Photon Netw Commun (2012) 23:109–122 DOI 10.1007/s11107-011-0341-z
Exploring online virtual networks mapping with stochastic bandwidth demand in multi-datacenter Gang Sun · Hongfang Yu · Lemin Li · Vishal Anand · Yanyang Cai · Hao Di
Received: 19 June 2011 / Accepted: 14 November 2011 / Published online: 26 November 2011 © Springer Science+Business Media, LLC 2011
Abstract Network virtualization serves as a promising technique for providing a flexible and highly adaptable shared substrate network to satisfy the diversity of demands and overcoming the ossification of Internet infrastructure. As a key issue of constructing a virtual network (VN), various state-of-the-art algorithms have been proposed in many research works for addressing the VN mapping problem. However, these traditional works are efficient for mapping VN which with deterministic amount of network resources required, they even deal with the dynamic resource demand by using over-provisioning. These approaches are obviously not advisable, since the network resources are becoming more and more scarce. In this paper, we investigate the online stochastic VN mapping (StoVNM) problem, in which the VNs are generated as a Poisson process and each bandwidth demand xi follows a normal distribution, i.e., xi ∼ N (μi , σi2 ). Firstly, we formulate the model for StoVNM problem by mixed integer linear programming, which with objective including minimum-mapping-cost and load balance. Then, we devise a sliding window approachbased heuristic algorithm w-StoVNM for tackling this NP-hard StoVNM problem efficiently. The experimental results achieved from extensive simulation experiments demonstrate the effectiveness of the proposed approach and superiority than traditional solutions for VN mapping in terms of VN mapping cost, blocking ratio, and total net revenue in the long term. G. Sun (B) · H. Yu · L. Li · Y. Cai · H. Di School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China e-mail:
[email protected] V. Anand Department of Computer Science, The College at Brockport, State University of New York, Brockport, NY 14420, USA
Keywords Virtual network · Online algorithm · Stochastic bandwidth demand · Mapping · Multi-datacenter
1 Introduction As emerging technologies for providing distributed services and applications, cloud computing [1], large-scale simulation and peta-scale scientific experiments [2] require the coordination of multiple geographically distributed server resources that are networked together. Network virtualization technology enables these distributed applications or services, since it allows multiple heterogeneous virtual networks (VNs) share the same underlying substrate network [3,4]. In this research work, we abstract either a distributed application (such as multi-datacenter application) or a service request submitted to substrate network as a virtual network (VN) request, which consists of a set of VN nodes and VN edges. Each VN node requires a certain amount of server resources (we call server resources as node resources in this work) for executing the applications; and each VN edge needs a mount of communication bandwidth for the purpose of data and information exchanging between VN nodes. Therefore, there are two important steps in mapping a VN onto a substrate network: VN node mapping and VN edge mapping. In VN node mapping, we map the VN nodes from the same VN onto different substrate nodes (we assume that each substrate node can host the VN node), such that the demand on node resource of each VN node is satisfied. In VN edge mapping, each VN edge is mapped onto a substrate path (set of substrate links) which has (have) the residual bandwidth capacity not less than bandwidth requirement of this VN edge.
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Since multiple virtual networks (VNs) share the resources of the same underlying substrate network in network virtualization environment (NVE), efficient algorithm for mapping the VNs onto the substrate network is essential. From the perspective of substrate network, an efficient VN mapping algorithm will increase the utility of substrate network and consequently produce more revenues. Furthermore, InPs also expect efficient VN mapping for reducing the operation costs and enhancing the quality of service (QoS). In order to address the key issue of constructing a VN, extensive state-of-the-art algorithms for addressing the NP-hard VN mapping problem have been proposed in many research works [5–12]. All of these works research the problem in which the VN has deterministic values of network resources requirement; however, this might raise a fundamental drawback as recent measurement studies show that a certain of network traffic changes dynamically, such as the traffic in production data centers [13–15]. It is impossible for adopting the traditional methods to make a reliable, deterministic estimate of bandwidth usage. Although we can employ a conservative strategy by provisioning bandwidth much higher than the actual bandwidth demand, obviously, it results in over-provisioning and wastes the scarce network resources. In this work, we investigate the online multiple stochastic VN mapping (StoVNM) problem, in which the VNs are generated as a Poisson process. Instead of using deterministic values, we employ random variables for better capturing the dynamic characteristics of actual bandwidth demand (VN edge) of each VN. In our work, each random variable xi follows normal distribution (Gaussian distribution), i.e., xi ∼ N (μi , σi2 ). Such a probabilistic characterization can better represent the uncertainty of the future bandwidth usage. With VN edge (bandwidth demand) size being a random variable, the VN edge mapping issue can be formulated as stochastic link packing problem, which states that VN edges with sizes following a normal distribution must be packed into certain substrate links (i.e., the links which VN edges are mapped onto them), such that each substrate link capacity violation probability does not exceed a given probability α.In this mapping problem, we try to minimize the mapping cost for reducing the operation cost of InP, as well as considering load balance for improving the VN acceptance ratio. To our knowledge, this is the first work that addresses the StoVNM problem. Just as the traditional VN mapping problem, the key challenge of the StoVNM problem is that effective and joint allocation of node and bandwidth resources. We model the StoVNM problem as an optimization problem. In this problem, if we let σi = 0, then the StoVNM problem researched in this work can be reduced to the traditional NP-hard VN mapping problem; thus, the StoVNM is also NP-hard. Consequently, we devise a window-based
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heuristic algorithm called w-StoVNM, for solving the StoVNM problem. The remainder of this paper is organized as follows. Section 2 discusses the related works. We elaborate on problem formulation in Sect. 3. This problem is modeled by using MILP in Sect. 4. Section 5 presents heuristic algorithm for solving this NP-hard problem. The simulation environment to evaluate the performance of our approach and associated experiment results are given in Sect. 6. Sect. 7 makes the conclusion of this paper.
2 Related work 2.1 Virtual network mapping In research literatures related to VN mapping (VNM) problem, most work [5,16–19] formulates the VN mapping as an optimization problem with the mapping cost as the objective. A number of constraints imposed by substrate network components are often associated with this optimization problem, such as link bandwidth capacity, node computing resource amount, node memory resource capacity, etc. However, the VN mapping problem with constraints on link and node capacity can be reduced NP-hard problem [5,20,21]. As a result, in order to solve the NP-hard optimization problem, these works use various heuristic-based algorithms with the common purpose of solving the NP-hard problem efficiently. In literature [22], the authors restudy the VN mapping problem from another perspective and model the interactions between InPs and SPs by using non-cooperative game theory. In this model, the authors establish a bandwidth allocation scheme by using the concept of the Nash Equilibrium. These existing works consider the constraints imposed by link bandwidth capacity and node resource capacity, in which the bandwidth demand is denoted by a deterministic value. However, the network traffic with characteristics of highly fluctuation and burstiness in realistic applications, such as the traffic in production data centers [13–15]. In contrast, in this paper, we model the communication bandwidth demand of VN as a random variable which follows normal distribution, which better captures the dynamics of bandwidth requirement. 2.2 Stochastic load balance (SLB) Consider n containers that need to be transported by m identical vehicles. The weight of the ith container is X i , where the X i s are mutual independent random variables. Let V j refer to the vehicle to which the ith container gets loaded. The stochastic load balancing (SLB) problem is the problem of loading containers to vehicles so as to minimize the expected maximum weight on any vehicle, i.e., the quantity
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⎧ ⎡ ⎧ ⎫⎤⎫ n ⎨ ⎨ ⎬ ⎬ L = min E ⎣ max Xi ⎦ ⎩ ⎭ ⎭ j=1,2,...,m ⎩ i=1:X i ∈V j ⎫⎫ ⎧ ⎧ n ⎬⎬ ⎨ ⎨ E [X i ] = min max ⎭⎭ ⎩ j=1,2,...,m ⎩
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(1)
i=1:X i ∈V j
gets minimized. The issue of stochastic load balance has attracted much interest from the research community. In Ref. [23], the authors have studied the problems of load balancing when the jobs have stochastic processing requirements or sizes. The scenarios that job sizes follow Poisson distribution and Exponential distribution are considered in this work. The study of allocating bandwidth for bursty connections has been undertaken in [24], in which the authors deal with the load balance problem under the scenario that random variables are Bernoulli trial. In literature [25], the authors research the dunamic bandwidth demand virtual machine cosolidating problem in data centers and model this problem as a stochastic bin packing (SBP) problem. The problem studied in this paper can be described as follows: Given a positive constant α ∈ (0, 1), the goal is to efficiently pack multiple connections (the bandwidth demand of each connection follow normal distribution) together on a link for minimizing the expect value of maximum load such that the capacity violation probability of each link does not exceed α.
3 Problem formulations In this section, we provide the model of VN mapping with link bandwidth and node resource constraints imposed by the substrate network. We will reuse partial of the framework proposed in our earlier work [18] to describe the VN mapping problem and MILP model. 3.1 VN request The user demands of link bandwidth, node resource, and QoS can be abstracted into a VN request, which can be modeled as a weighted undirected graph G V = (N V , E V ), where N V denotes the set of VN nodes, and E V indicates the set of VN edges (i.e., communication demands among the VN nodes). Each VN node n v ∈ N V requires a deterministic value of node resource for executing the applications, which denoted by ε(n v ); the link bandwidth required by each virtual edge ev ∈ E V for data and information changing is a random variable that follows normal distribution denoted by xi , i.e., xi ∼ N (μi , σi2 ), where the mean μi ∈ (0, 1)(this value achieved by normalized the expected value of bandwidth demand according to substrate link capacity) and the
standard deviation σi are small enough compared with μi . Then, we denote all of the link bandwidth demands of each VN as D = {x1 , x2 , . . ., x|E V | }, note that xi and x j are mutual independent if i = j. In the follow-up content, we may also use xev to denote the bandwidth required by virtual edge ev ∈ E V . 3.2 Substrate network Optical WDM networks are playing a major role as substrate networks in multi-datacenter to interconnect datacenters, since their advantages such as high speed, high signal to noise ratio (SNR), transparent transmission, and abundant bandwidth resources [26]. We model a substrate network as an undirected graph G S = (N S , E S ), where N S and E S represent the set of substrate nodes and the set of substrate links, respectively. Each substrate node n s ∈ N S has the ability for provisioning node resource for any VN node. We denote the available amount of node resource and the cost of per unit node resource of substrate node n s as c(n s ) and p(n s ), respectively. For each substrate link es ∈ E S , the available bandwidth capacity and the cost of per unit bandwidth are represented as b(es ) and p(es ), respectively. 3.3 Stochastic link packing Different from previous research works that bandwidth demand xi with a deterministic value, in this paper, we assume that the bandwidth demand xi follows a normal distribution N (μi , σi2 ). For a packing strategy, we use Des to denote the set of indices of communication demands carried by substrate link es ∈ E S . Since bandwidth demand xi independently follows a normal distribution, the total amount of bandwidth demand of the demands carried by a link follows
normal distribution too, which with mean i∈Des μi and
variance i∈Des σi2 . Given the link capacity violation probability α ∈ (0, 1), and from the definition of normal distribution, we have Prob x > μ+−1 (1−α)σ = α. Thus, Prob [x > b(es )] ≤ α holds if and only if μ + −1 (1 − α)σ < b(es ), where −1 is the inverse of cumulative distribution function of the standard normal distribution, called quantile function. Here, we let β := −1 (1 − α); thus, if and only if Eq. (2) holds for a given link capacity violation probability α, we say a link packing strategy is feasible for link es . μi + β σi2 ≤ b(es ) (2) i∈Des
i∈Des
Note that, we assume that each communication demand can be provisioned by one substrate link throughout this paper, i.e., μi + βσi ≤ b(es ), ∀i, ∀es , so as to make sure
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that there exists a feasible link packing solution for a given link capacity violation probability α.
2 2 Since i∈Des μi +β i∈Des σi ≤ i∈Des (μi +βσi ), if we use constant μi + βσi as the amount of bandwidth of communication demand i and reduce the problem to traditional link packing problem, thus any packing solution for traditional link packing is also feasible for our stochastic link packing problem. However, the resources used by latter are less than former. We elaborate on the difference by using an example as follows: Example Assume that there are 100 communication demands which all traverse the link es , their bandwidth demands follow an independent identical normal distribution, i.e., xi ∼ N (1, 0.42 ), ∀i. Link capacity violation probability is α = 0.0013, thus β = −1 (1 − α) = 3. Since μi = 1, σi = 0.4 for all i, thus the total bandwidth provisioned by link √ es for these demands can be computed as 1 × 100 + 3 × 100 × 0.4 = 112 units. If we use constant μi + βσi as the bandwidth demand for all these communication demands and reduce the stochastic link packing problem to traditional link packing problem, then the total bandwidth required can be computed as (1 + 3 × 0.4) × 100 = 220 units. In general, if use random variables which follow normal distribution to capture the dynamics of bandwidth demand, rather than use constant μi + βσi as the bandwidth demand, for n (here n is a given number) communication demands n 2 which all traverse link es , we can save β(nσi − i=1 σi ) units of bandwidth resources, which benefit from statistical multiplexing of the bandwidth on link es . 3.4 Residual resources and load balancing Residual resources include the residual bandwidth on each substrate link and residual node resource on each substrate node. We consider substrate link load balance while mapping a VN request onto the shared substrate network. The residual bandwidth capacity of substrate link es , Re (es ) is defined as the total amount of bandwidth available on link es . ⎛ ⎞ Re (es ) = b(es ) − ⎝ μi + β σi2 ⎠ (3) i∈Des
i∈Des
We define the link load balance problem as minimize the maximum link load ratio, which is formulated as: ⎧ ⎧ ⎞⎫⎫ ⎛ ⎬⎬ ⎨ 1 ⎨ ⎝ μi + β σi2 ⎠ (4) min max ⎭⎭ ⎩es ∈E S ⎩ b(es ) i∈Des
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i∈Des
Similarly, the residual node resource capacity of substrate node n s , Rn (n s ) is defined as the node resource available on node n s . Rn (n s ) = c(n s ) − ε(n v ) (5) n v ∈Sn s
where Sn s denotes the set of VN nodes hosted on substrate node n s . We define the node load balance problem as minimize the maximum node load ratio, which is formulated as: ⎧ ⎧ ⎞⎫⎫ ⎛ ⎬⎬ ⎨ 1 ⎨ ⎝ ε(n v )⎠ (6) min max ⎭⎭ ⎩n s ∈N S ⎩ c(n s ) n v ∈Sn s
3.5 VN mapping The substrate network has to make a decision that whether accept or reject a VN request, while the VN request arrives. If a VN request is accepted, then the substrate network takes a suitable VN mapping and allocates the network resource to VN nodes and VN edges such that the resource demand of this VN is satisfied. The allocated resources are not released until the VN is expired. Map a VN onto the substrate network can be implemented by two steps: VN node mapping and VN edge mapping. The VN nodes from the same VN are hosted on different substrate nodes, such that for each n v ∈ Nv , there must hold ε(n v ) ≤ Rn (n s )|n v ∈ Sn s . Each virtual edge ev is mapped to a substrate path Pev (i.e., a set of substrate links) between the corresponding substrate nodes that host the virtual nodes Sc(ev ) and Dt (ev ), such that for each substrate link es ∈ Pev , there must hold xi ≤ Re (es )|i ∈ Des . Where Sc(ev ) and Dt (ev ) represent the source VN node and sink VN node of virtual edge ev , respectively. 3.6 Objectives We focus on proposing online mapping algorithms to map multiple VN requests onto a shared substrate network, with network resource constraints imposed by the substrate network. We expect to increase the net revenue of InP in the long run thus we consider load balance and mapping cost while implementing VN mapping. Since load balance contributes to enhance the VN acceptance ratio, and for an accepted VN, lower mapping cost contributes to increase net revenue. We define the revenue R(G V ) of an accepted VN request as: R(G V ) = x i pb + ε(n v ) pn (7) xi ∈D
n v ∈N V
where pb and pn , respectively, refer to the prices of per unit bandwidth resource and node resource the user pay to InP.
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The cost C(G V ) of mapping a VN request defined as the total cost of network resource allocated to that VN. f leesv p(es )+ r n nn vs p(n s ) C(G V ) = ev ∈E V es ∈E S
n v ∈N V n s ∈N S
(8) where f leesv represents the amount of bandwidth allocated to virtual edge ev on substrate link es . And r n nn vs denotes the amount of node resource allocated to virtual node n v on substrate node n s . Thus, we have the net revenue produced by an accepted VN request, RNet (G V ), as follows: RNet (G V ) = R(G V ) − C(G V )
(9)
Note that, for an accepted VN request, maximizing net revenue RNet (G V ) equals to minimizing mapping cost C(G V ). So we will use minimizing mapping cost C(G V ) as the objective as well as considering load balancing in our MIP model described in next section.
4 MILP model 4.1 Problem description Given: A shared substrate network, G S = (N S , E S ), with resource capacity limitations and a VN request G V = (N V , E V ) with deterministic value of node resource requirement and random variable value follow normal distribution of bandwidth requirement. Problem: How to jointly allocate bandwidth and node resources of substrate network to the VN request considering load balancing, such that the total mapping cost of of the VN request, i.e., sum of node resource and bandwidth resource cost is minimized?
Based on the above graph transformation, the VNM problem can be formulated as a mixed integer Multi-Commodity Flow (MCF) problem, in which the bandwidth requirement on each virtual edge ev ∈ E V of VN request G V is considered as a commodity. We present the detailed MILP formulation for VNM problem as follows. The key notations used in our model summarized as follows: f leesv
Amount of bandwidth allocated to virtual edge ev on substrate link es ; Cost of per unit bandwidth of substrate p(es ) link es ; Amount of node resource allocated to VN r n nn vs node n v on substrate node n s ; Cost of per unit node resource of substrate p(n s ) node n s ; LLRmax Maximum link load ratio of substrate network; NLRmax Maximum node load ratio of substrate network; The set of links of augmented graph G ∗ ; E∗ ∗ The set of nodes of augmented graph G ∗ ; N Binary variable denotes that whether link e e is used, 1 if is used, and 0 otherwise. Sc(e) Source node of link e; Dt (e) Sink node of link e; Residual bandwidth capacity of link e; Re (e) Residual node resource capacity of substrate Rn (n s ) node n s ; Node resource required by VN node n v ; ε(n v ) Total bandwidth capacity of substrate link es ; b(es ) Band width required by virtual edge ev , x ev which follows a normal distribution. Objective function: ⎧ ⎫ ⎨ ⎬ min f leesv p(es )+ r n nn vs p(n s ) ⎩ ⎭ es ∈E S ev ∈E V
4.2 MILP model To formulate MIP model for VN mapping problem, we apply the following graph transformation similar to [5] to G S . We add |N V | virtual nodes into G S ; each virtual node corresponds to a VN node in the VN request. Each virtual node is set to with an infinite resource capacity. We assume that each virtual node is connected to all the substrate nodes n s ∈ N S . We call the links connecting virtual nodes and substrate nodes as virtual links. We assume that the bandwidth resources on each virtual link are unlimited. By means of graph transformation, we achieve the augmented graph as G ∗ = (N ∗ , E ∗ ), where N ∗ = Ns ∪ N V , and E ∗ = E S ∪ {e |Sc(e) ∈ VF , Dt (e) ∈ VL } ∪ {e |Sc(e) ∈ VL , Dt (e) ∈ VF }, where Sc(e) and Dt (e) denote source and sink node of link e, respectively.
n s ∈N S n v ∈N V
(10) min LLRmax
(11)
min NLRmax
(12)
The objective function (10) tries to minimize the total mapping cost (i.e., cost of node resource and bandwidth resources). It can be used to guarantee that the solution be achieved for virtual network mapping (VNM) problem with minimum total cost. Objective function (11) and (12) minimize the maximum the link load and node load of each substrate link and node, respectively, for avoiding bottle neck link and node. Constraints: e ev f le1v + f le2 ≤ 2 × Re (e1) × e1 , (13) ev ∈E V
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∀e1, e2 ∈ E ∗ , Sc(e1) = Dt (e2), Sc(e2) = Dt (e1) e × ε(Sc(e)) ≤ Rn (Dt (e)),
r n nn vs = (14)
∗
∀e ∈ E , Sc(e) ∈ N V , Dt (e) ∈ N S 1 LLRmax ≥ f leesv , ∀es ∈ E S b(es )
(15)
ev ∈E V
NLRmax
1 ≥ r n nn vs , ∀n s ∈ N S c(n s )
e∈E ∗ , Dt (e)=m ∗
∀ev ∈ E V , ∀m ∈ N \{Sc(ev ), Dt (ev )} f leev − f leev = xev , ∀ev ∈ E V (18)
e∈E ∗ , Dt (e)=Sc(ev )
f leev −
e∈E ∗ ,
e∈E ∗ ,
Dt (e)=Dt (ev )
f leev = xev , ∀ev ∈ E V (19)
f leev = 0, ∀ev ∈ E V
(20)
f leev = 0, ∀ev ∈ E V
(21)
Dt (e)=Sc(ev ) e∈E ∗ , Sc(e)=Dt (ev )
Constraint (17), (18), (19), (20), and (21) are flow conservation constrains; these flow conservations show the fact that the net flow of a node n ∗ ∈ N ∗ \{Sc(ev ), Dt (ev )} is zero, where ∀ev ∈ E V . e = 1, ∀n v ∈ N V (22) e∈E ∗ , Sc(e)=n v ,Dt (e)∈VS
Here, if we let σi = 0 for each bandwidth demand xi (i.e.,0 bandwidth required by each virtual edge ev ), then xi has a deterministic value xi = μi ; thus, the StoVNM problem researched in this work can be reduced to the traditional NP-hard VN mapping problem which with deterministic value of requirement on network resource is studied in [5,6,18,19]. Obviously, StoVNM is also NP-hard. In order to solve this intractable NP-hard problem efficiently, consequently, we propose heuristic algorithms for StoVNM to reduce the computational complexity. This is the subject of this section. In this section, we propose a window-based heuristic algorithm for StoVNM problem, called w-StoVNM. The framework of w-StoVNM is shown in Fig. 1. 5.1 Sliding window technique Since the arrival of VN requests are Poisson process and dynamic, there are VN requests arrive in the VN requests collection V N Req continuously. If we map a VN request once it arrives, the mapping solution is not advisable; obviously, if we map all the VN requests simultaneously till all the VN requests arrives, we can achieve the optimal mapping solutions, but this strategy cannot timely response to each VN requests. Consequently, we adopt the sliding window technique to address this issue. We define a window w as a set of VN requests. Initially, we use window w to select VN requests from the beginning of the collection V N Req Begin
e ≤ 1, ∀n s ∈ N S
(23) Mapping the VN requests in window w onto substrate network
e∈E ∗ , Dt (e)=n s ,Sc(e)∈N V
Constraint (22) makes sure that there is only one substrate node selected for hosting each virtual node, and constraint (23) ensures that no more than one virtual node is hosted on a substrate node. e1 = e2 , ∀e1 ∈ E ∗ , Sc(e2 ) = Dt (e1 ), Dt (e2 ) = Sc(e1 ) (24) Constraint (24) ensures that the binary variables for the two links, which in the opposite directions between any node pair of the augmented graph G ∗ , have the same value.
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Equation (25) denotes the amount of resources required by virtual node n v on substrate node n s .
Sc(e)=Dt (ev )
e∈E ∗ ,
(25)
5 Algorithms for StoVNM
Constraint (13) and (14) are link and node capacity constraints; they ensure that the total required bandwidth and node resources must remain within the residual link bandwidth and node resource capacities, respectively. Constraint (15), (16), objective function (11) and (12) jointly balance the load of substrate network. f leev − f leev = 0, (17)
e∈E ∗ , Sc(e)=Sc(ev )
e × ε(Sc(e)), ∀n s ∈ Ns
e∈E ∗ ,Sc(e)∈N V , Dt (e)=n s
(16)
n v ∈N V
e∈E ∗ , Sc(e)=m
Sliding the window w
Are there VN requests in window w need mapping ? N End
Fig. 1 The framework of w-StoVNM algorithm
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Slide window
VNReq VN1 VN2 VN3 VN4 VN 5 VN6 w , size=3
w , size=3
Fig. 2 The sliding window technique
according to the size of window and map these VN requests included in w onto substrate network simultaneously. Then, we slide the window w according to window size for mapping new VN requests. This process effectively divides the dynamic multiple VN mapping problem into a set of static multiple VN mapping (MVNM) problems. The basic idea of sliding window is shown in Fig. 2. 5.2 Algorithm for MVNM In MVNM algorithm, we map the VN requests included in window w onto substrate network (allocate network resources to these VN requests) one by one, by calling NSVIM algorithm proposed in our early work [18]. The objective of MVNM algorithm is that minimizing the mapping cost while considering the load balance of substrate network. The objective function used in MVNM algorithm is defined as follows: min
γ × p(es ) f leesv Re (es ) + δ
es ∈E S
ev ∈E V
λ × p(n s ) + r n nn vs Rn (n s ) + δ n s ∈N S
(26)
n v ∈N V
where γ and λ are used to adjust the weight/importance of link and node load balancing, respectively. To ensure that the resources (bandwidth and node resources) with more residual capacities are preferred over the resources with less residual capacities, we divide the cost by residual capacities. Where δ is a small positive (i.e., δ → 0+ ) constant used for avoiding the denominator divided by zero. The algorithm for MVNM problem is described as Fig. 3. 5.3 w-StoVNM algorithm In w-StoVNM algorithm, initially, we locate the left marginal of window w at the beginning of collection V N Req and mapping the VNs in w onto substrate network by calling MVNM algorithm. Then, we slide the window by updating the left marginal Wleft of window w, for mapping new VN requests. The incremental Wincre of left marginal Wleft of window w computed as (27), while sliding the window w.
Algorithm 1: MVNM algorithm Input: 1. Substrate network GS = ( N S , ES ) ; 2. A set of VN requests (window w ⊂ VNReq ). Output: Mapping results M w , blocked VN VN blo . 1: Sort the VN requests in w in a descending order according to the resource required by each VN, let w* be this ordered set of VN request; let M w = ∅ . 2: for all VNi ∈ w* do 3: if ExpiredVN ≠ ∅ , then 4: Updating resources, ExpiredVN = ∅ 5: end if 6: Call NSVIM algorithm [18] for mapping VNi 7: if find a mapping M i for VNi such that γ × p (e ) λ× p(n ) e n ∑ R ( e ) +δ ∑ fle + ∑ R ( n ) +δ ∑ rnn is minimized, s
es ∈ES
e
s
s
v
ev ∈EV
s
ns ∈N S
n
s
v
nv ∈NV
s
8: updating the mapping results M w . 9: else VN blo = VN blo ∪ {VN i } 10: Updating resources of substrate network 11: end for 12: return M w and VN blo
Fig. 3 The pseudo code of MVNM algorithm
Algorithm 2: w-StoVNM algorithm Input: 1. Substrate network GS = ( N S , ES ) ; 2. VN requests collection VNReq ; 3. Window size WS . Output: Mapping cost CostM , net revenue Rnet and blocked VN VN blo . 1: Initialization. Let Wleft = VNReqbegin , VN blo = ∅ and CostM = 0 . 2: Call MVNM algorithm for mapping the VN requests included in window w ; 3: Compute CostM and Rnet according to M w ; 4: Slide the window (i.e. Wleft ← Wleft + Wincre ) for mapping new VN requests; 5: If w ≠ ∅ , then go to step 2; 6: return M cost , Rnet and VN blo .
Fig. 4 The pseudo code of w-StoVNM algorithm
Wincre = min {W S , |w|}
(27)
where W S denotes the given window size and |w| denotes the number of VN requests included in window w. Detailed w-StoVNM algorithm is described in Fig. 4.
6 Simulations We evaluate the performance of our proposed framework and algorithms by using detailed simulation experiments. In this section, we give the detailed description for simulation environment and present our main experimental results.
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6.1 Simulation environment To evaluate the effectiveness of our approaches, we conduct extensive simulation studies on three different network topologies as shown in Fig. 5. The ITALYNET, USANET, and CHINANET are well-known network topologies that have traditionally been used for conducting network and simulation experiments. The three network topologies chosen vary in terms of the number of nodes, links, and connectivity and thus provide a good basis for evaluation of our approaches. In these four substrate networks, all link bandwidth capacities
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4
(a) Net -1: ITALYNET with 21 nodes
are assumed to be 1 unit and all node resource capacities are assumed to be 100 units. We assume that per unit node resource cost and per unit link bandwidth cost to be equal to 1 unit. In our simulations, when given an upper bound of link capacity violation probability α, we can compute β as β = −1 (1 − α), e.g., for α = 0.0013, one can compute β as β = −1 (1 − α) = 3. The virtual networks (VN requests) are generated randomly as a Poisson process, such that the number of VN nodes is equal to a given number N and the average probability of connectivity of any VN node pair is about 0.5. We assume that the capacities of node resource required by each VN node are uniformly distributed between 1 and 10; the means of each bandwidth demand are uniformly distributed between 0 and 0.16; and the variances of each bandwidth demand are uniformly distributed between 0 and 0.08. The resource allocated to a VN requests will be released while this VN is expired. We have implemented our algorithms by using Microsoft Visual Studio 2005 and C++ programming language. In our simulations, each MILP problem is solved by CPLEX solver. The notations used for each of the approaches compared in our simulation experiments are enumerated in Table 1, where “over-provisioning” to denote the traditional conservative over-provisioning strategy for network resources provisioning, in which we use deterministic value μi + βσi as the bandwidth demand for each VN edge, and w-StoVNM is notation of the approach proposed in this paper.
6.2 Performance metrics
(b) Net-2: USNET with 46 nodes
To evaluate the effectiveness of our approach, we use the following three performance metrics in our simulation experiment. Since the VN request arrives successively, we calculate these performance metrics in a fixed period, such as every 100 VNs in this paper.
(1) Mapping Cost: It is the sum of the node resource cost on all substrate nodes and the bandwidth cost on all substrate links for mapping the accepted VN requests. Total mapping cost C defined as:
C(G V ) , where C(G V ) defined as (8) and C= G v ∈V Nacc
(c) Net-3: CHINANET with 55 nodes Fig. 5 Substrate networks used for simulation experiments. a Net-1: ITALYNET with 21 nodes, b Net-2: USNET with 46 nodes, c Net-3: CHINANET with 55 nodes
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V Nacc denotes the set of accepted VN requests. (2) Blocking Ratio: Ratio of the number of accepted VN requests to the number of total arrived VN requests. The blocking ratio B, defined as: blo | , where |V Nblo | and |V N Req| denote the B = |V|VNNReq| number of blocked VN requests and total VN requests, respectively.
Photon Netw Commun (2012) 23:109–122 Table 1 Algorithms compared
117
Notations
Brief description
Over-provisioning
Traditional conservative over-provisioning strategy for network resources provisioning Algorithm proposed in this work
w-StoVNM
(b) 0.325 0.300 0.275 0.250 0.225 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 0.000 0
0.21 0.18
w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
Blocking Ratio
Blocking Ratio
(a)
0.15 0.12 0.09
w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
0.06 0.03 0.00 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
Number of VN Request
(d)
(c)
0.30 0.14 0.25
0.10 0.08 0.06
w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
0.04 0.02
Blocking Ratio
Blocking Ratio
0.12
0.20 0.15
w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
0.10 0.05
0.00
0.00 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0
Number of VN Request
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
(f)
(e) 0.20
0.16
0.10
0.14 0.12 0.10 0.08
w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
0.06 0.04 0.02 0.00 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request Fig. 6 Comparison of blocking ratio for different networks. a Simulation for Net-1 with various window size. b Simulation for Net-2 with various window size. c Simulation for Net-3 with various window size.
Blocking Ratio
Blocking Ratio
w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2
0.12
0.18
0.08 0.06 0.04
w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
0.02 0.00
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of Request d Simulation for Net-1 with various β. e Simulation for Net-2 with various β. f Simulation for Net-3 with various β
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Total Mapping Cost
5
(b)
x104 w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
4
3
2
x104
w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
6
5
Total Mapping Cost
(a)
4
3
2
1 1
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
Number of VN Request
7
5
w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
6
Total Mapping Cost
(d)
x104
5 4 3 2
x104 w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
4
Total Mapping Cost
(c)
3
2
1 1
0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
6
Total Mapping Cost
(f)
x104
5
4
7
w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
3
2
1
x104
w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
6
Total Mapping Cost
(e)
Number of VN Request
5 4 3 2 1 0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
Fig. 7 Comparison of mapping cost for different networks. a Simulation for Net-1 with various window size. b Simulation for Net-2 with various window size. c Simulation for Net-3 with various window size.
(3) Net Revenue: Total income produced by accepted VN requests subtracts the total mapping cost for these VN requests. The net revenue R, defined as:
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Number of VN Request
d Simulation for Net-1 with various β. e Simulation for Net-2 with various β. f Simulation for Net-3 with various β
R =
G V ∈V Nacc
RNet (G V ), where V Nacc denotes the set
of accepted VN requests, and the definition of RNet (G V ) please see (9).
Photon Netw Commun (2012) 23:109–122
w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
9 8
Total Revenue
(b)
x104 10
7 6 5 4
11
x104 w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
10 9 8
Total Revenue
(a)
119
7 6 5 4
3
3
2
2
1
1
0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request 13
10
w-StoVNM: β=3, Ws=2 w-StoVNM: β=3, Ws=4 w-StoVNM: β=3, Ws=6 over-provisioning: β=3, Ws=2 over-provisioning: β=3, Ws=4 over-provisioning: β=3, Ws=6
12 11 10
Total Revenue
(d)
x104
9 8 7 6 5 4
x104 w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
9 8
Total Revenue
(c)
Number of VN Request
3
7 6 5 4 3 2
2 1
1 0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
Number of VN Request 11 10 9
Total Revenue
8 7
x104 w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
6 5 4
(f) 12 10
Total Revenue
(e)
8
x104 w-StoVNM: Ws=4, β=3 w-StoVNM: Ws=4, β=2 w-StoVNM: Ws=4, β=1 over-provisioning: Ws=4, β=3 over-provisioning: Ws=4, β=2 over-provisioning: Ws=4, β=1
6
4
3 2
2
1 0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number of VN Request
Fig. 8 Comparison of total revenue for different networks. a Simulation for Net-1 with various window size. b Simulation for Net-2 with various window size. c Simulation for Net-3 with various window size.
d Simulation for Net-1 with various β. e Simulation for Net-2 with various β. f Simulation for Net-3 with various β
6.3 Numerical results of simulation
VN request number. The simulation results show that our approach performs well and leads to lower blocking ratio than over-provisioning. This is due to fact that the w-StoVNM algorithm proposed in this work can provide
Impact on VN requests blocking ratio. Figure 6 depicts the performance of VN requests blocking ratio against
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bandwidth resources to VN requests on demand rather than over-provisioning and save resource for accepting more VN requests (i.e., leading lower blocking ratio). Furthermore, Figure 6a–c shows that larger window size leads to lower blocking ratio while fixing the link capacity violation probability parameter β, and Fig. 6d–f shows that smaller parameter β (i.e., lager link capacity violation probability) leads to lower blocking ratio of VN requests. Impact on total VN mapping cost. Figure 7 shows the numerical results of total VN mapping cost against VN request number under different substrate networks. The simulation results show that our approach w-StoVNM has better performance which leads to lower VN mapping cost than over-provisioning. This is because that our approach proposed in this paper benefit from statistic multiplexing while allocating bandwidth resources for VN requests whereas the traditional over-provisioning strategy uses more resources than w-StoVNM and leads to a higher VN mapping cost. With the increasing window size, the achieved VN mapping cost will be reduced (please see Fig. 7a–c), since more VN request be mapped optimally. Another interesting observation is that when ones are sensitive to the probability of link capacity violation (the upper bound of violation probability is lower), the achieved VN mapping cost is higher (please see Fig. 7d–f), since more bandwidth has to be allocated to VN requests. Impact on total net revenue. Figure 8 shows the experimental simulation results of total revenue against VN request number for different substrate networks. It turns out that our approach w-StoVNM produces more revenue than over-provisioning does. This is due to that our approach proposed in this paper uses less bandwidth resources and incurs a lower VN mapping cost than traditional over-provisioning strategy. From Fig. 8a–c, we can see that the revenue increased with the increasing of window size while the link capacity violation probability is fixed, this is because increase window size is helpful for reducing VN mapping cost. Given an window size, bigger parameter β results in higher total revenue (please see Fig. 8d–f), since more bandwidth resource has to be allocated. In this section, we conduct extensive simulation experiments on different substrate networks to evaluate the performance of our approach in terms of blocking ratio, total mapping cost, and total revenue against VN request number. The simulation results show that our approach performs well and leads to lower total mapping cost and blocking ratio as well as higher revenue than over-provisioning does. This is due to fact that our approach proposed in this work can capture the dynamics of bandwidth demands and provision network resource on demand for avoiding waste network resources, rather than over provisioning. Furthermore load balancing attempts to avoid congestion on links and nodes, leaving more critical resources available for latter VN
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requests. As a result, w-StoVNM accepts more VN request while using less network resources and thus generates more revenue, lower blocking ratio, and lower VN mapping cost than traditional over-provisioning strategy.
7 Conclusions Network virtualization serves as a promising technique for providing a flexible and highly adaptable shared substrate network to satisfy the diversity of demands and overcoming the ossification of Internet infrastructure. In such a network virtualization environment, the problem of efficiently allocating the network resources of shared substrate network to multiple virtual networks is increasingly important and challenging. Different with the traditional VN mapping research works which take the conservative strategy—overprovisioning for dynamic demands of network resources, we conduct the first work to study the multiple stochastic virtual networks mapping (StoVNM) problem by using random variables to capture the dynamics of network resource demands. We model this problem as an optimization problem by using mixed integer linear programming (MILP). However, this problem is intractable since it is NP-hard. Consequently, we devise a window-based heuristic algorithm called w-StoVNM, for solving the StoVNM problem efficiently. We evaluate the effectiveness of our approach by conducting an extensive simulation experiments under various realistic substrate network topologies. The experiment results show the effectiveness and superiority of our approach than existing solutions for virtual network mapping in terms of VN mapping cost, blocking ratio, and total net revenue in the long run. Acknowledgments This research was partially supported by Natural Science Foundation of China grant (No. 60872032, 60972030 and 61001084), the National Grand Fundamental Research 973 Program of China under Grant No. 2007CB307104, and the Fundamental Research Funds for the Central Universities (ZYGX2010J002, ZYGX2010J009).
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Author Biographies Gang Sun received his M.En. degree in Signal and Information Processing in 2009 from Chengdu University of Technology. Currently, Gang Sun is pursuing his Ph.D. degree in Communication and Information System at University of Electronic Science and Technology of China. His research interests include network survivability and next generation network.
Hongfang Yu received her B.S. degree in Electrical Engineering in 1996 from Xidian University, her M.S. degree and Ph.D. degree in Communication and Information Engineering in 1999 and 2006 from University of Electronic Science and Technology of China, respectively. From 2009 to 2010, she was a Visiting Scholar in the Department of Computer Science and Engineering, University at Buffalo (SUNY). Her research interests include network survivability and next generation Internet, cloud computing, etc. Lemin Li graduated from Jiaotong University, Shanghai, China in 1952, majoring in electrical engineering. From 1952 to 1956, he was with the Department of Electrical Communications at Jiaotong University. Since 1956, he has been with Chengdu Institute of Radio Engineering (now the University of Electronic Science and Technology of China). From August 1980 to August 1982, he was a Visiting Scholar in the Dept. of Electrical Engineering and Computer Science at the University of California at San Diego, USA, doing research on digital and spread spectrum communications. His present research work is in the area of communication networks including broadband networks and wireless networks.
Vishal Anand is an associate professor at The College at Brockport, SUNY. He received his B.S. degree in Computer Science and Engineering from the University of Madras, Madras (Chennai), India in 1996, and the M.S. and Ph.D. degrees in Computer Science and Engineering from the University at Buffalo, SUNY in 1999 and 2003. He has worked as a research scientist at Bell Labs, Lucent technologies and Telcordia Technologies (ex-Bellcore), where he investigated issues relating to traffic routing and survivability in optical networks. He is the recipient of the “Rising Star” and the “Promising Inventor Award” award from the Research Foundation of The State University of New York (SUNY), and the recipient of the “Visionary Innovator” award from the University of Buffalo (SUNY). His research interests are in the area of wired and wireless computer communication networks and protocols, cloud and grid computing.
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Photon Netw Commun (2012) 23:109–122 Yanyang Cai is pursuing his M.En. degree in Communication and Information System at University of Electronic Science and Technology of China. His research interests include network survivability and next generation network.
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Hao Di is a Ph.D. candidate and a research group member in University of Electronic Science and Technology of China. His research interests include next generation network.
