Ipact of Processing Costs on Service Chain Placeent in Network Functions Virtualization Marco Savi, Massio Tornatore, Giacoo Verticale Dipartiento di Elettronica, Inforazione e Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 3, Milano, Italy firstnae.lastnae@polii.it Abstract The Network Functions Virtualization (NFV) paradig is the ost proising technique to help network providers in the reduction of capital and energy costs. The deployent of virtual network functions (s) running on generic x hardware allows higher flexibility than the classical iddleboxes approach. NFV also reduces the coplexity in the deployent of network services through the concept of service chaining, which defines how ultiple s can be chained together to provide a specific service. As a drawback, hosting ultiple s in the sae hardware can lead to scalability issues, especially in the processing-resource sharing. In this paper, we evaluate the ipact of two different types of costs that ust be taken into account when ultiple chained s share the sae processing resources: the upscaling costs and the context switching costs. Upscaling costs are incurred by s ulticore ipleentations, since they suffer a penalty due to the needs of load balancing aong cores. Context switching costs arise when ultiple s share the sae CPU and thus require the loading/saving of their context. We odel through an ILP proble the evaluation of such costs and we show their ipact in a s consolidation scenario, when the x hardware deployed in the network is iniized. I. INTRODUCTION Current fixed and obile network operators are struggling to reduce capital and energy costs and increase their revenues by sartly upgrading their network infrastructure. In the last years, an innovative network architecture paradig called Network Functions Virtualization (NFV) [1] has eerged as a proising technique to help network operators in achieving this goal. NFV is based on the concept of network function. A network function is an abstract building block whose ai is to process the network traffic to accoplish a specific task. Exaples of network functions are Firewalls, Network Address Translators, Traffic Monitors etc. Nowadays, the network functions are usually ipleented on dedicated hardware, usually referred as iddleboxes. Such iddleboxes are able to handle heavy traffic loads, but they are expensive and inflexible, since they are designed to perfor liited operations on the traffic, and they ust be diensioned at peak loads, leading to waste of resources when the traffic is low, e.g., in off-peak hours. The NFV paradig consists in oving fro an hardware to a software ipleentation of network functions in a virtualized environent. This way, ultiple and heterogeneous virtual network functions (s) can be hosted by the sae generic x hardware. NFV adds flexibility to the network since it allows network operators to efficiently diension and dynaically consolidate the s. Another value added by NFV is the siplicity in the deployent of heterogeneous network services. In fact NFV can exploit the concept of service chaining [], according to which a service (e.g., web browsing, VoIP, video streaing, online gaing etc.) can be provided by one or ore service chains (SCs), i.e., a concatenation of appropriate s that ust be crossed by the traffic associated to that specific service. NFV has also soe drawbacks. Moving fro an hardware to a virtualized software ipleentation can lead to scalability issues in the resource sharing, especially concerning processing. In this paper, we evaluate the ipact of processing-resource sharing aong ultiple s in a consolidation scenario. We show how the size and the nuber of s sharing the sae x hardware influence the hardware processing perforance, and how the deployent in the network of ultiple SCs using such s is affected. Indeed, consolidating ultiple s in the sae hardware leads to inefficiencies due to the need of saving/loading the context (i.e., the state) of the s sharing the processing resources (context switching costs) [3][], as well as to inefficiencies due to the needs of ulti-core ipleentations (upscaling costs) [5]. To the best of our knowledge, this paper is the first study evaluating such processing-related costs in a consolidation scenario. The reainder of the paper is organized as follows. Section II discusses soe related works. In Section III we introduce our syste odel, showing how we odel the physical network, the s, the SCs and the processing-related costs. Section IV introduces our optiization proble for the evaluation of the effects of upscaling and context switching costs in a consolidation scenario. In Section V we show the nuerical results and Section VI draws the conclusion of our work. II. RELATED WORKS In our paper, we develop an optiization proble for SC and placeent that can be seen as an extension of soe well-known Virtual Network Ebedding (VNE) probles, as the ones shown in [][7][]. In our proble, the SCs can be seen as siple-chain virtual graphs, where the chained s are virtual nodes connected together by virtual links. Such SCs ust be ebedded in a physical network, where each virtual link can be apped to a physical path [], ultiple
NFV node NFV node SC1 SC n 1 3 s sharing the sae node v 3 shared by different SCs v 5 v 7 NFV node v 1 v Fig. 1. Physical topology where soe nodes are equipped with generic ulti-core x hardware (i.e., NFV nodes) v v v virtual nodes can be apped to the sae physical node [7], and the virtual nodes ust be consolidated []. With respect to Refs. [][7][], in our SCs ebedding proble it ust be also guaranteed that a virtual node (i.e., a ) can be shared aong ultiple virtual networks (i.e., SCs). Ref. [9] is the first work investigating the optial placeent of SCs in the network following a VNE approach. The authors first foralize the concept of SC and, then they forulate a Mixed Integer Quadratically Constrained Proble to evaluate the optial placeent of s and SCs. In our paper, we develop a siilar optiization proble, but we extend the analysis to cover also the processing-related costs for the physical nodes. Soe other studies deal with efficient placing of SCs and s in the network. Ref. [] proposes an efficient algorith for the counication aong different chained s that are hosted by the sae generic hardware, while Ref. [10] introduces an algorith to efficiently handle traffic load variations by dynaically instantiating s. Ref. [11] defines an online algorith for joint Virtual Machine placeent, routing and consolidation with the ai of iniizing traffic costs. Our approach is siilar, but we investigate the iplications of such strategies fro a processing-resource sharing perspective. III. SYSTEM MODEL A. Physical topology and NFV nodes odeling We odel the physical network as a connected directed graph G = (V, E). All the network nodes v V have forwarding capabilities. Then, the links (v, v ) E connecting the nodes are high-capacity fiber links. The network nodes v can also be equipped with standard x hardware (see Fig. 1), that can be used to host different s. The x hardware can have different capabilities in ters of storage and processing, and can potentially be hosted in every powered physical location (e.g., in a cabinet, in a central office, in a core exchange, etc.). In this paper, we focus on the processing aspect and we generally call the nodes hosting generic x hardware NFV nodes. We assue that the NFV nodes are ulti-core systes, and we easure the processing capability of each NFV node in ters of the nuber of CPU cores γ v it is equipped with. Clearly, if γ v = 0 the node v has not processing capability. Fig.. Exaple of SCs that ust be ebedded in the physical network, where different s can share the processing resources of the sae NFV node and ultiple SCs can share the sae B. Virtual network functions odeling In general, a f F can be seen as a black-box perforing soe operations on the network traffic. Every f hosted by a NFV node v ust be assigned a dedicated processing capability c f,v in order to fulfil all the operations it has to perfor on the input traffic. The processing capability c f,v represents the fraction of the overall processing capability γ v of the node v assigned to the f, expressed in ters of nuber of CPU cores. We say that a has a larger size when it is assigned a larger processing capability c f,v. We assue that a f can be shared aong different SCs (e.g., in Fig. ) each requiring a fraction π f of the processing capability c f,v. Note that heterogeneous s require a π f that can largely vary fro one to each other, depending on the coplexity of the operations perfored. In our work we assue that π f has been optially chosen, i.e., for each SC, traffic is queued for a negligible tie before it is processed. In other words, every is designed in such a way it does not introduce additional latency due to a bad or inadequate ipleentation. C. Service chains odeling When a service is required between two end-points, one or ore SCs c C ust be deployed in the network. In general, heterogeneous SCs can be deployed, and soe SCs can be shared aong different services. Without any loss of generality, we consider a one-to-one correspondence between a service and a SC. A SC c can be odelled by a siplechain graph C c = (X c U c, G c ), where X c is the set of start/end points u, U c is the set of requests u and G c is the set of virtual links (u, u ) chaining consecutive requests/end points u and u. Fro a topological point of view, both requests and end points are virtual nodes u X c U c. Note that, siilarly to [9], in our odel we decouple the concepts of f and of request u: for every SC c, every request u U c is apped to a specific f. We then introduce the apping paraeter τ c u F to specify the apping of the request u for the SC c to a specific f. Siilar considerations can be done for the end points u X c.
We assue that such end points are fixed in the network, and we introduce the apping paraeter η c u V to specify the apping between the end point u for the SC c to a specific physical node v. In our odel, every single SC c can serve an aggregate set of users. Such aggregate SCs are deployed when ultiple users require the sae service between the sae two end points. Every SC is also associated to soe perforance/qos paraeters such as: The requested bandwidth δu,u c, i.e., the bandwidth that ust be guaranteed between two requests/end points u and u to provide the service offered by the SC c. Note that every virtual link (u, u ) can be associated to a different bandwidth requireent because the chained s can lead to a change in the traffic throughput. The axiu tolerated latency ϕ c, i.e., the axiu delay that can be introduced by the network without affecting the service between the two end points of the SC c. D. Processing-related costs for the NFV nodes The processing-related costs (upscaling and context switching) for a NFV node v are ainly influenced by: The size of the s f hosted by the node v; The nuber of requests u apped to the s f, which share the processing resources of the NFV node v. Upscaling costs: When a f ust process a high quantity of traffic, it ight need ore processing resources than the ones provided by a single CPU core. In this case, a ulti-core ipleentation of the is needed and soe upscaling costs ust be taken into account. These costs arise because of the software architectural challenges that ust be addressed when a ulti-core is ipleented. In fact, the traffic that ust be handled by ulti-core s needs to be balanced aong different CPU cores by a load balancer. The needs of load balancing add a new layer in the syste architecture that can becoe a bottleneck, ust be carefully designed and leads to soe perforance penalties [1]. For this reason, the higher is the nuber of CPU cores used by the ulti-core, the higher are the upscaling costs. We then assue that such costs can be odelled as a step function of the nuber of CPU cores used by the. Context switching costs: When ultiple s are hosted by the sae NFV node, the CPU cores are shared aong such different s. This leads to another kind of costs, i.e., the context switching costs. These costs are related to the effort that ust be taken into account in order to execute ultiple s in the sae CPU, and we assue that they linearly increase with the nuber of requests apped into the s that share the CPU cores γ v of the NFV node v. These costs are strictly related to the needs of cache sharing aong different s and of saving/loading the context (i.e., the state) of a for CPU resource sharing [13][1]. These considerations show how a size/nuber trade-off on the s sharing the sae node exists, since a node TABLE I SUMMARY OF GRAPHS AND SETS CONSIDERED IN THE MODEL Graph/Set G = (V, E) C C c = (X c U c, G c ) F Description Physical network graph, where V is the set of physical nodes v and E is the set of physical links (v, v ) connecting the nodes v and v Set of the service chains c that ust be ebedded in the physical network G Siple-chain graph for the SC c, where X c is the set of fixed start/end point u, U c is the set of requests u, G c is the set of virtual links (u, u ) connecting the request (or start point) u and the request (or end point) u Set of s f that can be requested and deployed in the network TABLE II SUMMARY OF PARAMETERS CONSIDERED IN THE MODEL Paraeter Doain Description τu c c C requested by the request u ηu c u U c c C in the SC c (τu c F ) Physical node where the start/end point u X c u for the SC c is apped to (ηu c V ) π f f F Fraction of the CPU processing required by each request u for the f γ v v V Nuber of the CPU cores hosted by the node v β v,v (v, v ) E Bandwidth capacity of the physical link (v, v ) λ v,v (v, v ) E Latency of the physical link (v, v ) µ v v V Upscaling latency of the node v ω v v V Context switching latency of the node v κ v v V Upscaling processing of the node v ξ v v V Context switching processing of the node v δu,u c c C Requested bandwidth on the virtual link (u, u ) G c (u, u ) for the SC c ϕ c c C Maxiu tolerated latency by the SC c M Big-M paraeter hosting a big nuber of sall s ust deal with big context switching costs and sall upscaling costs, while a node hosting a sall nuber of big s leads to sall context switching costs but big upscaling costs. We assue that the upscaling and context switching costs lead to two different perforance degradation effects: Increase of the latency introduced by the NFV node. In fact, both deciding how to balance traffic aong different CPU cores and saving/loading the s context require soe coputational tie. We call µ v the upscaling latency and ω v the context switching latency for the NFV node v. Decrease of the actual processing capability of the NFV node. In fact, both taking a decision about how to balance traffic aong different cores and saving/loading the s context require soe dedicated processing capability, that in turn cannot be used by the s hosted in that node. We call κ v the upscaling processing and ξ v the context switching processing for the NFV node v.
TABLE III DECISION VARIABLES FOR THE ILP MODEL Variable Doain Description c u,v {0, 1} c C Binary variable such that u U c c u,v = 1 iff the v V request u for the SC c is apped to the node v, otherwise c u,v = 0 c f,v [0, γ v] v V f F fraction of the CPU cores in the node v used by the f Real variable indicating the i f,v {0, 1} f F Binary variable such that v V i f,v = 1 iff the f is hosted by the node v, otherwise i f,v = 0 e c v,v,x,y,u,u {0, 1} c C Binary variable such that (v, v ) E e c v,v x V,x,y,u,u = 1 iff the physical link (v, v ) y V belongs to the path between (u, u ) G c the nodes x and y, where the requests u and u for the SC c are apped to, otherwise e c v,v,x,y,u,u = 0 a v {0, 1} v V Binary variable such that a v = 1 iff the node v is active, otherwise a v = 0 IV. OPTIMIZATION PROBLEM FOR S CONSOLIDATION In this Section we forulate an Integer Linear Prograing (ILP) odel for consolidation. A suary of the sets, paraeter and variables used in the odel is reported in Tables I, II, III. Given a physical network topology and soe SCs, we want to optially decide the position and the size of the chained s while iniizing the nuber of active NFV nodes (i.e., the nodes hosting at least one ) in the network. The constraints are grouped in three categories: requests placeent constraints, routing constraints and perforance constraints. The requests placeent constraints ensure a correct apping of the s f on the NFV nodes v as well as a correct apping between the requests u and the s f. The routing constraints guarantee a correct apping of the virtual links (u, u ) on physical paths constituted by different links (v, v ). Finally, the perforance constraints are related to all the perforance requireents that ust be addressed in the network. 1) Objective function: in a v (1) v V The objective function iniizes the nuber of active NFV nodes. This way, we try to consolidate as uch as possible the s. This optiization proble can be useful for network operators to plan the best strategical placeent of the x hardware. The nuber of active nodes in the network is a easure of the cost for NFV ipleentation. ) Requests placeent constraints: c u,η = 1 c C, u u c Xc () c u,v = 0 c C, u X c, v V : v η c u (3) The fixed start/end point u of a SC c is apped to the node v specified by the paraeters η c u (constraint ) and it is not apped to any other node (constraint 3). c u,v = 1 c C, u U c () v V Every request u for each SC c ust be apped to exactly one node v. c C c u,v c f,v π f f F, v V (5) The overall nuber of requests u apped to the f hosted by the node v cannot overcoe c f,v /π f, i.e., the axiu nuber of requests u for the f that can be apped to that node, according to the processing requireent π f per request u and the fraction of CPU cores c f,v of the node v assigned to the f. c f,v M i f,v f F, v V () i f,v c f,v < 1 f F, v V (7) Constraints and 7 ensure that i f,v = 0 if c f,v = 0 and that i f,v = 1 if c f,v > 0. i f,v is a flag variable used to understand if the f is apped to the node v. M is a big-m paraeter, greater than the axiu value taken by c f,v, i.e., M > ax v V {γ v }. i f,v c C c u,v f F, v V () The f is placed on a node v only if there is at least one request u apped to it. c f,v γ v ψ v v V (9) f F For each node v, the overall CPU processing assigned to the s f cannot overcoe the actual processing capability of the node, expressed in ters of CPU cores. The actual CPU processing capability of the node v is the difference between the CPU processing capacity γ v of the node and the overall upscaling/context switching processing costs ψ v. Such costs are taken into account every tie a request u is apped to a f. We define ψ v in the following way: ψ v = c C, f F c u,v ( c f,v κ v + i g,v ξ v ) v V (10) g F The right-hand side of ψ v requires the product between the binary variable c u,v and the real variable c f,v κ v + g F i g,v ξ v. Such product can be linearized, as well as the ceiling function c f,v. 3) Routing constraints: e c v,v,x,y,u,u c u,x c u,y c C, (v, v ) E, x V, y V, (u, u ) G c (11) A physical link (v, v ) can belong to a path between two nodes x and y for a virtual link (u, u ) of the SC c only if the two consecutive requests or start/end points u and u are apped to these nodes. The product c u,x c u,y of binary
variables can be linearized. e c x,v,x,y,u,u c u,x c u,y = 1 (x,v) E, x,y V e c v,y,x,y,u,u c u,x c u,y = 1 (v,y) E, x,y V c C, (u, u ) G c (1) c C, (u, u ) G c (13) The virtual link (u, u ) between two consecutive requests or start/end points u and u starts in one of the links connected to the node x, where the request or start point u is apped to (constraint 1), and it ends in one of the links of the node y, where the request or end point u is apped to (constraint 13). These constraints are called source and destination constraints, and the products e c x,v,x,y,u,u c u,x c u,y and e c v,y,x,y,u,u c u,x c u,y of binary variables can be linearized. e c v,x,x,y,u,u = 0 (v,x) E, v V c C, x V, y V, x y, (u, u ) G c (1) e c y,v,x,y,u,u = 0 (y,v) E, v V c C, x V, y V, x y, (u, u ) G c (15) While apping the virtual link (u, u ) for the SC c on a physical path between the nodes x and y where the requests or start/end points u and u are apped to, no incoing link for the node x (constraint 1) and no outgoing link for the node y (constraint 15) is considered. e c v,w,x,y,u,u = (v,w) E, v V (w,v ) E, v V e c w,v,x,y,u,u c C, w V, x V, y V, x w, y w, (u, u ) G c (1) e c v,w,x,y,u,u 1 (v,w) E, v V c C, w V, x V, y V, x w, y w, (u, u ) G c (17) While considering a generic node w (other than the source node x and the destination node y of a virtual link (u, u )), if one of its incoing links belongs to the path between the nodes x and y, then also one of its outgoing links ust belong to the path (constraint 1). Without constraint 17 ultiple incoing/outgoing links could be considered, but we deal with unsplittable flows. Constraints 1 and 17 are called transit constraints. e c v,v,x,y,u,u = 0 c C, v V, x V, y V, x y, (u, u ) G c (1) e c v,v,x,x,u,u = 0 c C, x V, (u, u ) G c, (v, v ) E, v v (19) In the physical network G = (V, E), every node v is connected to a self-loop (v, v) with infinite bandwidth (β v,v = ) and zero latency (λ v,v = 0). This self-loop is used when two consecutive requests or start/end points u and u are apped to the sae node x, and cannot be used otherwise (constraint 1). Moreover, no physical link (v, v ) different TABLE IV REQUIREMENTS FOR THE DEPLOYED SERVICE CHAINS Service Chain Chained s δ ϕ Web Service NAT-FW-TM-WOC-IDPS 100 kbit/s 500 s VoIP NAT-FW-TM-FW-NAT kbit/s 100 s Video Streaing NAT-FW-TM-VOC-IDPS Mbit/s 100 s Online Gaing NAT-FW-VOC-WOC-IDPS 50 kbit/s 0 s NAT: Network Address Translator, FW: Firewall, TM: Traffic Monitor, WOC: WAN Optiization Controller, IDPS: Intrusion Detection Prevention Syste, VOC: Video Optiization Controller fro the self-loop is used when the requests or start/end points u and u are apped to the sae node x (constraint 19). ) Perforance constraints: e c v,v,x,y,u,u δc u,u β v,v (v, v ) E (0) c C, x,y V (u,u ) G c The overall bandwidth δu,u c requested by the virtual links (u, u ) of every SC c and apped to the physical link (v, v ) cannot exceed the capacity of the link β v,v. e c v,v,x,y,u,u l v,v + σc ϕ c c C (1) (v,v ) E, x,y V (u,u ) G c The latency introduced by the network between the start and end point of a SC c cannot overcoe the axiu tolerated latency ϕ c. The first ter of the left-hand side of constraint 1 is related to the latency introduced by all the physical links (v, v ) of the paths to which the virtual links (u, u ) are apped, while the ter σ c is related to the latency introduced by the nodes because of upscaling and context switching. We define σ c in the following way: σ c = c u,v ( c f,v µ v + i g,v ω v ) c C () g F f F, v V The right-hand side of σ c requires the product between the binary variable c u,v and the real variable c f,v µ v + g F i g,v ω v. Such product can be linearized, as well as the ceiling function c f,v. i f,v M a v v V (3) f F a v i f,v v V () f F Constraints 3 and assure that a node is arked as active (i.e., a v = 1) only if at least one f is hosted by that node. The big-m paraeter M ust be chosen such that M > F. V. SIMULATION RESULTS A. Description of the siulation settings We solved the optiization proble shown is Section IV using CPLEX. We consider the physical network topology shown in Fig. 1 with ten physical nodes ( V = 10). This network topology is taken for the Internet network [15], considering only the nodes with advanced layer 3 services. We defined four different types of SCs that can be deployed in the network (see Table IV). Such SCs represent four different services, i.e., Web Service, VoIP, Video Streaing and Online Gaing. For every SC the traffic flows through a given
Active NFV nodes SCs: κ/γ = 0% κ/γ = 3% κ/γ = 9% SCs: κ/γ = 0% κ/γ = 3% κ/γ = 9% 500 1,000 1,500,000 Fig. 3. Nuber of active NFV nodes as a function of the nuber of users in the network, while considering the ipact of different upscaling costs and different nubers of deployed SCs in the heterogeneous scenario Active NFV nodes SCs: ξ/γ = 0% ξ/γ = 3% ξ/γ = 9% SCs: ξ/γ = 0% ξ/γ = 3% ξ/γ = 9% 500 1,000 1,500,000 Fig.. Nuber of active NFV nodes as a function of the nuber of users in the network, while considering the ipact of different context switching costs and different nubers of deployed SCs in the heterogeneous scenario sequence of s and two end points; soe s are shared aong the different SCs. Table IV shows also the requireents in ters of bandwidth δ and axiu tolerated latency ϕ for each SC: we assue that every virtual link (u, u ) of a SC requires the sae bandwidth and that all the nodes can be both NFV nodes and start/end points for the SCs. Every SC can serve an aggregate traffic fro ultiple users. We assue that all the nodes can potentially host the sae nuber γ of CPU cores and that they incur in the sae upscaling costs κ and the sae context switching costs ξ. We show our results considering two different scenarios: an heterogeneous scenario and an hoogeneous scenario. In the heterogeneous scenario, we uniforly randoize the choice of start/end points (aong the 10 nodes of the network) and of the deployed SCs (aong the four different types of SCs). In the hoogeneous scenario, we only uniforly randoize the choice of start/end points, while we assue that only one type of SC is deployed in the network. B. Heterogeneous scenario Figure 3 shows the ipact of the upscaling costs κ (noralized to the overall processing capability γ of the node) as a function of the nuber of users and SCs deployed in the network when the context switching costs ξ are negligible. The total nuber of users in the network is equally split aong the nuber of deployed SCs. We consider, for each point in the graph, 50 randoized instances and the deployent of Active NFV nodes 10 WS: κ/γ = 0% κ/γ = 3% κ/γ = 9% OG: κ/γ = 0% κ/γ = 3% κ/γ = 9% Het: κ/γ = 0% κ/γ = 3% κ/γ = 9% 500 1,000 1,500,000 Fig. 5. Nuber of active NFV nodes as a function of the nuber of users in the network, while considering the ipact of different upscaling costs in the hoogeneous scenario, i.e., when only Web Service (WS) chains or Online Gaing (OG) chains are deployed ( SCs in the network) Active NFV nodes 10 WS: ξ/γ = 0% ξ/γ = 3% ξ/γ = 9% OG: ξ/γ = 0% ξ/γ = 3% ξ/γ = 9% Het: ξ/γ = 0% ξ/γ = 3% ξ/γ = 9% 500 1,000 1,500,000 Fig.. Nuber of active NFV nodes as a function of the nuber of users in the network, while considering the ipact of different context switching costs in the hoogeneous scenario, i.e., when only Web Service (WS) chains or Online Gaing (OG) chains are deployed ( SCs in the network) SCs and SCs. Each instance can be solved in few inutes. All the curves show a onotonically increasing trend. In fact, increasing the nuber of users in the network requires an additional processing capability and thus a higher nuber of active NFV nodes. If also the upscaling costs are negligible (κ/γ = 0%), the nuber of NFV nodes in the network is alost the sae as the nuber of SCs in the network. This eans that if no processing costs (upscaling and context switching) are considered, increasing the nuber of deployed SCs does not affect the nuber of active NFV nodes in the network, as long as the overall nuber of users reains the sae. When the upscaling costs are not negligible, increasing the nuber of SCs in the network has always an ipact on the nuber of active NFV nodes (κ/γ = 3%, κ/γ = 9%), but the relative difference, considering the deployent of SCs and SCs, is alost the sae for both κ/γ = 3% and κ/γ = 9% (around one active node). This happens because splitting the nuber of users aong a higher nuber of SCs does not strongly ipact on the size of the hosted s. Note that with upscaling costs κ/γ = 9% soe loads result in infeasible probles and, thus, no points are reported in the graph. Figure shows instead the ipact of the context switching costs ξ when the upscaling costs are negligible. Also in this
case, the curves have a onotonically increasing trend. We can see then how an increase in the nuber of SCs deployed in the network has an ipact not only in ters of the absolute nuber of active nodes in the network, but also in ters of the relative difference between the nuber of active nodes in the case of deployent of SCs or SCs. In fact, Fig. shows how the difference in the nuber of active nodes is uch higher for ξ/γ = 3% than for ξ/γ = 9%: for ξ/γ = 3% such difference is around one active node, while for ξ/γ = 9% this difference is around three active nodes. This happens because the nuber of the overall s requests is higher when SCs are deployed than when only SCs are deployed, and this leads to higher context switching costs. C. Hoogeneous scenario For the hoogeneous case, we consider two different cases: in the first case, only Web Service (WS) chains are deployed, in the second only Online Gaing (OG) chains. Figure 5 shows the ipact of the upscaling costs in such cases, in coparison with the heterogeneous (Het) scenario. The nuber of deployed SCs is fixed at SCs. The deployent of an hoogeneous type of SC does not ipact so uch on the nuber of active nodes, in fact the curves of the hoogeneous (WS and OG) and heterogeneous (Het) cases are ore or less overlapped while considering the sae value of κ/γ. In average, the deployent of OG chains ipacts a bit less than the deployent of WS chains, for all the considered values of κ/γ, ainly due to the looser bandwidth requireents. While coparing such hoogeneous cases with the heterogeneous case, we notice that the heterogeneous case behaves siilarly to an average of the different hoogeneous cases. Siilar considerations can also be ade while considering the context switching costs (Fig. ). Note that for both the heterogeneous and the hoogeneous scenarios we siulated also the deployent of 3, and 5 SCs. We do not plot such curves, since the results we got are analogous and do not provide additional insights to the discussion. VI. CONCLUSION In this paper, we investigated the ipact of processingresource sharing aong s and scalability costs in a NFV scenario, when ultiple SCs ust be deployed in the network. The s placeent and distribution on NFV nodes lead to two different types of costs: upscaling costs and context switching costs. Such costs lead to a size/nuber trade-off that ust be investigated. We first focused on the atheatical odeling of the NFV nodes, of the s and of the SCs. Then, we defined an ILP odel aied at the /SC ebedding on a physical network while consolidating the s in the iniu nuber of NFV nodes and taking into account placeent, routing and perforance constraints. Then, we evaluated the ipact of upscaling and context switching costs on the cost for NFV ipleentation. Results show that, as the nuber of SCs grows, the ipact of context switching costs on the cost for NFV ipleentation is aplified. On the other hand, the nuber of SCs does not change how the upscaling costs translate into the cost for NFV ipleentation. We also showed how different latency and bandwidth requireents of the SCs ipact on the network cost. Our odel shows that NFV can handle ultiple SCs with different requireents deployed on the sae network without incurring in significant additional costs with respect to a scenario with hoogeneous requireents. Several issues still reain open for future research. For exaple, ore detailed odels for upscaling and context switching costs can be investigated, including energetic aspects. ACKNOWLEDGMENT The research leading to these results has received funding fro the European Counity Seventh Fraework Prograe FP7/013-015 under grant agreeent no. 3177 COMBO project. REFERENCES [1] M. Chiosi, D. Clarke, P. Willis, A. Reid, J. Feger, M. 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