Boutheina Dab
email: boutheina.dab@orange.com

Ilhem Fajjari
email: il-hem.fajjari@orange.com

Nadjib Aitsaadi
email: nadjib.aitsaadi@esiee.fr

Optimized Wireless Channel Allocation in Hybrid Data Center Network based on IEEE 802.11ad

Keywords: Data Center Network, MSDC, IEEE 802.11ad, channel allocation, graph coloring, col

The massive adoption of Cloud services is entailing a tremendous traffic transiting over data center networks (DCNs). Despite the impressive wired throughput, DCNs are suffering from network congestion. To deal with such resource limitation, first, we put forward a novel Hybrid Data Center Network (HDCN) architecture, based on CISCO's Massively Scalable Data Center (MSDC) model. HDCN enables high throughput wireless (IEEE 802.11ad) and wired (IEEE 802.3) transmissions. Second, we focus on one-hop inter-rack communications in HDCN and propose a novel wireless channel assignment algorithm. Our goal is to maximize the total throughput over the wireless and/or wired infrastructure. We formulate the problem as a minimum graph coloring. To address this NPhard problem, we propound two novel i) exact and ii) heuristic wireless channel assignment algorithms in HDCN denoted respectively by GC-HDCN and GH-GC-HDCN. The latter make use of i) column generation and ii) branch and price optimization schemes to resolve the Integer Linear Programming assignment problem while reducing computation time. Based on extensive simulations with QualNet simulator considering the full network protocol stack, the results obtained for both: i) uniform and ii) real Facebook's workload, show that our proposals outperform the related prominent strategies in terms of latency and throughput.

Introduction

Whilst no official standardization is yet delivered for 5G, experts assure that, the impressive proliferation of smart devices will lead to the explosion of traffic demand. Billions of connected users are expected to deploy a myriad of applications. In this respect, recent statistics elaborated by CISCO Visual Networking Index [START_REF]Cisco Visual Networking Index: Forecast and Methodology[END_REF] highlight that the annual global IP traffic will roughly triple over the next 5 years to reach 2.3 zettabytes by 2020. Unfortunately, Cloud computing infrastructure is struggling to resist this vertiginous growth.

This unprecedented traffic explosion as well as the continuous evolution of Cloud services mandate efficient scalable Data Centers (DCs). Indeed, the latter hold computation and storage resources.

To that end, major Cloud providers ramped up their investment in DCs that start supporting hundreds of thousands of servers. For instance, Google is estimated to host 0.9 millions of servers while the number of servers deployed by Microsoft DCs doubles every 14 months [START_REF]Mega-datacenter battle update: Internet giants continue to increase capex[END_REF].

Despite the impressive evolution of wired throughput, legacy three-tier Data Center Network (DCN) is unable to accommodate the traffic growth and has became prone to link congestion. Particularly, link oversubscription is significantly observed for inter-rack communications in DCN. In fact, intra-DCN traffic between top-of-rack switches is estimated to be 4 times higher than the incoming/outgoing traf-fic (i.e., inter-DCN) [START_REF] Ferraz | A high-performance two-phase multipath scheme for data-center networks[END_REF]. Over the last few years, various novel DCN architectures such as the CLOS-based architectures (FatTree [START_REF] Leiserson | Fat-trees: Universal networks for hardwareefficient supercomputing[END_REF] and VL2 by Microsoft [START_REF] Greenberg | Vl2: A scalable and flexible data center network[END_REF]) have been propounded to balance the load using multi-gigabytes wired links. However, these topologies are still facing congestion challenges. Indeed, deploying redundant multiple-gigabit wired links entails redundancy and wiring complexity. On the other hand, further augmenting the wired link capacity induces extra high costs. In fact, CISCO claims that the current power consumption of a 40 Gbps optics is more than 10X a single 10 Gbps.

To get rid of the aforementioned challenges, another innovative solution consists in augmenting the wired DCN with wireless links in order to provide extra bandwidth and improve network performance. Basically, wireless interfaces rely on IEEE 802.11ad protocol using the 60 GHz bandwidth. The solution deploys wireless antennas on topof-racks for inter-rack communications. This augmented architecture, called Hybrid DCN, will boost the network performance, since traffic can be forwarded over wireless and/or wired infrastructure. Actually, the hybrid DCN infrastructure is able to alleviate the congestion of oversubscribed links and thus improve QoS of applications. Contrarily to wired infrastructures requiring complicated routing and/or forwarding mechanisms to handle multi-hop communications, one-hop inter-rack communications are easily be achieved thanks to the transmission area and broadcast nature of one-hop wireless links. Furthermore, being more flexible, a wireless infrastructure can be simply deployed compared with classic wired networks. Finally, wireless communication is suitable for real DCN environment. In fact, the latter is free of obstacles and servers are not mobile.

In this work, we make use of the widespread millimeter wave wireless technology, operating in the spectrum between 57-64 GHz. In fact, this unlicensed band, commonly known as 60 GHz band, provides high multi-Gbps data rates. In that way, as explained in ETSI white paper [START_REF]Microwave and Millimetre-wave for 5G Transport[END_REF], mmWave technologies are able to fulfil all 5G requirements, such as deploying new mMTC, uRLLC services, Ultra-low transmission latency, process automation, etc. Specifically, we exploit the IEEE 802.11ad standard, providing 4 wireless orthogonal channels with a capacity of 6.7 Gbps. We envision a HDCN architecture based on Cisco's Massively Scalable Data Center (MSDC) model [START_REF]Massively Scalable Data Center (MSDC) Design and Implementation Guide[END_REF]. We augment the HDCN architecture by deploying the 2D beamforming wireless antennas. The latter are smart directional transceivers capable of transmitting signal in one single beam by targeting only the direction of the destination. Servers in the HDCN are regrouped into racks. Each rack has i) one Top-of-Rack (ToR) switch interconnecting servers through wired links and ii) four Transmission Units (TU) supporting IEEE 802.11ad [START_REF]IEEE Standard for Information technology-Telecommunications and information exchange between systems-Local and metropolitan area networks-Specific requirements-Part 11[END_REF]. Note that only four wireless channels are available and, hence, each TU is configured with a dedicated channel.

To deal with inter-rack communications in HDCN, we have already proposed in our previous work [START_REF] Dab | Online-batch joint routing and channel allocation for hybrid data center networks[END_REF] [20] a joint routing and wireless channel assignment algorithm, that efficiently routes and allocates resources for multi-hop flows. However, when the communicating racks are in the same transmission range, unfortunately our previous proposals promote the use of a multi-hop wireless path even when one single link is available. Although the obtained solution path is optimal, our previous proposals require high computation time before converging. In this regard, we focus on this paper on one-hop inter-rack communications in HDCN. We propose a new wireless resource allocation algorithm in order to maximize the total throughput while considering computation time.

It is worth noting that most DCN applications, such as map-reduce [START_REF] Cui | Channel allocation in wireless data center networks[END_REF], causing unbalanced traffic distributions, usually generate huge traffic demand with only a few stressed nodes. Moreover, the latter are not deterministically distributed which makes the set up of additional wired links unpractical. To get rid of such challenge, the main idea of our proposal is to maximize the proportion of traffic transiting over the wireless infrastructure and to transmit the rest over the wired infrastructure. In doing so, stressed nodes are less congested and hence the communication end-to-end delay is alleviated.

The wireless channel allocation problem in HDCN is formulated as a Minimum Graph Coloring Problem (Min-GCP).

Formally, an interference graph is generated where each i) node represents one communication, ii) edge denotes the presence of interferences between two communications, and iii) color is assigned to one wireless channel. The key insight of Min-GCP is to compute the minimum number of colors (i.e., chromatic number) required to color all the nodes in the graph while two adjacent cannot be assigned the same color. In fact, two neighbor communications are not allowed to use the same channel (i.e., color), otherwise interferences will occur. The Min-GCP is formulated as an Integer Linear Programming (ILP) optimization problem.

In this paper, we put forward a Centralized Controller (CC) scheduler which monitors the traffic in the HDCN and assigns the channel for inter-rack communications. The major contributions of this paper are summarized as follows:

• HDCN architecture: We design a HDCN architecture based on CLOS topology, following the CISCO's MSDC model and deploying 2D beamforming technique.

• Greedy-based approach:

We propose a heuristic-based wireless channel allocation approach in HDCN, named resource allocation algorithm based on Greedy-based Heuristic Graph Coloring in Hybrid Data Center Network (GH-GC-HDCN). The main idea is to reduce the computation time by generating new improving columns based on greedy heuristic instead of solving the pricing problem with B&C algorithm.

• Comparison with the optimal resolution:

We compare our heuristic solution GH-GC-HDCN to our previous optimal wireless channel allocation approach in HDCN [START_REF] Dab | A novel wireless resource allocation algorithm in hybrid data center networks[END_REF], named resource allocation algorithm based on Graph Coloring in Hybrid Data Center Network (GC-HDCN) 2 . Due to its NPhardness [START_REF] Held | Maximum-weight stable sets and safe lower bounds for graph coloring[END_REF], Min-GCP is resolved based on a column generation technique as follows. First, the ILP is relaxed and then solved to compute the fractional chromatic number. Afterwards, GC-HDCN makes use of Column Generation process which generates new columns based on the exact resolution of the pricing problem. GC-HDCN ensures the integrality of the solution thanks to Branch and Price algorithm.

Based on extensive simulations with QualNet simulator considering the full protocol stack, we evaluate the performance of both GC-HDCN and GH-GC-HDCN. First, we compare the performance of our proposal in both scenarios: omnidirectional and beamforming antennas. Then, we analyze the results of our strategies compared to the related work. We consider different traffic patterns in our experiments: 1) uniform traffic pattern based on Poisson distribution, and 2) Facebook DCN traffic workload [START_REF] Arjun | Inside the social network's (datacenter) network[END_REF]. The obtained results show that our proposed solutions outperform the related strategies. The remainder of the paper proceeds as follows. Section 2 succinctly review related work dealing with one-hop communication in HDCNs. In Section 3, we highlight the main features of the HDCN architecture. In Section 4, we present the motivations for addressing one-hop communications in HDCN. Then, in Section 5, the wireless channel allocation problem within HDCN is formulated. Afterwards, Section 6 details our two proposals. Network simulation environment and performance evaluation results are presented in Section 7. Finally, Section 8 concludes this paper.

Literature Review

In the present section, we, first, succinctly summarize the related channel allocation approaches in the Wi-Fi based networks. Second, we review the main wireless resource allocation strategies dealing with one-hop communications in hybrid DCN from an optimization perspective.

Channel allocation approaches for Wi-Fi based networks

In [START_REF] Vardhan | 60 ghz wireless links in data center networks[END_REF], the authors propounded a new fully wireless DCN topology arranging all racks in a single hexagonal instead of the classical row arrangement. They made use of IEEE 802.15.3.c standard [START_REF]IEEE Standard for information technology -telecommunications and information exchange between systems -local and metropolitan area networks -specific requirements[END_REF] to build the 60 GHz wireless links. Their approach makes possible the communication between adjacent racks, and between servers in the same rack, by adequately positioning the transceivers to form a polygon. Indeed, they enabled transceivers rotation (i.e., beam steering mechanism) in order to communicate with racks in different orientations using only point-to-point links. Note that this approach assumes that each rack has only two transceivers which limits the number of communications that can be simultaneously performed by a node.

In [START_REF] Gong | Channel assignment in multirate 802.11n WLANs[END_REF], the authors propound a new channel assignment strategy for multi-rate 802.11n wireless networks. Basically, their aim is to maxmimize the network throughput, while minimizing interferences. Therefore, they formulate the channel allocation problem as a throughput optimization problem, and propose a distributed throughput-maximizing channel assignment algorithm, named TMCA to solve it. [START_REF] Abraham | Adaptive channel allocation algorithm for wifi networks[END_REF] proposes an adaptive channel assignment scheme for Wi-Fi based networks. Particularly, the authors address the interference challenge during the channel assignment. Their proposed solution favorises the use of available channels with reduced interference effects. Note that this approach does not consider scalability constraint and, hence, can not be adopted for large scale networks.

Although minimum graph coloring has been widely used in literature for channel allocation in the context of IEEE 802.11n and g networks, the existing solutions are not suitable for large scale DCN. In fact, usually an exact resolution is used to solve the problem which is very time consuming for real large scale hybrid DCN.

Channel Allocation strategies for hybrid DCNs

In [START_REF] Cui | Channel allocation in wireless data center networks[END_REF], the authors propose a Genetic-based approach that tackles the channel allocation problem while involving traffic demands. To do so, they propose improved crossover and mutation operators. Nevertheless, this approach randomly generates the initial population which may notably impact the quality of the final solution. Simulation results show that both strategies improve the performance compared to Wired-DCN strategy. Note that the latter corre-sponds to a fully-wired DCN infrastructure where traffic is transmitted only using Ethernet links as in [START_REF] Al-Fares | A scalable, commodity data center network architecture[END_REF]. This approach has been improved in [START_REF] Cui | Dynamic scheduling for wireless data center networks[END_REF] where the wireless transmission scheduling is formulated as an optimization problem. Then, an optimized channel allocation solution is computed based on a relaxation-rounding algorithm.

In [START_REF] Cui | Wireless data center networking[END_REF], the authors formulate the one-hop channel allocation problem in HDCN as an optimization scheme while maximizing the utility of the network. The latter is defined as the product of the traffic amount transiting through the wireless infrastructure and the distance between the source and the destination. Then, a heuristic approach that repetitively executes Hungarian Algorithm, denoted by MaximumWeighted-HDCN, is proposed. In doing so, the best solution is greedily reached.This approach assumes that a wireless communication can be transmitted simultaneously through multiple links and proceeds with a dynamic programming approach to address this challenge. Note that the channel allocation decision depends on the already transmitted traffic which may impact the quality of solution in case of sporadic traffic demand.

Several other omni-directional strategies have been proposed in [START_REF] Zhua | Efficient hybrid multicast approach in wireless data center network[END_REF] [START_REF] Kassing | Beyond fat-trees without antennae, mirrors, and disco-balls[END_REF] in order to enhance DCN performance.

Several methods deal with the augmentation of DCN with 60 GHz wireless links using beamforming technique. In [START_REF] Zhou | Mirror mirror on the ceiling: Flexible wireless links for data centers[END_REF], the authors formulate the wireless channel assignment problem based on graph coloring, and solve it using a greedy heuristic. We notice that their approach is nonpreemptive, as it keeps unchanged the channels of ongoing communications. The main limitation of this strategy is that it requires a mechanical rotation mechanism to frequently rotate antennas inducing, hence, an extra delay.

In [START_REF] Zhou | Mirror mirror on the ceiling: Flexible wireless links for data centers[END_REF][36], the authors evaluate the 3D beamforming primitive in data centers, and prove that it enhances wireless links capacity and reduces interference. Indeed, they show that this technique eliminates link blockage by using ceiling reflectors. Consequently, any two racks in the DCN can communicate directly with each other in only one hop link, with no need to routing.

In [START_REF] Shan | Relieving hotspots in data center networks with wireless neighborways[END_REF], the authors propose to augment the wired DCNs with wireless links in order to alleviate congestion for elephant traffics. Mainly, it combines wired and wireless links in the same communication path. However, they only add the wireless links in the neighborhood of the source node, while wired links are used to deliver traffic between relay ToRs of the final destination. Furthermore, this approach minimizes hotspots by providing a new logical topology with IP address assignment and traffic engineering scheme.

In this paper, we envision a new hybrid wireless/wired Data Center Network architecture built over a three-stage CLOS topology and following CISCO's MSDC model. Moreover, we propound a novel wireless channel allocation mechanism, denoted by GH-GC-HDCN, which addresses the onehop communication while enhancing network performance. Different from [START_REF] Kassing | Beyond fat-trees without antennae, mirrors, and disco-balls[END_REF], our DCN architecture is augmented with wireless communication links, to avoid the high optical deployment cost. Unlike [START_REF] Cui | Wireless data center networking[END_REF], we consider unsplittable DCN traffic carried through a single channel. Besides, in contrast to the channel scheduling mechanism in [START_REF] Cui | Dynamic scheduling for wireless data center networks[END_REF] which accords a high priority to ongoing traffics, our approach does not differentiate between incoming communications and aims to enhance overall QoS required by applications. Unlike [START_REF] Zhua | Efficient hybrid multicast approach in wireless data center network[END_REF], we focus, in this work, on unicast inter-rack communications, while considering the limited number of available wireless channels in HDCN. In addition, our approach does not limit the use of wireless links to only the source node as it is suggested by [START_REF] Shan | Relieving hotspots in data center networks with wireless neighborways[END_REF]. In fact, a single wireless link can handle the traffic between the source and destination. Moreover, unlike our previous work [START_REF] Dab | A novel wireless resource allocation algorithm in hybrid data center networks[END_REF], we make use in this paper of 2D beamforming technique to alleviate interference effects. To do so, we do not assume fixed antenna directions, in order to maximize wireless infrastructure use. Moreover, we overcome the rotation delay induced with horn antennas in [START_REF] Zhou | Mirror mirror on the ceiling: Flexible wireless links for data centers[END_REF] by using switched beam antennas.

Hybrid Data Center Network Architecture

Following on from our prior work [START_REF] Dab | A novel wireless resource allocation algorithm in hybrid data center networks[END_REF], we investigate, in this paper, the use of wireless infrastructure to augment the wired DCN architecture. In doing so, traffic can be carried over wireless and/or wired links.

In order to mimic real data center environment, we design a HDCN architecture based the CISCO's Massively Data Center (MSDC) model [START_REF]Massively Scalable Data Center (MSDC) Design and Implementation Guide[END_REF]. In fact, MSDC is a promising framework that has proven its capability of supporting huge volume of traffic [START_REF]Cisco's Massively Scalable Data Center -Network Fabric for Warehouse Scale Computer[END_REF]. MSDC hierarchy is built over a three-stage CLOS topology which is a specific kind of layered DCN topology. Note that CLOS inhibits the transition of traffic through horizontal links (i.e., inside the same layer). Typically, it comes with two layers of switches: i) Leaf layer (i.e., layer 1), where the leafs represent the ToRs switches, and ii) Spine layer (i.e., layer 2), formed by aggregation switches referred as spines. Each leaf switch is linked to all spines. Accordingly, in a DCN of leafs and spines, there are × wired links. The main reason behind this link redundancy is to enable multi-path routing and to mitigate oversubscription caused by the conventional link state OSPF routing protocol. To that end, our HDCN makes use of Equal Cost Multi-Path (ECMP) protocol to maximize the load balancing of wired links' usage by dividing the traffic through multiple equal cost routes.

The wireless infrastructure is based on IEEE 802.11ad. This standard, presented by the TGad working group as the enabler of next generation Multi-Gbps WiFi, benefits from the available spectrum in the unlicensed 57-66 GHz band. It offers 4 orthogonal wireless channels with center frequencies fixed at 58.32, 60.48, 62.64 and 64.8 GHz respectively. The capacity of each wireless channel reaches 6.7 Gbps over a short range. Consequently, the whole data center network can be considered as Personal Basic Service Set (PBSS). Indeed, PBSS is IEEE 802.11ad-specific wireless LAN in which stations directly communicate to each other (i.e., Ad hoc network, no need to access point) [START_REF]IEEE Standard for Information technology-Telecommunications and information exchange between systems-Local and metropolitan area networks-Specific requirements-Part 11[END_REF]. Specifically, each node in PBSS is denoted by Directional Multi-Gigabit Sta- Based on this specification, we propose to deploy a Wireless Transmission Unit (WTU) composed of a set of 4 wireless directional antennas. Each transceiver is, hence, assigned to one wireless channel. Note that the 4 transceivers in WTU are independent, due to channel orthogonality, and can be simultaneously exploited. In doing so, each rack in the data center can communicate using both the wired (i.e., ToR) and/or wireless interfaces. Moreover, we assume grid physical topology of racks within a DCN.

In order to mitigate interference effects in HDCN, we deploy a beamforming mechanism. Unlike omni-directional antennas radiating signal in uniform way (circle), smart directional transceivers are capable of transmitting signal in one single beam by targeting only the direction of the destination. The signal coming from potential interfering antennas is either not received or very weak. Consequently, the Signal Interference to Noise Ratio (SINR) is maximized. We assume the geometric signal propagation model [START_REF] Shin | On the feasibility of completely wireless datacenters[END_REF] based on a spherical coordinate system with origin the transmitting antenna as shown in Fig. 1(a). The receiver antenna is characterized by radius , azimuth as shown in Fig. 1.(a). Note that we assume 2D beamforming and hence elevation is equal to 0. In this work, we deploy switchedbeam antennas [START_REF] Zhu | Cutting the cord: a robust wireless facilities network for data centers[END_REF] to avoid the extra delay induced by me-chanical rotations such as array antennas [START_REF] Zhou | Mirror mirror on the ceiling: Flexible wireless links for data centers[END_REF]. As depicted in Fig. 1.(b), a switched-beam antenna is characterized by an array of beams. Each one covers an angle of 2Π∕ . According to the geographical position of the destination, the transmitting antenna switches to the beam covering the destination. The receiving antenna senses the signal on all the sectors and exploits only the one achieving the maximum gain. Fig. 2 illustrates our HDCN architecture.

Motivations of one-hop communications in HDCN

Traffic within data centers is rapidly growing to reach roughly 70% of the total network traffic [START_REF] Bilal | Energy aware network operations[END_REF]. Intra-DC traffic is known for its both unbalanced and unpredictable aspects. More specifically, for private data centers, a large range of racks may be placed closely enough to each other that communicating racks are in the same wireless transmission area. In such a case, inter-rack communications can be performed in one single hop.

Moreover, despite the relatively limited transmission range of the 60GHz communications, 10m, it is capable of supporting indoor wireless transmissions, as proven in [START_REF] Cui | Wireless link scheduling for data center networks[END_REF].

In fact, interference between 60GHz wireless links is quickly reduced when they operate in the same geographic proximity, as claimed in [START_REF]The Benefits of 60 GHz Unlicensed Wireless Communications[END_REF]. Therefore, this reduction enables higher spectrum space reuse and enhances consequently the wireless channel utilization.

In our previous work [START_REF] Dab | Online-batch joint routing and channel allocation for hybrid data center networks[END_REF][20], we have already proposed a joint routing and wireless channel assignment algorithm, that efficiently routes and allocates resources for multi-hop traffic demands. However, to deal with one-hop communications, our approach JRCA-HDCN promotes the use of multi-hop wireless path. Unfortunately, in spite of its effectiveness in terms of latency and throughput, this strategy may require high computation time. Indeed, experimentation analysis show that sometimes our previous multi-hop solution JRCA-HDCN needs few minutes to converge.

These results motivate us to address the time complexity challenge for one-hop communications in HDCN. In fact, it is crucial to evaluate the fastness level of one-hop approach while guaranteeing a close-to optimal solution. It is worth noting that the main novelty of this paper is not related to the graph coloring problem formulation which is already put forward in our previous work. However, compared to the latter, the added-value of this journal is threefold:

Firstly, we propose a Greedy-based heuristic to solve the column generation algorithm for graph coloring problem. Indeed, with the advent of 5G, traffic is very heavy, and hence many communication flows are likely to be carried simultaneously. Consequently, graph coloring problem, which is NP-hard, based on exact resolution requires high computation time. This is the main insight behind our proposed heuristic solution.

Secondly, we deploy in this paper 2D beamforming technique (not omnidirectional) in order to alleviate interference effects. In fact, switched-beam antennas are smart direc-tional transceivers capable of transmitting signal in one single beam by targeting only the direction of the destination. In doing so, transmission range is enhanced and interference is reduced.

Thirdly, to assess the efficiency of our proposal, we run our experiments under realistic conditions. Besides uniform workload, we use real workload collected from Facebook data centers [START_REF] Arjun | Inside the social network's (datacenter) network[END_REF]. Note that this paper includes 40% of extension compared with our previous preliminary work.

Problem Formulation

In this section, we formulate the one-hop wireless channel allocation problem in HDCN. We will first describe the inter-rack wireless communication model. Then, we will formulate the problem based on a Minimum Graph Coloring.

Hybrid Data Center Network Model

Each Wireless Transmission Unit (WTU) denoted by  possesses 4 IEEE 802.11ad transceivers/antennas denoted by { 1 , 2 , 3 , 4 }. It is worth noting that the number of antennas depends on the number of orthogonal available channels in IEEE 802.11ad standard. We recall that each  is deployed over the top of the rack. The inter-rack communications are performed using the { } and/or the gigabit wired (ToR) switches. Our goal is to maximize the number of communications transiting over wireless links so as to alleviate the congestion over the wired infrastructure.

We model the set  encompassing the ongoing wireless communications and the new incoming communication request (i.e.,  = { }, ≠ ), as an undirected graph = (, ). Each node ∈  refers to one communication from the transmitter  to the receiver  . It is obvious that for each communication = ( ,  ),  is located within the IEEE 802.11ad transmission range _ of  . An edge = ( , ) ∈  exists only if is susceptible to interfere with or vice-versa. We model the interference between two wireless communications and as follows: i) transmitter  interferes with receiver  or ii) transmitter  interferes with receiver  .

We make use of the Friis signal transmission model, while considering the directional (i.e., non-uniform) propagation model. In fact, we assume that the data center environment is free of obstacles and that radio antennas are deployed on the top of racks. Indeed, unlike wired DCNs, where row-based arrangement of racks is well adopted because of client space utilization, wireless DCNs require novel rack placement. In this regard, a recent work proposes a new rack placement solution [START_REF] Celik | Optical wireless data center networks: Potentials, limitations, and prospects[END_REF] to free the communication space inside optical DCN from obstacles. This obstacle-free DCN topology is also suitable for wireless DCN infrastructure where LOS is required between transceivers. Note that in this work, we conceive our DCN topology to be obstacle free, but we did not focus on rack placement.

The receiving signal power sent by to is equal to:

( , , ) = + ( ( , , )) + 20 log 10 4 --

(5.1)
where i) is transmitting signal power, ii) ( ( , , )) is the gain of transmitting and receiving antennas and ( , , ) refers to the azimuth angle between antennas, iii) (meter) is the wavelength, iv) (meter) is separating distance between and , v) is the path loss effects, and vi) and are respectively the noise factor and the implementation loss fixed in IEEE 802.11ad standard. Similarly to [START_REF] Cui | Channel allocation in wireless data center networks[END_REF], we adopt the interference disk model which is independent of the antenna technique. Indeed, it depends only on the transmitting node position and the active channels. The Signal to Interference Noise Ratio (SINR) between transmitter and destination on the channel is equal to: Given the static topology of racks in the HDCN, we initially compute the table containing all the signalto-noise ratio values between all the racks for different antenna's orientations (i.e., beams). It is worth noting that entries in this table are opportunistically refreshed, during the ongoing wireless traffic transmissions. In fact, we measure signal strength received from active sending racks at different antenna's orientations. Then, signal measurements and transmitter antennas' orientations are shared by the centralized controller using the wired infrastructure. Note that thanks to table, it is possible not only to compute interference at each beam, but also to determine the best antenna orientation for two communicating ToRs. Moreover, since in our architecture both data and ACK packets are transmitted over wireless infrastructure, then for each communicating ToR, two beams are identified. By the incoming of each new communication, the involved antennas are configured to the suitable beam that ensure the maximum gain. Then, the entries of table are refreshed while taking into consideration the new and ongoing antenna orientations of all the racks in the HDCN. The interference/communication graph  is, hence, re-constructed.

( , , ) = ( , , ) ∑ ≠ ( , , ) (5.2

Channel allocation problem

Our objective is to maximize the proportion of transiting over the wireless links in order to alleviate the conges-tion level of wired network. To do so, an efficient wireless channel allocation is required. Since the decision is made at the arrival of each new request, then our approach allows the switching of wireless channels at the physical layer for the wireless communications. In doing so, a wireless communication can change its wireless channel and continues transmitting over the new (wireless/wired) assigned one.

Taking interference between wireless communications into account makes channel allocation process more challenging. To deal with such a challenge, channels are dynamically assigned while considering the interference between communications. Moreover, because the number of available channels is limited (i.e., 4), then our approach carries all the remaining unassigned communications through the wired links.

We formulate the wireless channel allocation problem as a Minimum Graph Coloring Problem (Min-GCP) [START_REF] Mehrotra | A column generation approach for graph coloring[END_REF] in such a way that each node ∈  (i.e., communication) will have exactly one color, while ensuring that any two adjacent nodes have different colors. Note that, henceforth, the objective is to minimize the number of colors used to cover all graph nodes.

We define a stable set as a subset of  which is composed of pairwise non-adjacent nodes. We recall that each node in  corresponds to an inter-rack communication between two wireless transmission units, while an edge in  corresponds to the interference link between two communications. Therefore, a stable set is composed of non-interfering communications. A maximal stable set refers to a stable set that is not strictly included in any other stable set. Let  denote the set of all maximal stable sets in . It is straightforward to see that all the nodes belonging to the same stable set can be assigned one color since they are not interfering. In doing so, the group of wireless communications from the same stable set make use of the same wireless channel. Consequently, the interference between inter-rack communications is eliminated during channel allocation process. Accordingly, our objective is to calculate the minimum number of stable sets, covering all nodes in . This number, denoted by (), corresponds to -coloring and is called the chromatic number of .

To calculate (), we formulate our problem as an Integer Linear Programming (ILP) based on the independent set formulation.

( ) = min ∑ ̂ ∈  ̂ subject to: ∀ ∈ , ∑ ̂ ∈  ̂ .1 { ∈ ̂ } ≥ 1 ∀ ̂ ∈ , ̂ ∈ {0, 1}
Problem 1: Min-GCP -Wireless channel allocation problem where i) ̂ is a binary variable indicating whether ̂ ∈  is assigned a color or not and ii) 1 { ∈ ̂ } is the boolean function that is equal to 1 if ∈ ̂ and to 0 otherwise. The first constraint guarantees that each node ∈  (i.e., communication) has at least one color (i.e., wireless channel). Then, only one color will be selected among those assigned to .

Note that the number of variables can importantly scale up with the size of the graph. Moreover, computing the chromatic number (), and hence resolving Min-GCP, is an NP-Hard problem, as it is proven in [START_REF] Held | Maximum-weight stable sets and safe lower bounds for graph coloring[END_REF] [START_REF] Gualandi | Exact solution of graph coloring problems via constraint programming and column generation[END_REF]. Therefore, making use of computational exact methods would not be the efficient way to handle the scalability challenge. Although minimum graph coloring has been widely used in literature for channel allocation in the context of IEEE 802.11n and g networks, the existing solutions are not suitable for large scale DCN. In fact, usually an exact resolution is used to solve the Min-GCP problem which is very time consuming for real large scale hybrid DCN. Therefore, an efficient scalable solution is required in order to provide an optimized channel assignment (i.e., minimum number of colors) while alleviating convergence time.

Proposal: GH-GC-HDCN

As explained above, solving the minimum coloring problem using integer linear programming resolution is time consuming due to the high number of variables. In fact, the graph's size is proportional to the set of maximal stable sets, . Thus, the generation of  requires high computation time. To address such dimension challenge, our approach decomposes the problem by first addressing a subset of variables and then progressively generating new variables when needed. This is the main idea of column generation optimization technique [START_REF] Mehrotra | A column generation approach for graph coloring[END_REF].

In this section, we detail our proposed strategy named Graph Coloring in Hybrid Data Center Network (GH-GC-HDCN) based on the column generation optimization technique and Greedy heuristic.

The main objective is to converge to an optimized solution of the minimum coloring problem. The key insight of GH-GC-HDCN is to generate the minimum number of maximum-sized stable sets required to color the whole graph. Note that each stable set consists of a group of wireless communications using the same wireless channel (i.e., same color).

GH-GC-HDCN proceeds as following. First, Problem 1 (i.e., Min-GCP -Wireless channel allocation problem) is relaxed, by transforming the integer variables ̂ into linear ones (i.e., 0 ≤ ̂ ≤ 1), and then resolved while considering an initial subset of maximum stable sets  generated by a Greedy Heuristic (GH) method. We define Restricted Master Problem (RM-Problem) as the linear relaxation of Min-GCP problem, which assumes only a subset of maximum stable sets. Secondly, the above RM-Problem is solved based on an exact (i.e., simplex) method. Note that the optimal dual variables corresponding to the constraints of Min-GCP are used to define a new sub problem called Pricing problem. The latter is solved in order to determine whether it would be judicious to add new variables (i.e., stable set) to  . If the Pricing problem resolution provides an improving stable set, then the latter is added to  and the RM-Problem will be again resolved. It is worth pointing out that in large scale DCN, generating new potential stable sets based on the exact resolution of the aforementioned pricing problem, requires high computation time [START_REF] Held | Maximum-weight stable sets and safe lower bounds for graph coloring[END_REF] [START_REF] Barnhart | Branch-and-price: Column generation for solving huge integer programs[END_REF]. Therefore, the main novelty of this work, compared to our previous GC-HDCN approach [START_REF] Dab | A novel wireless resource allocation algorithm in hybrid data center networks[END_REF], , is to solve Pr-ILP in a combined heuristic/exact way. Typically, GH-GC-HDCN keeps generating new optimal stable sets as far as the number of ongoing communications in the HDCN is less or equal to a specific threshold value ℎ .

Otherwise, when the number of ongoing communication flows is greater than ℎ , the transmission/interference graph becomes dense. Therefore, our approach makes use of a greedy heuristic, GH+ in order to generate new columns.

The process will be repeated until no new improving columns (i.e., variables) can be found. If the final solution of RM-Problem is integer then Min-GCP converges. Otherwise, a Branch and Price algorithm is carried out to enforce integrality and thus find the best integer solution. GH-GC-HDCN is summarized in Fig. 3. Hereafter, we detail each stage of GH-GC-HDCN .

Generation of initial solution

This stage consists in generating an initial subset  of maximal independent sets required to color the graph.  is built using the Greedy Heuristic GH [START_REF] Mehrotra | A column generation approach for graph coloring[END_REF]. The key idea of GH is to sort the nodes ∈  in descending order according to their connectivity degree. Then, the highest weighted node in  is selected as an initial element of the first maximal independent set ̂ 0 . Afterwards, remaining nodes are sequentially added to ̂ 0 as long as the resulting set is still independent. Once ̂ 0 is built, it is added to  . Next, the list of remaining nodes is sorted again. The process is iteratively repeated to create the rest of maximal independent sets { ̂ } and stops when each node belongs to at least one stable set. Note that  = { ̂ } and a node ∈  may belong to several independent sets ̂ .

Resolution of the relaxed RM-Problem

Once  is generated, it is used as an input of relaxed RM-Problem. It is worth noting that the number of variables (i.e., columns) corresponds to the size of  . On the other hand, the number of constraints is equal to || (i.e., set of communications in ). Hereafter, the definition of the relaxed RM-Problem:

min ∑ ̂ ∈ ̂ subject to: ∀ ∈ , ∑ ̂ ∈ ̂ .1 { ∈ ̂ } ≥ 1 ∀ ̂ ∈  , ̂ ≥ 0 Problem 2: Relaxed Restricted Master Problem (RM-Problem)
Our approach resolves the aforementioned relaxed RM-Problem using Simplex algorithm [START_REF] Barnhart | Branch-and-price: Column generation fro solving huge integer programs[END_REF].

Resolution of the pricing problem

The main purpose is to gradually enrich the set of maximal stable sets  . The idea is to adequately generate and add new columns (i.e., maximal stable sets) in order to converge to the optimal solution. We determine whether it is interesting to expand  by adding new improving stable sets or not. To do so, we search in an iterative manner for the stable sets having negative reduced costs. Note that the reduced cost of a stable set  is defined as:

( ) = 1 -Π( ) = 1 - ∑ ∈  (6.3)
where the coefficients correspond to the optimal dual variables of the relaxed RM-Problem constraints calculated by Simplex algorithm in the previous stage (Section 6.2).

It is straightforward to see that generating a new stable set  with a negative cost is equivalent to the resolution of pricing problem with an obtained objective function greater than 1. Otherwise, we can conclude that there is no more improving independent sets. Consequently, solving relaxed RM-Problem over the current  is equivalent to solving Min-GCP over .

max ∑ ∈ ⋅ subject to: ∀( , ) ∈ , + ≤ 1 ∀ ∈ , ∈ {0, 1}
Problem 3: Pricing problem: Pr-ILP It is clear to see that the above pricing problem, Pr-ILP, is an Integer Linear Programming (ILP) problem, since the variables are integer and the objective function is linear.

The resolution of Pr-ILP aims to find the maximal stable set that might improve the relaxed RM-Problem. It is worth noting that in today's large-scaled data centers, traffic is very heavy, and hence many ongoing communications are likely to be carried simultaneously. Consequently, the wireless transmission/interference graph  scales up with the traffic density. In such a case, researchers claim that generating new potential stable sets based on the exact resolution of the aforementioned pricing problem, requires high computation time [START_REF] Held | Maximum-weight stable sets and safe lower bounds for graph coloring[END_REF] [START_REF] Barnhart | Branch-and-price: Column generation for solving huge integer programs[END_REF]. To get rid of this complexity challenge, we propose, in this work, to solve Pr-ILP in a combined heuristic/exact way. The key insight of our approach is to keep generating new optimal stable sets as far as the number of ongoing communications in the HDCN is less or equal to a specific threshold value ℎ . Actually, in such a case, the graph  is still small-sized, and hence optimal stable sets can be computed, in a reasonable time, based on Branch-and-Cut (B&C) algorithm. Otherwise, when the number of ongoing communication flows is greater than ℎ , the transmission/interference graph becomes dense. Therefore, our approach makes use of the greedy heuristic, GH+ in order to generate new columns. In fact, finding many feasible maximal stable sets with negative cost is sufficient. Since GH+ is fast, it is carried out recursively to generate the new columns. Hereafter, we will detail both GH+ and B&C algorithms.

B&C algorithm

To solve the ILP formulation of Pr-ILP problem for small instances of the graph , our approach makes use of B&C algorithm. To do so, it basically relies on two main techniques: i) Cutting planes and ii) Branch-and-Bound, to reach efficiently the optimal solution. First, the algorithm relaxes the ILP problem by transforming all the integer variables , ∈  into continuous ones. Second, the Relaxed Linear Problem (R-LP) is solved based on Simplex algorithm. When an optimal solution is obtained, the algorithm checks whether some variables have fractional values. If such variables exist, then the algorithm cuts away parts of the solution set by adding a new linear constraint which is satisfied by all integer variables but violated by the fractional ones. Afterwards, the relaxed problem is resolved again in order to eliminate the fractional solutions while keeping the integer ones. Note that the process is repeated to improve the problem relaxation and hence become closer to the integer solution. The algorithm stops when no cutting plane can be found, or a fully integer solution is obtained. If no additional cutting planes can be generated, but the obtained solution is not integer, then B&C resorts to Branchand-Bound (B&B). The latter searches for the cutting planes in an efficient way in order to rapidly reach the optimal solution. To do so, it proceeds by partitioning the problem into new restricted regions. Then, it constructs a tree enumerating all the possible variable settings. Only some specific branches of the tree, that are expected to produce optimal/close to optimal values, are explored. The new linear problems are hence solved with Simplex algorithm and the process is repeated.

Pricing Greedy heuristic

We make use of a variant of GH defined in section 6.1 denoted by GH+. The pseudo-algorithm of GH+ is summarized in Algorithm 1. The new Greedy heuristic generates at most promising (i.e., negative reduced cost) maximal independent sets (i.e., columns) and the weight function of each node takes into account the calculated cost in the relaxed RM-Problem (Lines 1 -4). Formally, the weight of a node is defined as:

= √ 2 ⋅ 2 (6.4)
Algorithm 1: Pricing stage:

GH+ 1 Inputs: , { }, 2 Output:  ← ∪ {  } 3 for ∈  do 4 ← Connectivity degree of 5 ← √ 2 ⋅ 2 6  ← Descending sort of nodes ∈  w.r.t weights 7  ← ∅ 8 ← 1 9 ← 10 while = do 11  ← ∅ 12 ← Head() 13  ← { } 14  ← ∖{ } 15 for ∈  do 16 if disjoint ( , ̂ ) then 17  ←  ∪ { } 18 if  ≠ ∅ then 19 Calculate (  ) 20 if (  ) < 0 then 21  ←  22  ←  ∪  23 ← + 1 24 if > then 25 ← 26 if ( = ∅) then 27 ← 28 else 29 ←
where is the connectivity degree of in  and denotes its dual value (Line 5). To do so, GH+ sorts the nodes ∈  in a descending order according to their weights (Line 6). Then, the highest weighted node in  is selected as an initial element of the maximal independent set ̂ , ∈ {1, ..., } (Line 12). Afterwards, remaining nodes are sequentially added to ̂ as long as the resulting set is still independent (Lines 15 -17). Once ̂ is built, it is added to  (i.e.,  =  ∪ { ̂ }). Note that GH+ is iteratively repeated to create the rest of maximal independent sets { ̂ }. The process stops if maximal independent sets have been generated (Line 24) or when no new column can be found (Line 28). Similarly to the initial solution, a node ∈  may belong to several independent sets ̂ .

Afterwards, selected stable sets are used as the input of relaxed RM-Problem.

The column generation process is repetitively performed until no new column with negative reduced cost can be generated. The column generation process is summarized Algorithm 2: Column generation process

1 Inputs: ,  , ℎ 2 Output:  , {  },  ∈  ,  ∈ [0, 1] 3 ← 0 4 ← 5 while = do 6  ∈  , ∈  7 {  }, { } ← Solve relaxed-RM-Problem (,  ) 8 if ( () ≥ ℎ ) then 9  ← GH+ (, { }, ) 10 else 11  ← Exact-Pricing-B&C (, { }) 12  ←  13 if (  = ∅) then 14 ← 15  ←  16 else 17  +1 ←  ∪  18 ← + 1
in Algorithm 2, which combines the resolution of relaxed RM-Problem (Line 7) and pricing problem (Line 9). Once the process converges (i.e., no more improving column, Lines 13 -15), if the resulting solution of RM-Problem is integer (i.e., ∀  ,  ∈ {0, 1}), then we can conclude that the solution is optimized [START_REF] Mehrotra | A column generation approach for graph coloring[END_REF]. Otherwise, the integrality has to be enforced. To do so, Branch-and-Price algorithm is performed to compute the integer solution.

Branch and Price stage

The key insight of this stage is to enforce the integrality of variables  . To do so, we make use of Branch and Price (B&P) [START_REF] Held | Maximum-weight stable sets and safe lower bounds for graph coloring[END_REF]. It is a combination of B&B and column generation [START_REF] Mehrotra | A column generation approach for graph coloring[END_REF] methods. It performs pretty good when the lower bound is tight which is the case of our problem [START_REF] Held | Maximum-weight stable sets and safe lower bounds for graph coloring[END_REF]. Note that B&P is carried out only if no new columns (i.e., stable sets) can be added and the solution of the relaxed RM-Problem is not integer. Branching rules are defined in a way ensuring that i) the sub-problem tackled at each node in the solution tree is itself a graph coloring problem solved by column generation method and ii) the integer optimal solution is exactly supported by one branch in the solution tree.

B&P proceeds as following. First, two overlapping stable sets  1 and  2 are considered.  1 is selected such as is typified by the highest fractional value of  1 . The highest fractional value corresponds to the value  1 -⌊  1 ⌋ which is close to 12 .  2 is randomly selected such as  1 ∩  2 ≠ ∅. Two nodes 1 and 2 are then randomly selected such as: It is clear to see that thanks to the above branching, resulting subproblems do not define any additional constraints compared to the master problem (Problem 2). The two subproblems (i.e.,  and  ) are added to the tree of branch and price in which the root is an abstract node. The resolution of coloring subproblems and may add new columns (i.e., maximal stable sets) to tighten the relaxation of the relaxed RM-Problem and hence enforce integrality. Thanks to column generation (see Algorithm 2), we resolve the coloring problem of  . If the solution is integer then the process is converged. Otherwise, we resolve the coloring problem of  . Similarly, the convergence is reached if the solution is integer. Otherwise two other graphs, denoted  and  , are generated from the graph  or  and added to the tree of branch and price algorithm. Thanks to Depth First Search algorithm, the leaf node (i.e., sub-problem) in the B&P tree having the lowest objective function value (i.e., ∑ (  )) is elected. The same process is recursively repeated to the elected node until integrality is reached. It is worth noting that the convergence is ensured thanks to the Branch and Bound algorithm. B&P stage is summarized in Algorithm 3.

Performance evaluation

In this section, we assess the performance of our two proposals, GC-HDCN and GH-GC-HDCN, based on extensive network simulations. First of all, we describe the main stages of our implementation, namely i) IEEE 802.11ad standard integration in network simulator QualNet, ii) HDCN architecture design, and iii) development of our proposed decision algorithms and simulation environment set up.

Then, we define the performance metrics to assess our proposals and the related strategies. Afterwards, we evaluate the network performance of IEEE 802.11ad.

Algorithm 3: Branch and price

1 Inputs: ,  , {  },  ∈  ,  ∈ [0, 1] 2 Output:  , {  },  ∈  ,  ∈ {0, 1} 3 ← 4  ←  5  ← abstract root node 6 while = do 7 Select  1 ∈  ∶ | | |  1 -⌊  1 ⌋ -1 2 | | | = min  ∈ ( | | |  -⌊  ⌋ -1 2 | | | ) 8 Select randomly  2 ∈  ∶  1 ∩  2 ≠ ∅ 9 Select randomly 1 ∈  1 ∩  2 10 Select randomly 2 ∈  1 ⧵  2 ∪  2 ⧵  1 11 ← Build-Same (, 1 , 2 ) 12 Column-Generation ( ,  ,  1 , {  }) 13 if {  } are integer then 14 ← 15  ←  1 16 else 17 ← Build-Diff (, 1 , 2 ) 18 Column-Generation ( ,  ,  2 , {  }) 19 if {  } are integer then 20 ← 21  ←  2 22 else 23  ← Add-Sub-Problem ( ) 24  ← Add-Sub-Problem ( ) 25 Node-Tree ← Leaf-Depth-First- Search ( , "minimal", ∑ (  )) 26  ← Father(Node-Tree,  )
Finally, we discuss the effectiveness of our proposals by comparing it with the most prominent related strategies, found in literature: i) Genetic-HDCN [START_REF] Cui | Dynamic scheduling for wireless data center networks[END_REF][16], ii) MaximumWeighted-HDCN [START_REF] Cui | Wireless data center networking[END_REF], and iii) Wired-DCN.

Note that the latter is a fully Ethernet strategy. In order to evaluate the effectiveness of our approaches, GC-HDCN and GH-GC-HDCN, and prove their efficiency in Hybrid DCNs, we proceed as follows. First, we implemented the main features of IEEE 802.11ad standard in QualNet 3 . Note that the latter is an event driven network simulator based on C++ language. It is widely used by both industrial and research community. The design and the development of new protocols in whether wireless, wired or hybrid network infrastructures are enabled thanks to its modularity and layer based architecture.

We set the propagation parameters and rate table based on the IEEE 802.11ad. We assume that all the antennas have the same transmission power which is fixed to 10 dBm. We configure the QualNet physical layer with the free-space propagation model, by setting the Friis parameter to 2. We set the receive sensitivity _ ℎ to -78 dBm and _ ℎ to 10, as indicated in IEEE 802.11ad standard [START_REF]IEEE Standard for Information technology-Telecommunications and information exchange between systems-Local and metropolitan area networks-Specific requirements-Part 11[END_REF]. Furthermore, 4 wireless channels are available according to IEEE 802.11ad specification, with a bandwidth of 2.16 GHz and running frequencies ranging from 57 GHz to 66 GHz. To deploy beamforming technique, we associate 4 switchedbeam antennas, composed of 8 beams, to each ToR. Besides, we build our large scale data center based on CISCO's MSDC model, containing 256 racks [START_REF]Massively Scalable Data Center (MSDC) Design and Implementation Guide[END_REF], in which we: i) use OSPF protocol for traffic routing and ii) implement ECMP protocol in order to balance the load over the wired network. Each rack typically contains from 20 to 40 servers and the overall infrastructure includes more than 5000 servers. The geographic dimensions are 60m×60m. Servers of the same rack are interconnected through a leaf switch (i.e., ToR). Each leaf is connected to 4 spine switches.

As in [START_REF]Massively Scalable Data Center (MSDC) Design and Implementation Guide[END_REF], ToRs (i.e., leaves) are connected to servers 3 http://www.scalablenetworks.com/products/Qualnet/ via 1 Gbps links. Moreover, spine and leaf switches communicate through 10 Gbps links. In fact CISCO has found out that using multiple 10 Gbps links instead of a single 40 Gbps link alleviates power consumption. Indeed, the current power consumption of a 40 Gbps optics is more than 10X a single 10 Gbps. Similarly to [START_REF] Cui | Channel allocation in wireless data center networks[END_REF], we set the propagation delay of wired links to 2 s. The noise factor and implementation loss values are respectively set to 10, and 5, as it is given by IEEE 802.11ad specification [START_REF]IEEE Standard for Information technology-Telecommunications and information exchange between systems-Local and metropolitan area networks-Specific requirements-Part 11[END_REF]. Finally, we implemented i) our two variants of wireless resource allocation algorithm, GC-HDCN and GH-GC-HDCN, based on C++ language and CPLEX4 solver and ii) the related strategies.

Simulation setup

We run our experiments under different workloads. The traffic follows a Constant Bit Rate (CBR) model for which we set the inter-arrival packet time to 6 -seconds and the CBR packet size to 6214 Bytes. Note that the latter value is calibrated in a way that no fragmentation occurs during the IP encapsulation process. In fact, the maximum size of IEEE 802.11ad frame is 7995 Bytes [START_REF]IEEE Standard for Information technology-Telecommunications and information exchange between systems-Local and metropolitan area networks-Specific requirements-Part 11[END_REF]. We make use of UDP transport protocol to transmit the inter-rack traffic. We run the simulation for 100 communications. Each performance value of the implemented strategies is equal to the average of 6 simulations. Furthermore, our simulation results are always presented with confidence intervals corresponding to a confidence level of 95%.

Performance metrics

To assess the efficiency of our proposal and thus demonstrate the effectiveness of utilizing wireless transmissions in DCN, we proceed in a two-step approach. First, we designed various simulation scenarios using Qual-Net in an effort to target the performances of IEEE 802.11ad under different conditions. To do so, we evaluate the transmission range _ . Then, in order to evaluate purposes in our experiments, we consider the following metrics:

1) R : is the Residual wireLess traffic. It corresponds to the remaining amount of traffic to be transmitted over the ongoing wireless communications. It is straightforward to see that R evaluates the capacity of a channel allocation algorithm to carry out its traffic over the wireless infrastructure. It reflects, hence, the efficiency of the resource allocation strategy in using the wireless infrastructure. Consequently, the higher the value of R , the more the use of wireless resources is efficient.

2) D: is the cumulative delay of the network. In other words, it defines the cumulative transmission delay of all finished communications in the network. Let  denote the number of finished communications in the network and the delay spent by a communication to be transmitted. D is formulated as follows:

D = ∑  =1
3) D a : is the Average Delay in the network, which defines the average transmission delay per traffic request.

4) T: is the total throughput of the network. It corresponds to the cumulative transmission throughput of the traffic carried through the hybrid DCN. Let be the ℎ finished communications in the network at the departure time . Let be the volume of traffic transmitted by . 0 is the arrival time of the first communication 0 in the network. If  represents the number of finished communications, T can be calculated as:

T = ∑  =1 (  -0 )
5) T a : is the Average Throughput in the network, which defines the average transmission throughput obtained per traffic request.

7) S : is the average Spatial Spectrum Reuse of the ℎ channel, ∈ {1, … , 4}.

8)  : represents the computation time of the algorithm.

IEEE 802.11ad performance

We target the performances of IEEE 802.11ad under: i) omni-directional antenna communication mode and ii) beamforming technique mode.

To do so, we evaluate the transmission range using the following scenario. We consider 2 WTUs  1 and  2 communicating through a wireless channel . We start by fixing the distance separating  1 and  2 to 1 meter. Then, we progressively increase this distance by moving  2 away from  1 while evaluating the loss rate of the established communication. We repeat this experiment for each wireless channel , ∈ [START_REF]IEEE Standard for information technology -telecommunications and information exchange between systems -local and metropolitan area networks -specific requirements[END_REF][START_REF]IEEE Standard for Information technology-Telecommunications and information exchange between systems-Local and metropolitan area networks-Specific requirements-Part 11[END_REF]. The main goal is to estimate the transmission range _ of each wireless channel. We notice that transmitted frames from  1 are correctly received by  1 as long as the distance separating the two WTUs is lower than a coverage distance. Moreover, it is worth noting that _ is enhanced when beamforming is deployed. Indeed, this technique concentrates the transmission energy on a single beam instead of uniformly radiating the signal in a circle. On the one hand, the minimum coverage distance, _ , for omni-directional antennas is 8.2 meter, which corresponds to the channel 4 (i.e., highest frequency), while the maximum _ is measured to 9.1 meter and corresponds to 1 (i.e., lowest frequency). On the other hand, when switched-beam antennas are used, we notice that value can reach 10.8 meter (channel 1 ) and is limited to 9.7 meter (channel 4 ).

GC-HDCN and GH-GC-HDCN performances

To assess the efficiency of our proposals, we consider three main scenarios. First, Omni-Beam scenario, in which we only consider our exact solution GC-HDCN and we study the HDCN performance for both: i) omni-directional and ii) directional antennas cases. The objective is to prove the importance of beamforming technique and motivate its use in our HDCN architecture. Secondly, Uniform-Load scenario in which traffic distribution follows a Poisson process. In the third scenario, Real-Load scenario in which we consider real workload traces of Facebook's DC. Note that in both the second and third scenarios, we asses the performance of our exact (i.e., GC-HDCN) and heuristic (i.e., GH-GC-HDCN) approaches, while considering directional switched-beam antennas on the communicating WTUs. Note that for all these scenarios, the sending WTU is randomly selected using a uniform distribution in the set of racks deployed in the HDCN. Then, the destination WTU is randomly chosen based on a uniform distribution among the racks that are in the same transmission range of the sender.

Omni-Beam scenario

In this scenario, similarly to [START_REF] Hamedazimi | Firefly: A reconfigurable wireless data center fabric using free-space optics[END_REF], the transmission demands arrival follows a Poisson process with set to 4 communications per second. First, we evaluate our proposal by considering one-hop inter-rack communications where ToRs are equipped with omni-directional antennas, radiating signals in a uniform way, as in the related approaches, Genetic-HDCN [START_REF] Cui | Channel allocation in wireless data center networks[END_REF] and Hungarian-HDCN [START_REF] Cui | Wireless data center networking[END_REF]. Next, we resort to deploying switched-beam directional antennas on each ToR of the HDCN and we study the impact of the beamforming technique on the efficiency of our approach GC-HDCN. The objective is to prove the utility of beamforming mechanism to enhance the wireless resources usage and improve the network performance.

For both of the aforementioned deployment cases, we calculate, first, at each communication departure the amount of residual traffic (i.e., R ) circulating in the network. In doing so, we evaluate the ability of the resource allocation strategies to efficiently hand out ongoing communications.

It is clear to see through Fig. 4(a) that GC-HDCN enhances the use of wireless resources, particularly when the beamforming technique is used. Note that for a communication departure, a higher amount of residual wireless traffic with a lower proportion of wired traffic proves an enhanced use of wireless resources. In doing so, we reduce the network delay and ensure a higher throughput by alleviating the hot wireless/wired links. Note that only when omnidirectional antennas are used, Hungarian-HDCN outperforms our approach. This proves the important role of beamforming in reducing interference and hence the channels use.

To explore the impact of the wireless channel allocation strategies on the cumulative network performances, we evaluate the cumulative delay of the network, D. The results are illustrated in Fig. 4(b). It is straightforward to see that, by the end of simulations, GC-HDCN alleviates the total transmission delay by respectively 26.21%, 67.77% and 88.59% compared with Hungarian-HDCN, Genetic-HDCN and Wired-DCN. On the other hand, we notice that the use of beamforming technique enables our approach to further alleviate  by 57.99%. However, we notice that at the beginning, our solution has similar results to related methods. This is due to the fact that Hungarian-HDCN approach basically allocates long alive requests on wired infrastructure, which take more time to finish. TABLE 1 illustrates the average transmission delay of the 100 communication demands.

We remark that our approach GC-HDCN improves D a by 70.11%, 10.17% and 94% compared respectively to Hungarian-HDCN, Genetic-HDCN and Wired-DCN. In addition, the use of beamforming mechanism further alleviates the average delay by 24.18%.

The obtained results corroborate those depicted in Fig. 4(c) and confirm that GC-HDCN maximizes the total network throughput. In fact, Fig. 4(c) depicts the total network throughput, T, according to the number of finished requests.

It is worth pointing out that GC-HDCN achieves a higher total throughput which is improved respectively by 11.74%, 31.46% and 51.34% compared with Hungarian-HDCN, Genetic-HDCN and Wired-DCN related strategies. Besides, the throughput evolution is more noticeable when switched-beam antennas are deployed, in which case T is further enhanced by approximately 38.55%. Note that the total throughput decreases by the end of the simulation, because wired communications leave lastly the network. This results in a high delay and consequently reduces the final throughput. We notice, moreover, that Hungarian-HDCN ensures by the beginning of simulations a higher throughput compared to GC-HDCN. Indeed, it basically allocates long alive requests on wired infrastructure, which take more time. Contrarily, our approach minimizes the total traffic on wired network. These results confirm those of the average network throughput presented in TABLE 1. It is clear to see that our strategy GC-HDCN improves T compared to the three related approaches, when beamforming is enabled by 12.42%. With omni-directional antennas, GC-HDCN enhances T compared to Genetic-HDCN and Wired-DCN.

In order to gauge the efficiency of the wireless resource use, we evaluate the Spatial Spectrum Reuse S for each channel .

TABLE 2 represents the average spectrum reuse. It shows that GC-HDCN ensures high S value varying between 3 and 6 for the four wireless channels, while it is equal to almost 2 for the Genetic-HDCN strategy. This weak channel re-utilization strongly impacts the performance of the communications as illustrated in the above results. Note that the re-use of the spectrum by GC-HDCN is, in general, enhanced thanks to the beamforming technique.

Uniform-Load scenario

Based on the first scenario results, it is clear that the beamforming technique notably improves the HDCN performance in terms of delay, throughput and spectrum reuse. Therefore, we deploy, in this scenario, only switched-beam directional antennas. Then, we consider a uniform load pattern generated based on the Poisson process, as in [START_REF] Hamedazimi | Firefly: A reconfigurable wireless data center fabric using free-space optics[END_REF], with set to 4 communications per second. First, we run experiments to gauge the efficiency of both our heuristic-based solution, GH-GC-HDCN, and exact solution, GC-HDCN, while evaluating the computation time. Second, we compare our approaches to the related strategies Hungarian-HDCN, Genetic-HDCN and Wired-DCN.

The computation time  of the column generation process is a key parameter of GC-HDCN since it impacts the solution quality in term of optimality. Therefore, it is very crucial to evaluate the fastness level of GC-HDCN while guaranteeing a close-to optimal solution. In this stage, we run experiments for 100 inter-rack communication requests, and evaluate the average computation time for two cases: i) GH-GC-HDCN, for which the pricing problem is generated based on the greedy heuristic and ii) exact resolution of the pricing problem. The results of the average computation time,  , are illustrated in TABLE 3. Deep experimental analysis of GH-GC-HDCN show that when the size of the graph scales up and becomes greater than 10, the computation time of the pricing problem using B&C algorithm dramatically increases. Therefore, we set the threshold ℎ to 10. GH-GC-HDCN keeps generating new columns based on an exact pricing problem as long as the size of graph is less than ℎ . Otherwise, our heuristic-based approach switches to the use of the Greedy algorithm GH+. It is straightforward to see that the use of the heuristic solution to generate new columns alleviates the time complexity. Hereafter, we will compare the network performance of the above approaches to the related strategies. Similarly to the above Omni-Beam scenario, we evaluate herein the residual resources as well as the cumulative throughput when beamforming mechanism is deployed in the HDCN. It is clear to see through Fig. 5(a) that both GC-HDCN and GH-GC-HDCN enhance the use of wireless infrastructure while reducing the traffic allocated through wired links. Consequently, we notice that, as shown in Fig. 5(b), GC-HDCN ensures the lowest cumulative delay compared to the other strategies. Moreover, it is worth noting that our heuristic-based solution keeps a lower network delay compared to the other related strategies. These results corroborate with those of the average delay, illustrated in TABLE 4. In fact, we remark that both GC-HDCN and GH-GC-HDCN ensures the lowest value of D .

Similarly, GC-HDCN further enhances the total throughput compared to the related approaches, thanks to the use of the switched-beam antennas.

Fig. 5(c) depicts the total network throughput, T, according to the number of finished requests.

It is obvious to see that our proposal GH-GC-HDCN improves the throughput respectively by 67.29% and 70.83% compared with Genetic-HDCN and Wired-DCN. Moreover, GH-GC-HDCN and Hungarian-HDCN perform equally throughout most of the simulation and then our approach enhances T by 53.33% at the end of simulation. Note that the total throughput noticeably drops by the end of simulations because the wired communications leave lastly the network. Since they entail high transmission delay, their departure dramatically decreases T.

In order to further study the impact of our methods on resource usage, we evaluate the average spectrum re-use per channel. TABLE 5 shows that GC-HDCN ensures an S varying between 6 and 3 for the four wireless channels, while it varies between 6 and 1 for our heuristic-based approach GH-GC-HDCN. Although the latter does not guarantee the same usage rate of different channels, it succeeds to enhance S compared to Genetic-HDCN approach. Note that this efficient channel re-utilization strongly impacts the performance of the communications as illustrated in Fig. 5(b) and Fig. 5(c). Moreover, the strong reuse of the spectrum is enhanced thanks to the beamforming technique. In this scenario, we consider a real load traffic, using the recent workload traces of Facebook's DC [START_REF] Arjun | Inside the social network's (datacenter) network[END_REF]. In fact, Facebook monitoring system, fbflow, has collected, in 2015 for a period of 24-hours, samples of traffic patterns inside the production clusters 5 . Facebook has made accessible flow workload of some applications, namely: Hadoop, Web-servers, and Database. The loads have been measured with a uniform per-packet sampling, with a rate of 1:30. Note that each line of the workload traces includes many information as, the traffic type (i.e., intra-rack, inter-rack, and inter-DCN), the IP addresses of communicating racks, the traffic volume, etc. We manipulate workload traces to obtain for each flow its corresponding information: i) total measured traffic, ii) start time, iii) traffic type. In our simulations, we consider the inter-rack traffic generated by Hadoop, since it is considered to be the heaviest [START_REF] Arjun | Inside the social network's (datacenter) network[END_REF]. For each flow, we associate the communicating racks in Facebook DC to a source and destination node randomly generated in our HDCN topology based on uniform distribution. The traffic volume corresponds to the total traffic measured from the workload traces.

Similarly, we proceed as follows. We evaluate first the residual wireless and wired traffic transiting in the HDCN. We notice through Fig. 6(a) that solution GC-HDCN promotes the use of wireless infrastructure which alleviates congestion on ToR switches.

In this regard, the total network delay of the Hadoop flows is reduced by our approach compared to the related strategies. Typically, Fig. 6(b) shows that GC-HDCN alleviates D in general. At the end of simulation, D is reduced 5 https://research.fb.com/data-sharing-on-traffic-pattern-insidefacebooks-datacenter-network/ by approximately 19.8% and 8.9% compared respectively to Hungarian-HDCN and Genetic-HDCN. Consequently, the cumulative network throughput T is enhanced by our proposal with a rate of 11.3%, 5.4% and 36.31% compared to Hungarian-HDCN, Genetic-HDCN and Wired-DCN.

The above results corroborate with the average delay and throughput illustrated in TABLE 6. Indeed, GC-HDCN reduces D by 6.69% and 93% compared respectively to Genetic-HDCN and Wired-DCN, while it ensures approximately similar average delay compared to Hungarian-HDCN. In addition, our strategy enhances T by 3.2%, 7.5%, and 75% respectively compared to Hungarian-HDCN, Genetic-HDCN and Wired-DCN.

Intuitively, the above results affirm those of the instantaneous spatial spectrum reuse. Actually, as depicted in TA-BLE 7, our approach ensures a higher spectrum reuse compared to Genetic-HDCN. Whereas, Hungarian-HDCN shows comparable S values to our method.

Conclusion

In this paper, we addressed the problem of one-hop wireless communications in hybrid data center networks. To alleviate traffic congestion, we augmented the CISCO MSDC wired topology of DCNs with wireless infrastructure based on IEEE 802.11ad so as to enhance network performance. Additionally, we have deployed the 2D beamforming technique in order to minimize interference effects and leverage wireless infrastructure use. We addressed the problem of wireless channel allocation for one-hop inter-rack communications in HDCN. Our objective is to optimize the wireless channel allocation by: i) maximizing the number of flows using wireless infrastructure, while ii) avoiding interference effects. In doing so, congestion is alleviated and network performance is enhanced in DCN. We formulated our NP-Hard problem as a Graph Coloring statement and we made use of both Column Generation technique and Branch-and-Price algorithm to resolve it while considering computation time constraint. To solve the optimization problem, we proposed a new wireless channel allocation mechanism, named GH-GC-HDCN. Extensive network simulations conducted within QualNet simulator, for both uniform and real Facebook's DC traffic patterns, show that our proposal enhances data center performances and outperforms the most prominent related strategies in terms of: i) total network delay, ii) total network throughput, and iii) spectrum spatial reuse.

Figure 1 :

 1 Figure 1: Switched-beam antenna model

Figure 2 :

 2 Figure 2: Hybrid Cisco MSDC architecture of a data center network

Figure 3 :

 3 Figure 3: Flowchart of GC-HDCN

Figure 4 :

 4 Figure 4: Omni-Beam scenario

Figure 5 :

 5 Figure 5: Uniform-Load

Figure 6 :

 6 Figure 6: Real-Load

Table 1

 1 Omni-WTU scenario: Average network metrics

		D	T
	GC-HDCN (beamforming)	6.96 ± 0.25%	178.50 ± 20.44%
	GC-HDCN (omni)	9.18 ± 1.36%	156.33 ± 21.42%
	Genetic-HDCN	30.72 ± 6.89%	117.96 ± 23.89%
	Hungarian-HDCN	10.22 ± 2.03%	168.45 ± 22.14%
	Wired-ECMP-HDCN	332.46 ± 3.15%	8.37 ± 0.18%

Table 2

 2 Omni-Beam scenario: Average Spectrum Spatial Reuse

		S 1	S 2	S 3	S 4
	GC-HDCN (beamforming) 6.11 ± 0.29% 4.54 ± 0.31% 1.54 ± 0.19% 3.27 ± 0.29%
	GC-HDCN (omni)	1.89 ± 0.18% 4.22 ± 0.29% 5.24 ± 0.23% 3.13 ± 0.25%
	Genetic-HDCN	2.75 ± 0.6%	2.89 ± 0.14% 2.80 ± 0.17% 2.97 ± 0.17%
	Hungarian-HDCN	0.98 ± 0.10% 2.81 ± 0.22% 4.52 ± 0.30% 5.54 ± 0.27%
	7.1. Simulation Environment and Methodologies
	7.1.1. Experiment Design

Table 3

 3 Average computation time 

	GC-HDCN	GH-GC-HDCN
	 (sec) 169.9 ± 36.17%	63.32 ± 5.63

Table 4

 4 Uniform-Load: Average network metrics

	D	T

Table 5

 5 Uniform-Load: Average Spectrum Spatial Reuse

		S 1	S 2	S 3	S 4
	GC-HDCN	6.11 ± 0.29% 4.54 ± 0.31% 4.54 ± 0.31% 3.27 ± 0.29%
	GH-GC-HDCN	4.64 ± 0.37% 6.92 ± 0.30% 2.64 ± 0.26% 0.91 ± 0.11%
	Genetic-HDCN	2.60 ± 0.15% 3.09 ± 0.15% 2.69 ± 0.17% 2.87 ± 0.15%
	Hungarian-HDCN	1.0 ± 0.11%	2.63 ± 0.24% 4.27 ± 0.29% 6.60 ± 0.32%

Table 6

 6 Real-Load: Average network metrics

		D	T
	GC-HDCN	3.90 ± 0.24%	13.11 ± 2.28%
	GH-GC-HDCN	4.1 ± 0.39%	13.61 ± 2.58%
	Genetic-HDCN	3.84 ± 7.41%	12.69 ± 2.30%
	Hungarian-HDCN	4.18 ± 0.14%	12.12 ± 2.28%
	Wired-ECMP-HDCN 57.45 ± 4.46%	3.19 ± 0.13%

Table 7

 7 Real-Load: Average Spectrum Spatial Reuse

		S 1	S 2	S 3	S 4
	GC-HDCN	0.39 ± 0.03% 0.82 ± 0.03% 0.59 ± 0.03% 1.04 ± 1.004%
	GH-GC-HDCN	0.48 ± 0.07% 0.54 ± 0.03% 0.83 ± 0.06%	0.79 ± 0.01%
	Genetic-HDCN	0.65 ± 0.03%	0.78 ± 0.01	0.64 ± 0.025	0.63 ± 0.02
	Hungarian-HDCN	0.31 ± 0.02	0.61 ± 0.03	0.82 ± 0.03	1.04 ± 0.004
	7.4.3. Real-Load scenario

Preliminary results of GC-HDCN are published in IEEE MASS'15

∈  1 ∩ 

and 2 ∈  1 ⧵  2 ∪  2 ⧵  1 .As in[START_REF] Mehrotra | A column generation approach for graph coloring[END_REF], we define the following new coloring graph subproblems:

http://www-01.ibm.com/software/commerce/optimization/cplexoptimizer


Sébastien Duchêne
email: sebastien.duchene@umontpellier.fr

Jeffrey D Sachs

Thye Wing

Naoyuki Woo

Farhad Yoshino

Taghizadeh

Handbook of Green Finance (1200 words) For Submission in Ecological Economics (word version)

Achieving the objectives of the Paris Agreement requires developing financial tools, and completely re-examining the way finance works, so as to reallocate capital to low-carbon projects that promote energy transition without delay. New green finance tools and concepts, such as green bonds, green banks, carbon markets, green central banking or community-based green fund, have been widely developed in these last few years to promote economic growth and improve the environment. Banks, asset management companies, pension funds, insurers and households are gradually including environmental criteria in their financial portfolio choices. However, investments in green assets are far from growing sufficiently to meet a two-degree global warming limit, and a key contradiction remains for the time being unbridgeable: How to attract financial investors to greener projects, often characterized by a higher risk profile and lower profitability than

polluting projects with no environmental constraints? How can governments, public agencies and economic agents reduce the risk and increase the return of green assets to encourage investments in green activities? The purpose of this Handbook of Green Finance is to discuss these issues by detailing the barriers to financing green assets and the energy transition, providing solutions through the discussion of new financial instruments, the appropriate role of financial institutions, fintechs and investors, and debating effective public policies and regulatory strategies to be implemented, with some focus on Asian countries. The book is divided into 12 parts with 29 chapters, and is written by a reputable mix of researchers, policy-makers and professionals.

The first, introductory, part points out the decline of investments in renewable energy in 2017, and notes that many countries continue to invest massively in fossil fuels, without any significant foreseeable change in trend. At a microeconomic level, financial institutions and private investors still favor profitability of investments and low risk, which makes them deviate from green projects. Therefore, the editors call for the promotion of innovative financial instruments and new public policies to support green finance, taking advantage of all the available financial avenues that could shift capital to ecological transition.

The second part describes the barriers in Asian countries that inhibit the expansion of green projects: underdeveloped financial markets and venture capital, limited governmental funds to support green transition, the reluctance of risk averse private banks constrained by stringent capital regulations to finance projects with lower returns and higher risks, the limited availability of advanced technologies and finally the high cost of switching from fossil to clean energies. Many of these obstacles are similar to those found in Europe and America.

Since public funds are not sufficient to lead the ecological transition, part 3, the most interesting section of the book in my opinion as a finance researcher, discusses opportunities to develop private investments in green assets. An exciting point is that authors reconsider the classical valuation models of financial assets by adding an environmental dimension to the traditional components of risk and return. That is, individual utility functions integrate a "green value". This paradigm shift is a major step to evaluate financial assets and provides the opportunity for green products to compensate for their low level of profitability, by valuing the positive externality that they produce. To compel investors to embrace this shift, another chapter proposes central banks to be the major players. In addition to fighting inflation or financial instability, developing sustainability could be part of their responsibilities by promoting low-cost green financing of commercial banks, and by restricting the funding of polluting assets. Central banks could develop micro-prudential regulations, integrating ecological dimensions into bank credit risk management, requiring banks' disclosure and transparency on their own climate risk, and decreasing reserve requirements of banks that would finance green assets. From a macro-prudential point of view, the introduction of climate-related stress tests, the requirements of differentiated bank capital based on environmental criteria, or lower interest rates to bank loans for green projects, could drastically change capital allocation towards ecological projects.

Part 4 notes the difficultly of financing green projects, due to an overestimation of credit risk, the complexity in understanding the projects and estimating the technological risks, the lack of information on and track record of these new businesses, and the regulatory constraints that restrict private bank lending. The authors note that new policies can lower this funding cost through credit enhancement, for instance in the use of guarantees or insurance. Part 5 outlines the way green bonds, which have grown significantly in recent years and show promise, are used to finance green projects on the financial markets. The authors econometrically illustrate that, despite a widespread belief, the cost of financing green bonds is often not higher than conventional bonds, which should stimulate the growth for green instruments.

Part 6 promotes carbon capture and storage and carbon pricing to support the transition to "zero emission energy" and details the financial barriers that need to be addressed to facilitate its deployment. Part 7, which is of special interest for environmental economists who are concerned about progress in green finance, makes a practical link with part 3 and surveys the changes in practice of many institutional investors. These stakeholders provide a significant portion of the capital invested in financial markets and are beginning to play a major role in financing the ecological transition. They now begin to include environmental criteria in their asset allocation strategy, through positive and negative screening of companies, sustainability ratings of assets or active ownerships, and they invest in new green asset classes. New policy initiatives are also discussed, such as the "standardization of environmental impact metrics" for companies, the differentiated capital requirement for bank loans depending on how green the enterprise is, public private partnerships or Green Investments Banks.

Parts 8 and 9 address how fintechs, blockchain technology or big data can contribute to the green transition, as well as public programs and financing schemes implemented by some countries to support green projects and technologies. Part 10 discusses Community-Based Green Finance as a new tool to finance small scale green energy projects.

Finally, Parts 11 and 12 analyze the impacts of public policies, such as feed-in tariffs and loans, which seem to be fairly effective in some situations, or taxes, grants and subsidies, on the increase of investments in green energy. Public policies pursued in Asian countries are discussed in depth, considering their efficiency, weaknesses, obstacles, and achievements. Potential solutions are also presented. These last two parts illustrate that there is still a huge amount of work to comply with the Paris Agreement. This book suggests that the increase in green private investments will be supported by reducing the cost of capital and the risk of these projects. For this to be achieved, central banks, governments, and international institutions need to work in a coordinated framework and use new monetary tools to be combined with public policies.

Overall, this book is an exciting contribution on how to achieve the challenging targets of the Paris Agreement. It can be read as a whole or only for some specific chapters, depending on the reader's preferences. Environmental economists, financiers, and more broadly all academics, students or practitioners, who wish to learn about recent developments in green finance, will be able to explore what will certainly be the future of finance.