Multilayer Routing and Resource Assignment in Spatial Channel Networks (SCNs): Oriented Toward the Massive SDM Era
11 Multilayer Routing and Resource Assignment inSpatial Channel Networks (SCNs): Oriented Towardthe Massive SDM Era
Mingcong Yang
Member, OSA , Qian Wu, Maiko Shigeno, Yongbing Zhang
Abstract —In the past few decades, optical transport networks(OTNs) have undergone significant evolution, from the earliestwavelength-division multiplexing (WDM) OTNs to elastic opticalnetworks (EONs) and later to space-division multiplexing (SDM)OTNs, to address the continuous growth of Internet traffic. By2024, Pbps-level OTNs are expected, far exceeding the capacitylimit of single-mode fibers. The massive SDM era is on thehorizon. In this context, newly designed OTNs called spatialchannel networks (SCNs), which achieve high cost efficiencyby means of practical hierarchical optical cross-connects, haverecently been proposed. However, the evolution of OTNs willsimultaneously present challenges related to resource allocationin networking. For instance, with the evolution from WDM-OTNs to EONs, the resource allocation problem was transformedfrom the routing and wavelength assignment (RWA) problemto the routing and spectrum assignment (RSA) problem due tothe additionally introduced constraint of spectrum contiguity.Similarly, specially designed algorithms are also expected tobe essential for addressing the resource allocation problem inSCNs. In this paper, we define this new problem as the routing,spatial channel, and spectrum assignment (RSCSA) problem.We propose an integer linear programming (ILP) model and aheuristic algorithm to solve the RSCSA problem. We examinethe performance of the proposed approaches via simulationexperiments. The results show that both proposed approachesare effective in finding the optimal solutions or solutions close tothe lower bounds. To the best of our knowledge, this is the firstwork to focus on the problem of resource allocation in SCNs.
I. I
NTRODUCTION
With the development and increasing popularity of cloudcomputing, video-on-demand (VoD), the Internet of Things(IoT), and other emerging Internet services, network traffic isgrowing at an extremely rapid rate [1]. Consequently, sincethe deployment of wavelength-division multiplexing (WDM)optical transport networks (OTNs) at the beginning of the2000s, OTNs have undergone several significant evolutionsto support the rapid increase in network traffic. The firstevolution, from traditional WDM-OTNs to elastic opticalnetworks (EONs), which was induced by the introduction ofadvanced technologies such as orthogonal frequency-divisionmultiplexing, Nyquist wavelength-division multiplexing, anddistance-adaptive modulation, greatly increased the spectrumefficiency [2]. Moreover, to overcome the capacity limit ofconventional single-mode fibers (SMFs), that is, the so-callednonlinear Shannon limit, space-division multiplexing (SDM)
This work has been submitted to the IEEE for possible publication.Copyright may be transferred without notice, after which this version mayno longer be accessible. technology was later proposed, which motivated the furtherevolution of OTNs from EONs to SDM-OTNs [3].From the networking perspective, the resource allocationproblem also changed several times in accordance with thenew features introduced by the evolution of OTNs. In WDM-OTNs, the resource allocation problem is called the routingand wavelength assignment (RWA) problem [4]. A lightpath,composed of a routing path and a wavelength, must beassigned to each connection request. The assigned wavelengthmust be consistent along the entire lightpath (unless wave-length conversion is allowed) and be nonoverlapping with theother wavelengths on each fiber link. These constraints are theso-called wavelength continuity and wavelength nonoverlapconstraints, respectively.Compared with WDM-OTNs, more flexible spectrum di-visions are possible in EONs, such as the 12.5 GHz fre-quency slices (FSs) that conform to the G.694.1 stan-dard recommended by the International TelecommunicationUnion Telecommunication Standardization Sector (ITU-T) [5].In combination with the application of bandwidth-variabletransceivers (BVTs) [6] and optical cross-connects (BV-OXCs)[7] in EONs, it is possible to satisfy connection requests withvarying bit rates and to flexibly establish lightpaths by usingdifferent numbers of FSs as needed, thus achieving higherspectrum efficiency. However, a disadvantageous consequenceof this approach is that the FSs assigned to each connectionrequest should be contiguous, which introduces an additionalconstraint of spectrum contiguity. Therefore, the resource al-location problem is transformed into the routing and spectrumassignment (RSA) problem in EONs [8, 9].Although EONs are promising OTNs that achieve moreefficient utilization of spectrum resources compared to tradi-tional WDM-OTNs, the growth in the transmission capacityof standard SMFs has dramatically slowed because the currenttransmission capacity per fiber is approaching the nonlinearShannon limit of the existing SMFs. Nevertheless, the volumeof Internet traffic is expected to continue to strongly increasein the future, inexorably reaching this capacity limit [10].Thus, as a viable solution for overcoming this limit, SDMtechnology has emerged, the basic concept of which is toexpand the available space lanes (SLs) from the current singleSL (i.e., an SMF) to multiple parallel SLs to increase theavailable spectrum resources [11]. This expansion will enableus to assign spectrum resources straddling both the spectraland spatial domains, which will again make the resourceallocation problem more complicated because the appropriate a r X i v : . [ c s . N I] F e b SL(s) should be assigned to each lightpath simultaneously withthe assignment of the routing path and spectrum. Therefore, inthis case, the resource allocation problem becomes the routing,spectrum, and space assignment (RSSA) problem [12].As reported in Ref. [1], the compound annual growth rates(CAGRs) of the aggregate router blade interface rate have beenapproximately 40% in recent years. By 2024, the implementa-tion of an optical interface rate of up to 10 Tbps is expected tobe required. Moreover, considering that the interconnectionsbetween adjacent nodes are expected to consist of dozensor even hundreds of SLs (fibers/cores) in the near future,Pbps-level OTNs are anticipated. The massive SDM era ison the horizon. However, considering that the total bandwidthof the C-band is approximately 4 THz per fiber/core forSMFs or multicore fibers (MCFs), for ultralong-haul opticaltransmission, the enormous bandwidth requirement of a 10Tbps connection request will exceed the entire C-band forsuch fibers [15]. Wavelength switching will no longer benecessary to transmit such connection requests because theentire fiber/core will become a logical end-to-end interfaceto serve an ultrahigh-capacity optical data stream, which willbe routed as a single entity by optical bypass technology.In this context, spatial channel networks (SCNs) [13–18]have been proposed as an economical and realistic solutionoriented toward the future massive SDM era. Nevertheless,similar to what has happened heretofore, the evolution ofOTNs will present further opportunities for addressing theresource allocation problem in networking and will also posecorresponding challenges. Dedicated algorithms consideringthe features of the newly designed OTNs will be essential toaddress the corresponding resource allocation problem [15].In this paper, reviewing the significant evolution of OTNsthat has occurred over the past few decades, we focus onthe static resource allocation problem in SCNs, which wedefine as the routing, spatial channel, and spectrum assignment(RSCSA) problem. In Section II, we identify the novel featuresof SCNs from the networking perspective. In Section III,we define the RSCSA problem and prove that it is NP-hard. We also clarify the constraints corresponding to thefeatures of SCNs in detail. In Sections IV and V, we proposean integer linear programming (ILP) model and a heuristicalgorithm, respectively, for solving the RSCSA problem. InSection VI, we evaluate the performance of the two proposedapproaches via simulation experiments. Finally, in Section VII,we conclude the paper and present prospects for future work.II. S
PATIAL CHANNEL NETWORKS
As stated in Section I, network traffic has grown at anextremely rapid rate over the past few decades, which hasinevitably compelled the development of optical transmissiontechnologies, as well. As shown in the bottom left of Fig. 1,spectral superchannel transmission technology, which com-prises several adjacent optical carriers (OCs) without switchingguard bands (SW-GBs) between them, has been effectivelyapplied in EONs, leading to higher spectrum efficiency [19].In addition, the expansion of the SLs in SDM-OTNs enablesus to allocate OCs that span multiple SLs but share a single laser source to create a spatial superchannel, leading to highercost efficiency. Of course, any suitable hybrid combination ofthe two types of superchannels above, a so-called spectral andspatial superchannel, as shown in Fig. 1, is also feasible foruse in SDM-OTNs [20].Considering the aforementioned 10 Tbps client interfacerate that is anticipated to be achieved by 2024, one hundred32 Gbaud DP-QPSK OCs (each supporting 100 Gbps) willbe required to establish such a connection request for long-haul transmission, or other combinations may be suitable fora shorter distance, such as twenty-five 64 Gbaud DP-16-QAMOCs [1, 15]. We can see that a total spectrum of 3.2 THzis required in the ideal case (i.e., with the ideal Nyquistshaping and a gridless spectrum) for a 10 Tbps DP-QPSKspectral superchannel, and the entire C-band can accommodateonly one such superchannel. This indicates that wavelengthswitching support will no longer be necessary for every SL,since after a few more years, the spectral superchannel usedto serve a single connection request may require the entire C-band spectrum. SCNs with hierarchical optical cross-connects(HOXCs) have therefore been recently proposed [13–18].
A. Spatial channels
First, we introduce the concept of spatial channels (SChs)in SCNs. An SCh is defined as an ultrahigh-capacity opticaldata stream that occupies a large amount of spectrum, and itcan be optically routed in an end-to-end manner as a singleentity through spatial cross-connects (SXCs) (called spatialbypass in SCNs) [13–18]. It should be noted that the conceptsof SChs and superchannels are different, although in someprevious works, the abbreviation SCh has also been used forsuperchannels. In this paper, SpCh is used as the abbreviationfor the term ‘superchannel’ to avoid confusion. As shownin Fig. 1, there are four types of SChs, which are listed asfollows: • Type I: An SCh that carries a single high-capacity spectralSpCh (shown in blue and purple in Fig. 1). SChs of TypeI can be routed in an end-to-end manner through spatialbypass without wavelength switching. • Type II: An SCh that carries multiple spectral SpChsestablished between the same source-destination pair(shown in red and green). SChs of Type II can alsobe end-to-end spatially bypassed, while multiple spectralSpChs belonging to such an SCh can be allocated withoutSW-GBs. • Type III: An SCh that carries multiple spectral SpChsestablished between different source-destination pairs(shown in orange and yellow). These spectral SpChs areadded/dropped by the wavelength cross-connects (WXCs)at intermediate node(s), and thus, SW-GBs are requiredbetween them. • Type IV: An SCh that carries a single ultrahigh-capacityspatial and spectral SpCh (shown in black), which oc-cupies multiple SLs. However, in this paper, we do notconsider SChs of Type IV because such an SCh can beequivalently treated as multiple SChs of Type I, each ofwhich can be routed independently.
Fig. 1. Illustration of the spectral and spatial SpChs in SDM-OTNs vs. the SChs in SCNs.
B. Hierarchical optical cross-connects
As shown in Fig. 1, in an SCN, the switching layer isdivided into an SDM layer and a WDM layer to achieve highercost efficiency. SChs of Type I, Type II, and Type IV arespatially bypassed without passing through the WDM layer.In Ref. [15], four different types of HOXCs, which supportdifferent degrees of cost efficiency, routing flexibility, andscalability, have been proposed to achieve this functionality.In fact, the concept of HOXCs was first proposed in the late1990s [21, 22], and some efforts were made in this directionbefore the concept of SDM-OTNs began to gain in popularity[23, 24]. This paper aims to identify the distinctive featuresof SCNs from the networking perspective but does not focuson explaining the detailed architectures of HOXCs for SCNsor comparing them with previous architectures. Readers canrefer to Ref. [15] for more detailed related information.Fig. 2 illustrates the HOXCs proposed for use in SCNs,which are implemented on the basis of full-size core-selectiveswitches (CSSs) [25], sub-CSSs, full-size matrix switches(MSs) [26], and sub-MSs. • Full-size CSS-based HOXC: The full-size CSS-basedHOXC is the most cost-efficient solution among the fourHOXCs. It also supports the scaling up of the nodal de-gree. However, space lane change (SLC) is not supportedby this HOXC. For example, as shown in Fig. 2.(a), if weassume that the logical indices of the SLs (fibers/cores)are the same on each link, then an SCh that enters an intermediate node can be switched only to output ports(including drop ports) with the same index. • Sub-CSS-based HOXC: The sub-CSS-based HOXC isalso a cost-efficient solution but costs more than the full-size CSS-based HOXC. However, it supports the scalingup of not only the nodal degree but also the number ofSLs per degree in compensation for its additional cost.In addition, it has the same features as the full-size CSS-based HOXC from the networking perspective, as shownin Fig. 2.(a). • Full-size MS-based HOXC: The full-size MS-basedHOXC is the solution that provides the highest routingflexibility among the four HOXCs. As shown in Fig. 2.(b),this HOXC allows an SCh to be switched to any outputport (including drop ports). However, it is also the costli-est solution and does not support the scalability of thenodal degree and the number of SLs (per degree). It isworth noting that since the full-size MS-based HOXCsupports full SLC, a single add/drop port can be usedto add/drop SChs to/from SLs with different indices (atdifferent time points). Therefore, the add/drop port countscan be reduced to some extent (an example is illustratedby the gray dotted arrow). • Sub-MS-based HOXC: The sub MS-based HOXC is acompromise solution relative to the full-size MS-basedHOXC. In this case, the SLs are divided into multi-ple groups (e.g., two groups in the example shown inFig. 2.(c)), and SLC is available within each group.
Fig. 2. Illustration of four HOXCs proposed for use in SCNs from the networking perspective. Solid arrow: active switching; dotted arrow: possible switching.
Compared to the full-size MS-based HOXC, this solutionsacrifices some routing flexibility in exchange for supportfor the scalability of the number of SLs per degree anda considerable cost savings. Nevertheless, it is still muchcostlier than either of the two CSS-based HOXCs.In summary, the above four HOXCs show various differ-ences in cost efficiency, routing flexibility, and scalability.However, all of them cost less than conventional OXCs, whichrequire wavelength switching support on each SL in SDM-OTNs. In this paper, we consider only SCNs implementedon the basis of full-size/sub-CSS-based HOXCs (as shownin Fig. 2.(a)) and defer the consideration of applications ofthe two MS-based HOXCs to future research. This is becausethe two CSS-based HOXCs offer significantly higher costefficiency – readers can refer to the cost assessments inRefs. [14] and [15] for more details – and scalability thanthe two MS-based HOXCs do and thus are considered moresuitable for use in future commercial SCNs.III. R
OUTING , SPATIAL CHANNEL , AND SPECTRUMASSIGNMENT (RSCSA)
PROBLEM
A. Introduction to the RSCSA problem
Similar to the RWA problem in WDM-OTNs, the RSAproblem in EONs, and the RSSA problem in SDM-OTNs,the RSCSA problem can be subdivided into two main cases:the dynamic case and the static case.In the dynamic case, which emerges during network opera-tion, it is assumed that the connection requests are unknown inadvance and that they stochastically arrive and disappear oneby one. The resources required to serve connection requestsare assigned dynamically in accordance with the current stateof the network. The objective of the dynamic RSCSA problemis to minimize the network blocking probability (BP) orto maximize the network throughput while maintaining an acceptable BP (e.g., 1%) [27, 28], which is the same as theobjectives of the previous dynamic RWA, RSA, and RSSAproblems.In the static case, which mainly relates to the network plan-ning phase, a traffic matrix that contains a set of connection re-quests that must be served in the network is known in advance,and resources must be assigned to all of these connectionrequests simultaneously. In the static RSCSA problem, themain objective is to minimize the number of SLs that areused/required in the network, for the following three reasons: • Minimizing the number of FSs that are used/required (orthe maximum index of these FSs) in the network, as isdone in the static RWA, RSA, and RSSA problems, ispointless in this case because in an SCN, each connectionrequest is transmitted by an SCh, which may occupy theentire C-band spectrum. • Minimizing the number of SLs used is equivalent tomaximizing the number of SLs in the network that arenot occupied and thus are available for future connec-tion requests – assuming that the network scenario issemidynamic, we optimize the network by reassigningthe currently established connections as a ‘static’ set,and any connection requests that subsequently arrive inthe network are handled dynamically [29]. Therefore,minimizing the number of SLs used reduces the levelof congestion in the network. • The last reason is that there are many different possibletypes of SCN systems. Note that scalability of the SLsis not supported by all types of HOXCs, and in general,a system with 20 SLs is much cheaper than one with40 SLs. Therefore, if we can reduce the number of SLsrequired to below 20 during the network planning phase,great cost savings can be achieved.Another objective of the RSCSA problem, although with a lower priority, is to minimize the number of SLs withwavelength switching support that are used/required, for thefollowing two reasons: • As stated before, compared to SDM-OTNs, the key fea-ture of SCNs is that wavelength switching support is notnecessary on every SL because some connection requestscan be transmitted by SChs of Type I and Type II, whichcan be spatially bypassed at intermediate nodes. As shownin Fig. 2, the number of SLs with wavelength switchingsupport has a one-to-one relationship with the number ofdeployed WXCs. Therefore, during the network planningphase, minimizing the number of required SLs withwavelength switching support is equivalent to minimizingthe number of required WXCs at HOXCs. Between twoHOXCs that support the same number of SLs, the onewith fewer WXCs will certainly cost less. • As introduced in Section II-A, there are four types ofSChs, and from the networking perspective, an SCh ofType IV can be treated as multiple independent SChsof Type I. SChs of Type I and Type II can be spatiallybypassed at intermediate nodes using SLs without wave-length switching support. However, if the available SLswithout wavelength switching support are inadequate,SLs with wavelength switching support can also be used.In contrast, SChs of Type III can pass only throughSLs with wavelength switching support. Therefore, in thesemidynamic scenario, for two solutions to the RSCSAproblem that require an equal number of SLs, the onethat uses fewer SLs that support wavelength switching ispreferred – the more idle SLs with wavelength switchingsupport there are, the higher the possibility of satisfyingmore subsequent connection requests.In summary, the static RSCSA problem is a multiobjectiveproblem in which the decision on how to allocate resources,such as routing paths, SLs, modulation formats, and spectrum,for each connection request should be jointly made in anoffline manner.
B. NP-hardness of the RSCSA problem
In this subsection, we prove the NP-hardness of the RSCSAproblem by reducing the RWA problem for traditional WDM-OTNs to the related RSCSA problem.The RWA problem is a well-known NP-hard problem [30].An instance of the RWA problem includes a set of connectionrequests r ∈ R and a set of wavelengths λ ∈ Λ . Theobjective is to assign a routing path p r and a wavelength λ r to each r ∈ R while minimizing the number of wavelengthsthat are used/required in the network ( λ max ). In addition, theassignments should comply with the wavelength continuityand nonoverlap constraints. To solve the RWA problem in aform that is equivalent to the RSCSA problem, we considerthe network scenario shown in Fig. 3.We assume that a set of connection requests r ∈ R and aset of SLs l ∈ L ( L → Λ ) are given. Note that the two CSS-based HOXCs considered in this paper do not support SLC.Therefore, the wavelength continuity constraint is convertedinto an SL continuity constraint in the RSCSA problem, Fig. 3. Comparison between the RWA problem and the RSCSA problem. and a corresponding SL nonoverlap constraint should also besatisfied. Here, we simplify the RSCSA problem by ignoringthe second (minor) objective and assume that each r ∈ R exactly occupies the entire C-band of a single SL. Thus, weshould assign a routing path p r and an SL l r ( l r → λ r ) toeach r ∈ R while minimizing the number of SLs that areused/required in the SCN ( l max ; l max → λ max ). In this case,if we were able to optimally solve the simplified RSCSAproblem, we would also obtain the optimal solution to theRWA problem. Therefore, since the RWA problem is NP-hard and the original RSCSA problem is more complex thanthe simplified one, we can conclude that the original RSCSAproblem is also NP-hard.IV. I NTEGER LINEAR PROGRAMMING MODEL FOR THE
RSCSA
PROBLEM
As introduced in Section III-A, the static problem is mainlyrelated to the network planning phase and hence is not subjectto strict computational time constraints. Accordingly, complexand time-consuming mathematical optimization approaches,such as ILP, can be applied to solve the static problem [12].Therefore, in this section, we propose an ILP model for thestatic RSCSA problem.
A. ParametersV the set of nodes v in the network. E the set of links e in the network. N P the set of node pairs np in the network. R the set of connection requests r = (cid:104) s r , d r , t r (cid:105) , where s r , d r , and t r represent the source node, destinationnode, and traffic volume [bps], respectively, of con-nection request r . R np the set of connection requests between node pair np ,which is defined as R np = { r ∈ R |(cid:104) s r , d r (cid:105) = np } . P r the set of k candidate routing paths for connectionrequest r , which is obtained using the k -shortest-pathalgorithm proposed in Ref. [8]. P np the set of k candidate routing paths between nodepair np , where P np = P r for each r ∈ R np . L the set of SLs l ∈ L (per link) in the network. L W the set of SLs with wavelength switching support inthe network. L NW the set of SLs without wavelength switching supportin the network. m p the highest feasible modulation level for routing path p based on its length [km]. t p OC the traffic volume [bps] that a single OC can supporton routing path p based on m p . F GB the number of FSs occupied by an SW-GB. F OC the number of FSs occupied per OC. F max the total number of available FSs on each SL. f max the maximum index of the FSs on each SL. Notethat the indices of the FSs start from 0; therefore, f max = F max − . B. Variablesu l a binary variable that is equal to 1 if SL l is usedand to 0 otherwise. x plr a binary variable that is equal to 1 if lightpath (cid:104) p , l (cid:105) is assigned to serve connection request r and to 0otherwise. o plr an integer variable that indicates the number ofOCs that are assigned to lightpath (cid:104) p , l (cid:105) to serveconnection request r . α plr an integer variable that indicates the starting indexof the FSs assigned to lightpath (cid:104) p , l (cid:105) to serveconnection request r . β plr an integer variable that indicates the ending indexof the FSs assigned to lightpath (cid:104) p , l (cid:105) to serveconnection request r . θ pp (cid:48) lrr (cid:48) a binary variable that is equal to 1 if β plr is smallerthan α p (cid:48) lr (cid:48) and to 0 otherwise. C. Objectives
Main objective:
Minimize (cid:213) l ∈ L u l (1)Minor objective: Minimize (cid:213) l ∈ L W u l (2)As stated in Section III, the static RSCSA problem is amultiobjective problem. The main objective, shown in Eq. (1),is to minimize the number of SLs that are used/required inthe network, while the minor objective, shown in Eq. (2), isto minimize the number of SLs with wavelength switchingsupport that are used/required. D. Constraints (cid:213) p ∈ P r (cid:213) l ∈ L t p OC · o plr ≥ t r ∀ r ∈ R (3) For a connection request r , multiple lightpaths (cid:104) p , l (cid:105) can beestablished to serve it. Constraint (3) ensures that the sumof the traffic volumes carried by the established lightpaths(i.e., the left-hand side) is no smaller than the required trafficvolume for connection request r ( t r ). F max · x plr ≥ F OC · o plr ∀ r ∈ R , p ∈ P r , l ∈ L NW (4)Constraint (4) ensures that x plr is equal to 1 if there is atleast one OC assigned to lightpath (cid:104) p , l (cid:105) to serve connectionrequest r ( o plr ≥ ) and is equal to 0 if no OC has beenassigned ( o plr = ). β plr = α plr + F OC · o plr − x plr ∀ r ∈ R , p ∈ P r , l ∈ L W (5)Constraint (5) ensures the relationship between the startingand ending indices of the assigned FSs. f max · x plr ≥ β plr ∀ r ∈ R , p ∈ P r , l ∈ L W (6)Constraint (6) ensures that if lightpath (cid:104) p , l (cid:105) is establishedto serve connection request r ( x plr = ), then the ending indexof the FSs assigned to the lightpath ( β plr ) is no greater thanthe maximum index of the FSs ( f max ). | R | · k · u l ≥ (cid:213) r ∈ R (cid:213) p ∈ P r x plr ∀ l ∈ L (7)Constraint (7) ensures that u l is equal to 1 if SL l has beenassigned to establish at least one lightpath. F max · ( − x plr ) ≥ F OC · o p (cid:48) lr (cid:48) ∀ r , r (cid:48) ∈ R , p ∈ P r , p (cid:48) ∈ P r (cid:48) , l ∈ L NW : p (cid:44) p (cid:48) , p (cid:209) p (cid:48) (cid:44) ∅ (8)Constraint (8) ensures that if lightpath (cid:104) p , l (cid:105) is establishedto serve connection request r ( x plr = ) and SL l belongs to L NW , then SL l cannot be used to establish another lightpath (cid:104) p (cid:48) , l (cid:105) ( o p (cid:48) lr (cid:48) = ) that has one or more common links withrouting path p ( p (cid:209) p (cid:48) (cid:44) ∅ ) for connection request r (cid:48) . Notethat this constraint applies only when p (cid:44) p (cid:48) . If this is not thecase ( p = p (cid:48) ), then these two lightpaths can be established onthe same routing path p and SL l by composing an SCh ofType II (refer to the following Constraint (9)). F max ≥ F OC · (cid:213) r ∈ R np o plr np ∈ N P , p ∈ P np , l ∈ L NW (9)Constraint (9) indicates that for connection requests with thesame source-destination pair ( r ∈ R np ), they can be transmittedby lightpaths that share a common routing path p ∈ P np andSL l ∈ L NW , composing an SCh of Type II. θ pp (cid:48) lrr (cid:48) + θ p (cid:48) plr (cid:48) r = ∀ r , r (cid:48) ∈ R , p ∈ P r , p (cid:48) ∈ P r (cid:48) , l ∈ L W : p (cid:209) p (cid:48) (cid:44) ∅ (10) α p (cid:48) lr (cid:48) + F max · ( − θ pp (cid:48) lrr (cid:48) ) ≥ β plr + x plr ∀ r , r (cid:48) ∈ R , p ∈ P r , p (cid:48) ∈ P r (cid:48) , l ∈ L W : p = p (cid:48) , p (cid:209) p (cid:48) (cid:44) ∅ (11) α p (cid:48) lr (cid:48) + ( F max + F GB ) · ( − θ pp (cid:48) lrr (cid:48) ) ≥ β plr + ( F GB + ) · x plr ∀ r , r (cid:48) ∈ R , p ∈ P r , p (cid:48) ∈ P r (cid:48) , l ∈ L W : p (cid:44) p (cid:48) , p (cid:209) p (cid:48) (cid:44) ∅ (12) Constraints (10) ∼ (12) ensure spectrum contiguity andspectrum nonoverlap – the requirement of spectrum continuityis naturally satisfied – for the lightpaths passing through SLswith wavelength switching support ( l ∈ L W ). Since these aregeneral constraints that have been widely applied in manyprevious works focusing on the static resource allocationproblem in OTNs, we will not explain them in detail. However,it should be noted that two lightpaths established between thesame source-destination pair can be allocated without an SW-GB in the case that their routing paths and SLs are the same(i.e., Constraint (11)). F max · u l ≥ (cid:213) r ∈ R (cid:213) p ∈ P r : e ∈ p F OC · o plr ∀ e ∈ E , l ∈ L (13)Constraint (13) is a redundant constraint. For each SL l and link e , this constraint stipulates that the total number ofFSs assigned to the lightpaths that traverse them should be nogreater than F max . As seen from the results of our simulationexperiments, this constraint is able to significantly improve theconvergence rate of the ILP model.V. H EURISTIC ALGORITHM FOR SOLVING THE
RSCSA
PROBLEM
In this subsection, we propose a heuristic algorithm to solvethe static RSCSA problem. First, we introduce two tables anda function used in the heuristic algorithm, as follows: • We define a table T SCh-II . Each entry (cid:104) np : p , l , B (cid:105) inthis table records an SCh established between node pair np passing through routing path p and SL l , where thespectrum on this SCh is currently not fully used – B represents the remaining available spectrum on this SCh.In the heuristic algorithm, the connection requests areassigned resources one by one. Therefore, the remainingavailable spectrum on an SCh recorded in T SCh-II isexpected to be assigned to subsequent connection requestswith the same source-destination pair (i.e., between thesame np ) to compose an SCh of Type II. • We also define a table T SCh-III . Each entry (cid:104) r : t r rem (cid:105) inthis table records the currently unsatisfied traffic volume t r rem for connection request r . The unsatisfied trafficvolumes of the connection requests recorded in T SCh-III are expected to be served by SChs of Type III. • We define a function named
First-Fit SL Allocation (FF-SLA) . This function takes a routing path p as input anddetermines the available SL with the lowest index along p (denoted by l p FF ). In this paper, we assume that theindices of SLs without wavelength switching support (i.e., l ∈ L NW ) are lower than those of SLs with wavelengthswitching support (i.e., l ∈ L W ). The output of thisfunction is (cid:104) p , l p FF (cid:105) .The heuristic algorithm is divided into three parts, and wewill explain each of them individually. To facilitate readers’understanding, we present a simple illustration in Fig. 4. Weconsider a 6-node network with 4 SLs (per link), of whichonly one, SL-4, supports wavelength switching. For simplicity,we assume that each SL has 4 THz of available spectrum and supports 8 Tbps of traffic volume regardless of the pathlength (i.e., without considering adaptive modulation). Then,we consider a set of connection requests R = { r , r , · · · , r } .These connection requests belong to different sets R np : R np = { r , r , r } , R np = { r } , R np = { r , r } , and R np = { r } . The connection requests will be assigned resources oneby one following a specified service sequence R seq . A. Assignment for SChs of Type I and Type II
Initially, we attempt to assign SChs of Type I and TypeII for all connection requests. Fig. 4.(a) shows the arrival ofthe first connection request, r = (cid:104) , , Tbps (cid:105) . Accordingto the output of the
FF-SLA function, SL-1 and routingpath (cid:104) (cid:105) , with the shortest distance, are first selectedto establish an SCh of Type I with support for 8 Tbps oftraffic volume. However, 2 Tbps of the traffic volume of therequest still needs to be satisfied. Therefore, by running the
FF-SLA function again, SL-1 and routing path (cid:104) (cid:105) areadditionally selected, and another SCh is established. Sincethe spectrum available on this SCh is not fully used, theentry (cid:104) np : 1-2-5-6, SL-1, 3 THz (cid:105) is appended to T SCh-II .The remaining 3 Thz of spectrum is expected to be assignedto the subsequent connection requests r and r , which belongto the same R np as r , to compose an SCh of Type II. Then,we remove r from R np .Subsequently, a connection request r = (cid:104) , , Tbps (cid:105) ar-rives, as shown in Fig. 4.(b). The remaining available spectrumon the SCh recorded in T SCh-II has the highest priority forassignment to subsequent connection requests. Therefore, wefirst check whether there is an SCh between np that is notfully used recorded in T SCh-II . In this case, it is obvious that r can be transmitted using the remaining 3 THz of spectrumon the SCh recorded in T SCh-II above by composing an SCh ofType II. Since this SCh is fully used after being assigned to r , the corresponding entry is removed from T SCh-II . Finally,we remove r from R np .Then, a connection request r = (cid:104) , , Tbps (cid:105) arrives, asshown in Fig. 4.(c). We call the
FF-SLA function becausethere is no SCh between np recorded in T SCh-II at this time.Thus, SL-2 and routing path (cid:104) (cid:105) are selected to establishan SCh of Type I. Similar to the case of r , 2 Tbps of thetraffic volume of the request remains to be satisfied. However,we will not establish another not fully used SCh in this casebecause if we were to establish such an SCh (represented bythe red dotted line), its remaining available spectrum wouldhave no chance to be used because no subsequent connectionrequest exists between np , and this would result in a wasteof spectrum on this SCh. Instead, an entry (cid:104) r : 2 Tbps (cid:105) isappended to T SCh-III . The unsatisfied 2 Tbps of traffic volumefor r is expected to be served by an SCh of Type III – sharingthe spectrum with other connection requests between differentsource-destination pairs.The procedures described above will be repeated for eachconnection request r ∈ R (e.g., Fig. 4.(a) ∼ (g) for this example).Notably, SChs of Type I and Type II will not result in anyspectrum fragmentation and offer SW-GB savings comparedto SChs of Type III. Therefore, SChs of Type I and Type II Fig. 4. Illustration of the proposed heuristic algorithm. always have a higher priority for establishment than SChs ofType III. Consequently, although it is preferable to use SLswithout wavelength switching support (i.e., l ∈ L NW ) whenestablishing SChs of Type I and Type II (this is the reasonwhy lower indices are assigned to the SLs without wave-length switching), SLs with wavelength switching support (i.e., l ∈ L W ) are also allowed to be used if the available SLswithout wavelength switching support are inadequate (see, forexample, the assignment of r in Fig. 4.(g)). The pseudocodefor this part of the algorithm is shown in Algorithm 1. B. Reassignment for SChs of Type I and Type II
Using Algorithm 1, we have assigned SChs of Type Iand Type II to each connection request and obtained a ta-ble containing a set of connection requests with currentlyunsatisfied traffic volumes (i.e., T SCh-III ), which are expectedto be served by SChs of Type III. Here, l maxNW-A1 denotes themaximum index of the currently used/required SLs withoutwavelength switching support (i.e., the number of such SLs)in the network after the execution of Algorithm 1. It is obviousthat l maxNW-A1 SLs without wavelength switching support may notbe used on every link. For example, as shown in Fig. 4.(g), l maxNW-A1 is equal to 3, but only 2 SLs are used on link 2-5.Note that these unused SLs (e.g., SL-3 on link 2-5) cannotbe used to establish SChs of Type III hereafter because theydo not support wavelength switching. Therefore, before weassign SChs of Type III to the connection requests recordedin T SCh-III , we will first attempt to assign them one by one– starting from the one with the largest unsatisfied trafficvolume – to the unused SLs whose indices are smaller than l maxNW-A1 . As shown in Fig. 4.(h), the connection request r canbe successfully assigned to pass through routing path (cid:104) (cid:105) , although this will result in a certain degree of spectrumwastage. Then, we remove (cid:104) r : 6 Tbps (cid:105) from T SCh-III . In thisway, we can somewhat reduce the number of entries in T SCh-III ,thus making it possible to use fewer SLs with wavelengthswitching support hereafter. Such an assignment will not resultin any negative effect on the optimization objective(s) becausethe (main) objective of the RSCSA problem is to minimize thenumber of SLs that are used/required in the network, not tominimize their sum over all links.The pseudocode for this part of the algorithm is shown inAlgorithm 2. The inputs to Algorithm 2 are T SCh-III and l maxNW-A1 ,which are obtained after the execution of Algorithm 1. The Algorithm 1
Assignment for SChs of Type I and Type II
Input: R seq , R np for each np ∈ N P
Output: T SCh-III
1: Create new tables: T SCh-II and T SCh-III .2: for each r = (cid:104) s , d , t (cid:105) ∈ R seq do
3: Remove r from R np sd .4: if an SCh for (cid:104) np sd : p , l , B (cid:105) is recorded in T SCh-II then t p ← calculate the supportable traffic volume on the SCh based onthe highest feasible modulation format m p for routing path p andthe remaining available spectrum B .6: if t p > t then
7: Assign routing path p , SL l , and the required spectrum to r –create a (not fully used) SCh of Type II.8: t ← B ← B minus the required spectrum for r .10: Go to the next connection request (line 2).11: else
12: Assign routing path p , SL l , and the remaining availablespectrum B to r – create an SCh of Type II.13: t ← t − t p .14: Remove (cid:104) np sd : p , l , B (cid:105) from T SCh-II .15: end if end if while t > do (cid:104) best - p r , best - l r (cid:105) ← call the FF-SLA function for each candidatepath p r ∈ P r and select the one with the smallest l p r FF .19: t best- p r ← calculate the supportable traffic volume on routing path best - p r and SL best - l r based on the highest feasible modulationformat for best - p r .20: if t best- p r > t then if R np sd is not an empty set then
22: Assign routing path best - p r , SL best - l r , and the requiredspectrum to r – create a (not fully used) SCh.23: B rem ← calculate the remaining available spectrum of theSCh.24: Append (cid:104) np sd : best - p r , best - l r , B rem (cid:105) to T SCh-II .25: else
26: Append (cid:104) r : t (cid:105) to T SCh-III .27: end if t ← else
30: Assign routing path best - p r , SL best - l r , and the entire C-bandspectrum to r – create an SCh of Type I.31: t ← t − t best- p r .32: end if end while end for return T SCh-III output of Algorithm 2 is the modified T SCh-III , in which thenumber of entries may be reduced.
C. Assignment for SChs of Type III
Finally, we begin to assign resources to the unsatisfiedconnection requests that are still recorded in T SCh-III afterAlgorithm 2 has been executed. Similar to the approach thathas been widely applied to the previous RSA and RSSAproblems, each connection request will be assigned using the
First-Fit Spectrum Allocation (FF-SA) function [8], as shownin Fig. 4.(i). The pseudocode for this part of the algorithmis shown in Algorithm 3, where l min W and l max W represent theminimum and maximum indices, respectively, of SLs withwavelength switching support (i.e., l ∈ L W ). As stated before,the indices of the SLs with wavelength switching support (i.e., l ∈ L NW ) are lower than those of the SLs without wavelengthswitching support. Therefore, l min W and l max W are actually equalto | L NW | + and | L | , respectively. Algorithm 2
Reassignment for SChs of Type I and Type II
Input: T SCh-III , l maxNW-A1 Output: T SCh-III
1: Sort T SCh-III by t r rem , from largest to smallest.2: for each (cid:104) r : t r rem (cid:105) in T SCh-III do while TRUE do (cid:104) best - p r , best - l r (cid:105) ← call the FF-SLA function for each candidatepath p r ∈ P r , and select the one with the smallest l p r FF .5: if best - l r ≤ l max NW − A then t best- p r ← calculate the supportable traffic volume on routingpath best - p r and SL best - l r based on the highest feasiblemodulation format for best - p r .7: if t best- p r > t r rem then
8: Assign routing path best - p r , SL best - l r , and the requiredspectrum to r – create a (not fully used) SCh of Type I.9: Remove (cid:104) r : t r rem (cid:105) from T SCh-III .10: break while - go to the next connection request (line 2).11: else
12: Assign routing path best - p r , SL best - l r , and the entire C-band spectrum to r – create an SCh of Type I.13: t r rem ← t r rem − t best- p r .14: end if else break while - go to the next connection request (line 2).17: end if end while end for Algorithm 3
Assignment for SChs of Type III
Input: T SCh-III
Output: T SCh-III l current W ← l min W
2: Sort T SCh-III by t r rem , from largest to smallest.3: while l current W ≤ l max W and T SCh-III is not empty do for each (cid:104) r : t r rem (cid:105) in T SCh-III do best - p r ← apply the FF-SA function [8] for each candidate path p r ∈ P r on SL l current W and select the one with the lowest endingindex of FSs.6: if best - p r (cid:44) None then
7: Assign routing path best - p r , l current W , and the required FSs asobtained by the FF-SA function to r .8: Remove (cid:104) r : t r rem (cid:105) from T SCh-III .9: end if end for l current W ← l current W + 112: end while if T SCh-III is not empty then
14: Call Algorithm 2 again while allowing best - l r > l maxNW-A1 in line 5.15: end if Notably, we may not be able to successfully serve all unsat-isfied connection requests recorded in T SCh-III if the availableSLs with wavelength switching support are inadequate. In thiscase, we will call Algorithm 2 again, now allowing the use ofSLs with indices greater than l maxNW-A1 (i.e., removing line 5 andthe corresponding lines 15 ∼
17 from Algorithm 2).
D. Iteration with the simulated annealing metaheuristic
Similar to previous works focusing on the static RWA, RSA,and RSSA problems, the service sequence R seq is very impor-tant to our heuristic algorithm for solving the RSCSA problembecause the heuristic algorithm assigns resources to the con-nection requests one by one. Different service sequences willlead to different assignment results. Therefore, we apply thesimulated annealing (SimAn) metaheuristic approach [8] tofind a good sequence that yields better results. Undoubtedly,other iterative approaches, such as simple random shifting,could also be applied for this purpose. VI. S
IMULATIONS AND PERFORMANCE EVALUATIONSFig. 5. Network topologies: (a) the simple 6-node, 18-directed-link n6s9network; (b) the realistic 14-node, 42-directed-link NSF network [31].
In this section, we evaluate the performance of the proposedILP model and heuristic algorithm based on two networktopologies: i) the simple 6-node, 18-directed-link n6s9 net-work, as shown in Fig. 5.(a), and ii) the realistic 14-node,42-directed-link NSF network, as shown in Fig. 5.(b) [31].The following assumptions are adopted in the simulationexperiments: • A bundle of weakly coupled 4-core multicore fibers(MCFs), as proposed in Ref. [32], is assumed for eachlink (i.e., | L | = ) in the networks for the followingreasons: i) full compatibility with conventional SMFswhile maintaining a 125 µ m cladding diameter; ii) low in-tercore crosstalk (XT), enabling ultralong-haul all-opticaltransmission; and iii) significant cost savings when com-bined with the application of cladding-pumped multicoreerbium-doped fiber amplifiers (MC-EDFAs) [33, 34]. • The total spectrum per core of a 4-core MCF is consid-ered to be 4 THz (C-band), that is, 320 FSs conformingto the ITU-T 12.5 GHz grid [5]. • Each subtransceiver operates at a fixed baud rate of 32Gbaud, supporting an OC that occupies 37.5 GHz (i.e., 3FSs) [15, 35]. In addition, a spectrum occupation of 12.5GHz (i.e., 1 FS) is assumed for each SW-GB. • We consider four modulation formats in the simulationexperiments, namely, double polarization (DP) BPSK,QPSK, 8-QAM, and 16-QAM. The supportable bit ratesper OC are 50, 100, 150 and 200 Gbps. Notably, thetransmission reaches for the different modulation formatsare bounded by two factors: i) the optical signal-to-noiseratio (OSNR) and ii) the XT [36]. However, since weconsider 4-core MCFs with low XT interference, thetransmission reaches are mainly bounded by the OSNR inthis case. Therefore, for the different modulation formatslisted above, the transmission reaches are 6300, 3500,1200, and 600 km, respectively [35]. • Three candidate shortest routing paths ( k = 3) are con-sidered for each connection request. Moreover, to compare the network performance of SDM-OTNs and SCNs, we consider three different OXCs, as fol-lows: • The first is the conventional OXC applied in SDM-OTNs,which is implemented using stacked WXCs as the basicsolution to achieve SDM. In such an OXC, wavelengthswitching is supported on each SL – i.e., L W = L and L NW = ∅ . • The second is an HOXC (i.e., SXC+WXC) proposed forSCNs, which is implemented using CSSs as shown inFig. 2. We assume that in such an HOXC, one-ninth of theSLs support wavelength switching, in accordance with theassumptions proposed in Ref. [15]. That is, | L W | = (cid:100) | L | (cid:101) and L NW = L − L W . • The last is an OXC that does not support wavelengthswitching on any of its SLs. In such an OXC, only SXCsare deployed at intermediate nodes – i.e., L W = ∅ and L NW = L .All simulation experiments were performed in a MicrosoftWindows 10 environment using a computer with an AMDRyzen 6-core 3.6 GHz CPU and 16 GB of memory. A. Simulation experiments involving the simple n6s9 network
In these simulation experiments, we considered the simple,small-scale n6s9 network with 20 SLs (i.e., one bundle of five4-core MCFs per link). Therefore, in the HOXC case, the set ofSLs with wavelength switching support, L W , was { , , } .We considered different numbers of connection requests rang-ing from 20 to 100 (in increments of 20), representing differenttraffic loads. Specifically, the total average traffic volumeswith which the network was loaded ranged from 0.11 to 0.55Pbps. For each traffic load, we randomly generated 50 differenttraffic matrices R . Considering that current traffic volumes areexpected to increase by 10 × in the future (by 2024) [1, 15],for each unidirectional connection request in R , the trafficvolume was randomly selected from among traffic profiles of { } with probabilities of { } [27, 28, 37–39].To solve the ILP model proposed in Section IV, we used theoptimization software GUROBI v8.0.1 [40]. Since the RSCSAproblem is an NP-hard problem, as proven in Section III-B, itmay not be possible to completely solve the ILP model withina reasonable amount of time for certain input matrices and/ortraffic loads. Therefore, we bounded the running time of theILP model to 1 hour for the main objective and 300 secondsfor the minor objective. Moreover, the solutions given by theheuristic algorithm were input into the ILP model as initialsolutions to improve the convergence rate.The simulation results, including the average values of theobjective(s) and the 95% confidence intervals (
T-distribution ),are shown in Fig. 6. The abbreviations ‘LB’, ‘ILP’, and ‘HA’in Fig. 6 represent the lower bound of the RSCSA problemgiven by the ‘
BestBound ’ of
GUROBI , the optimal or currentfeasible solution obtained by solving the ILP model with a1-hour running time limit, and the solution obtained using theheuristic algorithm with 1000 iterations of R seq , respectively.The abbreviations ‘WXC’, ‘SXC+WXC’, and ‘SXC’ represent Fig. 6. Simulation results for the simple 6-node, 18-link n6s9 network. the three OXCs introduced above, that is, the OXC withfull wavelength switching support (i.e., L W = L ) for SDM-OTNs, the HOXC with partial wavelength switching support(i.e., | L W | = ) for SCNs, and the OXC without wavelengthswitching support (i.e., L W = ∅ ), respectively. Moreover, thenumber over the data bar represents the number of inputmatrices R for which the corresponding ILP models did notyield optimal solutions within 1 hour.From Fig. 6, we can see that even though only approx-imately one-ninth of the SLs support wavelength switchingin the HOXC case, the results of ‘ILP - SXC+WXC’ and‘ILP - WXC’ are the same, while negligible gaps (within2.4%) exist between the results of ‘HA - SXC+WXC’ and‘HA - WXC’. Moreover, as we can see from the results of‘ILP - SXC+WXC - Obj2’ and ‘HA - SXC+WXC - Obj2’,the average numbers of used/required SLs with wavelengthswitching support for the solutions obtained using both theILP model and the heuristic algorithm are less than 1.2 for alltraffic loads in the HOXC case. These observations indicatethat the conventional OXC with full wavelength switchingsupport offers no remarkable advantages for future connectionrequests with large traffic volumes (e.g., several or dozensof Tbps) – or, equivalently, for multiple connection requestsbetween the same source-destination pair with smaller trafficvolumes typical of current network traffic that are groomedinto a single connection request with a larger traffic volume.Moreover, according to the cost assessments presented inRefs. [14] and [15], for the network with 20 SLs consideredin these simulation experiments, the cost of either a full-sizeCSS-based HOXC or a sub-CSS-based HOXC (see Fig. 2.(a))designed for SCNs is only 25% of that of a conventional OXCwith full wavelength switching support designed for SDM-OTNs. Therefore, full wavelength switching support may nolonger be necessary for the future massive SDM era.In contrast, relatively large gaps, ranging from 8% to 14%,can be observed between the results for OXCs without wave-length switching support (i.e., ‘SXC’) and those for the abovetwo (H)OXC cases with full/partial wavelength switching support. These findings indicate that completely removingwavelength switching support from the intermediate nodes willresult in some loss of network performance. However, fewercost savings (compared with the great cost savings between‘WXC’ and ‘SXC+WXC’) can be achieved, as well. The trade-off decision should be made by the network operators.Moreover, we can observe that the ‘ILP’ and ‘HA’ resultsare very similar in all cases. For the two (H)OXC caseswith full/partial wavelength switching support (i.e., ‘WXC’and ‘SXC+WXC’), the ILP model can be completely solvedwithin 1 hour for all or the majority of the input matrices R ,depending on the traffic loads, and the results of both ‘ILP’and ‘HA’ are close to the lower bounds of the problem. For theOXC case without wavelength switching support (i.e., ‘SXC’),the ILP model becomes difficult to solve within 1 hour if thetraffic load is heavy. In this case, the gaps between the ‘HA’results and the lower bounds range from 1.1% to 8.8%, whilethose between the ‘ILP’ results and the lower bounds rangefrom 0.7% to 8.2%, which are considered acceptable. TABLE IA
VERAGE RUNNING TIMES OF THE PROPOSED HEURISTIC ALGORITHMWITH
ITERATIONS FOR THE SIMPLE N S NETWORK
OXC Traffic load | R | Architecture 20 40 60 80 100WXC 11.95 14.03 15.81 17.06 18.49SXC+WXC 3.98 4.09 4.44 4.20 4.64SXC 0.67 1.05 1.43 1.79 2.17
Table I lists the average running times (in seconds) of theheuristic algorithm with 1000 iterations (on a single thread)for the simple n6s9 network. We can see that the runningtimes of the heuristic algorithm in the conventional OXC casewith full wavelength switching are much longer than thosein the HOXC case with partial wavelength switching, and theshortest running times are incurred in the OXC case withoutwavelength switching. The reason for this observation is thatfinding a set of continuous and contiguous FSs with the lowestending index along a routing path by means of the
FF-SA function is much more difficult than finding a feasible SLwith the lowest index along a routing path by means of the FF-SLA function. Therefore, in the conventional OXC casewith full wavelength switching, the
FF-SA function will becalled more times – for each SL with wavelength switchingsupport until all connection requests have been served – bythe heuristic algorithm, resulting in a longer running time. Incontrast, in the OXC case without wavelength switching, theheuristic algorithm will not call the
FF-SA function even once,since there are no SLs that support wavelength switching,resulting in the shortest running time.In summary, the simulation results show that the proposedILP model (with a 1-hour running time limit) and heuristicalgorithm both work well for small-scale problem instances,for which the optimal solutions or solutions close to the lowerbounds can be obtained.
B. Simulation experiments involving the realistic NSF network
Fig. 7. Simulation results for the 14-node, 42-link NSF network.
In these simulation experiments, we considered the realisticlarge-scale NSF network with 40 SLs (i.e., one bundle offive 4-core MCFs per link). Considering that one-ninth ofthe SLs support wavelength switching [15], the set L W was { , , · · · , } in this case. Moreover, we also consideredheavier traffic loads – ranging from 100 to 500 (in incrementsof 100) connection requests – and 50 different traffic matrices R for each traffic load. In this case, the total average trafficvolumes with which the network was loaded ranged from 0.55to 2.75 Pbps. In such large-scale instances, acceptable solu-tions become difficult to obtain within a reasonable amountof time by solving the ILP model. Therefore, we relaxed theoriginal ILP model by removing Constraints (4) ∼ (12) andthe minor objective to obtain the lower bounds for the RSCSAproblem, which we then used as the benchmarks to evaluatethe performance of the heuristic algorithm. This relaxationmeans that i) wavelength switching is allowed on all SLs,ii) lightpaths can be established without SW-GBs, and iii) thespectrum contiguity constraint is relaxed. Consequently, in thiscase, the lower bound obtained by solving the relaxed ILP TABLE IIA
VERAGE RUNNING TIMES OF THE PROPOSED HEURISTIC ALGORITHMWITH
ITERATIONS FOR THE REALISTIC
NSF
NETWORK
OXC Traffic load | R | Architecture 100 200 300 400 500WXC 87.59 117.44 131.90 151.45 163.48SXC+WXC 50.13 75.13 78.98 84.17 86.13SXC 4.92 9.80 15.09 21.21 27.10 model is not only the lower bound of the RSCSA problem inan SCN but also the lower bound of the RSSA problem inan SDM-OTN – if we transform the objective of the RSSAproblem into the minimization of the number of SLs, asopposed to the number of FSs, that are used/required in thenetwork.The corresponding simulation results are shown in Fig. 7.We can observe that the results in Fig. 7 are similar to thosepresented in Fig. 6. First, the gaps between the results of ‘HA- SXC+WXC’ and ‘HA - WXC’ are negligible, ranging from2.3% to 4.2%. This means that the conventional OXC withfull wavelength switching support is not a preferred solutionfor future Pbps-level OTNs because of the significantly highercost – for the network with 40 SLs considered here, theconventional OXC configuration is 5.8 times as costly as thefull-size CSS-based HOXC configuration and 4.2 times ascostly as the sub-CSS-based HOXC configuration [14, 15] –for similar performance. By contrast, we can see that the gapsbetween the results of ‘HA - SXC+WXC’ and ‘HA - SXC’are relatively significant, ranging from 10.1% to 19.6% fordifferent traffic loads. Therefore, the network operators arerequired to make a decision concerning the balance betweenthe additional cost and better performance.Moreover, the results of ‘HA - WXC’ are close to the lowerbounds obtained by solving the relaxed ILP model (i.e., ‘LB’).The gaps between them range from 9.6% to 11.4%. Comparedto the results shown in Fig. 6, these gaps are relatively largebecause the lower bounds for these simulation experimentsare not strict – they are obtained by solving the relaxed ILPmodel, in which almost all of the constraints of the originalILP model have been removed. In addition, it should be notedthat it is unfair to evaluate the performance of the heuristicalgorithm by comparing the results of ‘HA - SXC+WXC’ or‘HA - SXC’ against these lower bounds because wavelengthswitching is allowed on all SLs in the relaxed ILP model.Finally, Table II lists the average running times (in seconds)of the heuristic algorithm for the realistic NSF network, fromwhich it can again be observed that the results are similar tothose in Table I. The heuristic algorithm can yield reasonablesolutions within an acceptable running time. Thus, we cansee that the proposed heuristic algorithm is also efficient forsolving realistic large-scale problem instances.VII. C
ONCLUSION
In this paper, we focused on the resource allocation problemin SCNs, which we defined as the routing, spatial channel, andspectrum assignment (RSCSA) problem. First, we reviewedthe key features of SCNs from the networking perspective and described how these features are related to the RSCSAproblem. We proved the NP-hardness of the RSCSA problemand proposed two approaches for solving it: an ILP model forsmall-scale problem instances and a heuristic algorithm withhigher scalability. Simulation results show that the ILP model(with a 1-hour running time limit) and the heuristic algorithmboth work well for small-scale problem instances, for whichthe optimal solutions or solutions close to the lower boundscan be obtained. In addition, the heuristic algorithm is alsoefficient for solving realistic large-scale problem instances.Moreover, the results show that compared to conventionalOXCs with full wavelength switching implemented by meansof stacked WXCs, which are typically used in SDM-OTNs,the CSS-based HOXCs designed for SCNs can enable greatcost savings while providing similar network performance, andconsequently, these HOXCs are expected to be a promisingsolution for the future massive SDM era. However, someimportant challenges remain that have not been addressed inthis paper, such as the resource allocation problem for anSCN with SLC support implemented by means of MS-basedHOXCs and the dynamic resource allocation problem, whichwill require further investigation in future work.R EFERENCES[1] P. J. Winzer and D. T. Neilson, “From scaling disparities to integratedparallelism: A decathlon for a decade,”
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