Cost efficient virtual network mapping across multiple domains with joint intra-domain and inter- domain mapping

Optical Switching and Networking 14 (2014) 233–240

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Optical Switching and Networking journal homepage: www.elsevier.com/locate/osn

Cost efficient virtual network mapping across multiple domains with joint intra-domain and inter- domain mapping$ Hongfang Yu a, Tao Wen a, Hao Di a, Vishal Anand b, Lemin Li a a Key Lab of Optical Fiber Sensing and Communications (Ministry of Education), University of Electronic Science and Technology of China, Chengdu, China b Department of Computer Science, The College at Brockport, State University of New York, USA

a r t i c l e i n f o

abstract

Article history: Received 15 February 2014 Received in revised form 15 April 2014 Accepted 13 May 2014 Available online 8 July 2014

Network virtualization allows the coexistence of multiple virtual networks on a shared substrate optical network, which interconnects the geo-distributed data centers. A virtual network (VN) typically spans across multiple domains, which may be managed by different infrastructure providers (InPs). The topology and resource information on each domain is confidential and kept private by the InP. However, a domain-level (global) view is required to achieve cost efficient VN mapping across multiple domains. In this paper, we present a framework for the cost efficient VN mapping across multiple domains with joint intradomain and inter-domain mapping. In the framework, the VN mapping is accomplished by the mapping manager that is the broker between the SPs and the InPs. The mapping manager collects the mapping candidates for the VN request from the InPs, and then establishes the abstracted domain-level graph. Finally the candidates are selected on the domain-level graph with cost and quality of services (QoS) consideration. We formulate the problem of candidate selection as an mixed integer linear programming (MILP), and then relax the integer constraints to obtain a linear program to solve it. The simulation results show that our proposed framework can achieve low resource allocation cost and good QoS performance for the VN request. & 2014 Elsevier B.V. All rights reserved.

Keywords: Domain-level graph Multiple domains Network virtualization Virtual network mapping

1. Introduction To improve the quality of experience (QoE) and quality of service (QoS) using cloud computing, cloud service ☆ This research was partially supported by the National Grand Fundamental Research 973 Program of China under Grant (No. 2013CB329103), Natural Science Foundation of China grant (Nos. 61271171 and 61001084), Program for Changjiang Scholars and Innovative Research Team (PCSIRT) in University and the 111 Project B14039. The research of Dr. Vishal Anand is supported in part by the Provost Fellowship and Scholarly Incentive Grant at the College at Brockport, SUNY. E-mail addresses: [email protected] (H. Yu), [email protected] (T. Wen), [email protected] (H. Di), [email protected] (V. Anand), [email protected] (L. Li).

http://dx.doi.org/10.1016/j.osn.2014.05.020 1573-4277/& 2014 Elsevier B.V. All rights reserved.

providers are building geo-distributed networks of data centers [1,2]. This multi-datacenter system cannot only reduce latency but also increase survivability in the presence of outages affecting an entire site or its connections to the outside world. Thus, the substrate network, which interconnects multiple data centers (each with a large number of computing physical server) with networks, is deployed to support a variety of distributed applications and services. Optical network [3] is a natural choice for the substrate network because of its high speed, enormous bandwidth and protocol transparency. With the increasingly popular virtualization of both computing and networking resources in a geo-distributed optical networks of data centers (called a physical substrate), multiple

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virtual networks (VNs) can share the physical resources of the underlying substrate [4–6]. Using virtualization multiple heterogeneous network architectures can cohabit on a shared substrate network. Virtualization technology is also a business enabler that allows agile deployment of various applications and services with increased flexibility while optimizing the use of infrastructure resources as well as reducing the cost of maintaining them. Each request in a virtualized network environment is modeled as a virtual network (VN), consisting of virtual nodes interconnected by virtual links. It is essential to address the VN mapping problem for enabling network virtualization. The VN mapping problem is the allocation of substrate network resources to satisfy the constraints of the virtual nodes and links of the VN, e.g., satisfying the required computing resources on the virtual nodes and the bandwidth resources of the virtual links. The underlying substrate optical network typically consists of multiple administrative domains provided by Infrastructure Providers (InPs). Each InP deploys and manages the physical network resources in its domain and offers their resources to the Service Providers (SPs). Each SP leases the resources from multiple InPs to create and deploy the VN for end users [4]. An InP can map the VN request in its administrative domain using the intra-domain algorithms proposed in [7–16] as the InP has the complete knowledge of its domain. However, when the VN request is mapped across multiple domains (e.g., to satisfy location, resource efficiency constraints) there is no global view of the underlying substrate as InPs in different domains do not expose or share their topology and resource information with each other. In this work we propose a scheme that uses a broker between the InPs and the SPs to enable the InPs to cooperate and accomplish the VN mapping across multiple domains. So far, there have been only very limited works on virtual network mapping across multiple domains [17–19]. The work in [17] proposes a side-by-side framework to address VN mapping across multiple domains. The VN request firstly is partitioned into several VN sub-graphs, and each sub-graph is mapped by an InP in its domain. Meanwhile, the works in [18,19] propose a sequent VN mapping framework. An InP partially maps the VN request and forwards the residual part to other InPs in a recursive manner. These works can map the VN request across multiple domains without knowing the topology and resource information of the individual domains. However, separation of inter-domain mapping and inter-domain mapping restricts the solution space, and may result in the poor performance and low resource utilization. In this paper we present a framework of cost efficient VN mapping across multiple domains, called MD-VNM. The framework determines the intra-domain and inter-domain mapping in the same phase, and also considers the QoS performance. In our framework, the mapping manager receives the VN request from the SP, and cooperates with the InPs to find the VN mapping. In each domain the InP finds the candidates for mapping the virtual nodes and links while considering the intra-domain QoS performance. The InPs then offer the intra-domain candidates to the mapping manager, bidding for the virtual nodes and links in the VN request, and cooperate with the mapping manager to find inter-domain

link candidates that interconnect intra-domain node candidates. The mapping manager collects these candidates and pricing provided by InPs to establish a domain-level graph and selects the candidates from the graph with (intra-domain and inter-domain) cost and QoS consideration, i.e., intra-domain and inter-domain VN mapping is determined in the same phase. The rest of the paper is organized as follows. The related work is introduced in Section II. Section III formulates the multi-domain cost efficient VN mapping problem. Section IV describes our framework for cost efficient VN mapping across multiple domains. Section V presents the MILP and the integer-relaxed algorithm for candidate selection. Section VI presents the simulation results, and Section VII concludes the paper. 2. Related work Some heuristics for VN mapping in single domain under different objectives and constraints have been proposed recently [7–16]. Based on shortest path, k-shortest paths and multi-commodity flow algorithms, the work in [7] presented two-stage simple approaches to relax the tight coupling between node mapping in the first stage and link mapping in the second. However, separating node and link mapping stages may result in poor performance. The work in [8–12] developed integer linear programming (ILP) optimization, column generation, heuristics and meta-heuristics by coordinating the node and link mapping phases, and introduced better correlation between the two phases. Quality of Service (QoS) constraint during VI mapping is considered in [13].The works in [14–16] consider the special constraints of optical networks, such as wavelength continuity, transmission impairments and sub-carrier slot continuity. More recent works [17–19] have considered virtual network mapping across multiple domains, where papers [17,18] focus on the general networks, and paper [19] focuses on the optical network. In [17], the VN request is partitioned into several VN sub-graphs, and each subgraph is mapped by an InP in its domain. Finally, the inter-domain virtual links interconnecting the VN subgraphs (in different domains) are mapped onto interdomain paths. In [18], the VN request is sent to all InPs. InPs do the mapping in sequence. The first InP partially maps the received VN request and forwards the residual part to other InPs in a recursive manner. After the mapping of virtual nodes is determined in different domains, the inter-domain virtual links are mapped. These current works do well in distributed environment. However, separation of intra-domain and inter-domain mapping is hard to achieve the global-view optimization and meet the end-to-end QoS requirements. The authors in [19] investigated dynamic virtual network mapping over multipledomain SDN networks and proposed an efficient VN request scheduling mechanism. 3. Problem statement In this section we formulate the problem of cost efficient VN mapping across multiple domains WDM network with the QoS consideration.

H. Yu et al. / Optical Switching and Networking 14 (2014) 233–240

235

domain 1

domain 1 a

domain 3 domain 2

b1

domain 3 b

domain 2

c

domain 1

b

a

b a

c

domain 1

a1 domain 3

domain 2

a2

c3

b3

domain 3

c

domain 2 a

b

Fig. 1. Example of VN mapping across multiple domains with the proposed framework.

3.1. Substrate network The underlying substrate optical network is modeled as a weighted undirected graph GS ¼(VS, ES), where VS is the set of substrate nodes and ES is the set of substrate links. We assume that all substrate nodes are equipped with sufficient wavelength converters. The substrate consists of N domains interconnected by inter-domain links, as shown in Fig. 1(a). The domain managed by the k-th (1r krN) InP is also modeled as a weighted undirected graph GS,k ¼ (VS,k, ES,k), where VS,k is the set of substrate nodes in the domain k and ES,k is the set of intra-domain substrate links in the domain k. ES,I is the set of interdomain substrate links. Then GS is GS,1 [ GS,2 [ …[ GS,N [ ES,I. The available computing capacity (e.g., CPU) on the node nS AVS is represented by CS(nS). Each (intra-domain or inter-domain) substrate link is denoted by lS ¼(i, j) AES where i and j are the end nodes of lS. Note that, the end nodes of an inter-domain link are the border nodes in different domains. The available bandwidth on the substrate link l is represented by BS(lS). The unit capacity price (or cost to the SP) of substrate nodes and links is denoted by uN(nS) and uL(lS) respectively. We used (lS) to represent the delay on the substrate link lS A ES. 3.2. VI Request Similarly, we model a VN request as an undirected graph GV ¼(VV, EV), where VV and EV are the sets of virtual nodes and links (as shown in Fig. 1(b)). The required computing capacity (CPU) of the virtual node nV A VV is represented by CV(nV). Each virtual link is denoted by lV ¼(i, j) AEV, where i and j are the end nodes of lV. The required bandwidth on the virtual link l is represented by BV(lV). 3.3. VN mapping across multiple domains The VN request is mapped across multiple domains with the virtual nodes mapped onto different substrate nodes, and the virtual links mapped onto the substrate paths so as to satisfy the resource requirements of the virtual nodes and links. In the domain-level mapping, the fact that the virtual node nV ANV is mapped in the domain GS,k is represented

by D(nV)¼k. nV -GS;k : DðnV Þ ¼ k; nV A V V ; GS;k DGS The virtual links may be mapped within a domain or across domains. The substrate path on which the virtual link lV AEV is mapped is denoted by p(lV). We use D(lV) ¼k to represent the fact that lV is mapped in the domain GS,k, i.e., the links of the path are in GS,k. lV -GS;k : DðlV Þ ¼ k; lV A V V ; GS;k DGS For example, as shown in Fig. 1(e), the virtual node a is mapped in domain2, and the virtual nodes b and c and the virtual link (b, c) are mapped in domain3. When D(i) and D(j) are not the same (i.e., D(i)aD(j)), the virtual link (i, j) A EV is an inter-domain virtual link that passes through the inter-domain substrate links. Based on the domain-level VN mapping the VN request is partitioned into H sub-graphs which are interconnected by inter-domain virtual links. Each sub-graph contains the virtual nodes and links that are mapped in the same domain. The i-th (1 rirH) sub-graph of GV is represented by GV,i ¼(VV,i, EV,i), where VV,I(DVV) and EV,i(DEV) are the sets of the virtual nodes and links in the sub-graph. EV,C is the set of all the inter-domain virtual links, and GV is GV,1 [ GV,2 [ …[ GV,H [ EV,C. For the intra-domain mapping when the VN sub-graph GV,i(1 rirH) is mapped in GS,k(1 rkrN), M(nV) ¼nS represents the fact that the virtual node nV in GV,i is mapped onto the substrate node nS in GS,k. And M(lV)¼ p (lV) represents the fact that the intra-domain virtual link lV in GV,i is mapped onto the substrate path in GS,k. The mapping of virtual nodes is subject to the computing capacity constraints and location constraints, and the mapping of virtual links is subject to the bandwidth capacity constraints. The problem of intra-domain VN mapping is formulated in [5–11], so we do not present the formulation in this work specifically. In this work we consider both the cost of the allocated resources (of the VN request) to the SPs and the time delay on virtual links. The node mapping cost of nV, which is the product of the cost of unit capacity and the used capacity is defined as in (1). f N ðnV Þ ¼ uN ðMðnV ÞÞC V ðnV Þ

ð1Þ

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The mapping of the inter-domain virtual link lV is similar to that of an intra-domain virtual link, subject to the available bandwidth on the substrate. Similarly, M (lV) ¼p(lV) represents the fact that the inter-domain virtual link lV is mapped onto the inter-domain path p(lV). Then the cost fL(lV) of mapping the (intra-domain or interdomain) virtual link lV is defined as in (2), which is the sum of the bandwidth cost on the links of the path p(lV), where the bandwidth cost on a link is the product of the unit cost of bandwidth and the allocated bandwidth. f L ðlV Þ ¼ f d ðlV Þ ¼

∑ uL ðlS ÞC V ðlV Þ

ð2Þ

∑ dðlS Þ

ð3Þ

VN request step 1

Mapping manager step 5

VN request

Domain-level graph

step 2

InP1

Candidate selection … step 6

InP2

step 4

InPN …

Find candidates

Find candidates

Find candidates

Domain 2

Domain N

step 3

lS A pðlV Þ

Domain 1

lS A pðlV Þ

The QoS performance fd(lV) is defined in (3), where fd(lV) is the maximum of the time delays on all the substrate links used by virtual link lV, i.e., the delay of the mapped path. We use the utility function fd to define the delay influence of the virtual link lV, as in (4), by choosing the maximum delay of all virtual links. f d ¼ max f d ðlV Þ

ð4Þ

lV A E V

Then the VN mapping cost fc, which includes cost of mapping all the virtual nodes and links, can be defined in (5). Considering both the cost of VN mapping and the delay influence of virtual links, we define the weighted cost function f in (6). fc ¼

SP

∑ f ðnV Þ þ ∑ f L ðlV Þ

nV A V V

ð5Þ

lV A E V

f ¼ f c þαf d

Fig. 2. The framework of MD-VNM.

Our mapping framework is shown in Fig. 2, where the mapping manager acts as the broker between the SP and the InPs, aiming to minimize the VN mapping cost. After receiving the VN request from the SP (step 1 in Fig. 2), the mapping manager sends the request to all InPs (step 2). Each InP then computes the intra-domain mapping to find the candidates for the virtual nodes and links (step 3). The mapping manager collects the candidates from InPs and forms an abstracted domain-level view (step 4). Based on the domain-level graph the candidates are selected (step 5) to accomplish the VN mapping. Finally the mapping manager sends the selection to InPs for instantiation (step 6). 4.2. Finding candidates

ð6Þ

4. The framework In this section, we present the framework for cost efficient VN mapping across multiple domains with the QoS consideration, called MD-VNM. 4.1. Framework design For cost efficient VN mapping across multiple domains, the cost of mapping should be estimated to guide the VN mapping both in the intra-domain level and the domain level. For the intra-domain level, the cost of mapping can be evaluated by the InP with the complete knowledge of its administrative domain. A domain-level view is needed to guide the domain-level mapping based on the interdomain cost. Since the domain information is unavailable, our framework uses a bidding mechanism to get a domain-level view, i.e., InPs bid for the virtual nodes and links of the VN request. The bid and the QoS performance for a virtual node and/or link can be regarded as a mapping candidate. The set of all the candidates for both intradomain and inter-domain mapping then form the abstracted domain-level view. Both the intra-domain and inter-domain VN mapping is determined by selecting the candidates with the aim of minimizing the VN mapping cost for SPs.

When each InP receives the VN request, it needs to find the feasible mapping for the virtual nodes and links and send these mapping candidates to the mapping manager to bid for the virtual components. Actually, finding candidates in a domain is similar to the VN mapping in a single domain, which is studied in [7–13]. Thus, the existing works in VN mapping of single domain can be used by the InPs (with modification for partial mapping) to find candidates according to its policy, e.g., minimizing resource cost or load balancing. In this paper, we adopt the Improved-vnmFlib algorithm proposed in our previous work [10] to find candidates. Considering it is not necessary to map the whole VN to find candidates, the backtrack mechanism is removed and the partial mapping is allowed when there is no available mapping for a virtual node or link in the Improved-vnmFlib algorithm. The finding candidates algorithm based on the ImprovedvnmFlib is shown in Fig. 3. In each step of the algorithm, an unmapped virtual node nV adjacent to the iterated virtual nodes (recorded in the set T, line 13) is tried to map, traversing all substrate nodes in the InP's own domain to find out the one with the lowest cost of mapping nV and its relevant links that connect to the virtual nodes in T (line 10).The function try() here is used to test the mapping result of substrate node. If there is an available substrate node, nV is recorded in the set M (line 11), and the corresponding candidate information is added into the set S (line 12). When all the virtual nodes have been tried (i.e., iterated over), the

H. Yu et al. / Optical Switching and Networking 14 (2014) 233–240

1: T = Ø, M = Ø, S=Ø 2: while(|T| ≠ |VV|) 3: if T = Ø, then

4: select one from VV as nV

5: else

6: select one which has connection with nodes in T as nV ( nV T) 7: lV← (i, nV) EV 8: T←nV T 9: for all nS VS 10: if try(nV,lV), then 11: M←nV M 12:S←Info(nV,lV) S //the candidate information of nV and lV are added into S 13: T←nV T 14: end while 15: return S

237

1:VD = Ø,ED = Ø 2: record the number of InPs by N 3: for i=1 to N 4: for all nD from i-th InP 5: VD←nD VD 6: end for 7: for all lD from i-th InP 8: ED←lD ED 9: end for 10:end for 11: for all m,n VD (m≠ n) 12: if (CFN (m), CFN (n)) EV, then 13: lD =find(m,n) 14: ED←lD ED 15: end for 16: return GD = (VD,ED) Fig. 4. The algorithm for domain-level graph.

Fig. 3. The algorithm for finding candidates.

algorithm is over and the candidate information will be returned. For example, as shown in Fig. 1(c), the candidates for virtual nodes a and b and virtual link (a, b) are found in domain1, another candidate for a is in domain2, another candidate for b is in domain 3, and the candidate for link(b, c) is also in domain3.The InPs send these candidates and the corresponding mapping cost as defined in (1) and (4) to the mapping manager to build the domain-level graph. The information of candidates sent to the mapping manager is consisted of the corresponding virtual component, cost and delay. The information of a node candidate sent from the k-th(1rk rN) domain can be represented by CadN,k(nV, fN(nV)), in which the fN(nV) denotes the price of this mapping. Similarly, the information of a link candidate can be represented by CadL,k(lV, fL(lV),dV(lV)), and the dV(lV) is the delay of the mapping link.

Thus, the mapping is done without the knowledge of the topology and resource information of the domains. The domain-level graph is modeled by GD ¼ (VD,ED), where VD and ED are the sets of the candidate nodes and links. Each candidate node or link (nD AVD or lD A ED) for a virtual node or link (nV AVV or lV AEV) is represented by CFN(nD) ¼nV or CFL(lD)¼lV. For example, as shown in Fig. 1(d), CFN(a1)¼a and CFN(b3)¼b. The procedure of establishing the domain-level graph is shown in Fig. 4. There are two phrases: (1) the intradomain candidates provided by all InPs are added into the domain-level graph (lines 3–10) and (2) the candidates for inter-domain virtual links are found and added into the graph (lines 11–15). Note that, if in the VN request a pair of virtual nodes is connected by a virtual link, there must be a candidate link between the corresponding candidate nodes. 4.4. Candidate selection

4.3. The domain-level graph Similar with the work in [20], the mapping manager collects all the candidates to establish a domain-level graph to determine the intra-domain and inter-domain VN mapping in the same phase. The candidates found by each InP is a sub-graph of the VN topology (as in Fig. 1(c)), and the mapping cost of the candidates is known. All the sub-graphs from the InPs are then added into the domainlevel graph (as in Fig. 1(d)). If there is a virtual link lV between the virtual nodes whose candidates are in different domains, the node candidates are interconnected by a link candidate lD using dotted lines in Fig. 1(d). The InPs and the mapping manager cooperate to estimate the mapping cost fL(lV) for the candidate for the inter-domain virtual link lV. The cost of inter-domain paths between the border routers can be got by the border gateway protocol (BGP) with the link weights set based on (4), and the cost of the intra-domain path between the node candidates and the border routers can be got from the InPs. Thus the cost of inter-domain paths between the node candidates can be estimated. After all the candidates for the interdomain virtual links are added, the domain-level graph is established as in Fig. 1(d). Then the candidates are selected on the domain-level graph to accomplish the global optimum VN mapping along with cost considerations.

On the established domain-level graph, candidates are selected for the virtual nodes and links in the VN request (as shown in Fig. 1(e)) while minimizing the efficient function defined in (6). The VN mapping fails if there are no candidates for a virtual node or link in the VN request. After successful candidate selection the mapping manager informs the InPs of the respective candidates for instantiation, e.g., virtual link setup in [18]. The detailed algorithm for candidate selection will be discussed in the next section. 5. The algorithm for candidate selection In this section we formulate candidate selection as a mixed-integer linear programming (MILP) problem. Since the problem is NP-hard, we relax the integer constraints to obtain a linear program and also present a heuristic algorithm to address the problem. 5.1. MILP formulation The (given) is denoted by are fN(CFN(nD)) candidate link

cost of selecting the candidate nD or lD fN(nD) or fL(lD), where fN(nD) or fL(lD) or fN(CFL(lD)), respectively. The delay of is denoted by dD(lD).On the established

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domain-level graph, candidates are selected to support the VN request, aiming to minimize total candidate cost. Variables x(nD) (nD A VD): Binary variable denoting whether the node candidate nD is selected; 1 if selected and 0 otherwise. y(lD) (lD AED): Binary variable denoting whether the link candidate lD is selected; 1 if selected and 0 otherwise. dMax: Continuous variable denoting the maximum delay of all virtual links.

∑ xðnD Þf N ðnD Þ þ ∑ yðlD Þf L ðlD Þ þ αdMax

nD A V D

ð7Þ

lD A E D

Constraints ∑

nD :cadðnD Þ ¼ nV

∑

lD :cadðlD Þ ¼ lV

varying from 0-1

3: lp() //solve the relaxed model 4: for all nV VV 5: nD←select_max(x(nD)) 6: nV ←CFN(nD) 7: F= nD F 8: end for 9: for all i,j F 10: if lD = (i, j) ED,then 11: F= lD F 12: end for 13: return F Fig. 5. The algorithm for candidate selection.

Objective Minimize f¼

1: F= Ø 2: relax x(nD) and y(lD) as c ontinuous variables

xðnD Þ ¼ 1;

yðlD Þ ¼ 1;

8 nV A V V 8 lV A E V

ð8Þ ð9Þ

xðnD Þ Z yðlD Þ;

8 lD ¼ ðnD ; n0D Þ or ðn0D ; nD Þ A ED

ð10Þ

dMax Z dD ðlD Þ;

8 lD A E D

ð11Þ

xðnD Þ A f0; 1g;

8 nD A V D

ð12Þ

yðlD Þ A f0; 1g;

8 lD A E D

ð13Þ

dMax Z 0

between the selected nodes is also chosen for the corresponding virtual links. Obviously, corresponding candidate link will always exist, if there is a virtual link between the virtual nodes. Thus, the whole mapping solution can be finalized after finishing the selection of all virtual nodes. As shown above, Fig. 1(e) is the VN mapping solution of Fig. 1(d) by using this scheme.

6. Simulation results In this section, we present our simulation environment and the results of our simulation experiments and analysis.

6.1. Simulation environment ð14Þ

The objective function (7) minimizes the total cost of selected candidates, in which the parameter α is used to ensure the value of dMax and mapping cost is in the same order of magnitude. Constraints (8)–(10) are for the selected candidates composing the VN topology. Constraints (8) and (9) ensure that one candidate is selected for each virtual node and link. Constraint (10) ensures that the end (candidate) nodes are also selected when a candidate link is selected. Constraint (11) is used to record the maximum delay among the selected candidate links. Constraints (12)–(14) are the feasible region constraints for the variables. 5.2. The candidate selection algorithm Since the problem of candidate selection is NP-hard, we present the heuristic algorithm based on integer relaxation for candidate selection as shown in Fig. 5. F represents the set of selected candidate nodes and links, which is the result of VN mapping across multiple domains. Here we adopt integer relaxation method to deal with the model by relaxing x(nD) and y(lD) to continuous variables ranging from 0 to 1. Based on the solution of the relaxed model (a LP model in line 3), for every virtual node nV AVV, find out the candidate node with the maximum x(nD), select and push it into F. Finally, the candidate link

In our simulation we assume that the underlying substrate consists of 5 domains. There are 12 substrate nodes in each domain from which 4 nodes are randomly selected as edge nodes, and each pair of the nodes is interconnected by an intra-domain link with the probability of 0.5. The available computing capacity on each substrate node is 200 units, and the unit cost of computing capacity is 1. The available bandwidth on each intradomain substrate link is 400 units, and the unit cost of bandwidth and the delay on the intra-domain links is 1. Each pair of domains is interconnected by edge nodes with probability of 0.5. When a domain pair is interconnected, the number of links interconnecting the domains is one or two with probability of 0.5 and the unit cost of bandwidth and the delay on the inter-domain links is 5. For each virtual request, virtual node pairs are randomly connected by a virtual link with the probability of 0.5. The required computing capacity of each virtual node varies according to a uniform distribution from 10 to 50 units. There are location constraints on virtual nodes, where the virtual node should be mapped within the required geographical region. In the allowed location region for each virtual node we assume 8 random substrate nodes (may be in different domains) are available. For each virtual link lV the required bandwidth varies according to a uniform distribution from 10 to 50 units. The parameter α (from (6)) is set at 20|VV|, 50|VV| and 100|VV|, respectively (|VV| denotes the number of virtual nodes). Obviously, α is larger, the weight of maximum delay is more important.

H. Yu et al. / Optical Switching and Networking 14 (2014) 233–240

Fig. 6. Comparison of the cost of VN mapping by three methods under varying number of virtual nodes. Parameter α is set at 20|VV|, 50|VV| and 100|VV| in (a), (b) and (c), respectively.

6.2. Comparison method To verify the MD-VNM framework, we make comparisons with the framework in [18] (denoted by Fwd_VNM) and MDVNM_MILP which directly solves the MILP formulation without relaxation, respectively. In the Fwd_VNM, a SP send a VN request to all the InPs, then each InP does the VN mapping it can bear in its own domain and forward the rest to others. By recursive executing the above steps, the SP will get a total mapping bid from feedback information of the InPs. Since Fwd_VNM is just a framework without the specific algorithm, it will adopt the same intra-domain algorithm as MDVNM in our simulations for fair comparison. In the MD-

239

Fig. 7. Comparison of the maximum delay of virtual links by three methods under varying number of virtual nodes. Parameter α is set at 20|VV|, 50|VV| and 100n|VV| in (a), (b) and (c), respectively.

VNM_MILP, even though the problem is NP-hard, we can solve it under the limited scales (by using CPLEX 10.0). In our simulations, single VN request is mapped across multiple domains with Fwd_VNM, MD-VNM_MILP and MD-VNM. The number of virtual nodes in a VN request varies from 3 to 16. Based on different VN sizes, we compare two metrics for the VN mapping cost and the maximum delay of virtual links.

6.3. Analysis of results In the first experiment, we investigate the cost of resource required by VN mapping under different α.

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As shown in Fig. 6, the cost of VN mapping is larger with the number of VN nodes increases, as the mapping of larger VN requests requires more computing and bandwidth resources. Furthermore, MD-VNM achieves a nearoptimal cost of MD-VNM_MILP, outperforms Fwd_VNM. The main reason is that in MD-VNM and MD-VNM_MILP, the InPs use global knowledge to construct the domainlevel graph and VN can be mapped onto the multipledomains substrate network with lower cost. At the same time, in Fwd_VNM, each InP only does local optimization for a sub-graph of VN, which makes it poor performance. Fig. 6 also shows that the difference between MD_VNM and Fwd_VNM tends to be larger as the number of VN nodes increases. This is because larger VN requests tend to have more inter-domain paths, and the inter-domain mapping is more cost efficient in MD_VNM than in Fwd_QoS. Fig. 7 shows the maximum delay of virtual links under different α. As shown in Fig. 7, MD-VNM always has similar performance with MD-VNM_MILP and a better performance with Fwd_VNM in terms of the maximum delay of virtual links. This is because both MD-VNM_MILP and MD-VNM provide the domain-level graph, which is helpful for delay optimization. Further note with the increasing of parameter α, the gap between MD-VNM/MD-VNM_MILP and Fwd_VNM grows. In Fig. 7(a), the maximum delay of virtual links in MD-VNM/MD-VNM_MILP is close to that in Fwd_VNM. In Fig. 7(b), the maximum delay in MD-VNM and MD-VNM_MILP is smaller than Fwd_VNM up to 35%. In Fig. 7(c), the difference grows up to 40%. The reason is in that the increasing of parameter α means the domain-level mapping more concerned about the maximum delay of virtual links. This makes VN mapping solution pursuit to the maximum delay less. 7. Conclusion In this paper we propose a framework for cost efficient VN mapping across multiple domains optical networks with the QoS consideration. In the proposed framework, the mapping manager and the InPs cooperate to establish a domain-level graph for intra-domain and inter-domain mapping. By using the domain-level graph, the cost of VN mapping can be evaluated to guide both the intra-domain and the inter-domain VN mapping in the same phase. We also develop MILP-relaxation based algorithms that select node and link mapping candidates that minimize costs while giving good virtual link delay property. The simulation results show that the framework leads to low mapping coast and good QoS performance.

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