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Table 1 Glossary of symbols and functions related to the optimization model

From: Adaptive placement & chaining of virtual network functions with NFV-PEAR

Symbol Formal specification Definition
Superscripts and subscripts
P   Physical infrastructure entity
S   SFC entity
Sets and set objects
\(p \in \mathcal {P}\) \(p = \left (N^{P}, L^{P}, E^{P}\right)\) Physical infrastructure instance, composed of nodes and links
iNP NP={i | i is a N-PoP} Network points of presence (N-PoPs) in the physical infrastructure
(i,j)LP \(L^{P} = \left \{(i,j)\,\vert \, i,j \in N^{P}\right \}\) Unidirectional links connecting pairs of N-PoPs i and j
i, rEP \(E^{P} = \{\langle i, r \rangle \,\vert \, i \in N^{P} \land r \in \mathbb {N}^{*}\}\) Identifier r of the actual location of N-PoP i
\(m \in \mathcal {F}\) \(\mathcal {F} = \{m\,\vert \,m\,\text {is a function type}\,\} \) Types of virtual network functions available
\(j \in \mathcal {U}_{m}\) \(\mathcal {U}_{m} = \{j\,\vert \,j\,\text {is an instance of}\, m \in \mathcal {F}\,\}\) Instances of virtual network function m available
\(\mathcal {Q}\)   Set of Service function chaining (SFC) requests to be deployed
\(q \in \mathcal {Q}\) \(q = \left (N^{S}_{q}, L^{S}_{q}, E^{S}_{q}\right)\) A single SFC request, composed of VNFs and their chainings
\(i \in N^{S}_{q}\) NS={i | i is a VNF instance or endpoint} SFC nodes (either a network function instance or an endpoint)
\((i,j) \in L^{S}_{q}\) \(L^{S}_{q} = \left \{(i,j)\,\vert \, i,j \in N^{S}\right \}\) Unidirectional links connecting SFC nodes
\(\langle i, r \rangle \in E_{q}^{S}\) \(E^{S}_{q} = \{\langle i, r \rangle \,\vert \, i \in N^{S} \land r \in \mathbb {N}^{*}\}\) Required physical location r of SFC endpoint i
\(H^{S}_{q}\)   Distinct forwarding paths (subgraphs) contained in a given SFC q
\(H^{H}_{q,i} \in H^{S}_{q}\) \(H^{H}_{q,i} = \left (N^{H}_{q,i}, L^{H}_{q,i}\right)\) A possible subgraph (with two endpoints only) of SFC q
\(N^{H}_{q,i}\) \(N^{H}_{q,i} \subseteq N^{S}_{q}\) VNFs that compose the SFC subgraph \(H^{H}_{q,i}\)
\(L^{H}_{q,i}\) \(L^{H}_{q,i} \subseteq L^{S}_{q}\) Links that compose the SFC subgraph \(H^{H}_{q,i}\)
\(y^{\prime }_{i,m,j}\)   Denotes whether there was a previous VNF placement
\(\delta ^{\prime }_{i,q,j}\)   Denotes whether there was a previous assignment of flow to VNF
\(\lambda ^{\prime }_{i,j,q,k,l}\)   Denotes whether there was a previous flow chaining
Parameters
ϕ \(\phi \in \mathbb {R}_{+}, \phi \geq 0\) Percentage of capacity of VNFs that can be violated.
α, β, and γ   Weight factors of the ILP model.
\(C^{P}_{i} \in \mathbb {R}_{+}\)   Computing power capacity of N-PoP i
\(B^{P}_{i,j} \in \mathbb {R}_{+}\)   One-way link bandwidth between N-PoPs i and j
\(D^{P}_{i,j} \in \mathbb {R}_{+}\)   One-way link delay between N-PoPs i and j
\(C^{S}_{q,i} \in \mathbb {R}_{+}\)   Computing power required for network function i of SFC q
\(B^{S}_{q,i,j} \in \mathbb {R}_{+}\)   One-way link bandwidth required between nodes i and j of SFC q
\(D^{S}_{q} \in \mathbb {R}_{+}\)   Maximum tolerable end-to-end delay of SFC q
Functions
\(f^{type}_{m}\) \(f^{type} : N^{P} \cup N^{S} \rightarrow \mathcal {F}\) Type of some given virtual network function (VNF)
\(f^{cpu}_{m,j}\) \(f^{cpu} : (\mathcal {F} \times \mathcal {U}_{m}) \rightarrow \mathbb {R}_{+}\) Computing power associated to instance j of VNF type m
\(f^{delay}_{m}\) \(f^{delay} : \mathcal {F} \rightarrow \mathbb {R}_{+}\) Processing delay associated to VNF type m
Variables
yi,m,jY \(Y = \{\,y_{i,m,j}\,,\,\forall \,i \in N^{P}, m \in \mathcal {F}, j \in \mathcal {U}_{m}\,\}\) VNF placement
δi,q,jΔ \(\Delta = \left \{\,\delta _{i,q,j}\,,\,\forall \,i \in N^{P}, q \in \mathcal {Q}, j \in N_{q}^{S}\,\right \}\) Assignment of required network functions/endpoints
λi,j,q,k,lΛ \(\Lambda = \left \{\,\lambda _{i,j,q,k,l}\,,\,\forall \,(i,j) \in L^{P}, q \in \mathcal {Q}, (k,l) \in L_{q}^{S}\,\right \}\) Chaining allocation
\(\overline {y}_{i,m,j} \in \overline {Y}\) \(\overline {Y} = \{\,\overline {y}_{i,m,j}\,,\,\forall \,i \in N^{P}, m \in \mathcal {F}, j \in \mathcal {U}_{m}\,\}\) Denotes whether an VNF placement changes
\(\overline {\delta }_{i,q,j} \in \overline {\Delta }\) \(\overline {\Delta } = \left \{\,\overline {\delta }_{i,q,j}\,,\,\forall \,i \in N^{P}, q \in \mathcal {Q}, j \in N_{q}^{S}\,\right \}\) Denotes whether an assignment of flow to VNF changes
\(\overline {\lambda }_{i,j,q,k,l} \in \overline {\Lambda }\) \(\overline {\Lambda } = \left \{\,\overline {\lambda }_{i,j,q,k,l}\,,\,\forall \,(i,j) \in L^{P}, q \in \mathcal {Q}, (k,l) \in L_{q}^{S}\,\right \}\) Denotes whether a flow chaining changes