Dynamic Embedding of Virtual Networks in Hybrid Optical-Electrical Datacenters Soumya Hegde, Raksha Srinivas, Dinil Mon Divakaran, Mohan Gurusamy Email-id: [email protected], [email protected], {eledmd, elegm}@nus.edu.sg Department of Electrical and Computer Engineering, National University of Singapore

Abstract—A promising development in the design of datacenters is the hybrid network architecture consisting of both optical and electrical elements. In this context, the joint problem of bandwidth allocation and VM-placement, a problem that only recently received attention in all-electrical datacenter networks, poses new and different challenges not addressed yet in hybrid datacenters. In particular, we foresee two issues: (i) the number of edgeswitches that can be simultaneously reached using optical paths from an edge-switch is limited by the switch size, (ii) the dynamic creation of optical paths can potentially establish a constrained optical network topology leading to poor performance. We abstract the requests of tenants as virtual networks, and study the problem of embedding virtual networks on a hybrid datacenter, which translates to the joint problem of bandwidth allocation and placement such that the topology constraints of virtual networks are satisfied. We develop and analyse two algorithms for embedding dynamically arriving virtual network demands on a hybrid optical-electrical datacenter. Through simulations, we demonstrate the effectiveness of not only exploiting the already established optical paths, but also of using electrical network in embedding requests of virtual networks. Keywords—datacenter, bandwidth, optical, embedding, virtual network

I.

I NTRODUCTION

Datacenters today are hosting increasing number of services and applications, which in turn generate tremendous amounts of traffic. The global datacenter traffic growth rate is estimated to be more than 30% per year till 2016; and the annual global datacenter IP traffic is estimated to reach 6.6 zettabytes (1021 bytes) by the end of 2016 [1]. The annual traffic growth rate between datacenters as well as within datacenters are also predicted to be above 30%. To meet such traffic growth trends, optical switching based on WDM (wavelength division multiplexing) technology has recently been proposed as a promising approach to connect a datacenter [6], [15], [3]. Optical networks provide not only huge bandwidth, but also reduce the cabling complexity and power consumption significantly (in comparison to electrical networks). An important feature of optical networks is its ability to dynamically reconfigure optical paths between any pair of switches connected using an optical switch. We can leverage on this capability to solve one important challenge in datacenters—VM (virtual machine) placement problem. As optical paths between edge-switches (top-of-rack switches) can Soumya Hedge is currently with Hewlett-Packard, and Raksha Srinivas is with Finisar Corporation. They were both affiliated with the NUS while carrying out this work.

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be created on-demand, there is more flexibility in placing VMs of a request. However, building an all-optical datacenter that provides simultaneous connectivity between every pair of edge-switches is expensive and impractical for large datacenters hosting tens of thousands of servers. This, along with the fact that, electrical network is better suited for multiplexing short and bursty traffic, makes a hybrid optical-electrical network architecture the right choice for future datacenters [6], [15]. A hybrid datacenter gives flexibility in connecting edgeswitches with high communication demands dynamically using optical network, while maintaining connections between edgeswitches with bursty traffic using electrical network. Fig. 1 gives an illustration of a hybrid optical-electrical datacenter network (similar to [6]) considered in this work. The optical switch connects the edge-switches using optical fibers. The number of optical fibers from an edge-switch is limited by design, and defines the reachability factor k. Traffic from a fixed number of ports (on different wavelengths) of an edge-switch are multiplexed into an optical fiber, and switched through the optical switch to any other edge-switch, where they are demultiplexed. For a given number of ports at an edgeswitch, the value of k determines a trade-off between the size of the optical switch (in number of ports) and the number of edge-switches that can be simultaneously reached. The higher the value of k, the larger the size of the optical switch and more the number of simultaneous optical paths from an edgeswitch. In Fig. 1, k is set to two; hence the maximum number of edge-switches that can be reached from any edge-switch using (one hop) optical paths at a given time is limited to

two. This is a cost-effective simple optical network which can connect large number of edge-switches using one or a few optical switches. For example, for k equal to four, an optical switch with 400 ports can connect 100 edge-switches. We note that, although an edge-switch can reach only k other edgeswitches simultaneously, due to the dynamic reconfiguration capability of the optical switch it can reach different sets of edge-switches at different times. In this paper, we address the important problem of allocating bandwdith for communication between VMs of a tenant in a hybrid optical-electrical datacenter (for the architecture presented above). In a datacenter supporting multi-tenancy, guaranteeing bandwidth is important to deliver predictable performance to applications running on the VMs [2]. Recent research works addressed this problem in all-electrical datacenter networks (see Section II). But the challenges posed by a hybrid optical-electrical datacenter architecture are different and new; and not yet addressed. While the dynamic creation of optical paths gives flexibility in placing VMs, the reachability factor limits the number of edge-switches that can be reached simultaneously from one edge-switch (using optical paths). Besides, the optical network topology that gets established dynamically may also pose constraints. We abstract a request from a tenant in the form of a virtual network, where a node of a virtual network corresponds to a set of VMs (VM-cluster), and the weight of an edge connecting two nodes gives the bandwidth required between the two corresponding VM-clusters of a tenant. This is a natural abstraction for most applications in datacenters, such as mapreduce, communicating tasks (where each task is carried out by a set of VMs), etc. [11]. We focus on the problem of embedding of virtual network on hybrid datacenter networks, which translates to the joint problem of bandwidth allocation and placement of VM-clusters such that topology constraints of the virtual network are satisfied (Section III). An intuitive way to solve this problem is to create new optical path for each edge of the virtual network (until no more can be created), and then explore the existing optical paths and electrical network for embedding the remaining edges. But such an approach can potentially create a constrained optical network topology, which may not suit well for future virtual network demands. To investigate this, we develop an algorithm based on the above approach called NLFE (NewLink-First Embedding); in addition we develop another embedding algorithm called ELFE (Existing-Link-First Embedding) that embeds edges on the existing optical paths and electrical network, and only in the worst case will create new optical paths (Section IV). We define a control parameter for flexible control of the proportion of edges that can be embedded on electrical network. Using simulations, we evaluate these algorithms with different topologies for dynamically arriving virtual networks (Section V). Our results provide interesting insights. One, ELFE significantly outperforms NLFE, demonstrating the effectiveness of exploiting existing optical paths. Two, partial (but limited) embedding of edges on electrical links can decrease the rejection ratio while still increasing the utilization of optical network capacity. After discussing the related works below, we define the problem in Section III. The steps to solve the problem of embedding a virtual network are described in Section IV,

where we also define the two embedding algorithms — NLFE and ELFE. Performance studies are carried out in Section V. II.

R ELATED WORKS

Chen et al. proposed an optical switch architecture for an all-optical datacenter network [3]. In the absence of packetswitched network, and as optical paths cannot be created between every pair of edge-switches, the architecture uses multi-hop communication routing traffic through a sequence of optical paths with electrical-optical-electrical (o-e-o) conversions. Such o-e-o conversions are not only undesirable for applications, but also add load to the edge-switches. Hybrid optical-electrical network architectures were proposed recently [15], [6]. At any instant of time, only one or a few optical paths can be provisioned from an edgeswitch, limiting the number of simultaneously reachable edgeswitches. This means, optical paths have to be removed and created frequently requiring reconfigurations. Considering such factors, these works proposed methods to monitor and estimate traffic between racks, and create or reconfigure optical paths between racks with high communication traffic, so as to meet changing demands. Note that these works assume VMs to be already placed, and hence do not take into consideration of the impact of VM-placement on the network traffic. The importance of allocating bandwidth for requests in (allelectrical) datacenter was recently highlighted by the research community. In this direction, Seawall enforces link-bandwidth allocation to competing VMs based on the weights assigned to VMs [14]; whereas Gatekeeper achieves bandwidth-sharing among competing tenants [13]. Other works consider advance reservation of bandwidth, given input demands. Our previous work on providing flexible bandwidth guarantees to flows allows users to specify two bandwidth values between a VM pair. While the minimum bandwidth is guaranteed throughout the duration of request, the peak bandwidth is provided only for a fraction of a duration (the range of which is defined by the user) [4]. SecondNet [8] takes a bandwidth matrix specifying bandwidth demands between every VM pair as part of an input request. Oktopus abstracted bandwidth demands between VMs of a tenant into two topologies [2]. While one topology was a single cluster of VMs requiring a star communication pattern, another topology abstracted the communication demand of clusters of VMs (or VM-clusters) as a tree. Another recent works we did in [9], [5] solves the joint problem of VMplacement and bandwidth allocation in all-electrical network by considering a matrix of bandwidth demands between every VM pairs as input. But assuming the bottleneck to be in the core network connecting the switches of a datacenter, the results showed that, for efficient allocation VMs can be grouped to form small number of VM-clusters (often four, and not more than six), and bandwidth allocation on these core links is dictated only by the traffic demands between VMclusters [5]. Works in [12], [10], [11] also address the joint problem VM-placement and reduction of bandwidth between VM-clusters in all-electrical datacenters, with [11] explicitly focussing on abstracting input as a graph of VM-clusters (similar to virtual network defined below). To the best of our knowledge, the problem of bandwidth allocation to dynamically arriving demands (virtual networks,

A

Problem: Given an input request in the form of a virtual network, the problem is to embed the virtual network on to the hybrid datacenter network. This translates to the joint problem of bandwidth allocation and placement of VM-clusters, while satisfying the topology constraint of the input virtual network

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defined below in Section III) in a hybrid datacenter has not been addressed yet. We take the first step towards addressing the problem with this work. III.

P ROBLEM DEFINITION

Bandwidth demands between VMs of a tenant can be specified in multiple ways. At the finest granularity, traffic demand between every VM pair can be specified by a tenant, thereby giving a bandwidth matrix as input for each request. Another possibility is to consider a virtual network as an input, where the nodes of a virtual network are VM-clusters, and the weight between two connected nodes gives the bandwidth demand between the corresponding VM-clusters. Providing such an interface to tenants is useful to abstract the communication demands of the VMs, removing the difficulty of tenants in specifying traffic demand between every VM pair. Besides, it localizes the traffic between VMs of a cluster. Fig. 2 gives a simple example of a virtual network. The input request has 10 VMs (v1 through v10 ). A, B and C are the nodes (VMclusters) in the virtual network, with b1 being the bandwidth demand between A and B, and b2 being the bandwidth demand between A and C. We assume that a node of a virtual network can be mapped on an edge-switch such that the bandwidth requirements of the individual VMs (from VM to edge-switch) of the corresponding VM-clusters can be met using the links connecting the servers to the edge-switch. We also assume, not more than one node of a virtual network should be mapped to an edgeswitch; otherwise two such nodes can be merged to form a bigger node in the virtual network. Taking cue from [2], [11], we consider three topologies for input virtual networks—star, tree, and random graph (Section V-A)—which together cover various relevant application scenarios in a datacenter. Placement of a VM-cluster on an edge-switch implies placement of the VMs of the cluster on the servers connected to the edge-switch. Multiple VMs can be placed on a server depending on the capacity of the server, in terms of both computing resource capacities as well as server-to-switch link capacity. Similarly, placement of a virtual network means, to find an edge-switch for each VM-cluster such that the bandwidth requirement between two VM-clusters is satisfied by the link connecting the corresponding edge-switches. We now define the problem.

Observe that the placement of VM-clusters (nodes of virtual network) is closely tied to the embedding of the corresponding edges of the request on the physical network. As bandwidth is a scarce resource, we assume bandwidth will become a bottleneck much before computing resources; and hence do not consider computing resources for allocation here. IV.

V IRTUAL NETWORK EMBEDDING

A hybrid datacenter offers multiple choices to embed a virtual network. A straightforward approach to embed a virtual network on a hybrid datacenter is to create as many optical paths as the number of edges in the virtual network, giving lesser priority to exploring existing optical paths. But such an approach can potentially create a constrained optical topology that rejects more requests than it would have if it gave least preference to creating new optical paths. In Section IV-C, we develop two algorithms to investigate this. In the following two sections, we break the problem into two subproblems and describe the steps involved in the algorithms. For simplicity, we assume bandwidth demands and link capacities are symmetric. A. Degree-constrained maximum weighted subgraph As no edge-switch can simultaneously connect to more than k other edge-switches using optical paths, an input virtual network with maximum degree greater than k cannot be embedded on the optical network. Hence, the first problem is to extract a subgraph of the given input virtual network, such that maximum degree of any node is k. The remaining edges have to be embedded on electrical network. To maximize bandwidth allocated on optical network, we set the objective as to maximize the sum of the weights of the selected edges. If ξ r denotes the set of edges in an input virtual network r, with we denoting the weight of the edge e ∈ ξ r , this problem can be formulated as: maximize

X

xe we

e∈ξ r

subject to

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xe ≤ k;

e∈ξ r

where xe ’s are the binary variables selecting the edges of the virtual network. This is the well-studied degree-constrained maximum weighted subgraph problem, which is N P-hard in nature [7]. But since a virtual network has VM-clusters as its nodes, it can be assumed to be small in size [2], [5]; we therefore solve it using integer linear programming (ILP) solver. B. Subgraph embedding Consider the optical network as a graph G, where the nodes correspond to the edge-switches, and a link connecting two nodes is the optical path connecting the corresponding edgeswitches. The nodes in G can be partitioned into two sets, L

and F, such that the subgraph formed of nodes in L is a logical topology established by optical paths, and F forms a free set of nodes. Any node in the logical topology would have at least one link incident on it; whereas there is no link incident on any node in F. Observe, logical topology is dynamic and may change (by adding new optical paths) to accept a request. For example, in Fig. 3(b), L = {1, 3, 4}, and F = {2}, and the logical topology is the graph formed of nodes in L. Once we obtain the subgraph of the input virtual network after solving the first subproblem, the next task is to embed this subgraph on to the hybrid datacenter. The link on which an edge (of the subgraph) should be embedded, can either be an existing link or a new link created by establishing optical path. The various possible options are: Step 1: Create a new optical path using the free set of nodes F, and embed the edge on this link. Step 2: Find a link in the logical topology L that can satisfy the bandwidth (weight) of the selected edge. Step 3: Find an appropriate node in the logical topology L with degree less than k, from which a new optical path can be created. The other node can be either a node in the logical topology L, or one in the free set of nodes F. Step 4: Find an electrical link with sufficient bandwidth to satisfy the weight of the edge. All along each of the four steps, it is important that the constraints due the previously embedded edges (topology constraints of the virtual network) are not violated. By embedding an edge on a physical link, we are also doing the placement of the corresponding VM-clusters on the end-points of the link (switches). Below we describe solution for each step. 1) Step 1: Given an edge to embed, Step 1 is trivial, as the only action required is to randomly select two nodes from F and establish an optical path. These two nodes are then moved to the set L as they are now part of the logical topology. The capacity constraint is also checked. 2) Step 2: Algorithm 1 finds and embeds an edge of the subgraph, say r, on the logical topology in this step. Let B r and B l be the bandwidth matrices of the subgraph r and the logical topology, respectively (with −∞ denoting bandwidth on unestablished optical links). Let Er be the set of edges of subgraph r previously embedded, and Vr be the corresponding set of nodes of r that have been mapped to edge-switches (in implementation, they are ordered lists for easy access to the corresponding mapped element). We denote by El and Vl the set of links and switches on the logical network, on which the sets of Er and Vr , respectively, are embedded. Also, for each of the above set, say X, we denote the complement by X. For example, consider the subgraph of an input virtual network given in Fig. 3(a) for embedding. The graph of the optical network, with edge (a, b) embedded on link (1, 3), is shown in Fig. 3(b). At this time, Er = {(a, b)}, Vr = {a, b}, El = {(1, 3)}, Vl = {1, 3}, Er = {(b, c), (b, d)}, Vr = {c, d}, Vl = {2, 4}. Consider using Step 2, to embed the next edge. Algorithm 1 finds the edge to be embedded (u, v) and an existing link on the logical topology, (s, t) for embedding. The mapping between an edge of the subgraph and a link on the physical network is stored in

Algorithm 1 ExistingOpticalLinkEmbedding(r) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:

if ∃(u, v) ∈ Er , 3 (u ∈ Vr ∧ v ∈ Vr ) then Find s, t ∈ Vl corresponding to u, v respectively l r if (Bs,t ) ≥ (Bu,v ) then M.append([(u, v), (s, t)]) return else return FAIL end if end if if @(u, v) ∈ Er , 3 (u ∈ Vr ∨ v ∈ Vr ) then Pick least degree node u ∈ Vr Pick least degree node s ∈ Vl , 3 degree(s) ≥degree(u) else Let u be the node 3 (u, v) ∈ E r ∧ (u ∈ Vr ∧ v ∈ / Vr ) Find s ∈ Vl corresponding to u end if P P r l (Bu,j ) then (Bs,i )≥ if ∀i∈neigbours(s) l t = max(Bs,i );v i l r if (Bs,t ) ≥ (Bu,v )

∀j∈neigbours(u) r = max(Bu,j ) j

19: then 20: M.append([(u, v), (s, t)]) 21: else 22: return FAIL 23: end if 24: end if

the list M. The first seven lines in Algorithm 1 check if there is an unmapped edge (u, v) such that both the end-points u and v are already mapped; in such a case this link is chosen for embedding the edge. In the case where there is no mapped node in the subgraph, we select a least degree node u and a corresponding edge-switch s (lines 10-12); and then select the other node and edge-switch in lines 17-18. Lines 13-15 are for the case where there is only one of the two end-points of an unmapped edge in the subgraph is mapped. For our example, the two if conditions (line no. 1 and line no. 10) fail, and b is selected in line no. 14. The corresponding edge-switch in Vl is 3. Assuming edge (b, d) and link (3, 4) satisfy the constraints in line numbers 17 and 19, the edge (b, d) gets embedded on the optical link (3, 4), as depicted in Fig. 3(c). 3) Step 3: This step comes only after at least one edge of the subgraph r is embedded; i.e., when Er 6= {}. For the example, from Fig. 3(c) (after Step 2), Er = {(a, b), (b, d)}, Vr = {a, b, d} and Vl = {1, 3, 4}. Algorithm 2 finds an unmapped edge with the maximum weight, such that at least one incident node is already mapped, and finds the appropriate edge-switch to which a new optical path should be created. The algorithm will embed the edge (b, c) on the newly created optical link (3, 2), as depicted in Fig. 3(d). 4) Step 4: This step is invoked only after one or more edges of the subgraph have been mapped on to the optical network. This, as in Step 3, introduces constraints for selecting the edge-switches, based on the previously mapped edges and nodes. We proceed exactly as in Step 3, except that, instead of establishing an optical path, here we check if the aggregate electrical paths between the two edge-switches have bandwidth greater than or equal to the weight of the edge; and if so, this set of paths, identified as a logical link (assuming multipath routing will take care of load balancing traffic among the electrical paths), is selected for embedding the edge. To

Algorithm 2 NewOpticalLinkEmbedding(r)

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Find maximum weighted edge (u, v) ∈ Er , 3 u ∈ Vr Let s ∈ Vl be the node on which u is mapped if degree(s) ≥ k then return FAIL end if if v ∈ Vr then Let t ∈ Vl be the node on which v is mapped else if ∃t ∈ L, 3 degree(t) < k then Let t be the other edge-switch else Choose a node randomly from F as t end if end if Create a new optical path between s and t; Update all sets M.append([(u, v), (s, t)])

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(b) Graph of optical network, after embedding edge (a, b) of r on link (1, 3)

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(d) Logical topology after creating link (3, 2) and embedding the edge (b, c) on it Fig. 3.

ensure high optical utilization, the number of edges of a given subgraph that can be mapped to electrical network is restricted by a control parameter m. We define m as a decreasing step function of the current (or in the words, ongoing) acceptance ratio of the input requests. The basic idea is to allow more edges to be embedded on electrical network only if the current acceptance ratio is decreasing, and vice-versa. The number of edges of a subgraph (of a newly arriving request) that can be embedded onto the electrical network will be evaluated after processing (and deciding on) every request, as each processing would have on the current acceptance ratio. The exact definition (specifically, the values the function would take) is provided in Section V.

Example illustrating embedding of a subgraph

The order in which the solution space for embedding an edge is explored can have an impact on the number of requests being accepted by the system. We define two embedding algorithms that use the four Steps defined above, but differing in the order in which each explore the hybrid network to embed the subgraph of an input virtual network. The two algorithms solve the first subproblem (Section IV-A) using ILP. Besides, in both algorithms, the edges of a virtual network not selected by the ILP for the subgraph (due to the degree constraint of k) are all mapped to the electrical network (after the subgraph is embedded) using Step 4 described previously, but without being constrained by m. This is naturally the only way to embed the remaining links, as they cannot be embedded on the optical network. Also recall, no two nodes of a given virtual network can be placed on the same edge-switch. Definition 4.1: Existing-Link-First Embedding (ELFE) algorithm: After obtaining the subgraph, ELFE algorithm first explores the logical topology for an appropriate link (Step 2), and then explores the electrical network (Step 4). If ELFE cannot find an appropriate link in the electrical network, it tries to establish a new optical path using a node in the logical topology (Step 3). If all these options fail, the ELFA algorithm forgoes all previously selected links for embedding, and embeds the entire subgraph using the free set of nodes (F) by creating as many optical links as the number of edges in the subgraph.

be between five and ten. Recall, a node in a virtual network is a clusters of VMs, hence the virtual network is small in size. Previous works have either considered small number of VM-clusters as input [2], [11], or found grouping VMs into small number of VM-clusters (given bandwidth requirements between every VM pair) suffices for efficient allocation [5]. The bandwidth on the edge connecting any two nodes (two VM-clusters) is drawn from the Exponential distribution with a mean of 200 Mbps. The following are the three topologies:

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Fig. 4. The step function defining the control parameter m. For acceptance ratio ≤ 0.75, m = 7.

The ELFE algorithm gives least priority to creating a new optical path. The motivation for this is to exploit the already established optical paths. On the other hand, the next algorithm defined below, first attempts to create a new optical path. Definition 4.2: New-Link-First Embedding (NLFE) algorithm: The NLFE algorithm (after obtaining the subgraph) first explores Step 1 (create new link, both nodes from F, followed by Step 3 (create new link, one node from L), and then Step 2 (use existing optical link), for embedding an edge of the subgraph; and only if all these steps fail, will it explore the electrical network for an appropriate link (Step 4). Observe that NLFE algorithm iterates over each edge in the subgraph and performs the aboves steps in order to embed the entire subgraph on to the hybrid network. V.

P ERFORMANCE ANALYSIS

Here we study the performance of the proposed algorithms—ELFE and NLFE algorithms. As mentioned earlier, we study the algorithms for a hybrid datacenter architecture illustrated in Fig. 1. There were 100 edge-switches, each with 32 ports connected to the electrical network and another 32 ports connected to the optical switch (using k fibers). The bandwidth per port was 1 Gbps. The reachability factor, k, was set to four. This means, an optical link carries eight wavelengths, thus delivering a total optical capacity of 8 Gbps between two edge-switches connected using the optical switch. The step function for the control parameter m used here takes value seven for acceptance ratio below or equal to 0.75, and decreases by 1 at steps of 0.05, reaching a lower limit of 2 for acceptance ratio between 0.95 and one. Fig. 4 shows the step function for m. Next we explain the scenarios (topologies for virtual networks) considered, and then discuss the results. A. Input scenarios The input to the algorithms, as mentioned previously, are a set of virtual networks. We consider three scenarios, each having one particular topology for the set of input virtual networks. In each virtual network, irrespective of the topology, the number of vertices is chosen randomly and uniformly to

1) Star: This topology abstracts communication pattern of one master and multiple slaves (all connected only to master). 2) Tree: In this scenario, we have three-level trees as input virtual networks. The number of nodes in level two (with root being at level one) is fixed as three. Hence, as the number of nodes is in the range [5−10], the minimum number of nodes in level three is one. We allow over-subscription; the bandwidth on an edge connecting the nodes of levels one and two (say, n1 and n2 , respectively) is half the total bandwidth on the links connecting the node in level-two (n2 ) to its children. 3) Random: Randomly generated connected graphs are virtual networks, with the number of edges in a virtual network limited to twice the number of vertices. To generate a virtual network with one of the above topologies, a random number for the number of vertices (in the range [5−10]) is given as input. Note that, given the number of vertices, the number of edges is automatically limited for star and tree topology. For the random graph topology, the number of edges is a random number between [10 − 20]. B. Results We compare the algorithms using the following metrics: •

Rejection percentage: This is the percentage of input virtual networks rejected by an embedding algorithm.

•

Utilization: This is the utilization of the optical network. For this metric, we define the total capacity as the sum of the capacities of the optical links that are established. Utilization is the percentage of this capacity that is allocated for requests.

For each experiment, the embedding algorithm processes a set of dynamically and incrementally arriving virtual networks at input. The number of virtual networks at input is varied (for different experiments) from 25 to 200 at steps of 25. We plot the rejection percentage and utilization for each scenario. Each point on the graphs is the mean value from 15 runs. Below, we discuss results for each scenario. 1) Scenario 1 - Star topology: Here the input virtual networks have the star topology. Fig. 5(a) plots the percentage of rejected requests, for varying number of input virtual networks. ELFE algorithm is seen to reject significantly lesser number of requests as compared to NLFE. At the minimum, ELFE algorithm brings down the rejection by 7%, while reducing the rejection by ≈ 28% on average. As NLFE first attempts to embed the input subgraph by establishing new optical paths, it leads to the formation of a constrained logical topology. This eventually results in the rejection of future requests, even while bandwidth being available in the optical network; see the

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Scenario 1: Virtual networks with star topology

utilization plotted in Fig. 5(b). The ELFE, which establishes a new set of optical paths only in the worst case, increases the utilization by more than 18% on average. Observe that in a virtual network with star topology and having n edges, the master node has degree n. Hence, accepting a virtual network with many edges, such that some edges are mapped on to electrical network will leave the logical topology less constrained. On the other hand, NLFE always embeds an edge on a newly created optical path between two nodes in the free set of nodes (F), until there is none available. This leads to a more constrained optical topology. 2) Scenario 2 - Tree topology: The input virtual networks have a three-level tree topology in this scenario. The rejection percentage, as seen in Fig. 6(a), is lesser for both the embedding algorithms with tree topology than with the star topology. The maximum degree of any node in this scenario is lesser than that of any node in the virtual networks with star topology. Comparing plots in figures 6(a) and 6(b), the ELFE again performs better than NLFE. The ELFE algorithm brings down the rejection by a minimum of ≈ 30%, and by ≈ 70% on average. In comparison with NLFE, the ELFE also

Fig. 6.

Scenario 2: Virtual networks with tree topology

increases the utilization by more than 50% (on average). 3) Scenario 3 - Random graph topology: The input virtual networks are formed as random graphs. Over here, to see the impact of the control parameter m (the value that decides the number of the edges of a subgraph that can be mapped on to electrical network), we also simulate the NLFE algorithm with the value of m set as static and equal to three. We refer to this algorithm as NLFEm=3 . Fig. 7 plots the rejection percentage. As the topology of input virtual networks is completely random, the difference in performance of ELFE and NLFE has reduced, although ELFE still performs better. But observe that, having a lower and static value of m has resulted in much higher rejection ratio for NLFEm=3 . Recall, the mean number of edges on a virtual network is 15 in this scenario, and the maximum value of m is seven for both ELFE and NLFE algorithms. This scenario reveals the importance of electrical network in embedding virtual networks. An all optical datacenter with limited reachability may perform worse with high rejections. Though not plotted here, we made similar observations for the optical utilization, with ELFE performing slightly

Rejection percentage

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different virtual network topologies. Overall, ELFE algorithm outperformed NLFE algorithm, bringing down the rejection percentage significantly. This highlights the importance of exploiting existing optical paths (leading to the formation of a less constrained logical topology), in combination with electrical network, for reducing rejection ratio of requests while maintaining high utilization of optical network capacity. We also demonstrated the effectiveness of the control parameter (m) which dynamically decides the number of edges of a request that should be embedded on electrical network.

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VII.

ACKNOWLEDGEMENT

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Fig. 7.

This work was supported by Singapore Ministry of Education Academic Research Fund Tier 2 Grant No. MOE2013T2-2-135, NUS WBS No. R-263-000-B11-112.

Scenario 3: Rejection for virtual networks with random topology

R EFERENCES 30 Star, ELFEm Star, ELFEm 25

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[6] Fig. 8.

Comparing rejection percentage by varying m for ELFE

better than NLFE, but both performing significantly better than NLFEm=3 . 4) Impact of m: To further study the impact of the control parameter m, we also increased the maximum value of m from seven to 10. Correspondingly, to maintain the same number of steps in m, the minimum value was increased from two to five. We refer to this modified ELFE as ELFEmmax =10 . We compare this against the standard ELFE (referred to as ELFEmmax =7 ) for the scenarios with star and random topologies, which gave higher rejection than with the tree topology. Fig. 8 plots the results. Increasing the maximum value of m (from 7 to 10) reduced the overall average rejection by ≈ 85% and ≈ 50% with star and random topologies, respectively. We observed, the overall average utilization improved by more than ≈ 50% and ≈ 17%, respectively, for star and random topologies. This means, exploiting the electrical network not only reduces the rejection, but also increases the utilization of optical network capacity as more requests are accepted.

[7] [8]

[9]

[10]

[11]

[12]

[13]

[14]

VI.

C ONCLUSIONS

In this paper, we addressed the problem of embedding dynamically arriving virtual network demands from tenants on a hybrid optical-electrical datacenter network. We developed two algorithms, ELFE and NLFE, and evaluated them for

[15]

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