Survey of Resource Allocation Schemes and

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Survey of Resource Allocation Schemes and Algorithms in Spectrally-Spatially Flexible Optical Networking This is a preprint electronic version of the article submitted to Optical Switching and Networking Mirosław Klinkowski, Piotr Lechowicz, and Krzysztof Walkowiak

Abstract Space division multiplexing (SDM) is a promising optical network solution with the scaling potential to overcome the possible capacity crunch problem in backbone networks. In SDM optical networks, optical signals are transmitted in parallel through spatial modes co-propagating in suitably designed optical fibers. The combination of SDM with wavelength division multiplexing technologies, based on either fixed or flexible frequency grids, allows for a significant increase in the transmission system capacity and improves network flexibility through the joint management of spectral and spatial resources. The goal of this tutorial paper is to survey the resource allocation schemes and algorithms that aim at efficient and optimized use of transmission resources in spectrally and spatially flexible optical networks. There are several flavors of SDM, which results from available optical fiber and switching technologies; hence, we begin from classifying SDM optical network scenarios. As some SDM solutions have specific limitations, for instance, due to coupling of spatial modes, we discuss related constraints and their impact on resource allocation schemes. Along with defining possible SDM scenarios, we survey the proposed in the literature resource allocation algorithms and classify them according to diverse criteria. This study allows us to overview the state-of-the-art solutions in the scope of planning and operating SDM optical networks in a systematic way as well as to identify some open issues that lack solutions and need to be addressed.

I. I NTRODUCTION The overall Internet traffic grows quickly due to continuously increasing popularity of various network services. According to the recent Visual Networking Index report published by Cisco, the IP traffic will grow at a compound annual growth rate (CAGR) of 22 percent from 2015 to 2020. These trends lead to incremental exhaustion of available capacity in optical transport networks [2]. The currently deployed wavelength division multiplexing (WDM) optical communication systems rely on singlemode fibers (SMFs), in which optical signals are transmitted in parallel through non-overlapping, fixed-spaced channels in the frequency domain [3]. The refractive index profile and core dimension of an SMF is adequately tailored so that to allow for light guiding of a single spatial mode in the core region. For efficient utilization of scarce spectrum resources in SMFs, the concept of spectrally flexible (elastic) optical networking was developed in recent years [4]. The key features of elastic optical networks (EONs) are adaptive use of various modulation formats [5], which differ in spectral efficiency and transmission reach, and multiplexing of wavelengths (also called frequency slots or spectrum channels) of suitably tailored width within a flexible frequency grid, which extends the idea of fixed-grid WDM optical networks. EONs enable multi-carrier (i.e., super-channel, abbreviated as SCh) transmission, where a high-capacity SCh transmitted over the network consists of a number of optical carriers (OCs) [6]. The carriers are generated/terminated using transceiver devices, each making use of a certain modulation format and carrying a fraction of aggregated traffic. Although flexible-grid EONs support high bit-rate traffic demands more efficiently than conventional fixed-grid WDM optical networks, still their transmission capacity is approaching its upper bound due to the nonlinear Shannon limit of the conventional SMF [2]. SDM is a forthcoming optical network technology going beyond the capabilities of WDM/EON systems by enabling parallel transmission of several co-propagating spatial modes [7]–[10]. SDM operates with suitably designed fibers in which the modes are guided either through multiple cores placed within a single fiber cladding or in a single core of enlarged dimension and modified refractive index contrast [11]. SDM, when combined with WDM/EON, brings many benefits including enormous increase in transmission capacity, extended flexibility in resource management due to the introduction of the spatial domain, as well as potential cost savings thanks to the sharing of resources and the use of integrated devices, such as switches [12], amplifiers [13], or transponders [14]. Nevertheless, it poses also many challenges, with the main concerning development of practical, reliable, and cost-effective hardware for long-haul transmission systems, capable of competing with current optical Mirosław Klinkowski ([email protected]) is with National Institute of Telecommunications, 1 Szachowa Street, 04-894 Warsaw. Piotr Lechowicz and Krzysztof Walkowiak are with Faculty of Electronics, Wroclaw University of Science and Technology, Wrocław, Poland.

network solutions. General discussion on SDM can be found in [2], [12], [15]–[18], and for extensive overview of SDM enabling technologies refer to [9] and [19]. The main concern in optical networking is provisioning of lightpath connections for transmitted signals. A lightpath is an optical path established between a pair of source-destination nodes. In spectrally flexible optical networks, the lightpaths carrying SChs are routed through the network over SMF links in appropriately assigned spectrum channels, which are switched in network nodes with the aim of wavelength selective switches (WSSs). Having a set of traffic demands, the routing of lightpaths requires a contention-free allocation of spectrum resources on each link belonging to the routing path of each connection realizing a demand. It translates into the problem of routing and spectrum allocation (RSA), which consists of finding lightpath connections, tailored to the actual width of transmitted signals (i.e., SChs), for end-to end demands that compete for spectrum resources [20]. The RSA problem is present both in the phase of network planning and during its operation. The introduction of the spatial dimension (through the availability of parallel spatial modes in SDM) allows for carrying the OCs of SChs using not only continuous frequency slots in the frequency domain (as in EONs), but also to distribute them over different spatial modes. In spectrally-spatially flexible optical networking, RSA converts into the problem of routing, spatial mode, and spectrum allocation (abbreviated as RSSA), which adds new decision variables related to the selection of spatial modes in network links by means of which the optical signals will be transmitted. The handling of a much larger number of spatial modes than in WDM/EON optical networks increases dramatically the complexity of both hardware and control functions in SDM networks. It results in a large set of decision variables in network optimization, which makes RSSA more complex than the RSA problem. Moreover, as discussed in next sections, some SDM scenarios may have specific requirements, for instance, certain groups of modes may need to be transmitted and switched together due to mode coupling. Consequently, new dedicated resource allocation schemes and algorithms are required in such networks. A. Motivations and Contributions The main motivation behind this work is the lack of a comprehensive literature survey on spectral and spatial resource allocation schemes and RSSA algorithms in SDM networks. There is a need for such a survey and particular motivations are the following. First of all, the SDM technology is still in its infancy and there are many technological options considered for realization of SDM transmission in optical networks, each having specific properties, for instance, due to coupling of spatial modes. Since it is not clear which solution (or solutions) will be finally deployed, we try to identify the most promising SDM network scenarios based on the analysis of research works that appeared in the literature. Secondly, we aim at classifying SDM technologies according to their characteristics, which are reflected in proper resource allocation schemes to be applied when establishing/provisioning optical connections in the network. Such classification will help to understand the differences between particular solutions and their impact on network operation. In this scope, we also briefly summarize the so far performed comparative studies concerning different SDM scenarios. Eventually, although the research work in SDM network optimization is in its initial phase and efficient algorithmic solutions for more complex networking problem are yet to come, in the literature there have been proposed different approaches for solving specific network planning and operation problems that arise in SDM optical networks. Before developing new and more efficient algorithms as well as before addressing the RSSA optimization problems and the SDM optical network scenarios that lack solutions, the state-of-the-art methods presented in related works should be recognized, which is our another goal. The result of our study is compiled and reported in this work, which is to the best of our knowledge the only comprehensive literature survey for studies related to the RSSA problem. More specifically, the major contributions of the survey are as follows: (a) an overview of SDM technologies with identification of their main properties that have impact on network operation and spectral-spatial resource allocation process, (b) a classification of SDM network scenarios considered in the literature using diverse classification criteria allowing for identification of the most relevant ones, (c) a review of the studies that compare different SDM scenarios, (d) a comprehensive survey of the state-of-the-art RSSA-related algorithms in spectrally-spatially flexible optical networks, (e) the identification of the research gaps and the unanswered questions in the scope of planning and operating SDM optical networks. In this survey, our main focus is on optical backbone networks, and other applications of SDM, for instance, such as short-distance data center networks [21] or access networks [22], are left for further studies. Also, we do not include into analysis some older works, such as [23]–[26], which although address multi-fiber optical network scenarios, still do not fully exploit spectral and spatial flexibility offered by the SDM transmission and switching technologies that are discussed in this paper. Finally, we do not cover the issues strictly related to the control plane which, however, is responsible for performing connection setups/teardowns and thus for applying the discussed resource allocation schemes in the network, it may have different implementations, such as for instance based on the software defined network (SDN) concept [27], [28], which analysis is out of scope of this paper. In order to report the state-of-the-art and current proposals, we collected more than 40 recent papers, and considering a variety of classification criteria we reviewed the related papers. In each case where there existed some doubts how to classify given work with respect to specific criteria, for instance, if required information was not provided in the article or it was not clearly stated, then such work was not included in this specific classification. 2

TABLE I: Glossary. ACF A/D AoD CAGR DAC DSP EON FB FMF FMFB FM-MCF FrJ-Sw HC-PBGF ILP Ind-Sw IP IQ-MOD J-Sw LS MCF MIMO MIP MMF Mode-Sw OC RMBSA RMCSA RSA RSCA RSCMA RSSA RWA RWCA SA SCh SDM SDM MUX SMF SMFB SM-MCF SSA TX WDM WSS XT

Annular core fiber Add/drop Architecture on Demand Compound annual growth rate Digital-analog converter Digital signal processing Elastic optical network Fiber bundle Few-mode fiber Few-mode fiber bundle Few-mode multi-core fiber Fractional joint switching Hollow-core photonic band gap fiber Integer linear programming Independent switching Integer programming Modulator Joint switching Laser source Multi-core fiber Multiple-input multiple-output Mixed-integer programming Multi-mode fiber Spatial mode switching Optical carrier Routing, modulation format, baud rate, and spectrum allocation Routing, modulation format, core, and spectrum allocation Routing and spectrum allocation Routing, core, and spectrum allocation Routing, spectrum, core, and mode assignment Routing, spatial mode, and spectrum allocation Routing and wavelength assignment Routing, wavelength, and core allocation Simulated annealing Super-channel Space division multiplexing SDM multiplexer Single-mode fiber Single-mode fiber bundle Single-mode multi-core fiber Spatial mode, and spectrum allocation Transmitter Wavelength division multiplexing Wavelength selective switch Crosstalk

The rest of this survey paper is organized into six sections. In its first part, which spans from Section II to Section V, we aim at classifying the surveyed solutions according to particular SDM optical network scenarios. Namely, in Sections II and III, we focus on SDM suitable optical fibers and switching paradigms, respectively. Since all the considered options for realization of SDM are widely discussed in the literature (e.g., see [9] and [19]), including their pros and cons, we skip their detailed description and address specific technology-related constraints that influence the functioning of resource allocation schemes and algorithms. In Section IV, we classify different types of SChs supported by SDM optical networks, which lead to different spectral-spatial resource allocation schemes. As a summary, in Section V, we distinguish several complete network scenarios that assume jointly specific technological options. Next, in Section VI, which is the largest and consists of several subsections, we classify and discuss particular algorithmic approaches the are presented in the surveyed papers. Among others, we focus on the addressed network planning and operation problems, applied optimization methods, solving the RSSA problem, integer linear programming (ILP) modeling of RSSA, dealing with crosstalk, resource fragmentation, and considered performance/optimization metrics. Finally, in Section VII, we conclude our work and point out some possible directions for future research. II. O PTICAL F IBERS FOR SDM N ETWORKS There are several types of optical fibers suitable for SDM networks [73]–[75], and the most frequently invoked in the literature can be classified as following (see Fig. 1): • Single-mode multi-core fiber (SM-MCF or MCF) – involves multiple single mode cores embedded in a fiber cladding, with each core guiding one mode. Several core placement structures are proposed for MCFs, including one-ring, dual-ring, linear-array, two-pitched or hexagonal structure. The spacing and packing of cores, respectively, can be either even or uneven and either close or wide. 3

SMFB

SM-MCF

Coupling

no

no or weak

Crosstalk

no

yes

SM-MCF (closely packed)

strong (all modes)

FMF

FMFB

FM-MCF

strong (all modes strong (in groups) strong (in groups) or in groups)

yes

no

no

yes

SM-MCF (uneven spacing)

strong (in groups) yes

Fig. 1: Types and properties of SDM optical fibers. TABLE II: SDM optical fibers in the surveyed works. Fiber type

References

Single-mode fiber bundle Multi-core fiber Few-mode fiber Few-mode fiber bundle Few-mode multi-core fiber

[29]–[41] [29], [41]–[67] [40], [68]–[70] [40] [66], [71], [72]

Few-mode fiber (FMF) – a kind of a multi-mode fiber (MMF) that enables SDM transmission through a relatively low number of transverse modes propagating in a higher index core. FMFs are applicable for higher transmission distances, when compared to MMFs, since mitigating the impairments and interference (mixing) of the few co-propagating modes is less troublesome than in the case of MMFs supporting higher numbers of modes. • Few-mode multi-core fiber (FM-MCF) – combines propagation of traverse modes in multiple higher index cores. FM-MCF possess advantages of both MCF and FMF technologies, simultaneously cutting back their negative impact. • Fiber bundle (FB) – the simplest form of SDM, achieved with the aggregation of many fibers in a bundle of fibers. While the main representative in this group is a single-mode fiber bundle (SMFB), in which each fiber supports one spatial mode, other possible cases can be assumed, for instance a few-mode fiber bundle (FMFB), where each fiber guides a few modes. SMFB is a commercially available technology, which should facilitate a migration from currently deployed networks to SDM. Among other SDM fibers, which are less prominent in the literature, there are the fibers supporting modes carrying Orbital Angular Momentum (OAM) [76] and hollow-core photonic band gap fibers (HC-PBGFs) [77]. Apart from [16], in which the OAM technology is mentioned, we have not found any application of such types of fibers in the surveyed works. MCF is one of the most popular and efficient ways to realize SDM. Indeed, as shown in Table II, MCF is the most frequently considered technology in the surveyed papers. In the majority of reported cases, the core placement is even-wide-hexagonal and at least 7 cores were evaluated. In [62], also a two-core MCF is analyzed, in [48] and [45], there are 3 cores arranged in a triangle, while in [49] and [58], the MCF consists of 12 cores in a one-ring structure. We have not spot any works with more diverse MCF layouts. SMFB is the next most addressed SDM transmission media. We have found four works concerning FMFs and one analysing a FMFB-based SDM network. Eventually, there are three papers mentioning the use of a FM-MCF, however, only in one of them, i.e., in [72], the discussed below constraints related to mode coupling are taken into account in network analysis. SDM fibers can be further categorized with respect to the level of mixing of co-propagating spatial modes (see Fig. 2). In a strongly coupled transmission, the modes intermix, but the information is maintained within the set of modes. To unravel the mixed information at signal destination, multiple-input multiple-output (MIMO) digital signal processing (DSP) should be applied to the entire set of mixed modes; hence, these modes must remain together. Contrarily, an uncoupled or weakly-coupled transmission does not have such limitations, what allows for higher network flexibility, since individual spatial modes can be switched and routed independently over the network, and which reduces transceiver costs due to the lack of MIMO DSP. Accordingly, we can distinguish: • Uncoupled/weakly-coupled modes (see Fig. 2a)) – such transmission can be achieved in both the SMFBs and the MCFs of large enough inter-core pitch. In SMFBs, there is no coupling between the spatial modes thanks to their physical separation since each mode is guided in a separate fiber. In MCFs, preserving high enough distance between the fiber cores suppresses the coupling of spatial modes to very low values (i.e., the coupling is weak) so that there is no need for application of MIMO processing in order to correctly receive the optical signals transmitted over particular modes. Note that even in weakly-coupled MCF there is some interaction between spatial modes in a form of signal crosstalk, which impacts transmission quality and may require the use of signal regenerators in larger networks. • Strongly-coupled all modes (see Fig. 2b)) – the coupling concerns the entire set of spatial modes, as in FMFs and OAMs. It •

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a) uncoupled/weakly-coupled modes

b) strongly coupled all modes Zdž Zdž Zdž Zdž

c) strongly coupled groups of modes

D/DK^W

Zdž Zdž Zdž Zdž

D/DK^W D/DK^W

Zdž Zdž Zdž Zdž

Fig. 2: Mode coupling in SDM fibers.

spectral slices

b) Mode-Sw: spatial mode switching across all wavelength channels

c) J-Sw: wavelength switching across all spatial modes (joint)

d) FrJ-Sw: wavelength switching across spatial mode subgroups (fractional)

groups of modes

spatial modes

a) Ind-Sw: independent spatial mode & wavelength switching

superchannels

Fig. 3: Spectral-spatial switching paradigms.

is also proper to some MCF designs in which the cores are closely packed and the modes couple through evanescent fields. MIMO DSP needs to be applied to all the spatial modes together. • Strongly-coupled groups of modes (see Fig. 2c)) – in which multiple coupled mode subgroups exist, however, the coupling between the modes belonging to different groups is either weak or none. MIMO DSP techniques need to be applied only to the contents of each group. Here, we include the FM-MCFs with far enough spaced cores and FMFBs. In the former, strong coupling concerns the set of modes guided within each individual core while, in the latter, within each fiber. As well, we classify here some MCF designs in which certain subgroups of cores are closely packed, but these groups are spaced far enough from each other. Eventually, as discussed in [78] and utilized in [68], [70], strong coupling within sub-groups of spatial modes may exist in FMFs. There are some consequences of selecting strongly-coupled SDM fiber solutions, to be accounted for when planning and operating the network. In such case, each set of coupled modes, for given frequency and polarization, should be treated as a single entity and carried through the network together. The switching equipment should be capable of switching together the entire groups of modes which, as a positive effect, may lead to the reduction of its complexity [12], [19]. At the same time, each coupled group of modes can be handled (e.g., routed) independently to other groups. For efficient resource utilization, SDM transponders should be capable of transmitting/receiving optical signals (carriers) in parallel, using different spatial modes and, preferably, exploiting the entire utilized group(s) of modes within assigned spectrum window(s). The majority of the surveyed works assume either none (as in SMFBs) or weak (as in MCFs) coupling of spatial modes; hence, there is no need for MIMO DSP and the spatial modes can be routed independently over the network. In [16], [40], [69], [79], the coupling of all modes is considered, while strong coupling of groups of modes is taken into consideration in [40], [68], [70]. The application of MIMO DSP to strongly coupled modes is explicitly mentioned in [16], [40] in the case of all modes, and in [40], [68], [70] for groups of modes. Regarding MCFs, this type of transmission media is prone to excessive signal impairments due to inter-core crosstalk (abbreviated as XT), which may occur between adjacent cores whenever optical signals are transmitted in an overlapping spectrum segment [80]. XT expresses the level of disturbance caused by the optical signal power leaking from adjacent cores on the signal propagating in a specific core. The effect of XT intensifies with the transmission distance [81]; hence, larger networks are especially sensitive to this impairment. It was shown that the level of XT depends slightly on signal frequency [82] and is subject to stochastic changes in time [83]. Besides, XT in MCFs can be minimized by using trench-assisted cores (step-index), fiber bend, well-spaced cores, different refractive indexes of cores [9], [81], or by assigning bi-directional optical signals in adjacent cores [84]. Except for [54], [67], the surveyed works in which a MCF is used take into account the effect of crosstalk. Besides, in [68], [70], intermodal XT between different groups of strongly-coupled modes in a FMF is assumed. The application of MIMO DSP for crosstalk suppression is discussed in [49], [57], [58], and it is used in network performance analysis in [72]. Eventually, in Table III we report the main advantages and disadvantages of SDM fibers which are quoted in the literature and that we mention in this section and throughout this paper. For more extensive discussion of the characteristic of particular SDM fiber solutions refer to [9], [74], [75].

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TABLE III: Comparison of the main features of SDM fiber solutions. Fiber type

Advantages

Disadvantages

SMFB

- commercially available - the transmission not affected by crosstalk between the spatial modes - very large number of supported spatial modes in paraller SMF links - high switching flexibility - no need for SDM multiplexers/demultiplexers for component interconnections

- low space efficiency when compared to other SDM solutions due to large cable dimensions

MCF

- high number of supported spatial modes and high space efficiency - high switching flexibility - possible integration of SDM components allowing for cost reduction

- transmission quality affected by inter-core crosstalk - need for dedicated SDM components (such as multiplexers/demultiplexers) - higher fiber diameters when compared to SMFs, especially in the MCF supporting higher numbers of cores, which will require new standards (e.g., concerning connectors)

FMF

- integration of SDM components and associated cost reduction

- physical impairments due to mode coupling seriously affecting transmission quality - need for complex and costly MIMO-DSP to deal with mode coupling - need for dedicated SDM components - low switching flexibility

FM-MCF

- possible high number of supported spatial modes and very high spatial efficiency - integration of SDM components and associated cost reduction

- physical impairments due to mode coupling - need for MIMO-DSP, however, of reduced complexity when compared to the FMFs of the same overall number of supported modes - need for dedicated SDM components - moderate switching flexibility

III. S PECTRAL -S PATIAL S WITCHING To be viable and ready for deployment, SDM optical networking requires cost-effective, scalable, and flexible switching solutions. Several generic SDM switching architectures are considered in [12], and state-of-the-art technologies enabling their implementation are surveyed in [19]. As discussed briefly below, these solutions offer different levels of spectral-spatial switching flexibility and are suitable to operate with different SDM optical fibers: • Independent spatial mode and wavelength channel switching (i.e., space–wavelength granularity), also referred to as independent spatial/spectral switching [79], or independent switching [35] (Ind-Sw) – here, each spatial mode and spectrum channel can be switched independently, as shown in Fig. 3a). It provides the finest switching granularity, since all spectral slices (i.e., spectral resources) and spatial modes can be independently directed to any output port, however, at the cost of high hardware complexity. • Spatial mode switching across all wavelength channels (space granularity), sometimes called spatial switching [79] (ModeSw) – in which the entire communication band per mode is jointly switched, as in Fig. 3b). • Wavelength switching across all spatial modes (wavelength granularity), also called spectral switching [79] or joint switching [35] (J-Sw) – where all spatial modes are treated as a single entity, while spectral slices can be freely switched by the WSS; see Fig. 3c). • Wavelength switching across spatial mode subgroups (fractional space–full wavelength granularity), sometimes called fractional joint switching [35] or grouped spectral switching [79] (FrJ-Sw) – a kind of a hybrid approach in which groups of spatial modes (handled as one entity), as well as all spectral slices, can be independently switched to all output ports, as in Fig. 3d). Hybrid solutions combining these switching paradigms are also possible. In [33], [39], in which FrJ-Sw is implemented, some of the switching nodes are equipped with extra WSSs, which adds some spectral-spatial switching flexibility and allows for spatial group sharing. In [30], [48], [51], [61], a programmable switching node architecture implementing the concept of Architecture on Demand (AoD) is considered. AoD allows to setup flexibly the optical node structure by reconfiguring dynamically interconnections of building modules according to switching requests. Apart from cost [51] and power [48] savings, such approach enables hybrid switching. Eventually, in [67] a generic hybrid switching node allowing for either spatial mode or spatial mode with wavelength switching granularity is considered. In Table IV, we associate the surveyed works with particular switching paradigms which are considered in these studies. The large majority of works assume the finest space-wavelength switching granularity provided by Ind-Sw. There are also several papers concerning J-Sw and FrJ-Sw. In most cases, they focus on comparative performance analysis of joint-switching approaches with respect to the Ind-Sw paradigm. Apart from [61], [67], we have not found other works focusing on the 6

TABLE IV: Considered switching paradigms in the surveyed works. Switching paradigm

References

Ind-Sw J-Sw FrJ-Sw Mode-Sw Hybrid

[16], [29]–[39], [42]–[58], [60], [62]–[65], [67], [72], [79], [85], [86] [16], [31]–[41], [59], [69], [70], [79], [85] [31]–[35], [37]–[39], [68], [70], [85] [61], [67] [33], [39], [67]

a) spectral transponder

b) spatial transponder

c) spectral -spatial transponder

Client Interface

Client Interface

Client Interface

TX DSP

TX DSP

TX DSP

d) spectral -spatial transponder Client Interface TX DSP

DACs

DACs

DACs

DACs

DACs

DACs

DACs

DACs

DACs

DACs

DACs

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

IQ-MOD

f1 LS

SDM MUX

f1

f2

spectral SCh

Ĩ

f1

spatial SCh

LS f1

f2

SDM MUX

Ĩ

f1

f2

LS

N:1

LS

N:1

LS f2

N:1

f1

N:1

LS

Ĩ

spectral -spatial SCh (regular )

f2

LS f1

SDM MUX

f1

f2

Ĩ

spectral -spatial SCh (irregular)

Fig. 4: Exemplary SDM transponders (only transmitters shown), and generated super-channels (SChs); TX DSP – digital signal processor, DACs – digital-analog converters, IQ-MOD – modulator, LS – laser source, N : 1 – coupler, SDM MUX – SDM multiplexer. Mode-Sw paradigm (i.e., space granularity), what may result from its coarse granularity (related to the number of modes) [19] and, thus, low flexibility and limited applicability. Indeed, in [61] a kind of Mode-Sw is used in a programmable filterless SDM network, which utilizes passive components (splitters/couplers) to route and combine spatial modes between input and output ports of switching nodes. In [67], potential use of Mode-Sw is mentioned when discussing an SDM optical switch, however, it is not evaluated. Besides [71], where optical switching nodes with limited switching function (i.e., blocking) are considered, the surveyed works assume non-blocking switching architectures. IV. S PECTRAL AND S PATIAL S UPER -C HANNELS Spectrally-spatially flexible optical networking enables transmission of different types of SChs. Such SChs consist of multiple optical signals (carriers) distributed both in the spectral and spatial domains. Following the experimental results in [14] and the discussion in [79], we discern three general types of SDM suitable transponders, each capable of generating/terminating a specific SCh: • Spectral transponders – as in EONs, they generate spectral SChs consisting of multiple OCs that are co-routed using a single spatial mode in an SDM fiber; see Fig. 4a). To improve spectral efficiency, the OCs are placed near the Nyquist condition, i.e., without switching guard-bands between the OCs, and only the guard-bands between neighbouring SChs are necessary. • Spatial transponders – they allow for parallel transmission over SDM fibers in which all OCs of a SCh use the same frequency, but have assigned different spatial modes, as shown in Fig. 4b). Such transponder shares a single laser source and allows the simultaneous transmission/reception of multiple signals using an SDM (de)multiplexer. As a negative effect, a spatial SCh needs the guard-bands on each of its spatial modes. • Spectral-spatial transponders – they extend the concept of spatial transponders through generation/reception of spectral-spatial SChs, each consisting of spectral SChs transmitted over multiple spatial modes. Again, some optical components, such as laser sources, can be reused, and the guard-bands per each mode are needed. Within this class, we distinguish regular spectral-spatial SChs, in which the shape of spectral SCh is reproduced on all the used spatial modes and which has a regular block-like form (see Fig. 4c)), and irregular spectral-spatial SChs, which may take more arbitrary shapes (as shown in Fig. 4d)). The transmission of SChs over the network requires adequate allocation of spectral-spatial resources in network links and suitable configuration of switching equipment. Note that each of the above types of SChs makes use of a specific subset of transmission/switching resources, namely, either spectral or spatial or spectral-spatial, what leads to different resource allocation schemes. 7

TABLE V: Types of super-channels in the surveyed works. Type of SCh

References

Spectral Spatial Spectral-spatial (regular) Spectral-spatial (irregular)

[16], [16], [16], [42],

[29]–[31], [34], [36], [37], [42]–[48], [50], [51], [53]–[57], [60], [61], [64]–[67], [71], [79] [31]–[41], [69], [79] [36], [57], [59], [63], [66], [68], [70] [52], [62], [86]

TABLE VI: Achievable switching flexibility for each transmission type. Mode coupling

Ind-Sw Mode-Sw FrJ-Sw J-Sw

none/weak

strong (in groups)

strong (all)

yes yes yes yes

no no yes yes

no no no yes

As reported in Table V, the vast majority of the surveyed works consider spectral SChs, while both spatial and spectralspatial SChs are less frequently used. Some of the papers compare the SDM scenarios that operate with different types of SChs (see the references associated with at least two different types), while the others works are dedicated to specific SChs. There are three papers that consider irregular spectral-spatial SChs. In [52], only such SChs that occupy spectrally-spatially contiguous regions on a grid-like representation of spectral-spatial resources are allowed. On the contrary, [42] does not have such limitation and SChs can consist of non-contiguous OCs spread over either spectral (spectral SChs) or both spectral and spatial (spectral-spatial SChs) resources. Similarly, in [62] the sub-carriers of spectral-spatial SChs are allocated individually, with the spectrum contiguity constraint relaxed. One of the resource allocation schemes presented in [33], [39] allows to split large spatial SChs into smaller chunks that can be allocated at different frequencies. In [70], a specific kind of spatial contiguity is assumed, where groups of spatial modes can be accessed by SChs only in an inclusive way, i.e., the use of a higher indexed group forces the use of all lower indexed groups of modes. The results of the so far performed studies show that spatial SChs are less spectrally efficient than spectral SChs of equivalent capacity, since switching guard-bands need to be placed on every spatial dimension to separate neighbouring SChs. Reducing the width of guard-bands separating spatial SChs can improve the network performance in terms of bandwidth blocking probability as analysed in [59]. Besides, the shape of spectral-spatial SChs can be optimized by considering a metric that takes into account, on one hand, the adverse impact on spectrum usage caused by guard-bands in spatial SChs and, on the other, the effect of crosstalk arising between spectral SChs [63]. Another source of resource wastage is over-provisioning of spectral-spatial resources when transmitting/switching the SChs that do not fully fit these resources. For instance, amount of wasted resources due to joint switching (both J-Sw and FrJ-Sw) of ”shorter” spatial SChs has been evaluated in a function of offered traffic load in [32]. V. SDM N ETWORK S CENARIOS The SDM fiber, switching, and transponder solutions, which we have discussed in the previous sections, can be combined in different configurations. It leads to different SDM network scenarios with their proper resource allocation schemes and constraints. In this section, we aim at classifying such scenarios and identifying which are the most frequently considered in the literature, as well as we discuss some of their constraints. First of all, not all combinations of SDM solutions are possible since there are some limitations of certain SDM technologies that prevent their joint and efficient use. Indeed, as discussed in [32], [35] and summarized in Table VI, it is necessary to use adequate transmission media in order to fully-exploit given spectral-spatial switching flexibility in the network. Uncoupled/weaklycoupled SDM fibers (such as SMFBs or MCFs) can be utilized with any switching option. On the contrary, the SDM fiber solutions with strongly coupled modes (such as FMFs, FMFBs, and FM-MCFs) have strict requirements concerning joint routing of modes and, therefore, joint switching paradigms fit best such type of transmission, even though Ind-Sw might handle the switching of coupled modes, as long as they remained together. Mode-Sw requires uncoupled SDM fibers. As well, there is some correlation between the SCh types and the transmission media and switching solutions. Indeed, spectral SChs occupy single modes and, therefore, it is not efficient to utilize them in strongly coupled transmission and joint switching scenarios. The above relations are reflected in the various SDM network scenarios that are considered in the surveyed works, and which we classify in Fig. 5. As noted in Sec. II, almost all these scenarios make use of uncoupled/weakly-coupled fiber solutions (either SMFB or MCF). Spectral SChs are utilized only with this category of fibers and only with Ind-Sw. On the contrary, spatial and spectral-spatial SChs are suitable for any transmission type and any wavelength-capable switching option (i.e., Ind-Sw, J-Sw, and FrJ-Sw). 8

spectral SChs [29]–[31], [34], [36], [37] Ind-Sw

spatial SChs [31]–[35], [37]–[39] spectral-spatial SChs [36]

SMFB

spatial SChs [31]–[35], [37]–[41]

J-Sw

spectral-spatial SChs [36] FrJ-Sw

spatial SChs [31]–[35], [37]–[39]

Hybrid

spatial SChs [33], [39]

Ind-Sw

spectral SChs [29], [42]–[51], [53]–[58], [60], [64]–[67] spectral-spatial SChs [42], [52], [57], [62], [63], [66], [86]

SDM network scenarios

MCF J-Sw

spatial SChs [41], [59]

Mode-Sw

spatial SChs [40], [69]

J-Sw

spectral-spatial SChs [70]

FMF FrJ-Sw

FMFB

spectral SChs [61], [67]

spectral-spatial SChs [68], [70]

J-Sw

spatial SChs [40]

Fig. 5: Joint classification of the surveyed SDM network scenarios. TABLE VII: Space continuity constraint in the surveyed works. Spatial continuity

References

no yes

[29], [30], [34], [36], [37], [43]–[51], [54]–[58], [60], [66], [67], [71], [72] [29], [33], [36], [39], [52], [53], [57], [62], [66], [68]–[70], [79], [87]

Each of the SDM network scenarios is characterized by its proper resource allocation scheme that is the outcome of the applied SDM technology. For instance, the use of joint switching imposes the need for the allocation of all spatial modes within given spectrum range on all links of the routing path of a lightpath connection. Apart from that, some additional constraints related to the capabilities of utilized transmission devices (such as switches or signal regenerators) can be distinguished, including such important ones as space continuity constraint and spectrum continuity constraint. If we assume certain logical labeling of spatial resources (modes, cores, or fibers), the same in each network link, the space continuity constraint means the assignment of unique spatial resources to lightpaths, which do not change on the routing path. Spatial continuity is imposed, for instance, by some switching architectures, in which each independent spatial mode of the input fiber is mapped to the same mode on the output fiber, eliminating SDM lane change operation [12]. Nevertheless, as shown in Table VII, most of the surveyed works relax this constraint and allow for the spatial mode conversion, which can be achieved with the Ind-Sw/Mode-Sw architectures. In this scope, the impact of core switching on network performance is analyzed in an SDM network with MCFs in [29], [66]. The spectrum continuity constraint means that lightpath connections have uniquely assigned spectrum segments, which 9

are the same on all links belonging to their routing paths. In the surveyed works, we have not found any works exploiting spectrum conversion in switching nodes, i.e., the common assumption is that the SChs are transmitted with the spectrum continuity constraint. There are several lessons learn from the comparative studies of different SDM scenarios that have been undertaken in the literature. The analysis of SCh allocation schemes shows that spectral SCh transmission leads to superior network performance, due to its higher spectral efficiency, and spatial SChs bring cost savings thanks to the sharing of network elements among spatial dimensions [31], [34], [37]. Both the size and number of traffic demands as well as spectral switching granularity have impact on the performance of SDM switching paradigms [38]. As shown in [32], the Ind-Sw, FrJ-Sw and J-Sw paradigms offer almost similar performance under high traffic loads offered to the network. The impact of spectral-spatial switching granularity, including the frequency grid granularity and the size of spatial mode subgroups in FrJ-Sw, on spectrum utilization in the network is studied in [35]. The results show that the support of finer spectral granularity in an SDM networks with a joint switching paradigm (either J-Sw or FrJ-Sw) can lead to significant performance improvements that are close to the case of Ind-Sw. Eventually, comparative analysis of network performance under different types of fibers is presented in [40] for FMF, FMFB, and SMFB solutions, and in [41] for MCFs and SMFBs. The obtained results show that the MIMO-DSP for a FMF network consumes more than twice the power required for SMFB and FMFB networks with the same number of supported spatial modes. Besides, the network capacity scales better in an SMFB network than in an MCF network if the number of supported spatial modes increases, which is due to the negative impact of the inter-core crosstalk in MCFs. VI. R ESOURCE A LLOCATION A LGORITHMS In this section, we classify different algorithms and methods that have been proposed in the surveyed papers for efficient and optimized use of transmission resources in SDM optical networks. We begin with reporting the network planning and operation problems addressed in the literature so far (see Section VI-A) and, afterwards, we discuss main algorithmic approaches applied for solving these problems (in Section VI-B). In Section VI-C, we focus on the RSSA problem–a basic optimization problem in spectrally-spatially flexible optical networks, which is present both in the phase of network planning and during its operation. Next, in Section VI-D we present and discuss some basic integer linear programming models used in RSSA optimization, while in Section VI-E we describe an exemplary greedy heuristic algorithm for solving RSSA. In Section VI-F, we extend the general discussion concerning RSSA to the question of optimized selection of spectral-spatial resources for individual lightpaths, which emerges in dynamic network scenarios. Dedicated Sections VI-G and VI-H are devoted to the issue of crosstalk and resource fragmentation, respectively. Eventually, in Section VI-I, we complete our study with a report on the performance and optimization metrics that are used in the surveyed papers. A. Addressed Problems In general, network design and resource allocation problems encountered in optical networking can be classified as either static or dynamic. In the former case, mainly related to the network/connection planning phase, all traffic demands for which connections have to be established in the network are known in advance and must be allocated in the network at the same time. Moreover, the decision how to allocate the demands (i.e., select the required resources for each demand such as routing path, spatial mode, spectrum) is made in an off-line manner, without strict processing time constraints. Therefore, complex and time consuming optimization methods, such as mathematical programming or metaheuristics, can be applied for the static problems. In turn, in dynamic problems, which emerge during network operation, it is assumed that demands (connection requests) are not known in advance, but they arrive and disappear stochastically (i.e., one-by-one). The resources required to establish connections for the requests are chosen dynamically according to the current state of the network. In dynamic problems, the decisions must be taken in an online manner, almost immediately, based on the current availability of network resources [88], [89]. As shown in Table VIII, the surveyed papers address dynamic problems more frequently than static problems. The reason for this fact can be explained in several ways. First, a static optimization of optical networks does not efficiently utilize energy and network resources, especially for low traffic loads and dynamically changing traffic patterns, which become more and more popular in current networks according to growing popularity of data center oriented services. Moreover, this can in part follow from the fact that generating feasible solutions for dynamic problems is generally more simple than solving efficiently static problems, which require more advanced optimization methods. Another possible reason is the high complexity of optimization problems in SDM networks, which comes from the large number of decision variables and constraints related to the selection of routing, spatial mode, and spectrum resources jointly, for all demands. Indeed, as shown in the following subsections, the optimization approaches proposed so far for solving static problems are relatively simple in most of these works. The majority of dynamic scenarios in the surveyed papers focus on a basic lightpath provisioning problem. Similarly, in the context of static problems, most works concern lightpath planning. As summarized in Table IX, only several papers address more advanced problems/scenarios. Among them, the largest set of papers is related to equipment planning, which apart from the resource allocation subproblem embraces such issues as: planning of AoD nodes [47], [48], [51], minimizing the number

10

TABLE VIII: Types of optimization problems related to SDM networks addressed in the surveyed works. Problem type

References

static dynamic

[32], [35]–[37], [45], [48]–[51], [53], [55], [58], [61], [62], [68], [70], [72] [16], [29]–[31], [33], [34], [38]–[44], [46], [47], [52], [54], [56], [59], [60], [63]–[67], [69], [71], [79], [85]–[87], [90]

TABLE IX: Specific problems/scenarios in the surveyed works. Problem/Scenario

References

Equipment planning Virtual optical network Network resilience Software defined network Anycasting Virtual concatenation Filter-less network Traffic grooming

[37], [54], [58], [85], [53] [42] [61] [41],

[40], [47], [48], [51], [61], [68], [70], [72], [79] [56], [60], [69] [85] [90]

[67]

of active transmitters [79], minimizing the cost of installed transceivers [68], [70], addressing MIMO processing [40], [72], and minimizing the overall SDM network cost [37]. Most of the works concerning static optimization assume a fixed set of demands. Only in [51] a quasi-static traffic model is taken into account, where traffic demands, once established, stay permanently in the network, however, the capacity of the established demands increases with time. B. Algorithmic Approaches We begin from reviewing optimization methods applied for solving static optimization problems in SDM networks, afterwards, we focus on algorithmic approaches for dynamic resource allocation problems in such networks. A common approach to modeling and solving optimization problems in communication networks is mathematical programming (MP). Depending on the type of problem variables, such problems are formulated as either mixed-integer programming (MIP), integer programming (IP), or integer linear programming (ILP) problems. This class of problems can be solved in a straightforward way using dedicated MIP solvers (e.g., CPLEX Optimizer or Gurobi Optimizer) that provide an efficient implementation of state-of-the-art MP algorithms, including Simplex and branch-and-cut, as well as specialized heuristics. Another benefit of MP methods is that they yield exact (i.e., globally optimal) solutions. However, their key shortcoming is low scalability, i.e., MP methods cannot provide optimal or even feasible results in a reasonable time for larger instances of complex problems, which is the case of most of optimization problems in optical networks. On the contrary, heuristic and metaheuristic methods represent a set of approximate optimization techniques that can solve various large scale problems relatively quickly; however, the solutions do not have the optimality guarantee, which is only ensured in exact algorithms. The basic heuristic algorithm frequently used in resource allocation problems implements a greedy approach, in which decisions are made in a single run of the algorithm, e.g., the demands are processed in a certain sequence and are allocated one by one in the network according to a first-fit approach [88], [89], [91]. Note, an exemplary greedy RSSA algorithm is discussed with more details in Sec. VI-E. Table X presents various optimization approaches used in the surveyed works for solving static optimization problems. The most usual one is the ILP modeling, which is discussed in more detail below. Also simple greedy algorithms are quite frequently used. The least popular are meteheuristics, which are applied in only three papers and in all cases they are based on a simulated annealing (SA) approach. Eventually, in [72] (not included in the table), an IP formulation with non-linear problem constraints (related to physical layer impairments) is presented. It should be noticed that metaheuristics have been generally a popular and effective tool for various network optimization problems [88]. However, the very huge complexity of the RSSA problems in spectrally-spatially flexible optical networks may limit the applicability of metaheuristic methods. The recent research on application of metaheuristics for optimization of flexible-grid EONs have shown that in many cases such methods face huge scalability problems [91]. In particular, metaheuristic algorithms usually analyze a wide set of solutions generated by some disruptions of the current solution. But, considering RSA problems in flexible-grid EONs even a minor modification of a feasible solution can lead to infeasibility, since some constraints of the optimization problem are not fulfilled. This follows mostly from the fact that the overall solution space of RSA problems is very huge, while the number of feasible solutions in the overall space can be extremely small due the construct of the RSA problem, namely, spectrum continuity and contiguity constraints. Since originally metaheuristics are constructed to solve problems without constraints, additional techniques must be applied to adapt a particular metaheuristic method to a highly constrained problem like RSA, which does not always provide good performance [91]. Therefore, in our opinion the application of metaheuristic methods for RSSA problems, which are even more complex than analogous RSA problems, can 11

TABLE X: Static optimization in the surveyed works. Optimization approach

References

ILP modeling Greedy heuristic Metaheuristic

[36], [45], [48], [49], [53], [55], [58], [61], [68], [70] [32], [35], [37], [45], [50], [51], [53], [55], [62], [86] [32], [35], [55], [92]

TABLE XI: Applied routing and spectrum allocation modeling approach. Routing modeling

Spectrum modeling

References

Link-path

Channel-based

[36], [55], [70]

Slice-based

[45], [48], [53], [61], [68]

Channel-based

[49], [58], [72]

Node-link

encounter many challenges. One of the key challenge will be to develop problem coding approaches (solution representation) to enable effective and relatively fast performance of the algorithm. The ILP models presented in the surveyed works extend the models developed previously for spectrally flexible EONs [91]. As in EONs, there are two basic approaches applied to routing modeling, namely, link-path and node-link; see Table XI. Solving the link-path models is usually less time-consuming than solving the node-link models [91], what may be the reason for their more frequent application in the surveyed papers. Regarding spectrum allocation modeling, the surveyed papers make use of either slice-based or channel-based approach. Both these cases are represented in a similar number of studies. Note that in Section VI-D we present and discuss some basic versions of ILP models of the RSSA problem in spectrally-spatially flexible optical networks. The majority of proposed ILP models are dedicated to SDM networks with MCFs, and only two works consider the use of FMFs [68], [70]. In [36], several general ILP models suitable to various SDM network scenarios are formulated and their theoretical complexity is briefly discussed based on the estimated upper bounds on the number of problem variables and constraints. It should be underlined that only relatively small problem instances have been solved using MP methods in the analyzed papers, e.g., for network topologies consisting of only 6 nodes. Spectrally-spatially flexible optical networks introduce additional space dimension and it can be expected that ILP modeling of SDM networks will face even higher scalability issues when compared to the already complex optimization of EONs [91]. We have not found any works concerning advanced MP optimization methods, such as column generation or branch-andprice, which have been successfully applied in EONs (e.g., see [93], [94]). Moreover, the only metaheuristic used in the surveyed papers is the SA algorithm, whereas many various metaheuristics have been successfully implemented and worked well in the context of EONs. Eventually, only in [55] the authors try to perform a more deep analysis and comparison of some optimization methods in SDM networks. Regarding dynamic network scenarios, only simple heuristic approaches have been proposed for allocating resources for arriving connection requests. It follows mostly from the fact that in dynamic routing and resource allocation problems, the decisions must be made very fast. Also, in most cases, the dynamic problems concern individual demands and, therefore, resource allocation decisions are much simpler than in static optimization problems. In this context, in most of the surveyed papers, the proposed heuristics utilize a simple first-fit approach that intents to establish the demand on the first available routing path, spatial mode, and frequency slot. Also, there are some more sophisticated algorithms in which the selection of routing path or/and spatial mode and spectrum resources is supported by some cost-based metrics [43], [44], [46], [47], [52], [60], [87]. Finally, it is worth mentioning that the resource allocation algorithms developed for dynamic problems can be also utilized, after adaptation, for solving some static problems. Indeed, assuming that static demands are processed one by one according to a selected sequence (as in the above discussed greedy heuristic), the currently processed demand can be established in the network using a dynamic algorithm. C. RSSA Problem The basic optimization problem in the fixed-grid WDM optical networks is the routing and wavelength assignment (RWA) problem. RWA consists in selecting a routing path and a wavelength (frequency slot) for each traffic demand from the given set of demands. In RWA, the wavelength continuity constraint must be satisfied, namely, the same wavelength must be assigned to each demand on all links of the routing path, except when the use of wavelength converters is allowed. In the flexible-grid EONs, RWA converts into the RSA problem, in which the frequency slot is not fixed, but is adjusted to the actual bandwidth of the demand. Again, the spectrum continuity constraint is enforced on the routing path segments on which the spectrum conversion is not available. Additionally, the spectrum contiguity constraint must be satisfied, which enforces the assignment

12

of an adjacent subset of unoccupied frequency slices (forming the frequency slot) within the flexible frequency grid. The finer granularity of the frequency grid and the spectrum contiguity constraint make RSA much complex than RWA [20], [89], [91]. SDM introduces an additional spatial dimension that must be addressed in network optimization, i.e., besides routing and spectrum allocation, each demand must be assigned to one or more spatial resources that are available in the network. Therefore, the basic optimization problem in spectrally-spatially flexible optical networks is routing, spatial mode, and spectrum allocation (RSSA) [33], [38], [39], [79], [85], [90]. Depending on the type of SDM fiber in use, the RSSA problem may have a slightly different name and meaning. For instance, if MCFs are used, RSSA is usually referred to as the routing, core, and spectrum allocation (RCSA) problem [29], [43], [45]–[47], [52], [57], [64], [65]. Further extensions of RSSA are also possible whenever additional problem variables are involved, for instance, related to modulation format or baud rate selection. For example, such extended problems may concern: routing, modulation format, core, and spectrum allocation (RMCSA) [50], [51], [55], [61], or routing, spectrum, and core and/or mode assignment (RSCMA) [66]. Routing, modulation format, baud rate, and spectrum allocation (RMBSA) is studied in the SDM networks with FMFs [68], [70]. Eventually, the routing, wavelength, and core allocation (RWCA) problem is considered in fixed-grid WDM networks with MCFs [49]. In general, RSSA is an N P-complete problem. Indeed, even if only one spatial mode is available, the problem reduces to the RSA problem, which is N P-complete [20]. The previous research on RSA in EONs has shown that the RSA problem itself is very challenging [89], [91]. There are two general approaches that are applied to cope with the RSSA problem. In the first one, it is assumed that all available network resources and their related variables (representing routing, spatial mode, and spectrum selection decisions) are optimized jointly. This approach is a common one in the ILP modeling; see e.g., [36], [45], [48], [49], [53], [55], [58], [61], [68], [70]. The joint RSSA optimization is also present in some other works concerning static problems, but solved using heuristic algorithms, such as [32], [35], [50], [51], as well as in selected papers devoted to dynamic scenarios [37], [52], [54], [56], [60], [87]. All the remaining surveyed papers make use of the second general approach, which consists in decomposition of the RSSA problem into the routing subproblem and the spatial mode and spectrum allocation subproblem (R+SSA) that are solved separately. Namely, first lightpath routing is determined and next, for the selected routing path, the other resources (i.e., spatial mode and spectrum) are assigned.

C d 4 (2

b/s )

km

e1 :

500

500

0G

d5 (100 Gb/s)

E

A

e 2: 500 km (a) Network Topology

Gb/s)

km

(10

s) Gb/

d6

200

e 6: E

B

d3 (

km

d1 (400 Gb/s)

D

B

e : 5 80 9

00

s) Gb/

d2 (400

m

0k

50 e 4: D

C e: 3 50 0k m

A

(b) Traffic demands

Fig. 6: Network topology and traffic demands In order to visualize the RSSA problem in a spectrally-spatially flexible optical network we consider a simple example. Fig. 6a) presents a network topology with 5 nodes {A, B, ...} ∈ V and 6 bi-directional links {e1 , e2 , ...} ∈ E that connect pairs of network nodes. Fig. 6b) shows a set of demands {d1 , d2 , ...} ∈ D for which lightpath connections have to be established in the network. Each demand is defined by its source node, destination node, and bit-rate. A flexible grid of 12.5 GHz frequency slice granularity is considered. In this example, as in [95], we assume that the transceivers operate at a fixed baud rate and each transceiver transmits/receives an optical carrier that occupies 3 slices (i.e., 37.5 GHz). Moreover, a fixed guard-band of width equal to 1 slice is used to separate neighbor SChs in the spectral domain. We consider four modulation formats, namely BPSK, QPSK, 8-QAM, and 16-QAM, with the transmission reach and supported bit-rates as in [34] and presented in Table XII. For the sake of simplicity, we assume that each network link contains two spatial modes, the spectrum conversion is not allowed, and spatial modes can be freely switched in traversed nodes. The objective function of the considered RSSA optimization problem represents the maximum spectrum usage, which is defined as the width of spectrum (in terms of the number of frequency slices) required in the network to serve all demands. Fig. 7a) presents the initial state of the network in which there are already established lightpaths carrying spectral-spatial SChs for 3 demands, namely, d1 , d2 and d3 . The vertical axis represents the spatial modes in the network links. The spatial modes are denoted as k1 and k2 , and spatial mode k in network link e is identified by a tuple (e, k). The horizontal axis 13

TABLE XII: Transmission reach and supported bit-rate per one transceiver for different modulation formats.

Reach [km] Bit-rate [Gbps]

BPSK

QPSK

8-QAM

16-QAM

6300 50

3500 100

1200 150

600 200

represents the frequency slices in each network link. The unfilled ones (i.e., in white color) correspond to free slices, therefore, they can be used to allocate next demands. In the considered example, we want to establish a lightpath carrying a spectralspatial SCh for demand d4 of 200 Gb/s bit-rate between nodes C and D. For that purpose, we have to decide on a routing path, a modulation format, a spectrum segment, and spatial modes.

d1

d3

s5

demands d4

guardband d5

s9

d6 s1

e1 k 1 k2 e2 k 1 k2 e3 k 1 k2 e4 k 1 k2 e5 k 1 k2 e6 k 1 k2

edges and spa al modes

edges and spa al modes

s1

d2

f

frequency slices (b) Demand d4 on link e4 , spatial mode k1

(a) Current state of network

s5

s9

s1

e1 k 1 k2 e2 k 1 k2 e3 k 1 k2 e4 k 1 k2 e5 k 1 k2 e6 k 1 k2

edges and spa al modes

edges and spa al modes

s9

f

frequency slices

s1

s5

e1 k 1 k2 e2 k 1 k2 e3 k 1 k2 e4 k 1 k2 e5 k 1 k2 e6 k 1 k2

f

s5

s9

e1 k 1 k2 e2 k 1 k2 e3 k 1 k2 e4 k 1 k2 e5 k 1 k2 k e6 1 k2

frequency slices

f

frequency slices

(c) Demand d4 on link e5 , spatial modes k1 , k2 and link e6 , spatial modes k1 , k2

(d) Demand d4 on link e5 , spatial mode k1 and e6 , spatial mode k1

Fig. 7: RSSA problem in spectrally-spatially flexible optical network. Table XIII presents 3 (of many) candidate lightpaths for demand d4 . Note, that depending on the length of selected routing path, different modulation formats can be considered according to the transmission reach values in Table XII and, in consequence, different number of frequency slices is required. The network states resulting from the selection of each of these three candidate lightpaths are presented in Fig. 7b), Fig. 7c), and Fig. 7d), respectively. As we can see, the lowest spectrum usage (equal to 8) is obtained if the second candidate lightpath is selected. TABLE XIII: Candidate lightpath allocations for demand d4 and resulting spectrum usage. No

Path

Length [km]

Modulation format

Slices

Spectrum usage

#1

((e4 , k1 ))

500

16-QAM

7-11

11

#2

((e5 , k1 ), (e5 , k2 ), (e6 , k1 ), (e6 , k2 ))

1309

QPSK

4-8

8

#3

((e5 , k1 ), (e6 , k1 ))

1309

QPSK

4-11

11

Eventually, in Fig. 8 we provide some complete solutions for the presented example in which all demands are allocated. 14

In particular, Fig. 8a) shows an optimal RSSA solution found with the CPLEX solver solving an ILP model and Fig 8b) presents an RSSA solution yielded by the greedy heuristic algorithm described in [96]. Additionally, adequate RSA solutions corresponding to an EON scenario (i.e., with only one spatial mode available) are presented in Figures 8c) and 8d), respectively.

d1 s5

d3

demands d4

guardband d5

d6

s9

s1

e1 k 1 k2 e2 k 1 k2 e3 k 1 k2 e4 k 1 k2 e5 k 1 k2 e6 k 1 k2

edges and spa al modes

edges and spa al modes

s1

d2

f

s5

f

frequency slices

frequency slices

(a) SDM optimal solution

s5

s9

(b) SDM greedy heuristic solution

s13 edges

edges

s1 e1 e2 e3 e4 e5 e6

s9

e1 k 1 k2 e2 k 1 k2 e3 k 1 k2 e4 k 1 k2 e5 k 1 k2 e6 k 1 k2

f

e1 e2 e3 e4 e5 e6

s1

s5

s9

s13

f

frequency slices

frequency slices

(c) EON optimal solution

(d) EON greedy heuristic solution

Fig. 8: Exemplary solutions of RSSA and RSA, respectively, in SDM and EON network. D. Basic ILP Models for RSSA In this section, we formulate three alternative ILP models for the RSSA problem in spectrally-spatially flexible optical networks. The ILP formulations implement the modeling approaches presented in Table XI, namely: (a) link-path routing with channel-based spectrum modeling, (b) link-path routing with slice-based spectrum modeling, and (c) node-link routing with channel-based spectrum modeling. The models are formulated based on the information available in papers [36], [45], [48], [49], [53], [55], [58], [61], [68], [70]. To keep the presentation more consistent and understandable, we unified and simplified the notation. It should be stressed that various variations of the RSSA problem are addressed in the referenced papers (as discussed in Sec. VI-A). Here, we focus on a basic version of the RSSA problem and our main goal is to show how to model the specific constraints related to spectrally-spatially flexible optical networks in which each network link can consist of several spatial modes. As the objective function, we use the maximum spectrum usage, which is a frequently utilized metric in optimization of optical networks, including SDM networks (see Section VI-I for more information on this issue). The maximum spectrum usage is defined as the number of frequency slices required in the network to serve all demands. However, it should be noted that the presented ILP models can be modified to account for other objective functions and different networking scenarios. Moreover, in all three models we assume that the space continuity constraint is relaxed, namely, a particular lightpath can have allocated various spatial modes in the links of its routing path. Eventually, we consider that lightpath connections carry spectral SChs, in other words, only one spatial mode is associated with a lightpath on each link belonging to its routing path. Note that the presented models can be straightforwardly adapted to the scenarios with other types of SChs (such as spatial or spectral-spatial), which is discussed in [36]. The network is modeled as a directed graph G = (V, E), where V is a set of network nodes and E is a set of fiber links that connect pairs of network nodes. The set of spatial modes available on each link is denoted as K. On each spatial mode k ∈ K, for each network link e ∈ E, the available spectral resources are divided into frequency slices which are included in set S. By grouping a set of adjacent slices, frequency slots of different width can be created for supporting heterogeneous traffic demands. Let D denote the set of unicast traffic demands, where demand d ∈ D is defined by its source and destination nodes, and bit-rate hd . Here—again for the sake of simplicity and to keep the presented models simple—we do not apply the concept of distance adaptive transmission described in [36], [91]; hence, we assume that each demand makes use of the same modulation format (e.g., BPSK), which is independent on the selected routing path. Consequently, bit-rate hd can be translated into the 15

number of slices nd forming a spectral channel (also called frequency slot) that have to be allocated on a routing path for realizing demand d. There are two basic approaches that can be applied to modeling of routing in optical networks: link-path and node-link. The former one assumes that for each demand a set of precomputed paths connecting the end nodes of a demand is given and in the optimization process one of the paths is selected for each demand. The latter approach consists in determining a routing path using a so-called flow conservation law, which is represented in an ILP formulation by a set of specific constraints that concern the preserving of network flows in each network node. For more details on routing modeling see [88], [89], [91]. In turn, for spectrum allocation modeling in flexible-grid optical networks we can use two following approaches: channel-based or slice-based. The former one makes use of a precalculated set of spectral channels (frequency slots), where each channel is identified by a contiguous subset of frequency slices necessary to realize a particular demand. In this case, the ILP formulation uses a set of problem variables that represent the selection (or not) of certain channel for each demand. The latter approach models spectrum allocations by explicitly assigning the starting slice for each demand and by avoiding collisions between competing demands in the spectrum domain by means of suitable problem constraints. For a more detailed discussion of spectrum modeling in flexible-grid networks check [91]. The first presented model is based on the link-path routing with channel-based spectrum modeling. Note that the channelbased approach was first proposed in [97]. A (spectral) channel can be defined as a pre-computed set of spectrum contiguous frequency slices of a particular size (i.e., number of slices). Let C(d) denote a set of candidate channels for demand d of the size of exactly nd slices. To represent the association of channel c with its frequency slices in a formal way, let constant γdcs be 1 if channel c supporting demand d (i.e., included in set C(d)) uses slice s, and 0 otherwise. Let P (d) denote a set of routing paths, where each path p ∈ P (d) consists of a subset of network links connecting the end nodes of demand d. Having defined for each demand d both a set of candidate paths P (d) and a set of candidate channels C(d), the decision variable xdpc simply indicates whether channel c on candidate path p is used to realize demand d. Three other sets of variables are used to control the allocation of spectrum resources, namely: binary variables yes indicate if slice s is occupied in any spatial mode of link e, binary variables ys represent aggregate information if slice s is occupied in any spatial mode on any network link and, finally, integer variable y indicates the width of spectrum (in terms of the number of slices) required in the network to realize all demands and it is applied as an objective function that is subject to minimization. SDM/Link-path/Channel-based sets E network links K spatial modes available on each link S frequency slices D traffic demands P (d) candidate routing paths for flows realizing demand d C(d) candidate spectral channels of size nd for demand d constants δedp =1, if link e belongs to path p realizing demand d; 0, otherwise nd requested number of slices for demand d γdcs =1, if channel c associated with demand d on uses slice s; 0, otherwise variables xdpc =1, if channel c on candidate path p is used to realize demand d; 0, otherwise (binary) yes =1, if slice s is occupied in any spatial mode of link e; 0, otherwise (binary) ys =1, if slice s is occupied in any spatial mode on any network link; 0, otherwise (binary) y indicates the number of slices required to realize all demands (integer) objective minimize y constraints X X

(1a)

xdpc = 1,

d∈D

(1b)

p∈P (d) c∈C(d)

X X

X

γdcs δedp xdpc ≤ |K|yes ,

e ∈ E, s ∈ S

(1c)

d∈D p∈P (d) c∈C(d)

X

yes ≤ |E|ys ,

s∈S

(1d)

e∈E

16

y=

X

ys .

(1e)

s∈S

Optimization objective (1a) minimizes the number of slices actually used (equal to the sum of variables ys ). Constraint (1b) ensures the selection of a routing path and the selection of a spectral channel. The avoidance of collision in spectral domain is assured by inequality (1c), which imposes that slice s on link e can be used at most |K| times, what refers to the number of available spatial modes on each link. Constraint (1d) defines variable ys and the last equation (1e) defines variable y. The second presented model uses again the link-path concept, but for spectrum modeling it applies the slice-based approach. Therefore, we need to introduce some new notation. Let binary variable xdp represent the selection of routing path p for demand d. Furthermore, binary variable xdke indicates if demand d uses spatial mode k on link e. To avoid spectrum overlapping some additional variables are required. Let binary variable zdd0 ke indicate if demand d and/or demand d0 use spatial mode k on link e. In addition, integer variable fd defines the starting slice allocated for demand d. Finally, binary variable udd0 ke is set to 1, if the starting slice of demand d on spatial mode k of link e is lower thanPthat of demand d0 (i.e. fd < fd0 ). Moreover, e−1 we introduce a set of constants κe for each e ∈ E with values such that |K| l=1 κl < κe . Constants κe are required to ensure the spatial mode selection on each link of the selected routing path with the possibility to use differnet spatial modes on differnt links. SDM/Link-path/Slice-based constants (additional) Pe−1 κe constants having values such that |K| l=1 κl < κe , which are used for spatial mode selection P on different links of routing path M large number, M ≥ nd d∈D

variables (additional) xdp =1, if candidate path p is used to realize demand d; 0, otherwise (binary) xdke =1, if demand d uses spatial mode k on link e; 0, otherwise (binary) zdd0 ke indicates, if demand d and/or d0 go through spatial mode k on link e (integer) fd indicates the starting slice used for demand d (integer) udd0 ke = 1, if the starting slice of demand d is lower than that of demand d0 (i.e. fd < fd0 ) on spatial mode k of link e; 1, otherwise (binary) objective minimize y constraints X xdp = 1,

(2a)

d∈D

(2b)

p∈P (d)

XX

κe xdke =

e∈p k∈K

X

κe xdp ,

xdke + xd0 ke = zdd0 ke , z

dd0 ke

≤ 2(u

d ∈ D, p ∈ P (d)

(2c)

e∈p

dd0 ke

+u

d, d0 ∈ D, k ∈ K, e ∈ E

d0 dke

),

d, d ∈ D, k ∈ K, e ∈ E

fd − fd0 + M (udd0 ke + xdke + xd0 ke ) ≤ 3M − nd , fd0 − fd + M (ud0 dke + xdke + xd0 ke ) ≤ 3M − nd0 , fd + nd − 1 ≤ y,

(2d)

0

(2e)

d, d0 ∈ D, k ∈ K, e ∈ E 0

d, d ∈ D, k ∈ K, e ∈ E

d ∈ D.

(2f) (2g) (2h)

Optimization objective (2a) is the same as in model (1). Equation (2b) ensures that a single routing path is selected to realized each demand. Constraint (2c) selects a spatial mode on each link of the chosen routing path p for demand d. In more detail, the construct of constant κe guarantees that for each link e included in the selected path p (according to variable xdp ), exactly one spatial mode is used, i.e., variable xdke is set to 1 only for only one spatial mode k. Constraints (2d)-(2g) ensure the spectrum continuity and non-overlapping spectrum allocation to different demands that share the same spatial mode on a given link. In more detail, there are two different cases analyzed in constraints (2d)-(2g). First, when one (or both) of the demands (i.e., d and d0 ) does not pass through spatial mode k on link e, i.e., xdke or/and xd0 ke equal to 0. In this case, constraints (2f) and (2g) are deactivated, i.e., they both always hold regardless of fd and fd0 values, due to the larger value on the right-hand side of the constraints. In the second case, when both demands d and d0 use spatial mode k on link e, i.e., both variables xdke and xd0 ke are equal to 1, one of the constraints (either (2f) or (2g)) is activated according to the values of udd0 ke and ud0 dke . Finally, inequality (2h) defines the variable y.

17

The last model uses the node-link routing together with channel-based modeling. The following new binary variables are required to formulate the ILP model. First, let xdekc denote if demand d uses channel c on spatial mode k of link e. The next new variable is udc that indicates, if demand d is assigned to channel c. Finally, variable yesk denotes if slice s is occupied on spatial mode k of link e. SDM/Node-link/Channel-based sets (additional) V network nodes δ + (v) links leaving node v δ − (v) links entering node v constants (additional) sd source node of demand d td destination node of demand d variables (additional) xdekc = 1, if demand d uses channel c on spatial mode k of link e; 0, otherwise (binary) udc = 1, if demand d uses channel c; 0, otherwise (binary) yesk =1, if slice s is occupied on spatial mode k of link e; 0, otherwise (binary) objective minimize y constraints X X

(3a)

xdekc −

X

X

xdekc =

e∈δ − (v) k∈K

e∈δ + (v) k∈K

  +udc −udc =  0 X

udc = 1,

if v = sd if v = td , otherwise

(3b) v ∈ V, d ∈ D, c ∈ C(d)

d∈D

(3c)

c∈C(d)

X X

xdekc γdcs ≤ yesk ,

e ∈ E, s ∈ S, k ∈ K

(3d)

d∈D c∈C(d)

X

yesk ≤ |K|yes ,

e ∈ E, s ∈ S

(3e)

k∈K

X

yes ≤ |E|ys ,

s∈S

(3f)

e∈E

y=

X

ys .

(3g)

s∈S

Again the objective function (3a) represents the number of slices actually used. Equalities (3b) represent the flow conservation constraint and determine a routing path from the source node to the destination node of each demand. Constraint (3c) ensures that each demand is assigned to exactly one spectrum channel. Inequality (3d) controls the usage of slices in each spatial mode and link. Three last contraints (3e), (3f) and (3g) define variable y. Due to limited size of this survey, the presented above ILP models (1), (2), and (3) account for a basic version of the RSSA problem. However, these models are generic and can be used as a starting point to address more complex scenarios that arise in the context of SDM networks. Moreover, in Table XIV we show the computational complexity of the models in terms of the number of variables and constraints. We can notice that the size of link-path models can be tuned by selecting the number of candidate paths denoted by |P |, while the node-link model directly includes all possible routing paths, what can significantly increase its complexity. Similarly, we can tune the size of the channel-based models by decreasing the size of candidate channel set denoted by |C|. However, the default size of set C directly depends on the number of slices |S| available in the network. Since the objective function is to minimize the number of slices and in the beginning of the optimization process the size of set S can be difficult to estimate, a good approach is to solve first the problem by a heuristic method to obtain an upper bound on |S| and next use this value when solving the ILP models.

18

TABLE XIV: Complexity of the ILP models. Model

Number of Variables

Link-path/Channel-based

|D||P ||C| + |E||S| + |S| + 1

Link-path/Slice-based

|D||P | + |D||K||E| + 2|D|2 |K||E| + |D| + 1

Node-link/Channel-based

|D||E||K||C| + |D||C| + |E||S||K| + |E||S| + |S| + 1

Model

Number of Constraints

Link-path/Channel-based

|D| + |E||S| + |S| + 1

Link-path/Slice-based

2|D| + |D||P | + |D||K||E| + 3|D|2 |K||E|

Node-link/Channel-based

|V ||D||C| + |D| + |E||S||K| + |E||S| + |S| + 1

E. Greedy RSSA Algorithm Many of surveyed works make use of a variation of greedy RSSA algorithm, i.e., a deterministic heuristic algorithm that creates a solution in a constructive manner. At each iteration, such algorithm assigns a value to a selected decision variable and, once fixed, the variable remains unchanged until the algorithm processing is terminated and the complete solution is obtained. The variables and their values are selected using a local heuristic that makes locally optimal choices (from the point of view of current decisions) with the hope of finding sufficiently good solution in the end. The final solution is built up on local decisions undertaken during the algorithm processing The main advantages of greedy algorithms are their simplicity in design and lower complexity compared to metaheuristic methods [98]. In Fig. 9, we show a diagram of a greedy algorithm used for solving the RSSA problem. An input is the set of demands D which have to be served. In step 1, a preprocessing of demands is performed, e.g. demand sorting and calculation of various metrics for each demand. At the beginning, in step 2, the solution set S is set to be empty. In step 3, a condition is checked whether there exists a demand in set D. If yes, at each loop iteration, a local heuristic selects demand di from set D and determines the path, spectrum segment, and spatial mode(s) that will serve it (in step 4). Next, in step 5 demand di is allocated in the network using the selected resources. Finally, the undertaken decisions concerning demand di are included into solution set S (in step 6) and the demand is removed from set D (in step 7). When the loop meets the stopping criteria, i.e., there are no demands left in D for processing, the algorithm terminates and returns the final solution. In the preprocessing phase, demands can be sorted according to a selected metric, for instance, in descending order of their shortest-path transmission distance [62], or in descending order of the required number of frequency slices on their shortest routing path [45], [50], [55]. In the case of a draw, an additional metric can be used such as the required bandwidth or hop count. In [51], demands are sorted in descending order according to the values of the product of requested frequency slices on the shortest path and the physical length of that path. Moreover, if more than one candidate routing path is available, the paths are often sorted increasingly according to their physical distance. Analogously, the candidate lightpaths routed over the same physical path are frequently sorted in an increasing order of their associated spectrum segment [55]. It should be mentioned that the order of processed demands may have a crucial impact on the quality of generated solutions. In the local heuristic, one of pending demands is selected and processed. There are two general approaches for performing such selection, namely, it may be either fixed or adaptive. In the former one, demands are chosen in an order obtained after preprocessing [51], [53], [62], i.e., at each iteration the lightpath selection is performed for the first pending demand. Note that the selection of routing, spectrum, and spatial mode resources can be done either jointly or separately (see Sec. VI-C). In the latter approach, demands can be allocated in a different order than the one obtained after preprocessing. In particular, all the demands may be previewed according to the initial order, but then either the best one or the first one that satisfies certain condition may be selected for further processing. In [45], [50], [55], at each iteration the current value of an objective function, which in these works represents spectrum usage, is checked. Next, in first step, from the set of all unprocessed demands and considering their candidate lightpaths, there are allocated as many as possible demands that do not worsen the objective function. In second step, if some of the demands remain in the unprocessed set, the first available one is selected for processing which, in consequence, will lead to the increase of the objective function value. Then, the above-mentioned steps are repeated until all demands are served. Moreover, the local heuristic may use same additional constrains than the ones related to the basic RSSA process, for instance, such as checking if crosstalk levels do not exceed acceptable threshold values [45], [51], [53], [62]. Greedy heuristic is time efficient and effective approach for provisioning feasible solutions to the RSSA problem. It can be straightforwardly adjusted to more complex SDM network scenarios, for example, with such requirements as crosstalk awareness or resource fragmentation awareness, by modification of its local heuristic subroutines. Additionally, it can be easily hybridized by a combination with other more intelligent methods, such as metaheuristics. In that case, a metaheuristic may be responsible for optimizing the demand order, while the greedy heuristic looks for the best lightpath selection for each demand according to that order. 19

variables D: set of demands to process S : solution set di : demand selected at i-th iteration from set D si : solution obtained after i-th iteration

input: D

1: Preprocessing 2: S = ∅

no

3: D 6= ∅ subprocedures Preprocessing : sorting, metrics calculation, etc.

yes

4: 5: 6: 7:

Local-Heuristic(D): selection of the best candidate (demand) from set D for next allocation

di := Local-Heuristic(D) si := Allocate-Demand(di ) S := S ∪ si D := D \ di

Allocate-Demand(di ): reservation/allocation of frequency slices for demand di in the spatial modes throughout its route

output: S Fig. 9: Block diagram of greedy RSSA algorithm.

F. Selection of Spectral and Spatial Resources Here, we analyze the spatial mode and spectrum allocation (SSA) subproblem of RSSA in more details. Namely, we focus on the question of selecting spectral-spatial resources for individual lightpaths, which emerges mainly in dynamic network scenarios, but can be also utilized by greedy optimization techniques (see Sec. VI-B and Sec. VI-E). Upon a lightpath connection request, and having decided the routing path, the SSA process aims at assigning to the lightpath some available (i.e., not occupied) spectral and spatial resources on each link belonging to the path. In a lightly loaded network, there may exist a huge amount of free resources, which results in an immense set of possible SSA solutions. Even though given SSA strategy may not have an immediate impact on the considered network performance metric (e.g., such as connection blocking probability), it may lead to the deterioration of overall network performance in highly loaded conditions. Therefore, it makes the application of adequate SSA approach so important. The allocation of spectral-spatial resources can be represented by means of a matrix structure, for instance, a matrix of binary variables, where each matrix element denotes whether given frequency slice on the corresponding spatial mode is used or not. In [63], such matrix is utilized and, besides, another matrix representation that indicates the size of available contiguous spectrum blocks is proposed. To facilitate the analysis of continuous fragments of resources within the allocation matrix, the authors of [52] use a connected component labeling algorithm, which follows from pattern analysis of digital images. The idea of using image processing is further explored in [99], where the method of inscribed rectangle search is proposed. In EONs, one of the most frequently utilized spectrum allocation policies is first-fit (FF), which selects the frequency slot of the lowest frequency slice index [100]. In SDM, there is some ambiguity in term FF due to the existence of the spatial domain. Indeed, depending on the order in which the spatial and spectral domains are processed, resource selection may differ. 20

TABLE XV: SSA policies in the surveyed SDM scenarios. SCh type

SSA policy

Spectral

References

FC

[42], [45], [55], [62], [69]

CF

[29], [31], [34], [37], [44], [46], [66], [79]

AS

[29], [30], [47], [79]

Cost-based

[43], [51], [56], [60], [64]

Spatial

FC

[31]–[35], [79]

Spectral-spatial

AS

[57], [66]

In the surveyed papers that assume spectral SChs, we distinguished the following general SSA policies: • Spectral FC – at first, a spectral SCh with the lowest frequency slot is selected, and then with the lowest core index (see Fig. 10a)). • Spectral CF – at first, a spectral SCh with the lowest core index is selected, and then with the lowest frequency slot (see Fig. 10b)). • Spectral AS (align strict) – a pre-computed region is selected, which matches the connection bandwidth (see Fig. 10c)). • Spectral Cost-based – a spectral SCh is selected with a minimum value of a cost metric. The cost represents either the crosstalk effect, link mapping ability, or cost of selected node architecture deployment (see Fig 10d)). s5

k4

s9

4

k3

s13

7

3

6

k2

2

k1

s1

8

1

5

f

Alloca on direc on

spa al modes

spa al modes

s1

s5

k3

3

k2

1

k1

k3

s1

6

k2 k1

s13

7 5 3 1

4 2

4

2

f

frequency slices

f

Alloca on direc on

spa al modes

spa al modes

k4

s9

5

6

(c) Spectral AS

(a) Spectral FC

s5

s13

k4

frequency slices

s1

s9

s5

k4 k3

frequency slices

s13

4 2

k2 k1

s9 6

3 1

5

f

frequency slices

(b) Spectral CF

(d) Spectral Cost-based

Fig. 10: Spectral allocation policies Besides, all the above-mentioned policies are applicable for both spatial SChs and spectral-spatial SChs. However, in the surveyed papers, only the FC policy is examined for spatial SChs (see Fig. 11a)), while spectral-spatial SChs make use only of the AS approach (see Fig. 11c)). In Table XV, we link these SSA policies with the surveyed papers. As it can be easily noticed, the largest and most diverse set of different SSA policies is dedicated to spectral SChs; here, CF is the most utilized one. Much less solutions are proposed for spatial SChs, while only two papers present an SSA policy (i.e. AS) for spectral-spatial SChs. In [66], different SSA policies, such as FCM, FMC, MFC, etc., are studied in the context of FM-MCF fibers, where symbols F, M, and C represent, respectively, frequency slot, spatial mode, and fiber core. The order of letters determines the priority of processed resources, e.g., in CMF, first a fiber core is selected, then a spatial mode is chosen in this core, and finally, an optical channel of the lowest possible slice index is determined. The results of numerical experiments show that the CFM allocation strategy (which corresponds to presented above Spectral CF strategy) yields the best performance in terms of bandwidth blocking probability. However, the performance of different strategies may heavily depend on considered network scenario, traffic characteristics, and assumed network physical model. Regarding comparative studies, four distinct SSA strategies corresponding to different types of SChs are analyzed in [79]. Spectrum First is applied only to spectral SChs, and first, it finds the minimal core index on a selected path and then the lowest frequency slot. Space First (see Fig.11b) is used for spatial SChs, and again it implements the FF spectrum allocation policy. Degenerate Space First is similar to the previous one, but it enforces that only one connection might use spatial resources 21

within a given spectrum region. Finally, Align Strict, which is appropriate for spectral SChs, tries to fit the demand to a pre-computed spectrum block of a matching size. The performance results presented in [79] show that the Align Strict strategy achieves the lowest connection blocking probability in the considered dynamic network scenario. The next best performing one is Spectrum First, and both Spaceoriented policies are the worst ones. It is worth to mention, that spatial allocation policies have some overhead due to guardbands, and in consequence, it may be hard to achieve the same performance as in spectral strategies (again, it may be dependent on the network physical assumptions). In [34], different allocation strategies are considered in terms of the switching paradigms cost. Ind-Sw paradigm requires the highest number of WSSs, which results in the highest equipment cost. The second one is FrJ-Sw and the least costly one is J-Sw. In order to efficiently make use of spectral allocation strategies, the Ind-Sw paradigm has to be applied. In turn, for spatial strategies either J-Sw or FrJ-Sw approach may be used. An optimal scenario would be to use the less costly J-Sw paradigm with spatial SChs and achieve the same network performance as with the most flexible Ind-Sw paradigm and spectral SChs. However, to the best of our knowledge, such solution has not been presented in the literature so far. G. Dealing with Crosstalk

spa al modes

spa al modes

spa al modes

As mentioned in Section II, the SDM networks with weakly-coupled s3 s7 s1 s5 MCFs are prone to excessive signal impairments due to the inter-core k4 2 crosstalk effect (XT); hence, this important issue need to be addressed 6 8 k3 when planning and operating such networks. As reported in Table XVI, k2 there are two main approaches to cope with the XT effect in the surveyed 1 4 5 3 7 papers: best-effort and strict constrained, whereas the former concept can k1 f be further divided into two types, namely, best-effort avoidance and bestfrequency slices effort core prioritization. To visualize these concepts, in the following dis(a) Spatial FC cussion we will refer to an example related to a MCF with 7 hexagonallyarranged cores, with the order of cores presented in Fig. 12. The cores s3 s7 s1 s5 with indices 1 to 6 have 3 adjacent cores (e.g., core 1 is adjacent to cores k4 2, 6 and 7), and core number 7 has 6 adjacent cores. We assume that XT 8 k3 occurs only between adjacent cores when signals are transmitted in an 7 6 k2 overlapping spectrum range. 3 5 1 4 The best-effort avoidance approach tries to either avoid or minimize k1 2 f the XT between neighboring cores during the allocation of lightpaths. frequency slices Here, the applied SSA police aims at decreasing the spectrum overlap in (b) Spatial CF neighbor cores to minimize a crosstalk ratio, which is defined for each spectrum slice as a number of cores that cause the crosstalk for this slice s1 s5 s9 s13 [43]. In a consequence, if the network is not congested, the XT can be k4 avoided. k 3 approach. 3 5 6 Fig. 13a) illustrates the best-effort avoidance Spectrum resources are partially occupied by earlier established connections.kThe forthcoming demand has to be allocated 2 1 4 on one core and it requires 5 adjacent slices.kWe f 1 assume that 2the allocation starts from the 6 1 slice of index 5. There are available three candidate SChs — SCh1 on core 2, SCh2 on frequency slices core 5 and SCh3 on core 7, with crosstalk ratios equal to 3, 5 and 8, respectively. Hence, (c) Spectral-Spatial AS the SCh1 is selected to realize the demand. 5 7 2 In turn, the best-effort core prioritization approach generally has the same goal as the Fig. 11: Spatial allocation policies above concept, but additionally it implements a dedicated core prioritization mechanism. Specifically, during the allocation process, the cores are analyzed in a given sequence, according to their priority. The core priority is pre-determined and such that it reduces 4 3 the dominant XT in the given MCF structure [29]. Fig. 13b) shows an exemplary order of demand allocations according to that policy. At first, connections are realized on cores 1, 3 and 5 — signals are transmitted on non-adjacent Fig. 12: Cores order in a MCF cores. Next, if it is not possible to allocate demands on first order cores, the ones with with 7 cores. indices 2, 4 and 6 are used. Eventually, the connections make use of core 7. The second general approach to cope with XT, i.e., strict constrained, relies on estimation of XT during the resource allocation process. In particular, a lightpath is provisioned for a connection request (demand) only when the levels of XT of not only this new lightpath but also other already established lightpaths meet a predefined XT threshold, since the additional XT triggered by the new connection may influence the signal quality of other connections [57]. In the example in Fig. 14a), cores 1, 3, 4 and 5 are occupied at certain frequency intersection. A new request is allocated on core 2, aware that signal may be leaking from adjacent cores. However, the newly established connection also interferes 22

TABLE XVI: Dealing with crosstalk in the surveyed works. XT-aware approach

References

Best-effort avoidance Best-effort core prioritization Strict constrained

[42], [43], [61], [63] [29], [44], [46], [53], [57], [66] [45], [48]–[51], [53], [55], [57], [58], [60], [62], [66], [68], [70], [72], [86]

allocated demands s1

s5

1storder alloca on s1

candidates for next demand s9 s13

2 3

s5

1 SCh 1

spa al modes

spa al modes

1

2ndorder alloca on

4 5 6

SCh 2

7

SCh 3

f

5 6

s9

1

2 7

2 3 4

3rdorder alloca on s13

3

4

8

9 5

6 10

7

11 12

frequency slices

f

frequency slices

(a) Best-effort avoidance approach

(b) Best-effort core prioritization approach

Fig. 13: Best-effort XT-aware approaches

with the previously allocated ones on cores 1 and 3. In consequence, the transmission quality of these two connections may be degraded. The key disadvantage of the strict constrained approach is that it involves complex calculations depending dynamically on the current network state, which changes in time due to lightpath setups and tear-downs. Therefore, to simplify the problem, a worst-case approach is applied frequently. Here, a worst-case transmission reach of the optical signals (usually accounting for different bit-rates and modulation formats) across MCFs, which is estimated for the maximum possible XT that can occur in the fiber, is assumed in the allocation process. Accordingly, the lightpath are not established on routing paths with a length exceeding the obtained transmission reach [45], [48], [50], [51], [55]. In Fig. 14b), a new connection is going to be set up on core 2 (analogously to Fig 14a)). During allocation of each demand there is an assumption that there might be some transmission on all adjacent cores, therefore the maximum possible value of XT, resulting from all neighbouring cores, is taken into the account when estimating the transmission quality. Hence, after allocation, none of the connections exceeds acceptable XT threshold values. Various models are applied to estimate the transmission reach in XT-aware networks. Most of the surveyed papers that use the strict constrained approach, e.g., [9], [45], [51], [53], [60], [66], utilize the analytical formula proposed in [81] for XT estimation. Some other papers use a predefined value of the XT between two adjacent cores per a particular distance [49], [58], [72]. The authors of [50], [55] use data obtained by real laboratory XT measurements. Moreover, linear and nonlinear physical-layer impairments due to both intra-core (i.e., in a core) and inter-core (i.e. between cores) crosstalk are taken into account in [62]. All the above discussed papers concentrate on MCFs. There are also some papers that apply the strict constrained approach when estimating transmission quality with XT involved in SDM networks with FMFs [68], [70]. The presented XT-aware resource allocation approaches differ in terms of accuracy and complexity, which is summarized in Table XVII. Best-effort approaches do not involve high computational efforts, however, they provide medium allocation efficiency and there is a possibility that some of the connections exceed acceptable XT values. On the other hand, the strict constrained approach is complex in terms of XT calculations, nevertheless, it should result in the best allocation efficiency with the guarantee of acceptable XT levels of established connections. Its simplified version, i.e., the worst-case constrained approach, does not require any complex calculations during algorithm processing, since the XT impairments and resulting transmission reaches are estimated during planning phase. The disadvantage of this approach is that the spectral-spatial resources may not be efficiently utilized due to the over-estimation of XT.

23

free cores

6

5

0 connec ons on adjacent cores

1

7

4

2

6

2 connec ons on adjacent cores

5

1 connec on on adjacent cores

3

crosstalk threshold may be exceeded

new connec on

occupied cores

1 connec on on adjacent cores

1

7

4

2 2 connec ons on adjacent cores

3

(a) Strict constrained approach

free cores

6

5

assumes connec ons on all adjacent cores

1

7

4

crosstalk threshold may be exceeded

new connec on

occupied cores

6

2

5

3

1

7

4

XT threshold is not exceeded

2

3

XT threshold is not exceeded

(b) Worst-case constrained approach

Fig. 14: Constrained XT-aware approaches TABLE XVII: Comparison of different XT-aware approaches. XT-aware approach

Possibility of XT threshold exceeding

Computational effort

Allocation efficiency

yes yes no

medium small high

medium medium high

Best-effort avoidance Best-effort core-prioritization Strict constrained

H. Resource Fragmentation In dynamic optical network scenarios, the established lightpaths do not stay permanently in the network but they can be released after a certain period of time. Frequent changes in network state, together with varied bandwidth demands, may result in a resource fragmentation problem. Indeed, small portions of available frequency slices, released by previous connections and isolated from other non-occupied spectrum regions, become hard to reuse by upcoming lightpath requests, especially, if the spectrum continuity and contiguity constraints are imposed. Therefore, resource fragmentation leads to inefficient resource utilization and increased connection blocking probability. Depending on the type of SChs, fragmentation may concern spectral, spatial, or both domains. Some of the surveyed papers address this issue and present solutions to deal with that problem. A core classification method is proposed for MCF scenarios with spectral SChs in [29], [44], [46]. This approach assumes that there is a relatively small variety in the bandwidth demand volumes (i.e., the number of requested frequency slices), and their size is known in advance. Each fiber core is dedicated to realize the demands of a certain size; hence, its spectrum resources can be represented by a grid of a fixed granularity. After releasing a connection, there is enough space to support upcoming requests of the same size, which in turn reduces spectrum fragmentation. Moreover, there are some common fiber

24

TABLE XVIII: Optimization objectives in static network optimization problems in the surveyed works. Objective function

References

Maximum spectrum Overall spectrum Average spectrum Accepted traffic Network cost

[36], [49], [32], [49], [37],

[45], [55], [35] [51], [48],

[50], [53], [55], [61], [62] [58], [68], [70], [72] [58], [72] [51], [68], [70]

cores without predefined supported connection size, which are used whenever the dedicated cores are occupied or the demand volume is not a standard one. A similar approach to the above one is presented in [30], [47], [57], [66]. Here, instead of classifying the entire fiber cores, there are predefined prioritized areas in the frequency domain, which concatenate connections of a certain size. Connection alignment restrictions prevent fragmentation in the prioritized areas. As in the above approach, a common area for accommodating connections that are not accepted to their dedicated areas is defined. Also, prior knowledge about connection types is required. The prioritized and common areas are designed based on statistical characteristic of bottleneck links. It is allowed to split high bit-rate demands into smaller ones and accommodate them using adequate prioritized areas. Both methods, i.e., core classification and area prioritization, may additionally be combined with the best-effort core prioritization approach (see Sec. VI-G) to introduce some XT awareness. In [42], the concept of cross-core virtual concatenation is studied in an SDM optical network with the spectrum contiguity constraint relaxed. The considered spectral-spatial SChs have irregular shapes and their carriers can be distributed over different fiber cores. A fragmentation-aware method with congestion avoidance is proposed in [16]. Here, each new connection request is allocated in the network so that to minimize resource fragmentation and, at the same time, to reuse already fragmented spectrum segments. As in EONs [100], fragmentation-aware resource management techniques can be categorized as proactive and reactive. Proactive mechanisms try to minimize or prevent resource fragmentation at the time a request is admitted to the network. The above described methods fall into this category. On the other hand, reactive techniques focus on network defragmentation by means of rearranging already established connections, either by rerouting and/or reallocating them. Defragmentation algorithms can be triggered periodically or when a fragmentation metric oversteps certain threshold. They may be also invoked to release resources for a new connection request, which otherwise, might be rejected. To the best of our knowledge, reactive resource defragmentation algorithms have not been yet studied in SDM networks. I. Metrics The surveyed works make use of various performance and optimization metrics, which serve in the assessment of SDM network performance and in network optimization. First, we discuss different optimization objectives applied in static optimization problems, afterwards, we present the performance metrics used in dynamic network scenarios. As shown in Table XVIII, most of the surveyed papers focus on minimization of spectrum usage and, in this scope, three different objective functions are considered. The most frequently used—similarly as in optimization of EONs [91]—is the maximum spectrum that is defined as the maximum number of frequency slices required in the network to serve all demands. The overall spectrum expresses the total number of frequency slices allocated in all network links. Finally, the average spectrum represents average utilization of spectrum per link (i.e., spatial mode) per fiber. Another cost function used in optimization of SDM networks is related to accepted traffic, and it is defined as the maximum number of traffic demands, from the given set of demands, that can be established in the network. Furthermore, several works aim at minimizing network cost. The cost function proposed in [37] incorporates the cost of four network elements: the spectral/spatial SCh transceivers, the optical express (OE) nodes, add/drop (A/D) nodes, and the SDM amplifiers. In turn, the goal of the study in [48] is to minimize the number of switching modules (MUX/DEMUX), which also leads to maximizing fiber switching and to decreasing the power consumption. The static optimization problem addressed in [51] concerns minimizing the number of switching modules in the optical nodes. Eventually, the authors of [68], [70] include in their objective function the cost of transceivers to be installed in the network. Table XIX reports different performance metrics that are used in dynamic network scenarios in the surveyed works. The most popular metric is the connection blocking probability, defined as a ratio between the number of rejected and the number of all connection request offered to the network. In a few papers, a similar metric called the bandwidth blocking probability is applied, which typically expresses a ratio between the amount of rejected and offered demand volume (i.e., bandwidth or bit-rate). However, in most of these works the definition of the metric is not provided and, therefore, its meaning is not unambiguous. We want to underline that since in SDM optical networks the traffic volumes of connections might be very diverse, in our opinion the later metric might assess the performance of dynamic resource allocation algorithms in a more adequate and accurate way than the former one. Eventually, such performance metrics as spectrum utilization, algorithm runtime/connection setup time, 25

TABLE XIX: Types of performance metrics in dynamic network scenarios addressed in the surveyed works. Metric

References

Connection blocking probability Bandwidth blocking probability Spectrum utilization Runtime/setup time Throughput XT ratio Cost/hardware Fragmentation

[16], [52], [42], [71], [33], [29], [47], [44],

[29], [59], [60], [85], [39], [43], [54], [46]

[30], [63], [64], [90] [41], [44], [56],

[33], [34], [38]–[40], [42]–[44], [46], [47], [54], [56], [60], [64]–[66], [69], [79], [87] [66], [67], [71], [86] [65], [86] [79] [46], [52] [67], [79]

throughput, XT ratio, cost of involved network hardware, and spectrum fragmentation, are also present in selected works. However, we should mention that in both static and dynamic scenarios there have not been papers that consider optimization of SDM optical networks in the context of energy efficiency. One of the reasons behind this fact can be a lack of credible models of energy consumption in SDM networks that accounts for network devices specific to SDM. VII. C ONCLUDING R EMARKS Parallel transmission of multiple spatial modes in suitable designed optical fibers along with the use of spatially-spectrally flexible switching architectures increases network capabilities and gives rise to specific resource allocation schemes, which concern either spectral, or spatial, or spectral-spatial transmission resources. Efficient management of spectral and spatial resources, including their allocation for lightpath connections serving traffic demands, requires new algorithms capable of solving the challenging problem of routing, spatial mode, and spectrum allocation. Such additional issues as coupling of modes, which may require joint treatment of entire groups of modes, or the effect of inter-core crosstalk in MCFs, which may have a severe impact on transmission quality, have to be taken into account when planning and operating SDM optical networks. In this work, a comprehensive survey of resource allocation schemes and algorithmic approaches considered for SDM optical networks was presented. First, we classified the surveyed papers according to the types of SDM optical fibers and switching paradigms assumed in these studies. Then, we described different types of transponders and super-channels supported by SDM networks, which result in different spectral-spatial resource allocation schemes. Moreover, we focused on several complete network scenarios that assume jointly specific SDM technological options. Finally, we classified and discussed the resource allocation schemes and algorithms that are presented in the surveyed papers addressing some issues specific for SDM optical networks such as solving the RSSA problem, dealing with crosstalk, resource fragmentation, and considered performance and optimization metrics. Out study shows that the most frequently analyzed SDM network scenarios in the literature concern either SMFB or MCF solutions, while the SDM fibers guiding strongly-coupled modes are considered only in few works. Ind-Sw is the most utilized switching paradigm, whereas Mode-Sw is rarely encountered. Besides, spectral SChs dominate in the surveyed papers. There are several works that compare network performance under different resource allocation schemes. Eventually, both static and dynamic resource allocation problems have been addressed in the literature, using different algorithmic approaches (mainly heuristic) and taking into account various performance and optimization metrics. Concurrently, although we were able to collect over 40 papers related to resource allocation problems in spectrally-spatially flexible optical networks, still this subject is not deeply explored in the literature yet, especially, when comparing with relevant research on WDM and EON networks. We have noted that the set of addressed problem in the surveyed papers is not broad. Dedicated solutions for ”classical” network problems and scenarios, which were previously studied in the context of WDM/EON networks, for instance, such as network survivability or multi-layer optimization, are either barely studied or still missing. In our opinion, the most promising possible directions of further research in optimization of SDM networks are as follows: • Evaluation and comparison of performance gains from using different types of fibers and SDM technologies. • Developing efficient and scalable algorithms, e.g., based on advanced optimization decomposition methods or metaheuristic methods, capable of dealing with large RSSA problem instances in reasonable time and yielding results close to optimum. • Comparison of various ILP modeling approaches concerning routing modeling (link-path vs. node-link) and spectrum modeling (slice-based vs. channel-based) as well as accounting for different SDM scenarios. • Comprehensive and detailed analysis of the crosstalk influence on performance of SDM networks considering various types of transmission media. • Incorporating into the optimization framework various traffic grooming mechanisms indispensable in the context of very high bit-rates served by lightpaths in SDM networks. • Evaluation and comparison of protection mechanisms available in SDM networks with a special focus on such approaches as squeezed protection. 26

• •

Developing and evaluation of reactive defragmentation algorithms for SDM networks accounting for fragmentation of both spatial and spectrum resources. Addressing the problems of migration from classical optical system with a single spatial mode towards various scenarios of SDM systems including optimization of hybrid networks with both types of fibers. ACKNOWLEDGMENTS

The work of M. Klinkowski was supported by National Science Centre, Poland under Grant 2016/21/B/ST7/02212. The work of K. Walkowiak and P. Lechowicz was supported by National Science Centre, Poland under Grant 2015/19/B/ST7/02490. R EFERENCES R EFERENCES [1] M. Klinkowski, K. Walkowiak, A heuristic algorithm for routing, spectrum, transceiver and regeneration allocation problem in elastic optical networks, in: Proc. of IEEE ICTON, Trento, Italy, 2016. [2] P. J. Winzer, Spatial multiplexing in fiber optics: The 10x scaling of metro/core capacities, Bell Labs Techn. J. 19 (2014) 22–30. [3] I. P. Kaminow, T. Li, A. E. Willner, Optical Fiber Telecommunications Volume VIB, Sixth Edition: Systems and Networks, 6th Edition, Academic Press, 2013. [4] O. Gerstel, M. Jinno, A. Lord, S. J. B. Yoo, Elastic optical networking: A new dawn for the optical layer?, IEEE Comm. Mag. 50 (2) (2012) 12–20. [5] R. Go´scie´n, K. Walkowiak, M. 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