Throughput Maximization Leveraging Just-Enough SNR Margin and Channel Spacing Optimization Cao Chen, Fen Zhou, Senior Member, IEEE, Yuanhao Liu, and Shilin Xiao

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 20XX 1 Throughput Maximization Leveraging Just-Enough SNR Margin and Channel Spacing Optimization Cao Chen, Fen Zhou, Senior Member, IEEE, Yuanhao Liu, and Shilin Xiao

Abstract—Flexible optical network is a promising technology the resource allocation consists of finding a proper lightpath to accommodate high-capacity demands in next-generation net- and assigning adequate spectrum resources. Traditionally, each works. To ensure uninterrupted communication, existing light- channel in wavelength-division multiplexing (WDM) networks path provisioning schemes are mainly done with the assumption of worst-case resource under-provisioning and fixed channel has to follow a rigid fixed-sized grid, e.g. 50 GHz. Recently, spacing, which preserves an excessive signal-to-noise ratio (SNR) in the new paradigm of flexible optical networks (FONs), margin. However, under a resource over-provisioning scenario, each channel can choose multiple contiguous frequency slots the excessive SNR margin restricts the transmission bit-rate (FSs) with the size of 12.5 GHz, and freely adjust the center or transmission reach, leading to physical layer resource waste frequency. More recently, with the advance of coherent detec- and stranded transmission capacity. To tackle this challenging problem, we leverage an iterative feedback tuning algorithm tion technology and commercially available digital-to-analog to provide a just-enough SNR margin, so as to maximize converters (DACs), a large number of transceiver’s parameters the network throughput. Specifically, the proposed algorithm become possible to further improve the spectral efficiency, is implemented in three steps. First, starting from the high including baud-rate, modulation format (MF), forward error SNR margin setup, we establish an integer linear programming correction (FEC) overhead, probabilistic shaping, etc [2, 3]. model as well as a heuristic algorithm to maximize the network throughput by solving the problem of routing, modulation format, The development of optical networks enables us to reduce forward error correction, baud-rate selection, and spectrum the spectrum resource waste, increase spectral efficiency, and assignment. Second, we optimize the channel spacing of the obtain the higher transmission capacity [4, 5]. However, it is lightpaths obtained from the previous step, thereby increasing the not easy to jointly optimize these parameters. available physical layer resources. Finally, we iteratively reduce In addition, the selection criteria of these flexible parameters the SNR margin of each lightpath until the network throughput cannot be increased. Through numerical simulations, we confirm must satisfy the quality of transmission (QoT) requirement. the throughput improvement in different networks and with Each lightpath experiences not only the amplified spontaneous different baud-rates. In particular, we find that our algorithm emission (ASE) noise of optical amplifiers, but also the fiber enables over 20% relative gain when network resource is over- nonlinear interference (NLI) from other lightpaths that share a provisioned, compared to the traditional method preserving an common link. To avoid the complex calculation of NLI while excessive SNR margin. guaranteeing uninterrupted communication, the conventional Index Terms—Flexible Optical Networks (FONs); Flexible practice mostly assumes that each lightpath experiences the Baud-Rate; Throughput Maximization; Just-Enough SNR Mar- most NLI under resource underprovisioning scenario, which gin; Channel Spacing Optimization; preserves an excessive signal-to-noise (SNR) margin. The high SNR margin or the overestimation of NLI potentially inhibits I.INTRODUCTION the capacity or restricts the maximal transmission reach. To this end, NLI-aware resource allocation techniques have been CCORDING to recent traffic reports, network traffic proposed to reduce the overestimation to exploit the stranded (fueled by network services like video on demand, file A capacity [6]–[10]. This operation mode is also named as low sharing, online gaming, video conferencing, etc.) are still margin network in [11]–[13], or just-enough SNR margin in arXiv:2106.07536v2 [cs.NI] 17 Jul 2021 growing exponentially in today’s Internet [1]. The constant this paper. traffic growth relies on optical networks. In optical networks, Nevertheless, baud-rate was assumed to ideally match the The work is jointly supported by Eiffel Scholarship (No. 895145D), China occupied spectrum resource in most prior studies [6]–[10], Scholarship Council (No. 201806230093 and No. 202006960046), National thus they only need to consider the route, modulation format, Nature Science Fund of China (No.61775137, No.62071295, No.61431009, and spectrum assignment (RMSA). Limited by the frequency and No.61433009) and the National "863" Hi-tech Project of China. Cao Chen is with the State Key Laboratory of Advanced Optical Com- grid, different baud-rates (or the multi-baud-rates) can be ac- munication Systems and Networks, Shanghai Jiao Tong University, Shanghai, commodated by the same channel once the allocated spectrum 200240, China. He is also with the CERI-LIA in Avignon University, France resource is sufficient [14]. Recent studies have also shown that (email: [email protected]). Fen Zhou is with IMT Lille Douai, Institut Mines-Télécom, Univer- the same baud-rate using different bandwidths will experience sity of Lille, Center for Digital Systems, F-59000 Lille, France (email: baud-rate-related impairments, including the filter narrowing [email protected]). effect [15, 16], or the impairments caused by the resolution Yuanhao Liu is with the State Key Laboratory of Integrated Service Networks, School of Telecommunications Engineering, Xidian University, of DACs [17]. Thus, the resource allocation incorporating Xi’an, 710071, China. He is also with the CERI-LIA in Avignon University, the baud-rate selection maps into a new problem of routing, France (email: [email protected]). modulation format, FEC, baud-rate, and spectrum assignment, Shilin Xiao is with the State Key Laboratory of Advanced Optical Com- munication Systems and Networks, Shanghai Jiao Tong University, Shanghai, which was also investigated by [18]–[21]. Although some 200240, China (email: [email protected]). researchers have proposed the efficient heuristic algorithm CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 2 optimizing these parameters under the low SNR margin mode [22]–[24], there is no integer linear programming (ILP) model on the joint optimization of the route, MF, FEC, baud-rate, and spectrum assignment, especially aiming at improving the transmission capacity. This problem could become more complex when accounting for the various impairments. (by initial bit-rate list) (by adaptive bit-rate) Optimizing the physical layer parameter is also important to Traditional This paper mitigate the NLI and improve network capacity [12, 25, 26]. 1. excessive SNR margin 1. just-enough SNR margin - worst-case assumption - no assumption As reported by the recent studies of the physical layer (e.g., - fixed channel spacing - flexible channel spacing Gaussian Noise model on the Nyquist WDM channel [27, 28]), 100Gbps 2. fixed capacity =100Gbps 2. flexible capacity = 150Gbps the NLI between channels relates to the baud-rate, power, and ... Gbps different spectrum positions (channel spacing and channel order). There is a large volume of published studies on the power Fig. 1. Illustration of two operation modes in FON. optimization [5, 8, 29, 30] and the channel order optimization (the different relative orders of channels) [30, 31]. The last factor—channel spacing—is generally implemented by setting implement this function in large networks, we propose an a guard band, or more simply, by setting a fixed channel iterative feedback tuning algorithm to provide a just-enough spacing. Throughout this paper, we use the terms “guard band” SNR margin for each lightpath. Specifically, our algorithm is and “channel spacing” interchangeably. By setting a proper implemented in three steps. First, starting from a high SNR channel spacing, we can obtain the desired metrics, such as margin, we establish an ILP model and a heuristic algorithm improving the SNR performance [15, 32] or ensuring a proper that maximizes the network throughput. Unlike the existing security level [33]. Thus, it can be formulated as another lightpath provisioning that addresses the RMSA, the baud-rate optimization problem, channel spacing optimization problem, is also jointly optimized. Second, we provide a low-complexity as we will study in this paper. Although the transmission channel spacing optimization model to mitigate the NLI, which performance of different channel spacing has been widely is implemented by using the nearest neighbor channel. This studied [34] in the physical layer, a few studies concentrate on model also extends the application case from a ring network the optimization from the network operator’s view. While the into a mesh network compared to our previous study [36]. study in [35] has reported the best channel spacing strategy Finally, we iteratively reduce the SNR margin of each lightpath aiming at reducing the cost, it is still unclear the transmission to the just-enough level to increase the throughput. The main bit-rate gain of channel spacing optimization. contributions are summarized as follows, Traditionally, to obtain a higher transmission capacity, one We propose to maximize the throughput by leveraging • may take the evaluation metric, minimizing the maximum spec- just-enough SNR margin and channel spacing optimiza- tral usage, to spare spectrum resources for other lightpaths. tion. Through analyzing and evaluating the impact of dif- However, restricting the spectrum usage potentially encourages ferent lightpath’s parameters (route, MF, FEC, baud-rate, narrow channel spacing and neglects the benefit of large spectrum assignment), we determine the value of just- channel spacing on mitigating the filter narrowing effect or enough SNR margin accordingly. As far as we know, this NLI. Thus, the proposed channel spacing optimization strategy is the first time that the flexible transceiver’s parameters allows the channel to be freely adjusted among the available are jointly optimized while considering the NLI. spectrum resources without guard band constraints or spectrum We introduce an LP model based on the nearest neighbor • resource usage constraints, to earn the ultimate physical layer channel to optimize the channel spacing, resulting in a performance improvement. Recently, the method of individu- low-complexity method that applies to a large number ally optimizing the center frequency seems to be a powerful of lightpaths in both ring and mesh optical networks. technique [8, 31] to optimize the channel spacing rather than Besides, we compare the strategy with the widely used using the heuristic algorithm [32]. However, this novel method channel spacing strategies (fixed and candidate channel may face difficulty when increasing the number of lightpaths. spacing) in terms of the SNR performance and achievable To lower the complexity, we further propose a linear pro- throughput. gramming (LP) model based on the nearest neighbor channel, Next, for a given resource over-provisioning scenario, we • successfully extending the application case to the general mesh devise an iterative feedback tuning algorithm that can optical network with a large number of lightpaths. efficiently solve the throughput maximization leveraging In this paper, we aim at maximizing the network throughput just-enough SNR margin and channel spacing optimiza- in a static FON by leveraging just-enough SNR margin and tion problem. Specifically, we reduce the SNR margin channel spacing optimization. As shown in Fig. 1, only the by adjusting a slack parameter iteratively so that we can lowest transmission capacity can be adopted by the lightpath obtain the just-enough SNR margin. connecting (s, d) when considering the excessive SNR margin Finally, through extensive simulations in different net- • and fixed channel spacing, namely 100 Gbps in the traditional work topologies and with different baud-rates, we confirm operation mode. The transmission capacity can increase up the throughput improvement with the help of the just- to 150 Gbps or larger if leveraging the just-enough SNR enough SNR margin and channel spacing optimization. margin provisioning and channel spacing optimization. To Specifically, we observe that the relative throughput gain CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 3

ratio (no smaller than 20% in our case) is more evident in For each lightpath, the SNR will be degraded by the ASE resource over-provisioning scenarios, while the absolute noise and NLI. Supposing the transceiver of lightpath p uses throughput gain is more evident when network traffic load a launch power spectral density (PSD) Gp, its SNR is stated is medium. as follows [28, 38], The rest of this paper is organized as follows. In Sec. II, R G SNR = t p we present the preliminary information of the lightpath provi- p ·ASE NLI (1a) fgrid Bt (Gp + Gp ) throughput maximization leveraging · · sioning in FONs and the R G 1 just-enough SNR margin and the channel spacing optimiza- = t p · ASE GNLI (1b) fgrid Bt Gp · p tion problem 1 + ASE . Next, we propose an iterative feedback tuning · · Gp algorithm to solve this problem in Sec. III. To reduce the 1 1 = SNRbest = SNRbest complexity of the lightpath provisioning, we also present an p GNLI p (1c) · p p 1 + ASE efficient heuristic in Sec. IV. Illustrative numerical results are Gp X presented in Sec. V. Finally, Sec. VI concludes this paper. ASE where Gp is the accumulated PSD of ASE noise in optical NLI amplifiers, Gp is the accumulated PSD of NLI on fibers, II.THROUGHPUT MAXIMIZATION LEVERAGING Rt Gp best · SNRp = f B GASE is the received best-case SNR JUST-ENOUGH SNRMARGINAND CHANNEL SPACING grid· t· p GNLI PTIMIZATION ROBLEM p O P without NLI, p = 1 + GASE is the relative strength of NLI. X p In this section, we present the studied throughput max- By assuming a coherent receiver with adaptive digital signal imization leveraging just-enough SNR margin and channel processors and a matched receiver filter, the noise equivalent spacing optimization problem. First, we provide the necessary bandwidth of fgrid Bt is equal to the baud-rate Rt. Thus, we · information about the lightpath provisioning of FONs and can ignore the impact of bandwidth and baud-rate on the SNR explain how we calculate the SNR. Then, we describe the evaluation in the following. We also assume a uniform launch throughput maximization problem and present a benchmark PSD for all transceivers. method that utilizes the fixed baud-rate. Next, we present the The NLI strength in Eq. (1c) depends on the occupied Xp key problem to be solved in this paper. Finally, a case study channels of lightpath. For ease of discussion, we present the is used to illustrate the throughput improvement with the help log form of SNR as follows, of channel spacing optimization and just-enough SNR margin. SNR = SNRbest (2) p,[dB] p,[dB] − Xp,[dB] NLI A. Network model Gp where p,[dB] = 10 log10 1 + GASE . Next, we explain the We use G = (V,E) to denote an FON, where V and E are X p calculation of ASE noise and NLI. the node and link set, respectively. A link l E represents ∈ a fiber uv (u, v V ). On that fiber, the available spectrum ASE. We use the following equation to calculate the ASE ∈ • resource is F (unit: GHz), while the number of available FSs noise of optical amplifiers, is W (F =fgrid W , fgrid is 12.5 GHz bandwidth). A bit-mask ASE αL · G = N 10 span 1 n hν (3) b[1...W ] of size W is introduced to represent the occupied p p − sp th FS status of a fiber link, where b[w]=1 means the w FS is where Np is the number of spans, Lspan (km) is the occupied. We also use b [w] to represent whether the entity length of one span between two optical amplifiers, α th ∗ ‘*’ uses the w FS, where ‘*’ could be either a lightpath p, is the fiber attenuation factor, nsp is the noise figure of or a link l, or a channel ch. optical amplifiers, h is the Planck constant, and ν is the We assume that each lightpath p connecting a node pair absolute frequency of optical signal at 1,550 nm. (s, d) can adopt a transceiver t operating at a baud-rate Rt, NLI. We adopt the dilog method in [28, Eq. (11)] that • which needs to occupy a certain number of FSs denoted by calculates the NLI based on channel, Bt (refer to [37, Table 3]). The adopted transceiver is assumed X GNLI = N ηSCI G 3 + G ηXCI (f )N G 2 with the rectangular Nyquist spectrum on a channel ch. Note p p ch p p ch,ch0 ch,ch0 p,p0 p0 that the different baud-rates R could use the same amount ch0 t (4) of FS(s) Bt, or the same baud-rate could use the different amount of FS(s) only if B f R [14], we determine where ch and ch0 are the occupied channel of lightpath t · grid ≥ t the available bit-rate by the baud-rate Rt rather than occupied p and p0 respectively, Np,p0 is the number of common SCI XCI B . Also, the bit-rate depends on the spectral efficiency of spans of lightpath p and p0, η and η are the self- t ch ch,ch0 adopted MF and FEC (an example table that shows the SNR channel interference (SCI) efficiency and cross-channel threshold and bit-rate of transceiver at 32 Gbaud can refer to interference (XCI) efficiency respectively, fch,ch0 is the [38, Table I]). For ease of expression, we introduce the term channel spacing of two lightpaths (this paper uses the transmission mode to refer to a combination of MF and FEC. center frequency difference, i.e., f = f f ). ch,ch0 | ch − ch0 | Such a transmission mode can be adopted by the lightpath if An example curve of the XCI efficiency at different the SNR is no smaller than the SNR threshold of joint MF and channel spacing can refer to [39, Fig. 1]. Meanwhile, we FEC, i.e. QoT is satisfied. Next, we describe how we calculate ignore the impact of SCI, as it can be compensated by the SNR of each lightpath. digital back propagation [40] and has a weak impact on CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 4

the parameter of channel spacing [28]. QoT constraint, and subsequently focus on the lightpath provisioning by using a conservative transmission reach B. Throughput maximization problem or the conservative spectral efficiency. C2 - one baud-rate and one transmission mode con- Let us now look at the network throughput. We first define • straint: Each lightpath can adopt one and only one baud- a traffic demand matrix D, where each item Ds,d denotes the bit-rate demand between each node pair (s, d). We also define rate and one transmission mode. C3 - spectrum contiguity constraint: For each used another network provisioning capacity matrix T , where each • baud-rate Rt, we need to allocate Bt contiguous FSs, item Ts,d is the transmission capacity of the lightpaths between Bt node pair (s, d). To ensure that the traffic demand could be z }| { namely bch[1...W ] = [0 0 11 11 0 0]. It needs to fully accommodated, the bit-rate demand should be no bigger | ··· ···{z ··· } than the transmission capacity, i.e. D T . The network W s,d ≤ s,d ensure the size of spectrum allocated is sufficient, that is throughput TH in this paper is the sum of all bit-rate demands, f B R . P grid · t ≥ t i.e. TH = (s,d) Ds,d. C4 - spectrum continuity constraint: The occupied FSs • As this paper considers a static lightpath provisioning, we of the channel for a lightpath should be identical on each begin by assuming a constant traffic distribution, namely the link. ˆ ˆ normalized traffic demand matrix D is constant, where Ds,d = C5 - spectrum non-overlapping constraint: The occu- Ds,d ˆ • P D . Thus, we have TH Ds,d = Ds,d Ts,d. (s,d) s,d · ≤ pied FSs of two lightpaths should not overlap. Alterna- Given a normalized traffic demand matrix Dˆ and a trans- tively, each FS of a link can be used at most once. mission mode set , the objective function is expressed as Let us discuss the constraints (C1) and (C1*). Traditional M follows, method takes constraint (C1*) rather than (C1). The benefit is   that lightpath provisioning can be greatly simplified by split- X  ting the physical layer calculation and lightpath provisioning max Ds,d (5) (s,d)  when the SNR margin is enough for compensating the NLI. The limitation is that the improper SNR margin Mp requires In this problem, we need to individually design the number each lightpath to adopt the low order transmission mode with of lightpaths to implement the network provisioning capacity a low SNR threshold. As a result, the transmission bit-rate Ts,d for each node pair. For each lightpath, the route, MF, of each lightpath is strictly constrained, especially under the FEC, baud-rate, and spectrum assignment needs to satisfy the resource over-provisioning scenario where the actual NLI may following constraints. be far less than the worst-case NLI. C1 - QoT constraint: Each lightpath should guarantee a • Here, to make it clear, we present a benchmark method that satisfied QoT. The QoT constraint can be expressed either preserves excessive SNR margin [5]. It mainly includes two SNRp by the QoT metric p = SNRthreshold 1, or by its log parts, a lightpath precalculation based on constraint (C1*) and Q m ≥ form using Eq. (2), an ILP model based on constraints (C2)-(C5). Using the precalculated lightpaths, the ILP model calculates the maximal SNRbest SNRthreshold 0, (6) p,[dB] − Xp,[dB] − m,[dB] ≥ throughput by jointly solving the routing, transmission mode, threshold where SNRm is the SNR threshold of the adopted and spectrum assignment problem (on a fixed wavelength threshold transmission mode m. The SNRm of a transmis- grid). Additionally, we use another constraint to define the sion could vary with the baud-rates [17] or the aging mentioned scenario of resource over-provisioning, where the factor. Here, we assume ideal case that it only varies with actual NLI is much lower than the preserved SNR margin. the adopted transmission mode. Recall that the second This constraint limits the current maximum FS index Wcur, term in Eq. (6) depends on the route and spectrum Wcur W . Xp ≤ positioning of other undetermined lightpaths, which could 1) Lightpath precalculation: This part provides the can- dramatically increase the difficulty of lightpath provi- didate lightpaths to be used in the following optimization sioning. A simple way to eliminate in Eq. (6) is model. These candidate lightpaths are obtained by using the Xp using a high SNR margin Mp. Such a margin needs to K-shortest path algorithm [41] to generate the different routes, compensate all possible NLI of lightpath p. Thus, we give using the channel index method [42] to generate the possible worst the alternative constraint, SNR margin requirement. channel locations, and using Mp = to generate Xp C1* - SNR margin requirement: the available transmission modes. For a certain lightpath, its • transmission capacity will be zero if the Eq. (7) is not satisfied. SNRbest M SNRthreshold 0, (7) p,[dB] − p,[dB] − m,[dB] ≥ 2) Maximize throughput: This part maximizes the network where Mp is the SNR margin used for compensating throughput based on (C2)-(C5). the unknown NLI. Generally, the SNR margin M Parameters p ≥ worst, where worst is calculated under the worst- Dˆ , normalized traffic demand ratio of node pair (s, d). Xp Xp • s,d case assumption in which other spectrum resources of ch CH, channel index of different contiguous FSs. For • ∈ the links along lightpath are fully occupied by different example, for the transceiver using two contiguous FSs, channels. With the SNR margin requirement constraint, CH = [110 ], [0110 ], , [ 011] . { ··· ··· ··· ··· } we can neglect the complex physical layer calculation in b [w], equals 1 if the channel ch uses the wth FS, 0 • ch CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 5

otherwise. improve the global scaling factor of TH. Meanwhile, these pch,m , lightpath ID of the kth route of (s, d) using lightpaths will however introduce extra network cost and NLI. • s,d,k ∈ P channel ch and transmission mode m. Thus, the second question is that ii) how to remove these Cch,m, transmission bit-rate of the lightpath pch,m if it lightpaths while guaranteeing the current maximal network • s,d,k s,d,k satisfies Eq. (7), 0 otherwise. throughput? β , equals 1 if the kth route of node pair (s, d) uses • s,d,k,l Returning to the discussion of Eqs. (6) and (7), we can link l, 0 otherwise. now say that one important approach to increase the network Variables throughput is to allow the high-order transmission modes for each lightpath, i.e., increasing more lightpaths pch,m that Ds,d, bit-rate demand of node pair (s, d). s,d,k • pass the SNR margin requirement in precalculation process. Ts,d, provisioning capacity of node pair (s, d). • ch,m ch,m Mitigating the NLI strength of in Eq. (6) or lowering δ , equals 1 if the lightpath p is adopted, 0 Xp • s,d,k s,d,k the SNR margin M in Eq. (7) both provide the possibility. otherwise. p On the one hand, to mitigate NLI, prior studies have opti- TH, network throughput. • mized the power and channel order allocation [8, 30, 44]. Objective This paper will study it from another perspective, how to max TH (Max TH) use the idle spectrum resources in resource overpromising ch,m δs,d,k ,T H scenario, namely optimizing the channel spacing. Thus, the third question is that iii) how to minimize the NLI to allow s.t. Ds,d = TH Dˆ s,d, s, d = s (8a) X · ∀ 6 each lightpath to adopt a higher-order transmission mode? T = δch,m Cch,m, s, d = s (8b) s,d s,d,k · s,d,k ∀ 6 On the other hand, lowering the SNR performance has been k,ch,m an important subject of the prior studies [5, 8, 9, 23]. It seems Ds,d Ts,d, s, d = s (8c) that the low SNR margin provisioning is incorporated with X≤ ∀ 6 δch,m β b [w] 1, l, w the lightpath provisioning simultaneously, which could incur s,d,k · s,d,k,l · ch ≤ ∀ s,d=s,k,ch,m difficulty in lightpath provisioning. In addition, to the best 6 (8d) of our knowledge, the resource allocation with the flexible transceiver parameters has not been employed under the re- δch,m b [w] = 0. s, d, k, ch, W < w W, m s,d,k · ch ∀ cur ≤ source overprovisioning scenario [5]. Thus, the fourth question (8e) is that iv) how to find a properly preserved SNR margin to suit The objective is to maximize the network throughput. Con- the resource overprovisioning scenario to allow the lightpath straints (8a) ensure that the bit-rate demand follows the to adopt a higher-order transmission mode? distribution of the given normalized traffic demand matrix Dˆ Question i) is solved by incorporating the baud-rate into for each node pair. Constraints (8b) obtain the transmission the lightpath pre-calculation. Question ii) is solved by adding a capacity between (s, d). Constraints (8c) ensure that the traffic posterior optimization similar to Max TH but using a modified demand could be fully accommodated by the network provi- objective function that minimizes the total lightpaths. The sioning capacity. Constraints (8d) ensure that each FS of a exact technique for these two questions will be presented in link can be used at most once, which also guarantee only one Sec. III. The last two questions, iii) and iv) both involve the transmission mode for a lightpath. Constraints (8e) are used physical layer optimization. Here, we present the two problems to simulate the resource over-provisioning scenario by the FS sequentially. index Wcur. Finally, we see that the constraints (C3) and (C4) 1) Channel spacing optimization: Given a lightpath set are ignored due to the usage of pre-calculated lightpaths. with determined route, determined baud-rate, and de- Padopt termined transmission mode, we want to enhance the SNR by minimizing the NLI for each lightpath. When minimizing the C. Throughput maximization leveraging just-enough SNR NLI, the QoT metric of different lightpaths is considered. margin provisioning and channel spacing optimization Qp Thus, we leverage the three terms on the left side of Eq. (6) as The benchmark only allows optimization in networks con- the objective. The objective function is expressed as follows, figured by a single baud-rate. This function may be insufficient threshold best for the application in future optical networks that allows the min max p,[dB] + SNRm,[dB] SNRp,[dB] fch p adopt X − coexistence of flexible baud-rate optical networks [43]. Thus, ∈P the first question is that i) how to efficiently use the flexible s.t. (C1), (C5) baud-rate transceiver for lightpath provisioning? In addition, the benchmark method may inevitably generate redundant 2) Just-enough SNR margin provisioning: Given a normal- (s, d) lightpaths. For example, for some node pair with a short ized traffic demand matrix, a transmission mode set , and a M distance, their lightpaths have a high provisioning capacity, baud-rate set , the objective function is expressed as follows, easily satisfies the bit-rate demand, and probably leaves idle T spectrum resources on the connected links. It is inevitable to generate additional lightpaths that still connect the same max TH δch,m,R,f ,M node pair using these idle spectrum resources. Nevertheless, s,d,k ch p the local capacity improvement of several node pairs may not s.t. (C1), (C2) (C5) − CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 6

Unlike the benchmark method, the just-enough SNR margin Actual(lightpath) Threshold(lightpath) provisioning needs to determine a proper SNR margin Mp Threshold(16QAM) Threshold(64QAM) for each lightpath. Besides, the route (s, d, k), channel ch, (a) m R transmission mode , and baud-rate , as well as a proper 16 channel location fch for each lightpath are to be optimized. 3) A small instance: We provide a small instance to il- 15.5 lustrate the throughput improvement when adopting the just- enough SNR margin provisioning and channel spacing opti- 15 mization. This instance assumes 30 lightpaths in a point-to- point network with available spectrum resources of 4,000 GHz SNR [dB] 14.5 (F =4,000 GHz, W =320 FSs). In Fig. 2, we plot the SNR dis- min p = 0.61 dB 0dB tribution of 30 lightpaths provisioned by using three different Q ≥ 14 techniques: (a) traditional provisioning with an excessive SNR 0 50 100 150 200 250 300 margin, (b) provisioning with channel spacing optimization, (b) and (c) provisioning with just-enough SNR margin. Each red 16 circle denotes a lightpath that uses two contiguous FSs at 16 Gbaud, where x-axis is the spectrum position and y-axis is 15.5 the lightpath’s actual SNR. They can adopt either the transmission mode (16QAM, FEC = 0.92) or (64QAM, FEC = 0.68) 15 [38]. The bit-rates are set with 112.5 and 125 Gbps to match the 16 Gbaud transceiver in our assumption [38]. More details SNR [dB] 14.5 min p = 1.43 dB 0dB of the physical layer parameters (fiber type, EDFA noise Q ≥ figure, and span length) can refer to the simulation setup in 14 Sec. V-A. The uniform PSD Gopt (15.03µW/GHz) is given 0 50 100 150 200 250 300 by using a strategy similar to LOGON in [45, Eq. (6)] among (c) 4,000 GHz, in which we assume other spectrum resources are 16 fully occupied by all possible channels. In Fig. 2(a), 30 lightpaths all adopt 16QAM and provide 15.5 the bit-rate of 112.5 Gbps. The QoT is satisfied because all min p = 0.04 dB 0dB Q ≥ SNRs are over the SNR threshold( min = 0.61 dB > 0 dB). Qp 15 Next, if we apply the channel spacing optimization, as shown in Fig. 2(b), the SNRs of these lightpaths rise because of SNR [dB] 14.5 the less NLI( min = 1.43 dB). In Fig. 2(c), by using a Qp lower SNR margin, we can upgrade the transmission mode to 14 64QAM which provides a larger bit-rate of 125 Gbps while 0 50 100 150 200 250 300 guaranteeing the QoT for all lightpaths( min p = 0.04 dB). FS index Q Therefore, we can increase the network throughput from 30 112.5 Gbps in Fig. 2(a) to 30 125 Gbps in Fig. 2(c). Fig. 2. An example of lightpath provisioning leveraging just-enough SNR × × It should be also mentioned that the maximum SNR mar- margin and channel spacing optimization in a point-to-point network with 600 km (Wcur=60 FSs, W =320 FSs, PSD=15 µW/GHz in this example). gin improvement of the traditional fixed channel spacing is The lightpaths need to satisfy the QoT, i.e. Eq. (6) holds. (a) Traditional 0.02 dB (min =0.02 dB) in Fig. 2(c) (not shown in the provisioning through preserving an excessive SNR margin (throughput is Qp figure for the sake of clarity), which is achieved by setting 30 112.5 Gbps); (b) Provisioning with channel spacing optimization; (c) Provisioning× with just-enough SNR margin (throughput is 30 125 Gbps). 125 GHz frequency grid. This metric is still less than the × 0.04 dB of flexible channel spacing. Through the example in Fig. 2, we can upgrade the transmission mode and increase the bit-rate by properly setting and channel spacing optimization problem. The flowchart is the channel spacing and carefully adjusting the SNR margin. illustrated in Fig. 3. The first two phases mainly follow the However, it seems not easy to simultaneously adjust the benchmark method in Sec. II-B. The difference lies in that channel spacing and SNR margin, especially for a large the flexible baud-rates are incorporated. Then, we use Phase number of lightpaths. To tackle this challenge, we design an 3 to remove the redundant lightpaths of Phase 2. Next, iterative feedback tuning algorithm to iteratively adjust these in Phase 4, we optimize the channel spacing through an parameters until the just-enough SNR margin is found. LP model. Motivated by [5], we try to determine the just- enough SNR margin Mp with a unit gradient ratio step of ∆M . Compared to last iteration, more high-order transmission III.ITERATIVE FEEDBACK TUNING ALGORITHMAND modes are permitted for the precalculated lightpaths in Phase MATHEMATICAL MODELS 1, thus increase the possibility of Phase 2 using high-order In this section, we present the algorithm that solves the transmission mode, and consequently obtain higher network throughput maximization leveraging just-enough SNR margin throughput. Finally, the algorithm terminates once there exists CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 7 a lightpath in the network that violates the QoT constraint. δch,m,R b [w] = 0. s, d, k, ch,R,W < w W, m s,d,k · ch ∀ cur ≤ (10e)

G(V,E), Dˆ , Wcur, baud-rate set , TM T The description of these constraints can refer to the bench- transmission mode set M mark method in Sec. II-B. Different from that, we have ch,m ch,m,R replaced the variable δs,d,k with δs,d,k in order to optimize Phase 1: pre-calculate lightpaths (Sec. III-A) the baud-rate. Thus, the route (s, d, k), spectrum position ch,

Mp,dB = M ∆ , P transmission mode m, and baud-rate R in Max TH with p,dB − M,dB p Phase 2: Max TH with FB (ILP, Sec. III-B ) ∀ ∈ P FB are jointly optimized. THmax 2) Phase 3: (n) TH = THmax Phase 3: Remove redundant(ILP, Sec. III-B ) (n) = 0 Input: candidate lightpath set , capacity set , boolean Padopt Padopt (n) adopt • fp = fp P C ˆ P indicator βs,d,k,l, normalized trafﬁc demand matrix D, Phase 4: optimize channel spacing(LP, Sec. III-C ) n = n + 1 network throughput THmax. , fp Padopt0 Output: adopted lightpath set . • Padopt Process: this phase removes the redundant lightpaths of min p,dB 0? • p Q ≥ Yes ∈Padopt0 previous model. Based on the model Max TH with

No FB, we use the objective of minimizing the total number

(n) (n) (n) of lightpaths while maintaining the network throughput Output TH , adopt, fp P THmax.

X ch,m,R Fig. 3. Flowchart of our iterative feedback tuning algorithm. min δs,d,k (Remove Redundant) δch,m,R s,d,k s,d=s,k,ch,m,R 6 s.t. (10a) (10e) − A. Lightpath precalculation with flexible baud-rate TH THmax (11a) ≥ Input: topology G(V,E), maximal FS index W , • cur The objective is to minimize the total number of lightpaths transceiver set , transmission mode set , SNR margin T worst M in the network. Constraint (11a) maintains the throughput Mp (initialized by ). Xp THmax that is obtained from previous model. The redundant Output: lightpath set pch,m,R , capacity set • s,d,k ∈ P lightpaths can be removed through this model. ch,m,R , boolean indicator β . Cs,d,k ∈ C s,d,k,l Process: it stores the lightpath p that satisfies the SNR • C. Channel spacing optimization margin requirement in Eq. (7), where M acts as a slack p Input: adopted lightpath set . variable for each iteration. The satisfied lightpaths are • Padopt Output: optimal center frequency f , 0 on the stored into the candidate lightpath set and the capacity • p Padopt P optimal center frequency f . set . Different from the lightpath precalculation in Sec. p C ch,m,R II-B, here, we use p to denote the lightpath ID Process: min max ( + SNRthreshold SNRbest ) . s,d,k • p m,[dB] p,[dB] fp p adopt X − ch,m ∈P rather than ps,d,k . The channel index set CH is also extended to accommodate the different baud-rates. The objective function can be also expressed in the linear n n threshold oo SNRm form min max SNRbest p . We take the cen- fp p ·X B. Throughput maximization with flexible baud-rate ter frequency as a continuous variable among the available 1) Phase 2: spectrum resources [0,F ]. Thus, the center frequency of each Input: candidate lightpath set , capacity set , boolean lightpath can be individually optimized by an LP model. • P C ˆ Meanwhile, we have to resolve the issue that the relation- indicator βs,d,k,l, normalized traffic demand matrix D. Output: network throughput TH . ship between XCI efficiency and channel spacing in Eq. (4) • max Process: this phase calculates the maximal network is nonlinear. To this end, we use a piecewise-linear fitting • throughput through the following model, function [8]. Finally, the continuous variable fp is rounded to fit the frequency grid of fgrid (unit: GHz). The frequency max TH (Max TH with FB) mapping process in the final step is decomposed into two-step δch,m,R,T H s,d,k to accelerate the optimization. s.t. D = TH Dˆ , s, d = s (10a) The linear approximation function for fitting the XCI effi- s,d · s,d ∀ 6 X ch,m,R ch,m,R ciency is expressed as follows [8], Ts,d = δs,d,k Cs,d,k , s, d = s · ∀ 6 Bp,B Bp,B k,ch,m,R XCI p0 p0 η¯Bp,B (fch,ch0 ) = max aq fch,ch0 + bq (12) p0 1 q Q (10b) ≤ ≤ Bp,B Ds,d Ts,d, s, d = s (10c) Q Q a p0 ≤ ∀ 6 where is the number of fitting segments ( = 5), q X ch,m,R Bp,Bp δ β b [w] 1, l, w and bq 0 are the coefficients calculated by the algorithm in s,d,k · s,d,k,l · ch ≤ ∀ s,d=s,k,ch,m,R [46] based on the XCI efficiency of the channels from p0 to p 6 (10d) with frequency step of 1 GHz in Eq. (4). CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 8

Parameters can be dealt with independently. Thus, we place this constraint f , bandwidth of an FS (12.5 GHz). as a check function in the final step of our algorithm (see • grid xp,l, equals 1 if lightpath p uses link l, 0 otherwise. Sec. III-D). Here, the main variables to be optimized in Phase • 4 are the center frequencies f and η¯XCI . y , equals 1 if lightpath p and p0 share a common fiber p p,p0 • p,p0 link, 0 otherwise. One advantage of the above model is to obtain the ideal γ , spectrum position of a lightpath p. Here, we take the maximal performance improvement of individually optimizing • p starting FS index. the channel spacing. However, there are certain drawbacks u , equals 1 if γ is bigger than γ , 0 otherwise. associated with the method of fully estimating the NLI. For • p,p0 p p0 B , number of occupied FSs of a lightpath p. instance, assuming a simple case in a point-to-point link, the • p m , transmission mode of p. number of constraints in Eqs. (13c) and (13d) could approach • p Bp,B Bp,B 2 p0 p0 to (Q adopt ). Such a square relationship probably aq , bq , coefficients of piecewise linear fitting • O · |P | function. restricts the optimization to a small number of lightpaths, GASE, PSD of ASE between two optical amplifiers, i.e. which could be much difficult when facing a large number • span per one span. of lightpaths. To reduce the issue of complexity, we propose a nearest Variables neighborhood channel estimation method. This estimation f , center frequency of lightpath p that is relative to the • p method needs to use two parameters that describe the nearest optical signal frequency ν. channel relationship between lightpaths, y(1) and y(2) . y(1) η¯XCI , linear approximation of the XCI efficiency from p,p0 p,p0 p,p0 • p,p0 equals 1 if the lightpath p0 (shares the same link of p, i.e., lightpath p0 to p. yp,p = 1) has the minimum starting FS difference with , the strength of NLI of lightpath p on link l. 0 • Xp,l lightpath p on either side (left and right sides are separately network, the maximum NLI strength in a network (2) • employed), 0 otherwise. y equals 1 if there exists a third X p,p0 considering the QoT metric. (1) lightpath p00 that satisfies three conditions, i) y = 1, ii) Objective function p,p00 y(1) = 1, and iii) y = 1. Finally, with the obtained p00,p0 p,p0 min network (Phase 4) (1) (2) XCI parameter of y + y , we replace the parameter yp,p in fp,η¯ X p,p0 p,p0 0 p,p0 Eqs. (13c) and (13d). By doing so, we just need to consider SNRthreshold mp X network the NLI of the nearest four channels rather than all lightpaths s.t. p,l , p (13a) SNRbest · X ≤ X ∀ share a common link, thus lowering the complexity from p l 2 ( P 3 XCI ) (Q adopt ) to (Q adopt 4) for the point-to-point p =p xp0,l Gp η¯p,p O · |P | O · |P | · 06 · · 0 case. The effectiveness of this method has been experimentally xp,l 1 + ASE p,l, p, l Gspan ≤ X ∀ shown in the literature [34, Fig. 4], which shows that the (13b) impact of XCI efficiency could be negligible when the channel   h Bp,B Bp,B i spacing is sufficiently large (over two bandwidths).  a p0 (f f ) + b p0  q p p0 q Currently, the widely used channel spacing strategies are yp,p0 − 0,  u η¯XCI  ≤ (13c) still the fixed or candidate channel spacing strategies. To make p,p0 p,p0 · − a comparison with these strategies, we provide the following p, p0 = p, q  ∀ 6 constraints into Phase 4 to implement the function, respec- h Bp,B Bp,B i p0 p0  aq (fp0 fp) + bq  tively. Here, we refer to our optimization strategy as CSO, yp,p − 0, 0 XCI ≤ fixed channel spacing strategy as FIX, and candidate channel  (1 up,p ) η¯  (13d) · − 0 − p,p0 spacing strategy as CAN. In addition, CAN(opt) differs from p, p0 = p, q CAN(random) in whether to use the objective function of ∀ 6 fp Bp fp0 Bp0 minimizing the NLI. Also, FIX adopts no objective function. 0 + yp,p0 up,p0 ≤ fgrid − 2 − fgrid 2 (13e) For both FIX and CAN, the channel spacing set is denoted as . p, p0 = p H ∀ 6 FIX: Each channel follows a fixed channel spacing h, fp Bp fp Bp • + W, 0 . p (13f) h ( = 1). For the networks that support heteroge- fgrid 2 ≤ ≤ fgrid − 2 ∀ ∈ H |H| neous bandwidths, we allow the small bandwidth channel The objective is to minimize the maximum NLI strength to follow integer times of the channel spacing h, while for all lightpaths in the network. Such a ratio, denoted by the large bandwidth channel follows the given channel network, is calculated by constraints (13a). Constraints (13b) X spacing. Thus, for any p adopt, p0 adopt/ p , calculate the NLI strength for the lightpath p on link l. ∈ P ∈ P { } XCI (1) Constraints (13c) and (13d) calculate the XCI efficiency η¯ 0 (fp fp0 h)up,p0 yp,p (np,p0 1) h. (14) p,p0 ≤ − − · 0 ≤ − · + between lightpath p and p0 using fp fp0 . Constraints 1 n , n . (15) | − | p,p0 B p,p0 N (13e) guarantee that the occupied spectrum resources of two ≤ ≤ |T | ∈ Here, is the number of heterogeneous bandwidths lightpaths are non-overlapping if they use a common link. |TB| B in the transceiver set . Constraints (13f) restrict the spectrum resources among the t T spectrum interval [0,F ]. Note that the QoT constraint of (C1) CAN(opt): We allow the channel spacing f f to vary • p − p0 CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 9

among the interval [inf( ), sup( )] if the channel of p0 from the new lightpath, we give priority to option ii) and then H H is the nearest neighborhood channel of p. Thus, for any choose option iii). p , p0 / p , ∈ Padopt ∈ Padopt { } (1) Algorithm 1: Sequential loading algorithm for 0 (fp fp inf ( ))up,p y sup( ) inf ( ) ≤ − 0 − H 0 · p,p0 ≤ H − H throughput maximization (16) Input : G(V,E),W , W , Dˆ , lightpath set , cur s,d P CAN(random): Similar to CAN(opt), we allow the chan- capacity set , boolean indicator β • C s,d,k,l nel spacing fp fp0 to vary among the set [hp, sup( )] Output: TH, − H Padopt if the channel of p0 is the nearest neigbor channel of 1 Set the initial network throughput TH and step ∆TH , p, where hp is a random value. Thus, for any Terminate = false; ∈ H p adopt, p0 adopt/ p , 2 while !Terminate do ∈ P ∈ P { } (1) 3 for Node pair (s, d) do 0 (fp fp0 hp)up,p0 yp,p sup( ) hp (17) ≤ − − · 0 ≤ H − 4 Calculate current provisioning capacity Ts,d; 5 Option i): upgrade transmission mode m for mod existing lightpaths to obtain ∆TH ; 6 if T + ∆mod < (TH + ∆ ) Dˆ then D. Tuning SNR margin s,d TH TH · s,d 7 Try either one of the following: Finally, we deal with the excessive SNR margin. After 8 Option ii): adjust route k, baud-rate R, or trans the lightpath selection (Phase 2 and Phase 3) and the move the channel ch to gain ∆TH ; channel spacing optimization (Phase 4), the actual SNR 9 Option iii): establish a new lightpath to new performance of the lightpath in adopt0 could be higher than gain ∆TH ; P mod new trans its worst-case assumption. The enhanced SNR performance 10 if Ts,d + ∆TH + max(∆TH , ∆TH ) < allows the lightpath to choose higher-order transmission mode (TH + ∆ ) Dˆ then TH · s,d (see Fig. 2(b) and Fig. 2(c)). To allow the lightpath to adopt 11 Go to line 15 to terminate the a higher-order transmission mode, we can achieve it by either algorithm; of the options. Option i), upgrade the transmission mode of 12 else current lightpath set 0 , or option ii), lower the SNR Padopt 13 Upgrade the lightpaths of options ii) margin in the Phase 1. This paper takes the latter one to and iii) into lightpath set ; globally optimize the candidate lightpath set . To this end, Padopt P we create a control loop connecting the Phase 1 and store 14 TH = TH + ∆ ; the current feasible results. TH 15 Output TH and the lightpath set ; We describe the detailed process of tuning the SNR margin Padopt Mp. At each iteration, Mp decreases with a unit step ∆Mp (0.5 dB in this paper). As the lowered SNR margin Mp Based on the aforementioned discussion, we illustrate the employs in the pre-calculation process, all lightpaths are algorithm in Algorithm 1. The main idea is to sequentially promising to adopt a higher-order transmission mode, thus increase the bit-rate demand until the provisioning capacity increasing the average spectral efficiency of each lightpath cannot satisfy it. Specifically, in line 1, we manually set the in Phase 2 and obtaining a higher network throughput. throughput increment ∆TH (∆TH =25 Gbps is used, initial Through the iterations, the QoT metric min after Phase Qp network throughput TH=0 Gbps). From lines 3 to 13, we 4 also decreases. Once there exists a lightpath that violates try to increase the provisioning capacity to satisfy the bit-rate the QoT constraint ( p 0 , < 0), we output the ∃ ∈ Padopt Qp,dB demand of node pair (s, d) by sequentially using options i), ii), last stored results. and iii). Specifically, in line 5, we use option i) to increase the IV. HEURISTICFOR THROUGHPUT MAXIMIZATION provisioning traffic. Then, either option ii) or iii) is adopted if In this section, we devise a sequential loading algorithm to the provisioning capacity is smaller than the bit-rate demand, efficiently solve the throughput maximization problem in Sec. as shown from lines 6 to 9. Finally, the algorithm terminates if II-B. The baud-rate optimization is also incorporated. the maximum provisioning capacity is smaller than the bit-rate According to the common sense, we can increase the demand, as shown in line 11. It should be noted that the three provisioning capacity between (s, d) by using the following different options on the lightpath are all implemented among options, i) upgrade the transmission mode of the working the candidate lightpath set . Thus, the different network P lightpath, ii) adjust the baud-rate or the route of the working traffic loads can be implemented by restricting the available lightpath, iii) establish a new lightpath directly. We denote channel indexes ch. mod trans new by ∆TH , ∆TH , and ∆TH the bit-rate gain of these three options, respectively. As indicated by the adopted Gaussian V. SIMULATION RESULTS Noise [28] model, option i) increases the bit-rate without In this section, we use numerical simulations to verify the incurring additional NLI to other lightpaths, which is regarded throughput improvement by optimizing channel spacing and as the first choice. Options ii) and iii) both degrade the QoT using just-enough SNR margin. First, we compare the through- of other existing lightpaths. To reduce the extra interference put difference between just-enough SNR margin and excessive CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 10

SNR margin in single baud-rate networks. The comparison of SNR margin provisioning. The statistics value of the channel channel spacing strategies FIX, CAN(opt), CAN(random), and spacing for each lightpath is defined as the minimum channel the proposed CSO is also made in terms of throughput. Next, spacing with other lightpaths. we investigate the throughput difference in flexible baud-rate networks leveraging just-enough SNR margin provisioning and B. JP vs. traditional EP in single baud-rate networks CSO, where we highlight the throughput of different baud-rate selection strategies. Finally, we investigate the impact of power First, we compare the throughput difference between JP and setting for different channel spacing strategies, and analyze the EP for networks adopting a single baud-rate (16 Gbaud). The cost-performance ratio or the feedback tuning algorithm. latter is implemented by breaking the iterative loop of Fig. 3. The results from different lightpath provisioning algorithms are compared in order to validate the performance of JP, A. Simulation setup including the heuristic in Algorithm 1, and the ILP models The simulations use three network topologies, including 4- in Phase 2 and Phase 3. node ring (400 km per link), Cost239 network (11 nodes, 52 1) JP(CSO) vs. traditional EP(CSO) in a small network: links) [47], and NSF network (14 nodes, 44 links) [47]. It Figure 4 illustrates the throughput, the number of adopted should be noted that the NSF network are scaled with a factor transmission modes, the QoT metric, and the lightpath’s chan- of 0.5 to guarantee the traffic of the longest shortest path can nel spacing when network traffic load increases. Regardless of be transmitted. To guarantee that the ILP models in Phase the different lightpath provisioning algorithms, JP outperforms 2 and Phase 3 are solvable within a reasonable time, we the EP under the resource over-provisioning scenarios in terms assume that the available spectrum resource per link of 4-node of the throughput (network traffic load between 20% and 80% ring is F = 750 GHz (W = 60 FSs). While in the mesh network, in Fig. 4(a)). The observed increase in throughput is attributed we assume the available spectrum resource is F = 3,000 GHz to the more usage of high-order transmission modes with high (W = 240 FSs). Other network parameters are stated as follows. spectral efficiency (H-16QAM) (Fig. 4(b)). The network traffic load is varied with the available spectrum Another observation in Fig. 4(a) is that the relative through- resources, W from 20% W to 100% W . The number put gain of JP decreases with the network traffic load. This cur × × of candidate routes K equals 10. Similar to the prior studies trend could be explained using the conclusion of [12, Sec. 5 [5, 8, 10], we adopt the widely used assumption of uniform and 6] that the fraction ratio of achievable capacity is ap- ˆ 1 traffic for the normalized demand matrix, i.e. Ds,d = n (n 1) . proximately linearly with the overestimation of network SNR Our simulations run on an Intel Core PC with 3.6· GHz− margin. As network traffic load increases, less SNR margin CPU and 64 GB RAM. Specifically, the LP/ILP models and overestimation (Fig. 4(c)) between resource over-provisioning heuristic algorithm are solved with Gurobi 9.0 and MATLAB and resource under-provisioning means less throughput gain 2017a. For each LP/ILP model, we set the optimality gap (Fig. 4(a)). It should also be mentioned that the higher through- of 5% to guarantee the computational efficiency. The typical put of JP is achieved by using the high spectral efficiency single-mode fiber is assumed for physical layer calculation, i.e. transmission mode without increasing the number of lightpaths fiber attenuation ratio α=0.2 dB/km, second order dispersion (Fig. 4(b)). However, it cannot be ruled out that in some special 2 coefficient β2 =21.7 ps /km, frequency of optical signal ν = cases, the heuristic algorithm may generate extra lightpaths | | 1 192.5 THz, and fiber nonlinear coefficient γ=1.3 (W Km)− . even under the same network traffic load to obtain the higher · For the optical amplifiers, the noise figure nsp = 5 dB throughput. We will analyze the cost-performance ratio in and Lspan=100 km. The PSD for all transceivers is assumed Sec. V-D. with a constant 25 µW/GHz, which is slightly larger than In Fig. 4(d), we find that the channel spacing statistics value the power calculated by the LOGON strategy (15.3 µW/GHz (minimum channel spacing, average channel spacing, and among 3,000 GHz). The impact of different PSD settings maximum channel spacing) varies with the different network will be also discussed in Sec. V-E. In this paper, we have traffic loads. Such results imply that the traditional fixed adopted four different MFs (PM-QPSK, PM-16QAM, and PM- channel spacing probably lacks robustness when setting the 64QAM, PM-256QAM) and at most five FEC overheads for channel spacing. Next, we will discuss the resulting network each MF so that we can obtain the bit-rate from 50 to 375 Gbps performance difference of this characteristic. with a step of 25 Gbps [38]. Besides, the SNR threshold of 2) JP(FIX, CAN, CSO) vs. traditional EP(CSO) in large each transmission mode has been individually superimposed networks: Next, we investigate the throughput of different with a fixed penalty compared to the original data in [38] in channel spacing strategies. = 37.5 GHz for FIX, = H { } H order to avoid the unexpected electrical noise (receiver and 25, 37.5, 50 GHz for CAN(opt) and CAN(random). { } thermal shot noise, etc). Three baud-rates are also assumed In Fig. 5(a), the largest throughput gain ratio (50%) at [37, Table 3]: 16, 32, and 64 Gbaud, which occupies 2, 4, and the network traffic load of 20% is obtained by CSO among 6 FSs, respectively. Although the SNR threshold of higher the existing channel spacing strategies (FIX, CAN, CSO). baud rates could be introduced due to the current hardware This is expected because CSO uses no artificial constraint implementation [17], this paper assumes an ideal case that the of channel spacing and allows the transceiver to use the idle SNR threshold for different baud-rates is identical. spectrum resources to enhance the physical layer performance For ease of illustration, we use JP and EP to denote the just- and upgrade the high-order transmission mode. To gain more enough SNR margin provisioning and traditional excessive insight, we plot the number of adopted channel spacing in CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 11

(a) C. JP vs. traditional EP in multiple baud-rates networks 20 80% EP-CSO(ILP) JP-CSO(ILP) Next, we study the throughput of different baud-rates and compare the baud-rate selection policies (Fig. 6). The com- 15 EP-CSO(heu) JP-CSO(heu) 60% parison of different channel spacing strategies is also made (Table I). 10 40% 1) Different single baud-rates: The network throughput of three different single baud-rates is shown in Fig. 6(a). At 5 20% the network trafﬁc load of 100%, the largest throughput (295 Throughput [Tbps] Tbps) is achieved by the 64 Gbaud among three baud-rates 0 0 20% 40% 60% 80% 100% (210 Tbps for 16 and 32 Gbaud). It is expected because the spectral efﬁciency of 64 Gbaud transceiver is 2.7 bps/Hz, (b) while it is 2.0 bps/Hz for both 16 and 32 Gbaud (50 Gbps

L-16QAM(EP) for 16 Gbaud on 25 GHz, 100 Gbps for 32 Gbaud on 150 H-16QAM(EP) 50 GHz, 200 Gbps for 64 Gbaud on 75 GHz [37, 38]). An L-16QAM(JP) interesting observation in Fig. 6(b) is that the maximal absolute 100 H-16QAM(JP) gain (33/22/19.5 Tbps) is achieved in the medium network trafﬁc load for these single baud-rate networks. We provide a

#Lightpaths possible explanation by incorporating the previous finding on 50 the throughput gain ratio in Figs. 4 and 5. On the one hand, the available spectrum resources increase with the network traffic 0 20% 40% 60% 80% 100% load. On the other hand, the relative throughput gain ratio (or relative net spectral efficiency) decreases (see Figs. 5(a) (c) 3 and (d)). Therefore, their product, namely absolute throughput EP-CSO gain, could achieve the maximum when the network traffic JP-CSO load is at a certain value (medium in this case). 2 EP-Fix25 2) Baud-rate selection strategies (low baud-rate first, high [dB]

p baud-rate ﬁrst, vs. random baud-rate): Figure 6(c) shows

Q 1 the network throughput using different baud-rate selection policies. The utilization of prioritizing the higher baud-rate min 0 (higher baud-rate means higher spectral efficiency) has a QoT line higher throughput, which is similar to the study [21] that 20% 40% 60% 80% 100% priorities the maximum spectral efficient channels. Such a result validates the efficient selection strategy to achieve the (d) higher throughput from another perspective. Min(CSO) 3) JP(FIX, CAN, CSO) vs. traditional EP(CSO): We com- 150 Mean(CSO) pare the throughput gain ratio of different channel spacing Max(CSO) strategies in networks supporting flexible baud-rates. Table I illustrates the relative throughput gain ratio of different channel 100 spacing strategies. The performance of CAN (opt) is closer to the proposed CSO if the size of candidate channel spacing set 50 increases. However, the FIX and CAN (random) can be applied to increase the network throughput only in the scenarios of Channel spacing [GHz] 20% 40% 60% 80% 100% low network traffic load. These results show the advantage of CSO with a larger throughput gain ratio and with more Network traffic load Wcur/W network traffic load scenarios.

Fig. 4. Comparison of JP and EP with CSO for 4-node ring. (a) Throughput D. Cost-performance ratio vs. network traffic load. The relative gain ratio between JP and EP is also plotted on the right y-axis. (b) Adopted transmission modes vs. network In addition, we analyze the cost-performance ratio of the traffic load. L-16QAM includes 16QAM with FEC 0.62 and 0.72, while H- feedback tuning algorithm. Taken the Cost239 network as an 16QAM includes 16QAM with FEC 0.82 and 0.92. The spectral efficiency of H-16QAM is higher than L-16QAM; (c) Minimum QoT metric vs. network example, the relative increase of lightpath (all adopt 16 Gbaud) traffic load; (d) Channel spacing statistics of CSO. and throughput from EP to JP are shown in Fig. 7. The throughput gain ratio varies from 7% to 27%, which is always higher than the cost ratio that varies from -1% to 5%. Such Fig. 5(b) as well as the link statistics on each FS in Fig. 5(c). results reveal that the JP is more cost-effective. We see that the most significant channel spacing ( 50 GHz) ≥ is out of the scope of the other strategies (FIX and CAN). The E. Impact of physical layer parameters above findings can also be observed in the Cost239 network This section discusses the impact of different initial PSDs in Figs. 5(e). on channel spacing optimization. In Fig. 8, we see that the CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 12

80 80% EP-CSO JP-CSO JP-CAN(opt) JP-FIX37.5 44 CAN(opt) FIX37.5 CSO 60 JP-CAN(random) 60% 40

30 40 40%

# Links 20 Relative gain 20 20% Throughput [Tbps] 10

0 0% 20% 30% 40% 50% 60% 70% 80% 0 50 100 150 200 240 Network trafﬁc load Wcur/W FS index (a) Throughput (NSF) (b) Channel spacing under 20% load (c) Link spectrum usage (20% load) (NSF) (NSF)

80% EP-CSO JP-CSO CAN(opt) FIX37.5 CSO 200 JP-CAN(opt) JP-FIX37.5 52 JP-CAN(random) 60% 40

40% 30 100 # Links

Relative gain 20 20% Throughput [Tbps] 10 0 0% 20% 30% 40% 50% 60% 70% 80% 0 50 100 150 200 240 Network trafﬁc load Wcur/W FS index (d) Throughput (Cost239) (e) Channel spacing under 20% load (f) Link spectrum usage (20% load) (Cost239) (Cost239)

Fig. 5. Throughput with different channel spacing strategies.

(a) (b)

EP-SB16 JP-SB16 SB16 SB32 SB64 300 EP-SB32 JP-SB32 40 EP-SB64 JP-SB64 200 20 100 Throughput [Tbps] Absolute gain [Tbps] 0 0 0 20% 40% 60% 80% 100% 20% 30% 40% 50% 60% 70% 80% (c) (d)

EP-LB JP-LB LB RB HB 300 EP-RB JP-RB 40 EP-HB JP-HB 200 20 100 Throughput [Tbps] Absolute gain [Tbps] 0 0 0 20% 40% 60% 80% 100% 20% 30% 40% 50% 60% 70% 80%

Network trafﬁc load Wcur/W Network trafﬁc load Wcur/W

Fig. 6. Throughput in Cost239 network with different baud-rate configurations. (a) Different single baud-rates (SB16: only single 16 Gbaud); (b) Absolute throughput gain of JP with single baud-rate; (c) Different selection strategies of baud-rates (LB: low baud-rate first. RB: random baud-rate. HB: high baud-rate first); (d) Absolute throughput gain of JP with different selection strategies. CAO CHEN et al.: BARE DEMO OF IEEETRAN.CLS FOR IEEE JOURNALS 13

TABLE I RELATIVE THROUGHPUT GAIN RATIO IN COST239 NETWORKWITH VI.CONCLUSION SINGLEBAUD-RATE (ABOVE) AND WITH FLEXIBLE BAUD-RATES (BELOW) In this paper, we studied the problem of throughput maxi-

Single baud-rate (32 Gbaud) mization leveraging just-enough SNR margin provisioning and FIX CAN(random) CAN(opt) Load CSO channel spacing optimization. Based on the analysis of the 50 100 12.5,...,50 12.5,...,100 12.5,...,50 12.5,...,100 { }{ } { }{ } { }{ } 20% 17% 17% 17% 17% 17% 17% 33% lightpath’s parameters, we developed an iterative feedback 40% 0% 21% 0% 10% 0% 21% 24% tuning algorithm to provide a just-enough SNR margin. We 60% 0% - 0% - 0% 13% 13% 80% 0% - 0% - 0% 7% 7% also established an ILP model and a heuristic algorithm to 100% 0% - 0% - 0% 0% 0% maximize the network throughput by optimizing route, MF, Flexible baud-rates (16, 32, and 64 Gbaud) FIX CAN(random) CAN(opt) FEC, baud rate, and spectrum assignment. Furthermore, we Load CSO 75 150 12.5,..,75 12.5,..,150 12.5,..,75 12.5,..,150 { }{ } { }{ } { }{ } devised a low-complexity LP model that enables us to optimize 20% 24% 32% 24% 24% 20% 28% 36% 40% 10% - -- 1% 14% 14% the channel spacing for a large number of lightpaths. 60% -- -- 6% 14% 14% Through the simulations, we verify the throughput improve- 80% -- -- 0% 0% 4% 100% -- -- 0% 0% 0% ment of just-enough SNR margin provisioning compared to the conventional excessive SNR margin provisioning. The - It signifies the channel spacing strategy is infeasible due to the limited spectrum relative gain reaches the maximum (over 20%) under low resources. network traffic load, while the absolute gain reaches the maximum under medium network traffic load. These obser- vations highlight that the importance of just-enough SNR Cost:lightpath 40% margin provisioning under low and medium network traffic Performance:network throughput load. Compared to the existing channel spacing strategies (fixed and candidate), our channel spacing optimization also 20% demonstrates a clear advantage on two aspects: (1) it can be applied with more resource over-provisioning scenarios; (2) it has a larger relative throughput gain ratio. We also show that

Ratio: (JP-EP)/EP 0% the channel spacing becomes more significant when the NLI 20% 30% 40% 50% 60% 70% 80% from neighbor channels becomes severe, i.e. in the nonlinear regime. In addition, we find that the higher baud-rate with Network traffic load W /W cur larger spectral efficiency should be prioritized to gain a larger throughput when implementing the lightpath provisioning. Fig. 7. Cost-performance ratio of JP-CSO and EP in Cost239 network. Taken together, our findings highlight an important role for just-enough SNR margin provisioning and channel spacing 80 optimization in FONs with flexible baud-rates. This paper

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