Piotr Lechowicz
email: piotr.lechowicz@pwr.edu.pl

Massimo Tornatore

Adam Włodarczyk

Krzysztof Walkowiak

Fragmentation Metrics in Spectrally-Spatially Flexible Optical Networks

Keywords: spectrally-spatially flexible optical networks, network fragmentation, network optimization, routing, space, and spectrum allocation

Spectrally-spatially flexible optical networks (SS-FONs) are proposed as a solution for future traffic requirements in optical backbone networks. As SS-FONs operate within flex-grid, the provisioning of lightpaths spanning multiple frequency slots results in spectrum fragmentation, especially in presence of dynamic traffic. Fragmentation, in turn, may lead to blocking of dynamic requests due to the lack of sufficiently-large free spectral windows. In this paper, to reach a better understanding of fragmentation in SS-FON, we extend several metrics used in (single-core) elastic optical networks to measure the fragmentation in SS-FONs. Next, we apply these metrics to a dynamic-routing algorithm with the goal of minimizing bandwidth blocking. Finally, we analyze the impact of spatial continuity constraint (SCC) on the network fragmentation. Simulations run on two representative network topologies show that the root mean square factor metric yields the best performance in terms of blocking when compared to other analyzed metrics.

Introduction

According to various statistics, network traffic has been growing in last two decades 30% to 60% per year, depending on the nature and penetration of services offered by network operators. Since there is no indication that this growth trend will cease anytime soon, in near future we will face a 'capacity crunch' consisting in incremental exhaustion of available capacity in optical transport networks [START_REF] Winzer | Optical networking beyond wdm[END_REF][START_REF]Spatial multiplexing in fiber optics: The 10x scaling of metro/core capacities[END_REF]. One possible solution is space division multiplexing (SDM) -a new optical transmission approach designed to scale the capacity of wavelength division multiplexing (WDM) network and flex-grid elastic optical network (EON) technologies. SDM enables parallel transmissions of a number of co-propagating spatial modes (fibers, cores, or modes) in appropriately designed optical fibers. The concept of spectrally-spatially flexible optical networks (SS-FONs) combines SDM and EONs technologies, and offers several advantages such as substantial increase in transmission capacity, extended flexibility in resource management due to the introduction of spatial domain (spatial modes), and potential cost savings due to the possibility of resource sharing and the use of integrated devices [START_REF] Marom | Switching solutions for WDM-SDM optical networks[END_REF][START_REF] Saridis | Survey and evaluation of space division multiplexing: From technologies to optical networks[END_REF][START_REF] Shariati | Realizing spectrally-spatially flexible optical networks[END_REF][START_REF] Klinkowski | Survey of resource allocation schemes and algorithms in spectrally-spatially flexible optical networking[END_REF].

One of the potential drawbacks of the flex-grid is the fact that provisioning of lightpaths satisfying continuity and contiguity constraints and using multiple relatively-small (e.g., [START_REF] Liu | 3d elastic optical networks in temporal, spectral, and spatial domains with fragmentation-aware rssma algorithms[END_REF].5-GHz) frequency slices instead of a fixed larger channel spacing (e.g., 50 GHz) introduces spectrum fragmentation, especially in dynamic traffic context. The key negative consequence of high fragmentation is that dynamic requests can be blocked due to lack of sufficient spectral resources to provision the requests. Therefore, properly addressing the fragmentation problem can allow network operator to allocate more requests in the network. One possible way to mitigate spectrum fragmentation consists in performing periodic or ondemand spectrum defragmentation. However, defragmentation entails complex operational procedure to avoid traffic interruptions, and several studies have instead suggested to to limit fragmentation by performing fragmentation-aware routing, spatial mode, and spectrum allocation (RSSA) [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF][START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF][START_REF] Sugihara | Dynamic resource allocation for immediate and advance reservation in space-division-multiplexing-based elastic optical networks[END_REF].

Several research problems related to fragmentation in EONs have been already investigated (see survey papers [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF][START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF]). However, only few papers have so far focused on fragmentation in SDM optical networks. In [START_REF] Fujii | On-demand spectrum and core allocation for reducing crosstalk in multicore fibers in elastic optical networks[END_REF], a core classification method is proposed for multi-core fibers (MCFs), where each fiber core is dedicated to realize requests of certain size. A similar approach is presented in [START_REF] Tode | Routing, spectrum, and core and/or mode assignment on space-division multiplexing optical networks[END_REF], where prioritized areas in the frequency domain are used, instead of dedicated spatial mode resources. In [START_REF] Liu | 3d elastic optical networks in temporal, spectral, and spatial domains with fragmentation-aware rssma algorithms[END_REF], a fragmentation-aware routing algorithm with congestion avoidance is proposed. In [START_REF] Zhao | Crosstalk-aware spectrum defragmentation based on spectrum compactness in space division multiplexing enabled elastic optical networks with multicore fiber[END_REF], a crosstalk-aware defragmentation method is described, which uses 'spectrum compactness' metric to trigger and control the defragmentation process. In [START_REF] Sugihara | Dynamic resource allocation for immediate and advance reservation in space-division-multiplexing-based elastic optical networks[END_REF], some of the connections are established using advanced reservation, i.e., these requests can tolerate a certain delay before setup. To the best of our knowledge, even though in the context of EONs several fragmentation metrics have been studied (as external fragmentation, Shannon entropy, access blocking probability [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF][START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF], and root mean square factor [START_REF] Ye | Root mean square (rms) factor for assessing spectral fragmentation in flexible grid optical networks[END_REF], just to name few), there is a lack of investigation on the definition of fragmentation metrics in SS-FONs. Note that in SS-FONs the additional spatial dimension triggers new challenges, since each lightpath can be allocated on various spatial modes. A key issue here is the space continuity constraint (SCC). In particular, if we assume a precise indexing of spatial modes, the same in each network link, the SCC implies that a unique spatial mode shall be assigned to all the channels along a lightpaths. The SCC is enforced by some switching architectures, allowing for devices complexity and cost reduction, at the expense of limited resource allocation flexibility [START_REF] Marom | Switching solutions for WDM-SDM optical networks[END_REF][START_REF] Rumipamba-Zambrano | Space continuity constraint in dynamic flexgrid/sdm optical core networks: An evaluation with spatial and spectral superchannels[END_REF]. Since the SCC imposes to assign the same spatial mode index along the whole end-to-end routing path, the RSSA problem becomes more challenging and it may significantly increase requests blocking.

The main contribution of this paper is threefold. First, we formulate several fragmentation metrics that can be used in SS-FONs. To this end, we adapt fragmentation metrics proposed for EONs to SS-FONs. Second, we discuss simulation results showing performance of the proposed fragmentation metrics applied in dynamic routing in terms of the bandwidth blocking probability. Finally, we investigate the impact of SCC on the spectrum fragmentation in SS-FONs. To the best of our knowledge, this is the first paper that proposes and analyzes various fragmentation metrics in SS-FONs.

The rest of the paper is organized as follows. In Sec. 2, we present the considered network model. Sec. 3 discusses our proposed fragmentation metrics. Finally, in Sec. 4 numerical results are presented, and Sec. 5 concludes this work.

Network Model

The SS-FON is modeled as directed graph G(V , E) where V is the set of network nodes and E is the set of physical links that interconnects them. Each link e ∈ E comprises a set of spatial modes K(e), where each spatial mode may refer to fiber in a bundle, core in MCF or mode in multi-mode fiber, without loss of generality. Each spatial mode provides B THz of available spectrum, divided into F S small frequency slots (slices). By grouping several adjacent slices, a spectral super-channel (SCh) can be formed and used to provision the request. As in [START_REF] Rottondi | Optical ring metro networks with flexible grid and distance-adaptive optical coherent transceivers[END_REF], the network is equipped with coherent transceivers that use digital signal processing for distortion compensation and operate at fixed baud-rate of 37.5 GBaud. Each transceiver occupies 3 optical carriers of 12.5 GHz width. Table 1 presents supported bit-rates and transmission reaches for various MFs for single transceiver [START_REF] Rottondi | Optical ring metro networks with flexible grid and distance-adaptive optical coherent transceivers[END_REF][START_REF] Khodashenas | Comparison of spectral and spatial superchannel allocation schemes for sdm networks[END_REF]. If the requested bit-rate exceeds supported maximum capacity of a transceiver, the signal can be transmitted using several adjacent transceivers creating a SCh. Each SCh is separated from the adjacent one with 12.5 GHz-wide guardband. Request realization using single SChs allows for spectrum savings, as only one guardband is required. The highest considered request bit-rate is equal to 1 Tbps, thus possible SChs granularities are equal to 3n + 1, where n ∈ {1, ... , 20} [START_REF] Walkowiak | Dynamic routing in spectrally spatially flexible optical networks with back-to-back regeneration[END_REF].

For the sake of simplicity, we do not consider inter-core crosstalk (IC-XT) physical layer impairments, however the applied model can be easily modified to include them, e.g., through an application of worst-case IC-XT estimation [START_REF] Walkowiak | Dynamic routing in spectrally spatially flexible optical networks with back-to-back regeneration[END_REF]. We consider several metrics adapted from EONs to determine the link fragmentation in SS-FONs, namely, external fragmentation (EF), Shannon entropy (SE), access blocking probability (ABP) [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF], root of sum of squares (RSS), root mean square factor (RMSF) [START_REF] Ye | Root mean square (rms) factor for assessing spectral fragmentation in flexible grid optical networks[END_REF]. Let F m e denote a fragmentation of link e ∈ E, calculated according to some metric m defined further. The fragmentation of whole network using metric m (denoted as F m ) is calculated as the average value of link fragmentation F m e among all links e ∈ E as presented in Eq. [START_REF] Winzer | Optical networking beyond wdm[END_REF].

F m = e∈E F m e |E| (1)
It should be stressed that the link fragmentation metrics developed in the context of EONs cannot be directly applied in SS-FONs due to the introduction of the additional spatial dimension. Thus, below we modify the EON link fragmentation metrics in order to address multiple spatial modes. The general idea is to construct the SS-FON metric as the average of the corresponding EON metric calculated for each spatial mode separately.

Notation

Let introduce notation used to define fragmentation metrics. As described in Sec. 2, each link e ∈ E consists of K(e) spatial modes. Each spatial mode k ∈ K(e) contains S slots. Set Γ (e, k) denotes blocks of available contiguous slots on spatial mode k ∈ K(e) on link e ∈ E, and for each block γ ek ∈ Γ (e, k) we define |γ ek | as its size in slices. The highest allocated slice on the spatial mode k ∈ K(e) on link e ∈ E is represented by s max ek . Set G denotes types of granularities used by transceivers, i.e., what are the possible sizes of spectral SChs supported by the transceiver [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF] assuming guardbands. According to applied network model, the set G contains values equal to 3n + 1, where n ∈ {1, ... , 20}. Next, a /b is the integer division of a /b.

Fragmentation Metrics

The first considered metric is EF, originated in computer memory management (Eq. ( 2)). EF depends on the ratio of largest free contiguous fragment of memory to the total free memory on each core. As the main drawback, the EF metric does not reflect changes in smaller free portions of spectrum until the size of the largest free block is not changing. In consequence, when comparing two fragmented links, the result obtained by this metric is not always meaningful, as it does not prevent from fragmenting smaller portions of spectrum than the largest free block [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF][START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF].

F EF e = 1 - 1 |K(e)| • k∈K(e) max γ ek ∈Γ (e,k) |γ ek | γ ek ∈Γ (e,k) |γ ek | (2)
The SE in information theory is the amount of information available in the message. Eq. (3) presents adaptation for fragmentation in SS-FONs. In contrast to EF, SE takes into the account sizes and amount of smaller portions of free spectrum blocks, which makes it more meaningful to compare two links [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF][START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF][START_REF] Horota | Routing and spectrum assignment algorithm with most fragmented path first in elastic optical networks[END_REF][START_REF] Shen | Fragmentation aware routing and spectrum assignment algorithm for elastic optical networks[END_REF].

F SE e = 1 |K(e)| •
k∈K(e) γ ek ∈Γ (e,k)

|γ ek | S • ln S |γ ek | (3)
The ABP metric is formulated based on the available granularities of the transceivers (Eq. ( 4)). More specifically, it depends on the ratio between the sum of possible channels created with transceivers operating at various granularities in every free spectrum fragment, to the number of possible channels created with various transceivers granularities assuming that all slots are contiguous. Comparing to other metrics, ABP is aware of the possible sizes of SChs that can be created in the network [START_REF] Chatterjee | Fragmentation problems and management approaches in elastic optical networks: A survey[END_REF][START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF].

F ABP e = 1 - 1 |K(e)| • k∈K(e)
γ ek ∈Γ (e,k) g∈G

|γ ek | /g g∈G γ ek ∈Γ (e,k) |γ ek | /g (4)
Another possible metric is RSS, presented in Eq. ( 5). It depends on the sum of square roots of sum of sizes of all free segments on each spatial mode. As SE, this metric can reflect the changes in smaller free segments of the spectrum.

F RSS e = 1 - 1 |K(e)| • k∈K(e) γ ek ∈Γ (e,k) (|γ ek | 2 ) γ ek ∈Γ (e,k) |γ ek | (5)
The RMSF is defined in Eq. ( 6) as a function of the highest allocated slice on each core S max ek , number of free spectrum segments |Γ (e, k)| and their sizes γ ek ∈ Γ (e, k). The metric increases, when i) a higher slot is allocated in each spatial mode; ii) there are more free contiguous segments on spatial modes, i.e., available slices are divided into a larger set of free segments; iii) the size of large free segments decreases at the cost of increase of smaller segments size [START_REF] Ye | Root mean square (rms) factor for assessing spectral fragmentation in flexible grid optical networks[END_REF].

F RM SF e = 1 |K(e)| • k∈K(e) s max ek • |Γ (e, k)| γ ek ∈Γ (e,k) |γ ek | 2 |Γ (e,k)| (6)
Note that for metrics EF, ABP and RSS, in order to avoid the division by 0 when there are no available free segments on a spatial mode, the value inside the sum is replaced with 1. Contrarily, in RMSF in such case, the value is replaced with 0. Moreover, for all metrics except se and RMSF, values vary within 0, 1 range, while for SE and RMSF within 0, +∞).

In :

F EF e = 1 - 1 3 2 1 + 1 + 2 + 1 + 1 + 2 2 + 1 + 1 + 2 + 3 2 + 3 + 1 + 1 = 0.635 a2
F ABP e = 1 - 1 3 2 /2 + 2 /3 6 /2 + 6 /3 + 2 ( 2 /2 + 2 /3 ) 6 /2 + 6 /3 + 2 /2 + 2 /3 + 3 /2 + 3 /3 7 /2 + 7 /3 = 0.600 a4)
Root of sum of squares :

F RSS e = 1 - 1 3 √ 4 • 1 2 + 2 2 6 + √ 2 • 2 2 + 2 • 1 2 6 + √ 2 2 + 3 2 + 2 • 1 2 7 = 0.516 a5)
Root mean square factor :

F RM SF e = 1 3
11 • 5 (4 • 1 2 + 2 2 )/5 + 12 • 4 (2 • 2 2 + 2 • 1 2 )/4 + 11 • 4 (2 2 + 3 2 + 2 • 1 2 )/4 = 33.381 b1) External fragmentation : F EF e = 1 - 1
: F ABP e = 1 - 1 3 3 ( 2 /2 + 2 /3 ) 6 /2 + 6 /3 + 3 /2 + 3 /3 + 2 /2 + 2 /3 5 /2 + 5 /3 + 1 = 0.133 b4) Root of sum of squares : F RSS e = 1 - 1 3 √ 3 • 2 2 6 + √ 3 2 + 2 2 5 + 1 = 0.234 b5)
Root mean square factor :

F RM SF e = 1 3 10 • 3 (3 • 2 2 )/3 + 12 • 2 (3 2 + 2 2 )/2
+ 0 = 8.138

Fig. 1: Example of fragmentation metrics calculations.

Fragmentation-Aware Dynamic RSSA Algorithm

In this work, we consider a dynamic routing, i.e., requests arrive dynamically to the network and for each request a lightpath is selected, otherwise the request is rejected. For this purpose, we propose a fragmentation minimizing k -shortest path (FM-kSP) heuristic, where the objective is to minimize the amount of rejected traffic. For each request, FM-kSP selects one path out of k-shortest paths, in order to realize the lightpath aiming for the minimum increase of the network fragmentation F m , considering one of link fragmentation F m e definitions. In more detail, when a new request arrives to the network, the algorithm investigates all precomputed candidate shortest routing paths. For each path, the lowest possible SCh is found, i.e., a SCh with the lowest ending slice index forming a candidate lightpath. Note, the selection of SCh respects the network constraints, i.e., SCC, continuity and contiguity constraints. For each obtained lightpath, the variation in terms of overall network fragmentation ∆F is calculated as follows:

∆F m = e∈E F m e -F m e |E| , (7)
where F m e is the fragmentation measure of link e when the new lightpath is allocated, and F m e is the link fragmentation in current state of network. To calculate F m e and F m e , one of the introduced above metrics is used. Next, the lightpath with the lowest fragmentation difference ∆F m is selected, and request is realized on that lightpath. Note that if the difference is negative, the network fragmentation has improved. If there is no lightpath satisfying all constraints, the request is blocked.

Numerical Experiments

In this section, we evaluate the performance of proposed allocation strategies based on the fragmentation metrics under dynamic traffic requests in SS-FON. We study two network topologies: i) European network (Euro28) -28 nodes, 82 links and average link length of 625 km; ii) United States network (US26) -26 nodes, 84 links and average link length of 755 km (see Fig. 2). The considered SS-FON operates with a flexible grid of 12.5 GHz slices and each network link comprises 7 spatial modes. Each spatial mode provides 4 THz bandwidth divided into 320 slices, each of 12.5 GHz width. In this study, dynamic traffic requests arrive following a Poisson process with an average arrival rate of λ = 10 requests per unit time. The lifetime of each request follows a negative exponential distribution with an average of 1/µ. Therefore, the traffic load is λ/µ normalized traffic units (NTUs). The requested bit-rate of each request is selected randomly within 50 Gbps to 1 Tbps range, with 50 Gbps granularity. Therefore, traffic of 100 NTUs refers to around 52.5 Tb/s of overall traffic load. We investigate two types of requests distributions: i) traffic A -the source and destination nodes are selected with uniform distribution; ii) traffic B -the source and destination nodes are selected proportionally to the distance (in kilometers) between them, i.e., the nodes which are located more closely to each other are more often selected to realize the request. For each analyzed value of traffic load expressed in NTUs, one set with 60 000 requests is processed, however, first 10 000 requests are excluded from the analysis to enable the network to achieve a steady state. For each request, the number of candidate routing (shortest) paths is 10. Additionally, we consider two scenarios, with SCC and with SCC relaxed (denoted as no SCC). The main performance metric is bandwidth blocking probability (BBP), defined as the sum of bit-rates of rejected requests, over the sum of bit-rates of all requests offered to the network. As an additional performance metric, the accepted traffic for 1% threshold of BBP is used. In more detail, for each scenario the maximum value of traffic load expressed in NTUs is selected, for which the BBP is not larger than 1%, which is a commonly accepted threshold for BBP.

To allocate requests, the FM-kSP algorithm is used (see Sec. 3.3) considering all presented fragmentation metrics. As a reference, first-fit k-shortest path (FF) method is used, which allocates requests on the first possible SCh, processing candidate paths in increasing order of lengths in km. The first experiment presents relation between values of various fragmentation metrics and BBP as a function of NTUs for FF methods, fragmentation metrics are evaluated afterwards traffic allocation. Fig. 3 presents results for traffic A, with and without SCC. The offered traffic load varies from 500 to 1 000 NTUs with 50 NTUs granularity. As the values of RMSF (Eq. 6) can be higher than 1, they are normalized to fit the range of other metrics, i.e., from 0, 1 . Thus, these results are divided by 600. The first observation is that BBP increases exponentially (linear growth on logarithmic axis) with the linear increase of NTUs offered to the network. Furthermore, the value of fragmentation metrics also increases, suggesting that for higher values of NTUs, the network is more fragmented, and explaining the rapid increase of BBP. When SCC is assumed, the BBP is 120.5 % higher in average between 600 and 1 000 NTUs. Note, most of the fragmentation metrics are higher, i.e., 19.4%, 14.8%, 10.3%, and 16.6% for EF, SE, ABP and RSS, respectively. Surprisingly, the RMSF is 10.7% lower assuming SCC. Similar trends were obtained for traffic B and for US26 network topology and are not reported.

In the next experiment, performance of proposed FM-kSP algorithm using various fragmentation metrics is compared in terms of BBP. Fig. 4 presents BBP for various algorithms as a function of NTUs. In the following results, the names of metrics correspond to the FM-kSP algorithm utilizing that metric, exceptd FF which denotes the reference first fit k-shortest path algorithm. The first observation is that the RMSF algorithm provides the lowest BBP for most values of NTUs. For instance, for 1 000 NTUs the RMSF allows to decrease BBP by 28.8% and 53.8% in regard to the second best strategy, ABP, and the reference one, FF, respectively, for traffic A and SCC relaxed (Fig. 4a). On average RMSF, allows accepting 5.9 % and 8.8% more NTUs for 1% BBP threshold, in reference to ABP and FF, respectively. Further, SCC relaxation allows allocating larger NTUs achieving the same BBP, e.g., 11.1% of gain considering 1% of BBP and the RMSF strategy in Euro28 network (compare Figs 4a and4b). Next, requests following traffic B provide lower BBP when compared to traffic A, e.g., with a difference of 50.0% for BBP of 1% for RMSF (compare Figs. 4a and4c). Finally, the US26 network can accommodate smaller overall traffic with the same performance than Euro28 network. For BBP of 1% and RMSF, the difference between allocated NTUs is equal to 21.2% (compare Figs 4a and4d). Comparing RMSF to other metrics, the conclusions are as follows: i) as all other metrics, it is aware of the number of free segments; ii) as all other metrics except EF, it is aware of the size of free segments; iii) contrarily to other metrics, it is aware of the highest allocated slice on each spatial mode. The last property might be the main factor determining the best performance of the RMSF metric. In context of EONs, RMSF outperforms two other metrics, namely, spectrum compactness and utilization entropy [START_REF] Ye | Root mean square (rms) factor for assessing spectral fragmentation in flexible grid optical networks[END_REF]. However, to the best of our knowledge, RMSF has not been compared to metrics proposed in this paper. Moreover, as EON is a special case of SS-FON assuming one spatial mode, the difference in performance of proposed metrics should be valid also for EON. Next, we provide a detailed analysis of performance differences between considered scenarios for the RMSF metric, which -as shown above -provides the best results in terms of BPP. Fig. 5 presents percentage of served requests as a function of length of selected routing path in hops for Euro28 network for accepted traffic for 1% threshold of there is no significant difference between SCC and SCC relaxed case. Finally, the higher values of traffic can be accepted in traffic B comparing to traffic A, as the distances between source-destination nodes in traffic B are more often shorter. Requests with shorter distances are easier to realize, especially in highly fragmented network, as the continuity constraint is more challenging on path involving higher amount of links. The last Fig. 6 presents percentage of MFs usage for the RMSF strategy for 1% BBP threshold. For both network topologies, more efficient MFs are more often selected comparing traffic B to traffic A because average length of shortest path for requests is lower in traffic B. Moreover, when SCC is present, higher amount of more efficient MFs is selected, but achieving least amount of accepted traffic for 1% BBP threshold. When SCC is relaxed, increased flexibility allows to allocate additional traffic using longer paths. Finally, in US26 less efficient MFs are more often selected, regardless of traffic profile or SCC. The reason for this is that US26 network has higher average link length when compared to Euro28 network.

Concluding Remarks

In this work, we have focused on the performance of dynamic allocation algorithm applying various fragmentation metrics used to provision dynamic traffic in SS-FONs. According to results of simulations, the RMSF metric allows to decrease BBP and in consequence allows to accept around 8.8% more traffic (NTUs) assuming 1% threshold of BBP, when compared to the reference method. Additionally, the relaxation of space continuity constraint allows to allocate 11.1% more traffic comparing to the case without relaxation. Finally, the traffic profile (i.e., source-destination node selection probability distribution) and the number of available candidate paths have a significant impact on BBP. In the future work, we plan to develop defragmentation methods in SS-FONs based on proposed fragmentation metrics.

Fig. 1 ,

 1 we present an example of metrics calculation for two different links. The gray squares indicate slices utilized by lightpaths, while white squares are free slices. The operator a /b denotes the integer division a /b. In the example, the available granularities of transceiver are G = {2, 3}, therefore, free segments of size 1 are ignored in the F ABP e

Fig. 2 :

 2 Fig. 2: Network topologies.

Fig. 3 :

 3 Fig. 3: BBP and value of fragmentation metrics as a function of NTUs for FF heuristic, Euro28 network topology, traffic A.

  US26 network, traffic A, space continuity constraint relaxed.

Fig. 4 :

 4 Fig.4: BBP for FM-kSP heuristic using various fragmentation minimizing strategies and FF method as a function of NTUs.

  path [hops] no SCC, |P|=10, 1000 NTUs SCC, |P|=10, 900 NTUs no SCC, |P|=1, 650 NTUs SCC, |P|=1, 625 NTUs (a) Traffic A.

Fig. 5 :

 5 Fig. 5: Percentage of requests as a function of selected path length for Euro28 1% threshold of BBP, RMSF strategy.

Fig. 6 :

 6 Fig. 6: MFs percentage usage for 1% BBP threshold as a function of traffic profile and SCC presence for RMSF strategy.

Table 1 :

 1 Transmission reach and supported bit-rate for MFs.

		BPSK	QPSK	8-QAM	16-QAM
	Transmission reach [km]	6300	3500	1200	600
	Bit-rate [Gbps]	50	100	150	200
	3 Fragmentation Metrics and Fragmentation-Aware
	Dynamic RSSA Algorithm

Acknowledgments

The work of P. Lechowicz was supported National Science Centre, Poland under Grant 2015/19/B/ST7/02490. The work of A. Włodarczyk and K. Walkowiak was supported by National Science Centre, Poland under Grant 2017/27/B/ST7/00888.


Nikos Terzenidis

Miltiadis Moralis-Pegios

Stelios Pitris

Charoula Mitsoli- Dou

George Mourgias-Alexandris

Apostolis Tsakyridis

Christos Vagionas

Konstantinos Vyrsokinos

Theoni Alexoudi

Nikos Pleros
email: npleros@csd.auth.gr

Optics for Disaggregating Data Centers and Disintegrating Computing

Keywords: computing architectures, disintegrated computing, Network-on-Chip, silicon photonics

We present a review of photonic Network-on-Chip (pNoC) architectures and experimental demonstrations, concluding to the main obstacles that still impede the materialization of these concepts. We also propose the employment of optics in chip-to-chip (C2C) computing architectures rather than on-chip layouts towards reaping their benefits while avoiding technology limitations on the way to many-core set-ups. We identify multisocket boards as the most prominent application area and present recent advances in optically enabled multisocket boards, revealing successful 40Gb/s transceiver and routing capabilities via integrated photonics. These results indicate the potential to bring energy consumption down by more than 60% compared to current Quick-Path Interconnect (QPI) protocol, while turning multisocket architectures into a single-hop low-latency setup for even more than 4 interconnected sockets, which form currently the electronic baseline.

Introduction

Workload parallelism and inter-core cooperation are forcing computing to rely at a constantly growing degree on data movement. That led to an upgraded role for the onchip and off-chip communication infrastructures that support low-power and highbandwidth interconnect technologies. This came almost simultaneously with the revolutionary advances triggered in the field of optical interconnects [START_REF] Bergman | Photonic Networks for Intra-Chip, Inter-Chip, and Box-to-Box Interconnects in High Performance Computing[END_REF] and silicon photonics [START_REF] Lipson | Guiding, modulating, and emitting light on Silicon-challenges and opportunities[END_REF]. The last 20 years, optical interconnects were transformed to a mature technology for rack-to-rack [START_REF]Intel SiP 100G PSM4 Optical Tx[END_REF] and board-to-board communications [START_REF]Luxtera 2x100G-PSM4 OptoPHY Product Family[END_REF], supporting also the emerging concepts of disaggregated computing [START_REF] Zervas | Optically disaggregated data centers with minimal remote memory latency: Technologies, architectures, and resource allocation[END_REF] and leaf-spine Data Center architectures [START_REF] Bielski | dReDBox: Materializing a Full-stack Rack-scale System Prototype of a Next-Generation Disaggregated Datacenter[END_REF]. However, the on-chip and chip-to-chip photonic technologies are still far away from commercialization, despite the fact that various photonic Networkon-Chip (NoC) architectural concepts have already proposed [START_REF] Shacham | Photonic Networks-on-Chip for Future Generations of Chip Multiprocessors[END_REF]. In parallel, computing has also experienced some radical advances by turning from simple dual-and quad-core layouts into a highly heterogeneous environment both at chip-and system-level. As shown in Fig. 1, General-Purpose Graphic Processing Units (GP-GPUs) [START_REF] Kider | NVIDIA Fermi architecture[END_REF] can host more than 4000 CUDA cores on the same die, offering, however, only a 2 Gflop per core processing power. Processing power per core increases in manycore architectures, where up to 1000 cores can be employed [START_REF] Bohnenstiehl | KiloCore: A 32-nm 1000-Processor Computational Array[END_REF]. However, when high-performance cores are required as in the case of Chip Multiprocessor (CMP) configurations [START_REF]Intel Xeon Platinum 8180 Processor[END_REF] only a number of up to 32 cores can fit on the same die. The ideal scenario towards boosting processing power would of course imply a die that employs as many cores as a GPU does, but with core capabilities similar to the high-performance cores available in CMPs.

The number of high-performance cores performing as a single computational entity can scale to higher values only through multi-socket designs with 4 or maximum 8 interconnected sockets. The most recent top-class Intel Xeon 8-socket board yields a total number of up to 224 cores [11], requiring, of course, the use of high-bandwidth off-chip inter-socket interconnects. Going one step beyond the multisocket scheme, disintegration of processor dies has been coined in the recent years as a way to form macrochips that will synergize a high amount of high-performance cores, usually exploiting optical inter-die links [START_REF] Raj | Macrochip" computer systems enabled by silicon photonic interconnects[END_REF]. This versatile environment at chip-scale suggests a diverse set of requirements that has to be met by optics, depending on the application. However, it creates also a new opportunity to rethink the role of optics in on-and off-chip computing, building upon the proven capabilities of optical hardware towards strengthening the compute architecture/technology co-design perspective.

In this paper, we attempt to investigate the new perspectives for optics in computing, reviewing the high-priority challenges faced currently by the computing industry and evaluating the credentials of state-of-the-art photonics to address them successfully. We provide a review of the work on photonic NoCs, highlighting the bottlenecks towards their materialization. Building on the state-of-art pNoC implementations , we conclude to a solid case for employing integrated photonics in inter-chip multisocket and disintegrated layouts rather than in Network-on-Chip (NoC) implementations, proposing at the same time a flat-topology chip-to-chip multisocket interconnect technology. We demonstrate experimental results for 40 Gb/s multi-socket boards (MSBs) operation, showing the potential to scale to >8-socket designs boosting the number of directly interconnected high-performance cores. Combined with the Hipoλaos Optical Packet Switch (OPS) that has been recently shown to support subμsec latencies [START_REF] Terzenidis | High-port low-latency optical switch architecture with optical feed-forward buffering for 256-node disaggregated data centers[END_REF], an optically-enabled rack-scale 256-socket disaggregated setting using a number of 32 interconnected optical 8-socket MSBs, could be implemented, forming in this way a powerful disaggregated rack-scale computing scheme.

The paper is organized as follows: Section II outlines the main challenges faced today in the computing landscape, providing also an overview of the research on pNoC architectures, concluding to their main limitations. Section III argues for the employment of optics in MSBs and provides experimental results on a 40Gb/s flattopology 8-node chip-to-chip (C2C) layout, using O-band integrated photonic transceiver and routing circuitry. Finally, Section IV concludes the paper.

Overview of the PNoC architectures

In order to define and refine the role of optics in the current computing landscape, it is critical to identify the main challenges currently experienced by the computing industry along the complete hierarchy from on-chip through multi-socket chip-to-chip computational modules. Fig. 2 provides an illustrative overview of the main bandwidth, latency and energy needs for different on-chip and off-chip interconnect layers and data transfer operations in a 20x20mm 2 processor chip fabricated by a 28nm Integrated Circuit (IC) CMOS technology.

Fig. 2. Energy, bandwidth and latency requirements at different on-chip and off-chip communication needs. The size of every cache memory is bigger for larger capacity caches and their distance from the core is higher as the cache hierarchy increases.

A digital processing operation performed by the core consumes only 20pJ/bit, but sending data across the chip requires 0.1pJ/bit for a 1mm long electrical link, 1pJ/bit for a 10mm link and goes up to 4pJ/bit for a link length of 40mm. When going offchip in order to access DRAM, a high amount of 30pJ/bit is consumed, while a chipto-chip interconnect link like QPI requires 16.2pJ/bit. Accessing L1 cache requires 0.2pJ/bit, while L2 and L3 access requires 1 and 2-4pJ/bit, respectively. Memory bandwidth reduces with increasing memory hierarchy, with L1 memory bandwidth approaching 20GB/sec and gradually decreasing when going to L2 and L3 access until an upper limit of 12.5GB/sec in the case of DRAM access. Latency follows the inverse path, starting from a high >55nsec value when fetching from DRAM and gradually reducing with increased memory hierarchy, with L1 access latency being around 1.3nsec. Having this overview, the main challenges today are formed around: i) Interconnect energy consumption: A modern CPU consumes around 1.7nJ per floating-point operation [START_REF] Parker | The Evolution of GPU Accelerated Computing[END_REF][START_REF] Dally | Challenges for Future Computing Systems[END_REF], being 85x higher than the 20pJ per floating point required for reaching the Exascale milestone within the gross 20MW power envelope. Current architectures rely to a large degree on data movement, with electronic interconnects forming the main energy consuming factor in both on-and off-die setups [START_REF] Dally | Challenges for Future Computing Systems[END_REF]. With the energy of a reasonable standard-cell-based, double-precision fusedmultiply add (DFMA) being only ~20 pJ, it clearly reveals that fetching operands is much more energy-consuming than computing on them [START_REF] Parker | The Evolution of GPU Accelerated Computing[END_REF][START_REF] Dally | Challenges for Future Computing Systems[END_REF].

ii) Memory bandwidth at an affordable energy envelope: The turn of computing into strongly heterogeneous and parallel settings have transformed memory throughput into a key factor for increasing processing power [START_REF] Parker | The Evolution of GPU Accelerated Computing[END_REF], with the most efficient way for improvement still being the use of wider memory buses and hierarchical caching. However, the highest memory bandwidth per core in modern multicore processors can hardly reach 20 GB/sec [START_REF] Saini | Performance Evaluation of the Intel Sandy Bridge Based NASA Pleiades Using Scientific and Engineering Applications[END_REF], with L1 cache latency values still being >1nsec.

iii) Die area physical constraints: The need to avoid the latency and energy burden of DRAM access has enforced a rich on-chip L1, L2 and L3 cache hierarchy that typically occupies >40% of the chip real-estate [START_REF] Borkar | The future of microprocessors[END_REF], suggesting that almost half of the die area is devoted to memory and interconnects instead of processing functions. iv) Cache coherency-induced multi-and broadcasting traffic patterns: The need for cache coherency at intra-chip multi-and manycore setups, as well as at inter-chip multisocket systems, yields communication patterns with strong multi-and broadcast characteristics, that have to be satisfied at a low-latency low-energy profile by the interconnect and network-on-chip infrastructure. Multibus ring topologies form a widely adopted multicast-enabling NoC architecture in current modern multi-core processors [START_REF] Kumashikar | 14nm Broadwell Xeon® processor family: Design methodologies and optimizations[END_REF], but still the cache coherency control messages may often account for more than 30% of the total available bandwidth, which may reach even 65% in multisocket settings [START_REF] Bull | An efficient server architecture for the virtualization of business-critical applications. white paper[END_REF].

The first attempts to exploit photonics for overcoming the on-chip bandwidth, energy and latency bottlenecks mainly inspired by the rapidly growing field of silicon photonics [START_REF] Lipson | Guiding, modulating, and emitting light on Silicon-challenges and opportunities[END_REF]. A number of breakthrough computing architectures relying on pNoC were demonstrated, proposing and utilizing novel silicon photonic transceiver and switching schemes. The pioneering work on photonic Torus [START_REF] Shacham | Photonic Networks-on-Chip for Future Generations of Chip Multiprocessors[END_REF] was followed by performance and energy advances in pNoC-enabled many-core designs, addressing even cache-coherency needs [START_REF] Kurian | ATAC[END_REF]. All this shaped a promising roadmap for the many-core computing architectures [START_REF] Shacham | Photonic Networks-on-Chip for Future Generations of Chip Multiprocessors[END_REF], [START_REF] Chen | Runtime Management of Laser Power in Silicon-Photonic Multibus NoC Architecture[END_REF][START_REF] Li | Aurora: A Cross-Layer Solution for Thermally Resilient Photonic Network-on-Chip[END_REF][START_REF] Bahirat | METEOR[END_REF][START_REF] Wang | RPNoC: A Ring-Based Packet-Switched Optical Network-on-Chip[END_REF][START_REF] Gu | MRONoC: A Low Latency and Energy Efficient on Chip Optical Interconnect Architecture[END_REF][START_REF] Werner | Efficient sharing of optical resources in low-power optical networks-on-chip[END_REF]. At the same time, it revealed the requirements to be met by silicon photonics towards materializing their on-chip employment in practical NoC layouts: transceiver line-rates between 1-40 Gb/s and optoelectronic conversion energies between a few tens to a few hundreds of fJ/bit were considered in the vast majority of pNoC schemes [START_REF] Shacham | Photonic Networks-on-Chip for Future Generations of Chip Multiprocessors[END_REF], [START_REF] Chen | Runtime Management of Laser Power in Silicon-Photonic Multibus NoC Architecture[END_REF][START_REF] Li | Aurora: A Cross-Layer Solution for Thermally Resilient Photonic Network-on-Chip[END_REF][START_REF] Bahirat | METEOR[END_REF][START_REF] Wang | RPNoC: A Ring-Based Packet-Switched Optical Network-on-Chip[END_REF][START_REF] Gu | MRONoC: A Low Latency and Energy Efficient on Chip Optical Interconnect Architecture[END_REF][START_REF] Werner | Efficient sharing of optical resources in low-power optical networks-on-chip[END_REF]. Driven by these efforts, photonic integration technology achieved the performance metrics required by pNoC architectures with silicon photonic modulators and SiGe Photo-diodes (PDs) operating at data rates up to 56Gb/s exhibiting an energy efficiency than a few tens of fJ/bit [START_REF] Pantouvaki | Active Components for 50 Gb/s NRZ-OOK Optical Interconnects in a Silicon Photonics Platform[END_REF].

Fig. 3 summirizes the most important pNoC and on-chip switches up to now . Silicon switches have witnessed a remarkable progress yielding high-port connectivity arrangements with a variety of underlying physical mechanisms like the thermo-optic (TO), electro-optic (EO) and opto-mechanical effects [START_REF] Lee | Silicon Photonic Switching: Technology and Architecture[END_REF], allowing for 32×32 EO Mach-Zehnder Interferometric (MZI)-based layouts [START_REF] Qiao | 32 × 32 silicon electro-optic switch with built-in monitors and balanced-status units[END_REF], 64×64 TO MZI designs [START_REF] Qiao | Ultra-large-scale silicon optical switches[END_REF] and up to 128×128 Microelectromechanical switches (MEMS) [START_REF] Kwon | 128×128 Silicon Photonic MEMS Switch with Scalable Row/Column Addressing[END_REF].

All these demonstrations indicate that integrated photonics can now indeed offer the line-rate, energy, footprint and connectivity credentials required by pNoC-enabled manycore computing architectures. However, the realization of a manycore machine that employs a pNoC layer seems to be still an elusive target, with the main reason being easily revealed when inspecting the non-performance-related co-integration and integration level details of a pNoC-enabled computational setting. Manycore architectures necessitate the on-die integration of a few thousands of photonic structures [START_REF] Shacham | Photonic Networks-on-Chip for Future Generations of Chip Multiprocessors[END_REF], residing either on 3D integration schemes [START_REF] Yoo | Heterogeneous 2D/3D photonic integrated microsystems[END_REF] or on monolithically co-integrated electronic and photonic structures, with transistors and optics being almost at the same layer [START_REF] Sun | Single-chip microprocessor that communicates directly using light[END_REF]. However, 3D integration has still not managed to fulfil the great expectations that were raised and is still struggling to overcome a number of significant challenges [START_REF] Li | Chip Scale 12-Channel 10 Gb/s Optical Transmitter and Receiver Subassemblies Based on Wet Etched Silicon Interposer[END_REF]. On the other hand, monolithic integration has recently accomplished some staggering achievements reporting on real workload execution over an opto-electronic die with optical core-memory interconnection [START_REF] Sun | Single-chip microprocessor that communicates directly using light[END_REF]. Nevertheless, this technology has still a long-way to go until reaching the complexity and functionality level required by a many-core pNoC design.

With almost the complete Photonic Integrated Circuit (PIC) technology toolkit being today available as discrete photonic chips, computing can reap the benefits of optics by employing photonics for off-die communication in i) multisocket and ii) disintegrated layouts. Both schemes can yield a high number of directly interconnected high-performance cores, unleashing solutions that cannot be met by electronics. At the same time, this approach is fully inline with the 2.5D integration scheme that employs discrete photonic and electronic chips on the same silicon interposer and has made tremendous progress in the recent years [START_REF] Zhang | Heterogeneous 2.5D integration on through silicon interposer[END_REF]. To this end, the employment of off-die communications via discrete photonic chips can form a viable near-term roadmap for the exploitation of photons in computational settings.

3

Optics for multi-socket boards MSB systems rely currently on electrically interconnected sockets and can be classified in two categories: i) "glueless" configurations, where point-to-point (P2P) interconnects like Intel's QPI [START_REF]Intel. An Introduction to the Intel QuickPath Interconnect[END_REF] can offer high-speed, low-latency, any-to-any C2C communication for a number of 4 or 8 sockets. A 4-socket setup can yield a cache-coherent layout with directly interconnected sockets and latency values that range between 60-240nsec. Scaling to 8-socket designs can only be met through dual-hop links, degrading latency performance but still comprising a very powerful cache-coherent computational setting: Intel's Xeon E7-8800 v4 was the first processor supporting 8-socket configurations and was by that time advertized as being suitable to "dominate the world" [START_REF] Intel | [END_REF]. Fig. 4(a) depicts a 4-socket (4S) and 8-socket (8S) layout, respectively, along with their respective interconnects. A typical interconnect like Intel's QPI operates at a 9.6 Gb/s line-rate and consumes 16.2 pJ/bit, while the total bandwidth communicated by every socket towards all three possible directions is 38.4 GB/s, i.e. 307.2 Gb/s [START_REF] Maddox | Weaving high performance multiprocessor fabric[END_REF].

ii) "glued" configurations, where scaling beyond 8-socket layouts is accomplished by exploiting active switch-based setups, such as PCI-Express switches, in order to interconnect multiple 4-or 8-socket QPI "islands" [START_REF] Maddox | Weaving high performance multiprocessor fabric[END_REF].

With latency and bandwidth comprising the main performance criteria in releasing powerful MSB configurations, "glueless" layouts offer a clear latency-advantage over the "glued" counterparts avoiding by default the use of any intermediate switch. Photonics can have a critical role in transforming "glued" into "glueless" architectures even when the number of interconnected sockets is higher than 8, enabling single-hop configurations, with Fig. 4(b) illustrating how the basic flat-topology can be accomplished for the case of an 8-Socket layout. This has been initially conceived and proposed by UC Davis in their pioneering work on Flat-Topology computing architectures [START_REF] Grani | Flat-Topology High-Throughput Compute Node With AWGR-Based Optical-Interconnects[END_REF] via Arrayed Waveguide Grating Router (AWGR) interconnects, utilizing low-latency, non-blocking and all-to-all optical connectivity credentials enabled by their cyclic-routing wavelength properties. UC Davis demonstrated via gem5 simulations the significant execution time and energy savings accomplished over the electronic baseline [START_REF] Grani | Flat-Topology High-Throughput Compute Node With AWGR-Based Optical-Interconnects[END_REF], revealing also additional benefits when employing bit-parallel transmission and flexible bandwidth-allocation techniques. Experimental demonstrations of AWGR-based interconnection for compute node architectures were, however, constrained so far in the C-band regime, limiting their compatibility with electro-optic Printed Circuit Board (PCB) technology that typically offers a low waveguide loss figure at the O-band [START_REF] Sugama | Development of high-density single-mode polymer waveguides with low crosstalk for chip-to-chip optical interconnection[END_REF]. As such, AWGR-based experimental compute node interconnect findings were reported so far only in pNoC architectural approaches, using a rather small line-rate operation of 0.3 Gb/s [START_REF] Yu | A scalable silicon photonic chip-scale optical switch for high performance computing systems[END_REF].

The European H2020 project ICT-STREAMS is currently attempting to deploy the necessary silicon photonic and electro-optical PCB technology toolkit for realizing the AWGR-based MSB interconnect benefits in the O-band and at data rates up to 50Gb/s [START_REF] Kanellos | WDM mid-board optics for chip-to-chip wavelength routing interconnects in the H2020 ICT-STREAMS[END_REF]. It aims to exploit wavelength division multiplexing (WDM) Silicon photonics transceiver technology at the chip edge as the socket interface and a boardpluggable O-band silicon-based AWGR as the passive routing element, as shown in a generic N-socket architecture depicted in Fig. 4(c). Each socket is electrically connected to a WDM-enabled Tx optical engine equipped with N-1 laser diodes (LD), each one operating at a different wavelength. Every LD feeds a different Ring Modulator (RM) to imprint the electrical data sent from the socket to each one of the N-1 wavelengths, so that the Tx engine comprises finally N-1 RMs along with their respective RM drivers (DR). All RMs are implemented on the same optical bus to produce the WDM-encoded data stream of each socket. The data generated by each socket enters the input port of the AWGR and is forwarded to the respective destination output that is dictated by the carrier wavelength and the cyclic-frequency routing properties of the AWGR [START_REF] Grani | Flat-Topology High-Throughput Compute Node With AWGR-Based Optical-Interconnects[END_REF]. In this way, every socket can forward data to any of the remaining 7 sockets by simply modulating its electrical data onto a different wavelength via the respective RM, allowing direct single-hop communication between all sockets through passive routing. At every Rx engine, the incoming WDM-encoded data stream gets demultiplexed with a 1:(N-1) optical demultiplexer (DEMUX), so that every wavelength is received by a PD. Each PD is connected to a transimpedance amplifier (TIA) that provides the socket with the respective electrical signaling.

The AWGR-based interconnect scheme requires a higher number of transceivers compared to any intermediate switch solution, but this is exactly the feature that allows to combine WDM with AWGR's cyclic frequency characteristics towards enabling single-hop communication and retaining the lowest possible latency. Utilizing an 8×8 AWGR, the optically-enabled MSB can allow single-hop all-to-all interconnection between 8 sockets, while scaling the AWGR to 16×16 layouts can yield single-hop communication even between 16 sockets, effectively turning current "glued" into "glueless" designs. The ICT-STREAMS on-board MSB aims to incorporate 50GHz single-mode O-band electro-optical PCBs [START_REF] Lamprecht | EOCB-Platform for Integrated Photonic Chips Direct-on-Board Assembly within Tb/s Applications[END_REF], relying on the adiabatic coupling approach between silicon and polymer waveguides [START_REF] Dangel | Polymer Waveguides Enabling Scalable Low-Loss Adiabatic Optical Coupling for Silicon Photonics[END_REF] for low-loss interfacing of the Silicon-Photonics (Si-Pho) transceiver and AWGR chips with the EO-PCB.

Next, the first 40Gb/s experimental results of demonstration with the fiberinterconnected integrated photonic building blocks is presented, extending the recently presented operation of the 8-socket architecture at 25 Gb/s [START_REF] Moralis-Pegios | Chip-to-Chip Interconnect for 8-socket direct connectivity using 25Gb/s O-band integrated transceiver and routing circuits[END_REF]. The main integrated transmitter, receiver and routing building blocks that were used, comprise three discrete chips, i.e. a Si-based RM [START_REF] Pantouvaki | Active Components for 50 Gb/s NRZ-OOK Optical Interconnects in a Silicon Photonics Platform[END_REF], a Si-based 8×8 AWGR routing platform [START_REF] Pitris | Silicon photonic 8 × 8 cyclic Arrayed Waveguide Grating Router for Oband on-chip communication[END_REF] and a co-packaged PD-TIA [START_REF] Moeneclaey | A 40-Gb/s Transimpedance Amplifier for Optical Links[END_REF], which are depicted in Fig. 4(d), (e) and (f), respectively.The silicon O-band carrier-depletion micro-ring modulator is an all-pass ring resonator fabricated on imec's active platform with demonstrated 50 Gb/s modulation capabilities [START_REF] Pantouvaki | Active Components for 50 Gb/s NRZ-OOK Optical Interconnects in a Silicon Photonics Platform[END_REF]. The RM can be combined with a recently developed low-power driver [START_REF] Ramon | Low-Power 56Gb/s NRZ Microring Modulator Driver in 28nm FDSOI CMOS[END_REF], leading to an energy efficiency of 1 pJ/bit at 40 Gb/s. For the routing platform, the demonstration relied on an O-band integrated silicon photonic 8×8 AWGR device [START_REF] Pitris | Silicon photonic 8 × 8 cyclic Arrayed Waveguide Grating Router for Oband on-chip communication[END_REF] with 10 nm-channel spacing, a maximum channel loss nonuniformity of 3.5 dB and a channel crosstalk of 11 dB. Finally, the Rx engine employed a co-packaged uni-traveling InGaAs-InP PIN photodiode (PD) connected with a low-power TIA implemented in 0.13 μm SiGe BiCMOS [START_REF] Moeneclaey | A 40-Gb/s Transimpedance Amplifier for Optical Links[END_REF]. The PD-TIA energy efficiency for operation at 40 Gb/s is 3.95 pJ/bit.

The energy efficiency of the proposed 40 Gb/s chip-to-chip (C2C) photonic link is estimated at 5.95 pJ/bit, assuming a 10% wall-plug efficiency for the external laser. This indicates that the proposed architecture has the credentials to lead to 63.3% reduction in energy compared to the 16.2 pJ/bit link energy efficiency of Intel QPI [START_REF] Maddox | Weaving high performance multiprocessor fabric[END_REF]. Fig. 5 (a)-(h) show the eye diagrams of the signal at the 8 outputs of the AWGR coresponding to the 8 routing scenarios for all possible input-output port combina- tion, indicating clear eye openings and successful routing at 40 Gb/s with ER values of 4.38±0.31 dB and AM values of 2.3±0.3 dB, respectively. The RM was electrically driven with a peak-to-peak voltage of 2.6 Vpp, while the applied reverse DC bias voltage was -2.5 V. The optical power of the CW signal injected at the RM input was 8 dBm, with the modulated data signal obtained at the RM output having an average optical power level of -6.3 dBm.

Going a step further, the proposed optically-enabled MSBs can be beneficially employed in rack-scale disaggregated systems when equipped with an additional transceiver lane for dealing with the off-board traffic and are combined with the recently demonstrated Hipoλaos high-port switch architecture [START_REF] Terzenidis | High-port low-latency optical switch architecture with optical feed-forward buffering for 256-node disaggregated data centers[END_REF]. Recently in [START_REF] Terzenidis | Dual-layer Locality-Aware Optical Interconnection Architecture for Latency-Critical Resource Disaggregation Environments[END_REF], it was shown that rack-scale disaggregation among a 256-node system can be successfully accomplished for a variety of communication patterns with an ultra-low mean latency value of < 335 nsec for 10 Gb/s data rates. The disaggregated architecture are expected to improve drastically when scaling Hipoλaos data-rates to 40Gb/s, making this compatible with the 40Gb/s silicon photonic transmitter reported in this paper.

Conclusion

We reviewed the pNoC-enabled manycore architectures proposed over the last decade. After analyzing the co-integration aspects as the main limitation for the realization of pNoC-based computing, we have defined a new role for photonics in the landscape of computing related to off-die communication. We discussed how optics can yield single-hop low-latency multisocket boards for even more than 4 interconnected sockets, demonstrating experimental results for 40Gb/s C2C interconnection in a 8node setup via integrated photonic transmitter and routing circuits. Combining 8socket optical boards with a Hipoλaos optical packet switch shown in [START_REF] Terzenidis | High-port low-latency optical switch architecture with optical feed-forward buffering for 256-node disaggregated data centers[END_REF], photonics can yield a powerful 256-node compute disaggregated system with latency below the sub-μs threshold considered for memory disaggregation environments.

Fig. 1 .

 1 Fig. 1. Evolution from single-to many-core computing architectures.

Fig. 3 .

 3 Fig. 3. Evolution of photonic Network-on-Chip and on-chip photonic switches.

Fig. 4 .

 4 Fig. 4. (a) C2C routing in current electronic 4S and 8S MSBs, (b) Flat-topology 8S layout using AWGR-based routing, (c) proposed N×N AWGR-based optical C2C interconnect for MSB connectivity. Photonic integrated circuits employed as the basic building blocks in the 40Gb/s experimental demonstration: (d) Ring Modulator, (e) 8×8 cyclic-frequency AWGR and (f) PD-TIA module. (blue-highlighted areas: part of the architecture demonstrated experimentally, white-highlighted areas: basic building blocks used for the demonstration).

Fig. 5 .

 5 Fig. 5. Eye diagrams (a)-(h) after routing via the respective In#iOut#j I/O ports of the AWGR.

Acknowledgment: This work is supported by the H2020 projects ICT-STREAMS (688172) and L3MATRIX (688544).


Alan A Díaz-Montiel

Marco Ruffini
email: marco.ruffini@tcd.ie

A Performance Analysis of Supervised Learning Classifiers for QoT Estimation in ROADM-Based Networks

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Transport metro-access networks need fast and dynamic reconfiguration procedures to overcome the challenges imposed by 5G-service driven requirements [START_REF] Ruffini | Multidimensional Convergence in Future 5G Networks[END_REF]. While several advancements at the hardware level (e.g., higher capacity transceivers) are being achieved, the disaggregation between such components and their control systems remains a challenge [START_REF] Ruffini | Moving the Network to the Cloud: the Cloud Central Office Revolution and Its Implications for the Optical Layer[END_REF]. With the emergence of the software-defined networking (SDN) paradigm a decade ago, multiple consolidated projects have tackled the openness and programmable features of optical network elements (i.e., switches) [START_REF]OpenROADM Project[END_REF]4]. Due to the heterogeneous and complex nature of equipment at deployment sites today, the development of robust protocols and standards is ongoing but is a lengthy process. In the meantime, the community has been evaluating different control system strategies, which have resulted in a set of recommendations well documented in the ITU-T G.7702 [5]. More recently, the application of artificial intelligence (AI) for multiple use cases in the optical domain has been widely evaluated. The latter has been surveyed by Mata et al.,[START_REF] Mata | Artificial intelligence (AI) methods in optical networks: A comprehensive survey[END_REF]. Particular attention is being paid to the utilization of estimation/classification algorithms that could predict the performance of signals traversing networks at various domains and configuration settings (e.g., heterogeneous equipment).

Commercial optical networks are today rather static, and their evolution towards a fully flexible and dynamic system will face control and management challenges. For instance, state-of-the-art wavelength division multiplexed (WDM) networks consider reconfigurable add/drop multiplexers (ROADMs), built up with colorless wavelength selective switches (WSS) and optical cross-connects (OXCs) as the de facto switching and routing networking elements, enhancing higher levels of flexibility. Additionally, in order to reach longer distances with fewer resources, the consideration of automatic gain controlled (AGC)-EDFA systems is the default approach. However optical reconfigurations are still carried out slowly (e.g., in terms of several minutes), mainly due to the effects of power excursions in gain-controlled optical amplifiers [START_REF] Li | tSDX: Enabling Impairment-Aware All-Optical Inter-Domain Exchange[END_REF][START_REF] Junio | Channel power excursions from single-step channel provisioning[END_REF].

Hence, the development of tools capable of handling such impairments at low cost without compromising the current performance is of high interest. Thus, the inclusion of ML-based classification algorithms in the control system of optical networks has been a promising area of research in recent years. For instance, Morais and Pedro [START_REF] Morais | Machine learning models for estimating quality of transmission in DWDM networks[END_REF] reviewed the models of K-nearest neighbors (KNN), logistic regression (LR), support vector machines (SVM), and artificial neural networks (ANN), analysing their performance in terms of predicting the QoT of unestablished lighpaths using simulated data. Their results suggest that ANN outperforms the other models with 99.9% accuracy. On a different study [START_REF] Mata | Supervised Machine Learning Techniques for Quality of Transmission Assessment in Optical Networks[END_REF], Mata et al., evaluated the models of SVM, LR, classification and regression trees (CART), bagging trees (TREEBAG) and random forests (RF), comparing them in terms of estimation performance and prediction time. In their results, SVM, RF and TREEBAG achieved an accuracy performance up to 99.9%, while keeping their prediction time low. While these studies suggest that the use cases associated with the optical domain may be successfully addressed with ML, we believe further research is required that considers extra physical layer phenomena in the training of the ML-models, including EDFA power excursions and non-linear physical impairments. In this study, we analyze the performance of various classification algorithms, when considering both topological features (e.g., number of nodes, fibre distance, amplifier span, etc.) and wavelength load [START_REF] Díaz-Montiel | Active Wavelength Load as a Feature for QoT Estimation Based on Support Vector Machine[END_REF] (e.g., the active wavelengths in the system before the new test channel is added). Indeed, as shown in [START_REF] Junio | Channel power excursions from single-step channel provisioning[END_REF], the wavelength load configuration is one of the main parameters affecting power excursions in EDFAs. With regard to the comparison of the ML models, two crucial metrics are considered: the accuracy at classifying different classes of traffic (based on OSNR levels), and the computational time required to train (build) the models, together with the size of the data samples. For the execution of the experiments, we used the Scikit-learn: Machine Learning in Python software [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF], which provides powerful APIs for ML classification models. Then, for the data generation, we used the Optical-MAN simulator first introduced in [START_REF] Díaz-Montiel | Performance Analysis of QoT Estimator in SDN-controlled ROADM networks[END_REF], which allows for the creation of large scale optical networks deployed with WSSbased ROADMs and EDFA-amplified links, and the simultaneous transmission of 90 channels in the C-band spectrum (1529.6 nm -1565.2 nm).

The remainder of the paper is structured as follows. In section 2, we introduce the system setup and the data generation process with our customized tools. Then, in section 3 we briefly describe the classification algorithms used for the performance analysis. The experiments and results are discussed in section 4. Finally, we present our conclusions in section 5.

2 System setup and data generation

System setup

Experimentation procedures with ML-models typically require large data sets in order to achieve good performance when tackling complex systems. In the optical domain, common network-QoT metrics for supervised learning models are the OSNR and bit error rate (BER) of the transmission systems. Hence, in order to collect large amount of samples from a physical testbed, constant monitoring procedures are needed at least at one point of the network. Alas, the collection of physical layer data in the optical domain (e.g., signal power levels) is an expensive task, mainly due to the high costs and complexities of the monitoring equipment. In addition, the deployment of large-scale, topology-variable optical networks is an infeasible task for many of the academic laboratories today. Because of that, the exploration of optical networks via simulation tools has been used in several cases [START_REF] Morais | Machine learning models for estimating quality of transmission in DWDM networks[END_REF][START_REF] Sartzetakis | Formulating QoT Estimation with Machine Learning[END_REF][START_REF] Tremblay | Machine Learning Techniques for Estimating the Quality of Transmission of Lightpaths[END_REF][START_REF] Barletta | QoT estimation for unestablished lighpaths using machine learning[END_REF]. Although these virtual environments enable the low cost, fast generation of physical layer optical data, the experimentation tools (ML-models) would always require further trial with real testbed data, which would account for their validation.

For the purposes of this study, we used the customized Optical-MAN simulator first introduced in [START_REF] Díaz-Montiel | Performance Analysis of QoT Estimator in SDN-controlled ROADM networks[END_REF], which allows to deploy WDM network topologies and collect synthetic labeled data to train ML models, as we recently presented in [START_REF] Díaz-Montiel | Active Wavelength Load as a Feature for QoT Estimation Based on Support Vector Machine[END_REF] for the analysis of support vector machines (SVM). Our tool simulates WDM networks considering: optical nodes consisting of ROADMs equipped with WSSs, optically-amplified AGC-EDFA links, and also AGC-EDFAs for pre-/post-signal amplification, together with fiber non-linear effects such as stimulated Raman scattering. An abstracted representation of a linear network topology composed of the network elements described is presented in Fig. 1. End-to-end transmission is enabled for up to 90 wavelength channels in the C-band (1529.6 nm -1565.2 nm). For the EDFAs in our system it is assumed gain flattening filtering (GFF) is used for gain equalization. While accounting for GFFs at line-amp sites covers for the smoothing of unequal signal intensities, a residual wavelength-dependent gain (WDG) still remains after the filtering due to imperfect equalization. The latter, may result in the detrimental performance of the transmitted signals.

Parting from the models proposed by Junio and Kilper in [START_REF] Junio | Channel power excursions from single-step channel provisioning[END_REF], we take under consideration the power excursions, enabling the analysis of the effects these cause in the system, and how ML-models could learn from these.

For all the simulated network scenarios, we assume optical performance monitoring is implemented at the receiver end, which enables for the tracking of the signal performance in terms of signal power and noise. Then, as first introduced in [START_REF] Díaz-Montiel | Performance Analysis of QoT Estimator in SDN-controlled ROADM networks[END_REF], we compute the OSNR levels of all the individual signals being transmitted.

Data generation

Our goal is to analyze the performance of multiple supervised learning classifiers at predicting the QoT of newly established lightpaths in a network with a given topology and wavelength load configuration. Our approach consists in extending the system parameters used to train the ML models, to not only consider network topological settings (i.e., number of nodes, number of links, etc.) as it is the common case, but to also include the number of active wavelength channels (WCs) and their position in the spectrum before installing a new WC. For the latter, we introduce the concept of a 1x10 data-array, which represents 10 segments of the transmission spectrum (i.e., the C-band in our study). In a 90-WC WDM network, each slot (bin) of the 1x10 data-array would correspond to a number of active-WCs in the range 1-10. A graphic representation of this concept is depicted in Fig. 2.

We further exploit the capabilities of the classifiers by attempting to predict the performance of newly established WCs in a network considering WLscenarios for which we do not train the ML models. That is, in a 90-WC WDM system, for the training data sets we only consider WL-scenarios for n active-WCs so that {n|n ∈ {1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70}}. Subsequently, we would register the labeled OSNR levels of the (n+1)-WC to be installed. On the opposite, for the testing data set we consider WL-scenarios where n can take any value so that {n|n ∈ {1 -90}}. Hence, the classifiers would attempt to predict the OSNR levels of the (n+1)-WC, which it may have not been trained for before.

The input labeled data used to train our models consisted in: the wavelength (measure in nm) of the monitored WC, 5 topology configuration settings (depicted in Table 1), and the 1x10 segmented spectrum data-array described above, accounting for 16 parameters in total. The data generation process with the Optical-MAN simulator consist in the following steps: Set a linear WDM network topology and configuration settings by randomly selecting a combination of values from Table 1.

(i)

Randomly select n-WCs to add to the network as per WL case, and add them to the network.

(ii)

Randomly select another (n+1)-WC and add it to the network. (iii)

For the (n+1)-WC, monitor power and noise levels, compute the OSNR levels, and generate a register (labeled sample).

(iv) Through this sequence of steps we can study the effects of the various topological and network configuration settings perceived on the transmitted signals. Also, because the deployed networks may be significantly different from one another, the generated data used to train the ML classifiers enable the performance analysis of such estimation tools in generic use cases. In our case, we attempt for the classifiers to learn about the physical effects resulting of the interaction of the WCs (i.e., power excursions) and the correlation that exists between the number of WCs active in the system, their position in the signal spectrum, and the transmission performance. Furthermore, we were interested in studying the scalability performance of the ML classifiers. For this we used six sets of independent data samples of different size. In total, we generated 294,343 data samples, which we split as 22.7 K, 42.3 K, 45 K, 51 K, 57.7 K, 75.6 K, for the six sets, respectively. Also, we balanced the input data in order to mitigate misperformance of the classifiers.

Classification algorithms and optimization techniques

This section describes the classification algorithms analyzed in this study. Only supervised learning-based algorithms were used for the development of this study. We have reviewed 11 algorithms, which we have split in two classes: normal and ensemble-based. The normal class is composed by K-Near Neighbors (K-NN), Linear-Support Vector Machine (L-SVM), Radial Basis Function SVM (RBF-SVM), Logistic Regression (LR), Decision Tree (DT), Artificial Neural Network (ANN), Naive Bayes (NB), and Linear Discriminant Analysis (LDA). The ensemble-based class is composed by Random Forest, Ada Boost, and Bagging. We have used the Scikit-learn: Machine Learning in Python API [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] for the development of the experiments. We briefly describe the main components of each algorithm and their functionality, for details of the models we redirect to the documentation. Naive Bayes this algorithm considers the assumption of conditional independence between the multiple features given the values of the various classes. Our implementation of this algorithm considered the likelihood of the sample features to be Gaussian.

Normal Classifiers

K-Near Neighbors this algorithm attempts to classify values in multiple classes by clustering the data and comparing individual values to their

Time complexity: O(nK), for K number of classes.

Linear Discriminant Analysis this algorithm operates by finding linear correlations of the input features. Time complexity: O(n 2 f eatures n samples ).

Ensemble Classifiers

These types of classifiers are the result of the combination of various classifiers altogether. These can be understood as optimization techniques for algorithms that do not perform well for given use cases, such as overfitting the data or performing weakly due to the lack of data. For the purposes of this study, we reviewed the ensemble classifiers of random forests, boosting and bagging.

Random Forest this ensemble learning method for classification consists in the combination of multiple decision trees at training time, and operates the classification of various classes by selecting the classification/prediction mode among the decision trees.

Ada Boost this algorithm, adaptive boosting (AdaBoost), operates by combining multiple "weak" classifiers in order to create a much stronger/accurate tool.

Bagging this algorithm, bootstrap aggregation (bagging), operates by bootstrapping (random sampling with replacement) multiple models in parallel, and come up with hypothesis of more accurate classifications, making a decision based on the most accurate hypothesis.

In order to speed up the execution time, our experiments were run in a Linux x86 64 server with 10 Intel(R) Xeon(R) CPU E5-2699 v4 @ 2.20 GHz processors. We used the Scikit-learn: Machine Learning in Python software [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] because it allows for a simplistic and variable implementation of the various classification algorithms. For instance, it enables the declaration of multiple parameters for training the models. Additionally, it enables a dynamic search of the best possible parameters to train specific models for given use cases with the GridSearchCV function. Thus, after a wide search of such parameters for our use case, we found those that enhanced a mean tolerable prediction accuracy among the various classifiers, which are depicted in Table 2.

The visual representation of the results of the normal classifiers are depicted in Fig. 3(a) and 3(b), showing the training computational time and the F1-score against the multiple sample sizes, respectively. We can demonstrate the tradeoff between high classification accuracy and high training computational time, which is a well-known feature among these type of classifiers.

However, the novelty of this analysis focuses on the ability for these statistical tools to learn from our selected optical domain features (i.e., active wavelength load) and the topological configurations (i.e., number of nodes and fiber spans) together. Taken these under consideration, the comparison shows that some algorithms perform better than the others, in some cases achieving up to 90% score. While in the literature we can find similar work achieving higher prediction accuracy [START_REF] Morais | Machine learning models for estimating quality of transmission in DWDM networks[END_REF], [START_REF] Tremblay | Machine Learning Techniques for Estimating the Quality of Transmission of Lightpaths[END_REF], [START_REF] Barletta | QoT estimation for unestablished lighpaths using machine learning[END_REF], our analysis considers EDFA power excursions and nonlinearities (SRS) that introduce power dynamics that are harder to predict. Thus, we believe the results can be further improved in the future by better depicting the physical layer parameters for training the algorithms.

In addition, there are two main patterns we can identify in Fig. 3(a). i) Contrary to an expected passive incremental accuracy performance as the volume of the sample size increases, our results show a decreased performance in the classifiers that can achieve the highest accuracy after 45,000 samples, and an increased/convergent performance in the classifiers that achieve a higher accuracy up to 70%. ii) In the figure, we can also perceive a group-like behaviour among some of the classifiers. Neglecting the RBF-SVM model (due to its deficient performance), we can see two groups, one consisting of the classifiers KNN, L-SVM, DT and NB; and the second with LR, ANN and LDA. While the F1score achieved by the former does not go higher than 70% in our experiments, the latter group show an F1-score performance of almost 90%. By our last sample data (75.6K), all of the classifiers seem to begin convergence. Intrinsically, each of these are mathematically-defined in different forms, however, we require further exploration that could justify a correlation with the manipulation of the input data to each of them and their performance when attempting to predict unestablished lightpaths as in our use case.

The visual representation of the results of the ensemble classifiers are depicted in Fig. 4(a) and 4(b). As for the normal classifiers, we can also see a trade-off between training computational time and the classification performance. While the results from the AdaBoost and the Bagging models were expected, the low performance of the Random Forest algorithm was not such. However, these results claim to be implementation-dependent, and due to the (almost) reasonable performance of the Decision Tree classifier in the previous analysis, we would further explore the implementation of Random Forest.

We have summarized these results in Table 3. In general, the common tradeoff between training computational time and accuracy performance suggests that these algorithms may be suitable for enhancing the speed of metro-access transport networks. Further research is required in the categorization of the physical layer features, so that we could build models performing with higher accuracy. In addition, more complex techniques such as deep-/reinforcement-/transferlearning, seem promising candidates to tackle the weaknesses of the algorithms presented in this study.

Conclusions

In this paper, we have introduced a performance analysis of classification algorithms to address the prediction of OSNR levels of newly established lightpaths in optical networks with different configuration settings and dimensions. Our results suggest that the implementation of such algorithms to complement the decision-making processes of control systems is promising, but requires further analysis of the physical layer features in the optical domain. Furthermore, we have found an interesting behavioural correlation between some of the classifiers that we will further explore in the future. Also, while in our implementation none of the classifiers achieved a F1-score performance above 90%, we believe that through finer tuning and with the implementation of more complex techniques, such as deep-/reinforcement-/transfer-learning, we could achieve better performance and find a better suitability for the metroaccess transport networks scenario.

Fig. 1 .

 1 Fig. 1. Linear topology. Deployment may vary in different network settings as per features in Table1.

Fig. 2 .

 2 Fig. 2. Segmented spectrum of EDFA wavelength-dependent gain.

Fig. 3 .

 3 Fig. 3. (a) Training time of each classifier. (b) F1-score of each classifier.

Fig. 4 .

 4 Fig. 4. (a) Training time of each ensemble classifier. (b) F1-score of each ensemble classifier.

Table

  .

Table 1 .

 1 Topology configuration parameters.

	Feature	Value
	Number of ROADMs	2 to 8
	Number of fiber spans	2 to 6
	Length of fiber span	80 to 150 km
	Launch channel power	-10 to 4 dBm
	EDFA Preamp gain Fiber compensation adjustment dB

  Decision Tree this algorithm consists in identifying relations between the input parameters and their output values building comparison points (tree branches), to subsequently perform as a binary tree.Time complexity: O(n f eatures n samples log(n samples )).Artificial Neural Network this algorithm is based on a multilayer perceptron, which learns correlations between inputs and outputs and generates 'weights' for future inputs that are used for minimizing the error of classification. Time complexity: O(nmh k oi), for n training samples, m features, h neurons per layer, k hidden layers, o output neurons, and i number of iterations.

	Radial Basis Function SVM Same as before.
	Time complexity: O(n f eatures n 2 samples ), with radial basis kernel function.
	Logistic Regression this algorithm utilises a logistic function to build a model
	of dependent variables, more commonly used for binary scenarios. However, by
	applying optimization algorithms it is possible to use logistic regression for classi-
	fying multiple classes. Our implementation considers the Limited-memory Broy-
	denFletcherGoldfarbShanno (L-BFGS) optimization algorithm.
	Time complexity: O(n 2 ), with L-BGFS.

K number of immediate neighbours. Time complexity: O(kn 2 ), for k nearest neighbors. Linear-Support Vector Machine this algorithm attempts to learn from the input data by categorising and building different classes (hyperplanes), which are subsequently used for future classification. It relies on regularization parameters C and gamma to avoid misclassification and correlation of individual points, respectively. It also transforms the problem with linear algebra -kernel functions. Time complexity: O(n f eatures n 2 samples ), with linear kernel function.

Table 2 .

 2 Parameters used for training the algorithms.

	Algorithm		Model Parameters
	K-Nearest Neighbours			K:1
	Linear SVM	kernel:linear, gamma:100, C:0.0001
	RBF SVM	kernel:rbf, gamma:100, C:0.0001
	Logistic Regression	solver:lbfgs, multiclass:multinomial, random˙state:1
	Decision Tree			max depth:5
	Artificial Neural Network		alpha:1, max iter:10000
	Naive Bayes			default
	Random Forest	max depth:5, estimators no.:10, max features:1
	AdaBoost			estimators no.:10
	Bagging	estimators no.:100, max samples:0.8, max features:0.8
	1.2 1.6 0.8 0.4 0.0 0 5 10 15 20 Seconds 22.7K 42.3K 45.0K 51.0K 57.7K 75.6K DT NB LDA KNN LR Sample size 0 150 300 450 600 L-SVM RBF-SVM ANN	F1-score (%)	0.9 0.8 0.7 22.7K 42.3K 45.0K 51.0K 57.7K 75.6K Sample size 0.0 0.3 0.4 0.5 0.6 0.1 0.2 KNN L-SVM RBF-SVM LR DT ANN NB LDA
	(a)			(b)

Table 3 .

 3 Results of the F1-score and training computation time for each classifier against the sample sizes.

		KNN	L-SVM	RBF-SVM	LR
	*SS F1-S (%) TT F1-S (%) TT F1-S (%) TT F1-S (%) TT
	22,723	0.34	2.09s	0.41	36.03s	0.16	56.05s	0.38	3.24s
	42,320	0.64	4.24s	0.6	1.44m	0.25	2.32m	0.76	9.1s
	44,952	0.66	4.45s	0.66	1.66m	0.25	2.82m	0.85	6.31s
	50,984	0.62	5.14s	0.59	1.96m	0.25	3.33m	0.75	11.49s
	57,740	0.66	5.72s	0.61	2.20m	0.29	3.99m	0.73	8.26s
	75,624	0.65	8.6s	0.63	3.95m	0.29	7.87m	0.72	13.27s
		DT	ANN	NB	LDA
	*SS F1-S (%) TT F1-S (%) TT F1-S (%) TT F1-S (%) TT
	22,723	0.46	0.27s	0.39	1.55m	0.45	0.23s	0.39	0.37s
	42,320	0.59	0.44s	0.79	2.17m	0.54	0.37s	0.75	0.60s
	44,952	0.6	0.48s	0.87	1.97m	0.6	0.42s	0.81	0.61s
	50,984	0.57	0.55s	0.77	2.62m	0.52	0.40s	0.73	0.69s
	57,740	0.58	0.90s	0.74	2.16m	0.58	0.41s	0.7	0.69s
	75,624	0.59	1.04s	0.74	2.92m	0.58	0.61s	0.69	0.86s
	* Note: SS: sample size; F1-S: F1-score; TT: training time.

Acknowledgements

Financial support from Science Foundation Ireland (SFI) 15/US-C2C/I3132 and 13/RC/2077 (CONNECT) are gratefully acknowledged.