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Unformatted text preview: UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008 1 EE 550: Problem Set # 7 Solution I. SLOTTED TIME AND PERIODIC SERVICE OPPORTUNITIES 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 MT ) W = W q + T = MT 2(1 MT ) + T The system is stable whenever < 1 / ( MT ) . II. COMPARISON OF MULTIACCESS PROTOCOLS IN A NUSER SYSTEM 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 i N ) for i { 1 ,...,N } The system is stable for all ( 1 ,..., N ) that satisfy 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 i E { V } ) W i = 1 + E V 2 2 E { V } (1 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 dont remember the value of E V 2 for a geometric. Lets 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 ( 1 ,...,,....
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This note was uploaded on 12/21/2010 for the course EE 550 taught by Professor Neely during the Fall '08 term at USC.
 Fall '08
 Neely

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