20092ee132B_1_hw4sol

20092ee132B_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 ( 29 ( 29 ( 29 ( 29 { } ( 29 0 0 (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 R R n R R n n k R N N p p E N nP N n n p p p N = = = = = - = - ( 29 { } 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 k k R R R k k k M N M N N k M = = = = ( 29 { } 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 = { } ( 29 ( 29 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 P N P N = = = = = = = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1 1 2 3 1 2 1 1 2 1 2 3 2 , 0,. .., 0 ... 0,. .., 0, 2 1, 1, 0,. .., 0 ... 0,. .., 0, 1, 1 2, 0,. .., 0 1, 1, 0,. .., 0 1 2 1 1 1 M M M R R R R R M M M M R R R R R R R R M M R R R R R R R N N P N N N P N N N N P N N N N M M P N N N P N N N N M p p - - - = = + + = = = + = = = = + + = = = = = = = = + = = = = = - - ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 2 1 1 1 2 1 1 . 2 M M M M p p p p p p M M 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 ( 29 ( 29 ( 29 ( 29 ( 29 1 c c r P A A B P A P A B p p q = = + = + - I U I 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: ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 R p p q r E N r p q + - = = - - - (2) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 1 2
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20092ee132B_1_hw4sol - UCLA Electrical Engineering...

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