1
Energy Aware Virtual Network Embedding
Sen Su Zhongbao Zhang
∗
Alex X. Liu
∗
Xiang Cheng Yiwen Wang Xinchao Zhao
Abstract—Virtual network embedding, which means mapping
virtual networks requested by users to a shared substrate
network maintained by an Internet Service Provider, is a key
function that network virtualization needs to provide. Prior
work on virtual network embedding has primarily focused on
maximizing the the revenue of the Internet Service Provider
and did not consider t he energy cost in accommodating such
requests. A s energy cost is more than half of the operating cost
of the substrate networks, while trying to accommodate more
virtual network requests, minimizing energy cost is critical for
infrastructure providers. In thi s work, we make the first effort
towards energy aware virtual network embedding. We first pro-
pose an energy cost model and formulate the energy aware virtu al
network embedding problem as an integer linear programming
problem. We then propose two efficient energy aware virtual
network embedding algorithms: a heuristic based al gorithm and
a particle swarm optimization technique based algorithm. We
implemented our algorithms in C++ and performed side-by-side
comparison with prior algorithms. The simulation results show
that our algorithms significantly reduce the energy cost by up
to 50% over th e existing algorithm for accommodating the same
sequence of virtual network requests.
Index Terms—Network virtualization, virtual network embed-
ding
I. INTRODUCTION
A. Background and Motivation
Network virtualization is the key technology that allows
multiple heterogeneous Virtual Networks (VNs) to coexist on
the same shared Substrate Network (SN). It brings three major
benefits. First, it enables resource sharing among these VNs
and makes most efficient use of the SN. Second, it offers
opportunities to design and evaluate new network protocols
and architectures. Third, it provides more flexibility to expand
or shrink the VN as needed.
This paper concerns the problem of VN embedding. Net-
work virtualization involves one Internet Service Provider
(ISP) and multiple users, where the ISP manages the physical
SN infrastructure while each user requests VNs from the ISP.
Each VN request consists of a network topology where each
node and edge have some constraints. The node constraints are
typically on capacity (such as CPU computing power, memory
∗
Zhongbao Zhang and Alex X. Liu are the corresponding authors of this
paper.
Some preliminary results of this paper were published in the First IEEE
INFOCOM Workshop on Communications and Control for Sustainable En-
ergy Systems: Green Networking and Smart Grids (INFOCOM WS-CCSES),
Orlando, FL, March 25-30, 2012 [1].
Sen Su, Zhongbao Zhang, Xiang Cheng, Yiwen Wang are with the State
Key Laboratory of Networking and Switching Technology, Beijing University
of Posts and Telecommunications, Beijing, China. Email: {susen, zhongbaozb,
chengxiang, wangyiwen}@bupt.edu.cn.
Alex X. Liu is with is with the Department of Computer Science and
Engineering, Michigan State University, East Lansing, MI, USA. E-mail:
alexliu@cse.msu.edu.
Xinchao Zhao is with the School of Science, Beijing University of Posts
and Telecommunications, Beijing, China. Email: xcmmrc@gmail.com.
and storage capacity, etc) and location (
i.e.
, the location in
the topology of the substrate network of the ISP). The edge
constraints are typically on communication bandwidth. When
the ISP receives a VN requests from users, the ISP needs to
map the VN to the physical nodes and links in its network,
which is called VN embedding.
B. Limitation of Prior Art
The VN embedding has received significant attention in
recent years. The primary goal of prior work is to maximize
the revenue of the ISP by accommodating more VN requests
on the same SN [2]–[8]. The key limitation of prior studies
is that they did not consider the energy cost for serving
VN requests. However, energy is a major cost for ISPs.
For example, in US, Akamai, one of the world’s leading
providers of content delivery networking services, has an
annual electricity cost of about $10 Million [9]. In China,
China mobile Communications Corporation, the largest mobile
service provider in the world, consumes over 13 TWH power
consumption in 2011 [10]. Telecom Italia, the second largest
consumer of electricity in Italia, consumes more than 2 TWh
per year, which is equivalent to the energy consumed by
660,000 families in one year [11]. Thus, to maximize the
net profit, the ISP needs to strike the right balance between
accommodating more VN requests and minimizing energy
costs for serving VN requests.
C. Proposed Approach
In this paper, we propose to tradeoff between maximizing
the number of VNs that can be accommodated by an ISP
and minimizing the energy cost of the whole system. For
each VN request, the ISP maps the VN to some physical
nodes and links in its network in such a way that the
amount of additional energy cost caused by accommodating
the VN request is minimized. This approach is based on two
observations. The first observation is that the substrate nodes
are usually geographically distributed to deploy and deliver
service to end users, and the electricity price may differ for
different locations and may fluctuate over time [9], [12]. Based
on this observation, an ISP should try to map the virtual nodes
of a VN to the physical nodes that have the lowest electricity
price while satisfying the location constraint of the VN. The
second observation is that the power consumption of a server
is approximately in linear to its CPU utilization with a large
offset, which equals up to nearly 50% of the peak power [13].
Based on this observation, an ISP should try to map the virtual
nodes of a VN to the physical nodes that are already actively
running; thus we can maximize the number of nodes that do
not have any load and therefore can be put to sleep to save
energy.
2
D. Technical Challenges a nd Proposed So lutions
The first technical challenge is on modeling and quantifying
the energy cost of the complex physical network infrastructure
of an ISP. Specifically, we need to model both electricity
price and energy consumption. For electricity price, we use
a discrete-time model to characterize the spot dynamics of
electricity price. For energy consumption, we first classify
substrate nodes into host nodes, which need to execute some
computational tasks, and router nodes, which need to forward
packets to and from host nodes. We further classify them
into active nodes, which need to be powered up, and inactive
nodes, which can be powered off to save energy. Then, for
different types of nodes, we build the corresponding energy
consumption models. Based on such models, we carry out
the quantitative analysis of the overall energy consumption,
including the energy consumption of virtual nodes and virtual
links for accommodating a VN request. After the modelings,
we can quantify the electricity cost by calculating the time
integral of the electricity price and the power consumption.
The second technical challenge is on designing energy
aware VN embedding algorithms. To address this challenge,
first, we model our energy aware VN embedding problem as
an integer linear programming problem. Second, we propose
a heuristic algorithm called EA-VNE for solving this problem.
This algorithm consists of two steps: node mapping and link
mapping. The node mapping step further consists of two
substeps: router node mapping and host node mapping. In
the router node mapping, we exploit the location- and time-
varying diversities of electricity prices to save energy cost.
To maximize the probability of performing successful link
mapping in the next step, we design a worst-fit strategy for the
bandwidth resources. In the host node mapping, we design a
best-fit strategy to minimize the number of hosting nodes and
make the best use of the resource while satisfying the node
requirements of the VN request. In the link mapping step,
we design an active nodes and router ports preferred shortest
path algorithm that tries to minimize the number of forwarding
nodes and ports. To further minimize energy cost, we design
an approximation algorithm called EA-VNE-EPSO, which is
based on the well known particle swarm optimization (PSO)
technique. Specifically, we treat a VN embedding solution
as a particle in PSO and thus each particle will achieve a
better and better embedding solution through the iteration
process by learning from the experience of other particles.
To accelerate the convergence of this iterative algorithm,
we propose an energy aware local selection strategy based
on the characteristics of VN embedding. Furthermore, we
propose a non-uniform mutation strategy to prevent premature
convergence.
E. Summary o f Experimental Results
We carry out extensive simulation and show that our algo-
rithms outperform the state-of-the-art algorithm in terms of
long-term average energy cost while gaining competitive rev-
enues for ISPs. While maintaining nearly the same revenues,
our algorithms EA-VNE and EA-VNE-EPSO save up to 40%
and 50% of energy cost than prior art, respectively.
F. Key Contributions
We make the following key contributions in this paper:
1) We make the first attempt to incorporate the energy
factor in performing VN embedding. We formulate an
energy cost model for studying the energy aware VN
embedding problem.
2) We design two VN embedding algorithms to reduce the
energy cost while keeping nearly the same revenue so
as to maximize the profit for the ISPs.
3) We conducted side-by-side comparison between our
algorithms and the state-of-the-art algorithm. We show
that our algorithms outperform the state-of-the-art algo-
rithm in terms of both long-term average energy cost
and revenues for ISPs.
The rest of the paper is organized as follows. In Section
II, we present the energy cost model and the energy aware
VN embedding problem formulation. In Section III and IV,
we present our heuristic and meta-heuristic energy aware
VN embedding algorithms, respectively. We evaluate our VN
embedding algorithms in Section V. Section VI reviews related
work. Finally, Section VII concludes the paper.
II. SYSTEM MODELING
In this section, we first present a network model. Second, we
formulate an energy model for VN infrastructure. Third, we
quantitatively analyze the energy consumption for accommo-
dating a VN request. Finally, we formulate the energy aware
VN embedding problem based on the model. The notations
used in this paper are summarized in Table I.
TABLE I: Notations
Notation Description
s, t Substrate nodes.
u, v Virtual nodes.
i(j) Substrate router (host) node.
r(h) Virtual router (host) node.
R
i
(R
r
) The residual (demanding) number of virtual routers.
C
j
(C
h
) The residual (demanding) CPU value.
M
j
(M
h
) The residual (demanding) memory value.
S
j
(S
h
) The residual (demanding) storage value.
B
st
(B
uv
) The residual (demanding) bandwidth value.
Dis(i, r) The Euclidean distance between r and i.
W The maximum accepted distance value for mapping a
virtual node to a substrate node.
x
r
i
(y
h
j
) A binary variable. x
r
i
represents router node mapping
while y
h
j
represents host node mapping. x
r
i
(y
h
j
) = 1 if
mapping r (h) to i (j) and 0 otherwise.
f
uv
st
A binary variable. f
uv
st
= 1 if mapping virtual link l
uv
to the physical link l
st
and 0 otherwise.
P S
i
(P S
j
) A binary variable. P S
i
(P S
j
) = 1 if i (j) is in active
state and 0 otherwise.
A. Network Modeling
A substrate network (SN) is represented by a weighted
graph G
s
= (N
s
, L
s
), where N
s
denotes the set of physical
nodes and L
s
denotes the set of physical links. The substrate
nodes can be classified into two categories: router nodes and
host nodes. That is, N
s
= (N
sr
, N
sh
), where N
sr
denotes the
set of routers and N
sh
denotes the set of hosts. Similarly, the
substrate links can also be classified into two categories: the
backbone link and the local link, denoted by L
sr
and L
sh
,
respectively. Fig. 1 (b) shows an SN example where circles
and rectangles denote router and host nodes, respectively.
3
(a) VN Request
18
H
A1
R
C
R
D
H
A2
H
C1
H
D1
R
a
H
a1
H
a2
R
b
H
b1
R
A
(b) Substrate Network
(12,20,15)
(1)
(20,35,15)
(1)
(25,18,30)
100
20
R
B
(75,79,35)
80
(65,97,48)
28
240
320
90
400
800
60
(15)
(20)
(25)
(20)
(32,45,56)
(54,32,58)
45
Fig. 1: Example of VN embedding
For router nodes, we consider the following three attributes.
The first attribute is the number of virtual routers that the
physical router can support for deploying and running different
personalized network protocols. In Fig. 1 (b), the numbers that
are in parentheses and besides each circle are such numbers.
The second attribute is the location that the router is located as
routers are generally geographically distributed. For example,
in Fig. 1 (b), R
A
may be located in Los Angeles, R
B
in
Chicago, R
C
in New York, and R
D
in New Jersey. We use
a 2-dimensional coordinate Loc(i) = (x
i
, y
i
) to denote the
location of node i. The third attribute is electricity price. The
power market of different locations are managed by different
Independent System Operators (ISOs) and the ISOs are under
competitive electricity market structure. Therefore, different
locations often have different electricity prices. Even for the
same location, the electricity price may vary frequently over
time [9], [12]. Fig. 2 shows the hourly electricity price of
the first week of Sep 2011 available at [14] for five regions
in the day-ahead market, including Eastern Hub of PJM
(Pennsylvania-Maryland-New Jersey), NP-15 Hub of CAISO
(California), Capital Hub of NYISO (New York), Mass Hub of
ISO-NE (New England) and Illinois Hub of MISO (Midwest).
We observe that the electricity price varies over both location
and time. To characterize the spot price dynamics, in this
paper, we use a discrete time model, which has a time window
(
e.g.
, an hour) of interest t ∈ 0, 1, ..., T . We use P r
i
(t) to
denote the electricity price for node i at time slot t, t
a
to
denote the arriving time of a VN request, and t
d
to denote the
duration of the VN request being served in the SN. Thus, the
expiration time t
e
of the VN request is t
e
= t
a
+ t
d
.
For host nodes, we consider three attributes: CPU speed,
memory size, and storage capacity. In Fig. 1 (b), the triple
besides each host node denotes the values of these attributes.
0
20
40
60
80
100
1 2 3 4 5 6 7 8
Electricity Price ($/MWh)
Day of Sep, 2011
Eastern Hub, PJM
NP-15, CAISO
Capital, NYISO
Mass, ISO-NE
Illinois, MISO
Fig. 2: Hourly electricity price of five locations in one week
For links, we consider bandwidth. In Fig. 1 (b), the number
besides each link is the bandwidth. Note that the modeling,
analysis, and algorithms in this paper can be easily extended
to incorporate other attributes, such as latency and jitter
constraints, as well.
A virtual network (VN) is represented as a weighted graph
G
v
= (N
v
, L
v
). Here N
v
= (N
vr
, N
vh
), where N
vr
and N
vh
denote the set of virtual router and host nodes, respectively;
and L
v
= (L
vr
, L
vh
), where L
vr
denotes the set of virtual
links between any two virtual routers and L
vh
denotes the
set of virtual links between host nodes and routers. Fig. 1 (a)
shows the VN requested by a user on the SN in Fig. 1 (b).
We now formally define the VN embedding problem. Given
a VN request G
v
with a set of virtual nodes N
v
= (N
vr
, N
vh
)
and a set of virtual links L
v
= (L
vr
, L
vh
), and a SN G
s
with
a set of physical nodes N
s
= (N
sr
, N
sh
) and a set of physical
links L
s
= (L
sr
, L
sh
), embed G
v
on G
s
, which means to find
two one-to-one mappings: M
n
and M
l
. Here M
n
is a one-
to-one mapping from N
v
to a subset of N
s
. This mapping
includes two sub-mappings, M
nr
and M
nh
, where M
nr
is
from N
vr
to a subset of N
sr
, and M
nh
is from N
vh
to a subset
of N
sh
. For each virtual router node n
vr
and the physical node
M
nr
(n
vr
) that it maps to, M
nr
(n
vr
) satisfies both virtual
router quantity and location requirements of n
vr
. For each
virtual host node n
vh
and the physical node M
nh
(n
vh
) that it
maps to, M
nh
(n
vh
) satisfies the node requirements on CPU
speed, memory size, and storage capacity of n
vh
. Here M
l
is from L
v
to a subset of P
s
, which denotes all loop-free
paths composed by the physical links in L
s
. This mapping also
includes two sub-mappings, M
lr
and M
lh
, where M
lr
is from
L
vr
to a subset of P
sr
, denoting all loop-free paths between
any two routers, and M
lh
is from L
vh
to a subset of P
sh
,
denoting all loop-free paths between hosts and routers. For
each virtual link l
v
and the physical path M
l
(l
v
) that it maps
to, the bandwidth of each physical link in M
l
(l
v
) is no less
than the bandwidth requirement of l
v
. Take the embedding in
Fig. 1 as an example. The node mapping solution is {{R
a
→
R
A
, R
b
→ R
C
}, {H
a1
→ H
A1
, H
a2
→ H
A2
, H
b1
→
H
C1
}} and the link mapping solution is {{(R
a
, R
b
) →
(R
A
, R
B
, R
C
)}, {(R
a
, H
a1
) → (R
A
, H
A1
), (R
A
, H
a2
) →
(R
A
, H
A2
), (R
B
, H
b1
) → (R
C
, H
C1
)}}.
Note that, we focus on considering one ISP in this paper.
When multiple ISPs collaborate to provided VN services, as
each ISP knows only the characteristics of his own infrastruc-
ture, there are many technical challenges to address this issue.
The readers can refer to [15] for more discussion on this topic.
B. Energy Cost Modeling
1) Node Energy Cost: Let ∆P R
r
i
denote the additional
power consumption for mapping a virtual router r ∈ N
vr
to
a substrate router i ∈ N
sr
, and ∆P H
h
j
denote the additional
power for mapping a virtual host node h ∈ N
vh
to a substrate
host node j ∈ N
sh
. The node energy cost can be calculated
as follows:
∆EN =
X
r∈N
vr
X
i∈N
sr
x
r
i
∆P R
r
i
Z
t
e
t
a
P r
s
(t)dt+
X
h∈N
vh
X
j∈N
sh
y
h
j
∆P H
h
j
Z
t
e
t
a
P r
s
(t)dt (1)
This formula requires two inputs: ∆P R
r
i
and ∆P H
h
j
.
We first discuss the calculation of ∆P R
r
i
. As a typical
router usually consists of four main components as shown in
4
Table II, we estimate P R
i
, the power consumption of mapping
a router node, based on the model proposed in [16]:
P R
i
= P
f
+ L · P
l
+ P · P
p
. (2)
Here L and P denote the number of linecards and ports of the
routers, respectively. Thus, the incremental power consump-
tion for mapping a virtual router node r to i is calculated as:
∆P R
r
i
=
(
P
f
+ ∆L · P
l
+ ∆P · P
p
(if P S
i
= 0)
∆L · P
l
+ ∆P · P
p
(otherwise)
(3)
TABLE II: Components of a typical router
Components Function Power Notations
Chassis Cooling equipments and others. The sum of these
two parts: P
f
.
Switching
fabric
Learning and maintaining the
switching tables.
Line-
cards
Forwarding packets between the
switching fabric and ports.
P
l
Ports Transceiving packets in and out. P
p
We now discuss the calculation of ∆P H
h
j
. Many studies
have reported that the full-system average power consumption
of a typical server is approximately in linear with CPU utiliza-
tion [17], [18]. The power consumption of other components,
such as memory and storage, is very small [19]. We use
the following equation to estimate the power consumption of
mapping a host node:
P H
j
= P
b
+ P
l
· util, (4)
where P
b
is the server’s baseline power without any CPU
load, and P
l
represents the energy proportion factor for CPU
utilization util. Thus, the additional power consumption for
mapping a virtual host node h to j is:
∆P H
h
j
=
(
P
b
+ P
l
· C
h
(if P S
j
= 0)
P
l
· C
h
(otherwise)
. (5)
2) Link Energy Cost: We consider both long physical
links, which span over a large geographical region and there-
fore require repeaters that consume power, and short physical
links, which span a small geographical region and require no
repeaters. We use P L
uv
st
to denote the power consumption of
the repeaters on a long link l
st
∈ L
sr
when mapping a virtual
link l
uv
∈ L
vr
. The overall link cost can be calculated as:
∆EL =
X
l
uv
∈L
vr
X
l
st
∈L
sr
f
uv
st
∆P L
uv
st
P r
s
(t)dt (6)
We set ∆P L
uv
st
to be linear with the traffic volume of the l
uv
and the distance between s and t based on the findings in [11]:
∆P L
uv
st
= Dis(s, t) · P
r
·
B
uv
OB
st
, (7)
where P
r
denotes the power density of the repeaters over
distance and OB
st
denotes the overall bandwidth capacity of
substrate backbone link l
st
.
3) Switching Cost: Powering up a router (or a server)
incurs a onetime energy consumption for transiting from the
power-saving state into the active state, which is called a
switching cost. We use E
sr
(or E
sh
) to denote this cost. The
overall switching cost can be calculated as:
∆ES =
X
r∈N
vr
X
i∈N
sr
x
r
i
· (1 − P S
i
) · E
sr
·
Z
t
e
t
a
P r
i
(t)dt+
X
r∈N
vh
X
j∈N
sh
y
h
j
· (1 − P S
j
) · E
sh
·
Z
t
e
t
a
P r
j
(t)dt.
(8)
C. Proble m Statement of Energy Aware VN Embedding
With the goal of minimizing the overall energy cost ∆E =
∆EN + ∆EL + ∆ES and the binary variables of x
r
i
, y
h
j
and f
uv
st
, we next formulate the energy aware VN embedding
problem as an integer linear programming (ILP):
Objective:
Min ∆E = ∆EN + ∆EL + ∆ES (9)
Capacity Constraints:
(∀r ∈ N
vr
)(∀i ∈ N
sr
) :
(
x
r
i
· R
r
≤ R
i
x
r
i
· Dis(i, r) ≤ W
(10)
(∀h ∈ N
vh
)(∀j ∈ N
sh
) :
(
y
h
j
· C
h
≤ C
j
, y
h
j
· M
h
≤ M
j
y
h
j
· S
h
≤ S
j
, y
h
j
· Dis(j, h) ≤ W
(11)
(∀l
st
∈ L
s
)(∀l
uv
∈ L
v
) : f
uv
st
· B
uv
≤ B
st
(12)
Connectivity Constraint:
(∀s ∈ N
s
)(∀l
uv
∈ L
v
) :
X
l
st
∈L
s
f
uv
st
−
X
l
ts
∈L
s
f
uv
ts
=
1, if x
u
s
= 1 or y
u
s
= 1
−1, if x
v
s
= 1 or y
v
s
= 1
0, otherwise
(13)
Variable Constraints:
(∀i ∈ N
sr
) :
X
r∈N
vr
x
r
i
≤ 1; (∀r ∈ N
vr
) :
X
i∈N
sr
x
r
i
= 1
(∀j ∈ N
sh
) :
X
h∈N
vh
y
h
j
≤ 1; (∀h ∈ N
vh
) :
X
j∈N
sh
y
h
j
= 1
(14)
(∀i ∈ N
sr
)(∀r ∈ N
vr
) : x
r
i
∈ {0, 1},
(∀j ∈ N
sr
)(∀h ∈ N
vh
) : y
h
j
∈ {0, 1},
(∀l
st
∈ L
s
)(∀l
uv
∈ L
v
) : f
uv
st
∈ {0, 1} (15)
5
D. Performance Metrics
We define the long-term average revenue, earned by the ISP
for accommodating VN requests, as follows:
lim
T →∞
P
N
i=1
R
i
(G
v
)
T
, (16)
where R
i
(G
v
) = (
P
h∈N
vh
(C
h
+ M
h
+ S
h
) + W
R
·
P
r∈N
vr
R
r
+
P
l
v
∈L
v
B(l
v
))·t
d
represents the revenue for accommodating the
ith VN request with the pay-as-you-go billing model. Here
W
R
denotes the market pricing weight of the resource of
virtual routers over host node resources. To calculate energy
cost in the long run, we define the long-term average energy
cost, which the ISP must pay, as follows:
lim
T →∞
P
N
i=1
E
i
(G
v
)
T
, (17)
where N is the number of VN requests accepted by the SN
successfully in time T and E
i
(G
v
) denotes the energy cost for
the ith VN request. In this paper, we aim to both maximize
the revenue and minimize the energy cost at the same time.
III. ENERGY AWARE HEURISTIC VN EMBEDDING
ALGORITHM
Solving ILP is well known to be NP-hard [20]. Although
standard exact algorithms such as branch and bound (BB)
and cutting plane (CP) guarantee optimal solutions, they may
incur exponential running time. Thus, they are not practical for
online VN embedding when the problem size is large. In this
section, we present a simple heuristic yet efficient algorithm,
called EA-VNE, to produce an energy aware solution. EA-
VNE is a two-step algorithm: the first step handles node
mapping and the second handles link mapping.
A. Node Mapping
In this step, we first perform the router node mapping, then
the host node mapping.
1) Router Node Mapping: We have two goals to achieve
in this mapping. First, we want to optimize the energy cost in
mapping virtual router nodes. Second, as router node mapping
affects both host node mapping and link mapping, we want to
perform router node mapping so that the host node mapping
and link mapping will be successful and optimized.
Towards the first goal, we need to map the virtual router
node on such substrate router nodes that have low electricity
price and are in the active state. For each substrate router node
i, we calculate the additional energy cost when we map the
virtual router node r on it as follows: E
r
i
=
R
t
e
t
a
P r
i
(t)P R
r
i
dt.
We then sort the candidate substrate nodes according to the
values of E
r
i
in non-decreasing order. We use NR
E
to denote
the set of ranking values of substrate router nodes.
To achieve the second goal, we design a worst-fit scheme for
the bandwidth constraint by selecting the substrate nodes with
higher degree and bandwidth. This will make the subsequent
host node mapping and link mapping easier.
Inspired by Google’s Pagerank algorithm, we rank each
substrate router node s based on itself and it’s neighbors, called
Noderank. We first measure the bandwidth resource of each
substrate node s by H(s) =
P
l∈L(s)
B
l
, where L(s) denotes the
set of all the outgoing links of s, and B
l
denotes the available
bandwidth resource of link l. Next, we compute the initial
N
(0)
R
value for node s by N
(0)
(s) =
H(s)
P
w∈N
sr
H(w)
. In Pagerank,
from current page, a random walker can follow it to another
page that it links to and can also jump to any other page. We
name these two operations as forward and jump operations. Let
s, t ∈ N
sr
be two different nodes. Let p
J
st
=
H(t)
P
w∈N
sr
H(w)
, p
F
st
=
H(t)
P
w∈nbr
1
(s)
H(w)
, where p
J
st
denotes the jumping probability from
node s to land on node t, nbr
1
(s) = {t | (s, t) ∈ L
sr
}, and
p
F
st
the forward probability from node s to node t. Clearly,
∀s ∈ N
sr
:
P
t∈N
sr
p
J
st
= 1,
P
t∈nbr
1
(s)
p
F
st
= 1. For any node
t ∈ N
sr
, let
NR
(i+1)
(t) =
X
s∈V
p
J
st
· p
J
s
· N R
(i)
(s) +
X
s∈nbr
1
(t)
p
F
st
· p
F
s
· NR
(i)
(s), (18)
where p
J
s
+ p
F
s
= 1, p
J
s
≥ 0, p
F
s
≥ 0, and i = 0, 1, · · · .
Noderank represents both the bandwidth resource and the
topological attribute of substrate nodes. See details in our
previous work [5]. Finally, we sort these nodes according to
the values of Noderank in non-increasing order. We use NR
R
to denote the set of ranking values for these nodes. We choose
the substrate nodes with higher ranking.
Next, we prove that our algorithm converges.
Theorem 3.1: The N R algorithm eventually converges at a
steady state.
Proof: Each value of N R
(i)
can be treated as a state of
the Markov chain. First, the number of states is finite since
the number of the substrate router nodes is limited. Second,
the chain is irreducible since each node is strongly connected.
Third, the chain is aperiodic since each node affects its own
rank value. Thus, the theorem follows.
To tradeoff between the above two potentially conflicting
goals, we design a comprehensive measurement for substrate
nodes as follows:
NR = α · NR
E
+ (1 − α) · NR
R
, (19)
where α denotes the ranking weight (0 < α < 1). Thus, we
choose the substrate node with the highest overall ranking for
mapping virtual nodes.
The benefits of such ranking measure are two-fold. First, it
can save energy cost by exploiting the diversity of electricity
price and reducing the number of active router nodes. Second,
it increases the possibility of accepting the VN requests and
hence producing more revenues for ISPs.
2) Host Node Mapping: In this step, the goal is to search a
substrate host node for each virtual host node while meeting
their node constraints on CPU speed, memory size, and
storage capacity. As host node mapping affects link mapping,
besides satisfying the above node constraints, we also need
to consider link mapping in this host node mapping step.
By treating virtual nodes as items and substrate nodes as
