Notations

The Evolved Packet System (EPS) architecture depicted in Figure 3.1 presents the different elements of the LTE network for the flat access network and the core network; it also shows the links between those elements. In fact, Figure 3.1 represents the graphical representation of the notation which is composed of sets, decision variables, traffic variables, and cost parameters.

Figure 3.1: Graphical representation of the notation

3.3.1.1 Sets

 _Β, the set of eNodeBs that are already installed in different locations.

- µ_b^c, the traffic of type c∈C for an eNB∈B.

 C, the set of traffic characteristics such that C={c₀,c₁,...,c₄} and where:

- c : Number of subscribers. ₀

- c : Busy Hour Session Attempt (BHSA). ₁ - c : Traffic bandwidth (Mbps). ₂

Access Network Evolved UTRAN

(E-UTRAN)

Core Network Evolved Packet Core

(EPC)

- c : Number of attached subscribers. ₃

- c : Simultaneous Evolved Packet System Bearers (EPSB). ₄

 M₀₁,M₀₂,M₁₂,M₁₃,M₂₄, and M₄₅ are respectively the sets of links that can be used to connect eNBs with MME, eNBs with S-GW, MME with S-GW, MME with HSS, S-GW with P-GW, and P-GW with PCRF.

- w^m: Capacity of the link or the interface of type m. For example, link of type Fast Ethernet or Gigabit Ethernet.

 S₁,S₂,S₃,S₄, and S are respectively the sets of potential sites to install the MME, S-₅ GW, HSS, P-GW, and PCRF.

 T₁,T₂,T₃,T₄and T are respectively the set of types of network elements (i.e. MME, S-₅ GW, HSS,P-GW, and PCRF) where:

- SUB^t: Subscribers’ capacity for a node of type t.

- BHSA^t: BHSA capacity for a node of type t.

- β : Switching fabric capacity in Mbps for a node of type t. ^t

- EPSB^t : Simultaneous Evolved Packet System Bearers (EPSB) capacity for a node of type t.

- ASUB^t: Attached Subscribers’ capacity for a node of type t.

- SIG^t: Signaling capacity for a node of type t.

- EPSBW : Throughput per bearer.

- SUBW : Throughput per subscriber.

- TPSPB: Transactions per second per bearer.

- TPS: Transactions per second per subscriber.

- η₀^t,η₁^t,η₂^t,η₃^t,η₄^t,η₅^t,η₆^t,η₇^t and η are respectively the maximum number of interfaces ₈^t that can be installed in a node of type t to connect the eNB to the MME (S-GW), the MME to the eNB, the S-GW to the eNB, the MME to the HSS (S-GW), the HSS to the MME, the S-GW to the MME (P-GW), the P-GW to the S-GW, the P-GW to the PCRF, and the PCRF to the P-GW.

3.3.1.2 Decision Variables

Links

 ν₀₁^bi, a binary variable such that ν₀₁^bi =1if and only if the eNB (b∈Β) is connected to the MME installed at site i∈ . S₁

 ν , the number of links of type ₀₁^bim m∈M₀₁ connecting the eNB (b∈Β) to the MME installed at sitei∈ . S₁

 ν , a binary variable such that ₀₂^bj ν₀₂^bj =1 if and only if the eNB (b∈Β) is connected to the S-GW installed at site j∈ . S₂

 ν , the number of links of type ₀₂^bjm m∈M₀₂ connecting the eNB (b∈Β) to the S-GW installed at site j∈ . S₂

 ν , a binary variable such that ₁₂^ij ν₁₂^ij =1 if and only if the MME installed at site i∈ S₁ is connected to the S-GW installed at site j∈ . S₂

 ν₁₂^ijm, the number of links of type m∈M₁₂ connecting the MME installed at site i∈ S₁ is connected to the S-GW installed at site j∈ . S₂

 ν , a binary variable such that ₁₃^ik ν₁₃^ik =1 if and only if the MME installed at site i∈ S₁ is connected to the HSS installed at site k∈ . S₃

 ν , the number of links of type ₁₃^ikm m∈M₁₃ connecting the MME installed at site S1

i∈ to the HSS installed at sitek∈ . S₃

 ν₂₄^jl, a binary variable such that ν₂₄^jl =1 if and only if the S-GW installed at site S2

j∈ is connected to the P-GW installed at site l∈ . S₄

 ν , the number of links of type ₂₄^jlm m∈M₂₄ connecting the S-GW installed at site S2

j∈ to the P-GW installed at site l∈ . S₄

 ν , a binary variable such that ₄₅^lp ν₄₅^lp =1 if and only if the P-GW installed at site S4

l∈ is connected to the PCRF installed at sitep∈ . S₅

 ν , the number of links of type ₄₅^lpm m∈M₄₅ connecting the P-GW installed at site S4

l∈ to the PCRF installed at site p∈ . S₅

Nodes

 χ₁^it, a binary variable such that χ₁^it =1 if and only if a MME of type t∈ is installed Τ₁ at site i∈S¹.

 χ , a binary variable such that ₂^jt χ₂^jt =1 if and only if a S-GW of type t∈ is Τ₂ installed at site j∈ . S₂

 χ^3kt, a binary variable such that χ₃^kt =1 if and only if a HSS of type t∈ is installed Τ₃ at site k∈S₃.

 χ^4lt, a binary variable such that χ₄^lt =1 if and only if a P-GW of type t∈ is Τ₄ installed at site l∈ . S₄

 χ , a binary variable such that ₅^pt χ₅^pt =1if and only if a PCRF of type t∈ is Τ₅ installed at site p∈ . S₅

3.3.1.3 Traffic Variables

The traffic flow is asymmetric in both directions on a link between two network elements. In other words, the uplink and the downlink don’t have the same amount of traffic. The downlink is considered in this thesis since most of the traffic is sent on the downlink direction.

 f₀₁^cbi, the traffic of type c∈ on the link from the eNB C b∈ to a MME installed at B site i∈S₁.

 f₀₂^cbj, the traffic of type c∈ on the link from the eNB C b∈ to a S-GW installed at B site j∈S₂.

 f₁₂^cij, the traffic of type c∈ on the link from the MME installed at site C i∈ to a S-S₁ GW installed at site j∈ . S₂

 f₁₃^cik, the traffic of type c∈ on the link from the MME installed at site C i∈ to a S₁ HSS installed at site k∈ . S₃

 f₂₄^cjl, the traffic of type c∈ on the link from the S-GW installed at site C j∈ to a S₂ P-GW installed at site l∈ . S₄

 f₄₅^clp, the traffic of type c∈ on the link from the P-GW installed at site C l∈ to a S₄ PCRF installed at site p∈ . S₅

3.3.1.4 Cost Parameters

Links

 a₀₁^bim, the link and interface costs (including installation cost) for connecting an eNB installed at site b∈ to a MME installed at site B i∈ through a link and interface of S₁ type m∈M₀₁.

 a₀₂^bjm, the link and interface costs (including installation cost) for connecting an eNB installed at site b∈ to a S-GW installed at siteB j∈ through a link and interface of S₂ type m∈M₀₂.

 a₁₂^ijm, the link and interface costs (including installation cost) for connecting a MME installed at site i∈ to a S-GW installed at siteS₁ j∈ through a link and interface of S₂ type m∈M₁₂.

 a₁₃^ikm, the link and interface costs (including installation cost) for connecting a MME installed at site i∈ to a HSS installed at site S₁ k∈ through a link and interface of S₃ type m∈M₁₃.

 a₂₄^jlm, the link and interface costs (including installation cost) for connecting a S-GW installed at site j∈ to a P-GW installed at site S₂ l∈ through a link and interface of S₄ type m∈M₂₄.

 a₄₅^lpm, the link and interface costs (including installation cost) for connecting a P-GW installed at site l∈ to a PCRF installed at siteS₄ p∈ through a link and interface of S₅

The total cost of the network consists of two parts: the cost of the links and interfaces as well as the cost of the nodes. The cost of the links and interfaces, denoted by C is given by the _L