Optimization Algorithms for Data Center Location Problem in Elastic Optical Networks

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1 Optimization Algorithms for Data Center Location Problem in Elastic Optical Networks Mirosław Klinkowski 1, Krzysztof Walkowiak 2, and Róża Goścień 2 1 National Institute of Telecommunications, Warsaw, Poland. 2 Wroclaw University of Technology, Wrocław, Poland. ABSTRACT Modern optical networks must meet many challenges following from the quick and permanent development of IT industry. The latest trend that focuses much attention is cloud computing a concept that enables delivery of various IT services over the Internet. From the network perspective, the cloud computing implies a growth of the traffic volume and advent of new traffic patterns. In this work, we combine the cloud computing topic with a new concept of optical networks, namely, Elastic Optical Network (EON), which allows for efficient and scalable bandwidth provisioning. In more detail, we address the problem of data center location with lightpath provisioning in EONs. We focus on optimization algorithms for locating data centers with the objective to minimize the amount of spectrum required to serve anycast demands. We propose effective heuristic algorithms that utilize the wellknown optimization technique called Column Generation. We present results of numerical experiments run on a real topology of an European backbone network. Our heuristics perform well comparing to optimal results provided by the CPLEX solver. Moreover, we report the increase of spectrum requirement in years according to traffic demand patterns generated according to Cisco forecasts. Keywords: Data centers; elastic optical networks; ILP modelling, network planning. 1. INTRODUCTION According to many foresights including Cisco Networking Index and Cisco Global Cloud Index, the most voluminous traffic in the future Internet will be video as well as cloud computing data traffic. Both types of traffic are related to special network sites hosting data centers (DCs) and/or content servers. Data/content is replicated in DCs and a client can connect to the most convenient one (e.g., the nearest), by means of anycast routing, hereby decreasing latency and capacity requirements in the network [1]. Still, it is expected that the emerging network services supported by DCs will consume large amount of bandwidth due to the growing number of users and increasing capacity of demands. Concurrently, Elastic Optical Network (EON) technologies are a promising approach for improving the spectral efficiency and flexibility of optical networks [2]. The use of advanced modulation formats and techniques in bandwidth variable transponders and the operation within flexible frequency grids allows for adaptive bandwidth provisioning in accordance to traffic demands. The second pros of EON is the support for high capacity demands that cannot be transported efficiently in current WDM networks. The elastic and highly scalable bandwidth provisioning of EON architectures is considered one of technological pillars for building effective and costefficient cloud and anycastready transport networks [3]. In EON, the adaptation of modulation levels of transmitted signals according to optical path (i.e., lightpath) characteristics brings significant savings in terms of spectrum utilization. Since the transmission on shorter paths can be performed with higher modulation levels, the aggregated anycast traffic directed towards nearer DCs will require less spectrum resources allocated to lightpath connections [4]. The savings in the spectrum usage can be potentially translated to lower costs of the switching equipment installed in the network. Currently, cloud computing systems and transport networks are managed independently as well as both environments are agnostic of each other. To improve performance of cloudready transport networks it is highly required to enable cooperation between these two environments including, among others, coordinated network planning and design. In a crossstratum optimization approach a new problem arises, namely, where to locate data centers (servers) in EON so that to optimize selected objectives (e.g., spectrum usage). In the paper, we present an Integer Linear Programming (ILP) formulation of the joint problem of data center location with routing and spectrum allocation (DCLRSA) for aggregated anycast traffic demands. Since the RSA problem itself is N Pcomplete [5], we propose several heuristic algorithms to solve it. The DC location problem has been studied in WDM networks [6], however, to the best of our knowledge, it has not been addressed in EON. The remainder of the paper is organized as follows. In Section 2, we formulate the DCLRSA optimization problem. In Section 3, we propose optimization algorithms for solving DCLRSA. In Section 4, we provide numerical results to show the effectiveness of algorithms. Finally, in Section 5, we conclude the work. 2. PROBLEM FORMULATION In this Section, we present an ILP model of an offline problem of Data Centre Location with Routing and Spectrum Allocation (DCLRSA) in an EON with static traffic demands. We take a similar modelling approach as in [7] for formulating the ILP problem.
2 The considered network is modelled as a directed graph G = (N, E), where N denotes the set of nodes and E denotes the set of fiber links connecting two nodes in N. Let set V N denote the set of possible locations of data centers and let K be the number of DCs to be located in the network. A set D of static traffic demands is given and each demand d D is represented by a tuple (o d, h d ), where o d is the origin node and h d is the requested bitrate. We assume anycasting, i.e., every DC provides the same content and each demand can be served by any data center located in the network. A demand is connected to a DC by a lightpath unless a DC is located in the origin node of the demand  in such a case the demand is served locally. We assume a scenario when the DCs provide some content (e.g., video or other multimedia content) and the traffic is delivered from DCs to clients. The traffic in the opposite direction is significantly smaller and is not included in the model. The frequency spectrum, available in each link e E, is represented by an ordered set of frequency slices S = { s 1, s 2,..., s S }. Routing path p is identified with a subset p E and, consequently, p denotes the number of hops. Let t p denote the termination node of path p. Let P d be a set of admissible paths for demand d D. Channel c is defined as a contiguous (adjacent) subset of slices in ordered set S, i.e., c S. Let C dp be a set of admissible channels for demand d on path p P d ; each c C dp provides enough spectrum to support traffic volume h d. Eventually, lightpath l is represented by a tuple (p, c), where p is a routing path and c is a channel. Let L d be a set of admissible lightpaths for demand d; for each l = (p, c) L d we have p P d and c C dp. Let L = d D L d denote the set of all admissible lightpaths. We introduce a set of problem variables: x d = 1 if demand d is routed over the network using a lightpath, and 0 if its origin is in a node with a DC; x l = 1 if lightpath l is established, and 0 otherwise; x es = 1 if slice s S is used in link e E, and 0 otherwise; x s = 1 if slice s S is used in the network, and 0 otherwise; x dv = 1 if demand d is served by DC node v, and 0 otherwise; x v = 1 if a DC is located in node v, and 0 otherwise. The DCLRSA problem is formulated as following: min z = s S x s (1a) [ λ d ] x l = x d d D (1b) l L [ d η dv ] x l = x dv d D, v V (1c) [π es 0] l=(p,c) L d : t p=v d D,l=(p,c) L d : p e,c s x l x es e E, s S (1d) x es x s e E, s S (1e) x d + x v = 1 d D, v = o d (1f) x d, x l, x es, x s, x v, x dv {0, 1}. x v = 0 v N V (1g) x dv x v d D, v V (1h) x v K (1i) v where variables in brackets, i.e, λ d, η dv, and π es, are dual variables. The objective (1a) is to minimize the width of spectrum, expressed as the number of slices, required in the network (as in [5] and [8]). Constraint (1b) assures single path routing and spectrum allocation, i.e., whenever demand d is routed over the network (x d = 1) then exactly one lightpath is selected. Constraint (1c) assures that each demand has its lightpath terminated in a DC node (i.e., for which x dv = 1). Constraints (1d) guarantees that a slice on a particular link can be allocated to at most one lightpath. Constraint (1d) indicates that slice s is used if it is used at least in one link in the network. Equation (1f) assures that a demand is either terminated in its origin node, if the node is equipped with a DC (x v = 1), or is routed over the network (x d = 1). Equation (1g) means that nodes that are not candidate DC nodes cannot be equipped with a DC. Constraint (1h) assures that a DC has to be located in a node in which at least one demand is terminated. Constraint (1i) introduces a limit on the number of DCs located in the network. Finally, (1j) assures that variables are binary. 3. ALGORITHMS In this section, we present several approaches for solving the DCLRSA problem. (1j)
3 1) Joint DCLRSA optimization: A first approach is to run a BranchandBound (BB) algorithm, implemented in an ILP solver, to solve Problem (1). Since formulation (1) is not compact, a set of admissible (candidate) lightpaths L has to be provided; remind that each admissible lightpath in (1) is represented by variable x l. a) BB with complete set of admissible lightpaths: A common and frequently used approach is to include a large set of path selection variables (in our case, variables x l ) in the problem formulation. Set L d, d D, has to be large enough in order to not leave out good (optimal) solutions from the feasible solutions space. Therefore, in the algorithm evaluation in Section 4, for each pair of origin o and termination t nodes we precalculate k = 30 shortest paths and include them in set Q ot. Then P d consists of all paths between the origin of demand d and each possible DC location, i.e., P d = v V Q o d v. Concurrently, C dp includes all admissible channels on path p P d. Eventually, for each demand d, we form the set of all admissible lightpaths L d = {(p, c) : p P d, c C dp } and for each l L d we include variable x l into formulation (1). Since formulation (1) requires set S to be defined, we set S equal to the solution of the DCL+RSA/MSF heuristic algorithm (described in the next subsection), which is an upper bound on DCLRSA. We denote this approach as DCLRSA/BB. b) PriceandBranch (PB) algorithm: In DCLRSA/BB, the number of variables x l may be very large since all possible lightpaths are include into the formulation. Consequently, the solution may be not attainable for larger problem instances. A way to decrease the computational effort is to make use of optimization decomposition methods. Column Generation (CG) is one of the techniques that allows to reduce the amount of variables (referred to as columns) in Linear Programming (LP) formulations [9]. In CG, the problem is initiated with a small, feasible list of admissible columns, which is then extended iteratively with new columns. A key element of CG is to formulate and solve a pricing problem, which concerns finding such a new column that, when included into the problem formulation, it leads to the improvement in the objective function value of the LP problem relaxation. If such column exists, it is included and the pricing problem is solved again, otherwise, the CG procedure is terminated. For demand d, the pricing problem of (1) consists in finding lightpath l = (p, c) terminated in a candidate DC node v V for which the reduced cost calculated as zl d = λ d e p s c πes η dv, using dual variables of formulation (1), is positive (i.e., zl d > 0). In our implementation, in each CG iteration, we check all lightpaths considered in the described above DCLRSA/BB approach and, for each d D, we include into L d the one with the largest positive zl d. The initial, feasible set of columns is obtained by running the DCL+RSA/MSF algorithm. PriceandBranch (PB) is a common heuristic approach which consists in: 1) generating columns (with the CG technique) and, after that, 2) running a BB algorithm for the obtained set of columns. In the remainder, we denote this approach as DCLRSA/PB. 2) Decomposing the problem (DCL+RSA): For large networks the solution of joint DCLRSA optimization problem may be not attainable. Therefore, we break the DCLRSA problem into (i) the data center location (DCL) and (ii) the routing and spectrum allocation (RSA) subproblems and solve them separately. In particular, we find a set of DC nodes with a DCL algorithm and then we run a RSA algorithm so that to provide, for each demand, a lightpath connection to a DC node. a) DCL algorithm: To find a placement of DCs, we solve the following ILP problem: min z = d D v V,v o d C dv x dv v V x dv = 1 d D (2a) x dv x v d D, v V (2b) x v R (2c) v x v, x dv {0, 1}. where C dv denotes the cost of connecting demand d to node v. The objective of Problem (2) is to minimize the overall cost of serving all demands. Constraint (2a) assures that each demand is connected to a DC node. Constraint (2b) assures that a DC is located in node v whenever at least one demand is connected to this node. The maximum number of DCs located in the network is limited by constraint (2c). Eventually, (2d) assures that variables are binary. We consider three options for calculating C dv, namely: minl: as the length of the shortest physical path (denoted as p dv ) between nodes o d and v (in km), mins: as the width of an admissible channel on path p dv for demand d, minsl: as the product of the width of an admissible channel on path p dv and the number of hops of p dv. If not mentioned otherwise, in the evaluation Section we apply the minsl option. (2d)
4 TABLE I COMPARISON OF ALGORITHM PERFORMANCE; T IN SECONDS. Scenario DCL+RSA/MSF DCL+RSA/PB DCLRSA/PB DCLRSA/BB Year D K S T L S T L S T L S T , (60% gap) > 12h , , , , , , (34% gap) > 12h , , , , , , , , (42% gap) > 12h b) RSA algorithms: In our study, we analyse the following two RSA algorithms: Most Spectrum First (MSF)  the algorithm is based on a greedy heuristic presented in [8]. In MSF, the demands are processed onebyone in a decreasing order of the size of requested spectrum. For each demand, a lightpath allocating the lowest possible slice index is selected from a set of precalculated lightpaths; in evaluation, we consider the same set of lightpaths as in DCLRSA/BB. Since in our problem the demand volumes are expressed in terms of bitrates, when ordering the demands we take into account the amount of spectrum that would be required on the shortest routing path to the nearest DC in the network. RSA with PriceandBranch  similarly as in DCLRSA/PB, we generate columns and then run a BB algorithm. Note that since the DC nodes are given (by the DCL phase), x v variables are fixed in formulation (1) and, as a consequence, the formulation converts to an RSA formulation with anycast routing. By combining the above DCL and RSA algorithms, we obtain two methods for providing solutions to DCLRSA, namely, DCL+RSA/MSF and DCL+RSA/PB. 4. NUMERICAL RESULTS In this Section, we evaluate the algorithms presented in Section 3. The evaluation is performed for EURO28 (28 nodes, 82 links) network topology (see Fig. 1a)). We investigate several scenarios referring to different numbers of data centers and different traffic damands. We analyze K {1, 3, 5} and we consider each network node is a candidate DC node. Demands are created for period using the CAGR (Compounded Annual Growth Rate) of 31% under the forecast presented in Cisco Global Cloud Index report. We assume that the starting overall volume in year 2012 is 1 Tbps. As in [10], this overall volume is split among network nodes, representing cities, according to a multivariable gravity model that takes into account the population and GPD (Gross Domestic Product) of the country the city belongs to. It should be noted that the population denotes not the particular city population but the population of the region that the city covers (e.g., country). If the obtained demand exceeds the value of 400 Gbps, it is divided proportionally into smaller demands. We apply the half distance law of [11] for calculating the channel width of a lightpath. In particular, the spectral efficiency (SE) depends on the routing path length (L) and it is equal to 1, 2, 3, and 4, respectively, for L less or equal to 3000 km, 1500 km, 750 km, and 375 km (as in [8]). For L > 3000 km, we consider SE = 0.8. Without loss of generality, we neglect the presence of guard bands separating two spectrum adjacent connections. We assume the frequency slice width is equal to s = 12.5 GHz. Consequently, the requested number of slices for a lightpath for demand d is calculated as n = h d /SE s. In Table I, we compare the algorithm performance. We report the objective value (S, i.e., the number of slices), computation times (T ), and the number of admissible columns (lightpaths), denoted as L. We can see that DCLRSA/BB, which involves a common BBbased approach for solving ILP problems, allows us to solve only small problem instances (with smaller demands) in moderate times (below 12 hours). The increase of traffic demands in subsequent years leads to the increase of spectrum requirements in the network (S). As a result, the number of x s and x es variables and, consequently, x l variables in formulation (1) increases. We can also see that the use of column generation in both DCLRSA/PB and DCL+RSA/PB approaches allows to reduce considerably the number of columns ( L ) when comparing to DCLRSA/BB and, by these means, to provide solutions for larger problem instances. Still, the PB approach achieves results close to the optimal ones. Moreover, the DCL+RSA decomposition of the problem, especially in the case of DCL+RSA/PB, provides good results in shorter times than in DCLRSA/PB. The difference between joint DCLRSA and decomposed DCL+RSA approaches is more pronounced in the scenarios with lower number of DCs and higer traffic demands.
5 Figure 1. a) EURO28 topology (link length in km); b)c) Comparison of cost functions in DCL. In Fig. 1b) and c) we focus on a comparison of different cost functions that can be used in the DCL heuristic algorithm. The results correspond to the number of slices (S) optimized with the DCL+RSA/PB algorithm. We can see that in both scenarios with 3 and 5 DCs the optimization based only on the physical distance between origin and DC nodes (i.e., minl) leads to worst results. As a conclusion, when optimizing spectrum usage in the network, the placement of DCs should also take into account spectrum requirements of lightpath connections. 5. CONCLUSIONS The main challenge in the application of anycasting in transport networks supporting the cloud computing traffic is that usually the data centers and network operators operate independently as separate business parties. Therefore, a multilayer oriented network management and crossstrata capabilities are indispensable to make anycasting efficient in cloud computing scenarios In this paper, we have addressed the problem of data center location with lightpath provisioning in Elastic Optical Networks. In particular, we have formulated an offline optimization problem in the form of the ILP model and proposed several heuristic algorithms to solve it. The default ILP solution method, i.e., Branchand Bound algorithm, in most cases has not been able to provide feasible solutions in a very long execution time (> 12 hours). However, the proposed heuristic algorithms making use of a Column Generation technique have yielded feasible and good (close to optimal) results for all experiments in relatively low time. The involvement of traffic demands when taking a decision regarding the data center placement (as in our DCLRSA problem formulation) may lead to different results in different years since traffic volumes are subject to change. Therefore, such optimization should be performed jointly for a number of consecutive time periods. The formulation and solution of such a problem is left for further study. ACKNOWLEDGMENTS This work has been supported by the Polish National Science Centre under grant agreement DEC2011/01/D/ ST7/05884 and grant Algorithms for optimization of routing and spectrum allocation in content oriented elastic optical networks which is being realized in years REFERENCES [1] Q. Zhang et al., Cloud computing: stateoftheart and research challenges, Journal of Internet Services and Applications, vol. 1, no. 1, pp. 7 18, [2] O. Gerstel et al., Elastic optical networking: A new dawn for the optical layer? IEEE Comm. Mag., vol. 50, no. 2, pp , [3] L. Contreras et al., Toward cloudready transport networks, IEEE Comm. Mag., vol. 50, no. 9, [4] K. Walkowiak and M. Klinkowski, Joint anycast and unicast routing for elastic optical networks: Modeling and optimization, in Proc. ICC, Budapest, Hungary, [5] M. Klinkowski and K. Walkowiak, Routing and spectrum assignment in spectrum sliced elastic optical path network, IEEE Commun. Lett., vol. 15, no. 8, pp , [6] B. Jaumard et al., Selecting the best locations for data centers in resilient optical grid/cloud dimensioning, in Proc. of IEEE ICTON, Coventry, England, [7] L. Velasco et al., Modeling the routing and spectrum allocation problem for flexgrid optical networks, Phot. Netw. Commun., vol. 24, no. 3, pp , [8] K. Christodoulopoulos et al., Elastic bandwidth allocation in flexible OFDM based optical networks, IEEE J. Lightw. Technol., vol. 29, no. 9, pp , [9] M. Pióro and D. Medhi, Routing, Flow, and Capacity Design in Communication and Computer Networks. Morgan Kaufmann, [10] A. Deore et al., Total cost of ownership of WDM and switching architectures for nextgeneration 100Gb/s networks, IEEE Comm. Mag., vol. 50, no. 11, pp , [11] A. Bocoi et al., Reachdependent capacity in optical networks enabled by OFDM, in Proc. of OFC, San Diego, USA, 2009.
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