Modeling and Optimization of Virtual Networks

in Multi-AS Environment

Yong Xue, Alexander Brodsky and Daniel Menasce

George Mason University, Fairfax, Virginia, U.S.A.

Keywords:

Internet Architecture, Virtual Network Optimization, Virtual Network Synthesis (VNS), Virtual Network

Embedding (VNE), Multi-Domain VNE, Heuristic Approximation Algorithm.

Abstract:

An evolutionary approach to the Internet Ossiﬁcation problem is to adopt a pluralistic architectural paradigm

by leveraging the existing Internet infrastructure and developing multiple virtual service networks on top of it

to satisfy diverse and ever-demanding new service and application requirements from end users. However, op-

timal provisioning or mapping of virtual network requests (VNR) to shared underlay network resources across

multiple autonomous network systems (i.e., Multi-AS) is still an open problem. This paper investigates issues

related to the multi-domain virtual network embedding (MD-VNE) problems and proposes a novel Multi-AS

Virtual Network Synthesis (VNS) paradigm that closely mimics real-world network settings. Proposed in this

paper is a model-based approach to optimally solving the VNS problem through formal Multi-AS network

modeling and optimization using Integer Linear Programming (ILP) with deﬁned optimal objectives for de-

riving exact optimal solution. Besides, the paper also proposes a simple greedy-heuristic (GH) approximation

algorithm to the optimization solution to address implementation complexity concerns. The experiment and

evaluation results of the exact optimization solution and the approximation solution are presented and com-

pared based on a number of deﬁned evaluation metrics.

1 INTRODUCTION

Internet backbone consists of hundreds of intercon-

nected independent networks known as autonomous

systems (AS), each of which is under the control of

a single technical and administrative authority from

governance, routing and network management per-

spectives. Internet Service Provider (ISP) networks

are an example of AS. The current Internet offers only

best effort services and has been greatly challenged

by the ever-increasing needs of emerging functions

and performance capabilities to meet diverse user ser-

vice and applications requirements. Due to the nature

of sheer size and scale of the current Internet infras-

tructure and enormous investments made so far by

ISPs, networking equipment vendors, and enterprise

users alike, any future Internet architecture develop-

ment that requires disruptive or revolutionary changes

becomes extremely difﬁcult, thus so called Internet

Ossiﬁcation problem (Anderson, 2005).

An evolutionary approach to the above Internet ar-

chitecture dilemma is to leverage the existing Inter-

net infrastructure and develop multiple virtual service

networks on top of it to meet new network service

needs, which meanwhile will also allow for differ-

ent Internet architectures to be experimented and val-

idated in an operational network environment. This

pluralistic paradigm of Internet infrastructure has

been widely embraced by both academia and indus-

try communities to become a de facto choice for the

next generation Internet architecture as discussed in

papers (Chowdhury, 2010), (Anderson, 2005), (Duan,

2012) and (Duan, 2020). However, optimal provision-

ing or mapping of virtual network requests (VNR)

to shared underlay network resources across multiple

autonomous network systems is still an open prob-

lem.

A virtual network (VN) is a logical network de-

signed as an overlay on top of the existing phys-

ical networks by leveraging the ritualized network

link and node resources using network virtualiza-

tion technologies. Network Virtualization (NV) pro-

vides an environment in which multiple logically sep-

arated virtual networks can be developed and de-

ployed on a shared physical network infrastructure.

An Overlay Service Network (OSN) is a virtual net-

work that provides speciﬁc network services with

Quality of Service (QoS) guarantee, such as Video

Streaming Network, High Resiliency Network, Vir-

tual Private Network (VPN) and software deﬁned

Xue, Y., Brodsky, A. and Menasce, D.

Modeling and Optimization of Virtual Networks in Multi-AS Environment.

DOI: 10.5220/0011799000003396

In Proceedings of the 12th International Conference on Operations Research and Enterprise Systems (ICORES 2023), pages 211-220

ISBN: 978-989-758-627-9; ISSN: 2184-4372

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

211

Wide-Area networks (SD-WAN), by leveraging the

network resources in the underlay network and pro-

viding required resource and trafﬁc control and man-

agement functions to meet the speciﬁed set of service

and QoS requirements to its end users (Chowdhury,

2010), (Song, 2012), (Sitaraman 2014) and (Yang

2019). Note that modern IP-based Internet infrastruc-

ture has gone beyond terrestrial networks to encom-

pass 5G mobile networks and high-speed SATCOM

networks, which signiﬁcantly extend the global reach-

ability of the Internet with diverse network infrastruc-

tures and provides more opportunities for OSN devel-

opment.

A VN can be realized using speciﬁc set of node

and link resources explicitly made available by the

underlay network infrastructure providers. The prob-

lem of mapping the nodes and links of a virtual net-

work topologically to a given set of underlay net-

work resources in an optimal way is called Virtual

Network Embedding (VNE) problem (Fischer, 2013).

When the underlay network is composed of multi-

ple network domains, we call it multi-domain VNE

(MD-VNE). The mapping optimization can involve

multiple metrics and parameters such as QoS param-

eters, network cost, and network performance mea-

sures. Solutions to the VNE problems provide tech-

niques for provisioning the virtual networks (VN)

by allocating proper network resources in the un-

derlay networks to meet VN’s topology and capac-

ity requirements. Figure 1 below illustrates the con-

cept of mapping two virtual networks (VNET1 and

VNET2) to a shared network topology from two in-

frastructure network providers (AS1 and AS2). For

example, VNET1 can be instantiated using physical

node and link resources by one provider (AS1) based

on network topology and capacity constraints, while

VNET2 needs resources from both AS networks (AS1

and AS2) for the mapping.

Figure 1: Multi-AS VNE Illustration.

The VNE problem requires full knowledge of un-

derlay network topology and capacity, which is rela-

tively straightforward in single domain environment.

But the problem becomes much more challenging in a

multi-AS environment due to the autonomous nature

of the Internet and its distributed control and manage-

ment model. Inter-AS VNE poses some special chal-

lenges in terms of resource dissemination and discov-

ery, as well as resource allocation and network map-

ping. The autonomous characteristics of the multi-AS

is manifested in the following adverse ways:

1. There is no visibility between adjacent AS net-

works because each AS, like an Internet Service

Provider (ISP), normally will not reveal its inter-

nal network resources and topology information

to any outside entities due to business competi-

tion. Without full-knowledge of all the underlay

networks, it is hard to engineer a cross-AS virtual

service network optimally.

2. There is a lack of shared control and manage-

ment between adjacent AS networks, thus making

it hard to provision and optimize the network re-

source allocation and utilization for Multi-AS VN

on a global basis to meet the end-to-end cost or

performance requirement.

Most existing MD-VNE solutions are top-down

and faced with challenges in two fronts: 1) how to

decompose a VNR into smaller VN requests to map

to different underlay network domains and combine

the mapped solutions thereafter; 2) how to gain full

topology and resource knowledge of all network do-

mains in a easy and salable. Note that the complexity

of the current approaches can be signiﬁcantly reduced

if the problem is paraphrased in a bottom-up way in

terms of network synthesis from a network resource

pool from multiple ISPs using a distributed manage-

ment and centralized control resource dissemination

and management paradigm following a common net-

work abstraction approach.

This paper proposes a model-based approach to

the formal modeling and optimization of the Multi-

AS VNS problem. The contributions of this research

paper include the following: 1) we ﬁrst developed

the concept of virtual network synthesis (VNS) to

simplify solving virtual network mapping problem

in Multi-AS environment and proposed a dynamical

resource discovery and dissemination paradigm for

Multi-AS VN provisioning problem; 2) we then de-

veloped a formal network model for a gateway-based

virtual network provisioning reference model in solv-

ing the Multi-AS VNS problem and formally formu-

late the Multi-AS VNS problem to an ILP optimiza-

tion problem using a cost utility objective function,

and two QoS parameters constraints: capacity and

end-to-end delay; 3) ﬁnally we developed a simple

greedy heuristic (GH) approximation algorithm to the

Multi-AS VNS optimization problem to show a ap-

proximation method to the Multi-AS VNS optimiza-

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212

tion and conducted experimental evaluation to com-

pare the performance of the ILP-based exact solution

and GH algorithm-based approximate solution.

The rest of the paper is organized as follows. Sec-

tion II provides a brief review of some related re-

searches. After that, the paper describes the for-

mal models and the proposed resource dissemination

framework for the Multi-AS VNS problem in Section

III. The formulation of the formal Mutli-AS VNS op-

timization problem is described in Section IV. Then

Section V describes a simple GH approximation algo-

rithm to the optimization method. Finally, in Section

VI, we describe the experiment and performance eval-

uation results of the ILP optimization and the GH ap-

proximation algorithm. Section VII provides a brief

summary of the research results and identify some po-

tential future works.

2 RELATED WORK

VN Embedding (VNE) problems have been exten-

sively studied in the research community and there

are many different techniques, mostly based on opti-

mization theory for solving this problem as summa-

rized in several comprehensive survey papers, includ-

ing those by (Belbekkouche, 2012), (Fischer , 2013)

and (Cao, 2019). The optimization objective can be

based various business or technical metrics including

cost, revenue and proﬁt, QoS, reliability and availabil-

ity. The majority of the VNE research results are lim-

ited to single domain and VNE across multiple sub-

strate network domain has gained more attention for

the past decade or so in both industry and academic

communities as industry has been trying to develop

new networking capabilities across Internet to meet

ever demanding service needs.

The solutions to the MD-VNE problem mostly

use top-down approach and are explored under dif-

ferent restrictive assumptions as discussed in (Houidi,

2011), (Hong, 2014) and (Yang, 2019). Resource

discovery and dissemination across multiple domain

have been recognized as a core challenge to any

practical solution for the MD-VNE problem as dis-

cussed in (Belbekkouche, 2012), (Dietrich, 2013)

and (Figueria, 2015). Some examples of the pro-

posed approaches include the policy-based frame-

work for multi-domain VNE by (Samuel, 2013), re-

cursive hierarchical embedding by Vaishnavi (2015),

integrated approach across network and clouds in

(Sonkoly, 2015), multi-domain connection stitching

in (Li, 2016) and PSO meta-heuristic approach in

(Hou, 2020). Note that, all the mentioned works to

some degree demonstrate some ideas to address part

of the gaps in the MD-VNE problems under some re-

strictive conditions, but they all fall short of being a

complete and practical solution to the MD-VNE prob-

lem in real-world multi-AS environment.

3 NETWORK MODELS AND

RESOURCE MANAGEMENT

To facilitate the development of solutions to the multi-

AS VNS problem, we adopted a common layered net-

work and service model (Fischer, 2013) and devel-

oped a robust Multi-AS VNS network reference as

discussed below.

3.1 Multi-AS VNS Network Modeling

A 3-tier network and service model based on the con-

cepts of service providers, virtual network providers

and infrastructure providers as in Figure 2 can be used

to model networks and services for the multi-AS VNS

problem. A brief account of each layer of the network

providers and their roles are as follows:

1. Service Providers (SP): A SP at top wants to build

a service-speciﬁc virtual network (VN) meeting

its topological, performance, budget, and QoS re-

quirements. The SP normally submits a virtual

network request (VNR) to its serving virtual net-

work provider (VNP) for the required virtual net-

work (VN).

2. Infrastructure Network Providers (InP): InPs at

the bottom are network infrastructure service

providers that can provide point-to-point network

services to the VNP in the form of network seg-

ments or virtual circuits between their edge ser-

vice access points. There can be zero, one or more

segments between a pair of access points offered

by a given InP depending upon their service of-

fering and available resource at a given time. As-

sociated with each segment are end points, costs,

capacity and QoS parameter bonded by a service

level agreement (SLA).

3. Virtual Network Provider (VNP): The VNP in the

middle deploys its service gateways at locations

where the VN services are expected and build

needed local connections to all or a subset of InPs

that are within proximity of its VNP gateways.

VNP will utilize the gateways and network re-

source pools provided by the underlay InPs to in-

stantiate the VNR.

Note that a VNP determines a set of strategic net-

work hub locations to deploy virtual network gate-

ways that are physically connected to a set of InPs

Modeling and Optimization of Virtual Networks in Multi-AS Environment

213

Figure 2: Tiered Network Model for Multi-AS VNS.

within its geographic proximity. Then the VNP will

utilize the network segments between the gateway

nodes to synthesize a virtual network based on VNR

from SP.

Formal modeling of Multi-AS VNS optimization

problem involves two level of models: 1) formal rep-

resentation of networks and 2) a logical reference

model for Multi-AS InP networks interconnected by

VNP gateways. We formally model a network using a

property graph model called an Attributed Relational

Graph (ARG) that is extremely powerful in modeling

networks for which its nodes and links have associ-

ated properties. Speciﬁcally, a network G can be de-

ﬁned by a 4-tuple G = (N, L, A, B) where

N = {n

, n

, ··· , n

} is a ﬁnite set of m nodes,

L = {l

i j

=< n

, n

> ∧i, j = 1, 2, ··· , n} ⊆ N × N,

is a set of links of the network.

A = {a

, a

, ··· , a

} and B = {b

, b

, ··· , b

} each is

a set of deﬁned attributes for nodes and links respec-

tively.

We use G



, L

, A

, B



to represent a

VN in SP’s VNR and G



, L

, A

, B



to rep-

resent the combined substrate InP networks plus

VNP gateway nodes as illustrated in Figure 5. Be-

sides, the network attributes include node/link capac-

ity (NCap/LCap), Cost and delay (D).

For formulating the Multi-AS VNS optimization

problem we developed a logical network reference

model shown in Figure 3, characterizing the sub-

strate network environment in the Multi-AS problem.

Note that the substrate network nodes are VNP gate-

ways and gateway-to-gateway connections consisting

of two gateway-InP connections plus one or more seg-

ments InP network in between. Together they form

the network resources pool for the VNP controller to

use to synthesize the VN from SP.

Figure 3: Multi-AS VNS Network Reference Model.

3.2 Resource Management

A practical Multi-AS VNS solution depends on the

capability to dynamically update the network re-

source pool available to the VNP per contract be-

tween VNP and the InPs. This paper proposes a

resource discovery and management framework in

which distributed control and dissemination of net-

work resources from InPs to VNP and central man-

agement of the resources by VNP for network re-

source allocation and synthesis. As shown in Figure

4, each InP has a local resource manager (LRM) re-

sponsible for available dissemination and VNP has a

global resource manager (GRM) responsible for man-

aging and allocating all the network resources pro-

vided by InP. The GRM notiﬁes LRMs of use of a

speciﬁc network segments in its VNS decision. Note

that, this framework for resource management can be

implemented using different technologies and meth-

ods, for example Open API for data federation.

Figure 4: Multi-AS VNS Resource Management Frame-

work.

4 VNS OPTIMIZATION

In this section, we develop a formal ILP optimization

formulation for the Multi-AS VNS problem.

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214

Table 1: Notation Description.

Notation Description

, n

virtual network nodes of SP

i j

virtual link between VN nodes n

and n

, g

network gateway nodes of VNP’s substrate network

number of infrastructure providers InPs

gateway link from gateway node g

to InP

klmu

path segment m of InP

between gateway nodes g

and g

klmu

link between gateway nodes g

and g

via path segment s

klmu

klu

set of segments between gateways g

and g

in InP

unit link cost for connecting gateway g

to InP

unit node capacity cost for the gateway g



the number of nodes and links in the requested virtual network from SP



the number of nodes and links in the substrate networks formed by VNP and InP

set of all paths between gateway nodes g

and g

across all InP network topology

kln

nth path in set P

between VNP nodes g

and g

i j

a mapped path in P

for VN link l

i j

determined by VNP’s path selection policy

max

cost budget of a virtual network request (VNR)

max

maximum delay allowed between any node pair of a virtual network request

A binary parameter indicating if gateway g

is connected to InP

klmu

A binary parameter indicating if the g

and g

are connected via path segment s

klmu

and gateway links

A binary parameter indicating if the n

and g

are co-located

A decision binary variable indicating if the virtual gateway node g

is selected to host SP node n

i j

klmu

A decision binary variable indicating if a gateway hop link h

klmu

is on the

mapped path of VN link l

i j

4.1 Notations and VNS Mapping

Intuitively, our Multi-AS VNS problem is for VNP

to ﬁnd an optimal mapping between the VN and the

substrate network resources such that the each node

in VN is instantiated as a virtual node (in virtual ma-

chine) in a VNP gateway (node level mapping) and

for each link in VN the VNS algorithm will map it to

a path consisting of sequence of instantiating gateway

nodes and InP segments. The objective is to minimize

he total cost of the mapped virtual network that meet

all the capacity, end-to-end delay and total cost con-

straints. Note that the VN network costs depends on

the aggregated costs of the mapped gateway VMs and

InP segments in the substrate network layer. The for-

mula and equations in this section describes the math-

ematical formulation of the Multi-AS problem as an

integer programming (ILP) problem. Figure 5 below

illustrate the mapping concepts from VN to the InP

segment resources and VNP gateways. Notations and

variables used are deﬁned and described in Table 1.

4.2 Optimization Formulation

In our ILP optimization formulation, the VNR’s

topology, node/link capacities as well as link delays

are speciﬁed and provided by SP to the VNP. The goal

is to select a set of gateways and path segments that

Figure 5: Multi-AS VNS Optimization Optimization.

result in a lowest cost network that satisﬁes the end-

to-end packet delay, node/link capacity constraints

and total network cost budget requirements. Figure 6

and the following equations formally deﬁne the cost,

delay and capacity of the mapped path. For each VN

link l

i j

, the mapped path capacity and cost for p

i j

can

be calculated below:

LCap(p

i j

) = min

s∈p

i j

LCap(s)

Cost(p

i j

) =

∑

s∈p

i j

Cost(s)

where M

) = g

and M

) = g

Input: Virtual network G

work cost budget C

max

, and end-to-end network delay

Modeling and Optimization of Virtual Networks in Multi-AS Environment

215

Figure 6: Multi-AS VNS Optimization Mapping.

bound

max

= min

i j

∈L

, 1≤i< j≤

D(l

i j

)

Objective Function:

Minimize Cost(G

) =

∑

i=1

∑

k=1

∑

i=1

∑

j<i

∑

k,l=1

∑

u=1

klu

∑

m=1

(β

+ β

+Cost(s

klmu

))z

i j

klmu

where K

= NCap(n

) for 1 ≤ i ≤



Subject to the Following Constraints:

1) Auxiliary Variables:

klmu

determines if direct connections exist between

gateway nodes g

and g

klmu

= f

klmu

(1)

for 1 ≤ k, l ≤



and 1 ≤ u ≤

and 1 ≤ m ≤

klu

determines if node n

and gateway g

are co-

located.

= 1 if Loc(n

) = Loc(g

) (2)

for 1 ≤ i ≤



and 1 ≤ k ≤



2) Each virtual SP node is matched to only one gate-

way node and each gateway node is matched to no

more than one SP node.

∑

k=1

= 1 (3)

∑

i=1

≤ 1 (4)

(5)

for 1 ≤ i ≤



and 1 ≤ k ≤



3) Node and link capacity

(NCap(g

) − NCap(n

))x

≥ 0 (6)

for 1 ≤ i ≤



and 1 ≤ k ≤



In addition, for k, l = 1, 2, · · · ,



and u =

1, 2, ··· , N

and m = 1, 2, ··· ,

klu

(LCap(L

) − LCap(s

klmu

))Z

klmu

≥ 0 (7)

(LCap(L

) − LCap(s

klmu

))Z

klmu

≥ 0 (8)

Then, for each gateway hop link s

klmu

∑

i j

∈L

LCap(l

i j

klmu

≤ LCap(s

klmu

) (9)

4) The path preservation rules constrain the substrate

network level routing for mapped path of M

i j

)

from the gateway node g

to g

. For k ̸= l and

for u = 1, 2, ··· , N

and m = 1, 2, ··· ,

klu

we should

have:

A. For given l

i j

∈ L

and mapped g

and g

∈ N

There should be exact one segment starting at g

and

one ending at g

. i.e. :

∑

h=1

∑

u=1

klu

∑

m=1

i j

khmu

= 1 (10)

∑

h=1

∑

u=1

klu

∑

m=1

i j

hlmu

= 1 (11)

B. For all intermediate gateway nodes on the path, the

total number of segments coming in should equal to

total number of segments coming out:

∑

h=1

∑

u=1

klu

∑

m=1

i j

htmu

∑

h=1

∑

v=1

klu

∑

n=1

i j

thnv

for t ̸= k, l (12)

C. For given l

i j

∈ L

and mapped g

and g

∈ N

There should be no more than one segment used on

intermediate gateway link s

thmu

for t ̸= k, h ̸= l, i.e. :

klu

∑

m=1

i j

thmu

≤ 1 (13)

5) Each VN link should be mapped to a path with de-

lay no more than the given network delay bound D

max

i.e.,for a mapping of l

i j

∈ L

onto the mapped path p

i j

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216

D(p

i j

) =

∑

xymu

∈p

i j

(D(L

) + D(s

xymu

) + D(L

)) ≤ D

Max

(14)

6) Variable domain constraints

i j

∈ {0, 1} (15)

where i = 1, 2, ··· ,



and j = 1, 2, ··· ,



i j

klmu

∈ {0, 1} (16)

where i, j = 1, 2, ··· ,



and k, l =

1, 2, ··· ,



, u = 1, 2, ··· , N

and m = 1, 2, ··· ,

klu

7) Network cost constraint: total VN costs should be

no more than the cost budget

Cost(G

) ≤ C

max

(17)

Decision Variables:

i j

} and {z

i j

klmu

}

5 A VNS APPROXIMATION

ALGORITHM

Solving the Multi-AS VNS ILP optimization prob-

lem is expected to incur high computing costs or pos-

sibly become NP-hard because its combinatorial na-

ture, which means the solution may not be tractable

or calculated efﬁciently when the size of the problem

is large. A viable approach to addressing this com-

plexity is to ﬁnd an approximate algorithm to the ex-

act optimization solution. The goal is to develop a

heuristic algorithm that can yield a good-enough so-

lution close to the exact solution but with signiﬁcantly

lower computational cost.

This section describes an approximation algo-

rithm using simple greedy heuristic (GH). A heuris-

tic is a problem-domain speciﬁc knowledge that can

be used to facilitate a quick and “optimal” solution

development. The greedy heuristic approximation al-

gorithm trades optimality and solvability of the prob-

lem (if an optimal solution exists) for lower comput-

ing cost (time). We assume the VN node mapping to

gateway node is already determined based on the lo-

cation constraint which is normally a easy step for our

VNS problem.

Below is the pseudo code of our GH-based ap-

proximation algorithm that ﬁnds an approximation

solution to the Multi-AS VNS optimization problem.

Greedy Heuristics (GH) Algorithm:

procedure GH_VNS (vnr)

L = LinkSort (vnr) as input link list

solution = {}

while not NotEmpty (L) do

l = SelectNextBestLink (L)

map l to next feasible kth-shortest path

if SatisfyConstraints(solution, l) then

solution = solution U {l}

endif

endwhile

return solution

end

Heuristics used include:

• Mapping each VN link to the lowest possible cost

path between two mapped gateway nodes ﬁrst

tend to yield lower mapping cost for each link,

thus better total cost across all the VN links.

• Map each link in VN in descending capacity order

tends to generate the closest solution to the opti-

mal one and reduce blocking probability across all

the VN links.

Some of the implementation strategies and observa-

tions are in order below:

1. To save time, we pre-calculate all K-shortest paths

(select K=2 or 3) between gateway nodes using

segment resources across all InPs.

2. A solvable problem using optimization approach

can fail to produce an approximate solution due to

blocking in the GH algorithm.

3. Different sequence of VN links input could yield

different blocking behavior (which one is blocked

and how many), thus can lead to different approx-

imate solutions, particularly for larger size VNS

problem.

6 EXPERIMENT AND RESULTS

This section brieﬂy describes our experiments with

generating exact solution of the VNS problem using

ILP and approximate solution using greedy heuristics

(GH) algorithm.

6.1 Experiment Setting and Evaluation

Methods

The experimental testbed uses the following hardware

and software: 1) Hardware: A HP laptop with Intel

Core i7 CPU, and 16 G RAM, running Window 10; 2)

Software: Python 3.1 package and AMPL optimiza-

tion tool and CPLEX solver are utilized.

Modeling and Optimization of Virtual Networks in Multi-AS Environment

217

The primary objective of the experiment is to eval-

uate and compare the performance of the VNS exact

optimal solution and approximate solutions in terms

of run-time and network cost produced by the VNS

ILP optimization and approximate GH algorithm. We

deﬁne the following two basic metrics plus two de-

rived metrics for quantitative performance analysis.

1. Run time denoted by T (alg): The time in seconds

taken to calculate an exact or approximate solu-

tions by the optimization or approximation algo-

rithms,

2. Cost denoted by C(alg): The total cost of the ex-

act or approximate solutions by the optimization

or approximation algorithms.

Other than these two basic metrics, we deﬁne two de-

rived metrics to assist in the quantitative analysis. The

following normalized metrics are deﬁned where A

opt

represents the optimization algorithm and A

apr

repre-

sents approximation algorithm.

1. Approximation Error Rate (AER): Measure how

close the cost of the solution generated by approx-

imation algorithm is to the cost of the optimal so-

lution.

AER(A

apr

) =

(C(A

apr

) −C(A

opt

))

C(A

opt

)

Since C(A

opt

) is always less than or equal to

C(A

apr

), we have AER(A

apr

) ≥ 0 and a larger

value means a worse performance.

2. Speed-Up Factor (SF): The ratio between the time

taken by optimal algorithm over the time taken by

the approximate algorithm.

SF(A

apr

) =

T (A

opt

)

T (A

apr

)

Since T (A

opt

) is normally greater than or equal

to T (A

apr

), we have S F(A

apr

) ≥ 1 and a larger

value means a better performance. This is speed

up for an approximation algorithm in reference to

the optimal algorithm.

6.2 Experiment Data Set

To evaluate our Multi-AS VNS algorithms, a pair of

input data of SP VNR and InP networks intercon-

nected the VNP gateways are required to run the algo-

rithm, whether it’s optimization algorithm or approx-

imation algorithm.

The test data for each algorithm run requires the

following:

1. SP network: The VNR network is speciﬁed by SP

in terms of network topology, node capacity, and

link QoS parameters including capacity and delay.

2. VNP network: The set of the VNP gateway nodes

and associated links to the InP edge nodes within

a given proximity. The links assume enough ca-

pacity.

3. InP network resource pool: The set of path seg-

ments provided by a set of InP network. Each

segment includes capacity, cost and delay values.

A sample SP VNR network and InP network

with aggregated path segments from multiple InPs are

shown below in Figures 7 and 8.

Figure 7: A Sample VNR Network Topology.

Figure 8: A Sample InP Network Topology.

The experiments use real-world like synthetic data

generated using following rules:

• The test VNR network and InP network topology

are from a ﬁxed set of deﬁned VNR and InP net-

work topology as a base network data set, namely

the building blocks.

• From the base set of the base set of the VNR and

InP networks , we can combine a subset of net-

works from the base to generate a large number

of VNR and InP test networks with with varying

topological complexity and sizes.

• For this experiment, 1000 compatible VNR-InP

network pair data set (i.e., matching at the node

level) are generated as the population and we ran-

domly select 50 of them as sample for the experi-

mental evaluation.

• Each test run is for the optimization algorithm and

the GH approximation algorithm to run on a select

VNR-InP input data pair.

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6.3 Experiment Results and Analysis

This section presents analysis results in the context of

two questions we are trying to answer:

1. How does the GH approximation algorithm per-

form compared to the optimization algorithm in

terms of basic performance metrics, namely run

time and cost of the VNS algorithms for a given

VNR run.

2. How well does the GH approximation algo-

rithm perform in terms of decrease in run-time

(i.e.,speed gain) vs. increase in cost of the solu-

tion (i.e., loss in optimality) for each VNR run.

The following line charts in Figures 9,10,11 and

12 show the performance comparison between the

ILP based optimal algorithm and GH approximation

algorithms across all the test data based on two ba-

sic metrics and two derived metrics deﬁned in section

6.1.

Figure 9: Run-time for Approximate GH Algorithm.

Figure 10: Cost for Approximate GH Algorithm.

From the result charts, the following observations

can be made:

1. The run times for the GH approximation algo-

rithm are fractional compared to those of the opti-

mization algorithm for all the VNR-InP test data.

It is at least one order of magnitude lower.

2. Note that the gain in speed for the approximation

algorithms is at the cost of very small optimality

Figure 11: Speed-up for Approximate GH Algorithm.

Figure 12: Error for Approximate GH Algorithm.

loss for the VNS solution.

In summary, the GH approximation algorithm’s per-

formance seems to be pretty stable and consistent, and

have a speed up of 1-2 order of magnitude compared

to the optimization algorithm depending on the VNR-

InP test data. However the cost increase of the GH ap-

proximate solution is practically negligible for major-

ity of the test case. Thus GH approximation algorithm

is recommended in practical system implementation.

7 CONCLUSION AND FUTURE

RESEARCH

This paper addresses a challenging and open prob-

lem, called multi-AS virtual network synthesis (VNS)

problem that is aimed at effectively and optimally pro-

visioning virtual networks over shared substrate net-

work resources across multiple ISP network infras-

tructure. The proposed approach is novel and tied

to real-world virtual network provisioning problem

across multiple ISPs, such as Over-The-Top (OTT)

virtual networks, and software deﬁned WAN (SD-

WAN). The research efforts include network model-

ing, optimization formulation, a heuristic approxima-

tion algorithm, and the performance experiment and

quantitative evaluation of the optimal and approxima-

tion algorithms based on a random set of synthetic

test data. The evaluation results are promising and

Modeling and Optimization of Virtual Networks in Multi-AS Environment

219

have demonstrated the validity and effectiveness of a

GH-based approximation algorithm for practical use.

Beside the research results reported in this paper,

some follow-on and extended researches can be per-

formed, which include: 1) develop additional heuris-

tic and meta-heuristic approximation algorithms. For

example, the GRASP and ILS heuristic algorithms

can be developed and evaluated; 2) Multi-objective

optimization for Multi-AS VNS can be formulated to

address the need to optimize generic multiple con-

straints and multiple objective optimization problems,

e.g., combined cost and end-to-end QoS optimization.

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