Design of Multi-Layer Networks

This completes our presentation of the restoration/protection design problems. Of course, many other valid variants of the presented problems can be considered. For example, such elements as modular link capacity, modular flows or single-path routing can be taken into account.

7.1.3. Design of Multi-Layer Networks

In this subsection we shall present selected problems on multi-layer design, involving more than one layer of resources (one layer of resources has been assumed in Subsections 7.1.1 and 7.1.2). In fact, we will consider the case of two resource layers plus the demand layer.

7.1.3.1. Nominal design of multi-layer networks

In this section we will present two nominal design problems for three-layer networks: a dimensioning problem and an allocation problem.

The following formulation is a counterpart of the simple single-layer design problem DP1-LP (7.1.1). Although the considered model actually involves only two layers of resources (Layers 1 and 2) we refer to it as a three-layer network, including the auxiliary Layer 3 used for modelling the demands. This allows for the unified interpretation of the constraints.

TLDP-LP (Three-Layer Design Problem - Linear Programme) indices

d=1,2,...,D demands (links of Layer 3)

j=1,2,...,m(d) allowable paths in Layer 2 for flows realizing demand d e=1,2,...,E links of Layer 2

k=1,2,...,n(e) allowable paths in Layer 1 for flows realizing link e g=1,2,...,G links of Layer 1

constants

h_d volume of demand d

aedj = 1 if link e of Layer 2 belongs to path j realizing demand d, 0 otherwise

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b_gek = 1 if link g of Layer 1belongs to path k realizing link e of Layer 2, 0 otherwise cg unit cost of link g of Layer 1

Because of the presence of two upper layers there are two sets of demand-flow constraints (7.1.19b and 7.1.19d), and because of two lower layers there are two sets of load-capacity constraints (7.1.19c and 7.1.19e). The rule is that the capacity of links of the upper layer are realized by means of the path flows in the neighbouring lower layer; this is expressed by the demand constraints. Also, both resource layers (Layers 1 and 2) are networks on their own, so the link capacity constraints must be obeyed in each of them.

link g = 1 with marginal cost ξ_g = 1 and capacity w_g

flow through link e cannot exceed its capacity y_e flow x₁₂

flow through link e cannot exceed its capacity y_e flow x₁₂

Problem TLDP-LP is illustrated in Figure 7.2 which shows that the volume h₁ of demand d = 1 between nodes 1 and 2 can be realized by means of two Layer 2 flows (direct flow x11 and flow x12 on path 1-4-3-5-2). Then the capacity of link e = 1 resulting from its load (the sum of all flows through the link) can be realized by means of two Layer 1 flows (flow z₁₁ on path 1-6-2 and flow z12 on path 1-4-3-5-2). The resulting loads of the Layer 1 links determine their capacities and hence the network cost.

As DP1-LP, Problem TLDP-LP can be easily and effectively solved using the generalized version of the shortest path allocation rule described in Section 7.1.1 for the former problem.

(In fact in can be generalized to arbitrary number of layers.)

In practice, certain requirements on links’ modularity in one of the layers, or in both layers can be imposed, as well as on the integral flows. The IP problem resulting from such full integrality requirements is obtained when the constraints

ΣdΣj a_edjx_dj ≤ Mye e=1,2,...,E (7.1.19f)

Σe Σk bgekzek ≤ Nug g=1,2,...,G (7.1.19g)

all variables are non-negative integers (7.1.19h)

are used. Above, M (Layer 1) and N (Layer 2) are link capacity modules. The integrality of variables makes the considered problems NP-complete.

The next generalization is the counterpart of the single-layer allocation problem AP1-LP (7.1.8) for the case of two layers of resources.

TLAP-LP (Three-Layer Allocation Problem - LP) constants

h_d volume of demand d

aedj = 1 if link e of Layer 2 belongs to path j realizing demand d, 0 otherwise bgek = 1 if link g of Layer 1belongs to path k realizing link e of Layer 2, 0 otherwise u_g capacity of link g

variables

x_dj continuous flow allocated to path j realizing volume of demand d ye continuous capacity of link e

zek continuous flow allocated to path k realizing capacity of link e constraints

Σj x_dj = h_d d=1,2,...,D (7.1.20a)

ΣdΣj a_edjx_dj≤ y_e e=1,2,...,E (7.1.20b)

Σk zek = ye e=1,2,...,E (7.1.20c)

Σe Σk bgekzek ≤ ug g=1,2,...,G. (7.1.20d)

Note that in TLAP-LP capacities of the Layer 1 links are fixed, whilst the Layer 2 links’

capacities are variables. TLAP-LP as an LP problem and can be solved accordingly. In general, there are instances of TLAP-LP with only bifurcated feasible solutions.

Observe that in the considered case, modularity of the link capacity variables in Layer 2 can be required (and possibly integrality of the flow variables in both layers). The IP problem resulting from such full integrality requirements is obtained when the constraint

ΣdΣj δ_edjx_dj ≤ Mye e=1,2,...,E. (7.1.20e)

is used.

7.1.3.2. Restoration design for three-layer networks

In this paragraph we present an example of a restoration design three-layer problem. The concerns designing a three-layer network robust to failures, where flows of Layers 1 and 2 are assumed to be reconfigurable, and the reconfiguration is unrestricted. As will become clear in a while, this assumption implies that the capacities of the links of the upper layer (Layer 2) are flexible, and in general situation-dependent. Links of the lowermost Layer 1 are not flexible: what can only happen is that a part or entire of their nominal capacity can be lost in a failure situation.

TLRDP-LP (Three-Layer Restoration Design Problem - LP) indices as in TLDP-LP (7.1.19), and

s=1,2,...,S failure-demand situations (including the nominal state) constants

hds volume of demand d in situation s

a_edj = 1 if link e of Layer 2 belongs to path j realizing demand d, 0 otherwise

b_gek = 1 if link g of Layer 1 belongs to path k realizing link e of Layer 2, 0 otherwise α_gs fractional availability coefficient of link g of Layer 1 in situation s (0 ≤ αgs ≤ 1) cg unit cost of link g of Layer 1

variables (all variables are continuous and non-negative) xdjs flow allocated to path j of demand d in situation s yes capacity of link e in situation s

z_eks flow allocated to path k realizing capacity of link e in situation s ug capacity of link g

objective

minimize C(u)= Σg cgug (7.1.21a)

constraints

Attention: This is not an ITU publication made available to the public, but an internal ITU Document intended only for use by the Member States of the ITU and by its Sector Members and their respective staff and collaborators in their ITU related

Σj xdjs = hds d=1,2,...,D s=1,2,...,S (7.1.21b)

ΣdΣj aedjxdjs ≤ yes e=1,2,...,E s=1,2,...,S (7.1.21c)

Σk zeks = yes e=1,2,...,E s=1,2,...,S (7.1.21d)

Σe Σk bgekzeks ≤ αgsug g=1,2,...,G s=1,2,...,S. (7.1.21e)

The demand-flow constraints assure that the situation-dependent demand volumes h_ds (which are equal to the capacities of the corresponding Layer 3 links) are realized by the situation-dependent flows in Layer 2 (constraints (7.1.21b)), and that the demand imposed on Layer 1 and specified by the Layer 2 links’ capacities y_es is realized by the situation-dependent Layer 1 flows (constraints (7.1.21d)). The load-capacity constraints (7.1.21c) and (7.1.21e) take care about the feasibility of flows in Layers 2 and 1, respectively, i.e. they assure that the situation-dependent loads of links do not exceed their capacities. Observe that although for any optimal solution to TLRDP-LP (which is an LP problem) in constraints (7.1.21c) equalities will always hold, this is not the case for constraints (7.1.21e). In the latter case come links may not be saturated in some situations even for an optimal solution.

Problem TLRDP-LP is illustrated in Figure 7.3. You can notice that the volume hds of demand d in situation s can be realized by means of two flows xd1s and xd2s in Layer 2. In the

considered situation two of the Layer 1 links fail totally, so the capacity y_es can be realized only by flow z_e3s in Layer 1 since the two remaining flows, z_e1s and z_e2s, use failed links.

As before, many other valid variants of the problems presented in this subsection can be considered. Besides such elements as modular link capacity, modular flows or single-path routing, the extensions to more layers, the use of different restoration mechanisms, etc., can be taken into account. demand d with volume h_ds

link g with marginal demand d with volume h_ds