Distributed Consensus-based Auctions for Wireless Virtual Network Embedding
Flavio Esposito
Francesco Chiti
fesposito@exegy.com
Advanced Technology Group
Exegy, Inc., St. Louis, MO
francesco.chiti@unifi.it
Department of Information Engineering
University of Florence, Italy
Abstract—Software-Defined Networks (SDNs) based approaches represent an opportunity for easing the deployment
and the management of wide-area wireless network services.
In this paper, we focus on a particular SDN management
mechanism that wireless infrastructure providers need to adopt
to support such services: the wireless virtual network (VN)
embedding problem. We formulate the problem leveraging on
optimization theory, analyzing its complexity, and proposing a
general distributed auction mechanism. This mechanism leverages on the max-consensus literature to guarantee bounds on the
embedding time and on the performance with respect to a Pareto
optimal solution.
Using extensive simulations, we confirm superior resource
utilization when compared with existing distributed VN embedding solutions, proving to be an attractive and flexible resource
allocation approach for wireless SDNs.
Many VN embedding solutions have been proposed for
wireless [8], [9] and wired distributed service architectures,
although most of them with embedding goals tailored to
specific applications or providers’ goals [10]. Only recently,
distributed policy-based embedding solutions were designed
as an effective way to customize the embedding, based on the
Infrastructure Providers (InPs) wishes (see e.g., [11]–[14].) In
a wireless settings however, to our knowledge, a distributed
VN embedding that enables InPs to choose their own policies
does not yet exist. Moreover, to our knowledge, this paper
is the first to propose a wireless virtual network embedding
solution that provides guarantees on both the convergence time
to a VN embedding, and on the performance of their heuristics.
I. INTRODUCTION
To this aim, we present a Consensus-based Auction for
Wireless (CAW) virtual network embedding. Auctions are an
effective resource allocation mechanism when the value of the
resources is unknown. In distributed settings, the bandwidth
available at neighbor nodes can be at best estimated. Our CAW
mechanism allows wireless physical nodes bid on the virtual
resources to be embedded, providing guarantees on a Paretooptimal VN embedding solution, and on the time to reach
such solution — a critical metric especially in delay-sensitive
SDN applications. Moreover, wireless nodes running CAW can
instantiate their own (private) policies, and adapt the VN embedding based on their goals. For example, physical nodes can
(collectively or individually) choose a bidding function that
would balance their load, or a adapt a “bin packing” bidding
strategy to leave idle a higher number of nodes to save energy.
We already presented a non-wireless version of CAW for the
virtual network embedding problem [11], [12]. In this paper,
we extend the problem formulation to the wireless scenario,
and we show how not only the approximation algorithm, but
even the resulting complexity becomes harder (from NP-hard
to strongly NP-hard), but it can be reduced to polynomial in
a particular case with only two wireless interfaces.
Managing a distributed control plane in a wireless network
is challenging, even for a WLAN with a small number of access points. Allocating radio resources on a limited spectrum,
implementing efficient handover mechanisms, or balance the
load between cells are only a few of these challenges. The
network management complexity is then acerbated by the scalability requirements of such networks, usually centralized but
deployed by several communicating processes; for example, a
typical IEEE 802.11 [1] enterprise WLAN can have thousands
of access points, while cellular networks serve hundreds of
thousands of customers.
Software Define Networks (SDNs) based approaches represent an opportunity for easing the deployment and management of such wireless network. Recent work have demonstrated how traditionally hard-to-implement features are indeed becoming a reality with the SDN-based networks. Few
applications include, for example, seamless mobility through
efficient handovers [2], [3], load balancing [2], [4], on demand
virtual access points creation [2], [5], downlink scheduling
(e.g., an OpenFlow switch can do a rate shaping or time
division) [5], dynamic spectrum usage [5], or inter-cell interference coordination [3], [5]. Not only in the business
community, but even in the research community wireless SDN
are used to manage wide-area virtual network testbeds [6], [7],
where the physical channel is sliced to allow simultaneous
access to multiple researchers. Before running any of these
virtual network (VN or slice) services, wireless infrastructure
providers need to create such slices, solving a VN embedding
problem, i.e., the (NP-hard) problem of finding, for each
virtual node, a physical hosting node, and for each virtual
link, at least one loop-free physical path.
The rest of the paper is organized as follows: in Section II
we define the wireless VN embedding problem, and we study
its complexity, showing how a centralized solution can be
found in O(1.344n ), where n is the number of wireless
physical nodes, and only using polynomial space. We then
present CAW, our policy-based distributed auction mechanism
in Section III, and give examples of such policies in Section IV. We further compare the performance of representative
policy instantiations with two existing distributed embedding
solutions in Section VI and conclude our paper in Section VII.
2
II. T HE W IRELESS V IRTUAL N ETWORK E MBEDDING
In this section we formulate the wireless virtual network
embedding problem as a centralized optimization problem,
and we show that it can be reduced from the edge coloring
problem, therefore justifying the need for an efficient heuristic
or approximation algorithm. By leveraging a recent result by
Kowalik [15], we also show how, a centralized solution to the
problem can be found in O(1.344n ), where n is the size of
the physical wireless network, using only polynomial space.
maximize
Np Nv
X
X
Uijk (xij , fk )
i=1 j=1
subject to
Nv
X
Cj xij ≤ Ci
∀i ∈ VG
j=1
Np
X
xij ≤
i=1
Np Nv
X
X
1
∀j ∈ VH
xij = Nv
A. Centralized Wireless Virtual Network Embedding
Given a wireless virtual network H = (VH , EH ) and a
physical network G = (VG , EG ), the wireless virtual network
embedding problem is the problem of finding, for each virtual
node in VH , one wireless hosting physical node in VG , and for
each virtual link in EH , at least one loop-free physical path,
where the physical links composing the path are chosen from
the set EG . A wireless (e.g., Wi-Fi) virtual network embedding
has a crucial additional constraint: every physical node in the
multigraph G representing the physical network can only have
a limited (three in the case of Wi-Fi) outgoing edges, as there
are only limited (three) wireless network interfaces in the
wireless (IEEE 802.11) standard that allow interference-free
transmissions. 1
We assume that each wireless physical node i ∈ VG , where
|VG | = Np belonging to some InP, has a positive utility Uijk
when hosting a virtual node j ∈ VH , where |VH | = Nv , or
when relaying traffic within the flow fk , where k ∈ K is
the set of the loop-free physical paths in G. We also assume
that InPs cooperate to provide a wide-area service, seeking a
Pareto optimality, i.e., they aim to maximize the summation
of their utilities, without behaving selfishly. We then model
the wireless virtual network embedding problem with the
centralized optimization Problem (1a), where xij = 1 if virtual
node j is assigned to physical node i and 0 otherwise. The
vector xi ∈ {0, 1}VH , whose elements are the xij variables,
represents the assignments for physical node i. The first
three constraints express, respectively, the requirements that
the number of virtual node capacity assigned to a wireless
physical node i should not exceed its physical rate capacity Ci ,
each virtual node should be assigned to at most one wireless
physical node, and all virtual nodes should be assigned to
at least a physical node. The fourth constraint ensures that
no two adjacent interfaces use the same frequency z ∈ Φ,
to avoid interference: Iz (u) is an indicator variable equal to
one if physical node u transmits using frequency z, and zero
otherwise. The remaining equations are the existential, and the
flow conservation constraints. Such constraints ensure that the
net flow on each physical link is zero, except for the source
sk and the destination tk , that have virtual link demand dk .
1 In
the rest of the paper we use wireless and Wi-Fi interchangeably.
i=1 j=1
Φ
X
(1a)
Iiz ≤ 1
∀ i ∈ VG
z=1
K
X
fk (u, v) ≤ C(u, v)
∀ (u, v) ∈ VG
k=1
X
fk (u, w) = 0
∀k ∀ u 6= si , ti
w∈W
fk (u, v) = −fk (v, u)
∀k ∀ (u, v) ∈ VG
X
X
fk (sk , w) =
fl (w, tk ) = dk ∀ sk , tk
w∈W
w∈W
xij = {0, 1}, fk ≥ 0 ∀k, ∀i ∈ VG , ∀j ∈ VH
B. Wireless Virtual Network Embedding Complexity
From the optimization problem defined in the previous
section we note how the problem is at least NP-hard. This
is because the first set of constraints (the first inequality)
is equivalent to the capacity constraint of the set packing
problem, known to be an NP-hard problem. Moreover, the
flow constraints represent the classical multi-commodity flow
constraints, which is known to be solvable in polynomial time,
if each virtual link (flow) is allowed to be split among two
or more loop-free physical paths, and otherwise known to be
strongly NP-hard.
We assume that our physical network runs the one of the
IEEE 802.11 b/g/n protocols whose frequency spectrum is 100
MHz wide and made up of 11 channels centered 5 MHz apart.
Each 2.4 GHz channel is 20 - 22 MHz wide, making the
spectrum crowded. In the case of Wi-Fi, channel overlapping
is undesirable. This is why, to be able to transmit data without
interference, once a virtual network is formed, each physical
nodes can only typically use at most three channel interfaces:
the interface to channel 1,6, and 11.
Given a physical network where each physical node has
these three interfaces, each virtual node needs to be mapped
to an interface such that the physical outgoing edges do not
use the same channel. Therefore the following result holds:
Theorem II.1. The 3-channels wireless VN embedding problem is NP-complete. Moreover, it is possible to test whether
a 3-channels embedding exists in time O(1.344n ), where n is
the size of the physical network, only using polynomial space.
3
Initialization
Bundle Construction
(Sub-modular Utility)
Communication
Winners & bundles
Conflict Resolution
(Assignment Rules Table)
N
Consensus
reached?
Y
End
Fig. 1. CAW workflow: a virtual link embedding phase (k-shortest path) follows the virtual node bidding and agreement phases (virtual node embedding).
Proof. (sketch) Let us first show that the problem is NPComplete. Given a multigraph G = (VG , EG ), with a set
of vertices VG , and a set of edges EG , the graph coloring
problem is the NP-complete problem of assigning a color to
each edge so that no two adjacent edges share the same color.
Let us consider G to be the physical hosting network and let
three colors representing channel 1, 6 and 11 of the IEEE
802.11 b/g/n spectrum. In polynomial time, an algorithm A
takes each virtual link of the virtual network and assigns one
of the three colors to it at random. Given a virtual network
embedding solution, we can check in polynomial time if the
edge coloring problem is satisfied on the hosting physical
network G, hence our problem is NP-complete. The remaining
part of the theorem can be proved by construction, and we omit
it for lack of space, since it is a direct corollary of the recent
Theorem 2 of Kovalik [15].
For delay-sensitive wireless applications, maximizing the
convergence time —the time it takes to find an embedding
may be more important than having the physical network fully
utilized. If we are willing to avoid transmissions on one of the
three channels, then the wireless virtual network embedding
problem can be solved in polynomial time. It is in fact trivial to
verify if a graph can be edge-colored with (one or) two colors
only — we only have to verify the capacity constraints.
III. C ONSENSUS - BASED AUCTIONS FOR W IRELESS
N ETWORK E MBEDDING
In this section we present CAW network embedding mechanism. Auctions are a well known approach to allocate goods
when the value of such goods is unknown to the bidders,
or when the bidders’ budgets are unknown to the auctioneer.
In distributed settings, the congestion level of the network is
unknown a priori, or at best, estimated. A distributed auction
on the flows (virtual links) to be embedded (allocated) on a
possibly congested software-defined wireless network is hence
an effective congestion avoidance mechanism.
The mechanism. CAW consists of two phases: a bidding, and
an agreement phase. The two phases iterate synchronously,
that is, a second bidding phase for a virtual link (flow) does
not start until the first agreement phase on a previous flow
terminates. Physical nodes act upon messages received at
different times during each consensus phase; therefore, the
consensus phase is asynchronous. This is also different from
the results presented in [11] and [12], where the consensus
phase was assumed to be synchronous. In the rest of the paper,
we denote such rounds or iterations with the letter t and we
omit t when it is clear from the context.
Bidding phase (Algorithm 1.) Each wireless physical node
i ∈ I uses a private utility function (see Equation ??) to bid
on a {flow, frequency} tuple j ∈ F, and stores all bids in
|F |
a vector bi . Each element bij ∈ bi ∈ R+ is a positive
real number representing the highest bid known so far on the
{flow, frequency} tuple j. The mechanism is general enough
to work with any private utility function, but in this paper we
assume that the wireless physical nodes bid higher values if
their residual congestion on a given physical flow is higher,
and if they have higher residual capacity. In particular, we
consider the notion of stress as the product of the physical
node capacity, times the residual available capacity on all the
adjacent physical outgoing links. The lower is the stress of a
wireless physical node, the higher will be its bid.
The wireless physical nodes store the IDs of the flowfrequency tuple on which they are bidding in a list (bundle
vector) mi ∈ F Ti , where Ti is the target number of flows
that i can allocate, F the set of available frequencies, and
|mi | is the number of flows that a node are allowed to bid
simultaneously. Wireless physical nodes need to be also aware
|F |
of a mapping vector fi ∈ R+ (with size equal to the number
of flows to be embedded) between the frequencies (currently)
assigned by the winner physical nodes and their ID.
Agreement or Consensus phase (Algorithm 2.) After the
private bidding phase, each physical node exchanges the bids
|F |
with its neighbors, updating the assignment vector ai ∈ VG
and the flow vector f with the latest information on the current
assignment of all flows, for a distributed auction winner
determination. We assume that nodes are aware of its firsthop neighbors, and the routing table.
A. Conflicts resolution
When it receives a bid update, a node i has three options:
(i) ignore the received bid leaving its bid vector and its
allocation vector as they are, (ii) update according to the
information received, i.e. wij = wkj and bij = bkj , or (iii)
reset, i.e. wij = ∅ and bij = 0. When |mi | > 1, the bids are
not enough to determine the auction winner as virtual nodes
can be released, and a node i does not know if the bid received
has been released or is outdated.
Bids should be ignored or reset if they are outdated, and
subsequent bids to an outbid virtual node should be released,
to maximize the summation of the utilities.
To avoid storing bids computed with out-of-date utility
values, physical nodes simply reset their own bundle at the
beginning of every bidding phase (Algorithm 1, line 4).
Remark (reducing the communication and runtime complexity.) In order to reduce the communication cost, we let the
nodes send their values updates only when they first learn
4
Algorithm 1: bidding for wireless node i at iteration t
1: Input: ai (t − 1), bi (t − 1)
2: Output: ai (t), bi (t), mi (t), fi (t),
3: ai (t) = ai (t − 1), bi (t) = bi (t − 1)
4: mi (t) = ∅, fi (t) = ∅
5: if biddingIsNeeded(ai (t), Ti ) then
6:
if ∃ j : hij = I(Uij (t) > bij (t)) 6= 0 then
7:
η = argmaxj∈F {hij · Uij }
8:
mi (t) = mi (t) ⊕ η // append η to bundle
9:
biη (t) = Uiη (t)
10:
update(η, ai (t))
11:
update(η, fi (t))
12:
Send / Receive bi to / from k ∀k ∈ Ni
13:
Send / Receive ai to / from k ∀k ∈ Ni
14:
Send / Receive fi to / from k ∀k ∈ Ni
15:
end if
16: end if
Algorithm 2: agreementPhase for node i at iteration t
1: Input: ai (t), bi (t), mi (t), fi (t)
2: Output: ai (t), bi (t), mi (t), fi (t)
3: for all k ∈ Ni do
4:
for all j ∈ F do
5:
if IsUpdated(bkj ) then
6:
update(bi (t), ai (t), mi (t), fi (t))
7:
end if
8:
end for
9: end for
about them, not at every round. This strategy is, however,
not sufficient to decrease the order of magnitude of the worst
case message overhead [16]. In a system implementation, the
message overhead could be further reduced by compressing
the attributes to be sent.
IV. T HE P OLICIES
One of the design goals of CAW is its flexibility, i.e., the
ability to create customizable allocation algorithms to satisfy
desired policies, rules, and conditions in any wireless settings.
We describe here few representative examples of such policies,
and later in this section we show how CAW can be instantiated
to satisfy providers’ embedding goals.
A straightforward example of policy is the (normalized)
utility function U that InPs use to bid on virtual resources
(nodes.) In our evaluation (Section VI) we use:
X
Ti − Sij
Ti = Ci +
Cik ,
Uij =
(2)
Ti
k∈Ni
where Ti is the target virtual (node and links) capacity that
is allocatable on i, and Sij the stress on physical node i,
namely, the sum of the virtual node capacity already allocated
on i, including virtual node j on which i is bidding, plus
the capacity of the virtual links allocated on the adjacent
physical links in the previous round. Note that, due to the
max consensus definition [16], the bid bij at physical node i
on virtual node j is the maximum utility value seen so far. The
normalization factor T1i ensures that such bids are comparable
across physical wireless nodes.
We have seen from related work, e.g. [17], [18], how
embedding protocols may require SPs to split the VN. CAW
is able to express this requirement by enforcing a limit on the
length of the bid vector bi , so that physical nodes bid only on
the released VN partition. Each InP can also enforce a load
target Ti on its resources, by limiting the bundle size.
Another policy is the assignment vector ai , that is, a vector
that keeps track of the current assignment of virtual nodes.
ai may assume two forms: least and most informative. In
its least informative form, ai ≡ xi is a binary vector where
xij is equal to one if physical node i hosts virtual node j
and 0 otherwise. In its most informative form, ai ≡ wi is
a vector of winning physical nodes so far on virtual nodes;
wij represents the identifier of the physical node that made
the highest bid so far to host virtual node j. Note that when
ai ≡ wi the assignment vector reveals information on which
physical nodes are so far the winners of the auction, where
if ai ≡ xi physical node i only knows if it is winning each
resource or not. As a direct consequence of the max consensus,
this implies that when the assignment (allocation) vector is
in its least informative form, each physical node only knows
the value of the maximum bid so far without knowing the
identity of the bidder. We also leave as a policy whether the
assignment vector is exchanged with the neighbors or not. In
case all physical nodes know about the assignment vector of
the virtual nodes, such information may be used to allocate
virtual links in a distributed fashion. Instead, if ai ≡ xi , to
avoid physical nodes flooding their assignment information, i
asks the SP about the identity of the physical node hosting the
other end of the virtual link and attempts to allocate at least
one loop-free physical path.
V. C ONVERGENCE AND P ERFORMANCE G UARANTEES
All wireless nodes need to be aware of the mapping, by
exchanging their bids with only their first-hop neighbors,
therefore a change of information needs to traverse all the
physical network, which we assume has diameter D. This
result states that D hops are also enough, that is, a necessary
and sufficient condition to reach max consensus on a single
virtual node allocation is that each node is visited once.
An interesting observation that follows from the result
is that the number of steps for CAW to converge on the
embedding of a slice of |VH | virtual nodes is always D · |VH |
in the worst case, regardless of the size of the bundle vector.
This means that the same worse-case convergence bound is
achieved if CAW runs on a single or on multiple virtual nodes
simultaneously. Formally, we have the following result:
Proposition V.1. (Convergence of CAW.) Given a virtual network H with |VH | virtual nodes to be embedded on a physical
wireless network with diameter D, the utility function of each
physical node is sub-modular, and the communications occur
over reliable channels, then the CAW mechanism converges in
a number of iterations bounded above by D · |VH |.
5
We omit the proof of as it is equivalent to the proof obtained
for wired VN embedding [11].
We showed in Section III that the problem of embedding a
wireless virtual network becomes intractable as the size of the
physical or virtual network grows. In general, this means that
the sum of the utility of the providers after an embedding could
be arbitrarily low. However, because we assumed a decreasing
marginal gain in adding new virtual nodes to the previously
allocated, this does not happen as we inherit the following
bound:
Proposition V.2. (CAW Approximation.) The CAW embedding
algorithm yields an (1 − 1e )-approximation w.r.t. the optimal
node assignment solution, and w.r.t. the VN embedding, if we
consider physical networks whose diameter D is at most one.
We omit also this proof of as it is equivalent to the proof
obtained for wired VN embedding [12].
Note how, although virtual nodes and links are tightly
connected, the wireless node embedding is the only computationally intractable subproblem if we allow path splitting.
CAW splits the embedding in two phases (see Figure 1), so
once there is agreement on which wireless physical nodes
will host the source and the destination virtual node, for each
virtual link, the loop-free physical path assignment phase (run
as a k-shortest path) can be found in polynomial time, as a
result of a fractional multi-commodity flow problem. A virtual
link can be in fact mapped onto multiple physical paths [19].
VI. E XPERIMENTAL R ESULTS
To test the proposed distributed consensus-based auction
algorithms, we developed our own trace-driven simulator.
Our results confirm the theoretical bounds and show how a
representative CAW policy instantiations outperform similar
(policy-based) distributed virtual network embedding solutions. Although they were not designed for wireless networks,
we have chosen to compare CAW against the first VN
embedding solution [18], and against PolyVine [14], to our
knowledge, the first distributed policy-based VN embedding
solution.
A. Simulation Environment
Physical Network Model: We use the BRITE topology generator [20], to build a flat topology using either the Waxman
model [21], or the Barabasi-Albert model [22] with incremental growth and preferential connectivity, respectively. We tested
our algorithms with various physical network sizes, varying
the number of physical nodes and physical links (as in [19].)
Our simulations do not consider delay constraints, while link
capacity constraints are discussed later in this section. The
results are similar regardless of the topology generation model
and the physical network size, and we only show the results
obtained for physical networks of size n = 50 and a BarabasiAlbert physical topology.
Virtual Network Model: we use a real dataset of 8 years
of Emulab [23] requests. For each simulation run, we sample
a request from a dataset of 61968 requests; the average size
of a request is 14 with standard deviation of 36 virtual nodes;
99% of the requests have less than 100 virtual nodes, and 85%
have at most 20 virtual nodes. Excluding the 10% long-lived
requests that cause the standard deviation of VN lifetime to
exceed 4-millions seconds, the duration of the requests is on
average 561 with 414 seconds of standard deviation. As the
dataset does not contain neither the number of virtual links
nor the virtual network topology, we connect each pair of
virtual nodes at random with different average node degree
(Figure 2b,c.) All our simulation results show 95% confidence
intervals; the randomness comes from both the virtual network
topology to be embedded, and the virtual constraints, that
is, virtual node and link capacity requirements. Similarly to
previous work [19], we let capacity requirements be real
numbers uniformly distributed between 1 to 100 units. We
impose both virtual-physical node and link average capacity
ratio R = {50, 100, 500}. The results are similar and we only
show plots for R = 100. Moreover, we add a node capacity
physical constraint to make sure each physical node supports
at least the capacity of its adjacent physical links.
Comparison Method: we compare our CAW mechanism,
instantiated instantiated with different policy configurations,
with another policy based distributed virtual network embedding algorithm, PolyViNE [14], and with the first published
distributed virtual network embedding algorithm [18], that we
call Hub and Spoke due to the adopted heuristic, that partition
the VN to be embedded in hub and spoke partitions, and tries
to embed sequentially all the VN partitions. A VN is rejected
as soon as one partition cannot be embedded
B. Simulation results
Our main simulation results show how CAW policies that
balance the load on the wireless physical network (e.g., SAD,
as in Single-item Auction for Distributed embedding) are more
effective than bin packing policies (e.g., MAD, as in Multiitem Auction for Distributed embedding.) This is because
the only three available interfaces (channels 1,6 and 11 of
IEEE 802.11) constraint the physical network multigraph (a
graph with multiple edged between the nodes) so that the
outgoing link capacity is quickly exhausted, if too many
virtual nodes are mapped to the same wireless physical node
(Figure 2b.) This result is confirmed by the physical node
utilization (Figure 2d), lowest for the SAD policy.
We also show results for a path bidding policy (PAD), in
which physical nodes bid on virtual paths of size p, to be
embedded on physical loop-free paths of size less or equal
than p, to avoid physical node relays (a drawback of the SAD
and MAD policies).
The efficiency of a wireless embedding is also evaluated
with the VN allocation ratio — ratio between VN successfully
embedded and requested, and with the resource utilization —
physical node and link capacity utilized to embed the VN
requests, as well as with the endurance of the algorithm, i.e.
the number of successfully allocated requests before the first
VN request is rejected (Figure 2c). Finally, we evaluate the
convergence time, — number of algorithm steps until a valid
6
60
40
20
0
5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
size of VN request
0.8
SAD
MAD
PAD
Hub & Spoke
PolyVINE
0.6
0.4
0.2
0
3
4
5
6
Average Virtual Node Degree
(a)
7
(b)
SAD
MAD
Hub & Spoke
PolyVINE
100
80
60
40
20
0
5
10
15
Virtual Network Size [# of nodes]
(c)
20
Physical Node Utilization
80
Continuously Allocated VNET
1
Slice Allocation Ratio
Convergence Time Steps
100
1
0.8
0.6
0.4
0.2
0
S MH P S MH P S MH P S MH P
Distributed Slice Embedding Algorithm
(d)
Fig. 2. (a) The number of steps required for CAW to convergence is linear in the size of the VN as shown theoretically. (b) CAW’s instantiated with the
SAD policy allocates more slices since it balances the load on the physical network. A single shortest path is available. (c) CAW’s instantiated with the SAD
policy is able to allocate more VN before the first one is rejected. (f) CAW’s instantiated with the SAD policy better balances the load on physical nodes. S,
M, H and P indicate SAD, MAD (the two CAW’s policies), Hub and Spoke and PolyViNE, respectively.
embedding is found (Figure 2d), confirming our theoretical
results (Proposition V.1).
VII. C ONCLUSIONS
We presented the problem of embedding a virtual network
on a wireless physical network, via a centralized optimization
problem, and we studied its complexity. We then focus on
a specific wireless protocol, the IEEE 802.11, that has only
three available transmissions channels. This assumption led to
a simplification, with respect to the general virtual network
embedding problem. In particular, the multigraph representing
the wireless physical network can have at most three edges.
This restriction allowed us to show, leveraging on a known
solution to the edge-coloring problem, that the time complexity
of finding a centralized solution is O(1.344n ), where n is the
size of the physical network.
After motivating the need for a polynomial time algorithm
to solve the problem, we proposed CAW, a distributed approximation algorithm, that leverages our previous results on
VN embedding to converge to a conflict-free assignment of
virtual nodes and links on the wireless physical network. We
then developed our own event-driven simulation and showed
that CAW outperforms existing distributed VN embedding
solutions, in terms of VN allocation ratio and in terms of
physical network utilization.
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