Routing, Modulation Level, and Spectrum Assignment in Elastic Optical Networks—A Serial Stage Approach with Multiple SubSets of Requests Based on Integer Linear Programming
Abstract
:1. Introduction
 i
 Proposal of five new ILP alternatives based on RML+SA strategy, as follows:
 Linkoriented routing, with split traffic flow, and one set of requests.
 Linkoriented routing, with split traffic flow, and multiple subsets of requests.
 Linkoriented routing, with unsplit traffic flow, and multiple subsets of requests.
 Pathoriented routing, with split traffic flow, and multiple subsets of requests.
 Pathoriented routing, with unsplit traffic flow, and multiple subsets of requests.
 ii
 Performance of simulations to study the advantages and limitations of the proposed RML+SA approaches.
2. Related Works
 Optimization approaches:
 Twostage serial optimization: In this approach, algorithms obtain the solution in two stages; the RML problem is solved first, and then the SA problem is approached as a coloring problem. This approach has been called RML+SA, where the RML phase calculates an ideal cost for the RMLSA problem [6]. On the other hand, it is also possible to calculate the solution in an SA+RML approach [30,31]; however, this approach is outside the scope of this study.
 Request management:
 Multiple subsets of requests: The set of requests is divided into several smaller subsets, and then the lightpaths are calculated and installed consecutively for each subset [28].
 Routing strategies:
 Pathoriented routing: Routing algorithms select a path for a request from precomputed paths [6], typically the k shortest paths.
 Linkoriented routing: Routing algorithms have all possible routes available [14]; all network links are candidates to be part of a route. No precalculated path is necessary.
 Traffic flow division:
3. Mathematical Programming Models
 1LM = One set of requests, linkoriented routing, and multiple traffic subflows.
 MLM = Multiple subsets of requests, linkoriented routing, and multiple traffic subflows.
 ML1 = Multiple subsets of requests, linkoriented routing, and one traffic flow.
 MPM = Multiple subsets of requests, pathoriented routing, and multiple traffic subflows.
 MP1 = Multiple subsets of requests, pathoriented routing, and one traffic flow.
3.1. Problem Statement
 Satisfy all the source—destination connection demands, determining the route, the modulation format, and the spectrum assignment for each traffic request;
 Optimize spectrum usage minimizing the maximum index of slot used on all optical fibers in the network.
 The spectral bandwidth of each optical fiber is divided into slots.
 The fiber optic capacity in slots terms is equally limited in all links.
 Connection demands are bidirectional, and RMLSA algorithms calculate a lightpath for each request.
 Between two lightpaths using the same link, there is at least one slot as a guardband.
 A request is represented by a 3tuple: s = (${v}_{o}$, ${v}_{d}$, $\lambda $), indicating ${v}_{o}$ the source node, ${v}_{d}$ the destination node and $\lambda $ the requested bandwidth (data rate).
 Multiple subsets imply a scheme to approach the problem studied in this work, which does not imply management requirements in the upper layers of the network.
 For the division of traffic flows we consider sliceable bandwidthvariable transponders (SBVTs) mentioned in [13].
3.2. Calculation Process for the Multiple SubSets of Requests Approaches
Algorithm 1: $RMLSA+SA$ for Multiple Subsets of Requests Process 

3.3. RML Phase Formulation for MPM
3.4. RML Phase Formulation for MLM
3.5. SA Phase Formulation
3.6. ILP Models Summary
 RML phase: the following particular cases should be taken into account
 One set of requests: in Equations (4) and (11), we have that ${H}_{tl}$ = 0 when the value of ${\mathsf{\Omega}}_{t}$ = ∅, since there was no previous request; and therefore it represents only the maximum slot used by incoming requests.
 Multiple traffic subflows: inequality (12) limits to one the number of paths to be used per request considering $K=1$. This affects Equation (23), causing it to assign all the required slots to a single path for each request.
 SA phase:
4. Simulation Test
 To determine the benefits of dividing the requests into subsets and the advantages in term of computational time of pathbased routing over linkbased routing.
 To study the performance of the proposed models (MLM, MPM, ML1, MP1, and 1LM) compared to the stateoftheart models (1L1, 1PM, and 1P1).
4.1. Computational Environment
4.2. Network Topologies
4.3. Simulation Scheme
 Research Question 1: What would happen as the number of subsets of the requests increases as they are divided into smaller subsets?
 Research Question 2: Are unsplit traffic flow models worse concerning spectrum efficiency and computational time than split traffic flow models? Up to how many subdemands is convenient to split each request?
 Research Question 3: For pathoriented routing, how many available paths ($Kmax$) are advisable to use to improve spectrumusage efficiency without deteriorating computational time?As the available paths ($Kmax$) increase, the spectrum efficiency should increase at the expense of computation time. To answer this question, the performance of solutions were averaged according to the pathoriented routing for the Abeline and Nobeleu network topologies, considering the following parameters:
 
 MP1 $(Kmax=1,\phantom{\rule{4pt}{0ex}}2,\dots ,\phantom{\rule{4pt}{0ex}}10,$$K=1,\phantom{\rule{4pt}{0ex}}RG=5).$
 
 MPM $(Kmax=3,\phantom{\rule{4pt}{0ex}}4,\dots ,\phantom{\rule{4pt}{0ex}}10,$$K=3,\phantom{\rule{4pt}{0ex}}RG=5).$
 
 $100\%$ load (40 requests for Abeline and 80 requests for Nobeleu).
 Research Question 4: Between the pathoriented and linkoriented routing, which strategy would be the best in terms of used spectrum and computational time?Linkoriented routing could obtain better results than pathoriented routing in terms of spectrum efficiency at the cost of computational time. To answer this question, the following parameters were considered:
 
 1P1 $(Kmax=4,K=1,RG=1).$
 
 MP1 $(Kmax=4,K=1,RG=3).$
 
 1PM $(Kmax=4,K=3,RG=1).$
 
 MPM $(Kmax=4,K=3,RG=3).$
 
 1P1 $(K=1,RG=1).$
 
 ML1 $(K=1,RG=3).$
 
 1LM $(K=3,RG=1).$
 
 MLM $(K=3,RG=3).$
 
 Loading percentage: $20\%,40\%,60\%,80\%,100\%$ (8, 16, 24, 32, 40 requests for Abeline and 16, 32, 48, 64, 80 requests for Nobeleu).
Considering the complexity of the Nobeleu network topology, the following parameters were also used: 
 1P1 $(Kmax=8,K=1,RG=1).$
 
 MP1 $(Kmax=8,K=1,RG=3).$
 
 1PM $(Kmax=8,K=3,RG=1).$
 
 MPM $(Kmax=8,K=3,RG=3).$
5. Results and Discussion
5.1. Results for Research Question 1
5.2. Results for Research Question 2
5.3. Results for Research Question 3
5.4. Results for Research Question 4
5.5. General Discussion
 Dividing the requests into multiple subsets up to a certain threshold reduces computation time. The threshold will vary according to the topology considered. Particularly, in both studied topologies, an RG between 3 and 5 is recommended.
 Splitting the traffic stream helps improve the used spectrum at the cost of increased computational time. We recommend splitting the traffic stream into no more than three substreams for the studied topologies and loads.
 The number $Kmax$ of precomputed shortest paths may help to improve the efficient use of spectrum and computation time. We observe that threeshortest paths are suitable in the topologies under study.
 Pathbased models are more convenient compared to linkbased models when the complexity of the problem increases. Determining the appropriate number of routes is necessary to perform simulations since it depends on optical resources and network structure.
6. Conclusions and Future Work
 use the multisubset request scheme with different heuristics;
 extend this study to the Routing, Baud rate, Code, Modulation Level, and Spectrum Allocation (RBCMLSA) approach;
 use the multisubset request scheme considering dynamic and reserved traffic; and
 consider other quality metrics such as requests blockage, power consumption, and optical channel impairments.
Author Contributions
Funding
Conflicts of Interest
Appendix A
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Optimization Approach  Request Management  Routing Strategies  Traffic Flow Division  Contributions Reported in the Literature  

Onestage RMLSA  Twostage RML+SA  (1) One Set of Requests  (M)ultiple Subsets of Requests  (P)athoriented routing Precalculated Path Table  (L)inkoriented routing All paths available  (1) One Traffic Flow  (M)ultiple Traffic subflows  
✓  ✓  ✓  ✓  [13]  
✓  ✓  ✓  ✓  [13,14,15,16,17,18,19]  
✓  ✓  ✓  ✓  [20]  
✓  ✓  ✓  ✓  [5,6,17,21,22,23,24,25,26,27]  
✓  ✓  ✓  ✓  [16,28]  
✓  ✓  ✓  ✓  1L1 [8]  
✓  ✓  ✓  ✓  1PM [29]  
✓  ✓  ✓  ✓  1P1 [6]  
The proposed RML+SA models in this work  
✓  ✓  ✓  ✓  1LM  
✓  ✓  ✓  ✓  MLM  
✓  ✓  ✓  ✓  ML1  
✓  ✓  ✓  ✓  MPM  
✓  ✓  ✓  ✓  MP1 
Parameters  

$RG$:  Number of subset of requests, $RG\in \{1,2,\dots ,\mathcal{S}\left\right\}$. 
t:  Iterator of the global optimization process, $t\in \{1,2,\dots ,RG\}$. 
$\mathcal{V}$:  Set of network nodes, $\mathcal{V}=\{{v}_{i}:i\in \{1,2,\dots ,\mathcal{V}\left\right\}\}$. 
$\mathcal{L}$:  Set of network links, $\mathcal{L}=\{{l}_{i}:i\in \{1,2,\dots ,\mathcal{L}\left\right\}$ where $l=({v}_{i},{v}_{j})$. 
${\mathcal{J}}_{t}$:  Status of slots at iteration t, ${\mathcal{J}}_{t}=\{{J}_{tli}:l\in \mathcal{L},i\in \{1,\dots ,Fmax\}\}$, if the ith slot of the link l is busy, then ${J}_{tli}$ = 1, otherwise 0. 
$Fmax$:  Number of slots available per link in the network. 
$\mathcal{S}$:  Set of requests, $\mathcal{S}=\{{s}_{i}:i\in \{1,2,\dots ,\mathcal{S}\left\right\}$ where $s=({v}_{o}^{s},{v}_{d}^{s},{\lambda}^{s})$, ${v}_{o}^{s}\in \mathcal{V}$ is source node, ${v}_{d}^{s}\in \mathcal{V}$ is destination node, and ${\lambda}^{s}$ is requested bandwidth. 
${\mathcal{S}}_{t}$:  Subset of requests to be installed at iteration t, ${\mathcal{S}}_{t}\subset \mathcal{S}$. 
${\mathsf{\Omega}}_{t}$:  Subset of requests already installed up to iteration $t1$, ${\mathsf{\Omega}}_{t}={\mathcal{S}}_{1}\cup \dots \cup {\mathcal{S}}_{t1}$. 
$\mathcal{M}$:  Set of available modulation format, $\mathcal{M}=\{{m}_{i}:i\in \{1,2,\dots ,\mathcal{M}\left\right\}\}$ where $m=(T,R)$, T is the transmission range and R is the modulation rate. 
$Kmax$:  Number of available paths per requests. 
K:  Maximum number of traffic subflows allowed per request, 1 ≤ K ≤ $Kmax$. 
${\mathcal{P}}_{s}$:  Set of $Kmax$ paths available for the request s, ${\mathcal{P}}_{s}=\{{p}_{1},{p}_{2},\dots ,{p}_{Kmax}\}$ where $p=\{{l}_{1},{l}_{2},\dots ,{l}_{n}\}$. 
${\mathcal{P}}_{s}^{l}$:  Set of available paths of the request s that use the link $l\in \mathcal{L}$, ${\mathcal{P}}_{s}^{l}\subset {\mathcal{P}}_{s}$. 
$GB$:  Number of slots for guardband. 
C:  Bandwidth of one slot with BPSK base modulation. 
${L}_{l}$:  Length of a link l, ${L}_{l}\ge 0$. 
$L{P}_{sp}$:  Length of a path $p\in {\mathcal{P}}_{s}$, $L{P}_{sp}={\sum}_{l\in {\mathcal{P}}_{s}}{L}_{l}$. 
${\mathsf{\Lambda}}_{sp}$:  Number of slots assigned to path $p\in {\mathcal{P}}_{s}$, ${\mathsf{\Lambda}}_{sp}=\u2308{\textstyle \frac{{\lambda}^{s}}{R\xb7C}}\u2309$. 
${\mathsf{\Lambda}}_{sm}$:  Number of slots assigned to request $s\in \mathcal{S}$ with modulation $m\in \mathcal{M}$, ${\mathsf{\Lambda}}_{sm}=\u2308{\textstyle \frac{{\lambda}^{s}}{{R}_{m}\xb7C}}\u2309$. 
${H}_{tl}$:  Number of slots busy in the link l at the iteration t, ${H}_{tl}={\sum}_{i=1}^{Fmax}{J}_{tli}$. 
$\mu $:  Weight of the sum in objective function in RML phase, $\mu \ge 0$. 
${\mathcal{L}}_{v}^{out}$:  Set of outgoing links from node v, ${\mathcal{L}}_{v}^{out}\subset \mathcal{L}$. 
${\mathcal{L}}_{v}^{in}$:  Set of incoming links to node v, ${\mathcal{L}}_{v}^{in}\subset \mathcal{L}$. 
${\Phi}_{sp}$:  Number of slots assigned to the already installed request $s\in {\mathsf{\Omega}}_{t}$ in the $p\in {\mathcal{P}}_{s}$, ${\Phi}_{sp}\in \{0,1,\dots ,Fmax\}$. 
Variables  
${C}_{RML}$:  Maximum slot obtained in the RML phase, ${C}_{RML}\in \{0,1,\dots ,Fmax\}$. 
${C}_{SA}$:  Maximum slot obtained in SA phase, ${C}_{SA}\in \{0,1,\dots ,Fmax\}$. 
B:  Total number of slots used in the network, $B\in \{0,1,\dots ,\mathcal{L}\xb7Fmax\}$. 
${F}_{l}$:  Number of slots used in the link l, ${F}_{l}\in \{0,1,\dots ,Fmax\}$ 
${X}_{sp}$:  Traffic flow rate of request s on the path $p\in {\mathcal{P}}_{s}$, ${X}_{sp}\in [0,1]$. 
${X}_{skm}$:  Traffic flow rate of request s on the path $k\in \{1,2,\dots ,K\}$ with modulation format m, ${X}_{skr}\in [0,1]$. 
${h}_{sp}$:  Binary variable, if the path $p\in {\mathcal{P}}_{s}$ is used, then ${h}_{sp}$ = 1, otherwise 0. 
${h}_{skm}$:  Binary variable, if the path $k\in \{1,2,\dots ,K\}$ with modulation format m is used, then ${h}_{skm}$ = 1, otherwise 0. 
${Y}_{skml}$:  Binary variable, if the path $k\in \{1,2,\dots ,K\}$ with modulation format m uses the link l, then ${Y}_{skml}=1$, otherwise 0. 
${Z}_{skm}$:  Binary variable, if modulation format m is assigned to the path $k\in \{1,2,\dots ,K\}$, then ${Z}_{skm}=1$, otherwise 0. 
${N}_{skm}$:  Number of slots used by the path $k\in \{1,2,\dots ,K\}$ with modulation format m, ${N}_{skm}\in \{0,1,\dots ,Fmax\}$. 
${N}_{skml}$:  Number of slots used by the link l of the path $k\in \{1,2,\dots ,K\}$ with modulation format m,${N}_{skml}\in \{0,1,\dots ,Fmax\}$. 
${f}_{sp}$:  First index of a slot block assigned to the path $p\in {\mathcal{P}}_{s}$, ${f}_{sp}\in \{1,2,\dots ,Fmax\}$. 
${\delta}_{sp}^{{s}^{\prime}{p}^{\prime}}$:  Binary variable, if ${f}_{{s}^{\prime}{p}^{\prime}}>{f}_{sp}$, then ${\delta}_{sp}^{{s}^{\prime}{p}^{\prime}}$ = 1, otherwise 0. 
Phases  Models  

MPM  1PM  MP1  1P1  
RML  (1)–(7)  (1)–(7)  (1)–(7)  (1)–(7) 
SA  (26)–(38)  (26), (27), (29)–(33)  (26)–(38)  (26), (27), (29)–(33) 
Model Parameters  
$RG$  $>1$  $=1$  $>1$  $=1$ 
K  $>1$  $>1$  $=1$  $=1$ 
Phases  MLM  1LM  ML1  1L1 

RML  (8)–(24)  (8)–(24)  (8)–(24)  (8)–(24) 
SA  (26)–(38)  (26), (27), (29)–(33)  (26)–(38)  (26), (27), (29)–(33) 
Model Parameters  
$RG$  $>1$  $=1$  $>1$  $=1$ 
K  $>1$  $>1$  $=1$  $=1$ 
Setting  Variables and Constraints  Proportional Amount  Asymptotic Notation 

RMLMPM  Integer Variables  $\propto \left\mathcal{L}\right+\left\mathcal{S}\right\xb7Kmax$  $O\left(\right\mathcal{S}\xb7\mathcal{L}\left\right)$ 
Constraints  $\propto 2\xb7\left\mathcal{L}\right+2\xb7\left\mathcal{S}\right+\left\mathcal{S}\right\xb7Kmax$  $O\left(\right\mathcal{S}\xb7\mathcal{L}\left\right)$  
RMLMLM  Integer Variables  $\propto \left\mathcal{L}\right+2\xb7\left\mathcal{S}\right\xb7Kmax\xb7\left\mathcal{M}\right\xb7\left\mathcal{L}\right+3\xb7\left\mathcal{S}\right\xb7Kmax\xb7\left\mathcal{M}\right$  $O\left(\right\mathcal{S}\xb7\mathcal{L}{}^{2})$ 
Constraints  $\propto 2\xb7\left\mathcal{L}\right+2\xb7\left\mathcal{S}\right+2\xb7\left\mathcal{S}\right\xb7\left\mathcal{V}\right+\left\mathcal{S}\right\xb7Kmax\xb7\left\mathcal{M}\right\xb7(4\xb7\mathcal{L}+6\xb7\mathcal{V})$  $O\left(\right\mathcal{S}\xb7\mathcal{L}{}^{2})$  
SA  Integer Variables  $\propto 2\xb7\left\mathcal{S}\right\xb7Kmax+2\xb7{\left\mathcal{S}\right}^{2}\xb7{\left(Kmax\right)}^{2}$  $O\left(\right\mathcal{S}{}^{2}\xb7\left\mathcal{L}{}^{2}\right)$ 
Constraints  $\propto 2\xb7\left\mathcal{S}\right\xb7Kmax+2\xb7{\left\mathcal{S}\right}^{2}\xb7{\left(Kmax\right)}^{2}$  $O\left(\right\mathcal{S}{}^{2}\xb7\left\mathcal{L}{}^{2}\right)$ 
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Maidana Benítez, L.V.; Villamayor Paredes, M.M.R.; Colbes, J.; BogadoMartínez, C.F.; Barán, B.; PintoRoa, D.P. Routing, Modulation Level, and Spectrum Assignment in Elastic Optical Networks—A Serial Stage Approach with Multiple SubSets of Requests Based on Integer Linear Programming. Math. Comput. Appl. 2023, 28, 67. https://doi.org/10.3390/mca28030067
Maidana Benítez LV, Villamayor Paredes MMR, Colbes J, BogadoMartínez CF, Barán B, PintoRoa DP. Routing, Modulation Level, and Spectrum Assignment in Elastic Optical Networks—A Serial Stage Approach with Multiple SubSets of Requests Based on Integer Linear Programming. Mathematical and Computational Applications. 2023; 28(3):67. https://doi.org/10.3390/mca28030067
Chicago/Turabian StyleMaidana Benítez, Luis Víctor, Melisa María Rosa Villamayor Paredes, José Colbes, César F. BogadoMartínez, Benjamin Barán, and Diego P. PintoRoa. 2023. "Routing, Modulation Level, and Spectrum Assignment in Elastic Optical Networks—A Serial Stage Approach with Multiple SubSets of Requests Based on Integer Linear Programming" Mathematical and Computational Applications 28, no. 3: 67. https://doi.org/10.3390/mca28030067