Advance computer Communication

ECE 533: Advanced Computer Communication Networks (Fall '21)

Take-Home Quiz #6 due on 11/12/2021 @ 1pm

1) Consider the network in the following. Assume that the link capacities over L1, L2, L3, L4, and L5 are equal and 1packet/sec. Sources S0, S1, S2 are associated with transmission rates x0, x1, x2. Considering log utility, find the optimal rate allocation.

2) Prove that at time slot t and state m, 𝑚𝑎𝑥𝛼 ∑ 𝜀𝑞𝑖 𝑁 𝑖=1 (∑ 𝜋𝑚(∑ 𝛼𝑚,𝑟𝑟𝑖𝑟∈𝑅𝑚 )𝑚 ) is equivalent to

𝑚𝑎𝑥𝑟∈𝑅𝑚 ∑ 𝑟𝑖(𝑡)𝑞𝑖(𝑡) 𝑁 𝑖=1 .

3) Consider the line network in the following, where both links (1, 2) and (2, 3) are ON- OFF channels such that the probability to be ON is 0.5. One packet can be transmitted when a link is ON.

Assume the utility functions associated with the three flows are Ui(xi) = -wi/xi, where i = a, b, c

and wi is a constant. Please write down the network utility maximization problem, and solution

for the following cases.

(a) Links 1-2 and 2-3 are wired links (i.e., there is no interference between these two links). (b) Links 1-2 and 2-3 are wireless links (i.e., the two links interfere).