if you are familiar with probabilities.

(d) Suppose there are now three nodes, A, B, and C, and all are attempting to transmit using this same simplified retransmission scheme (picking, at random, to wait either 5 s or 10 s). Re-calculate your answer to (c) in this new situation. Make simplifying assumptions if needed. Hint: again, you can calculate this exactly if you are familiar with probabilities, or you can use a spreadsheet of the most likely scenarios, or even write a small program to simulate the scenario. Justify your answer.

UPDATE: For this question, we are interested in knowing how well (or poorly) this simplified ethernet protocol would behave in the situation where A, B, and C all collide. With A, B, and C each trying to send one packet, does it take a very long time before they are all eventually successful? Or do they quickly resolve the collision and finish sending after just a few attempts? There are lots of subtle nuances here: what if A, B, and C all pick 5 s as their waiting period? What if A picks 5 s but both B and C pick 10 s (meaning that A has now finished, but B and C still need to try again to resolve their collision). What if A and C pick 5 s, but B picks 10 s? Make some simplifying assumptions, try to figure out what the likely scenarios might be, and justify your reasoning. It's okay to not have a fully precise/quantitative answer here, or to approximate generously.