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Shortest Path Tour Problem Based Integer Linear Programming for Service Chaining in NFV Networks Masahiro Sasabe Graduate School of Science and Technology Nara Institute of Science and Technology 8916-5 Takayama-cho, Ikoma, Nara 630-0192, Japan [email protected] Takanori Hara Graduate School of Science and Technology Nara Institute of Science and Technology 8916-5 Takayama-cho, Ikoma, Nara 630-0192, Japan [email protected] Abstract—Network functions virtualization (NFV) is a new paradigm to achieve flexible and agile network services by decoupling network functions from proprietary hardware and running them on generic hardware as virtual network functions (VNFs). In the NFV network, a certain network service can be modeled as a sequence of VNFs, called a service chain. Given a connection request (origin node, destination node, and service chain requirement, which is a sequence of functions), the service chaining problem aims to find an appropriate service path, which starts from the origin and ends with the destination while executing the VNFs at the intermediate nodes in the required order. Some existing work noticed that the service chaining problem was similar to the shortest path tour problem (SPTP). To the best of our knowledge, this is the first work that exactly formulates the service chaining problem as the SPTP-based integer linear programming (ILP). Through numerical results, we show the SPTP-based ILP formulation can support about two times larger scale systems than the existing ILP formulation. Index Terms—Network functions virtualization (NFV), service chaining, integer linear programming (ILP), shortest path Tour problem (SPTP). I. I NTRODUCTION Network function virtualization (NFV) can realize flexible and agile network services by decoupling network functions (e.g., routing, firewall, deep packet inspection, and load bal- ancing) from dedicated hardware (e.g., appliances and mid- dleboxes) and running them on generic hardware as virtual network functions (NFVs) [1], [2]. In addition to the flexibility and agility, the NFV can also reduce both capital expenditures (CAPEX) and operating expenditures (OPEX) because it can be built on the generic hardware and liberate operators from learning knowledge for the conventional dedicated hardware. The resource allocation is one of the challenging issues in the NFV networks. We can view a certain network service as a sequence of VNFs, called a service chain. Given a connection request with origin node, destination node, and service chain requirement, which is a sequence of functions, the service chaining problem aims to find an appropriate service path, which starts from the origin node and ends with This work was supported in part by the Japan Society for the Promotion of Science (JSPS) KAKENHI (C) under Grant 19K11942, Japan. the destination node while executing the corresponding VNFs at the intermediate nodes in the required order. Some existing work noticed the similarity between the shortest path tour problem (SPTP) [3] and the service chaining problem [4], [5]. The SPTP is a variant of the shortest path problem (SPP) and tries to find a shortest path from an origin node to a destination node such that the path must visit a sequence of disjoint node subsets T 1 ,..., T K in this order. For example, T k (k =1,...,K) can be regarded as sightseeing spots in a certain tour. Bhat and Rouskas first showed the possibility of modeling the service chaining problem as the SPTP [4]. Gao and Rouskas proposed the SPTP-based heuris- tic algorithm for the online service chaining [5]. Recently, Andrade and Saraiva proposed the integer linear programming (ILP) formulation for the constraint SPTP [6]. The constrained SPTP is a special case of the SPTP such that the path does not include repeated edges. In this paper, combining this approach and a novel network model called augmented network, we will show that the service chaining problem can exactly be modeled as the SPTP-based ILP. Although there are various types of ILP formulations for the service chaining [7]–[12], to the best of our knowledge, this is the first work that exactly formulates the service chaining problem as the SPTP-based ILP. Through numerical results, we show the proposed ILP formulation can support about two times larger scale systems than the existing ILP formulation. The rest of the manuscript is organized as follows. Section II gives the related work. After introducing the system model in Section III, we develop the SPTP-based ILP for the service chaining in NFV networks. We demonstrate that the SPTP- based ILP is more scalabe than the existing ILP formulation in Section V. Finally, Section VI gives the conclusions and future work. II. RELATED WORK A. NFV and Its Resource Allocation Problems The resource allocation is one of the most challenging problems in NFV networks because it requires to deal with various types of requirements (e.g., service (VNFs) chaining, Submitted version

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Page 1: Shortest Path Tour Problem Based Integer Linear

Shortest Path Tour Problem Based Integer LinearProgramming for Service Chaining in NFV

NetworksMasahiro Sasabe

Graduate School of Science and TechnologyNara Institute of Science and Technology

8916-5 Takayama-cho, Ikoma, Nara 630-0192, [email protected]

Takanori HaraGraduate School of Science and Technology

Nara Institute of Science and Technology8916-5 Takayama-cho, Ikoma, Nara 630-0192, Japan

[email protected]

Abstract—Network functions virtualization (NFV) is a newparadigm to achieve flexible and agile network services bydecoupling network functions from proprietary hardware andrunning them on generic hardware as virtual network functions(VNFs). In the NFV network, a certain network service can bemodeled as a sequence of VNFs, called a service chain. Givena connection request (origin node, destination node, and servicechain requirement, which is a sequence of functions), the servicechaining problem aims to find an appropriate service path,which starts from the origin and ends with the destination whileexecuting the VNFs at the intermediate nodes in the requiredorder. Some existing work noticed that the service chainingproblem was similar to the shortest path tour problem (SPTP).To the best of our knowledge, this is the first work that exactlyformulates the service chaining problem as the SPTP-basedinteger linear programming (ILP). Through numerical results,we show the SPTP-based ILP formulation can support about twotimes larger scale systems than the existing ILP formulation.

Index Terms—Network functions virtualization (NFV), servicechaining, integer linear programming (ILP), shortest path Tourproblem (SPTP).

I. INTRODUCTION

Network function virtualization (NFV) can realize flexibleand agile network services by decoupling network functions(e.g., routing, firewall, deep packet inspection, and load bal-ancing) from dedicated hardware (e.g., appliances and mid-dleboxes) and running them on generic hardware as virtualnetwork functions (NFVs) [1], [2]. In addition to the flexibilityand agility, the NFV can also reduce both capital expenditures(CAPEX) and operating expenditures (OPEX) because it canbe built on the generic hardware and liberate operators fromlearning knowledge for the conventional dedicated hardware.

The resource allocation is one of the challenging issues inthe NFV networks. We can view a certain network serviceas a sequence of VNFs, called a service chain. Given aconnection request with origin node, destination node, andservice chain requirement, which is a sequence of functions,the service chaining problem aims to find an appropriateservice path, which starts from the origin node and ends with

This work was supported in part by the Japan Society for the Promotionof Science (JSPS) KAKENHI (C) under Grant 19K11942, Japan.

the destination node while executing the corresponding VNFsat the intermediate nodes in the required order.

Some existing work noticed the similarity between theshortest path tour problem (SPTP) [3] and the service chainingproblem [4], [5]. The SPTP is a variant of the shortest pathproblem (SPP) and tries to find a shortest path from an originnode to a destination node such that the path must visit asequence of disjoint node subsets T1, . . . , TK in this order. Forexample, Tk (k = 1, . . . ,K) can be regarded as sightseeingspots in a certain tour. Bhat and Rouskas first showed thepossibility of modeling the service chaining problem as theSPTP [4]. Gao and Rouskas proposed the SPTP-based heuris-tic algorithm for the online service chaining [5].

Recently, Andrade and Saraiva proposed the integer linearprogramming (ILP) formulation for the constraint SPTP [6].The constrained SPTP is a special case of the SPTP suchthat the path does not include repeated edges. In this paper,combining this approach and a novel network model calledaugmented network, we will show that the service chainingproblem can exactly be modeled as the SPTP-based ILP.Although there are various types of ILP formulations for theservice chaining [7]–[12], to the best of our knowledge, thisis the first work that exactly formulates the service chainingproblem as the SPTP-based ILP. Through numerical results,we show the proposed ILP formulation can support about twotimes larger scale systems than the existing ILP formulation.

The rest of the manuscript is organized as follows. Section IIgives the related work. After introducing the system model inSection III, we develop the SPTP-based ILP for the servicechaining in NFV networks. We demonstrate that the SPTP-based ILP is more scalabe than the existing ILP formulationin Section V. Finally, Section VI gives the conclusions andfuture work.

II. RELATED WORK

A. NFV and Its Resource Allocation Problems

The resource allocation is one of the most challengingproblems in NFV networks because it requires to deal withvarious types of requirements (e.g., service (VNFs) chaining,

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VNF placement, scheduling, and demand provisioning) as wellas their complex dependence. The recent survey on NFVnetworks can be found in [1], [2].

Service chaining is one of the major NFV resource allo-cation problems, which aims to find an appropriate servicepath under the constraints of function placement and ser-vice chain requirement (e.g., processing/bandwidth demandand a sequence of functions). It is a kind of combinatorialoptimization problems and there are multiple studeis on themathematical formulation of the service chaining problem [7]–[12]. Nguyen et al. formulated the service chaining problemas ILP over an expanded network to minimize the blockingprobability of service chain requests [7]. Huin et al. developedILP over a layered graph to minimize the total bandwidthusage [8]. Gupta et al. also formulated ILP to minimize thetotal bandwidth usage [9]. Savi et al. modeled ILP to minimizethe number of active nodes by considering the processing-related costs, i.e., context switching costs and upscaling costs,in a VNF consolidation scenario where multiple VNFs wereserved at the same hardware [10]. Nejad et al. [11] aimed tomaximize the total revenue. Hyodo et al. formulate ILP tominimize the placement cost and bandwidth usage under therelaxation of VNF order and non-loop constraints [12].

The path-selection based service chaining aims to find anappropriate service path from given path candidates. D’Oro etal. applied the non-cooperative game theory to achieve servicechaining a distributed manner [13]. They also proposed atwo-stage Stackelberg game composed of leading servers andfollowing users to achieve the distributed resource allocationand orchestration [14].

Some researchers noticed that the service chaining is similarto the conventional SPTP [4], [5]. Bhat and Rouskas showedthat service chaining could be modeled as SPTP [4]. Theyproposed a heuristic algorithm to solve the SPTP and com-pared it with existing algorithms. Gao et al. proposed an onlinepath-selection algorithm to minimize the maximum networkcongestion [5]. To the best of our knowledge, this is thefirst work that exactly formulates the SPTP-based ILP for theservice chaining.

B. Shortest Path Tour Problem

The SPTP is a variant of the SPP where some intermediatenodes (locations) should to be visited in a predefined order [3].

Festa et al. achieved polynomial-time reduction of the SPTPto a classical SPP over a modified digraph and proposedheuristics to solve the SPTP [15]. Ferone et al. studiedconstrained SPTP, a variant of the SPTP, where the path didnot include repeated links, proved that it belonged to thecomplexity class of NP-complete, and proposed a branch-and-bound based heuristic [16]. Ferone et al. further proposed amathematical model and new branch-and-bound heuristics forthe CSPTP [17]. Recently, Andrade and Saraiva developed theILP formulation for the constrained SPTP [6]. In this paper, weformulate the SPTP-based ILP for the service chaining, withthe help of the existing approach [6] and the development of

a novel network model, called augmented network. (We willgive the detail of the augmented network in Section III-B.)

III. SYSTEM MODEL

A. Service Chain Request

We consider an NFV network where connection requestsrandomly arrive as in [7]. The orchestrator waits for C(C = 1, . . . , Cmax) requests and solves the service chainingproblem for the collected requests C. The way of processingthe request(s) can be categorized into three types, i.e., online,batch, and offline, depending on the size of C, i.e., C. Incase of the online processing (C = 1), the orchestratorserves the connection request just after its arrival. If theorchestrator knows all the requests (C = Cmax) in advance, itsolves the service chaining problem for them at once, andthus this results in the offline processing. In other cases,i.e., C = 2, . . . , Cmax − 1, the orchestrator adopts the batchprocessing with size of C. In what follows, we focus on theonline processing type (C = 1) as in [5], [7]. Note thatthe proposed approach in this paper can also be extended tosupport other processing types.

A connection c has a service chain requirement that consistsof origin node oc, destination node dc, sequence of Kc

functions Rc = (fc,1, . . . , fc,Kc), bandwidth capacity perphysical link bc, processing capacity for packet transfer perphysical node pnodec , and processing capacity for executionof function fc,k, pfuncc,fc,k

. oc (resp. dc) is a physical node thatconnection c starts from (resp. ends with). Rc is a sequence ofKc functions (fc,1, . . . , fc,Kc) with which connection c willbe served in this order. In general, connection c may requirea more complex processing order rather than the sequentialorder, e.g., split and merge. In this paper, we mainly focus onthe sequential type of requirement, which results in the simpleyet novel ILP formulation of the service chaining problem.As for the bandwidth/processing capacity requirement, weuse the same assumptions in [7]. We consider bc is a fixedvalue, e.g., constant bit rate (CBR). Whenever connection cpasses through a physical node, it needs a processing capacitypnodec to process the packets through the node. In addition,connection c requires processing capacity pfuncc,fc,k

for executingkth function fc,k in Rc. An example of the service chainrequirement for connection c, i.e., (oc, dc) = (v1, v5), Rc =(fc,1, fc,2, fc,3) = (f2, f3, f1), bc, pnodec , and {pfuncc,f }f∈Rc

, isshown in the top layer of Fig. 1.

Given a connection request c with origin node oc, destina-tion node dc, and service chain requirement Rc, which is asequence of functions, we try to find an appropriate servicepath Sc, which starts from oc and ends with dc while executingfunctions in Rc.

B. Augmented Network

We consider a physical network G = (V, E), where V (resp.E) is the set of physical nodes (resp. links), as shown in themiddle layer of Fig. 1. G is a directed graph where an arrowfrom node i ∈ V to j ∈ V expresses the physical link (i, j).For example, the bidirectional arrow between nodes v1 and v2

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Page 3: Shortest Path Tour Problem Based Integer Linear

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augmented network<latexit sha1_base64="gHXL07yMQrqR+wqLdKXKJ54jKLg=">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</latexit><latexit sha1_base64="gHXL07yMQrqR+wqLdKXKJ54jKLg=">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</latexit><latexit sha1_base64="gHXL07yMQrqR+wqLdKXKJ54jKLg=">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</latexit><latexit sha1_base64="gHXL07yMQrqR+wqLdKXKJ54jKLg=">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</latexit>

logical subpath<latexit sha1_base64="qNskqaZ9YiCpoOBldcW9T/nbhtQ=">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</latexit>

physical subpath<latexit sha1_base64="egBPJNOVoRa9p5kP5AOGBL5rfZc=">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</latexit>

vf1<latexit 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service path (shortest path tour)<latexit 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2nd subpath<latexit 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1st subpath<latexit 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service chain requirement<latexit sha1_base64="BPXy6rE3KOvhg2Szak+WXoPM0ho=">AAAON3iczVfNbxtFFH8tUErBNIULiIshKeIA0Tg9FCG5apWmpaqKmuajlZIo8q7H9ij7YWbXiVMrIMEF8Q9w4ARSDwgJzpy58A9w6IkbAnFCReLCgd97u/5IdhybDyFseXf2ze/33rzfzM4be+3AJKlSD0+cfOzxJ049efqpM08/U3r27My559aTuGN9vebHQWzvebVEBybSa6lJA32vbXUt9AJ919tZ5P67u9omJo5W0/223gprzcg0jF9LYdqeeTHRdtf4uuy3aiYqW/1ux1gd6ijdnplV80o+5WKjkjdmKf/cjs+d+pA2qU4x+dShkDRFlKIdUI0SfDeoQorasG1RDzaLlpF+TQd0BtwOUBqIGqw7uDbxtJFbIzyzz0TYPqIE+Fkwy3Refa++UI/Ud+pL9bP6Y6yvnvjgsezj7mVc3d4++/ELK79PZIW4p9Qaso4dc0oNelPGajD2tlg4Cz/j797/5NHKW3fO915Vn6tfM 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</latexit>

function<latexit 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pfuncc,f2

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pfuncc,f3

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dlinkv1,v4

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dfuncvf2

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pnodec

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pnodec

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pnodec

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pnodec

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Fig. 1: Overview of service chaining: relationship betweenservice chain requirement, augmented network, and servicepath.

indicates that there are two physical links between them, i.e.,(v1, v2) and (v2, v1), in Fig. 1. Each physical link (i, j) ∈ E(resp. each physical node i ∈ V) has residual bandwidth Bi,j

(resp. residual processing capacity Pi) at arrival of c. The NFVnetwork supports a set of F functions, F = {f1, . . . , fF }, inthe whole network while each physical node i ∈ V can supportpart of functions, Fi ⊆ F . Each function f is supported byNf (Nf > 0) physical nodes. In actual systems, Nf shouldbe carefully determined according to the demand for networkservices. This results in the problems of demand provisioningand function placement, which will be considered in the futurework. In what follows, we assume that Nf is identical, i.e.,Nf = N (N > 0).

A connection c requests a service chain requirementRc, which includes a sequence of Kc functions, Rc =(fc,1, . . . , fc,Kc

), from origin node oc to destination node dc.For example, in Fig. 1, Rc = (f2, f3, f1) and (oc, dc) =(v1, v5). Service chaining for connection c is finding anappropriate service path Sc, which starts from oc and endswith dc while executing functions of Rc in this order andpassing through the corresponding physical links. Inspired bythe approach in [6], we can regard service chaining as SPTP.(The detail will be given in Section IV.) The SPTP tries to finda shortest path from an origin node to a destination node withthe constraint that the path should visit at least one node fromK (K > 0) given disjoint node subsets T1, . . . , TK [15]. In ourcase, Tk = {vfc,k} (k ∈ Kc), where vfc,k is an imaginary nodesupporting kth function fc,k in Rc. For example, in Fig. 1,T1 = {vf2}, T2 = {vf3}, and T3 = {vf1}. In our case, thedisjoint node subsets can be regarded as a set of imaginarynodes, V = {vc,1, . . . , vc,Kc

}, where kth imaginary node

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physical node<latexit sha1_base64="CT+9IJ3E1L8eOJZH/+CZ/+EQOJg=">AAAIp3icjZXbbtNAEIanLQc3nFq4QeLGIkICCUV2b+CyKG2E1DYJylFqI2Q7m8SqT9gOJYrgAXgAuIVX4YJ34GnKvxs7apNuNrZij2f/b2Z2Z5W1I89NUsP4t7G5dev2nbvaduHe/QcPH+3sPm4n4Th2WMsJvTDu2lbCPDdgrdRNPdaNYmb5tsc69nmZj3c+szhxw6CZTiLW861h4A5cx0rh6kWjSQLb04Owzz7uFI2SIS592TAzo0jZVQ93t/7QGfUpJIfG5BOjgFLYHlmU4D4lkwyK4OvRFL4YlivGGX2lAtgxVAwKC95zPIf4Os28Ab55zETQDrJ4+MUgdXqRafqwB8I7e/P8+hWtLMdUxOY1TvC2s5g+vCmN4FVxuXJ9LtfEGB3gJ58/145RP1+rGLO4WKm1oByKuboivk+vUdOyl6/7qjg2NKt7kqLut6IXLtY5Eh7eJedaRwaidh1joXhOoGSw+mI2DOoU45PMw3PMauT18c6NxD6xhI6J1Sogti68Q7GKX4zR/ItXwmsOoL0QffeFJkD0KfxXKFTJPXkfQtTFv2N48ujf5soz1OiK+Om8V6ocvJb1csyUyznETH/8bWqX2rTy/Wmtcbm9MvO7hXyO6PtU+FfX3KCahG0o2RYdStiWkpWRh0rySEIeKckTCXmiJI9x38weK9mqhKwqyY6E7CjJioSsKMmyhCwryaaEbCrJtoRsK8kDCXmgJLsSsqsk67hvZutr7N3F3ZuI/711dm91aSflbL6TCji3zcVTetlo75VMs2R+2CvuG9kJrtEzek4vcUq/oX16j5m0EPsT/aRf9Ft7pdW0ttadSTc3MuYJXbs06z8MaL+E</latexit><latexit sha1_base64="CT+9IJ3E1L8eOJZH/+CZ/+EQOJg=">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</latexit><latexit sha1_base64="CT+9IJ3E1L8eOJZH/+CZ/+EQOJg=">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</latexit><latexit sha1_base64="CT+9IJ3E1L8eOJZH/+CZ/+EQOJg=">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</latexit>

function<latexit 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1st layer (copy)

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2nd layer (copy)

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3rd layer (copy)

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layered graph / expanded network

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vertical link / artificial link

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Fig. 2: Structure comparison among network models.

TABLE I: Scale comparison among network models.

network model # of nodes # of linkslayered graph / expanded network (Kc + 1)V (Kc + 1)E +KcNaugmented network V +Kc E + 2KcN

vc,k supports kth function fc,k in Rc. Note that imaginarynodes are not actual virtual nodes serving the correspondingfunctions but they play a key role in formulating servicechaining as the SPTP-based ILP. (The detail of formulationwill be given in Section IV.) We further introduce two sets ofvirtual links, E in and Eout. E in is a set of links incoming toimaginary nodes, E in = {(v, vf ) | v ∈ V, vf ∈ V, f ∈ Fv}.On the other hand, Eout is a set of links outgoing fromimaginary nodes, Eout = {(vf , v) | v ∈ V, v ∈ V, f ∈ Fv}.For example, in Fig. 1, the virtual link (vf2 , v1) indicates thatphysical node v1 can support function f2. We call the finallyobtained network the augmented network G+ = (V+, E+)where V+ = V ∪ V and E+ = E ∪ E in ∪ Eout. An example ofthe augmented network of G is shown in the middle layer ofFig. 1. For simplicity in description, the neighbors of node iin G+ is defined as V+

i ⊂ V+.One of the main contributions of this paper is the proposal

of the augmented network, which is much smaller than theexisting network models: layered graph [8] and expandednetwork [7], and contributes to the simple yet exact SPTP-based ILP formulation. Fig. 2 illustrates the structure com-parison among the three network models. The layered graph,which is given in the right of Fig. 2, consists Kc + 1layers of the original physical network and multiple verticallinks connecting two successive layers. A vertical link existsbetween node i ∈ V at kth layer and that at (k + 1)th layer(k = 1, . . . ,Kc) only when node i supports kth functionof the chain request c, fc,k. The expanded network is anextension of the layered graph. In the expanded network,an vertical (artificial) link only exists between nodes whenthe corresponding link and both the end nodes have enoughresidual bandwidth and processing capacity supporting theconnection c.

Table I presents the scale comparison among the network

Submitted version

Page 4: Shortest Path Tour Problem Based Integer Linear

models. Note that the number of links in case of the expandednetwork is the upper limit where all the nodes and links areavailable for the connection request. We can confirm that theaugmented network is much smaller than the layered graphand expanded network.

C. Service Path

According to the SPTP, the service path Sc for Rc =(fc,1, . . . , fc,Kc

) with origin oc and destination dc can be ex-pressed by a sequence of Kc+1 subpaths, (Sc,1, . . . ,Sc,Kc+1),where kth subpath Sc,k has origin node ac,k and destinationnode bc,k, which are given as follows:

(ac,k, bc,k) =

(oc, vfc,1), k = 1,

(vfc,k−1, vfc,k), k = 2, . . . ,Kc,

(vfc,Kc, dc), k = Kc + 1.

Sc,k starts from its origin node ac,k and ends with its destina-tion node bc,k through appropriate physical and virtual linksin G+. (Finding an optimal service path S∗c is our goal andwill be discussed in Section IV.) Note that there is no loopin each subpath but loop(s) may occur in the whole path,that is, some links may be used multiple times. For example,the bottom layer of Fig. 1 shows an example of servicepath Sc = (Sc,1, . . . ,Sc,4) where Sc,1 = ((v1, v4), (v4, vf2))(red arrow), Sc,2 = ((vf2 , v4), (v4, v3), (v3, vf3)) (orangearrow), Sc,3 = ((vf3 , v3), (v3, v2), (v2, vf1)) (green arrow),and Sc,4 = ((vf1 , v2), (v2, v3), (v3, v5)) (blue arrow). We canalso confirm service path Sc in the augmented network, asshown in the middle layer of Fig.1. In this case, each subpath(i.e., Sc,1,Sc,2,Sc,3, and Sc,4) does not have any loop whilethe whole service path Sc has a loop. (Physical link (v2, v3)is used twice.)

We consider that the optimality of service path Sc is evalu-ated by total delay consisting of processing delay at nodes andpropagation delay at physical links included in Sc. (The detailwill be given in Section IV.) Each physical link (i, j) ∈ Ehas propagation delay dlinki,j (dlinki,j > 0). As mentioned above,connection c requires packet processing at each physical nodethat it uses, and thus the corresponding processing delay atphysical node i ∈ V is given by dnodei (dnodei > 0). In Sc,each function f ∈ Rc is executed at physical node v ∈ Vwith processing delay of dfuncvf ,v

(dfuncvf ,v> 0).

Table II summarizes the notations used in this paper.

IV. MODELING SERVICE CHAINING AS SHORTEST PATHTOUR PROBLEM BASED INTEGER LINEAR PROGRAMMING

Inspired by the SPTP, we formulate the service chaining asthe following ILP P, which has the binary decision variablesxc,ki,j (c ∈ C, (i, j) ∈ E+, k ∈ K+

c ):

xc,ki,j =

1, if physical/virtual link (i, j) is used in kth

subpath of service path for connection c,0, otherwise.

TABLE II: Notations in the model.

Notation DefinitionG Physical network G = (V, E)V Set of physical nodes, V = |V|E Set of physical links, E = |E|F Set of functions, F = {f1, . . . , fF }, F = |F|Fi Set of functions contained in physical node i ∈ Vc Connection with origin oc and destination dcRc Sequence of functions (fc,1, . . . , fc,Kc ) required by c

Kc,K+c Kc = {1, . . . ,Kc},K+

c = {1, . . . ,Kc + 1}V Set of imaginary nodes, V = {vf}f∈F ,

where imaginary node vf supports function f

E in Set of links incoming to imaginary nodes,E in = {(v, v) | v ∈ V, vf ∈ V, f ∈ Fv}

Eout Set of links outgoing from imaginary nodes,Eout = {(v, v) | vf ∈ V, v ∈ V, f ∈ Fv}

G+ Augmented network of G, G+ = (V+, E+)

V+ Set of nodes in G+, V+ = V ∪ V , V + = |V+|E+ Set of links in G+, E+ = E ∪ E in ∪ Eout

Sc Service path for c, (Sc,1, . . . ,Sc,Kc+1)Sc,k kth sub service path with origin ac,k and

destination bc,kbc Bandwidth requirement of cpnodec Processing requirement of c for traversing a nodepfuncc,fc,k

Processing requirement of fc,k ∈ Rc at a nodeBi,j Residual bandwidth of link (i, j) at arrival of cPi Residual processing capacity of node i

at arrival of cdlinki,j Propagation delay of link (i, j) ∈ Ednodei Traversal delay of node i ∈ Vdfuncvf ,v Processing delay of function f of node v ∈ Vxc,ki,j Binary decision variables:

1: if link (i, j) is included in Sc,k , 0: otherwise

min∑

(i,j)∈E+di,j

∑k∈K+

c

xc,ki,j (1)

s.t. xc,ki,j = {0, 1}, (i, j) ∈ E+, k ∈ K+c , (2)∑

(ac,k,j)∈E+xc,kac,k,j

= 1, k ∈ K+c , (3)

∑(j,bc,k)∈E+

xc,kj,bc,k= 1, k ∈ K+

c , (4)

∑(i,j)∈E+

xc,ki,j =∑

(j,i)∈E+xc,kj,i ,

∀i ∈ V \ {ac,k},∀j ∈ V \ {bc,k}, k ∈ K+c , (5)

xc,ki,vfc,k= xc,k+1

vfc,k ,i,

∀(i, vfc,k) ∈ E in,∀(vfc,k , i) ∈ Eout, k ∈ Kc, (6)

xc,ki,vfc,m= 0, ∀(i, vfc,m) ∈ E in, k ∈ K+

c ,m 6= k, (7)

bc∑

k∈K+c

xc,ki,j ≤ Bi,j , ∀(i, j) ∈ E , (8)

pnodec

∑(v,j)∈E

∑k∈K+

c

xc,kv,j +∑

(vf ,v)∈Eoutpfuncc,f

∑k∈K+

c

xc,kvf ,v≤ Pv,

∀v ∈ V. (9)

Equation (1) is the objective function where di,j is given as

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follows:

di,j =

dnodei + dlinki,j , if (i, j) ∈ E ,dfunci,j , if (i, j) ∈ Eout,0, otherwise.

(10)

We first observe that the objective function (1) is the sameas that of the SPTP. From the viewpoint of service chaining,equation (10) indicates that passing each physical link (i, j) ∈E suffers both forwarding delay dnodei and propagation delaydlinki,j . The service path also suffers the execution delay of eachfunction i ∈ Rc at physical node j. As a result, the objectivefunction (1) can be rewritten by∑(i,j)∈E

(dnodei + dlinki,j )∑

k∈K+c

xc,ki,j +∑

(vf ,v)∈Eout

dfuncvf ,v

∑k∈K+

c

xc,kvf ,v,

where the first (resp. second) term corresponds to the physicalforwarding and propagation delay (resp. function execution de-lay). For example, in the bottom of Fig. 1, the horizontal (resp.vertical) arrows correspond to the physical (resp. virtual) linksincluded in service path Sc. Note that the objective functioncan be replaced with conventional ones, e.g., minimization ofthe total bandwidth usage.

Equations (2)–(9) give the constraints. Constraint (2) rep-resents the decision variables. Constraints (3)–(5) present theflow rules in each subpath Sc,k (k ∈ K+

c ). Constraint (3) (resp.Constraint (4)) indicates that the origin (resp. destination) nodeac,k (resp. bc,k) of the subpath k has incoming (resp. outgoing)flow. Equation (5) is the flow conservation constraint in thesubpath k (k ∈ K+

c ). For example, focusing on the 1st subpathin Fig. 1, we observe that the flow occurs at physical node v1,i.e.,

∑(v1,j)∈E+ x

c,1v1,j

= 1, goes through any physical nodev ∈ V , i.e,.

∑(v,j)∈E+ x

c,1v,j =

∑(j,v)∈E+ x

c,1j,v , and finally ends

with imaginary node vf2 , i.e.,∑

(j,vf2 )∈E+xc,1j,vf2

= 1.Constraint (6) guarantees the connectivity between two suc-

cessive subpaths Sc,k and Sc,k+1 (k ∈ K+c ). More specifically,

(k+1)th subpath should start from the same physical node asthe last physical node of kth subpath. For example, focusingon the 1st and 2nd subpaths in Fig. 1, we observe thatxc,1i,vfc,1

= xc,2vfc,1 ,i(∀(i, vfc,1) ∈ E in,∀(vfc,1 , i) ∈ Eout) where

fc,1 = f2. Constraint (7) prohibits imaginary node vfc,m frombeing used in mth subpath (m 6= k). For example, focusingon the 1st subpath in Fig. 1, we observe that xc,1i,vfc,m

= 0

(m 6= 1,∀(i, vfc,m) ∈ E in).Equation (8) gives the physical link capacity constraint

where the bandwidth consumption of physical link (i, j) ∈ Eshould be equal or less than the residual bandwidth capacityBi,j . Similarly, equation (9) shows the processing capac-ity constraint where the total processing load of physicalnode v ∈ V should be equal or less than the processingcapacity Pv . The processing load consists of the traversalcost, pnodec

∑(v,j)∈E

∑k∈K+

cxc,kv,j , and the processing cost,∑

(vf ,v)∈Eout pfuncc,f

∑k∈K+

cxc,kvf ,v

.

V. NUMERICAL RESULTS

In this section, we compare the proposed SPTP-based ILPformulation and the existing ILP formulation [7] in termsof the computational complexity. We solve the both ILPformulations by using the existing solver CPLEX 12.8 [18]running on the server with Intel Xeon E7-8895v3 (18 coresand 2.60 GHz) and 2 TB memory.

A. Evaluation Scenario

To evaluate the fundamental characteristics of the proposedscheme, we first use the physical network consisting of 200physical nodes where each physical node i ∈ V has thesame processing capacity of Pi = 1.71. We then generatephysical links between two arbitrary physical nodes at theprobability of π = 0.032 by using the existing scheme [19].All physical links between nodes i and j (i, j ∈ V) havethe same link capacity of Bi,j = 1.14. We set the physicallink delay between nodes i and j, dlinki,j , to be 10 [ms]. Eachphysical node i has the same traversal and processing delay,i.e., dnodei = 1 [ms] and dfunci = 50 [ms], respectively.

We set the total number of functions, F , to be 20 and eachfunction can be supported by five physical nodes (N = 5). Inthis paper, we focus on the online scenario (C = 1) wherethe orchestrator immediately performs the service chaining perconnection request, as in [7]. For each connection c, the originnode oc and destination node dc are randomly chosen from thephysical nodes. Each connection c requires Kc functions, eachof which is randomly chosen from the F functions. We set thebandwidth requirement, processing requirement for traversinga node, and processing requirement of fc,k ∈ Rc at a node asfollows: bc = 0.1, pnodec = 0.05, and pfuncc,fc,k

= 0.1.We evaluate the computational complexity in terms of the

execution time and CPLEX deterministic time. The executiontime is the actual time required to solve the problem. Notethat CPLEX supports the parallel optimization and we set thenumber of threads to be 32. Since the execution time dependson the hardware spec of the server, we also use the CPLEXdeterministic time to evaluate the substantial complexity ofthe problem. CPLEX provides us with a choice between de-terminism and opportunism algorithms [20]. The determinismalgorithm gives us a solution path in a deterministic mannerwhile the opportunism one takes advantage of opportunities toimprove performance. We adopt the determinism algorithm.

To show the scalability of the SPTP-based ILP in terms ofthe computation complexity, we compare it with the existingILP for the service chaining in [7]. Note that the authorsproposed not only the ILP formulation but also several heuris-tic algorithms to overcome the computation complexity, in[7]. In addition, the objective function is the minimizationof the link and node utilization to alleviate the blockingprobability of connection requests. In what follows, focusingon the computation complexity of the ILP formulation itself,we slightly replace the objective function of the existing ILPwith that of the SPTP-based ILP, i.e., the minimization of thetotal delay given by Equation (1).

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10−1 100 101

Execution time limit [s]

0

200

400

600

800

1000

1200

1400

Max

imu

mnu

mb

erof

physi

cal

nod

esV

Proposed scheme (Augmented network)

Nguyen+ (Expanded network)

Fig. 3: The maximum number of physical nodes to be solvedwithin the execution time limit.

500 1000 1500 2000 2500 3000

Number of nodes V

0

5000

10000

15000

20000

25000

30000

35000

40000

CP

LE

Xd

eter

min

isti

cti

me

Nguyen+ (Expanded network)

Proposed scheme (Augmented network)

Fig. 4: Impact of the number of nodes on deterministic time.

In what follows, we show the average of 100 independentnumerical experiments.

B. Impact of The Number of Nodes on Computational Com-plexity

Fig. 3 illustrates the maximum number of physical nodesthat the SPTP-based ILP and existing ILP can solve within theexecution time limit. We observe that the SPTP-based ILP cansolve 1.33–2.19 times larger scale systems than the existingILP under the same execution time limit. In particular, theSPTP-based ILP can support 440 (resp. 1440) nodes within 1(resp. 10) [s].

Fig. 4 presents how the deterministic time increases withthe number of nodes, V . Note that we limit the deterministictime to 40,000 [ticks]. We observe that the deterministic timeof both ILP exponentially grows with increase of V , but theincreasing rate of the SPTP-based ILP is much smaller thanthat of the existing ILP. As a result, the SPTP-based ILP cansolve the service chaining even in case of V = 3000 while theexisting ILP reaches the limitation of the deterministic time.As mentioned in Section III-B, the augmented network is more

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Connection size Kc

0

200

400

600

800

1000

CP

LE

Xd

eter

min

isti

cti

me

Nguyen+ (Expanded network)

Proposed scheme (Augmented network)

Fig. 5: Impact of connection size Kc on deterministic time.

compact than the expanded network, which contributes to thescalability.

C. Impact of Connection Size on Computational Complexity

Next, Fig. 5 shows the impact of the connection size Kc onthe deterministic time when V = 200. We first observe thatthe deterministic time of both ILP shows almost linear growthwith increase of Kc. We also confirm that the deterministictime of the SPTP-based ILP is always smaller than that of theexisting ILP. Specifically, the SPTP-based ILP can reduce thedeterministic time by 34.4% (Kc = 1) and 63.3% (Kc = F )compared with the existing ILP.

D. Impact of Objective Function on Computation Complexity

Finally, we examine how the difference of the objectivefunction affects the computation complexity. Figs. 6, 7, and8 are the results in case of the minimization of node and linkutilization, each of which corresponds to Figs. 3, 4, and 5,respectively. Note that the SPTP-based ILP can also supportthe minimization of node and link utilization by replacing di,jas follows:

di,j =

bc

Bi,j, if (i, j) ∈ E ,

pfuncc,f(i)

Pi, if (i, j) ∈ Eout,

0, otherwise,

(11)

where f(i) represents the function supported by the imaginarynode i. We observe that the results in case of the minimizationof node and link utilization are similar to those in case of theminimization of total delay.

VI. CONCLUSIONS

In this paper, we have developed a simple yet novel ILPformulation for service chaining in NFV networks, which isan extension of the ILP formulation for the SPTP with thehelp of the augmented network model. Through numericalexperiments, we have showed that the SPTP-based ILP cansupport about two times larger scale systems than the existingILP.

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10−1 100 101

Execution time limit [s]

0

200

400

600

800

1000

1200

1400

Max

imu

mnu

mb

erof

physi

cal

nod

esV

Proposed scheme (Augmented network)

Nguyen+ (Expanded network)

Fig. 6: The maximum number of physical nodes to be solvedwithin the execution time limit (blocking probability mini-mization case).

500 1000 1500 2000 2500 3000

Number of nodes V

0

5000

10000

15000

20000

25000

30000

35000

40000

CP

LE

Xd

eter

min

isti

cti

me

Nguyen+ (Expanded network)

Proposed scheme (Augmented network)

Fig. 7: Impact of the number of nodes on deterministic time(blocking probability minimization case).

As future work, we will extend the SPTP-based ILP for-mulation for both service chaining and function placement.In addition, we also plan to apply heuristic algorithms for theoriginal SPTP to tackle the computation complexity problems.

REFERENCES

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Connection size Kc

0

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400

600

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CP

LE

Xd

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