#  A Security-aware Network Function Sharing Model for 5G Slicing

Authors’ draft for soliciting feedback, March 6, 2023
Mohammed Mahyoub, AbdulAziz AbdulGhaffar1, Emmanuel Alalade, and Ashraf
Matrawy
School of Information Technology, Carleton University, Canada
1Department of Systems and Computer Engineering, Carleton University, Canada

###### Abstract

Sharing Virtualized Network Functions (VNFs) among different slices in Fifth
Generation (5G) is a potential strategy to simplify the system implementation
and utilize 5G resources efficiently. In this paper, we propose a security-
aware VNF sharing model for 5G networks. The proposed optimization model
satisfies the service requirements of various slices, enhances slice security
by isolating their critical VNFs, and enhances resource utilization of the
underlying physical infrastructure. The model tries to systematically decide
on sharing a particular VNF based on two groups of constraints; the first
group of constraints is common assignment constraints used in the existing
literature. The second group is the novel security constraints that we propose
in this work; the maximum traffic allowed to be processed by the VNF and the
exposure of the VNF to procedures sourced via untrusted users or access
networks. This sharing problem is formalized to allow for procedure-level
modeling that satisfies the requirements of slice requests in 5G systems. The
model is tested using standard VNFs and procedures of the 5G system rather
than generic ones. The numerical results of the model show the benefits and
costs of applying the security constraints along with the network performance
in terms of different metrics.

###### Index Terms:

5G Security, Network Slicing (NS), VNF Sharing, Optimization

##  I Introduction

5Gnetworks are visioned to support various applications and services with
diversified requirements [1]. One distinct concept in 5G architecture is the
Network Slicing (NS)which was not present in previous generations of cellular
networks. NS enables 5G operators to deploy multiple logical networks on
shared physical resources to serve traffic segments with different demands [2,
3]. This is achieved using different technologies integrated with 5G
architecture such as, most notably, Network Function Virtualization (NFV)
technology. NFV allows the deployment of Virtualized Network Functions
(VNFs)in software or a virtualized environment on commodity hardware. Both NS
and NFV help 5G operators to reduce the overall Capital Expenditure (CAPEX)
and Operational Expenditure (OPEX) by deploying VNFs efficiently and flexibly
to optimize the utilization of network resources [4].

In this paper, we propose a security-aware VNFs sharing model for 5G networks.
The proposed optimization model not only satisfies the service requirements of
various slices but also enhances security by isolating their critical VNFs
while enhancing resource utilization of the underlying physical
infrastructure. This goal is achieved by sharing as many noncritical VNFs as
possible to efficiently utilize resources and satisfy the latency limitations
of the procedures composing 5G slices. Although some literature studies
considered the sharing property of VNFs in the mapping process, they
subjectively decide on this property and use it as an input to their model.
This work tries to fill this gap by following a systematic way to decide
whether a particular VNF is critical and, if so, to avoid sharing it among
slices. In the proposed model, two novel security constraints are considered
to define the VNF criticality. The first constraint is the maximum traffic
that can be processed by a particular VNF. If a VNF has to process large user
and control traffic, it could become a bottleneck which makes it critical, and
thus, should not be shared between slices. The second one is exposure to
procedures initiated by untrusted entities (i.e. user devices or networks). If
a VNF is exposed to procedures coming from untrusted parties, this VNF should
not be shared among slices too. In case of the exposed VNF is compromised,
this can impact all other slices that share it. To this end, providing
isolation to critical VNFs is very crucial in 5G network slicing. In light of
the above discussion, the contributions of this paper are four-fold:

  * •

Proposing a multi-objective Mixed-Integer Nonlinear Programming (MINLP) model
aiming at minimizing the processing capacity needed and procedures’ latency of
all requested slices.

  * •

Providing a systematic way to decide on the sharing property of a particular
VNF by introducing new security constraints that define the VNF’s criticality.

  * •

Considering the procedure level granularity instead of abstracting a slice as
a unit. To the best of our knowledge, this is the first work to consider
procedure-level details in the optimization model.

  * •

The proposed model is tested using standard procedures and VNFs of 5G
architecture that are discribed in 3rd Generation Partnership Project (3GPP)
standards [5] rather than using generic VNFs or symbolic procedures.

The rest of this paper is structured as follows. Section II discusses the
related works and Section III explains the proposed model. The system setup
and model parameters are presented in Section IV. The standard 5G procedures
implemented to test the model are defined in Section V. The proposed model is
evaluated in Section VI and the limitations of this work are provided in
Section VII. Finally, Section VIII concludes this study.

##  II Related work

This section reviews the related literature studies that attempted to solve
the VNF placement and allocation problem using optimization approaches.
Although there are many studies that considered sharing a physical node
between multiple VNFs, our work mainly focuses on sharing the a VNFs
themselves between multiple 5G slices.

Leyva et al. in [6] proposed an Integer Linear Programming (ILP)-formulated
optimizing model for User Plane Functions (UPFs) chaining and placement in
Multi-access Edge Computing (MEC) system of 5G. Their model targeted the
provisioning cost and Quality of Service (QoS) optimization. It considered
several aspects such as resource capacity, service latency, UPF-specific
requirements, and the order of VNFs in the Service Function Chains (SFCs).
UPFs placement and routing are modelled as SFC embedding problem in which
active Protocol Data Unit (PDU) sessions are modelled as SFC requests. There
is no restriction on sharing a particular VNF except its capacity limit. To
solve the problem in a polynomial time, a customized heuristic along with
simulating annealing algorithm has been proposed in their work. Our work, on
the other hand, in addition to the data-plane function (i.e., UPFs), considers
control-plane functions as well.

Coelho et al. in [7, 8] modeled the provisioning of the NS requests at the
service level as an optimization problem. The model considered functional
splitting in the radio access domain and also the separation of the control
and data-plane functions. The authors assumed that the network slice request
might impose constraints on VNFs that can not be shared between NSs due to
their criticality or their belonging to different tenants. They tested
different sharing policies such as sharing Data Plane Services (DPS) only,
Control Plane Services (CPS) only, some of DPS, some CPS, or without sharing
constraints. These sharing policies are given to the model as input, however,
our model decides whether to share VNF systematically based on different
security constraints.

Malandrino et al. [9] studied reducing the cost of the 5G service deployment
through sharing VNFs subject to end-to-end delay requirements. With the
assumption that there is no isolation needed for the new service request, VNFs
are shared if convenient (i.e. meet the delay requirements). The authors
focused more on how to assign priorities for traffic flows that share the same
VNF. For this, they randomly assigned flows priority upon entering VNFs. To
reduce the time complexity, they proposed FlexShare as an assignment
algorithm.

Tang et al. [10] proposed a dynamic scaling approach for VNF based on traffic
analysis and VNF placement. They analyzed the traffic characteristic of
operator networks and then proposed an organizational approach for VNF
placement in a common data center. Their model aims to achieve high service
availability and save computational resources depending on the traffic
estimation to scale in/out the VNF instance. The authors considered general
VNF and only user traffic. Our work, however, considers the actual VNFs of the
5G core, data plane, and control plane traffic.

Truong-Huu et al. in [11] leveraged the VNF’s sharing property in their
optimization model to minimize the bandwidth and computational resources
required to serve slice requests. A VNF is identified as shareable depending
on its functionalists so that it can be assigned to serve different slices.
The network address translation function is an example of shareable VNF,
however, firewall service is non-shareable. In their work, the sharing
property of a VNF is set in advance and provided to the model as input.
Additionally, their work considered random traffic flows and generic VNFs that
serve these traffics.

Another work leveraging the shareable VNFs criteria to enhance resource
utilization is presented by Chengli et al. in [12]. Their enhancement is
evaluated in terms of the slice acceptance ratio. They claimed that some
common functions such as mobility management and network address translation
functions can be shared across multiple slices. Similar to the work in [11],
Chengli et al. [12] randomly set the sharing property in their experiments and
used it as an input for their model.

A queuing-based system model is proposed by Agarwal  et al. [13] for
optimizing the VNFs placement and allocation in physical hosts taking into
account the VNFs sharing. Authors in [13] utilized the concept of queuing
theory and considered random procedures with a sequence of random and generic
VNFs. In our paper, we consider 5G VNFs and multiple standard 5G procedures.
Other models were presented in [14, 15] to optimize the utilization of the
underlined physical infrastructure considering different slicing requirements.
However, both of them assumed that VNFs can’t be shared among slice or service
requests.

Finally, Sattar et al. proposed an optimal slice allocation model in 5G core
networks [16] and extended it to propose a security-aware optimization model
to protect the 5G core network slices against Distributed DoS (DDoS) attacks
in [17]. The model tried to isolate the network slices at the hardware level.
The authors considered both inter-slice and intra-slice isolation and
evaluated the performance of their proposed solution on a testbed which
involved both simulation and experimental parts. Their results confirmed the
benefits of utilizing a security-aware network slice optimization model to
mitigate the impact of DDoS attacks. Our work focuses on sharing and isolation
of 5G VNFs whereas the work presented in [17] only considers sharing of
physical resources and not the VNFs. Furthermore, our work considers the
standard VNFs of the 5G core along with several procedures used in the 5G
network.

To sum up, it can be observed that the capacity limit of the VNFs is the thing
that we have in common with most of the literature studied which is standard
in this area. To the best of our knowledge, this is the first study
incorporating security aspects into the optimization model. Not only that, but
our work also proposes a systematic way to decide on whether to share or not
to share a particular VNF installed in a specific physical node. Additionally,
our work considers procedure-level rather than slice requests or traffic
flows, and some standard VNFs and procedures of 5G rather than generic or
symbolic ones.

##  III System Model and Problem Formulation

In this section, the proposed model is given. The VNF sharing problem
considered in this study is formalized and solved as an MINLP. The proposed
model aims to optimize computational processing costs and the latency of
slices’ procedures.

###  III-A Model Description and Notations

The modeling of the virtual and physical networks is defined in this
subsection. In this model, each slice request $s\in\mathcal{S}$ is composed of
a set of procedures, $\mathcal{P}_{s}$. The virtual network is modeled in this
work by a set of directed graphs. Each graph
$(\mathcal{V}^{s}_{p},\mathcal{R}_{p}^{s})$ corresponds to a particular
procedure $p\in\mathcal{P}_{s}$ that belongs to a specific slice
$s\in\mathcal{S}$, where $\mathcal{V}^{s}_{p}$ is the set of VNFs serving the
procedure $p$ and $\mathcal{R}_{p}^{s}$ denotes the set of the virtual links
used by that procedure. Each procedure $p\in\mathcal{P}_{s}$ requires a
specific data rate, $\lambda_{p}^{s}~{}$, and a maximum tolerated delay,
$\delta_{p}^{s,max}$. Each VNF $v\in\mathcal{V}$ is represented by a tuple
$\langle
v_{i},~{}I_{v},~{}\delta^{n}_{v_{i}},~{}\zeta^{max}_{v},~{}\mu_{v},~{}\omega_{v}\rangle$
where $v_{i}$ is the deployed $i^{th}$ instance of VNF $v$ type, $I_{v}$
denotes the set of all instances of VNF $v$ type deployed across all physical
nodes, $\delta^{n}_{v_{i}}$ is the processing delay for $i^{th}$ instance of
type $v$ deployed in node $n$, $\zeta^{max}_{v}$ is the maximum accepted
processing capacity to which the VNF type $v_{i}$ can be extended, $\mu_{v}$
is the per unit processing capacity required by the VNF of type $v$, and
$\omega_{v}$ denotes the number of processed data units per unit processing
time by the VNF of type $v$. Finally, the physical infrastructure network is
modeled as a directed graph $\mathcal{G}=(\mathcal{N},\mathcal{L})$, where
$\mathcal{N}$ is the set of physical nodes and $\mathcal{L}$ denotes the
physical links between these nodes. Each physical node $n\in\mathcal{N}$ has a
finite processing capacity, $C^{max}_{n}$. A physical link $(n,m)$, between
node $n$ and node $m$, entails a deterministic delay $d{(n,m)}$ proportional
to its length and also a maximum bandwidth capacity $b{(n,m)}$. Table I
summarizes the notations and variable definitions used throughout this paper.

TABLE I: Used notations summary Parameter |  The definition
---|---
$\mathcal{S}$ |  The set of all slices
$\mathcal{P}$ |  The set of all procedures
$\mathcal{V}$ |  The set of all VNF types
$\mathcal{N}$ |  The set of all physical nodes
$\mathcal{L}$ |  The set of all physical connections between nodes in $\mathcal{N}$
$(n,m)$ |  The physical link beginning at node $n$ and ending at node $m$
$d{(n,m)}$ |  The delay of the physical link $(n,m)$
$b{(n,m)}$ |  The bandwidth capacity of the physical link $(n,m)$
$\mathcal{R}$ |  The set of virtual links used by all procedures in the network
$\mathcal{P}_{s}$ |  The set of all procedures belong to the slice $s$
$\mathcal{V}^{s}_{p}$ |  The set of all VNFs belong to procedure $p$ in slice $s$
$v_{i}$ |  The deployed $i^{th}$ instance of VNF $v$ type
$I_{v}$ |  The set of all instances of VNF $v$ type deployed over all physical nodes
$(v_{i},z_{j})$ |  The virtual link between VNF instances $v_{i}$ and $z_{j}$
$\mathcal{R}_{p}^{s}$ |  The set of virtual links used by procedure $p$ of slice $s$
$\lambda_{p}^{s}$ |  The packet rate of the procedure $p$ given one User Equipment (UE) is connected to the $s$
$\delta^{n}_{v_{i}}$ |  The processing delay for $i^{th}$ instance of the type $v$ deployed in node $n$
$\delta_{p}^{s,max}$ |  The maximum tolerated delay of the procedure $p$
$C^{max}_{n}$ |  The maximum processing capacity of the node $n$
$\zeta^{max}_{v}$ |  The maximum accepted processing capacity to which the VNF type $v_{i}$ can be extended
$\mu_{v}$ |  Per unit processing capacity required by the VNF $v$ type
$\omega_{v}$ |  Number of processed data units per unit processing time by the VNF $v$ type
$\delta_{p}^{s}$ |  The delay of the procedure $p$ of slice $s$ based on the existing configuration
$\zeta^{n}_{v_{i}}$ |  The total required processing capacity of VNF instance $v_{i}$ deployed in node $n$
$\psi_{p}^{s}$ |  Indicates whether the procedure $p$ is sourced by external entity
$\eta^{s}_{v,p}$ |  indicates whether the VNF-type $v$ is the first VNF traversed by procedure $p$
$\theta^{s}_{v,p}$ |  Indicates whether a procedure must traverse a VNF type $v$
Variable |  The definition
$\upgamma^{n,s}_{v_{i},p}$ |  Binary variable indicates whether VNF $v_{i}$ used by the procedure $p$ of slice $s$ is deployed at the node $n$
$\chi^{(n,m),s}_{(v_{i},z_{j}),p}$ |  Binary variable indicates whether the virtual link $(v_{i},z_{j})$, used by procedure $p$ of slice $s$, is mapped to the physical link $(n,m)$
$\beta^{n}_{v_{i}}$ |  Binary variable indicates whether the VNF instance $v_{i}$ is deployed at the node $n$
$\Omega^{n,s}_{v_{i}}$ |  Binary variable that indicates whether the VNF instance $v_{i}$ deployed in physical node $n$ is exposed to the outside by the slice $s$

###  III-B Model Assumptions

Few assumptions are considered in this work as outlined in this subsection.
The standard 5G VNFs considered in the model such as Access and Mobility
Function (AMF), Session Management Function (SMF), Network Repository Function
(NRF), etc. could be VM-based or container-based VNFs. It is not important
that the number of VNFs type per slice is the same as that for another slice.
Additionally, multiple instances of the VNF type can be initiated if required
as assumed in [14]. VNFs are required to dynamically support scale-in and
scale-out with minimal impact on the service quality offered [18]. Physical
nodes are geographically distributed and each of them can deploy any VNF type.
It is assumed that all traffic units need the same computational capability
for processing. Although in this work we focus on CPU or computational
capacity, the storage and memory could be accommodated.

In this model, the delay raised by the load balance in the case of muli-core
VNF is minimal to be considered. The load balancer is needed when a VNF
requires processing capabilities that cannot be fulfilled by a single core. In
this case, multiple cores are needed to satisfy the processing requirement of
that VNF. The load balancer will be used to balance the traffic between the
cores and may lead to some performance penalties. Additionally, the context
switching delay caused by sharing the CPU’s cores between multiple VNFs is not
taken into account. This delay comes in a form of cache sharing and
saving/loading the context of different VNFs. It is linearly increased with
the number of procedures using those VNFs.

###  III-C The Objective Function

The first part of the objective function of this model is to minimize the
total processing capacity needed to serve all slices. This part is satisfied
by sharing as many noncritical VNFs as possible while considering the security
constraints imposed to mitigate the risks that raise by such sharing. The
second part is to minimize the delay of all procedures. These two parts can be
formulated in Eq.(1). It is worth mentioning here that we use common
optimization goals in the literature which are minimizing delay and resource
consumption. Although we focus on minimizing the processing capacity and
procedures delay, our model can be extended to consider additional key
performance indicators seamlessly.

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to4.99pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip
2.49304pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.49304pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\delta^{s}_{p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.1pt{\vbox to5.97pt{\pgfpicture\makeatletter\hbox{\hskip
2.0486pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\delta^{s}_{p}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\leavevmode\hbox
to215.32pt{\vbox to34.66pt{\pgfpicture\makeatletter\hbox{\hskip
141.66624pt\lower-33.06078pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-138.33324pt}{-31.66078pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\vbox{\halign{#\hfil\cr\cr\vskip-12.00002pt\cr\hbox{{The
total required computational }}\cr}}\cr\vskip 0.0pt\cr\hbox{{capacity for all
\lx@glossaries@gls@link{main}{vnf}{\leavevmode VNFs}}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{{}}{{{}}{{}}}{}{}{}{}{{ {\pgfsys@beginscope{} {} {} {}
\pgfsys@moveto{1.99997pt}{0.0pt}\pgfsys@lineto{-1.19998pt}{1.59998pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{-1.19998pt}{-1.59998pt}\pgfsys@fill\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.32185pt}\pgfsys@lineto{0.0pt}{-32.86078pt}\pgfsys@lineto{-140.57458pt}{-32.86078pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.32185pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-21.69443pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\vbox{\halign{#\hfil\cr\cr\vskip-2.0pt\cr\hbox{{The
total delay}}\cr}}\cr\vskip 0.0pt\cr\hbox{{of all procedures}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.35551pt}\pgfsys@lineto{0.0pt}{-22.89442pt}\pgfsys@lineto{73.4578pt}{-22.89442pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.35551pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1)

Subject to constraints (6) to (19).

####  III-C1 Computational Capacity and Procedure Delay

In the following, we show how the computational capacity needed for a
particular VNF and a procedure delay are calculated.

  * •

VNF computational capacity: Generally, the more services/procedures a VNF
provides/hosts, the more physical resources are required. The processing
capacity $\zeta^{n}_{v_{i}}$, that is needed by a particular VNF, comes in two
forms; operational or base processing capacity $\zeta^{n,B}_{v_{i}}$ and
traffic processing capacity $\zeta^{n,T}_{v_{i}}$ as shown in Eq. (2). Based
on the number of procedures that a particular VNF $v$ instance serves, we can
calculate its $\zeta^{n,T}_{v_{i}}$. If the VNF type $v$ requires $\mu_{v}$
processing capability to process one unit of traffic, then the
$\zeta^{n,T}_{v_{i}}$ calculated as in Eq. (3).

$\zeta^{n}_{v_{i}}=\mathchoice{\leavevmode\hbox to13.82pt{\vbox
to11.12pt{\pgfpicture\makeatletter\hbox{\hskip 6.91023pt\lower-2.26079pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.91023pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\zeta^{n,B}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to13.82pt{\vbox to11.12pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.91023pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle\zeta^{n,B}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to9.81pt{\vbox to8.14pt{\pgfpicture\makeatletter\hbox{\hskip
4.90463pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.90463pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\zeta^{n,B}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.93pt{\vbox to6.76pt{\pgfpicture\makeatletter\hbox{\hskip
4.46713pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.46713pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\zeta^{n,B}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>to9.02pt{\vbox to10.41pt{\pgfpicture\makeatletter\hbox{\hskip
4.50877pt\lower-2.26077pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.50877pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\beta^{n}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to9.02pt{\vbox to10.41pt{\pgfpicture\makeatletter\hbox{\hskip
4.50877pt\lower-2.26077pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.50877pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle\beta^{n}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.36pt{\vbox to7.64pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\beta^{n}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to5.23pt{\vbox to6.25pt{\pgfpicture\makeatletter\hbox{\hskip
2.61453pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{$\scriptscriptstyle\beta^{n}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\mathchoice{\leavevmode\hbox
to13.34pt{\vbox to11.12pt{\pgfpicture\makeatletter\hbox{\hskip
6.67108pt\lower-2.26079pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.67108pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\zeta^{n,T}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{$\textstyle\zeta^{n,T}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to9.47pt{\vbox to8.14pt{\pgfpicture\makeatletter\hbox{\hskip
4.7338pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.7338pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\zeta^{n,T}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to6.76pt{\pgfpicture\makeatletter\hbox{\hskip
4.2963pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.2963pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\zeta^{n,T}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}~{}~{}~{}~{}\forall
v\in\mathcal{V},~{}~{}i\in I_{v},~{}n\in\mathcal{N}\leavevmode\hbox
to264.11pt{\vbox to45.6pt{\pgfpicture\makeatletter\hbox{\hskip
91.65154pt\lower-17.25002pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-88.31853pt}{-15.85002pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip-15.00002pt\cr\hbox{{The
$v$'s base capacity}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.32185pt}\pgfsys@lineto{0.0pt}{-17.05002pt}\pgfsys@lineto{-89.69876pt}{-17.05002pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.32185pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-13.57275pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip-2.0pt\cr\hbox{{Is
the \lx@glossaries@gls@link{main}{vnf}{\leavevmode VNF} instance $v_{i}$
activated?}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.32185pt}\pgfsys@lineto{0.0pt}{-14.77275pt}\pgfsys@lineto{172.2547pt}{-14.77275pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.32185pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-0.85pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip
20.00003pt\cr\hbox{{The traffic's processing capacity}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{5.4889pt}\pgfsys@lineto{0.0pt}{-2.05pt}\pgfsys@lineto{135.30789pt}{-2.05pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{5.4889pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2)

The total processing capacity than can be calculated as in Eq. (2)

$\zeta^{n,T}_{v_{i}}~{}=~{}\sum_{s\in\mathcal{S}}\sum_{p\in\mathcal{P}_{s}}~{}\mathchoice{\leavevmode\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle~{}\lambda_{p}^{s}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to11.98pt{\vbox to10.44pt{\pgfpicture\makeatletter\hbox{\hskip
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle~{}\lambda_{p}^{s}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.43pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle~{}\lambda_{p}^{s}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.59pt{\vbox to5.97pt{\pgfpicture\makeatletter\hbox{\hskip
3.29706pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.29706pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle~{}\lambda_{p}^{s}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathchoice{\leavevmode\hbox
to16.05pt{\vbox to7.81pt{\pgfpicture\makeatletter\hbox{\hskip
8.02455pt\lower-2.29443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.02455pt}{0.0pt}\pgfsys@invoke{
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}\hbox{{$\displaystyle~{}\upgamma^{n,s}_{v_{i},p}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
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8.02455pt\lower-2.29443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle~{}\upgamma^{n,s}_{v_{i},p}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to11.34pt{\vbox to5.79pt{\pgfpicture\makeatletter\hbox{\hskip
5.67105pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{$\scriptstyle~{}\upgamma^{n,s}_{v_{i},p}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to9.64pt{\vbox to4.93pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathchoice{\leavevmode\hbox
to12.27pt{\vbox to6.25pt{\pgfpicture\makeatletter\hbox{\hskip
6.13539pt\lower-1.94443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{$\displaystyle~{}\mu_{v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to12.27pt{\vbox to6.25pt{\pgfpicture\makeatletter\hbox{\hskip
6.13539pt\lower-1.94443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.13539pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle~{}\mu_{v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.63pt{\vbox to4.37pt{\pgfpicture\makeatletter\hbox{\hskip
4.31558pt\lower-1.36108pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.31558pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle~{}\mu_{v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.76pt{\vbox to3.12pt{\pgfpicture\makeatletter\hbox{\hskip
3.38005pt\lower-0.9722pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.38005pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle~{}\mu_{v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\\\
~{}~{}~{}~{}~{}\forall v\in\mathcal{V},~{}~{}i\in
I_{v},~{}n\in\mathcal{N}\leavevmode\hbox to282.35pt{\vbox
to53.44pt{\pgfpicture\makeatletter\hbox{\hskip
163.6663pt\lower-25.09444pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-0.85pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip
20.00003pt\cr\hbox{{The procedure's packet rate}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{5.4889pt}\pgfsys@lineto{0.0pt}{-2.05pt}\pgfsys@lineto{118.48285pt}{-2.05pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{5.4889pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-160.3333pt}{-23.69444pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip-12.00002pt\cr\hbox{{The
capacity needed for one traffic unit}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.35551pt}\pgfsys@lineto{0.0pt}{-24.89444pt}\pgfsys@lineto{-162.57465pt}{-24.89444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.35551pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (3)

  * •

Procedure delay: The experienced delay by a particular procedure $p$ is
calculated from two parts. The first part is the processing delay incurred by
VNFs that the procedure passes through. The second part is the propagation
delay of the links that the procedure uses. The total delay can be calculated
as in Eq. (4)

$\begin{split}\delta^{s}_{p}=\sum_{v\in\mathcal{V}^{s}_{p}}\sum_{i\in
I_{v}}\sum_{n\in\mathcal{N}}\mathchoice{\leavevmode\hbox to7.81pt{\vbox
to10.41pt{\pgfpicture\makeatletter\hbox{\hskip 3.90288pt\lower-2.26077pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.90288pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\delta^{n}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.81pt{\vbox to10.41pt{\pgfpicture\makeatletter\hbox{\hskip
3.90288pt\lower-2.26077pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.90288pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle\delta^{n}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to5.51pt{\vbox to7.64pt{\pgfpicture\makeatletter\hbox{\hskip
2.75601pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.75601pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\delta^{n}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.62pt{\vbox to6.25pt{\pgfpicture\makeatletter\hbox{\hskip
2.31157pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.31157pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\delta^{n}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\upgamma^{n,s}_{v_{i},p}\\\
\\\
~{}+~{}\sum_{(v_{i},z_{j})\in\mathcal{R}_{p}^{s}}\sum_{(n,m)\in\mathcal{L}}\mathchoice{\leavevmode\hbox
to32.21pt{\vbox to10pt{\pgfpicture\makeatletter\hbox{\hskip
16.10478pt\lower-2.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.10478pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle
d(n,m)$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to32.21pt{\vbox to10pt{\pgfpicture\makeatletter\hbox{\hskip
16.10478pt\lower-2.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.10478pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle d(n,m)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to23.05pt{\vbox to7pt{\pgfpicture\makeatletter\hbox{\hskip
11.52333pt\lower-1.75pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.52333pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle
d(n,m)$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to16.94pt{\vbox to5pt{\pgfpicture\makeatletter\hbox{\hskip
8.46902pt\lower-1.25pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.46902pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
d(n,m)$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\chi^{(n,m),s}_{(v_{i},z_{j}),p}~{}~{}~{}~{}~{}\forall
s\in\mathcal{S},p\in\mathcal{P}_{s}\leavevmode\hbox to260.91pt{\vbox
to22.8pt{\pgfpicture\makeatletter\hbox{\hskip
144.85164pt\lower-25.06078pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-141.51863pt}{-23.66078pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip-12.00002pt\cr\hbox{{The
delay of the instance of type $v$}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.32185pt}\pgfsys@lineto{0.0pt}{-24.86078pt}\pgfsys@lineto{-143.75998pt}{-24.86078pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.32185pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-22.69444pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip-10.00002pt\cr\hbox{{The
delay of the link (n,m)}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.35551pt}\pgfsys@lineto{0.0pt}{-23.89444pt}\pgfsys@lineto{115.85785pt}{-23.89444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.35551pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{split}$ (4)

Where the delay that is incured by a particular VNF instance $v_{i}$ is
calculated by Eq.(5).

$\begin{split}\delta^{n}_{v_{i}}=1/\omega_{v}~{}+~{}1/(\omega_{v}-\sum_{s\in\mathcal{S}}\sum_{p\in\mathcal{P}_{s}}\lambda_{p}^{s}~{}\upgamma^{n,s}_{v_{i},p})\\\
~{}~{}~{}~{}~{}\forall s\in\mathcal{S},p\in\mathcal{P}_{s}\end{split}$ (5)

The remaining part of this section explains the model constraints. They are
categorized under two groups; assignment, and security constraints as
explained in the following two subsections.

####  III-C2 Assignment constraints

We use common assignment constraints existing in many literature papers such
as [6] and [12]. We develop assignment constraints along the same lines as
other references, however, we develop these constraints to consider procedure
level, specific VNFs type, and both control- and data-plane functions. In the
following points, the assignment constraints are given.

  * •

Firstly, Eq. (6) defines the value constraint of variables used in this model.

$\chi^{(n,m),s}_{(v_{i},z_{j}),p}~{},~{}\beta^{n}_{v_{i}}~{},~{}\upgamma^{n,s}_{v_{i},p}~{},~{}\psi_{p}^{s},~{}\Omega^{n,s}_{v_{i}}~{}\in\\{0,1\\}$
(6)

  * •

Let’s denote by $\mathcal{P}_{s}$ the set of VNFs required by the procedure
$p$ of the slice $s$. Constraint (7) guarantees that each procedure and its
respective VNFs are mapped.

$\sum_{n\in\mathcal{N}}\sum_{i\in
I_{v}}\upgamma^{n,s}_{v_{i},p}=\mathchoice{\leavevmode\hbox to7.32pt{\vbox
to10.44pt{\pgfpicture\makeatletter\hbox{\hskip 3.65971pt\lower-2.29443pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.65971pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\theta^{s}_{v,p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.32pt{\vbox to10.44pt{\pgfpicture\makeatletter\hbox{\hskip
3.65971pt\lower-2.29443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.65971pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle\theta^{s}_{v,p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to5.16pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip
2.58055pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.58055pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\theta^{s}_{v,p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.22pt{\vbox to5.97pt{\pgfpicture\makeatletter\hbox{\hskip
2.1111pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.1111pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\theta^{s}_{v,p}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}~{}~{}~{}~{}\forall
s\in\mathcal{S},~{}p\in\mathcal{P}_{s},v\in\mathcal{V}\leavevmode\hbox
to209.3pt{\vbox to16.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-19.09444pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-17.69444pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip-6.00002pt\cr\hbox{{Must
the procedure $p$ traverse the \lx@glossaries@gls@link{main}{vnf}{\leavevmode
VNF} $v$?}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.35551pt}\pgfsys@lineto{0.0pt}{-18.89444pt}\pgfsys@lineto{208.89655pt}{-18.89444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.35551pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (7)

  * •

Constraint (8) and (9) ensure that a VNF instance will not be initiated unless
there is at least one procedure using it.

$\sum_{s\in\mathcal{S}}\sum_{p\in\mathcal{P}_{s}}\upgamma^{n,s}_{v_{i},p}\leq\mathcal{M}~{}\beta^{n}_{v_{i}},~{}~{}~{}~{}~{}\forall
v\in\mathcal{V},~{}i\in I_{v},~{}n\in\mathcal{N}$ (8)

Where $\mathcal{M}$ is a parameter greater than the maximum number of
procedures that will be mapped to the instance $v_{i}$.

$\beta^{n}_{v_{i}}-\sum_{s\in\mathcal{S}}\sum_{p\in\mathcal{P}_{s}}\upgamma^{n,s}_{v_{i},p}<=0,~{}~{}~{}~{}~{}\forall
v\in\mathcal{V},~{}i\in I_{v},~{}n\in\mathcal{N}$ (9)

  * •

Constraint (10) ensures that each instance $i\in I_{v}$ of a VNF type $v$ is
installed on one physical node at most.

$\sum_{i\in I_{v}}\beta^{n}_{v_{i}}\leq 1,~{}~{}~{}~{}~{}\forall
v\in\mathcal{V},~{}n\in\mathcal{N}$ (10)

  * •

The total capacity of a particular VNF instance, needed to process all
procedures mapped to it, cannot exceed the absolute maximum capacity assigned
to that VNF. This constraint has been considered in other papers in the
literature such as [9] and [6].

$\zeta^{n}_{v_{i}}\leq\mathchoice{\leavevmode\hbox to15.45pt{\vbox
to10.09pt{\pgfpicture\makeatletter\hbox{\hskip 7.72626pt\lower-1.94443pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.72626pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\zeta^{max}_{v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to15.45pt{\vbox to10.09pt{\pgfpicture\makeatletter\hbox{\hskip
7.72626pt\lower-1.94443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.72626pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle\zeta^{max}_{v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.98pt{\vbox to7.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.4875pt\lower-1.3611pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.4875pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\zeta^{max}_{v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.1pt{\vbox to5.31pt{\pgfpicture\makeatletter\hbox{\hskip
5.05pt\lower-0.97221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.05pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\zeta^{max}_{v}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},~{}~{}~{}~{}~{}\forall
n\in\mathcal{N},~{}~{}v\in\mathcal{V},~{}~{}i\in I_{v}\leavevmode\hbox
to164.26pt{\vbox to28.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-31.09444pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-29.69444pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\vbox{\halign{#\hfil\cr\cr\vskip-10.00002pt\cr\hbox{{The
maximum computational capacity }}\cr}}\cr\vskip 0.0pt\cr\hbox{{assigned to the
\lx@glossaries@gls@link{main}{vnf}{\leavevmode VNF} $v$}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.35551pt}\pgfsys@lineto{0.0pt}{-30.89444pt}\pgfsys@lineto{163.85794pt}{-30.89444pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.35551pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (11)

  * •

Constraint (12) ensures that the total capacity used by all VNFs deployed in a
physical node $n$ does not exceed the maximum processing capacity of that
node.

$\sum_{v\in\mathcal{V}}\sum_{i\in
I_{v}}\zeta^{n}_{v_{i}}~{}.~{}\beta^{n}_{v_{i}}~{}~{}\leq
C^{max}_{n},~{}~{}~{}~{}~{}\forall n\in\mathcal{N}$ (12)

  * •

Constraint (13) ensures that a physical link $(n,m)$ is used by a particular
procedure, to map virtual link $(v_{i},z_{j})$, iff the two VNFs $v_{i}$ and
$z_{j}$ are mapped to nodes $n$ and $m$, respectively. This constraint is a
non-linear constraint.

$\begin{split}\chi^{(n,m),s}_{(v_{i},z_{j}),p}\leq\upgamma^{n,s}_{v_{i},p}~{}.~{}\upgamma^{m}_{z_{j},p}~{}\\\
\forall(m,n)\in\mathcal{L},~{}(v_{i},z_{j})\in\mathcal{R},~{}p\in\mathcal{P}\end{split}$
(13)

  * •

Constraint (14) ensures that the total bandwidth required by all procedures
that move between VNFs through a particular link, $(n,m)$, are limited by the
finite capacity of that link, $\zeta^{max}_{(n,m)}$

$\begin{split}\sum_{s\in\mathcal{S}}\sum_{p\in\mathcal{P}_{s}}\sum_{(v_{i},z_{j})\in\mathcal{R}_{p}^{s}}\lambda^{s}_{p}~{}~{}\chi^{(n,m),s}_{(v_{i},z_{j}),p}~{}\leq~{}b(n,m)\\\
~{}~{}\forall(n,m)\in\mathcal{L}\end{split}$ (14)

  * •

Constraint (15) certifies that the latency introduced by nodes’ processing and
network propagation can’t exceed the maximum tolerated latency of a particular
procedure.

$\delta^{s}_{p}\leq\delta^{s,max}_{p}~{}~{}~{}~{}~{}~{}\forall
s\in\mathcal{S},~{}p\in\mathcal{P}_{s}$ (15)

####  III-C3 Security constraints

Two security constraints are formulated in the model; VNF’s maximum traffic
and VNF exposure constraints. These constraints are explained as follows:

  * •

The VNF’s maximum traffic: This constraint ensures that the traffic processing
capacity $\zeta^{T,n}_{v_{i}}$ of a VNF instance $v_{i}$ should not exceed the
predefined maximum traffic processing capacity $\zeta^{T,max}_{v}$. Using this
constraint, the $\zeta^{T,max}_{v}$ for a critical VNF instance can be set at
a lower value, and hence it will not be shared which will protect the critical
VNF. This is represented in constraint (16).

$\zeta^{n,T}_{v_{i}}~{}\leq\zeta^{T,max}_{v},~{}~{}~{}~{}~{}\forall
v\in\mathcal{V},~{}~{}i\in I_{v},~{}n\in\mathcal{N}$ (16)

  * •

The VNF exposure: The VNF that is exposed to the outside network cannot be
assigned to more than one slice. A VNF is exposed to the outside network if it
the first VNF in the VNFs chain serving a procedure that is initiated by the
UE or the Radio Access Network (RAN). First, let $\Omega^{n,s}_{v_{i}}$
denotes to that VNF instance $v_{i}$ deployed in physical node $n$ is exposed
to the outside by the slice $s$. The $\Omega^{n,s}_{v_{i}}$ is calculated by
Equations (17) and (18)

$\begin{split}\sum_{p\in\mathcal{P}_{s}}\mathchoice{\leavevmode\hbox
to7.59pt{\vbox to7.81pt{\pgfpicture\makeatletter\hbox{\hskip
3.79515pt\lower-2.29443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.79515pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\eta^{s}_{p,v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.59pt{\vbox to7.81pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{$\textstyle\eta^{s}_{p,v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to5.35pt{\vbox to5.51pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\eta^{s}_{p,v}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.36pt{\vbox to4.65pt{\pgfpicture\makeatletter\hbox{\hskip
2.17882pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.17882pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\eta^{s}_{p,v}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\mathchoice{\leavevmode\hbox
to9.14pt{\vbox to10.44pt{\pgfpicture\makeatletter\hbox{\hskip
4.56946pt\lower-2.29443pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\displaystyle\psi^{s}_{p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to9.14pt{\vbox to10.44pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.56946pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\textstyle\psi^{s}_{p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.43pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip
3.21736pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.21736pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\psi^{s}_{p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to5.13pt{\vbox to5.97pt{\pgfpicture\makeatletter\hbox{\hskip
2.56598pt\lower-1.63889pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.56598pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\psi^{s}_{p}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}~{}\upgamma^{n,s}_{v_{i},p}\leq\mathcal{C}~{}\Omega^{n,s}_{v_{i}}\\\
\forall s\in\mathcal{S},v\in\mathcal{V},~{}i\in
I_{v},~{}n\in\mathcal{N}\leavevmode\hbox to172.53pt{\vbox
to82.66pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-44.3108pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\vbox{\halign{#\hfil\cr\cr\vskip-25.00005pt\cr\hbox{{Is
the \lx@glossaries@gls@link{main}{vnf}{\leavevmode VNF} $v$ the first on the
}}\cr}}\cr\vskip 0.0pt\cr\hbox{{\lx@glossaries@gls@link{main}{vnf}{\leavevmode
VNFs} sequence of the procedure?}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.32185pt}\pgfsys@lineto{0.0pt}{-44.11081pt}\pgfsys@lineto{172.133pt}{-44.11081pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.32185pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-8.84999pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\vbox{\halign{#\hfil\cr\cr\vskip
30.00005pt\cr\hbox{{Is the procedure $p$ sourced }}\cr}}\cr\vskip
0.0pt\cr\hbox{{externally?}}\cr}}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{5.4889pt}\pgfsys@lineto{0.0pt}{-10.04999pt}\pgfsys@lineto{115.56094pt}{-10.04999pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{5.4889pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{split}$ (17)

Where $\mathcal{C}$ is a parameter greater than the maximum number of
procedures mapped into the $v_{i}$ and sourced externally.

$\begin{split}\Omega^{n,s}_{v_{i}}-\sum_{p\in\mathcal{P}_{s}}\eta^{s}_{p,v}~{}\psi^{s}_{p}~{}\upgamma^{n,s}_{v_{i},p}\leq
0\\\ \forall s\in\mathcal{S},v\in\mathcal{V},~{}i\in
I_{v},~{}n\in\mathcal{N}\end{split}$ (18)

Then the constraint (19) ensures that the VNF instance $v_{i}$ must not
assigned to more than one slice.

$\sum_{s\in\mathcal{S}}\mathchoice{\leavevmode\hbox to14.76pt{\vbox
to10.3pt{\pgfpicture\makeatletter\hbox{\hskip 7.38205pt\lower-2.26079pt\hbox
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}\hbox{{$\displaystyle\Omega^{n,s}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{$\textstyle\Omega^{n,s}_{v_{i}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
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5.2213pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.2213pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptstyle\Omega^{n,s}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to9pt{\vbox to6.19pt{\pgfpicture\makeatletter\hbox{\hskip
4.49908pt\lower-1.91635pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.49908pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scriptscriptstyle\Omega^{n,s}_{v_{i}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\leq
1~{}~{}~{}~{}~{}\forall v\in\mathcal{V},~{}i\in
I_{v},~{}n\in\mathcal{N}\leavevmode\hbox to177.58pt{\vbox
to15.8pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-18.06078pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0} {
{}{}{}{}{}}{}{{}}{}{{{}{}}{}}{{}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{-16.66078pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\hbox{\small{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\vbox{\halign{#\hfil\cr\cr\vskip-5.00002pt\cr\hbox{{Indicating
whether $v_{i}$ is exposed externally}}\cr}}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}{}{}}{}{{}}{}{
{}}{{{}}{{}}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-5.32185pt}\pgfsys@lineto{0.0pt}{-17.86078pt}\pgfsys@lineto{177.17969pt}{-17.86078pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-5.32185pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (19)

##  IV System Setup

This section provides details about the system setup and values of the
parameter used to test the model.

The used solver: The proposed model is implemented using the JuMP modeling
language [19] which is embedded in Julia [20]. As our model contains a
combination of linear as well as non-linear constraints, the Solving
Constraint Integer Programs (SCIP) solver [21, 22] is employed to solve the
modeled problem. The SCIP is currently one of the fastest non-commercial
solvers available to solve problems of Mixed Integer Programming (MIP) and
MINLP classes [21]. The experiments are performed on a Linux machine that has
an Intel processor with $32$ cores and $32$ GB of RAM.

The environment set-up: A total of three simulated physical nodes with a
maximum of $30$ capacity units each are considered in these experiments. Each
VNF requires one capacity unit of the physical node to be deployed (or
activated) and one more capacity unit to serve one procedure in each traverse.
For instance, if a procedure needs to use a VNF more than once, then the VNF
will require the same capacity as the number of times the VNF is visited by
that procedure. Multiple VNFs can be deployed in a physical node and the total
used computational capacity of all the VNFs deployed in that physical node
cannot exceed its maximum capacity units (i.e. $30$ units). Physical nodes are
connected as a mesh topology. The links’ propagation delay between physical
nodes is set to $5ms$ for all links. Whereas the processing time each VNF
takes to process each request of the procedure is randomly assigned between
$0.5ms$ to $1ms$ based on uniform distribution. The parameters used in the
model along with their corresponding values are summarized in Table II.

TABLE II: Parameters used in the Model

Parameter | Value
---|---
Number of physical nodes | $3$
Maximum capacity of nodes |  $30$ capacity units
Network connectivity | Mesh topology
Physical link delay | $5ms$
Physical link maximum bandwidth |  $40$ bandwidth units
Number of VNFs | $14$
Maximum capacity of VNF instance |  $10$ capacity units
VNFs base capacity |  $1$ capacity unit
Maximum VNF traffic allowed |  $2$ (variable in some experiments)
VNFs delay unit | Random between $1000$ and $2000$ packets/sec
Number of instances per VNF | $4$
Number of Procedures | $4$
Allowed delay for procedure |  $1$ second
Number of slices | $2$

The simulation time limit: The SCIP solver is used with two parameters, the
maximum number of threads used by the solver and the time limit to solve the
model. In these experiments, the maximum number of threads is set to $6$
threads in order to run multiple experiments at the same time. However, we
noticed that SCIP only used a single thread at any given time while switching
between these threads during the run (i.e. SCIP did not use all $6$ threads
concurrently). The time limit, on the other hand, is set in order to obtain a
sub-optimal solution from the model in a timely manner. Limiting the time is
also considered by previous studies [7] to avoid the long time that the model
could take to solve a problem with a high number of input parameters. Figure 1
compares the objective value obtained as a function of multiple values of the
time limit. When the time limit is set to $30$ minutes, the model provided the
highest objective value. As the time limit increases, the objective value
starts to saturate. So in light of these results, the time limit is set to $3$
hours for all subsequent experiments and this same value is already used in
the literature [7].

Figure 1: Objective value as a function of different time limits

The implemented scenario: A simple network scenario is implemented in order to
obtain and analyze the results from the proposed optimization model. A total
of two slices are considered and each slice consists of three procedures.
Slice one requires registration with AMF re-allocation, handover, and
authentication procedures. However, slice two requires general registration,
handover, and authentication procedures. These procedures are described in the
next section. The number of procedures sourced externally and the maximum VNF
traffic are varied across the conducted experiments. Table III summarizes the
network configuration of the studied scenario.

TABLE III: Implemented Scenario

Number of slices | Two
---|---
Procedures for Slice# $1$ | 1) Registration with AMF re-allocation procedure
| 2) Handover procedure
| 3) Authentication procedure
Procedures for Slice# $2$ | 1) General registration procedure
| 2) Handover procedure
| 3) Authentication procedure
Number of external procedures | Variable
Maximum VNF traffic capacity | Variable

##  V The Implemented 5G Procedures

Although our model can support all existing 5G procedures defined by 3GPP in
[23], only four procedures are implemented in this work to test the visibility
and correctness of the model. These procedures are the general registration
procedure, registration with AMF re-allocation procedure, handover procedure,
and authentication procedure. In fact, implementing more procedures would
enlarge the time needed to get results out of the model. In this section,
those implemented procedures are described briefly and the sequence of their
serving VNFs is provided. More details on these procedures can be found in
3GPP technical specification 23.502 [23].

General Registration Procedure: This procedure enables the UE to register with
the 5G network to receive services. The UE can perform this procedure in
different scenarios like the initial registration to join the network, the
emergency registration to use the emergency services, etc. The sequence of
VNFs used by this procedure is as follows:
$UE{\rightarrow}\,RAN\,{\rightarrow}\,New\,AMF\,{\rightarrow}\,Old\,AMF\,{\rightarrow}\,New\,AMF\,{\rightarrow}\\\
AUSF\,{\rightarrow}\,UDM\,{\rightarrow}\,New\,AMF\,{\rightarrow}\,UDM\,{\rightarrow}\,New\,AMF\,{\rightarrow}\\\
SDM\,{\rightarrow}\,New\,AMF\,{\rightarrow}\,SDM\,{\rightarrow}\,New\,AMF{\rightarrow}PCF{\rightarrow}\\\
New\,AMF{\rightarrow}\,SMF{\rightarrow}New\,AMF{\rightarrow}UE{\rightarrow}New\,AMF$.
Since the UE and the RAN are not actual VNFs but do appear in the sequence, we
remove the UE and the RAN from the beginning of the procedures’ sequence while
implementing the procedures in our model. More details on this limitation are
explained in section VII.

Registration with AMF Re-Allocation Procedure: In this procedure, the initial
AMF redirects the registration-related traffic to the target AMF. For
instance, this can happen when the initial AMF cannot serve the UE, so a
change in the AMF is required in this case. One important thing to mention
here is that multiple types of AMFs can be seen in the sequence of VNFs, for
instance, initial AMF, target AMF, etc. In our model, we consider these
variants of AMFs as different VNFs to ensure that these VNFs are deployed
separately from each other. The sequence of VNFs used by this procedure is as
follows:
$RAN{\rightarrow}\,Initial\,AMF\,{\rightarrow}\,UDM\,{\rightarrow}\,Initial\,AMF\,{\rightarrow}\,NSSF\,{\rightarrow}\\\
Initial\,AMF{\rightarrow}\,Old\,AMF\,{\rightarrow}\,Initial\,AMF\,{\rightarrow}\,NRF\,{\rightarrow}\\\
Initial\,AMF\,{\rightarrow}\,RAN\,{\rightarrow}\,Initial\,AMF\,{\rightarrow}\,Target\,AMF$.

Handover Procedure: Handover is another important procedure that takes place
in cellular networks due to the mobility of the UEs. It can also be carried
out due to other reasons like load balancing or achieving QoS requirements.
Despite there being other more complex variants of the handover procedures
specified by 3GPP in [23], however, we only consider the simple version of the
handover in this work which is called ”Xn based inter NG-RAN handover without
UPF re-allocation”. The sequence of VNFs used in this procedure is as follows:
$Target\,RAN\,{\rightarrow}\,AMF\,{\rightarrow}\,SMF\,{\rightarrow}\,UPF\,{\rightarrow}\,SMF\,{\rightarrow}\,Source\\\
RAN\,{\rightarrow}\,Target\,RAN\,{\rightarrow}\,SMF\,{\rightarrow}\,AMF\,{\rightarrow\,}Target\,RAN\\\
\,{\rightarrow}Source\,RAN$.

Authentication Procedure: 3GPP defines two protocols to be used for
authentication procedure, Extensible Authentication Protocol-Authentication
and Key Agreement (EAP-AKA’) and 5G-Authentication and Key Agreement (AKA). In
this study, we only implement the EAP-AKA’. The selection of the
authentication protocol is performed by Unified Data Management
(UDM)/Authentication credential Repository and Processing Function (ARPF)
depending on the Subscription Permanent Identifier (SUPI) of the UE [24]. The
authentication procedure can be performed as part of other procedures such as
the registration procedure or UE-triggered service request procedure. The
sequence of VNFs serving this procedure is as follows:
$ARPF\,{\rightarrow}\,UDM\,{\rightarrow}\,AUSF\,{\rightarrow}\,SEAF{\rightarrow}\,UE\,{\rightarrow}\,SEAF\,{\rightarrow}\\\
AUSF\,{\rightarrow}\,SEAF\,{\rightarrow}\,UE$.

##  VI Results and Discussion

This section presents the results obtained from the proposed model. Various
performance metrics are considered to test the correctness and effectiveness
of the model including the impact of security constraints, the used capacities
of the physical nodes, and the number of activated VNF instances.
Additionally, the delay to complete the procedures based on the mapping of
VNFs to the physical nodes is computed. The main goal of this analysis is to
convey the benefits of the proposed security-aware optimization model along
with the cost or overhead of prioritizing security. These results only apply
to the parameters used and the environment tested. Hence, these results are
not a general trend but show how the model could be used.

###  VI-A Impact of the Exposure Constraint

The impact of the exposure security constraint is analysed in terms of the