UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008 1 EE 550: Problem Set # 8 Due: Monday, November 24, 2008 I. SLOTTED ALOHA WITH CAPTURE Book problem 4.7. II. SLOTTED CSMA Book problem 4.21. III. UNSLOTTED ALOHA Book problem 4.12. IV. SPLITTING ALGORITHMS Book problem 4.13(a)(b). V. MORE ON SPLITTING ALGORITHMS Book problem 4.15. VI. STABILIZING A SWITCH WITH UNIFORM TRAFFIC Consider an N × N packet switch with virtual input queueing. Suppose trafﬁc arrives according to a continuous time Poisson process with uniform rates, so that λ ij = ρ/N for all input/output pairs ( i,j ) (for some value ρ < 1 ). Slot sizes are normalized to one unit of time. There are N ! possible permutation matrices M 1 ,M 2 ,...,M N ! . Consider the randomized scheduling policy of scheduling according to a randomly chosen permutation matrix (chosen with equal probability over all possible permutations). (a) For a single virtual input queue ( i,j ) , compute the probability that a server is allocated to this queue on a given slot. (b) Show that all virtual input queues are stable, and compute an exact expression for average delay (Hint: Use a slotted
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