# Offline Joint Network and Computational Resource Allocation for Energy-Efficient 5G and beyond Networks

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## Abstract

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## 1. Introduction

**only**when the selected routed path passes through the VNFs comprising the specific SFC in exactly the order specified; for example, if an SFC is described by the ordered sequence NAT→Firewall→NAT, then all packets belonging to this SFC must be processed by only these three VNFs in the specified order.

#### 1.1. Related Work

#### 1.2. Research Gap

#### 1.3. Our Contributions

- We formulate a concrete joint user association, traffic routing, and VNF placement optimization problem targeting at the
**overall**E2E network performance optimization, with minimal assumptions, that accounts for both communication and computation resources in**all**segments of a mobile network (i.e., AN, edge, and core) and explicitly account for the AN segment, typically ignored in the literature. - Due to the NP-hardness of the resulting problem, we also propose a heuristic algorithm, evaluate its performance via simulations and demonstrate its superior performance compared with other State-of-the-Art (SoA) algorithms. The proposed solutions can be applied for internode network optimization, in conjunction with approaches targeting intranode optimization for maximum performance.
- Expanding upon our previous work in [29], we also describe the proposed orchestration solution, which, integrated with the proposed algorithm, enables the automated and optimized provisioning and configuration of heterogeneous computational and network resources across
**all**network segments, targeting the orchestration of virtualized services according to the expected performance requirements and the specified SFC.

## 2. System Model and Problem Statement

**not**included in $\mathcal{V}$ but rather in $\mathcal{J}$ (we also define $\tilde{\mathcal{V}}\stackrel{\Delta}{=}\mathcal{V}\cup \mathcal{J}$). Each UE $j\in \mathcal{J}$ connects to a BS $a\in $ ${\mathcal{A}}^{\left(j\right)}\subseteq $ $\mathcal{V}$, where the dependence on j captures the typical signal level-based association rules. We define $\mathcal{A}\stackrel{\Delta}{=}{\cup}_{j\in \mathcal{J}}{\mathcal{A}}^{\left(j\right)}$ as the set of all BSs and ${\mathcal{S}}^{\left(a\right)}\stackrel{\Delta}{=}\left(\right)open="\{"\; close="\}">j:a\in {\mathcal{A}}^{\left(j\right)}$ as the set of UEs which

**may**be served by BS $a\in \mathcal{A}$. We focus on Downlink (DL) and consider the set of DL AN links ${\mathcal{E}}_{AN}$ $\stackrel{\Delta}{=}$ $\left(\right)$ with $\tilde{\mathcal{E}}$ $\stackrel{\Delta}{=}$ $\mathcal{E}\cup {\mathcal{E}}_{AN}$. To capture the underlying Physical layer constraints in the air interface, we also assume that each BS $a\in \mathcal{A}$ has a

**maximum**number of ${\overline{N}}_{a}^{\left(RB\right)}$ Resource Blocks (RBs) to allocate to the UEs in ${\mathcal{S}}^{\left(a\right)}$.

**complete software stacks**) that can be deployed and allow for multiple

**instances**of a given VNF in the same or different nodes depending on network traffic. Each VNF $f\in \mathcal{F}$ is described by the tuple $f\stackrel{\Delta}{=}({T}_{f},\text{}{\pi}_{f},\text{}{w}_{f},\text{}{\tau}_{f})$, where ${T}_{f}$ is an identifier of the VNF’s functionality (e.g., NAT etc.), ${\pi}_{f}>0$ is the data processing capacity of the VNF (in Mbps), ${w}_{f}>0$ is the amount of CPU resources (in GFLOPS) required for the VNF’s operation, and ${\tau}_{f}>0$ is the data processing

**delay**experienced by an individual data packet as it passes through the VNF. To ensure proper service endpoints, we also introduce the set ${\mathcal{F}}_{dum}\stackrel{\Delta}{=}\{{f}_{dum1},\text{}{f}_{dum2}\}$ of two “dummy” VNFs and include it into $\mathcal{F}$. The VNFs $f\in {\mathcal{F}}_{dum}\subseteq \mathcal{F}$ are characterized by ${\pi}_{f}=\infty $, ${w}_{f}=0$, ${\tau}_{f}=0$, which implies that the “dummy” VNFs work transparently, w.r.t., our model.

**ordered**sequence $\rho \stackrel{\Delta}{=}\langle {f}_{dum1},\text{}{f}_{1}^{\left(\rho \right)},\dots ,$ ${f}_{{N}_{\rho}}^{\left(\rho \right)},\text{}{f}_{dum2}\rangle $, where ${N}_{\rho}$ is the number of non-dummy VNFs in $\rho $ and ${f}_{i}^{\left(\rho \right)}\in \mathcal{F}\setminus {\mathcal{F}}_{dum}$. The traffic of $\rho $ is

**properly served only if**it passes through the VNFs in $\rho $

**exactly matching the specified order**. We also write $f\u21dd\rho $ to state that VNF f is contained in $\rho $ and define ${\mathcal{R}}_{\rho}\stackrel{\Delta}{=}\{f\in \mathcal{F}:f\u21dd\rho \}$. An equivalent description for $\rho $ is via a

**directed**graph ${\mathcal{G}}^{\left(\rho \right)}({\mathcal{V}}^{\left(\rho \right)},\text{}{\mathcal{E}}^{(\rho})$ with the virtual node set ${\mathcal{V}}^{\left(\rho \right)}$ $\stackrel{\Delta}{=}\{{f}_{dum1},\text{}{f}_{1}^{\left(\rho \right)},\text{}{f}_{2}^{\left(\rho \right)},\dots ,\text{}{f}_{{N}_{\rho}}^{\left(\rho \right)},\text{}{f}_{dum2}\}$ (i.e., each VNF $f\u21dd\rho $ is a node of ${\mathcal{V}}^{\left(\rho \right)}$) and the virtual edge set ${\mathcal{E}}^{\left(\rho \right)}=\left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{f}_{dum1},\text{}{f}_{1}^{\left(\rho \right)}\left(\right)open="("\; close=")">{f}_{{N}_{\rho}}^{\left(\rho \right)},\text{}{f}_{dum2}$ describing the relative order of the VNFs (see bottom part of Figure 2 for an example of such a virtual graph). For any virtual edge ${e}^{\prime}\in {\mathcal{E}}^{\left(\rho \right)}$, we denote with $h\left({e}^{\prime}\right)$, $t\left({e}^{\prime}\right)\in \mathcal{F}$ the respective VNFs at the head and tail of ${e}^{\prime}$.

**properly serve**the requested SFCs, as well as the E2E routing path for each request, so that the total system power consumption is minimized (equivalently, the energy efficiency in bits/Joule over all requests is maximized). Computation of the selected routing path also includes determination of the BS to which each UE attaches to.

#### 2.1. Power Consumption Model and Problem Formulation

**physical**link $e=(u,\text{}v)\in \mathcal{E}$ belongs to the

**physical**path in $\mathcal{G}$ onto which the

**virtual**edge ${e}^{\prime}\in {\mathcal{E}}^{\left({\rho}_{q}\right)}$ is mapped for SFC ${\rho}_{q}$. Finally, ${N}_{f,\tilde{y}}$ is the number of

**instances**of VNF f deployed on node $\tilde{y}$.

**only**when needed. Specifically, collocated CPU resources at node y are employed only when y

**actually**runs deployed VNFs, as captured by the Boolean indicator variable ${\xi}_{y}$. Hence, the power consumed by CPU processing at node y is given by

**actually**carries traffic, in

**either**link direction, for any request. To examine whether link $(n,\text{}m)\in {\mathcal{E}}_{fi}$ carries any traffic in the

**specific**direction from n to m, we introduce the Boolean variable ${w}_{n,m}$, which implies that it must hold

**actually**used. There exist certain consistency relations between these variables as shown in (3) below, where ${C}_{1}>0$ is a sufficiently large constant.

**active**fiber links of the switch, with ${P}_{port}$ being the power consumed by each active port [15].

**piecewise-linear**function accounting for the nonlinear dependence between achieved throughput and power expenditure. Finally, ${\chi}_{e}$ is a Boolean indicator variable for whether link e is

**actually**used to serve any traffic; if not, the link’s transceiver is turned off.

**all**of the above control variables appear as linear terms in the constraints (2), (3), (8)–(15) and as linear (in (1), (4), (6), (7)) or piecewise linear terms (in (5)) in the components of (16) comprising the objective function of (17), combined with the fact that the control variables take only integer or Boolean values. Note that a piecewise linear objective function can be readily converted into a purely linear form by introducing auxiliary variables (see Section 4.3.1 of [30]). Due to the high complexity of solving (17), we next propose a low-complexity heuristic algorithm and use (17) as a yardstick against which the heuristic’s performance is evaluated. For the reader’s convenience, we have collected all introduced notation into Table 1 at the end of the paper.

## 3. Proposed Energy-Efficient Vnf Placement, Traffic Routing, and User Association (Hero)

## 4. Interaction with Orchestration Framework

## 5. Evaluation Methodology and Simulation Scenarios

- Holu: This algorithm first performed the VNF placement based on the node centrality (closeness) and the CPU utilization of computing nodes and then decided upon traffic routing targeting the minimization of power consumption, while satisfying the E2E service delay constraint.
- BCSP: It considered node centrality (betweeness) for the VNF placement and the shortest-path in terms of delay for routing, while meeting the E2E service delay constraint.

## 6. Simulation Results and Discussion

## 7. Conclusions

**online**joint computational and communications resource allocation policy), the computational complexity of the heuristic may still be too high to properly adapt to the small time scale of request arrivals. To this end, in the future, we plan to explore learning-based techniques (inspired from approximate reinforcement learning) to construct low-complexity online energy-efficient resource allocation algorithms.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

3GPP | 3rd Generation Partnership Project |

5G | Fifth Generation |

5G-NR | 5G-New Radio |

5G-PPP | 5G Infrastructure Public Private Partnership |

6G | Sixth Generation |

B5G | Beyond Fifth Generation |

BS | Base Station |

AN | Access Network |

CAPEX | Capital Expenditure |

CPU | Central Processing Unit |

C-RAN | Centralized-Radio Access Network |

DL | Downlink |

E2E | End-to-end |

eMBB | Enhanced Mobile Broadband |

ETSI | European Telecommunications Standards Institute |

FPGA | Field Programmable Gate Array |

FW | Firewall |

gNB | gNodeB |

GPU | Graphics Processing Unit |

ICT | Information and Communications Technology |

IDPS | Intrusion Detection/Prevention System |

LCM | Lifecycle Management |

MANO | Management and Orchestration |

MEC | Multi-Access Edge Computing |

MILP | Mixed Integer Linear Program |

mMTC | Massive Machine Type Communication |

NAT | Network Address Translation |

NFV | Network Function Virtualization |

OPEX | Operational Expenditure |

OSI | Open System Interconnection |

QoS | Quality of Service |

RAN | Radio Access Network |

RB | Resource Block |

RF | Radio Frequency |

SINR | Signal to Interference plus Noise Ratio |

State-of-the-Art | SoA |

SC | Small Cell |

SDN | Software Defined Networking |

SDO | Standards Developing Organization |

SFC | Service Function Chain |

TM | Traffic Monitor |

uRLLC | Ultra Reliable Low Latency Communication |

UE | User Equipment |

UHD | Ultra-High Definition |

VM | Virtual Machine |

VNF | Virtual Network Function |

VOC | Video Optimization Controller |

WOC | WAN Optimization Controller |

## References

- Ali, J.; Roh, B.H.; Lee, S. QoS improvement with an optimum controller selection for software-defined networks. PLoS ONE
**2019**, 14, e0217631. [Google Scholar] [CrossRef] [PubMed] - Ali, J.; Roh, B.H. Quality of Service improvement with optimal software-defined networking controller and control plane clustering. Comput. Mater. Contin.
**2021**, 67, 849–875. [Google Scholar] [CrossRef] - Barakabitze, A.; Ahmad, A.; Mijumbi, R.; Hines, A. 5G network slicing using SDN and NFV: A survey of taxonomy, architectures and future challenges. Comput. Netw.
**2020**, 167, 106984. [Google Scholar] [CrossRef] - Abbas, N.; Zhang, Y.; Taherkordi, A.; Skeie, T. Mobile Edge Computing: A survey. IEEE Internet Things J.
**2018**, 5, 450–465. [Google Scholar] [CrossRef] [Green Version] - Tang, L.; Yang, H.; Ma, R.; Hu, L.; Wang, W.; Chen, Q. Queue-aware dynamic placement of virtual network functions in 5G access network. IEEE Access
**2018**, 6, 44291–44305. [Google Scholar] [CrossRef] - Laghrissi, A.; Taleb, T. A survey on the placement of virtual resources and Virtual Network Functions. IEEE Commun. Surv. Tutor.
**2019**, 1, 1409–1434. [Google Scholar] [CrossRef] [Green Version] - Liu, J.; Lu, W.; Zhou, F.; Lu, P.; Zhu, Z. On dynamic Service Function Chain deployment and readjustment. IEEE Trans. Netw. Serv. Manag.
**2017**, 14, 543–553. [Google Scholar] [CrossRef] - Dong, L.; da Fonseca, N.; Zhu, Z. Application-Driven provisioning of Service Function Chains over heterogeneous NFV platforms. IEEE Trans. Netw. Serv. Manag.
**2021**, 18, 3037–3048. [Google Scholar] [CrossRef] - Cao, H.; Du, J.; Zhao, H.; Luo, D.; Kumar, N.; Yang, L.; Yu, F. Resource-Ability assisted Service Function Chain embedding and scheduling for 6G networks With virtualization. IEEE Trans. Veh. Technol.
**2021**, 70, 3846–3859. [Google Scholar] [CrossRef] - Cudak, M.; Ghosh, A.; Ghosh, A.; Andrews, J. Integrated access and backhaul: A key enabler for 5G millimeter-wave deployments. IEEE Commun. Mag.
**2021**, 59, 88–94. [Google Scholar] [CrossRef] - Ghribi, C.; Mechtri, M.; Zeghlache, D. A dynamic programming algorithm for joint VNF placement and chaining. In Proceedings of the 2016 ACM Workshop on Cloud-Assisted Networking, Irvine, CA, USA, 12 December 2016. [Google Scholar]
- Ma, Y.; Liang, W.; Huang, M.; Xu, W.; Guo, S. Virtual Network Function service provisioning in MEC via trading off the usages between computing and communication resources. (early access). IEEE Trans. Cloud Comput.
**2020**. [Google Scholar] [CrossRef] - Soualah, O.; Mechtri, M.; Ghribi, C.; Zeghlache, D. Energy efficient algorithm for VNF placement and chaining. In Proceedings of the 2017 17th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing (CCGRID), Madrid, Spain, 14–17 May 2017. [Google Scholar]
- Ayoubi, S.; Sebbah, S.; Assi, C. A logic-based Benders decomposition approach for the VNF assignment problem. IEEE Trans. Cloud Comput.
**2019**, 7, 894–906. [Google Scholar] [CrossRef] - Varasteh, A.; Madiwalar, B.; van Bemten, A.; Kellerer, W.; Mas–Machuca, C. Holu: Power-aware and delay-constrained VNF placement and chaining. IEEE Trans. Netw. Serv. Manag.
**2021**, 18, 1524–1539. [Google Scholar] [CrossRef] - Tajiki, M.; Salsano, S.; Chiaraviglio, L.; Shojafar, M.; Akbari, B. Joint energy efficient and QoS-aware path allocation and VNF placement for Service Function Chaining. IEEE Trans. Netw. Serv. Manag.
**2019**, 16, 374–388. [Google Scholar] [CrossRef] [Green Version] - Dietrich, D.; Papagianni, C.; Papadimitriou, P.; Baras, J. Network function placement on virtualized cellular cores. In Proceedings of the 2017 9th International Conference on Communication Systems and Networks (COMSNETS), Bangalore, India, 4–8 January 2017. [Google Scholar]
- Yang, S.; Li, F.; Trajanovski, S.; Chen, X.; Wang, Y.; Fu, X. Delay-aware Virtual Network Function placement and routing in edge clouds. IEEE Trans. Mobile Comput.
**2021**, 20, 445–459. [Google Scholar] [CrossRef] [Green Version] - Spinelli, F.; Mancuso, V. Toward enabled industrial verticals in 5G: A survey on MEC-based approaches to provisioning and flexibility. IEEE Commun. Surv. Tutor.
**2021**, 23, 596–630. [Google Scholar] [CrossRef] - Open Source MANO. 2021. Available online: https://osm.etsi.org/ (accessed on 7 October 2021).
- O-RAN Alliance. 2021. Available online: https://www.o-ran.org/ (accessed on 7 October 2021).
- Casetti, C.; Chiasserini, C.; Martín-Pérez, J.; Molner, N.; Deiss, T. The Vertical Slicer: Verticals’ Entry Point to 5G Networks. In Proceedings of the 27th European Conference on Networks and Communications (EuCNC 2018), Ljubljana, Slovenia, 18–21 June 2018. [Google Scholar]
- Perez, M.G.; Perez, G.M.; Giardina, P.; Bernini, G.; Neves, P.; Alcaraz-Calero, J.; Wang, Q.; Koutsopoulos, K. Self-Organizing Capabilities in 5G Networks: NFV & SDN Coordination in a Complex Use Case. In Proceedings of the 27th European Conference on Networks and Communications (EuCNC 2018), Ljubljana, Slovenia, 18–21 June 2018. [Google Scholar]
- Khalili, H.; Papageorgiou, A.; Siddiqui, M.; Meixner, C.C.; Carrozzo, G.; Nejabati, R.; Simeonidou, D. Network Slicing-aware NFV Orchestration for 5G Service Platforms. In Proceedings of the 28th European Conference on Networks and Communications (EuCNC 2019), Valencia, Spain, 18–21 June 2019. [Google Scholar]
- Mesodiakaki, A.; Zola, E.; Santos, R.; Kassler, A. Optimal user association, backhaul routing and switching off in 5G heterogeneous networks with mesh millimeter wave backhaul links. Ad Hoc Netw.
**2018**, 78, 99–114. [Google Scholar] [CrossRef] - ETSI ES 203 228: Environmental Engineering (EE): Assessment of Mobile Network Energy Efficiency, November 2020. v.1.3.1. Available online: https://www.etsi.org/deliver/etsi_es/203200_203299/203228/01.03.01_60/es_203228v010301p.pdf (accessed on 7 October 2021).
- ETSI EN 303 471: Energy Efficiency Measurement Methodology and Metrics for Network Function Virtualisation (NFV), January 2019. v.1.1.1. Available online: https://www.etsi.org/deliver/etsi_en/303400_303499/303471/01.01.01_60/en_303471v010101p.pdf (accessed on 7 October 2021).
- Li, X.; Garcia-Saavedra, A.; Costa-Perez, X.; Bernardos, C.; Guimaraes, C.; Antevski, K.; Mangues-Bafalluy, J.; Baranda, J.; Zeydan, E.; Corujo, D.; et al. 5Growth: An End-to-End Service Platform for Automated Deployment and Management of Vertical Services over 5G Networks. IEEE Commun. Mag.
**2012**, 59, 84–89. [Google Scholar] [CrossRef] - Gatzianas, M.; Mesodiakaki, A.; Kalfas, G.; Pleros, N. Energy-efficient joint computational and network resource planning in Beyond 5G networks. Proc. IEEE Globecom.
**2021**. to appear. [Google Scholar] - Boyd, S.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Vertical Slicer. Available online: https://github.com/nextworks-it/slicer (accessed on 7 October 2021).
- TR 28.541: Management and Orchestration; 5G Network Resource Model (NRM); Stage 2 and Stage 3 (Rel. 17), 2020. v. 17.0.0. Available online: https://www.3gpp.org/ftp//Specs/archive/28_series/28.541/28541-h01.zip (accessed on 7 October 2021).
- ILOG CPLEX Optimization Studio. 2020. Available online: https://www.ibm.com/products/ilog-cplex-optimization-studio (accessed on 7 October 2021).
- Heddeghem, W.V.; Idzikowski, F.; Vereecken, W.; Colle, D.; Pickavet, M.; Demeester, P. Power consumption modeling in optical multilayer networks. Photon. Netw. Commun.
**2012**, 24, 86–102. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Mobile network of heterogeneous communication and computation nodes along with middlebox functionality offered by deployed VNFs. Dashed color lines indicate 2 illustrative E2E paths selected for 2 highlighted UEs.

**Figure 2.**Embedding of virtual network (bottom part) into the physical network (top part). For illustration, 2 distinct SFCs are shown (in green and red) corresponding to 2 different UE requests, and each SFC is described by its own virtual network. ${\mathcal{G}}^{\left({\rho}_{1}\right)}$ (green color) contains 6 virtual nodes and 5 virtual edges, whereas ${\mathcal{G}}^{\left({\rho}_{2}\right)}$ contains 5 virtual nodes and 4 virtual edges. The virtual nodes will be mapped to actual physical nodes, while the virtual edges will be mapped to actual paths in the physical network.

**Figure 3.**Flowchart of the proposed energy-efficient VNF placement, traffic routing, and user association algorithm (HERO).

**Figure 6.**Energy efficiency (bits/Joule) of all algorithms for different numbers of UEs per gNB area.

**Figure 7.**Execution time (s) in logarithmic scale of all algorithms for different numbers of UEs per gNB area.

**Figure 9.**Percentage of active nodes and links of all algorithms for different numbers of UEs per gNB area.

Symbol | Interpretation | Notation for Element (In Case of Sets) | Symbol | Interpretation | Notation for Element (In Case of Sets) |
---|---|---|---|---|---|

Input parameters | |||||

$\mathcal{V}$ | Set of network nodes ( excluding mobile users) | $u,\text{}v$ | ${\mathcal{V}}_{sw}$ | Set of nodes equipped with switch | $u,\text{}v$ |

$\mathcal{E}$ | Set of network links ( excluding access links) | $e=(u,\text{}v)$ | $\mathcal{J}$ | Set of UEs | j |

${\mathcal{E}}_{fi}$, ${\mathcal{E}}_{wl}$ | Set of fiber (resp. wireless)non-access links | $(u,\text{}v)$ | ${\mathcal{E}}_{AN}$ | Set of access links | $(a,\text{}j)$ |

${c}_{e}$ | Communication capacity of link e | NA | ${\delta}_{e}$ | Delay (i.e., transmission + propagation) induced by link e | NA |

${\mathcal{A}}^{\left(j\right)}$ | Set of BSs that UE j can connect to | a | ${\mathcal{S}}^{\left(a\right)}$ | Set of UEs that can be served by BS a | j |

${\mathcal{V}}_{c}$ | Set of network nodes equipped with computational resources | y | ${c}_{y}$ | Amount of computational resources available at node y | NA |

$\mathcal{F}$ | Set of available VNFs | $f=({T}_{f},\text{}{\pi}_{f},\text{}{w}_{f},\text{}{\tau}_{f})$ where ${T}_{f}$: VNF id, ${\pi}_{f}$: VNF data processing capacity, ${w}_{f}$: amount of CPU resources required by VNF, ${\tau}_{f}$: delay induced by VNF processing | |||

$\mathcal{C}$ | Set of available SFCs | $\rho =\langle {f}_{dum1},\text{}{f}_{1}^{\left(\rho \right)},\dots ,\text{}{f}_{{N}_{\rho}}^{\left(\rho \right)},\text{}{f}_{dum2}\rangle $ | |||

${\mathcal{R}}_{\rho}$ | Set of VNFs comprising SFC $\rho $ | f | ${\mathcal{G}}^{\left(\rho \right)}$ | Virtual directed graph describing SFC via virtual node set ${\mathcal{V}}^{\left(\rho \right)}$ and virtual edge set ${\mathcal{E}}^{\left(\rho \right)}$ | NA |

$\mathcal{Q}$ | Set of requests by UEs | $q=({s}_{q},\text{}{r}_{q},\text{}{\delta}_{q},\text{}{\rho}_{q})$ where s: source node of service request, ${r}_{q}$: E2E requested rate, ${\delta}_{q}$: E2E requested latency, ${\rho}_{q}$: requested SFC | |||

${N}_{a,j}^{\left(RB\right)}$ | Number of RBs needed to be assigned by BS a to UE j to meet its requested rate | ||||

${P}_{idle}^{\left(sw\right)}$ | Switch idle power | NA | ${P}_{e}^{(mmW,i)}$ | Idle power of transmitter in mmWave link e | NA |

${P}_{a}^{(gNB,i)}$ | Idle power of gNB a | NA | ${P}_{a}^{(SC,i)}$ | idle power of SC a | NA |

${N}_{RF}^{\left(mmW\right)}$ | Number of RF chains used in transmitter of mmWave link | NA | ${N}_{RF}^{\left(gNB\right)}$, ${N}_{RF}^{\left(SC\right)}$ | Number of RF chains used by gNB, SC | NA |

${\Delta}_{e}^{\left(mmW\right)}$ | Slope parameter of transmitter in mmWave link e | NA | ${\Delta}_{a}^{\left(gNB\right)}$, ${\Delta}_{a}^{\left(SC\right)}$ | Slope parameter of transmitter in gNB, SC a | NA |

Decision variables | |||||

${x}_{j,a}$ | Boolean indicator of whether UE j actually connects to BS a | ${\varphi}_{\tilde{y},f,q}$ | Boolean indicator of whether VNF f requested by service q is deployed on node $\tilde{y}$ | ||

${\theta}_{e}^{{e}^{\prime},q}$, ${\theta}_{u,v}^{{e}^{\prime},q}$ | Boolean indicator of whether physical link belongs to the physical path onto which the virtual link ${e}^{\prime}\in {\mathcal{E}}^{\left({\rho}_{q}\right)}$ is mapped for SFC $rh{o}_{q}$ | ${N}_{f,\tilde{y}}$ | Number of instances of VNF f deployed on node $\tilde{y}$ | ||

${z}_{n,m}$ | Boolean indicator of whether fiber link $(n,m)$ actually carries traffic in either of its directions | ${w}_{e}$, ${w}_{n,m}$ | Boolean indicator of whether fiber link $e=(n,m)$ actually carries traffic from n to m | ||

${\xi}_{y}$ | Boolean indicator of whether the computational resources of node y are used | ${\psi}_{n}$ | Boolean indicator of whether the switch at node n is actually used to forward traffic | ||

${\mu}_{a}$ | Boolean indicator for whether BS a serves any UEs | ${\chi}_{e}$ | Boolean indicator of whether wireless link e is actually used to serve any traffic | ||

${\nu}_{a}$ | Auxiliary variable for ${\mu}_{a}$ [25] | ${\sigma}_{e}$ | Auxiliary variable for ${\chi}_{e}$ [25] |

**Table 2.**SFC details [15].

Type | VNF Ordering | Throughput (Mbps) | Delay (ms) | Share (%) |
---|---|---|---|---|

Web | NAT→FW→TM→WOC→IDPS | [0.6–1] | 500 | 20 |

VoIP | NAT→FW→TM→FW→NAT | [0.404–0.64] | 100 | 20 |

Streaming | NAT→FW→TM→VOC→IDPS | [5–24] | 100 | 39 |

Gaming | NAT→FW→VOC→WOC→IDPS | [0.24–0.5] | 60 | 6 |

Ultra RT AI/ML | NAT→NAT | [15–25] | 1 | 15 |

**Table 3.**VNF details [15].

Type | NAT | FW | TM | VOC | WOC | IDPS |
---|---|---|---|---|---|---|

Process Capacity (Mbps) | 500 | 400 | 200 | 578 | 300 | 600 |

GFLOPS Requirement | 110 | 440 | 55 | 110 | 110 | 440 |

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

${P}_{idle}^{\left(sw\right)}$ | 315 W | ${P}_{port}$ | 7 W | Packet length | 1.5 KB |

${N}_{RF}^{\left(gNB\right)}$ | 8 | ${N}_{RF}^{\left(SC\right)}$ | 4 | ${N}_{RF}^{\left(mmW\right)}$ | 64 |

${\Delta}_{a}^{\left(gNB\right)}$ | 4.7 | ${\Delta}_{a}^{\left(SC\right)}$ | 4 | ${\Delta}_{e}^{\left(mmW\right)}$ | 100 |

${P}_{a}^{(gNB,i)}$ | 130 W | ${P}_{a}^{(SC,i)}$ | 6.8 W | ${P}_{e}^{(mmW,i)}$ | 3.9 W |

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## Share and Cite

**MDPI and ACS Style**

Gatzianas, M.; Mesodiakaki, A.; Kalfas, G.; Pleros, N.; Moscatelli, F.; Landi, G.; Ciulli, N.; Lossi, L.
Offline Joint Network and Computational Resource Allocation for Energy-Efficient 5G and beyond Networks. *Appl. Sci.* **2021**, *11*, 10547.
https://doi.org/10.3390/app112210547

**AMA Style**

Gatzianas M, Mesodiakaki A, Kalfas G, Pleros N, Moscatelli F, Landi G, Ciulli N, Lossi L.
Offline Joint Network and Computational Resource Allocation for Energy-Efficient 5G and beyond Networks. *Applied Sciences*. 2021; 11(22):10547.
https://doi.org/10.3390/app112210547

**Chicago/Turabian Style**

Gatzianas, Marios, Agapi Mesodiakaki, George Kalfas, Nikos Pleros, Francesca Moscatelli, Giada Landi, Nicola Ciulli, and Leonardo Lossi.
2021. "Offline Joint Network and Computational Resource Allocation for Energy-Efficient 5G and beyond Networks" *Applied Sciences* 11, no. 22: 10547.
https://doi.org/10.3390/app112210547