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ps7sol_ee550_fall08

ps7sol_ee550_fall08 - UNIVERSITY OF SOUTHERN CALIFORNIA...

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UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008 1 EE 550: Problem Set # 7 Solution I. S LOTTED T IME AND P ERIODIC S ERVICE O PPORTUNITIES Define a frame as the interval of time (of size MT ) that starts with a service opportunity and lasts until the start of the next service opportunity. Note that any arriving packet arrives within a frame, waits for the end of the frame, then waits for an additional N q frames before it can start service. Thus: W q = R + N q MT where R = MT/ 2 is the residual time of a frame. Therefore: W q = MT 2(1 - λMT ) W = W q + T = MT 2(1 - λMT ) + T The system is stable whenever λ < 1 / ( MT ) . II. C OMPARISON OF M ULTI -A CCESS P ROTOCOLS IN A N -U SER S YSTEM a) Each queue i is equivalent to the queue in problem 1 with M = N and T = 1 . Thus, we have: W i = 1 + N 2(1 - λ i N ) for i ∈ { 1 , . . . , N } The system is stable for all ( λ 1 , . . . , λ N ) that satisfy λ i < 1 /N for all i . b) For a given queue i , the success probability is equal to q i , where: q i = (1 /N )(1 - 1 /N ) N - 1 Because this does not depend on i , just call this q (i.e., q i = q for all i ). For a given queue i , service slots occur i.i.d. every slot with probability q . Similar to problem 1, we can define a (variable length) frame as the duration of time between the start of a service slot and the start of the next service slot. Note that each frame k has an integer size V k , where { V k } are i.i.d. and geometrically distributed with success probability q . Thus: W q = R + N q E { V } where R = E V 2 / (2 E { V } ) . Thus: W q,i = E V 2 2 E { V } (1 - λ i E { V } ) W i = 1 + E V 2 2 E { V } (1 - λ i E { V } ) It remains only to compute E { V } and E V 2 . Since V is geometric with success probability q , we have: E { V } = 1 /q I don’t remember the value of E V 2 for a geometric. Let’s derive it: E V 2 = q (1) + (1 - q ) E (1 + V ) 2 = q + (1 - q )(1 + 2 E { V } + E V 2 ) = 1 + 2(1 - q ) /q + (1 - q ) E V 2 Thus: E V 2 = 1 + 2(1 - q ) /q q = 2 /q 2 - 1 /q The system is stable for any rate vector ( λ 1 , . . . , λ N ) such that: λ i < q for all i ∈ { 1 , . . . , N } c) We have: W q = R + N q + GV R + λ W q + NV
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UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008 2 where λ = λ 1 + . . . + λ N . Thus:
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