1-2. A student is taking two important classes. There is only
time t left for studying for both classes, so if the student
spends time t1 studying for the first class, this student
will have time t2 = t − t1 left for the
The student wants to maximize both grades f1(t1)
- The grade for the first class depends on the time
t1 as f1(t1) =
(t − t1)2.
- The grade for the second class depends on the time
t2 as f2 =
(t − t2)2, i.e., since
t2 = t − t1, as
- What is the solution to this multi-objective optimization
problem when we select the equal weights w1 = w2
- What is the solution to in the general cases,
when we select an arbitrary value w1 from the interval [0, 1], and
take w2 = 1 − w1?
- Why multi-objective optimization is important for cloud computing?
3. Let us now assume that every day, the company uses
at least m = 100 computations, and the probabilities of different
numbers of computations x is described by the power law
p(x) = A * x−α, with α = 3. Assume that
the cost of a unit in-house computation is c0 = 10 money
computation unit and the cost of computing in the cloud is
c1 = 20 money units per computation unit.
much computing power x0 should be purchased for in-house
- What is the resulting expected cost?
- Compute the expected costs when we do all the computations
exceeding m in the
cloud, and show that the optimal arrangement indeed saves money.
In these computations, you can use the formulas that we derived in
class for the power law case:
- the optimal amount of computing power to purchase is
x0 = m * (c1 /
c0)1 / (α − 1);
- the resulting expected cost is equal to
((α − 1) / (α − 2)) *
c0(α − 2) / (α − 1) *
c11 / (α − 1) * m;
- if we do all the computations exceeding m in the cloud, then
the resulting cost is
m * (c0 +
c1 / (α − 2)).
4. Estimate the costs and decide whether it is beneficial to
sign a contract with the cloud provider for T = 2 years:
- the cost of buying a unit of
computations on a year-by-year basis is c0 = 1;
- the contract
offer a discount price c1 = 0.9,
- the discount rate is q = 0.8, and
- the price of computing decreases yearly by a factor of v = 0.9.
5. Neural networks:
- Why neural networks are needed for cloud computing?
- Let us assume that we have are training a neural network with K = 2
learn multiplication. This means that for inputs x1 = 2.0 and
x2 = 3.0, the desired output is Y = 6.0. On a certain iteration, we have
wki = 0 for all k and i, so that the outputs of the non-linear
neurons are equal to 0.5: y1 = y2 = 0.5. Suppose
that at this iteration, W1 = W2 = 1.0 and
W0 = 0.0. Assuming that α = 0.1, describe the values
of all the weights on the next iteration.
For each pattern with the known desired output Y, once the
NN computed the input y, you can find the error Δy = y
− Y. Based on this error, you can compute the changes to all
the weights as follows:
- In the first
layer, we have K non-linear neurons. Each neuron k (k = 1, 2, ...,
K) transforms the input signals x1, ..., xn
into a signal
yk = s0(wk1 *
x1 + ... + wkn * xn −
where s0(z) = 1 / (1 +
- The last linear neuron then transform these
signals yk into a single output
y = W1 *
y1 + ... + WK
* yK − W0.
- ΔW0 = α *
Δ y, for some small number α > 0;
ΔWk = − yk *
- Δwk0 = −
ΔWk * Wk * (1 − yk);
- Δwki = − xi
6. Fuzzy techniques:
- Why fuzzy techniques are needed for cloud computing?
- Briefly describe the main steps of fuzzy techniques.
- Which "and"-operations are typically used in fuzzy techniques? If
the expert's degree of belief in a statement A is 0.6 and the expert's
degree of belief in a statement B is 0.8, what is the expert's degree
of belief in the statement "A and B"?
7. What is green computing? Explain the need for saving energy in cloud
8-9. Describe, in detail, the paper that you reviewed as a
project for this class:
- what problem is addressed in
- what solution is proposed for this problem, and
- (if applicable) what are the remaining open problems.
10. Briefly describe someone else's project for this class.