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ps6sol_ee550_fall08

# ps6sol_ee550_fall08 - UNIVERSITY OF SOUTHERN CALIFORNIA...

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UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008 1 EE 550: Problem Set # 6 Solution I. RANDOM WALK ON A FINITE CONNECTED GRAPH We guess p i = Ad i /v i , where A is a normalization constant, which can be calculated by N X i =1 p i = 1 A = 1 N i =1 d i /v i p i = d i /v i N i =1 d i /v i . Next we examine the detailed balance equations for any two node i , j on the graph. If i and j are not neighbors, then q ij = q ji = 0 and obviously the DBE holds. Otherwise, p i q ij = A d i v i v i d i = A, and similarly p j q ji = A . Thus the DBE holds. It concludes that the system is a reversible Markov chain. (The DBEs also conclude that { p i } is a steady state distribution of the random walk.) II. WAVELENGTH CONTINUITY CONSTRAINT (a) Consider the following example: (1) Session 1 arrives, using BLUE. (2) Session 2 arrives, using BLUE. (3) Session 3 arrives, using RED. (4) Session 2 departs. (5) Session 4 arrives. Then it is easy to see that session 4 will be blocked although there is a free wavelength on each link. (b) If session 4 is not ON, we can use any wavelength for S1, and use BLUE for S2 and RED for S3. If session 4 is ON, give it BLUE. Then we use RED for S1, and use RED for S2 and S3. (Note: S2, S3, and S4 can never be ON together, as this would violate the circuit switch bound at link 3.) (c)

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ps6sol_ee550_fall08 - UNIVERSITY OF SOUTHERN CALIFORNIA...

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