Virtualized network services extension algorithms Omar Houidi ∗ , Oussama Soualah ∗ , Wajdi Louati ‡ , Djamal Zeghlache ∗ , and Farouk Kamoun † ∗ Telecom SudParis, Samovar-UMR 5157 CNRS, University of Paris-Saclay, France ‡ ReDCAD Lab, University of Sfax, Tunisia † CRISTAL Lab, National School of Computer Sciences, University of Manouba, Tunisia Email: {omar.houidi, oussama.soualah, djamal.zeghlache}@telecom-sudparis.eu, wajdi.louati@redcad.org, farouk.kamoun@ensi-uma.tn Abstract—This work presents algorithms to extend tenant Virtualized Network Functions Forwarding Graphs (VNF-FG) to embed new networking functions and introduce gradually additional services into the already deployed tenant dedicated slices. The related service graph extension problem is addressed through an Integer Linear Programming (ILP) algorithm that serves as a reference for performance comparisons with a proposed heuristic algorithm. An eigendecomposition approach is proposed, evaluated and compared. Index Terms—Network Functions Virtualization, VNF For- warding Graph Extension, Integer Linear Programming, Eigen- decomposition. I. I NTRODUCTION Network Functions Virtualization (NFV) [1] decouples net- work functions (such as NATs, ﬁrewalls, load balancers, etc.) from dedicated hardware devices to enable hosting of network services, known as Virtualized Network Functions (VNFs), on commodity hardware (such as switches or servers). NFV facilitates and accelerates service deployment and management by providers. NFV coupled with Software Deﬁned Networks (SDN) is expected to transform the business to business to consumer relationships and lead to new business models and value chains. This expected evolution generates additional needs, especially the need to extend already deployed tenant slices as demand and business grow. This work addresses this foreseen extension need where tenants will request additional networking services and functions to expand their deployment and service coverage. Providers that host these services and network functions in their infrastructures require dedicated extension algorithms of previously deployed tenant slices. There are several reasons for these extensions such as growing customer demand, increasing trafﬁc, introduction of new services and networking functions, addition of security and protection slices next to the tenant slices or simply a need to expand geographical coverage and interconnecting the extensions with previous slices. The providers will thus need to easily expand, add, insert, replace and remove services in and from already deployed slices. The extensions must be seamless to applications and ser- vices that do not tolerate interruptions, migration or disruptions in quality of service and experience. In order to meet these requirements, this paper proposes algorithms to extend already deployed network services and functions graphs to respond to new demands while taking into account the constraint of minimizing the impact on the original service graphs. According to [2] and [3], Virtual Network Function scaling and extension are triggered by new client requirements and the rising of network load over time. This is especially true for services with periodic resource demands. In this paper, we refer to the extension as a VNF-FG extension to emphasize networking level extensions of already deployed forwarding graphs in addition to the fact that we inherently answer the needs for service function chaining extensions. The extensions can be triggered by new requests or by the VNF-FG life cycle management that may decide to extend the initial graph to adapt to network and trafﬁc load changes such as adding load balancers, spawning new VNFs to absorb increasing load, etc. We ﬁrst propose an ILP model as an exact formulation of the VNF-FG extension problem and use the ILP for small problem sizes and as a reference for performance evaluation and comparison with a proposed heuristic algorithm. To reduce complexity while ﬁnding good solutions, we resort to an Eigendecomposition based algorithm using network adjacency matrices of the requested and hosting graphs as a basis for optimal embedding. The ILP and the heuristic algorithm ensure that the speciﬁed connecting points between the previously described graphs and the extensions are respected as speciﬁed in the client requests and that there is no disruption to the initially deployed VNF-FG. We compare the performance of these algorithms in terms of rejection rate of new requests, quality of the solutions and system load. Section II of this paper presents the related work. The prob- lem formulation is described in Section III. The proposals are introduced in Section IV. Section V reports the performance evaluation results. Section VI summarizes the main ﬁndings. II. RELATED WORK Since we are addressing the extension of service function chains already deployed in hosting infrastructures, the related work review focusses on the virtual network topology change problem with emphasis on the dynamic expansion and adap- tation of VNF-FGs whenever relevant. Indeed, previous work addressed partially (often did not address extension of already deployed services but rather initial placement) adaptation to changes of an existing (already deployed) graph and reaction to faults. We consequently limit the description to work as close as possible to ours, being aware that extensions of Service Function Chains (SFCs) or VNF-FGs have not been fully addressed in the past. VNF-FG placement and chaining is formulated as an ILP in [4], [5], and [6] to ﬁnd exact hosting solutions for the requested service graph. As the addressed problem is NP- Hard, the exact solutions do not scale with size and require an excessive amount of time to ﬁnd the optimal solutions. Heuristic algorithms (e.g., [7], [8]) are typically proposed to scale better with problem size by solving the problem iteratively to ﬁnd solutions much faster. Unfortunately, this is sometimes accomplished at the expense of quality of 978-1-5386-7659-2/18/$31.00 c 2018 IEEE