Bandwidth allocation is one of the main problems in network virtualization. Mechanisms to allocate bandwidth may avoid bottlenecked virtual links. This paper proposes a model based on optimization theory, to distribute the bandwidth among
virtual links looking for the minimization of the spare bandwidth in the substrate network.Postprint (published version

Botero Vega, Juan Felipe

Hesselbach Serra, Xavier

English

Bandwidth allocation is one of the main problems in network virtualization. Mechanisms to allocate bandwidth may avoid bottlenecked virtual links. This paper proposes a model based on optimization theory, to distribute the bandwidth among

virtual links looking for the minimization of the spare bandwidth in the substrate network

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Optimization model for bandwidth allocation in a network virtualization environment

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Noname manuscript No.(will be inserted by the editor)Optimization Model for Bandwidth Allocation in aNetwork Virtualization EnvironmentJuan Felipe Botero · Xavier HesselbachReceived: date / Accepted: dateAbstract Bandwidth allocation is one of the main problems in network virtualization.Mechanisms to allocate bandwidth may avoid bottlenecked virtual links. This paperproposes a model based on optimization theory, to distribute the bandwidth amongvirtual links looking for the minimization of the spare bandwidth in the substratenetwork.Keywords Network Virtualization · Bandwidth Allocation · Optimization Theory1 IntroductionNetwork virtualization architectures [1,2], that allows multiple virtual networks (vn)to run on top of a shared physical substrate1, has been proposed as an alternative toface up Internet ossification [3,4].Some virtual links could become bottlenecked due to an aggressive bandwidth con-sume (e.g. UDP applications) in other virtual links belonging to the same physical one.This paper proposes an optimization model to distribute the bandwidth among virtuallinks accomplishing flow demands and trying, at the same time, to minimize the totalspare bandwidth of the physical network.The paper is organized as follows: Next section describes network virtualization’smain challenges. Section 3 shows the notation used to create the bandwidth allocationmodel. Section 4 presents the proposed model to allocate bandwidth to virtual links.Finally, conclusions and future work are presented in section 5.Juan Felipe BoteroJordi Girona Street, 1 and 3, Barcelona, SpainTel.: +34-663-501824E-mail: jfbotero@entel.upc.eduXavier HesselbachJordi Girona Street, 1 and 3, Barcelona, Spain1 This paper will use indifferently the terms substrate network and physical network.22 Research Challenges in Network VirtualizationIn a network virtualization environment, multiple vns run on top of a substrate networkby means of the interconnection of virtual routers (virtual instances of substrate’srouters) using virtual links (virtual instances of substrate’s links).The main challenges to deploy a robust network virtualization are, among others,the admission control performed by the substrate, which will have to decide if a newvn is able to be created without harming the QoS of the active vns, standardize thefunctionality that virtual routers must obey and bandwidth allocation in vns that is themain focus of this paper.Virtual links, instances of substrate’s links, share the bandwidth of a commonphysical link. Static allocation of bandwidth among virtual links is the easier way tocreate an illusion of an isolated and dedicated network for each virtual one. However,static allocation entails the waste of bandwidth that is unused, by a virtual link, incertain time instant and could be used by other(s) virtual link(s). On the other hand,if a best-effort distribution is allowed, one virtual link can be bottlenecked and somevns might be left inoperative.3 Modeling of Bandwidth AllocationA network (substrate in our model) is a directed graph G(V,E), where V is a setof elements called vertices and E is a set of elements called arcs or edges, withone or more numbers associated with each arc. Considering a ordered set of verticesV1, V2, ..., Vn, Vn+1; a directed path is any sequence of arcs ∈ E of the following type:{(V1, V2), (V2, V3), ..., (Vn, Vn+1))}.Given a network G(V,E), where edge (i, j) ∈ E, C(K) is a set of commodities,where Cl(k) is the commodity l of the vn k. A commodity [5] is defined by Cl(k) =(sl(k), tl(k), hl(k)), where sl(k) and tl(k) are the source and sink of commodity l, andhl(k) is the demand between the source and the sink in the vn k. The notation of themodel variables is defined in Table 1.Table 1 Definition of VariablesTerms DefinitionV Set of physical nodes (routers) belonging to the substrate networkE Set of links belonging to the substrate networkV N Set of vns, virtualized from the substrate networkV Nk V Nk ∈ V N represents the vn number kCl(k) Commodity number l of the vn kPl(k) Allowed subset of directed paths for the commodity l in the vn kP pl (k) Ppl (k) ∈ Pl(k) is the directed path p for the commodity l in the vn kF pl (k) Bandwidth allocated to the flow that uses path p of commodity l in the vn khl(k) Minimum bandwidth required to the commodity l of the vn k.ρpl (i, j, k) Binary variable it is 1 if the link (i, j), in the vn k, is part of the path p for thecommodity l. It is 0 elsewhere.34 Bandwidth Allocation Optimization ModelMinimize:F =|E|X(i,j)∈E0@B(i, j)− |VN|Xk=1|C(k)|Xl=1|Pl(k)|Xp=1ρpl (i, j, k)Fpl (k)1A (1)Subject to:B(i, j)−|VN|Xk=1|C(k)|Xl=1|Pl(k)|Xp=1ρpl (i, j, k)Fpl (k) ≥ 0 for(i, j) ∈ E (2)|Pl(k)|Xp=1Fpl (k) ≥ hl(k) for 1 ≤ k ≤ |V N | and 1 ≤ l ≤ |C(k)| (3)∃!Fpl (k) 6= 0 ∀p|Ppl (k) ∈ Pl(k) (4)|V N | ≥ 0 |E| ≥ 0|Fpl (k)| ≥ 0 for 1 ≤ k ≤ |V N |, 1 ≤ l ≤ |C(k)| and 1 ≤ p ≤ |Pl(k)||C(k)| ≥ 0 for 1 ≤ k ≤ |V N ||pl(k)| ≥ 0 for 1 ≤ k ≤ |V N | 1 ≤ l ≤ |C(k)|ρpl (i, j, k) ≥ 0 for (i, j) ∈ E, 1 ≤ k ≤ |V N |, 1 ≤ l ≤ |C(k)| and 1 ≤ p ≤ |Pl(k)|(5)The objective is to find the F pl (k) minimizing the objective function (OF) F (Equa-tion (1)), representing the spare bandwidth in the substrate network, subjected to a setof constraints. First group of constraints (2) are of capacity, they assure the sum of thebandwidths assigned to each virtual link does not exceed physical link capacity. Secondgroup of constraints are related with demand (3); they assure that each commodityobeys its assigned demand. Third set of constraints (4) assures the uniqueness of thechosen path for a commodity. Last set of constraints (5) force variables to be positive.5 Modeling of a Simple Network TopologyFigure 1 is formed by tree vns on top of a substrate. There are three commoditiescrossing the network with node 1 as source and 11 as the sink, this network environmentcarries out each commodity in a different VN. Example variables are presented in Table2.Fig. 1 Example of three VNs running on top of a substrate with three commodities, eachcarried out by each VN4Table 2 Definition of Example VariablesVariable ValueV {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}E {(1, 2), (2, 3), (3, 4), (4, 11), (1, 5), (5, 6), (6, 7), (7, 11), (1, 8), (8, 9), (9, 10),(10, 11)}V N {V N1, V N2, V N3}B(i, j) 1Mbps ∀(i, j) ∈ EC(k), k ∈ V N {C1(k)}, C1(k) = (s1(k), t1(k), h1(k)) = (1, 11, 1) k ∈ {1, 2, 3}P1(1) {P 11 (1)}, P 11 (1) = {(1, 2), (2, 3), (3, 4), (4, 11)}P1(2) {P 11 (2)}, P 11 (2) = {(1, 5), (5, 6), (6, 7), (7, 11)}P1(3) {P 11 (3)}, P 11 (3) = {(1, 8), (8, 9), (9, 10), (10, 11)}F 1k (1), k ∈ V N ?, k ∈ V N → k ∈ {1, 2, 3}h11(k), k ∈ V N Commodity demands, 1 Mbps, k ∈ V N → k ∈ {1, 2, 3}ρpl (i, j, k) ρ11(i, j, k) = 0 ∀(i, j, k)− {ρ11(1, 2, 1), ρ11(2, 3, 1), ρ11(3, 4, 1), ρ11(4, 11, 1),ρ11(1, 5, 2), ρ11(5, 6, 2), ρ11(6, 7, 2), ρ11(7, 11, 2), ρ11(1, 8, 3), ρ11(8, 9, 3),ρ11(9, 10, 3), ρ11(10, 11, 3)}The model for this example is the following one: Minimize: F = 12 − 4F 11 (1) −4F 11 (2)− 4F 11 (3), subject to: F 11 (1) ≥ 1, F 11 (2) ≥ 1 and F 11 (3) ≥ 1The OF (Equation (1)) is reduced to an easier expression. Capacity and demandrestrictions are the same, that’s why only one set of restrictions are shown. It is simpleto find the optimal values; F 11 (k) is 1 for k = 1, 2, 3.6 Conclusions and Future WorkThis paper has proposed an optimization model to overcome the consequences of band-width sharing in virtual networks. The model tries to minimize the spare bandwidthin the substrate while allocates bandwidth to each commodity by using a unique-pathflow accomplishing certain constraints, among them, the assurance of a minimal band-width allocation to each commodity. This model avoids the bottlenecking because eachvirtual link will receive scarcely the bandwidth it has demanded. Future work will beconcentrated in the search of efficient algorithms that allocate that bandwidth.Acknowledgements This work has been partially supported by national Spanish projectCICYT TSI2007-66637-C02 and European Network of Excellence Euro-NFReferences[1] N. M. M. K. Chowdhury.: Identity Management and Resource Allocation in the Net-work Virtualization Environment. University of Waterloo Thesis (2008)[2] L. Peterson, S. Shenker, J. Turner.: Overcoming the Internet impasse through virtu-alization. In III Workshop on Hot Topics in Networks (HotNets-III) (2004)[3] J.Turner, T.Anderson, L.Peterson, S.Shenker.: Virtualizing the Net: A strategy fornetwork de-ossification. http://www.arl.wustl.edu/∼jst/talks/hotI-9-04.pdf[4] N.Feamster, L. Gao, J. Rexford.: How to lease the Internet in your spare time. ACMSIGCOMM Computer Communication Review, pp. 61-64. (2007)[5] M. Pioro and D. Medhi.: Routing, Flow, and Capacity Design in Communication andComputer Networks. Morgan Kaufmann Publishers. pp. 41. (2004)

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Optimization model for bandwidth allocation in a network virtualization environment

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