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20105ee132B_1_hw8sol

# 20105ee132B_1_hw8sol - UCLA Electrical Engineering...

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UCLA Electrical Engineering Department EE132B HW Set Solution #8 Professor Izhak Rubin Solution to Problem 1 (a) This is a Geom/Geom/1 Queuing System with q = 0.85 msg/slot. (i) p = 0.4 msg/slot a = p(1-q)/(q(1-p))=0.11765 E(X) = a/(1-a) = 0.1333 E(Q) = E(X) a = 0.0157 E(W) = a/(q(1-a)) = 0.15686 slots (ii) p = 0.8 msg/slot a = p(1-q)/(q(1-p))=0.70588 E(X) = a/(1-a) = 2.4 E(Q) = E(X) a = 1.694 E(W) = a/(q(1-a)) = 2.82 slots (iii) p = 10.0 msg/slot Since a > 1, the queuing system is unstable. E(X) = E(Q) = E(W) = (b) P(W>5) = a((1-q)/(1-p)) 5 (i) P(W>5) = 1.15*10 -4 (ii) P(W>5) = 0.1675 (iii) P(W>5) = 1

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Solution to Problem 2 (a) Let L denote the number of bits per message. The random variable L has a geometric distribution with a mean of 500 bits such that P(L=k)=r(1-r) k-1 for k 1 and r = 1/500. Let R denote the transmission rate. A slot is set to be equal to г seconds. Assume that R г is a positive integer. Note that R г has units of bits/slot. Then, the number of slots required to transmit a message is L S R . Using the expression given above, we obtain 1 1 1 1 1 1 1 1 1 k R
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