20105ee132B_1_hw4sol

20105ee132B_1_hw4sol - UCLA Electrical Engineering...

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UCLA Electrical Engineering Department EE132B HW Set Solution #4 Professor Izhak Rubin Solution to Problem 1             00 (1) Let 1 denote the number of retransmissions. Then, the random variable 1 has a geometric distribution with parameter . Thus, we have 1 1 1 1 (2) Let denote RR n nn k R NN p p E N nP N n n p p p N          1 the number of retransmissions for the data frame, 1,. .., . Then, . Note that , 1,. .., is a set of independent and identically distributed random variables. These random variab th M kk R R R k k k M N M N N k M     les are governed by a geometric distribution with parameter . For to be equal to 2, there are two possible cases: (i) One of the random variables in , 1,. .., is equal to 2, and other rando R k R p N M N k M       1 1 m variables are all equal to 0. (ii) Any two of the random variables , 1,. .., are each equal to 1, and the other random variables are all equal to 0. Hence, we have: 2 2 2 k R R M k R k R N k M P N M PN                  2 1 1 1 2 3 1 2 1 1 2 1 2 3 2 , 0,. .., 0 ... 0,. .., 0, 2 1, 0,. .., 0 ... 0,. .., 0, 1 2, 0,. .., 0 0,. .., 0 12 11 1 M M M R R R R R M M M M R R R R R R R R MM R R R R R R R N N P N N N P N N N N P N N N N P N N N P N N N N M pp              2 1 1 1 2 1 1. 2 M M p p p p p p     

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Solution to Problem 2 (1) Note that when station A transmits a data frame, it may not receive an acknowledgement if the frame is delivered to station B with errors or if the acknowledgement for the frame which is received without errors is lost. Let r denote the probability that station A does not receive an acknowledgement. Let A and B denote the events that a frame is received with errors and that an acknowledgement is lost, respectively. Then, we have           1 c c r P A A B P A P A B p p q  Whenever an acknowledgement is not received, the data frame is retransmitted. Therefore, the number of retransmissions for a data-frame N R (1) has a geometric distribution with parameter r . The average number of retransmissions is given by:          1 1 1 1 1 R p p q r EN r p q   (2)            2 2 1 2 1 1 1 1 R P N r r p q p p q    
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This note was uploaded on 01/12/2011 for the course EE 132B taught by Professor Izhakrubin during the Fall '09 term at UCLA.

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20105ee132B_1_hw4sol - UCLA Electrical Engineering...

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