Optimal Virtual Network Embedding: Node-Link Formulation

356 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 10, NO. 4, DECEMBER 2013

Optimal Virtual Network Embedding:Node-Link Formulation

Marcio Melo, Susana Sargento, Ulrich Killat, Andreas Timm-Giel, and Jorge Carapinha

Abstract—Network Virtualization is claimed to be a keycomponent of the Future Internet, providing the dynamic supportof different networks with different paradigms and mechanismsin the same physical infrastructure. A major challenge in thedynamic provision of virtual networks is the efficient embeddingof virtual resources into physical ones. Since this problem isknown to be NP-hard, previous research focused on designingheuristic-based algorithms; most of them either do not considera simultaneous embedding of virtual nodes and virtual links, orapply link-path formulation, leading to non-optimal solutions.

This paper proposes an integer linear programming (ILP) for-mulation to solve the online virtual network embedding problemas a result of an objective function striving for the minimizationof resource consumption and load balancing. To this end 3different objective functions are proposed and evaluated. Thisapproach applies multi-commodity flow constraint to accomplisha node-link formulation that optimizes the allocation of physicalnetwork resources.

This proposal is evaluated against state of the art heuristics.The performance of the heuristics related to Virtual Network(VN) request acceptance ratio is, at least, 30% below the oneof the Virtual Network Embedding Node-Link Formulation(VNE-NLF) method. From the three cost functions evaluated,the Weighted Shortest Distance Path (WSDP) is the one whichembeds more VNs and also requires, on average, less physicalresources per embedding.

Index Terms—Virtual networks, embedding, mapping, NP-hard, optimization, ILP model, heuristics.

I. INTRODUCTION

NETWORK VIRTUALIZATION has gained an increas-ing prominence in networking and telecommunications

fields in the last few years. Initially, the interest in networkvirtualization was mainly pushed by Future Internet researchinitiatives [1]–[4], mainly with the objective to find a platformon which novel Internet architectures could be experimentedand evaluated without limitations or constraints, namely thoseassociated with the traditional IP model. Later on, it becameclear that virtualization could constitute a key component ofthe next-generation Internet architecture itself [5], and not justas a mere platform for experimentation. It also became clearfor network operators that network virtualization could providea number of short/medium term business advantages, with

Manuscript received January 2, 2013; revised June 24, 2013. The associateeditor coordinating the review of this paper and approving it for publicationwas D. Medhi.

M. Melo and J. Carapinha are with Portugal Telecom Inovacao, Aveiro,Portugal, (e-mail: {marcio-d-melo, jorgec}@ptinovacao.pt).

S. Sargento is with the Instituto de Telelcomunicacoes, University ofAveiro, Portugal (e-mail: [email protected]).

U. Killat and A. Timm-Giel are with the Institute of CommunicationNetworks, Hamburg University of Technology, Germany (e-mail: {killat,timm-giel}@tuhh.de).

Digital Object Identifier 10.1109/TNSM.2013.092813.130397

potential reduction of costs and increase of revenues, as aninteresting tool from an operational point of view [6], [7].

Although there is a large interest on virtualized networksboth from the research community and network operators,several challenges still prevent them from being deployed inreal environments [8]. One of the major obstacles lies in theefficient embedding1 of a Virtual Network (VN) onto a phys-ical network. Since this process requires the simultaneous op-timization of virtual nodes and links placement, it is complexin nature, both in formulation and computationally. Severalworks, such as [9]–[17], have already proposed solutions tothis problem, mostly based on heuristic approaches; however,they do not provide the optimal solution for each VN mapping.

This paper focuses on the online embedding of VN requestsin the physical network. An Integer Linear Programming(ILP) formulation, the Virtual Network Embedding Node-LinkFormulation (VNE-NLF), is used to solve the VN assignmentproblem on the basis of a minimization of resource consump-tion and load balancing strategy. The VNE-NLF includes linkdelay constraints and supports the specification of the max-imum distance between virtual nodes. In addition, differentcost functions are proposed and analyzed, which enforce loadbalancing of links and nodes, and shortest distance paths.Simulation experiments show how far the state of the artheuristics differ from an ILP method. If the VN requestacceptance ratio is used as a measurement metric, the solutionsobtained by the state of the art heuristics are, at least, 30%below the ones of the VNE-NLF (see Fig. 3). From thecost functions evaluated, the Weighted Shortest Distance Path(WSDP) is the one which embeds more VNs and also requires,on average, less physical resources per embedding. Comparedto our previous work in [18], this paper:

i extends the mathematical formulation to support two newconstraints, i.e. link delay and maximum distance betweennodes;

ii proposes three new cost functions, i.e. Load Balancingplus ε Shortest Path (LB+εSP), Shortest Distance Path(SDP), and WSDP;

iii defines a new evaluation metric, i.e. the embedding factor,which represents the amount of resources that have beenrequested over the amount of resources that have beenleased;

iv and it provides a performance comparison with 6 state-of-the-art heuristics.

The contributions of this paper can be summarized asfollows:

1The terms embedding, mapping and assignment are used interchangeablyin this paper.

1932-4537/13/$31.00 c© 2013 IEEE

MELO et al.: OPTIMAL VIRTUAL NETWORK EMBEDDING: NODE-LINK FORMULATION 357

i ILP optimization for virtual network embedding, whichis based on node-link formulation; it enables the simul-taneous embedding of virtual nodes and links, whichoptimizes the allocation of physical network resources, i.e.Central Processing Units (CPUs) on the physical nodesand bandwidth on the physical links.

ii Analysis and evaluation of different objective goals: loadbalancing objective function LB+εSP, shortest path ob-jective function SDP, and load balancing combined withshortest path objective function WSDP;

iii Comparison of the performance of existing VN embeddingsolutions, heuristics, with a pure ILP formulation.

iv Evaluation metrics that relate e.g. VN acceptance ratio andlink utilization. Moreover, a new metric, the embeddingfactor, is proposed.

The rest of the paper is organized as follows. After summa-rizing the related works in section II, section III describes thevirtual network embedding problem, the notations and param-eters used, and the embedding process. Section IV describesthe proposed VNE-NLF and the applied constraints, whilesection V-A presents and discusses the different proposed costfunctions. Section VI analyzes the performance of the VNE-NLF with different cost functions, and compares it with sixexisting heuristics. Finally, section VII concludes the paperand describes the future work.

II. RELATED WORK

This simultaneous node and link mapping optimization canbe formulated as an un-splittable flow problem [9], knownto be NP-hard. In order to solve this problem, severalapproaches have been suggested, mostly considering the off-line version of the problem where the VN requests are fullyknown in advance.

In [10] a backtracking method based on sub-graph isomor-phism was proposed; it considers the on-line version of themapping problem, where the VN requests are not known inadvance, and proposes a single stage approach where nodesand links are mapped simultaneously, taking constraints intoconsideration at each step of the mapping. When a badmapping decision is detected, a backtrack to the previous validmapping decision is made, avoiding a costly re-map.

The work in [11] defined a set of premises about the virtualtopology, i.e. the backbone nodes are star-connected and theaccess-nodes connect to a single backbone node. Based onthese premises, an iterative algorithm is run, with differentsteps for core and access mapping. However, the algorithmcan only work for specific topologies.

A distributed algorithm was studied in [19]. It considersthat the virtual topologies can be decomposed in hub-and-spoke clusters and each cluster can be mapped independently,therefore reducing the complexity of the full VN mapping.This proposal has lower performance when compared withcentralized approaches.

Zhu et al. [9] proposed a heuristic based on a centralizedalgorithm to deal with VN mapping. The goal of the algorithmis to maintain a low and balanced load of both nodes and linksof the substrate network. Yu et al. [12] proposed a mappingalgorithm which considers finite resources in the physical

network, and enables path splitting (i.e. virtual link composedby different paths) and link migration (i.e. to change theunderlying mapping) during the embedding process. However,this level of freedom can lead to a level of fragmentation thatis unfeasible to manage large scale networks.

In [13] a formal approach is taken to solve the on-lineVN mapping problem using a mixed integer programmingformulation. Chowdhury et al. applied a two step approachto embed VNs on the substrate. In the first step, the virtualnodes are assigned to physical nodes and in the second stepthe virtual links are assigned to physical paths. Comparedto the previous state of the art heuristics, i.e. [9], [12], theformulation proposed by Chowdhury et al. provides a bettercoordination of the two phases, since an “augmented substrategraph construction” is used.

The approach in [13] completely differs from the math-ematical formulation proposed in this paper, which appliesa node-link formulation. In our approach, the universe ofembedding solutions is considered within the ILP formulation,and the VN embedding problem is solved in a single step usingthe multi-commodity flow constraint and by considering thenotion of direction of the flows.

Butt et al. [14] proposed a topology aware heuristic for VNmapping, and also suggested algorithms to avoid bottleneckson the physical infrastructure, where they consider virtual nodereallocation and link reassignment for this purpose. Nogueiraet al. [15] proposed a heuristic that takes into account the het-erogeneity of the VNs and also of the physical infrastructure.The heuristic is evaluated by means of simulation and also ona small scale testbed, where it achieves mapping times of theorder of tens of milliseconds.

Botero et al. [16] proposed an algorithm to solve the VNmapping problem, which also considers the CPU demandof the hidden hops. Chowdhury et al. [17] extended hispreliminary results [13] and included a generalized window-based VN embedding to evaluate the effect of look ahead onthe mapping of VNs.

Alkmim et al. [20] proposed a mathematical formulationthat aims to: i) map virtual routers and virtual links; ii)minimize the bandwidth consumption; and iii) minimize thetime required to instantiate a virtual router. In contrast tothis work, we also aim to optimize link load and CPU loaddistribution.

Although all these algorithms provide a solution for the VNmapping problem, an optimal solution for the embedding taskand its efficiency is not provided. Also, some of them fail tosolve the assignment problem as a simultaneous optimizationof the virtual node and link placement, which leads to non-optimal solutions.

The VNE-NLF applies a node-link formulation to solvethe VN embedding problem in a single step using the multi-commodity flow constraint. This approach provides the opti-mal solution for the objective function used, since the universeof solutions is considered within the ILP formulation.

III. NETWORK DESCRIPTION AND PROBLEM

FORMULATION

In this section, we introduce the virtual network embed-ding problem. In addition, the VN embedding notations used


Fig. 1. VN Embedding System—topology example.

throughout the paper are presented, and the virtual networkembedding system is explained. Finally, the mapping goalsare introduced to support the mathematical formulation.

A. Network Description

We use superscript to distinguish the physical network fromthe virtual network, where p and v correspond to physical andvirtual, respectively.

1) Physical network: A physical network canbe described as a weighted undirected graphGp = {Np, Lp, Cp, Bp, Dp,Disp} composed by a setof physical nodes, Np, and a set of physical links, Lp. Eachphysical node i is characterized by its processing capacity,Cp

i , commonly referred to as the CPU, and by its physicallocation, which can be defined by x and y coordinates.The distance between virtual nodes, Disp, can be obtainedusing expression (1). With respect to the physical links, weconsider that each link ij has a given bandwidth, Bp

ij , anda given link delay, Dp

ij , and we also assume that each linkis an undirected link. The bottom-right of Fig. 1 illustrates aphysical network topology example composed of 6 physicalnodes and 8 physical links, and the corresponding capacitiesof the nodes and the links are presented on top of theelements.

Dispij =√(xj − xi)2 + (yj − yi)2 (1)

2) Virtual Network Request: VN request canbe described as a weighted undirected graphGv = {Nv, Lv, Cv, Bv, Dv,Disv} composed by a setof virtual nodes, Nv, and a set virtual links, Lv. Each virtualnode m is characterized by the amount of required CPU, Cv

m,and the virtual links mn are logical connections betweenvirtual nodes and characterized by the amount of dedicatedbandwidth, Bv

mn, and by the maximum link delay permitted,Dv

mn. We also assume that each virtual link is an undirectedlink. The maximum distance between virtual nodes, Disv, canbe used to limit the number of intermediate hops betweenvirtual nodes. The left part of Fig. 1 represent the example oftwo virtual network requests, VN request 1 on the bottom-leftand VN request k on the top-left. Each VN request has a

Fig. 2. VN request life cycle—activity Diagram.

given lifetime that is, in principle, independent from eachother, and each lifetime could have different time scales,since it is strongly dependent on the purpose of the virtualnetwork request itself. If we consider a VN request for a liverock concert, the time scale will be hours, but if we considera VN for a culinary workshop of one week, the time scalewill be days.

3) VN Assignment Notations: First, we start with the con-vention used for the index notation: Np represent the set ofnodes that belong to the physical network; Lp represent the setof links that belong to the physical network; and Lp

i representa subset of links ij that are directly connected to the nodei. The same type of notation is used to represent the VNusing the letters m and n in the virtual network. The notationsused throughout this paper for the VN assignment problemare presented in table I. The table is divided into three parts:the static parameters of the physical network, the dynamicparameters of the physical network, and the virtual networkrequests with the demanded capacities.

B. Unfilled Physical Network Resources

The remaining capacity of each physical node at a specifictime t is given by the difference between the total processingcapacity and the capacity consumed by all virtual nodes allo-cated on that physical node, and is presented in expression (2),where u represents the set of all virtual nodes allocated on thatprecise physical node and at time t.

∀i ∈ Np : Cpi (t) = Cp

i (0)−∑u

Cvu(t) (2)


TABLE IVN ASSIGNMENT PROBLEM NOTATION.

Gp Physical NetworkNp Set of Physical Nodesi, j Physical Nodesij Physical LinkLp Set of Physical LinksLpi Set of Physical Links directly connected to Physical

Node iCp

i (0) Total CPU of Physical Node iDispij Distance Between Physical Nodes ij

Bpij(0) Total Bandwidth of Physical Link ij

Cpi (t) Available CPU at time t on Physical Node i

Bpij(t) Available Bandwidth at time t on Physical Link ij

Gv(k) Virtual Network Request kNv(k) Set of Virtual Nodes of VN Request kLv(k) Set of Virtual Links of VN Request kLvm(k) Set of Virtual Links directly connected to Virtual node

m of VN Request km,n Virtual Nodesmn Virtual LinkCv

m(k) CPU of Virtual Node m of VN Request kDisvmn(k) Maximum Distance Between Virtual Nodes mn of VN

Request kBv

mn(k) Bandwidth of Virtual Link mn of VN Request kDv

mn(k) Delay of Virtual Link mn of VN Request k

In parallel, the available bandwidth of each physical link ata specific time t is given by the difference between the totalbandwidth and the bandwidth consumed by all virtual linksegments allocated on that physical link, and is presented inexpression (3), where w represents the set of all virtual linksegments allocated on that specific physical link and at timet.

A virtual link can be composed of one or more physicallinks, physical path. We consider that each virtual link has asingle physical path, and we do not consider link aggregation(i.e. virtual link composed by different physical paths).

One physical link could accommodate one or more virtuallink segments belonging to different virtual links.

∀ij ∈ Lp(k) : Bpij(t) = Bp

ij(0)−∑w

Bvw(t) (3)

C. VN Request Embedding Process

The VN request embedding process can be divided into twocomponents: the component that ensures the mapping of thevirtual nodes, and the one that handles the mapping of thevirtual links.

1) Virtual Node Mapping: Each virtual node needs to bemapped onto one physical node, this relation is given bythe mapping function M[m ∈ Nv(k)] = i, where virtualnode m is mapped onto physical node i. Each physical nodecandidate needs to have, at least, the same amount of availableCPU as required by the virtual node, which is represented inexpression (4).

∀i, ∀M[m ∈ Nv(k) = i] : Cvm(k) ≤ Cp

i (t) (4)

2) Virtual Link Mapping: Each virtual link can be mappedonto one or more physical links (i.e. physical path), thisrelation is given by the mapping function M[Lv

mn], wherethe virtual link mn is mapped onto one physical path. Eachphysical link candidate belonging to the physical path needs to

have, at least, the same amount of bandwidth available as re-quired by the virtual link which is presented in expression (5).

∀ij ⊆ M[mn ∈ Lv(k)] : Bvmn(k) ≤ Bp

ij(t) (5)

D. VN Request Life Cycle

The embedding process begins upon a new VN requestarrival, which is depicted in Fig. 2. A VN mapping method(e.g., heuristic) is used to embed the VN; it takes as inputsthe current status of the physical network (e.g. available CPUcapacity and existing bandwidth) and the VN request itself.If the result of the mapping process is a viable solution, themapping is considered to be feasible; if not, it is consideredto be unfeasible and the VN embedding process stops.

E. Mapping Metrics

In order to assess the performance of an embedding method,and at the same time to compare it with others, differentperformance metrics were defined.

1) VN Request Acceptance Ratio: The VN request accep-tance ratio, AVN, is given by expression (6) and defines theoverall performance of an embedding method: the number ofVN requests accepted, k′, over the number of all VN requests,k.

AVN =k′

k(6)

2) Embedding Factor: The embedding factor, EVN, is givenby expression (7) and represents the ratio between the amountof virtual resources that were requested for the VN and theamount of physical resources that were effectively provisionedto accommodate that VN, i.e. the efficiency on embedding.The parameters, α, β, γ and η, are used to weight the differenttypes of resources.

EVN =α∑

m Cvm + β

∑mn B

vmn

γ∑

iCpi + η

∑ij B

pij

(7)

IV. VIRTUAL NETWORK EMBEDDING - MATHEMATICAL

FORMULATION

This section describes the mathematical formulation devel-oped to solve the online VN embedding problem with thedefined constraints.

An Integer Linear Programming (ILP) approach is used tosolve the online VN embedding problem; we propose a node-link formulation, and two assignment variables are appliedduring the embedding process. The index notation used hereis the same as in section III-A3.

A. Assignment Variables

The binary variable x is used in the mapping of the virtualnodes and is defined in expression (8), where xm

i → NV ×NP

matrix. With respect to the virtual links, the binary variabley is used and it is represented in equation (9), where ymn

ij →(LV )2 × (LP )2 matrix.


1) Virtual Node Assignment:

xmi =

{1, virtual node m is allocated at physical node i

0, else(8)

2) Virtual Link Assignment:

ymnij =

{1, virtual link mn uses physical link ij

0, else(9)

B. Constraints

To assure the correct mapping of the virtual nodes and ofthe virtual links, and also to obey to the conservation law onthe capacities of the physical nodes and physical links, a setof constraints is defined.

1) Assignment of virtual nodes to physical nodes: Equation(10) ensures that each virtual node is assigned, and that it isassigned to just one physical node.

∀m :∑i

xmi = 1 (10)

2) One virtual node per physical node: Equation (11)guarantees that each physical node can accommodate in themaximum one virtual node per VN request, although eachphysical node can accommodate other virtual nodes fromdifferent VNs. This constraint is used to ensure that eachvirtual node is assigned to a different physical node per VNembedding, and can be suitable in application scenarios whereit is required to have physical node diversity for redundancyreasons.

∀i :∑m

xmi ≤ 1 (11)

3) CPU conservation: Equation (12) assures that the avail-able CPU capacity of each physical node is not exceeded.

∀i :∑m

xmi · Cv

m ≤ Cpi (12)

4) Virtual Node distance: Equation (13) assures that themaximum distance between virtual nodes, Dv

mn, is not vi-olated. The maximum distance between virtual nodes is aparameter of the VN embedding problem. The effect of thisparameter on the VN embedding will be studied on a separatesection (see VI-C).

This parameter is given in distance units and can be usedto express the maximum radius between virtual nodes (in thesimulated scenario the location of the physical nodes is setin a grid). The distance between physical nodes, i.e. Dispij , isobtained using formula (1), and K represents a large constantwhich is used only in situations where the virtual node n isnot mapped at physical node i, i.e. xn

i = 0.

∀m,n ∈ Lvm,m < n, ∀i :∑

j

Dispij · xmj ≤ Disvmn · xn

i + (1 − xni ) ·K (13)

5) Assignment of virtual links to physical links - multi-commodity flow conservation with node-link formulation:To simultaneously optimize the mapping of virtual links andvirtual nodes, the multi-commodity flow constraint [21] isapplied with a node-link formulation [22]; moreover, thenotion of direct flows on the virtual links is used, which isrepresented in Eq. (14), where Lv

m represents all the virtuallinks that are directly connected with the virtual node m, andLpi represent all the physical links that are directly connected

with the physical node i

∀mn ∈ Lvm,m < n, ∀i :∑

ij∈Lpi

(ymnij − ymn

ji ) = xmi − xn

i (14)

6) Bandwidth conservation: To ensure that the availablebandwidth at each physical link is not surpassed, Equation(15) is defined.

∀ij ∈ Lpi , i < j :∑

mn∈Lvm,m<n

Bvmn(y

mnij + ymn

ji ) ≤ Bpij (15)

7) Link delay limit: The virtual link delay, Dvmn, is a

parameter of the VN embedding problem, and is equal to thesum of the delay of all physical links that compose the virtuallink. To ensure that the constraint on the link delay is notviolated we apply equation (16).

∀mn ∈ Lvm,m < n, ∀i :∑

ij∈Lpi ,i<j

Dpij(y

mnij + ymn

ji ) ≤ Dvmn (16)

V. VIRTUAL NETWORK ASSIGNMENT - OBJECTIVE

FUNCTION

One of the major challenges when formulating an ILPmodel for VN assignment resides in the definition of the objec-tive function: the allocation of resources need to be optimizedin order to support the efficiency of the corresponding VNprocess.

Moreover, the correct specification of the VN mappingconstraints (see IV) is also a challenge of this approach. In thissection, we describe the main goals that need to be achievedwhen formulating an objective function for virtual networkembedding; three different objective functions are proposedto achieve these goals.

A. Objective Goals

A primary goal for the embedding algorithm is to minimizeresource consumption in order to have resources availablefor forthcoming VN embedding requests. Minimization ofresource consumption is only possible for the bandwidthconsumption depending on the number of links involved inan embedding process. The processing power has just to beinstalled exactly in the amount required by the VN request onsome physical nodes.

Resource minimization consequently means that the VN’sshould exhibit minimal hop counts on their paths. This in turn


means that almost every physical node should be available tohost a virtual node. As long as the resources required by VN’sare small compared to physical capacities of nodes and links,this availability is guaranteed with high probability by a loadbalancing strategy, which results in some spare capacity foreach physical node or link.

Therefore, the dominating aspects in the formulation of anobjective function for the ILP problem are the minimizationof bandwidth consumption and load balancing.

B. Load Balancing plus ε Shortest Path (LB+εSP)

The objective function LB+εSP is proposed in expression(17), and it achieves two goals: the primary goal is to minimizethe maximum load per physical resources; in the case of differ-ent mapping solutions with the same maximum utilization, thesecond part of the objective function is activated which willopt for the solution which consumes the lowest bandwidth.LCp

max represents the overall maximum node load; LBp

max

represents the overall maximum link load. The parametersCp

i (0), Bpij(0), C

vm(k), Bv

mn(k) were defined in Table I; theparameter ε represents a small constant, which should be smallenough to not affect the first objective; and the parametersα and β are used to weight the load cost of each type ofresources.

minimize α · LCp

max + β · LBp

max + ε ·∑mn

ymnij · Bv

mn(t),

∀i ∈ Np :Cp

i (t) +∑

m xmi · Cv

m(k)

Cpi (0)

≤ LCp

max

∀ij ∈ Lp :Bp

ij(t) +∑

mn ymnij ·Bv

mn(k)

Bpij(0)

≤ LBp

max (17)

C. Shortest Distance Path (SDP)

The previous objective function (17) works well in sit-uations where there are abundant resources in the physicalnetwork. Then, bandwidth consumption is of no concern andload balancing is beneficial because it gives a high degree offlexibility in the resource allocation process.

Nevertheless, in situations where the physical resources arescarce, it is desirable to reduce the number of physical linksconsumed to the minimum possible.

Therefore, the objective function SDP, proposed in expres-sion (18), aims to minimize the number of physical linksconsumed due to the VN embedding, while it prefers physicallinks with more available bandwidth, and at the same timechooses physical nodes with more available CPU power,thereby supporting the load balancing aspect. The parametersα and β are used to weight the cost of each type of resource.(Note that the first term in eq. (18) would result in a constant,if Cp

i (t) was missing in the denominator.)

minimize α

(∑m

∑i

xmi

Cpi (t)

)+β

⎛⎝∑

mn

∑ij

ymnij

Bpij(t)

⎞⎠ (18)

D. Weighted Shortest Distance Path (WSDP)

The objective function WSDP, proposed in equation (19), issimilar to equation (18), although here the demanded capacityby the VN is included in the objective function. This has theeffect that high demands are allocated to nodes or links witha large amount of free capacity.

minimize α

(∑m

Cvm(k)

[∑i

xmi

Cpi (t)

])+

β

⎛⎝∑

mn

Bvmn(k)

⎡⎣∑

ij

ymnij

Bpij(t)

⎤⎦⎞⎠ (19)

VI. EVALUATION RESULTS

In this section, we describe the simulation scenario, theevaluation metrics, and depict our major results. We comparethe VNE-NLF model in its several versions with six state ofthe art methods.

A. Simulation Parameters

To evaluate the VNE-NLF model, we have implementeda discrete event simulator in Matlab R©, with the proposedformulation using different objective functions.

The physical network topology is created using the GT-ITMtool [23], the number of physical nodes is set to 50, whichis representative of a medium scale infrastructure provider,and the link probability between two physical nodes is setto 0.5. The node CPU capacity and the link bandwidth arereal numbers uniformly distributed between 50 and 100. TheVNs requests are also representative of either small or mediumscale virtual networks, and are created using the same topologygeneration method; the number of virtual nodes is not fixed,but follows a uniform distribution, from 2 to 10 virtual nodesper VN topology; the virtual link probability is set to 0.5.The CPU capacity of the virtual nodes and the bandwidth ofthe virtual links are also real numbers uniformly distributedbetween 0 and 20, and between 0 and 50, respectively2. Theconsidered values for the bandwidth and for the CPU are nor-malized, since the objective function aims at simultaneouslyoptimizing the allocation of both types of resources.

We assume that VN requests arrive according to a Poissonprocess, and that each VN has an associated lifetime measuredin time units with an average of 1/μ = 1000, following anexponential distribution. The same assumption was also takenby the authors of [13]. The average number of VN requestsper time unit, i.e., value of λ, is started with 3 VN requestsper 100 time units, and increases by 1 VN request, up to10 requests. This can give an insight into two opposite casescenarios, with a very high and very low acceptance ratio. Foreach value of λ, 10 trials are performed. A new set of VNrequests and a new physical network topology are generatedfor each trial. All simulations are set to run up to 50000 timeunits to mitigate the transient phase effect [24] and to obtainthe steady-state. A confidence interval of 95% is used for allresults presented below.

2These values were also considered by the authors of [9], [12], [13]


TABLE IICOMPARED VN EMBEDDING METHODS.

Notation Method DescriptionG-SP [9] Greedy Node Mapping with Shortest Path

Based Link Mapping.G-MCF [12] Greedy Node Mapping with Splittable Link

Mapping using MCF.R-ViNE [13] Randomized Node Mapping with Splittable

Link Mapping using MCF.D-ViNE [13] Deterministic Node Mapping with Splittable

Link Mapping using MCF.D-ViNE-SP [13] Deterministic Node Mapping with Shortest

Path Based Link Mapping.D-ViNE-LB [13] Deterministic Node Mapping with Split-

table Link Mapping using MCF, whereαuv = βuv = 1, ∀u, v, w ∈ NS .

VNE-NLF-LB+εSP VN Embedding with node-link Formulationusing overall Load Balancing; in the caseof having more than one solution, it usesShortest Path, where ε = 1.0× 10−11.

VNE-NLF-SDP VN Embedding with node-link Formulationusing overall Short Distance Path.

VNE-NLF-WSDP VN Embedding with node-link Formulationusing overall Weighted Short Distance Path.

The evaluated embedding methods are Greedy Node Map-ping with Shortest Path based Link Mapping (G-SP) [9],Greedy Node Mapping with Splittable Link Mapping usingMulti-Commodity Flow Constraint (G-MCF) [12], Random-ized Node Mapping with Splittable Link Mapping using Multi-Commodity Flow Constraint (R-ViNE) [13], DeterministicNode Mapping with Splittable Link Mapping using Multi-Commodity Flow Constraint (D-ViNE) [13], DeterministicNode Mapping with Shortest Path based Link Mapping(D-ViNE-SP) [13], Deterministic Node Mapping with Split-table Link Mapping using Multi-Commodity Flow Constraintand Load Balancing based (D-ViNE-LB) [13], and the pro-posed linear programming formulation, i.e. VNE-NLF, with 3different cost functions which were described in the previoussection. All these methods are briefly summarized in Table II.

The state of the art methods are simulated using an existingimplementation [25]; to solve the mixed integer programmingon the methods G-MCF, R-ViNE, D-ViNE, D-ViNE-LB, andD-ViNE-SP, the GLPK [26] solver version 4.20 is used.

All the simulations for the different embedding meth-ods were performed using an Intel R© Xeon R© [email protected], and the time consumed per VN requestembedding was registered.

The CPLEX R© [27] version 12.2 was used to solve the linearprogramming problem of the VNE-NLF; a time limit of 600seconds is defined for each VN mapping, although most ofthe VNs are embedded in hundreds of milliseconds; and theCPLEX R© was set to use only one CPU core for comparisonpurposes with the remaining methods. The evaluation metricsare the ones defined in section III-E.

B. Impact of the Number of VN Requests

This subsection presents the evaluation results as a functionof the VN request rate, for all the previously described metrics.To increase the readability of all figures, we have considereddifferent x values for different strategies, e.g.: 3.4, 4.4, 5.4 forG-SP; 3.3, 4.3, 5.3 for G-MCF; 3.2, 4.2, 5.2 for R-ViNE.

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Fig. 3. Average VN acceptance ratio as a function of VN request rate.

Before comparing the different embedding methods and al-gorithms, we should group them into four different categoriesaccording to the nature of the method itself, i.e. heuristic,heuristic combined with mixed integer programming, andlinear programming:

i Heuristic - the VN embedding problem is solved using asimple algorithm; this method performs the VN embed-ding very fast and a possibly sub-optimal embeddingsolution is obtained. The method G-SP [9] fits into thiscategory;

ii Heuristic combined with Mixed Integer Programming(MIP) - the VN embedding problem is solved in two steps:in the first step a mathematical algorithm is used to mapvirtual nodes on physical nodes, and in the second step theMIP is performed to embed the virtual links. The methodG-MCF [12] fits into this category.

iii Heuristic combined with Mixed Integer Programming(MIP) and a better coordination between mapping phasesis added - the same principle is applied, as in the above cat-egory, to solve the VN embedding problem, although a bet-ter coordination between the mapping phases is achievedusing an augmented ”substrategraphconstruction” [13].The methods R-ViNE, D-ViNE, D-ViNE-SP, D-ViNE-LB [13] fit into this category;

iv Integer Linear Programming (ILP) - the VN embeddingproblem is solved using integer linear programming. Thismethod obtains an optimal solution for a given costfunction combining resource consumption minimizationwith a load balancing strategy. The method VNE-NLF andits different objective functions fit into this category.

1) VN Request Acceptance Ratio: One of the main aspectsof the performance of each embedding method is the VNrequest acceptance ratio, which is shown in Fig. 3 and isgiven by formula (6). As can be observed, all methods show alinear behavior with the variation on the VN requests, wherethe acceptance ratio decays linearly with the number of VNrequests, and the slope is approximately the same for allmethods. This decay represents the fact that there are noinfinite physical resources.


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Fig. 4. Average node utilization as a function of VN request rate.

The method VNE-NLF, with its different objective func-tions, achieves the highest performance, and it clearly out-performs the other approaches. This is expected since integerlinear programming is applied to solve the VN embeddingproblem, and the optimal solution, according to the objectivefunction considered, is obtained per VN embedding.

The reason for these results, not only resides in the usageof an integer linear programming approach, but also in the uti-lization of the node-link formulation by the VNE-NLF, whichconsiders the universe of all possible embedding solutions,instead of a few solutions. If we take, for example, the firstcase with only 3 VN requests per 100 time units, the VNE-NLF is able to accept nearly all requests, while the remainingmethods are able to accept only 70% of the requests. Theembedding method that has the lowest acceptance ratio isthe D-ViNE-SP, and the method with the highest VN requestacceptance ratio is the VNE-NLF-WSDP.

It is expected that the VNE-NLF method will performbetter in all cases. For instance, if the embedding problemis feasible, i.e., possible solutions exist, the VNE-NLF willfind out the optimal solution according to the cost function.Using a heuristic approach or even a combined approach, thisis not always the case: frequently only a feasible solution willbe presented.

2) Node Utilization: The average node utilization as afunction of the number of VN requests is depicted in Fig.4. With a small number of VN requests, i.e., 3 VN requests,the node utilization does not go beyond 20% and 35% for theheuristic group (i.e. groups i, ii, and iii) and for the VNE-NLF, respectively. The VNE-NLF group is consuming moreresources of the physical nodes than the heuristics, which isexpected according to the acceptance ratio. When the numberof VN requests is increased, the node utilization also increases,since we are trying to accommodate more VNs on the infra-structure, but with the same amount of available physicalresources.

An efficient embedding method in situations of high VNdemand would be able to load the nodes to their full capacity.The important aspect to retain here is how much node re-

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Fig. 5. Average VN acceptance ratio times average node utilization as afunction of VN request rate.

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Fig. 6. Average link utilization as a function of VN request rate.

sources can be loaded and what kind of embedding methodstends to saturate them firstly.

The node utilization shows a dependency on the VN ac-ceptance ratio, as it can be perceived from Fig. 3 and Fig. 4.To provide a better understanding on this issue, we plot theacceptance ratio metric times the node utilization, which isshown in Fig. 5. We observe that the methods that make use ofheuristics, e.g. G-SP, or heuristics combined with MIP, e.g. G-MCF and D-ViNE-LB, show the same behavior for all the VNrequests considered, i.e. the VN acceptance metric multipliedby the node utilization metric is nearly constant. The samedoes not apply to the VNE-NLF, since it increases per VNrequest considered, until 6 VN requests per 100 time units,and beyond the 6 VN requests per 100 time units, it shows thesame behavior as its counterparts. This means that, althoughthe VN request rate is increasing, the VNE-NLF approachis able to keep with this increase until the VN embeddingproblem moves from an optimization problem (i.e. there aresufficient physical resources for the demand), to a feasibilityproblem (i.e. there are no sufficient resources for the demand).


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Fig. 7. Average VN request acceptance ratio times average link utilizationas a function of VN request rate.

3) Link Utilization: The physical link utilization metricis plotted in Fig. 6. Here we do not have the same regularbehavior according to the number of VN requests for all themethods, as shown before for the node utilization. Also, thereis no consensus in terms of clearly identifying which groupcauses the highest utilization on the physical links due to theembedding process. Nevertheless, we can clearly state that, onaverage, either the G-MCF or D-ViNE-LB shows the highestutilization on the links; in the extreme case scenario (i.e.with 10 VN requests) they have an average link utilizationof 60% and 67%, respectively. With respect to the lowest linkutilization, we observe that the embedding methods R-ViNE,D-ViNE, D-ViNE-SP, and VNE-NLF-WSDP are the ones thattend to consume less bandwidth, reaching values of 50% ofaverage link utilization, for the same considered situation.

Having in mind that one virtual link could be mapped inseveral ways, it is reasonable to observe different behaviorsaccording to the strategy of the method. If the strategy is tosave bandwidth, i.e. SP, the embedding will consume the leastbandwidth possible per VN mapping; if the strategy is loadbalancing on the links, i.e. LB, it will tend to balance theutilization among all links in order to distribute the total load.

The the link utilization also shows a dependency on the VNacceptance ratio, as can be observed in Fig. 3 and Fig. 6. Toprovide a better understanding, we plotted the acceptance ratiometric times the link utilization in Fig. 7. In contrast to thenode utilization, the dependency factor on the link utilizationshows a more complex behavior: it still increases significantlyuntil reaching 6 VN requests for the case of the VNE-NLFgroup, although it starts to decrease after 7 VN requests, in anot so expressive way. For the other methods the dependencyon the VN request rate is less pronounced.

4) Embedding Factor: Fig. 8 shows the embedding factoras a function of the VN request rate, where the weightparameters, α, β, γ, and η of equation (7) are set to 1. Theembedding factor slightly decreases with the number of VNrequests, except for the case of the VNE-NLF-LB+εSP. Thebehavior obtained when using the VNE-NLF-LB+εSP is asexpected, since it performs an overall load balancing of the

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Fig. 8. Average embedding factor as a function of VN request rate.

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Fig. 9. VN solving time as a function of VN request rate.

physical nodes and links, choosing the solution that consumesthe least bandwidth. Therefore, in the situation of only a fewVN requests, the method will tend to allocate more resourcesthan required due to the nature of the load balancing; with ahigher VN request rate, this situation tends to disappear oncethe available resources are scarcer. Therefore, the embeddingfactor will increase with the number of VN requests. We canalso state that the efficiency of the heuristic group, in general,is very low, lower than 50%. With respect to the VNE-NLFgroup, it has a good efficiency, being in most of the caseshigher than 85%; the efficiency of the VNE-NLF-WSDP iscloser to 100% which means that, on average, this methodprovisions the same amount of resources as requested per VN.

5) VN Embedding Time: An important aspect of all the VNembedding methods is the time that they require to embed,on average, a VN request and how it varies with respect tothe different loads on the physical infrastructure, i.e. the VNrequest rate.

Figure 9 shows the solving time for each method as afunction of the number of VN requests per 100 time units.

Before analyzing the figure, one must consider five different


aspects: i) all methods have been simulated using the samemachine; ii) the time to embed a VN strongly depends onthe physical characteristics (e.g. CPU) of that machine; iii)the time to embed a VN strongly depends on the nature ofthe embedding method (i.e. a mathematical algorithm willtake just a few milliseconds, while linear programming isexpected to take hundreds of milliseconds); iv) two differentlinear programming tools (GLPK was used to solve the MIPof G-MCF, R-ViNE, D-ViNE,D-ViNE-SP, D-ViNE-LB; andCPLEX R© to solve the ILP of the VNE-NLF); v) methodsR-ViNE, D-ViNE,D-ViNE-SP, and D-ViNE-LB perform twolinear programming operations, i.e. one for the mapping ofthe virtual nodes, and another for the mapping of the links.

The fourth aspect, although important for the solving time,will not interfere with the curve behavior, e.g. polynomial orexponential, since the same method, i.e. branch and cut, isapplied by both solvers (i.e. GLPK and CPLEX) to solve theVN embedding problem.

From the figure, we can observe two types of behaviors: themethod VNE-NLF with three different costs functions shows adecaying behavior with the VN request rate; for the remainingmethods we observe a nearly constant behavior.

For the first behavior, i.e. method VNE-NLF with threedifferent cost functions, one should take into considerationthat the VN request acceptance ratio is considerably higher,e.g. 90% until 6 VN requests: more than one mapping solutionper VN request is expected to exist; therefore, the optimizationprocess takes place and will consume the majority of thesolving time to obtain the optimal solution.

For the remaining methods, the VN request acceptance ratiois lower and below 70%: usually there is not more thanone mapping solution per VN request, on average, whichsignificantly reduces the solving time. This is the case of themethods G-MCF, R-ViNE, D-ViNE, D-ViNE-SP and D-ViNE-LB.

We can also add that the methods R-ViNE, D-ViNE, D-ViNE-SP, and D-ViNE-LB take twice the time on average toembed a VN compared to G-MCF. This is related with thenumber of MIP problems solved per VN embedding. The latteronly considers one MIP problem per VN embedding, whilethe former ones consider two MIP problems.

The method that performs the embedding in the shortesttime has the poorest performance (the G-SP), which solveseach VN request embedding problem in an average of 20milliseconds.

The method that requires the longest time to perform theembedding for the case of 3 VN requests has the highestperformance (the VNE-NLF), using the WSDP cost function,which takes less than 2 seconds on average for that case.However, if we increase the load, the situation significantlychanges, and the methods R-ViNE, D-ViNE and D-ViNE-LBtake more time to obtain the embedding solution. On average,they take 1 second to embed a VN, while the VNE-NLF-WSDP consumes less than 200ms.

C. Impact of the Maximum Distance Between Virtual Nodes

To evaluate the impact on the overall performance of thedifferent embedding methods due to the restriction on the

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Fig. 10. Average VN acceptance ratio as a function of the distance betweenvirtual nodes.

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Fig. 11. Average node utilization as a function of the distance betweenvirtual nodes.

maximum allowed distance between virtual nodes representedin expression (13), a new set of simulation experimentswas performed. The VN request arrival rate was fixed to4 VN requests per 100 time units; the maximum distancebetween virtual nodes was set to vary between 5 and 20within intervals of 2.5 distance units; for each consideredvalue of maximum distance, the same set of VN requestswas used. The remaining parameters i.e., virtual network size,link probability, and number of nodes were maintained. Toincrease the readability of all figures, only the best method ofeach group is presented in this section: G-SP (heuristic), G-MCF (mixed integer programming - link-path), D-ViNE-LB(mixed integer programming with better node-link embeddingcoordination - link-path ), and VNF-NLF-WSDP (integerlinear programming - node-link).

1) VN Request Acceptance Ratio: Fig. 10 depicts the VNrequest acceptance ratio as a function of the distance betweenvirtual nodes. Two different behaviors can be observed:

i The acceptance ratio increases with the distance between


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Fig. 12. Average link utilization as a function of the distance between virtualnodes.

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Fig. 13. Average embedding factor as a function of the distance betweenvirtual nodes.

virtual nodes: this is the case of the VNE-NLF group. Thisbehavior is expected if we consider the cases where VNrequests were initially not mapped due to the distance con-straint; increasing the permitted distance between virtualnodes will in principle result in more accepted VNs.

ii Increasing the distance between virtual nodes decreasesthe acceptance ratio: this is the case of the assignmentmethods G-SP, G-MCF, D-ViNE-LB.

2) Node Utilization: Fig. 11 depicts the average physicalnode utilization as a function of the distance between virtualnodes. The behavior observed for each method can be com-pared to that of the VN acceptance ratio.

3) Link Utilization: The average physical link utilizationis shown in Fig. 12, and different behaviors are observedaccording to the embedding method:

i G-SP and G-MCF methods reduce slightly the link uti-lization with the distance, and the method D-ViNE-LBmaintains the link utilization, despite some fluctuationsmay be observed.

ii The group of VNE-NLF demonstrates to slightly increase

the link utilization with the maximum distance betweennodes. This is expected, considering the increase on theVN request acceptance ratio with the distance.

4) Embedding Factor: The embedding factor is depicted inFig. 13, where three distinct behaviors can be observed:

i The embedding factor slightly decreases with the maxi-mum distance: this is the case of methods G-SP and G-MCF, where these demonstrate to loose efficiency with theconsidered distance.

ii The embedding factor does not vary with the maximumdistance: this is the case of method D-ViNE-LB.

iii The embedding factor increases with the maximum dis-tance: this is the case of the VNE-NLF group. Thisdemonstrates that the VNE-NLF group is able to be moreefficient with the relaxation on the distance constraintexpression (13).

VII. CONCLUSION

This paper proposed the VNE-NLF to solve the VN em-bedding problem. The model applies optimization theory tosimultaneously embed the virtual nodes and the virtual links.

Three new cost functions are proposed: the LB+εSP whichaims to minimize the overall load on the network per VNembedding; the SDP which aims to minimize the numberof physical links consumed, and at the same time it choosesphysical nodes with higher availability of resources; and theWSDP which includes the demanded capacity by the VN inthe objective function.

Simulation experiments show how far the state of the artheuristics are from an ILP based optimization method. Thedifference between the performance of the heuristics and theVNE-NLF approach is, at least, 30% for the VN requestacceptance ratio (see Fig. 3). The node utilization is alsohigher when comparing with the existing heuristics, which isexpected since we are accommodating more virtual nodes onthe network. However, the link utilization is similar to the onesof the heuristics, and in some cases (e.g. G-MCF, D-ViNE-LB)it is lower, which reflects the good efficiency of the embeddingwhen using the VNE-NLF approach. The embedding factor ofthe VNE-NLF is very high (i.e. it is close to 1). The resultsalso show that the maximum allowed distance between virtualnodes seems to affect differently the performance of eachembedding method: for the case of R-ViNE and D-ViNE withits three variants, it does affect negatively the performanceof the VN embedding; for the case of G-SP and G-MCF, itdoes not seem to cause a direct impact on the embedding; andregarding the VNE-NLF approach, it does affect positively theVN embedding.

The VNE-NLF with all its different cost functions provesto be an efficient embedding method. Not only it providesbetter results, but it also performs the VN embedding fasterthan the compared heuristics (with exception of G-SP) fora large number of VN requests per time unit. From all theproposed and simulated cost functions, the WSDP is the onethat demonstrates the best overall performance.

Future work will endorse VN embedding with multi-objective optimization support, and re-configuration of virtualnetworks. The definition of cost functions which take into


account energy parameters will be also addressed, through thestudy of the impact on energy saving, i.e. the shutdown of aninterface or even an equipment, by applying optimization tosolve the VN embedding problem.

ACKNOWLEDGMENT

The author was partially supported by the Portuguese Foun-dation for Science and Technology (FCT) with a scholarshipNo. SFRH/BDE/33751/2009. The research leading to theseresults has also received funding from the European UnionSeventh Framework Programme (FP7) under grant agreementnumber 257448. We would like to thank the anonymousreviewers for their insightful and constructive comments.

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Ms.C. Marcio Melo is a Ph.D. student at theUniversity of Aveiro. He received his Ms.C. degreein Electronics and Telecommunications Engineeringfrom the University of Aveiro in 2008. Since 2009 hehas been with PT Inovacao in the division of Tech-nological Coordination and Exploratory Innovation.He has been involved in FP7 European projects,such as 4WARD – Future Internet and SAIL (Scal-able and Adaptive Internet soLutions), and alsoEurescom research studies. His research interestsare in the area of Next Generation Networks, more

specifically Network Virtualization and Software Defined-Networking.

Dr. Susana Sargento received her Ph.D. in 2003 inElectrical Engineering. She joined the Departmentof Computer Science of the University of Portoin September 2002, and is in the Universidade deAveiro and the Instituto de Telecomunicacoes sinceFebruary 2004. She was also a Guest Faculty ofthe Department of Electrical and Computer Engi-neering from Carnegie Mellon University, USA, in2008/2009, in the scope of the Carnegie MellonPortugal Program.

She has been involved in several national andEuropean projects, taking leaderships of several activities in the projects, suchas the QoS and ad-hoc networks integration activity in the FP6 IST-DaidalosProject. She has been recently involved in several FP7 projects (4WARD,Euro-NF, C-Cast, WIP, Daidalos, C-Mobile), national projects, and CarnegieMellon Portugal research projects (DRIVE-IN with the Carnegie MelonUniversity). She has been TPC-Chair and organizing several conferences, suchas MONAMI’11, NGI’09, IEEE ISCC’07, and will be organizing NTMS’12and IEEE FEDNET (with IEEE NOMS’12). Her main research interests arein the areas of Next Generation and Future Networks, more specifically QoS,mobility, self- and cognitive networks.

Dr. Ulrich Killat studied physics in Hamburg andHeidelberg, Germany. He was then with the researchlaboratories of Philips in Hamburg and Aachen.Since 1991, he has been a professor and head ofDepartment of Communication Networks, HamburgUniversity of Technology (TUHH). His researchinterests include traffic theory, network planning,and modelling and simulation of communicationnetworks. He is a professor emeritus since 2009.


Dr. Andreas Timm-Giel was group leader from1994–1999 at the University of Bremen in the areaof mobile and satellite communications and involvedin several EU funded projects. After receiving hisPhD he moved to MediaMobil GmbH and M2SATLtd. as Technical Project Leader and Manager Net-work Operations. In December 2002 he joined theCommunication Networks group at the Universityof Bremen as senior researcher and lecturer. He wasleading several industrial, national and EC fundedresearch projects and from 2006 he was additionally

directing the interdisciplinary activity “Adaptive Communications” of TZI(Center of Computing and Communication Technologies). In November2009 he was appointed full professor at Hamburg University of Technology(TUHH) and is heading the Institute of Communication Networks. Since2012 he is coordinator of the research cluster SOMSED “Self-organizingmobile sensor and data networks” at TUHH and since 2013 deputy headof the TUHH’s school of electrical engineering, computer science andmathematics. His research interests are mobile and wireless communications,sensor networks and the Future Internet.

Ms.C. Jorge Carapinha is a Senior Research Engi-neer at PT Inovacao. He graduated in Electrical andComputer Engineering at the University of Coimbrain 1984 and received an MSc in Electronics andTelecommunications from the University of Aveiroin 1998. Since 1985 he has been with PT Inovacao(formerly CET). He has worked in several fieldsrelated to operator backbone networks and tech-nologies, including MPLS Virtual Private Networksand Quality of Service. He has a long record ofparticipation in international collaborative research

projects. Presently his activity is mainly focused on Network Virtualizationand Cloud Networking. He has authored/co-authored several papers onthese topics, published in technical journals or presented in internationalconferences, as well as Book Chapters.

Documents

Optimal Virtual Network Embedding: Node-Link Formulation