Consider the somewhat unrealistic feedback assumption for unslotted Aloha in which all nodes are informed, precisely τ time units after the beginning of each transmission whether or not that transmission was successful. Thus, in the event of a collision, each node knows how many packets were involved in the collision, and each node involved in the collision knows how many other nodes started transmission before itself. Assume that each transmission lasts one time unit and assume that m =∞. Consider a retransmission strategy in which the first node involved in a collision waits one time unit after receiving feedback on its collision and then transmits its packet. Successive nodes in the collision retransmit in order spaced one time unit apart. All new arrivals to the system while these retransmissions are taking place wait until the retransmissions are finished. At the completion of the retransmissions, each backlogged node chooses a time to start its transmission uniformly distributed over the next time unit. All new arrivals after the end of the retransmissions above start transmission immediately.
(a) Approximate the system above as a reservation system with reservation intervals of duration 1+ τ (note that this is an approximation in the sense that successful transmissions will sometimes occur in the reservation intervals, but the approximation becomes more accurate as the loading becomes higher). Find the expected packet delay for this approximation (assume Poisson arrivals at rate λ).
(b) Show that the delay above remains finite for all λ <>