A scalefree network is heterogeneous, as the degrees of the nodes, have a negative expo
nential distribution:
p
(
k
) =
1
ζ
(
γ, k
min
)
k

γ
.
(21)
In this expression
k
min
is the smallest degree of any vertex and for the applications we discuss
in this paper
k
min
= 1;
ζ
is the Hurvitz zeta function.
p
(
k
) is the probability that a vertex
has degree
k
, in other words it is connected to other
k
vertices.
Thus, the number of nodes with a high degree of connectivity is relatively small as we can
see from Table 6.
Table 6: A powerlaw distribution with degree
γ
= 2
.
5; the probability,
p
(
k
), and
N
k
, the
number of vertices with degree
k
, for a network with a the total number of vertices
N
= 10
8
.
k
p
(
k
)
N
k
k
p
(
k
)
N
k
1
0.745
74
.
5
×
10
6
6
0.009
0
.
9
×
10
6
2
0.131
13
.
1
×
10
6
7
0.006
0
.
6
×
10
6
3
0.049
4
.
9
×
10
6
8
0.004
0
.
4
×
10
6
4
0.023
2
.
3
×
10
6
9
0.003
0
.
3
×
10
6
5
0.013
1
.
3
×
10
6
10
0.002
0
.
2
×
10
6
An organization where these nodes are selected as controllers has several advantages:
•
There is a natural way to choose controllers,
c
i
is a controller if deg(
c
i
)
> κ
, with
κ
a
relatively large number, e.g.
κ
= 10.
•
There is a natural way to cluster the nodes around the controllers; a node
q
joins the
cluster built around the controller
c
i
at the minimum distance
d
(
q, c
i
). Thus, the average
distance between the controller and the servers in the cluster is minimal.
•
A number of studies have shown that scalefree networks have remarkable properties
such as: robustness against random failures [12], favorable scaling [2, 3], resilience to
congestion [32], tolerance to attacks [80], small diameter [18], and small average path
length [11].
Problem 7.
Use the starttime fair queuing (SFQ) scheduling algorithm to compute the
virtual startup and the virtual finish time for two threads
a
and
b
with weights
w
a
= 1
and
w
b
= 5
when the time quantum is
q
= 15
and thread
b
blocks at time
t
= 24
and wakes up at
time
t
= 60
. Plot the virtual time of the scheduler function of the real time.
51
As in Problem 6, we consider two threads with the weights
w
a
= 1 and
w
b
= 5 and the
time quantum is
q
= 15, and thread
b
blocks at time
t
= 24 and wakes up at time
t
= 60.
Initially
S
0
a
= 0,
S
0
b
= 0,
v
a
(0) = 0, and
v
b
(0) = 0. The scheduling decisions are made as
follows:
1. t=0
: we have a tie,
S
0
a
=
S
0
b
and arbitrarily thread
b
is chosen to run first; the virtual
finish time of thread
b
is
F
0
b
=
S
0
b
+
q/w
b
= 0 + 15
/
5 = 3
.
(22)
2. t=3
: both threads are runnable and thread
b
was in service, thus,
v
(3) =
S
0
b
= 0; then
S
1
b
= max[
v
(3)
, F
0
b
] = max(0
,
3) = 3
.
(23)
But
S
0
a
< S
1
b
thus thread
a
is selected to run. Its virtual finish time is
F
0
a
=
S
0
a
+
q/w
a
= 0 + 15
/
1 = 15
.
(24)
3. t=18
: both threads are runnable and thread
a
was in service at this time thus,
v
(18) =
S
0
a
= 0
(25)
and
S
1
a
= max[
v
(18)
, F
0
a
] = max[0
,
15] = 15
.
(26)
As
S
1
b
= 3
<
12, thread
b
is selected to run; the virtual finish time of thread
b
is now
F
1
b
=
S
1
b
+
q/w
b
= 3 + 15
/
5 = 6
.
(27)
4. t=21
: both threads are runnable and thread
b
was in service at this time, thus,
v
(21) =
S
1
b
= 3
(28)
and
S
2
b
= max[
v
(21)
, F
1
b
] = max[3
,
6] = 6
.
(29)
As
S
2
b
< S
1
a
= 15, thread
b
is selected to run again; its virtual finish time is
F
2
b
=
S
2
b
+
q/w
b
= 6 + 15
/
5 = 9
.
(30)
5. t=24
: Thread
b
was in service at this time, thus,
v
(24) =
S
2
b
= 6
(31)
S
3
b
= max[
v
You've reached the end of your free preview.
Want to read all 96 pages?
 Summer '17
 ALBERT DOMINIC
 The Land, cloud service, Cloud delivery models, Cloud Computing delivery