1 Introduction
2 Related work
3 Base optimization problem
3.1 Prerequisite data
-
catalogue of active and passive equipment,
-
topology of the infrastructure (duct) network,
-
infrastructure paths selected for every distribution and access node,
-
signal demand of each access node,
-
infrastructure nodes (sites) selected for installation of passive and active equipment.
3.2 Main decision variables
-
utilized OLT types (OLT locations are given),
-
utilized splitter types and their locations,
-
utilized cable types (network topology is given),
-
utilized cabinet types (network nodes are given),
-
splicing locations and utilized splice closures.
3.3 Problem structure
3.4 Objective function
3.5 Problem description layout
4 FTTH network model
4.1 Network fragment
4.1.1 Infrastructure layer model
4.1.2 Cable layer model
4.1.3 Signal layer model
-
the single head-end signal node \(n^{sh}\in \mathcal {N}^{SH}\) must be assigned to the single CO site \(n^{iso}\in \mathcal {N}^{ISO}\), i.e., \(si\_site(n^{sh})\equiv n^{iso}\),
-
every head-end signal distribution point \(n\in \mathcal {N}^{SDH}\) must be assigned to the single CO site \(n^{iso}\), i.e., \(\forall _{n\in \mathcal {N}^{SDH}}, si\_site(n)\equiv n^{iso}\),
-
each distribution signal distribution point \(n\in \mathcal {N}^{SDD}\) can be assigned to either the CO site \(n^{iso}\) or to any DP site \(s\in \mathcal {N}^{ISD}\), i.e., \(\forall _{n\in \mathcal {N}^{SDD}}, si\_site(n) \in \{ n^{iso}\} \cup \mathcal {N}^{ISD}\),
-
each access signal distribution point \(n\in \mathcal {N}^{SDA}\) can be assigned to an infrastructure site of any type but the CP, i.e, \(si\_site(n)\in \mathcal {N}^{ISO} \cup \mathcal {N}^{ISD} \cup \mathcal {N}^{ISA}\),
-
finally, each access signal node \(n\in \mathcal {N}^{SA}\), must be assigned to one of CP infrastructure sites \(s\in \mathcal {N}^{ISP}\).
4.1.4 Bundle layer model
-
singleton set \(\mathcal {N}^{BH}\equiv \{n^{bh}\}\) that aggregates the head-end signal node \(n^{sh}\) and every head-end signal distribution point \(n_{sdh}\in \mathcal {N}^{SDH}\), that are located in CO infrastructure site \(n^{iso}\). Referring to the sample network presented in Fig. 7, single bundle head-node \(n^{bh}\) aggregates head-end signal node \(n^{sh}\) as well as two head-end signal distribution points \(n_{sdh1}\) and \(n_{sdh2}\);
-
set \(\mathcal {N}^{BD}\) of distribution bundle nodes; each distribution bundle node \(n_{bd}\in \mathcal {N}^{BD}\), corresponds to the CO or a single DP infrastructure site \(n_{is}\in \mathcal {N}^{ISO}\cup \mathcal {N}^{ISD}\) and aggregates every distribution signal distribution points \(n_{sdd}\in \mathcal {N}^{SDD}: si\_site(n_{sdd})= n_{is}\), located in that site. In the sample network (see Fig. 7), there are two distribution bundle nodes \(n_{bd1}\) and \(n_{bd2}\) that aggregate, respectively, a subset of distribution signal distribution points \(\{n_{sdd1}, n_{sdd2}, n_{sdd3}\}\) and \(\{n_{sdd4}\}\);
-
set \(\mathcal {N}^{BA}\) of access bundle nodes; each access bundle node \(n_{ba}\in \mathcal {N}^{BA}\), relates to a single infrastructure site \(n_{is}\in \mathcal {N}^{IS}\), and aggregates every access signal distribution point \(n_{sda}\in \mathcal {N}^{SDA}: si\_site(n_{sda})=n_{is}\) as well as every access signal node \(n_{sa}\in \mathcal {N}^{SA}: si\_site(n_{sa})=n_{is}\) located at that site. In the sample network (in Fig. 7), there are three access bundle nodes \(n_{ba1}\), \(n_{ba2}\), and \(n_{ba3}\) that aggregate, respectively, subsets \(\{n_{sda1}, n_{sda2}, n_{sa1}\}\), \(\{n_{sda3}, n_{sa2}\}\), and \(\{n_{sda4}, n_{sa3}\}\) of the signal nodes.
4.1.5 Network fragment concluding remarks
4.2 Equipment catalogue fragment
Name | Catalogue | Section |
---|---|---|
\(\mathcal {T}^a\)
| Optical cables | |
\(\mathcal {T}^c\)
| OLT cards | |
\(\mathcal {T}^e\)
| Cabinets | |
\(\mathcal {T}^f\)
| Sites | |
\(\mathcal {T}^g\)
| Segment preparations | |
\(\mathcal {T}^o\)
| OLT devices | |
\(\mathcal {T}^q\)
| Fiber splices | |
\(\mathcal {T}^r\)
| Optical splitters | |
\(\mathcal {T}^{3r}\)
| Splitter combinations | |
\(\mathcal {T}^u\)
| Splice closures |
4.2.1 OLT cards
4.2.2 OLT devices
4.2.3 Optical splitters
4.2.4 Admissible splitter combinations
4.2.5 Sites
4.2.6 Hardware cabinets
4.2.7 Fiber splices
4.2.8 Cable closures
4.2.9 Cables
4.2.10 Infrastructure segment preparations
4.2.11 Common constants
-
common fiber attenuation per kilometer, denoted by a,
-
common signal power expected at the input port ONT/ONU device, denoted by \(t_c\). The value of this constant can take into account not only the sensitivity of the receiving port but also some additional reserve to compensate aging of optical fibers and components, thermal fluctuations, manipulation, etc. This constant can be easily differentiated between particular access nodes not introducing any additional complexity to the optimization problem.
5 Partial problems detail formulations
5.1 Bundle layer dimensioning partial problem
5.1.1 Assumptions
-
there is exactly one head-end bundle node \(n^{bh}\),
-
distribution cone \(bb\_c(n)\) of each distribution bundle node \(n\in \mathcal {N}^{BD}\), is given,
-
the physical length of every infrastructure trail is given; thus, the distance from each access bundle node \(a\in \mathcal {N}^{BA}\), to the head-end bundle node is also known,
-
every signal network connection that satisfies demand of a given access bundle node \(n\in \mathcal {N}^{BA}\), shares the same distribution infrastructure trail \(p\in \mathcal {P}^{ISD}\), as well as the same trunk infrastructure trail \(q\in \mathcal {P}^{ISH}\),
-
only symmetrical splitters with uniform splitting ratios are admissible.
5.1.2 Decision variables
5.1.3 Feasible set
5.1.4 Induced cost
5.2 Cable partial problem
5.2.1 Assumptions
-
distribution bundle links of different distribution bundle nodes do not share infrastructure segments,
-
there is at most one trunk cable at an infrastructure segment,
-
there is at most one distribution cable at an infrastructure segment,
-
infrastructure segment can be attributed with at most one segment preparation,
-
trunk and distribution fiber segments cannot share a cable segment meaning that a distribution fiber and a trunk fiber cannot be buried in the same cable.
5.2.2 Linking variables
5.2.3 Decisions variables
5.2.4 Feasible set
5.2.5 Induced cost
5.3 Splices and closures partial problem
5.3.1 Assumptions
5.3.2 Decision variables
5.3.3 Feasible set
5.3.4 Induced cost
5.4 Site equipment partial problem
5.4.1 Assumptions
5.4.2 Linking variables
5.4.3 Decision variables
-
to designate site type \(f\in \mathcal {T}^f\) assigned to site \(s\in \mathcal {N}^{IS}\), we use binary variable \(F_{sf}\in \{0,1\}\),
-
the number of cabinets of type \(e\in \mathcal {T}^e\), installed at infrastructure site \(s\in \mathcal {N}^{IS}\), for hosted in-there head-end (\(n^{bh}\)), distribution \(\mathcal {N}^{BD}\subseteq \mathcal {N}^B\), and access \(\mathcal {N}^{BA}\subseteq {N}^B\), bundle nodes is represented, respectively, by integer variables \(E^{BH}_{ue}\in \mathbb {Z}_+\), \(E^{BD}_{ue}\in \mathbb {Z}_+\), and \(E^{BA}_{ue}\in \mathbb {Z}_+\),
-
the number of OLT devices of type \(o\in \mathcal {T}^o\) installed at CO site \(n^{iso}\) is represented by integer variable \(O_o\in \mathbb {Z}_+\).
5.4.4 Feasible set
5.4.5 Induced cost
6 Numerical results
6.1 Testing methodology
Min | Mean | Max | |
---|---|---|---|
Demands | 321 | 994 | 1440 |
Clients | 13,152 | 23,308 | 36,849 |
Edges | 740 | 2107 | 3179 |
Edges total length | 38.1 km | 67.8 km | 106.0 km |
6.2 Simplifications and settings
Equipment | Cost |
---|---|
OLT with 8 card slots | 6000 |
Card with 8 B-class lasers | 8000 |
Splitter 1:2 | 10 |
Splitter 1:64 | 120 |
Single splice | 2 |
1 km of a 6-fiber cable | 3000 |
Closure accommodating up to 12 splices | 20 |
Cabinet serving up to 500 fibers | 1500 |
6.3 Gain
-
\(t_{heur}=0,\;t_{MIP}=t_{total}\)
-
\(t_{heur}=t_{total},\;t_{MIP}=0\)
-
\(t_{heur}=\frac{1}{2}\cdot t_{total},\;t_{MIP}=\frac{1}{2}\cdot t_{total}\)
-
\(t_{heur}=\frac{1}{5}\cdot t_{total},\;t_{MIP}=\frac{4}{5}\cdot t_{total}\)
-
\(t_{heur}=\frac{4}{5}\cdot t_{total},\;t_{MIP}=\frac{1}{5}\cdot t_{total}\)