On optimal trajectory for the multi-period evolution of FTTx networks

The infrastructure layer of the FTTx–OAN network (see Fig. 4) is the bottom layer of the model. It is characterized by FTTx–OAN infrastructure layer network\(G^{I}\!=\!(\mathcal {N}^{I},\mathcal {L}^{I})\) with infrastructure nodes\(\mathcal {N}^{I}\) (circles in the figure) representing, e.g., in-building compartments, manholes, masts or street cabinets, interconnected by infrastructure links\(\mathcal {L}^{I}\subseteq \mathcal {N}^{I}\times \mathcal {N}^{I}\) (edges connecting circle pairs) that stand for either trenches or overhead lines.

Within infrastructure nodes \(\mathcal {N}^{I}\) we distinguish subset \(\mathcal {N}^{IS}\subseteq \mathcal {N}^{I},\) of infrastructure sites (solid line circles) equipped to hold either active (OLT, ONT) or passive (optical splitters) devices. Set of infrastructure sites \(\mathcal {N}^{IS}\) is further partitioned into central sites \(\mathcal {N}^{ISC}\) (which is the singleton, i.e. \(\mathcal {N}^{ISC}\equiv {\{n^{ISC}\}}\)), distribution sites \(\mathcal {N}^{ISD}\), and access sites \(\mathcal {N}^{ISA}\), hereafter referred to as classes of infrastructure sites.

Let \(\mathcal {P}^{I}\subseteq 2^{\mathcal {L}^{I}},\) denote a set of paths in the infrastructure layer and let \(ii_{pathNodes}:\;\mathcal {P^{I}}\Rightarrow 2^{\mathcal {N^{I}\times \mathcal {N^{I}}}}\) designate a function that returns a pair of end nodes of path \(p\in \mathcal {P}^{I}\). We distinguish a subset \(\mathcal {P}^{IS}\subseteq \mathcal {P}^{I}\), of infrastructure paths, hereafter referred to as infrastructure trails, having both their end nodes in the set of infrastructure sites, i.e., \(t\in \mathcal {P}^{IS}\Rightarrow ii_{pathNodes}(t)\in 2^{N^{IS}\times \mathcal {N}^{IS}}\).

The fibre layer network (see Fig. 5) is modeled by graph \(\mathcal {G}^{F}=(\mathcal {N}^{F},\mathcal {L}^{F})\) with set of fibre nodes \(\mathcal {N}^{F}\) (optical splitters) and set of fibre links \(\mathcal {L}^{F}\) (segments of a fibre enclosed in cables).

Let \(\mathcal {P}^{F}\subseteq 2^{\mathcal {L^{F}}},\) designate a set of paths in the fibre layer, hereafter referred to as fibre paths, where each represents a continuous sequence of fibre links joint by means of splices or switched in optical distribution frames (ODF). An infrastructure trail \(t\in P^{IS},\) can accommodate a group of fibre paths distinguished by function \(if_{fiberPaths}:\;\mathcal {P}^{IS}\Rightarrow 2^{\mathcal {P}^{F}}\) that interconnects optical splitters installed in trail’s end infrastructure sites. Fibre paths are further split into trunk fibre paths \(\mathcal {P}^{FT}\), which support trunk bundle links, and distribution fibre paths \(P^{FD}\) supporting distribution bundle links.³

The FTTx–OAN signal layer network (see Fig. 6) plays the crucial role in our network model and provides bidirectional signal connections between ports of OLT and ONT devices, i.e., between S / R and R / S reference points of the architecture presented in Fig. 2. We characterize this layer network using graph \(\mathcal {G}^{S}=(\mathcal {N}^{S},\,\mathcal {L}^{S})\) with set of nodes \(\mathcal {N}^{S}\) (represented by rounded rectangles in the figure) and set of links \(\mathcal {L}^{S}\), where signal node \(n\in \mathcal {N}^{S}\), stands for an optical splitter and signal link \(l\in \mathcal {L}^{S},\) represents exactly one fibre path between a pair of connected splitters.

As already mentioned, we consider the architecture, where each signal connection goes through exactly three signal nodes (splitters),⁴ referred to—counting from the OLT’s side—as central signal node, distribution signal node, and access signal node; due to a splitter belonging to exactly one FTTx-ODN network, this rule partitions set \(\mathcal {N}^{S}\) into, respectively, \(\mathcal {N}^{SC}\), \(\mathcal {N}^{SD}\), and \(\mathcal {N}^{SA}\) subsets referred to as classes of signal nodes.

There are two distinguished classes of signal links: trunk signal links\(\mathcal {L}^{ST}\subseteq \mathcal {L}^{S},\) between the central and distribution signal nodes and, distribution signal links\(\mathcal {L}^{SD}\subseteq \mathcal {L}^{S},\) between distribution and access signal nodes. These two classes constitute the partition of set \(\mathcal {L}^{S}\).

To facilitate further analysis of inter-layer dependencies, we introduce function \(si_{site}:\,\mathcal {N}^{S}\rightarrow \mathcal {N}^{IS}\) that distinguishes infrastructure site \(i\in \mathcal {N}^{IS},\) that hosts signal node \(s\in \mathcal {N}^{S}\) (exemplary layout of infrastructure nodes that could host presented signal nodes is represented in Fig. 6 by grayed-out polygons). The following cases are admissible:We attribute each access signal node \(s\in \mathcal {N}^{SA},\) with a number \(h^{Sts}\) denoting the number of ONTs connected to the node at R / S reference point, i.e., the signal demand of this node in time period \(t\in \mathcal {T}\).

The considered bundle layer model (see Fig. 7) consists of directed graph \(\mathcal {G}^{B}=(\mathcal {N}^{B},\mathcal {L}^{B})\) with set of bundle nodes \(\mathcal {N}^{B}\) and set of bundle links \(\mathcal {L}^{B}\). Graph \(\mathcal {G}^{B}\) constitutes a contraction of graph \(\mathcal {G}^{S}\) of the signal layer. Thus each bundle node \(b\in \mathcal {N}^{B},\) represents a subset of signal nodes (distinguished by function \(bs_{nodes}:\mathcal {N}^{B}\mathcal {\rightarrow }2^{\mathcal {N}^{S}}\)) such that:We emphasize that the above assignment is exhaustive, i.e., within a single infrastructure site there is at most one bundle node of each particular class. According to the latter requirement, we introduce the partitioning of bundle nodes \(\mathcal {N}^{B}\) into central bundle nodes \(\mathcal {N}^{BC}\) (which is a singleton, i.e., \(\mathcal {N}^{BC}\equiv \{{n^{BC}\}}\)), distribution bundle nodes \(\mathcal {N}^{BD}\), and access bundle nodes \(\mathcal {N}^{BA},\) hereafter referred to as classes of bundle nodes. Similarly to the signal layer case, each bundle node of the access class \(b\in \mathcal {N}^{BA}\) representing subset \(\mathcal {N}^{SB}=bs_{nodes}(b)\) of access signal nodes, is attributed with signal demand \(h^{Btb}=\sum _{s\in \mathcal {N^{SB}}}h^{Sts}\), i.e., its signal demand in period \(t\in \mathcal {T},\) is equal to the sum of demands of these signal nodes.

Bundle link \(l\in \mathcal {L}^{B},\) represents in turn every signal link connecting signal nodes aggregated to the end bundle nodes of this bundle link (we use function \(bs_{links\!}:\mathcal {L}^{B}\mathcal {\rightarrow }2^{\mathcal {L}^{S}}\) that distinguish such set of signal links). Certainly, every signal link aggregated into a bundle link belongs to the same signal link class. Taking advantage of this observation, we partition also bundle links \(\mathcal {L}^{B}\) into trunk bundle links\(\mathcal {L}^{BT}\) between the central and a distribution bundle nodes and distribution bundle links\(\mathcal {L}^{BD}\) between a distribution and an access bundle node. We refer to this partitioning as bundle link classes. We assume that all fibre paths that support a given bundle link use the same infrastructure trail.

Figures 6 and 7 show configurations of signal and bundle layers that are mutually compatible. The signal layer model (see Fig. 6) comprises signal nodes and signal links of every particular class. Nodes co-located within the same infrastructure sites are enclosed within gray polygons. The bundle counterpart of this node is illustrated in Fig. 7. According to rules set in Sect. 3.4, a bundle node in a given infrastructural site aggregates every signal node of the same class located in this site and, within an infrastructure site, there is at most one bundle node of each particular class. Thus, in the example, the following contraction rules are applied:

This subsection completes the network model with catalog sets defining types of equipment admissible for installation at infrastructure sites and links. Every catalog set is denoted by C with a lowercase upper index; the same lowercase index is used for distinguishing properties of an instance of a particular type. Parameters common to every type, like cost (i.e., the capital expenditure—CAPEX), period lease cost or capacity, are denoted by lowercase letters c, \(\varsigma \), and \(\eta \) with appropriate upper indices.

From the business perspective, as illustrated in Fig. 8, resources of the FTTx network are split into hierarchy of business layers (c.f. [7, 13, 14]). The lowest passive infrastructure layer (corresponding to the infrastructure and fibre layers of the model introduced in Sect. 3) comprises all passive elements of the network including cable placeholders of every kind (trenches, ducts, sewage systems, and overhead lines), equipment placeholders (central office buildings, street cabinets, manholes, and pole masts), optical cables, individual optical fibres (dark fibres), and finally, individual wavelengths within a each fibre. The next active network layer (corresponding to the signal and bundle layers of our network model—see Sect. 3) embraces hardware (e.g., OLT devices, optical splitters, ONU/ONT devices) and software components (IT tools) that connect and orchestrate resources exposed by the passive network layer and provides data transport (broadband IP or Ethernet access) between SNI and UNI reference points. The retail service layer consumes the wholesale transport service and provides retail services offered to the end-users (these are common triple-play services like broadband internet access, voice communication, TV-broadcasting, and Video-on-Demand) complemented with operator/location specific services (like e-health, elderly care, video-conferencing, entertainment, teleworking, e-gov, e-education, e-commerce, smart monitoring, internet of things, cloud computing, and so on—cf. [14]). Finally, the end-user layer (not presented in the figure) represents every subscriber of every kind—domestic users, small companies (e.g., SOHO), large companies, and institutional (government, municipal), police, fire guard, medical, etc.

The layered architecture paves a way for identification of roles played by business entities in the context of FTTx networks (see Fig. 8). These are:In Fig. 8, two business reference points are also displayed, i.e., points of the interaction between pairs of roles. Both infrastructure to network (I–N) point and network to service (N–S) point are of the customer to provider type, where the upper role buys products, either the resources or services, exposed by the lower role.

Traditionally, every of these roles is played, in so called vertically integrated (or stovepipe) model, by a single business entity (the leftmost case in Fig. 8). Nowadays, this gradually—as argued at the begining of this section—ceases to be the case. Business entities cooperate in different configurations and at various levels of the architecture; a couple of distinguished common architectures—referred to, respectively, as passive layer open model (PLOM), active layer open model (ALOM), and 3-layer open model (3LOM)—are illustrated in respective subsequent parts of Fig. 8.

We consider the PIP business entity playing the passive infrastructure provider role that organizes and operates communication infrastructure, which can support one or more access networks. Without losing generality,⁶ we assume for a while that the entity owns all the resources comprising the infrastructure. The service the entity offers is leasing of infrastructure resources for a number of consecutive time periods. The price for the leasing is calculated as follows. First, the entity estimates its per-period total cost of ownership (TCO) on the basis of known per-period OPEX expenditures (related to salaries, insurance, energy consumption, etc.) and per-period cost of resource deprecation (to account CAPEX expenditures). Second, it allocates the TCO cost to every resource is offers for lease, possibly adds its profit margin, and thus, receives the required per-period lease prices.

To imagine how depreciation (sometimes referred to also as amortization) can be computed consider set \(\mathcal {A}^{e}\) of tangible assets owned by the PIP entity. Each asset \(a\in \mathcal {A}^{e},\) is attributed with capital expenditure (CAPEX), denoted by c(a),  a PIP-entity had to spend for its deployment (duct preparation, cable rollout) or procurement (OLT, OLT card, splitter, cable segment). Asset \(a\in \mathcal {A}^{e},\) has also defined the useful lifetime l(a), expressed in the number of time periods (as introduced in Sect. 2), optionally, the salvage value s(a) that defines a price the asset can be sold for after its useful lifetime has expired. The capital expenditure c(a) is converted to in-period depreciation \(\zeta ^{t}(a),\;t\in 1,2,\ldots ,l(a)\), in such a way that for each asset the following equation holds:In our experiments, we apply the linear deprecation model, where depreciation in every time period \(t\in 1,2,\ldots ,l(a)\) is constant, i.e., \(\zeta ^{t}(a)=\zeta (a)\) and estimated with the formula \(\zeta (a)=\frac{c(a)-s(a)}{l(a)}\).

Let \(\mathcal {P}^{e},\,e\in \mathcal {E},\) denote a set of resources the PIP entity offers for leasing. The PIP-entity can apply a couple of known methods, e.g., long run incremental cost (LRIC) or fully distributed cost approach (FDCA) [11] (also referred to as fully allocated cost in [30]), to allocate the TCO cost to these resources. In our investigations, we selected the most obvious FDCA method where costs allocated to every resource sum up to exactly the total TCO. Depending on the business context, e.g., resources are leased to divisions of the same or different companies, the PIP-entity can add some profit margin to the cost allocated to the leased resource to receive its lease price (we denote this by \(\zeta (p),\,p\in \mathcal {P}^{e}\) ).

In this subsection, we introduce a cost model for the NP role; the model is necessary to support the formulation of the multiperiod optimization problem in Sect. 5.

Consider the architecture presented in Fig. 9 that complements the architecture introduced in Sect. 4.2 with equipment provider role (EP) and the associated E-N business reference point. The raison d’etre of the NE role is to support a set of (one or more) NPs with a possibility to lease the active (OLT, OLT cards) or passive (optical splitters) equipment. Having such a possibility, an NP role can install/extract the equipment on period-to-period basis paying in each period for the equipment actually installed in its network. The example justification of such model is the business entity that performs the NP role simultaneously against many FTTx networks—the entity is free to move equipment (thus the equipment related cost) from one network to another. Certainly, in this example, the entity plays both NP and NE roles.

Finally, in our model, there are three reference points:We consider costs associated with the NP role. According to the model introduced in the previous sections, we identify three categories of cost—we call them according to the business interface they are presented at—I–N cost, E–N cost, and NP cost (i.e., the cost generated by the NP role itself).⁷

Consider a single time period \(t\in \mathcal {T},\) (see Sect. 2). Let \(x^{atw}\!\!\in \!\!\mathbb {Z}_{+},\) denote an integer variable specifying the number of optical splitters of type \(w\in \mathcal {C}^{r},\) installed in access bundle node \(a\in \mathcal {N}^{BA},\) that in time period t must be fed (through splitters of distribution and central bundle nodes) with an optical signal from any OLT port. Such connected splitters we hereafter call real splitters. Complementarily, let \(\hat{x}^{atw}\in \mathbb {Z}_{+},\) denote a similar integer variable specifying the number of splitters that are installed in time period t, but not necessarily connected in this period to OLT ports—we refer to them as virtual splitters.⁸ Continuing with the same naming scheme, we introduce pairs \((x^{dts},\,\hat{x}^{dts})\) and \((x^{Ctr},\hat{x}^{Ctr})\) of integer variables representing, respectively, the number of real and virtual splitters of type \(s\in \mathcal {C}^{r},\) installed at period \(t\in \mathcal {T},\) in every distribution bundle node \(d\in \mathcal {N}^{BD},\) and the number of real and virtual splitters of type \(r\in \mathcal {C}^{r},\) installed at period \(t\in \mathcal {T},\) in the central bundle node \(n^{BC}\). Finally, let \(y^{Ct}\) and \(z^{Ct}\) denote, in order, a variable representing the number of OLT cards and the number of OLT devices installed at period t.

Taking advantage of these variables, for each time period \(t\in \mathcal {T},\) we introduce the following vectors describing equipment remaining installed in the network during this time period: Then we introduce vector in the form (5) and refer to it as the configuration of the FTTx–OAN network in time period \(t\in \mathcal {T}\) .Let \({\mathscr {X}}^{\mathbf{t}}\) denote the set of all configurations \(\mathbf {X}^{\mathbf{t}}\) that satisfy every intra-period constraint of period \(t\in \mathcal {T}\) (i.e., that are compatible—see Sect. 2—with a demand vector \(d_{t}\)); hereafter we refer to this set as an intra-period feasible set of periodt (constraints defining this set are described in detail in Sect. 5.2).

In the next step, consider a period transition \(j\in \mathcal {J},\) between periods k and t, i.e., \(j=(k,\,t)\) (\(\mathcal {J}\) denotes a set of period transitions—see Sect. 1). Pair of variables \((x_{ins}^{ajw}\in \mathbb {R}_{+},\hat{x}_{ext}^{ajw}\in \mathbb {R}_{+})\) counts the number of splitters of type \(w\in \mathcal {C}^{r},\) which need to be installed/extracted from access bundle node \(a\in \mathcal {N}^{BA},\) at the transition \(j\in \mathcal {J}\); thus, expressions \(x_{ins}^{ajw}=max(\,0,\;x^{atw}+\hat{x}^{atw}-x^{akw}-\hat{x}^{akw})\) and \(x_{ext}^{ajw}=max(\,0,\;x^{akw}+\hat{x}^{akw}-x^{atw}-\hat{x}^{atw})\) hold true. In the similar way, we introduce pairs \((x_{ins}^{djs},x_{ext}^{djs}),\;d\in \mathcal {N}^{BD},\,s\in \mathcal {C}^{r}\) and \((x_{ins}^{Cjr},x_{ext}^{Cjr}),\;c=n^{BC},\,r\in \mathcal {C}^{r}\) that represent, respectively, the number of splitters of type \(s\in \mathcal {C}^{r},\) installed/extracted from distribution bundle node \(d\in \mathcal {N}^{BD},\) and the number of splitters of type \(r\in \mathcal {C}^{r},\) installed/extracted from the central bundle node \(N^{BC}\). Continuing, we apply the same rule introducing pairs \((y_{ins}^{Cj},y_{ext}^{Cj})\) and \((z_{ins}^{Cj},z_{ext}^{Cj})\) to denote variables counting, respectively, OLT cards and OLT devices that need to be installed\extracted at transition j. Finally, we introduce variables \(b^{aj}\in \{0,1\}\), \(b^{dj}\in \{0,1\}\), and \(b^{Cj}\in \{0,1\}\) to indicate if a particular (either access \(a\in \mathcal {N}^{BA}\), distribution \(d\in \mathcal {N}^{BD}\) or central \(n^{BC}\)) bundle node, requires site-survey at transition \(j\in \mathcal {J}\). The site survey is necessary if any change of the nodal equipment is required in it (either installation or extraction). Having all this defined, we introduce the following vectors: For any period transition \(j\in \mathcal {J},\) we refer to vector (7) as the reconfiguration between configurations \(X^{k=a(j)}\) to \(X^{t=z(j)}.\)Then, we define set \(\mathscr {H}^{j}\)—we refer to it as inter-period feasible setat transitionj—containing every feasible reconfiguration between these configurations (detailed description of the set is presented in Sect. 5.4).

Now we can formulate the multi-period optimization problem in the following way: Observe that optimization problem (8) has a special structure—one can distinguish the intra-period specific part (we refer to it as intra-period problem) defined by constraints (8c and 8d), and the inter-period specific part (inter-period problem)—constituted by constraint (8e). The intra-period part has the associated objective function components representing the \(E\!\!-\!\!N\) (see Sect. 4.4.1) and \(I\!\!-\!\!N\) (see Sect. 4.4.2) elements of the NP role’s cost (8a). The inter-period part has only the objective component (8b) associated with the NP cost component (see Sect. 4.4.3). Constraints (8d) are just to indicate that the green-field deployment is considered, as the initial configuration \(X^{0}\) has no equipment installed. Although we do not exploit this special structure to reformulate the problem, we use it to structuralize the problem description—feasible intra-period polyhedron and the associated components of the objective function are described in Sects. 5.2 and 5.3; the same information related to the inter-period part can be found in Sects. 5.4 and 5.5

The feasible polyhedron \(\mathscr {X}^{t}\) (see Sect. 5.1) of a single time period \(t\in \mathcal {T}\setminus \{0\},\) is described by means of a number of constraints. To facilitate reading, we split these constraints into groups related to particular levels of the network hierarchy—central, distribution, and access. Taking into account the amount of variables and parameters necessary to formulate the problem we decided to introduce them in places when they are used (each time we refer a variable introduced in the preceding sections, we try to put a related cross-reference to its definition).

Formulating the intra-period problem, we assume the following:We start the description of constraints from the central node \(n^{BC}\) and continue downward the network hierarchy through distribution nodes \(d\in \mathcal {N}^{BD},\) down to access bundle nodes \(a\in \mathcal {N}^{BA}\).

The intra-period cost, see (8b) in formulation (8), depends solely on the single state \(\mathbf {X}^{t}\) and represents the cost of leasing resources at \(E\!\!-\!\!N\) and \(I\!\!-\!\!N\) interfaces (see Sect. 4.2) in this period. It is expressed by the following equations: We split the component into two parts related, respectively, to \(E\!\!-\!\!N\) and \(I\!\!-\!\!N\) business interfaces. First, at \(E\!\!-\!\!N\) interface, between the network provider and the equipment provider role (see Sect. 4.4), we consider costs of OLT devices and OLT cards (13a) as well as costs of optical splitters installed in the central (13b), distribution (13c), and access (13d) nodes. Second, at \(I\!\!-\!\!N\) interface, between the network provider and the passive infrastructure provider role, we cost the exploitation (by installed splitters) of cabinet’s space at the central (14a), distribution (14c), and access nodes (14e), as well as exploitation of PIP’s communication infrastructure—installation of OLT port (14b) and access to a trunk (14d) and distribution (14f) fibre paths.

This section is to describe polyhedrons \(\mathscr {H}^{j},\,j\in \mathcal {J},\) (see Sect. 5.1) that comprise admissible values for vectors \(\mathbf {H}^{j}\). We emphasis that we consider a green-field design—we identified the initial period \(t=0\) with no equipment installed in any node of the network—see constraint (8d). Similarly to the intra-period problem, constraints are grouped according to the level of the considered network they pertain to.

To facilitate description of the transition cost (8b) we introduce the following symbols related to the particular cost components. We emphasise that all these components are discrete in opposition to lease cost identified in the intra-period problem.Having the symbols defined, we can finally write the related cost expression for every transition \(j\in \mathcal {J},\) as follows: Expressions (18a) and (18b) represent costs of installation/extraction of OLT devices and OLT cards. Remind that in our research we assume only one type of devices and cards. Expressions (18c), (18e), and (18g) represent cost components related to installation/extraction of optical splitters of particular types in, respectively, central, distribution, and access bundle nodes. Finally, expressions (18d), (18f), and (18h) indicate costs of site-surveys for the central, distribution, and access bundle nodes.

The formulation presented in Sect. 5 takes advantage on numerous parameters related to the cost of particular resources as well as to the customer demand (i.e., the time, location, and the number of optical signals the network is expected to provide). As these parameters can have the crucial impact on the solution, their values must be set with the utmost consideration.

In the reported experiments, we consider a single trajectory of the customer arrival process. Information on the final (observed in the sixteenth period) values of optical signal demand in each access point we took from our previous research devoted to one-step network deployment, see [23], while the arrival times are modeled using a logistic function, which is commonly used to model diffusion of innovations [27]. The aggregated result of the estimation, i.e, the total number of optical signal the network is expected to deliver in particular time periods, is presented in Fig. 10.

The cost parameters define charges for one-period exploitation of product resources exposed at the \(E\!\!-\!\!N\) and \(I\!\!-\!\!N\) business interfaces (see Sect. 4) as well as OPEX expenditures carried out by the NP role on the period-by-period basis. To estimate one-period charges for the product resources offered at the \(E\!\!-\!\!N\) business interface, i.e., OLT devices, OLT cards, and optical splitters, we take advantage of price information from the equipment catalog (see Sect. 3.6). Under the assumption that the considered set of periods \({{\mathcal {T}}}\) covers the whole lifetime of passive resources (and, its is twice as long as the expected lifetime active ones), applying the linear depreciation model (see Sect. 4.3), zero salvage value, and neglecting any other (like insurance, etc.) cost, we obtain an estimation of one-period resource charge dividing the catalog price (the procurement cost) of the resource by the number of time-periods it is expected to last.

Estimation of one-period charges for product resources offered at the \(I\!\!-\!\!N\) interface—i.e., trunk and distribution fibre paths; cabinet space in central, distribution, and access sites; allocation of a individual OLT ports—is more complicated due to the fact that these resources are build on the top of shared supporting infrastructure, and there is no one-to-one relation between each of them and any individual network element. To cope with this issue, we apply a two-step approach—first, we estimate reasonable deployment costs of individual network elements; second, we project these costs onto product resources they support. To complete the first step, we solve, with advantage of methods and tools elaborated in our previous research [23], an auxiliary problem that aims at minimizing the total cost of one-step network deployment. To estimate one-period charge for trunk and distribution fibre paths, we take advantage of the assumption that there is defined at most one infrastructure trail between a pair of sites; the trail is then used for realization of every fibre paths between bundle nodes located in these sites. Consider an infrastructure link, e.g., duct or overhead line, \(l\in \mathcal {L}^{I}\). A solution of the auxiliary problem gives us the cost of preparation of this link, the number and types of optical cables that are hosted in-there, and the number of active fibres they carry. With this information, we can estimate a fraction of the link cost associated with a single active fibre. Consider a set of distribution fibre paths that enter bundle node \(a\in \mathcal {N}^{BA}.\) Every of these paths uses the same way to the same distribution bundle node, hence every path accumulates a per-active-fibre cost of every traversed infrastructure link. Finally, we receive the one-period charge dividing the result cost by the number of time periods in the considered network lifetime. Figure 11 presents a distribution of distribution fibre one-period charges; the x-axis is denominated in cost ranges, while the y-axis shows a fraction of bundle nodes that are exposed to fibre costs fitting to particular ranges (expressed in parts per thousand). The one-period charges for trunk fibre paths can be estimated in a similar way.

The one-period charge for space of sites, which host central, distribution, and access bundle node, that can be used to accommodate optical splitters is estimated in a similar way. For each site \(n\in \mathcal {N}^{IS},\) the solution of the auxiliary optimization problem identifies its type \(f\in \mathcal {C}^{f},\) (building, in-building cabinet, street-cabinet, etc.), preparation cost \(c^{f}\) as well as cost \(c^{e}\) and capacity \(\eta ^{e}\) of every hardware rack (cabinet) of type \(e\in \mathcal {C}^{e},\) installed in-there. In our cost-projection scheme, we simply divide the total site-related cost by the number of ports of splitters installed in its hosted cabinets (and by the number of time periods in the considered network lifetime) consequently receiving an estimation of the single-period per-port charge (distribution of the single-period distribution port charge, received for the considered case, is presented in Fig. 12).

The only site where active equipment (OLT and OLT cards) is installed is the central site \(n^{ISC}\); presence of the OLT device—requiring power supplying, cooling, security—generates additional OLT-specific costs that the PIP role has to project onto product resources. To cope with this cost, we introduce a product referred to as OLT port access that expresses per-period cost of the OLT site preparation; it is computed by dividing a cost of the central site preparation by the number of active ports installed in the OLT device according to the optimal solution of the auxiliary problem.

It has to be emphasized that the single-period optimization problem takes at its input the same price-list as used for costing products at the \(E\!\!-\!\!N\) business interface (equipment and labor costs were taken from [1]) and thus, prices at the \(E\!\!-\!\!N\) and \(I\!\!-\!\!N\) interfaces constitute a consistent costing system.

Estimation of transition-related costs is more complicated—its every component, i.e., site-survey cost and equipment installation/extraction cost has to be set in consistency with \(E\!\!-\!\!N\) /\(I\!\!-\!\!N\) system—as otherwise one of the configuration-related and transition-related components might overweight results. Provided the price-lists expose the real-world costs, it is relatively easy to estimate reasonable cost of site-surveys (this is the way we did it for our experiments). Estimation of the cost of equipment installation/extraction, as it reflects both cost of the related manual intervention as well as complexity of operation to be done, is vaguer. For our experiment, we set these parameters arbitrarily, with the assumption that installation/extraction incurs constant (proportional to the catalogue price of the involved equipment) and varying (proportional to the number of external ports of the equipment) cost components.

In our implementation, we exploit AMPL modeling language [15] with the CPLEX solver. Both input data and the optimal solution are kept in Microsoft SQL database; catalog data is stored in the Microsoft Excel files. The general scenario of the computations is presented in Algorithm 1. First (steps 1 and 2), data describing physical and logical organization of the network is read-in from the database and Excel files. In step 3, we solve the auxiliary one-step deployment problem that gives us an optimal solution and allows for estimation of the cost parameter values (step 4). Then, in step 5, we read (from the database) the trajectory of the customer arrival process that describes number of optical signals the network is required to provide in every particular access point in every time period (preparation of these data is briefly described in Sect. 6.1).

As considered problem instances were too large (in step 15 of the algorithm, the problem passed to Cplex contains about 240000 variables—4864 binary and 68802 integer—and 135000 constraints) to be approached directly—doing in that way we were not able to receive even root-node relaxations—we adopted the following solution. First, in steps 6—9, we solve, one-by-one, a set of auxiliary single-period problems. A solution of each auxiliary problem \(\mathcal {P}^{t},\,t\in \mathcal {T},\) is then used to fix the t-period-related variables of the original problem \(\mathcal {P}\). Taking advantage of the assumption that demand of each particular access point does not fall in consecutive time periods, to speed up the computations, we solve the auxiliary problems (in the reverse order of elements of set \(\mathcal {T}\)) starting from the last one with the highest demand. Consequently, an optimal solution of problem \(\mathcal {P}^{t}\) constitutes a feasible solution of problem \(\mathcal {P}^{t-1}\). This fact can be exploited by the used MIP solver in the warm-start procedure. In step 10, we receive a feasible solution of the complete problem \(\mathcal {P}\) with variables fixed according to the solutions of the auxiliary problems; these restrictions are gradually (period by period) removed in steps 11—14 (we observed that performing in this way, we receive faster progress than in the case all the restrictions are removed simultaneously). In step 15, we solve whole problem \(\mathcal {P}\) with all the previously introduced restrictions removed until optimality gap of 5% has been reached (thanks to the described tricks, Cplex is able to perform the warm-start with known integer solution). Finally, the received solution is written into the database and then undergoes the final processing.

In our preliminary experiments, we considered one network topology consisting of a single central bundle node, two distribution bundle nodes, and 304 access bundle nodes; the lifespan has been divided into sixteen time periods of equal length. All the reported experiments were conducted on Hewlett-Packard HP DL380 G9 server (Xeon 10C processors) using AMPL and CPLEX 12.7 accessing up-to eight logical processors and up-to 64 GB RAM. We present results for three selected experiments—referred hereafter to as: CC experiment, CTC experiment, and CC-PIR experiment—that shared the same customer arrival trajectory differing in terms of objective functions applied and/or presence of transition-related constraints. Every experiment was terminated when a 5% gap has been reached.

The CC experiment aims at minimizing solely configuration costs (see Sect. 5.3) and permitting for arbitrarily large reconfigurations between consecutive periods (see the stub release approach in Sect. 1). This experiment is expected to identify the lower bound for the sum of the configuration-related costs. Next, the CTC experiment minimizes every cost component, i.e., configuration (see Sect. 5.3) as well as reconfiguration related (see Sect. 5.5), providing a reference optimal solution that can be used to evaluate the quality of results provided by simpler, thus computationally more effective, approaches. Finally, the CC-PIR experiment, similarly to the CC experiment, considers configuration-related costs, and lacks the transition related cost component; however the scale of reconfigurations between consecutive periods is limited in this case by the assumption (enforced by additional constraints) that only pure incremental reconfigurations are admissible (see Sect. 1), i.e., that equipment, once installed, cannot be extracted from the network.

Raw results of each of the three experiments are presented in a dedicated table pair—(Tables 4, 5), (Tables 6, 7) and (Tables 8, 9) for, respectively CC, CTC, and CC-PIR experiments. The first table of a pair (denoted as part A) contains a subset of columns which is complemented by columns of the second table (part B). All these tables have 16 rows—each row is associated with a single time-period indicated in the first column, denoted by t. Columns h, hi, and hvi represent: total demand (required number of optical signals), total installed service capacity (number of tributary ports of connected access splitters), and total installed virtual service capacity (number of tributary ports of installed virtual access splitters—see Sect. 5), respectively. Next three column quadruples, i.e., (cs, csv, csI, csE), (ds, dsv, dI, dE), and (as, asv, asI, asE), expose: (i) total cost of installed splitters, (ii) installed virtual splitters, (iii) installation of new splitters, (iv) extraction of installed splitter in, respectively, central (c), distribution (d), and access (a) sites. Next two column triples—(o, oI, oE) and (oc, ocI, ocE)—show: (i) cost of installed equipment, (ii) installation cost, (iii) extraction cost associated with, respectively, OLT devices and OLT cards. The three columns—cS, dS, and aS—denote expenditures related to site survey to be executed in, respectively, central, distribution, and access site. Columns df and af show charges to be paid for exploitation of trunk and distribution fibre paths, then column op indicates charges for allocation of OLT ports. Finally, column triple (ocp, dcp, acp) shows costs associated with occupied cabinet space in central, distribution, and access sites.

Information presented in these tables, despite being detailed and illustrative, does not directly show general trends. To facilitate this, we prepared more synthetic Tables 10, 11, and 12 that present accumulated costs grouped into four categories. The first category, denoted as I–N, illustrates total charge for infrastructure resources (trunk and distribution fibres; central, distribution. and access cabinet space; central OLT port allocation) the NP role has to pay at the I–N interface. Second category of costs, denoted as E–N, describes accumulated charge the NP role pays at the E–N interface for leasing passive (optical splitters) and active (OLT devices and OLT cards) equipment. The third category, denoted by H, indicates accumulated costs of configuration transitions (installation/extraction of equipment and site surveys). Finally, the fourth category shows the accumulated total expenditures.

As we mention in the beginning of this section, we consider results of the CTC experiment as a reference solution for the remaining two. Following this way, we prepared Figs. 13 and 14 that illustrate relations between results of, respectively, the CC experiment and the CC-PIR experiments to the reference solution. Each figure shows four curves that refer to particular cost categories identified above; within each category, series of data points are referring to points returned by the CTC experiment.

Results illustrated in Figs. 13 and 14 lead to the following conclusions. First, results of the CC experiment (see Fig. 13) perfectly adhere to our expectations—the experiment gives the best infrastructure cost (c.a. 90% of infrastructure cost provided by CTC) and very costly reconfiguration (c.a. 230% w.r.t. the same cost of the reference) what leads to the total cost about 10% greater than in the reference case. Second, analysis of the CC-PIR experiment (in Fig. 14) indicates that the approach (no transition cost in the objective function, solely pure incremental reconfiguration) can lead to surprisingly good results. The real transition costs are generally at the level 10% below this of the reference solution, infrastructure costs (I–N and E–N categories) are a bit higher, still the total cost, in the long term run, approaches the optimal (reference) one. This suggest that the CC-PIR approach could be considered as a simpler, but still very attractive, alternative for the reference approach.