ps8sol_ee550_fall08

ps8sol_ee550_fall08 - UNIVERSITY OF SOUTHERN CALIFORNIA...

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UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008 1 EE 550: Problem Set # 8 Solution I. S LOTTED A LOHA WITH CAPTURE — B OOK PROBLEM 4.7 (a) Note that one packet successfully leaves the system each slot in which one or more packets are transmitted. Thus if all waiting packets attempt transmission in every slot, a successful transmission occurs in every slot in which packets are waiting. Since the expected delay is independent of the order in which packets are successfully transmitted (since each packet requires one slot), we see that the expected delay is the same as that of a centralized slotted FCFS system. Now compare this policy with an arbitrary policy for transmitting waiting packets; assume any given sequence of packet arrival times. Each time the arbitrary policy fails to attempt a transmission in a slot with waiting packets, the FCFS system (if it has waiting packets) decreases the backlog by 1 while the other policy doesn’t decrease the backlog. Thus the backlog for the arbitrary system is always greater than or equal to that of the FCFS system (a formal proof of this would follow by induction on successive slots). Thus, by Little’s theorem, the arbitrary system has an expected delay at least as great as the FCFS system. (b) This is just the slotted FDM system of section 3.5.1 with m = 1 (i.e., a slotted M/D/1 queueing system). From Eq. (3.58), the queueing delay is 1 / [2(1 - λ )] slot times. The total delay, including service time, is then 1 + 1 / [2(1 - λ )] . (c) The solution to (b) can be rewritten as 1 + 1 / 2 + λ/ [2(1 - λ )] , where the first term is the transmission time (i.e., 1 slot), the second term is the waiting time from an arrival to the beginning of a slot, and the third term is the delay due to collisions with other packets. If each subsequent attempt after an unsuccessful attempt is delayed by k
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