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per is organized as follows: Section II discusses re-
lated work on NFEP and the characteristics of the
optimization models introduced in the related papers.
Section III introduces the optimization model, its con-
straints, variables, and objective function. Section IV
describes the modeling environment and results. We
conclude the work in Section V.
2 RELATED WORK
Solving NFEP is known to be NP-hard (Sahhaf et al.,
2015). Designing a heuristic algorithm can be a solu-
tion for this matter. As our aim is to study the effect
of a multi-hop wireless network on the placement of
NFs, we only consider exact solutions in this paper.
The exact solution in most of the proposed approaches
formulates NFEP using Linear Programming and can
be differentiated based on the constraints and the ob-
jective function.
The amount of work on NFEP is considerable.
Most of the works in this domain are related to wired
networks. In (Bouet et al., 2015), the authors used
Integer Linear Programming (ILP) in order to find
an optimum solution for placing Deep Packet Inspec-
tion (DPI) as a VNF in the network. In the proposed
method, the objective function is to minimize the acti-
vation and maintenance cost of the virtual DPI (vDPI)
and the considered constraint is the network’s avail-
able bandwidth. In (Mohammadkhan et al., 2015)
the authors used Mixed Integer Linear Programming
(MILP) to find an optimum solution. The proposed
optimization is based on maximizing the number of
services that can be supported in a switch. In this so-
lution, the constraints are based on the number of free
cores, tolerable delay of flows and links’ bandwidth.
The objectives of (Mohammadkhan et al., 2015) are
minimizing maximum link utilization and maximum
core utilization, which leads to the distribution of load
between available resources. Leivadeasa et al. pro-
posed a Mixed Integer Programing (MIP) formula-
tion with the nodes’ capacity and the bandwidth of
the links as the constraints in (Leivadeas et al., 2017).
It considers minimizing activation, maintenance cost,
and load balancing among the resources as the objec-
tive function (Leivadeas et al., 2017). (Botero et al.,
2012) is another proposed method based on an ILP
which aims at minimizing the resource consumption
and energy saving by turning off unused resources.
Finally, as mentioned earlier, (Sahhaf et al., 2015) is
considering the available resources of the nodes, the
available bandwidth of the links and the requested
QoS as constraints and minimizes the resources us-
age.
The topic of NFV in wireless networks has re-
ceived a significant attention in the literature, where
most of the focus is on wireless network virtualiza-
tion. (Riggio et al., 2015) talks about virtual WiFi
where kernelbased virtual machines are used as a vir-
tual wireless LAN device. The authors of (Riggio
et al., 2015) provide an integer linear programming
for placing the VNFs in a hybrid wireless network
were there are forwarding nodes, some with process-
ing capacity, and some are access points. In this
paper, the optimization method is designed without
considering the effect of interference. Its authors as-
sumed that Orthogonal Frequency Division Multiple
Access (OFDMA) is being used in order to handle the
problem of interference. (Lv et al., 2012) considers
the embedding of virtual wireless mesh gateways and
the virtual links between them. The problem of inter-
ference between the wireless links has been solved by
considering multi-radio multi-channel networks. Its
authors assign orthogonal channels to the neighbor-
ing links. The same method has been used in (Park
and Kim, 2009) where the interference is being han-
dled separately from the optimization model. To our
knowledge, none of the papers considering NFEP in
the wireless networks include the effect of interfer-
ence in their optimization model. We model the inter-
ference and provide the related formulation in order
to consider interference as one of the constraints.
3 OPTIMIZATION METHOD
As the service requests arrive over time, the embed-
ding algorithm decides where to place the NFs in the
physical network subject to various constraints. Each
request has an associated duration. If the request is
accepted, the required resources will be assigned and
when the request expires the used resources will be
released.
We are using Integer Linear Programming (ILP)
as the optimization method. ILP consists of two parts,
an Objective function which calculates the cost of
each mapping and chooses the one with the lowest
cost, and the constraints, which apply the limitations
we have with regard to the resources in the physi-
cal network. ILP will choose the mapping that sat-
isfies all of the constraints and minimizes the objec-
tive function. In this section, we define the variables,
constraints and the objective function similar to the
optimization method in (Sahhaf et al., 2015) and then
introduce the extension of the model and the added
constraint for multi-hop wireless networks.
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