COMPUTER networks nowadays rely on various middleboxes,

Transcription

1 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER Effcent Algorthms for Throughput Maxmzaton n Software-Defned Networks Wth Consoldated Mddleboxes Metan Huang, Wefa Lang, Senor Member, IEEE, Zchuan Xu, Member, IEEE, and Song Guo, Senor Member, IEEE Abstract Today s computer networks rely on a wde spectrum of specalzed mddleboxes to mprove network securty and performance. A promsng emergng technque to mplementng tradtonal mddleboxes s the consoldated mddlebox technque, whch mplements the mddleboxes as software n vrtual machnes n software-defned networks (SDNs), offerng economcal, and smplfed management for mddleboxes. Ths however poses a great challenge, that s, how to fnd a cost-optmal routng path for each user request such that the data traffc of the request wll pass through the mddleboxes n ther orders n the servce chan of the request, wth the objectve to maxmze the network throughput, subject to varous resource capacty constrants n SDNs. In ths paper, we study the network throughput maxmzaton problem n an SDN under two dfferent scenaros: one s the snapshot scenaro where a set of requests at one tme slot s gven, we am to admt as many requests n the set as possble to maxmze the network throughput; another s the onlne scenaro n whch requests arrve one by one wthout the knowledge of future arrvals. Gven a fnte tme horzon consstng of T equal tme slots, the system must respond to the arrved requests n the begnnng of each tme slot, by ether admttng or rejectng the requests, dependng on the resource avalabltes n the network. For the snapshot scenaro, we frst formulate an nteger lnear program (ILP) soluton, we then devse two heurstcs that strve for fne tradeoffs between the qualty of a soluton and the runnng tme of obtanng the soluton. For the onlne scenaro, we show how to extend the proposed algorthms for the snapshot scenaro to solve the onlne scenaro. We fnally evaluate the performance of the proposed algorthms through expermental smulatons, based on both real and synthetc network topologes. Expermental results demonstrate that the proposed algorthms admt more requests than the baselne algorthm and the qualty of the solutons delvered by heurstcs s comparable to the exact soluton by the ILP n most cases. Index Terms Software-defned networkng, network functon vrtualzaton, consoldated mddleboxes, throughput maxmzaton, routng algorthms, onlne algorthms, network resource allocatons. Manuscrpt receved March 13, 2017; revsed July 5, 2017; accepted July 6, Date of publcaton July 11, 2017; date of current verson September 7, The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was V. Fodor. (Correspondng author: Wefa Lang.) M. Huang and W. Lang are wth the Research School of Computer Scence, Australan Natonal Unversty, Canberra, ACT 2601, Australa (e-mal: u @anu.edu.au; wlang@cs.anu.edu.au). Z. Xu s wth the School of Software, Dalan Unversty of Technology, Dalan , Chna (e-mal: zchuanxu.mal@gmal.com). S. Guo s wth the Department of Computng, Hong Kong Polytechnc Unversty, Hong Kong (e-mal: song.guo@polyu.edu.hk). Dgtal Object Identfer /TNSM I. INTRODUCTION COMPUTER networks nowadays rely on varous mddleboxes, ncludng frewall, Intruson Detecton Systems (IDSs), WAN optmzers, and Deep Packet Inspectons (DPIs), to enhance the performance and securty of dfferent network servces [10], [14], [28]. Unfortunately, the management and deployment of these hardware mddleboxes are complex and costly [28]. For example, statstcs ndcated that large networks (10k-100k nodes) spent over a mllon dollars on deployng and mantanng hardware mddleboxes whle medum and small networks (1k-10k nodes) spent between $5,000 and $50,000 n the last fve years [28]. Wth the advancement of the Network Functon Vrtualzaton (NFV), mddleboxes can be mplemented n Vrtual Machnes (VMs) that run n Physcal Machnes (PMs) [9], [26], [28]. The NFVs can be relocated and nstantated at servers located at dfferent locatons n a network wthout needs of purchasng and nstallng expensve mddleboxes. By decouplng network functons from the hardware platform on whch network functons are executed, NFV has the great potental to lead to sgnfcant reductons n operatng expenses (OPEX) and captal expenses (CAPEX) of network servce provders and facltate the deployment of new servces wth ncreased aglty and faster tme-to-value [25]. We refer to the software mplementaton of mddleboxes as the consoldated mddleboxes. Along wth the technque of Software-Defned Networkng (SDN), consoldated mddleboxes offer a promsng alternatve way to provde cheap and smplfed management of mddleboxes [12], [27]. In ths paper we deal wth realzng user requests wth each specfyng a sequence of mddleboxes n SDNs wth the am to maxmze the network throughput. Ths problem poses great challenges. One challenge s that dfferent types of resources n SDNs have dfferent capactes. For nstance, the forwardng table of an SDN-enabled swtch usually s made by Ternary Content-Addressable Memory (TCAM) to facltate fast, parallel lookups of forwardng rules. TCAM however s expensve and energy hungry, ts capacty thus s restrcted to a few thousand table entres [18]. Meanwhle, the computng resource of the PM attached to an SDN-enabled swtch s lmted too. Another challenge s that all resources n an SDN are dynamcally allocated, causng sgnfcant fluctuatons n ther consumptons and avalabltes. The tme-varyng nature of resource demands and consumptons complcates the cost c 2017 IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 632 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER 2017 modelng of resource usages. In addton, each user request requres ts traffc to traverse a specfed sequence of mddleboxes that s referred to the servce chan of the request. In ths paper we wll address the aforementoned challenges. In spte of several studes of consoldated mddleboxes [5], [12], [26], none of the studes has taken the forwardng table sze nto consderaton. Almost all exstng solutons adopt a strategy that decomposes the routng path fndng and the servce chan executon nto two separate subtasks [12], the solutons thus are suboptmal. To the best of our knowledge, we are the frst to formulate a novel routng optmzaton problem wth consoldated mddleboxes n SDNs by jontly takng nto account both routng path fndng and consoldated mddlebox placement whle meetng dfferent user QoSs, by provdng effcent heurstc solutons. The man contrbutons of ths paper are as follows. We consder the network throughput maxmzaton problem of realzng user requests wth servce chans n SDNs, subject to varous network resource capacty constrants. We frst formulate an Integer Lnear Program (ILP) soluton to the problem when the problem sze s small. We then devse a heurstc by provdng a novel cost model to capture resource consumptons. We also propose a faster heurstc to quckly respond to user requests, by explorng non-trval tradeoffs between the accuracy (qualty) of a soluton and the runnng tme of obtanng the soluton. Furthermore, we consder dynamc admssons of user requests where user requests arrve one by one wthout the knowledge of future arrvals, by showng how to extend the proposed algorthms for dynamc admssons of requests. We fnally evaluate the performance of the proposed algorthms through smulatons, based on real and synthetc network topologes. Expermental results demonstrate that the proposed algorthms are very promsng compared to a baselne algorthm and the ILP, whch delvers optmal solutons. The rest of the paper s organzed as follows. Secton II wll revew related work. Secton III wll ntroduce the system model and notatons, and defne the problem. Secton IV wll formulate an ILP soluton to the problem. Sectons V and VI wll present two heurstc algorthms. Secton VII wll devse an onlne algorthm for dynamc request admssons. Secton VIII wll evaluate the performance of the proposed algorthms through smulatons, and Secton IX wll conclude the paper. II. RELATED WORK Whle mddleboxes are wdely used to guarantee securty and performance of routng traffc n contemporary computer networks, the deployment of tradtonal hardware mddleboxes ncurs hgh captal nvestment and operatonal costs [27], [28]. To tackle these ssues, recent efforts on new frameworks and archtectures of consoldated mddleboxes [2], [9], [11], [23], [27], have been demonstrated as promsng alternatves to tradtonal hardware mddleboxes. For example, Sekar et al. [27] devsed an archtecture CoMb that focused on consoldatng software-based mplementatons of mddlebox functons on a shared hardware platform. Qaz et al. [26] developed SIMPLE that enforces hgh-level routng polces for mddlebox-specfc traffc. Fayazbakhsh et al. [8] proposed FlowTags, because tradtonal flow rules do not suffce n the presence of dynamc modfcatons performed by mddleboxes. Martns et al. [23] ntroduced a vrtualzaton platform to mprove network performance by revsng exstng vrtualzaton technologes to support the deployment of modular, vrtual mddleboxes on lghtweght VMs. One fundamental problem under such archtecture s network throughput maxmzaton of realzng user routng requests wth specfed servce chans whle meetng varous resource constrants and user QoS requrements. A few recent studes nvestgated ths ssue [5], [12], whch however nether consdered resource constrants such as the forwardng table sze constrant on swtches, nor took global optmzaton, thereby the solutons delvered are suboptmal specfcally, Charkar et al. [5] assumed that every swtch n a network can perform mddlebox functons wthout consderng forwardng table szes. Gushchn et al. [12] assumed that the routng traffc of a request can be splt nto multple paths, and proposed a two-stage local optmzaton (before and after the vrtual mddleboxes). Zhang et al. [33] presented a routng scheme that reduces TCAM space usage wthout causng network congeston. However, they dd not consder user requests wth servce chan requrements. Kuo et al. [19] studed a problem of VM placement and path selecton, strvng for a tradeoff between lnk and server usage. Ths work, however, s dfferent from ours because they assumed that multple requests of the network functon can be satsfed usng a sngle VM that mplements the network functon. On the other hand, L et al. [20] presented the desgn and mplementaton of a system that dynamcally provsons resources to provde tmng guarantees wth the objectve of maxmzng the number of requests admtted to the cloud, whle meetng the deadlne requrements of admtted requests. In another related paper [15], Huang et al. consdered a jont optmzaton problem of mddlebox selecton and routng wth the objectve to maxmze the throughput or a specfed set of sessons n an SDN, and proposed a polynomal algorthm based on the Markov approxmaton technque. Cao et al. [4] studed the problem of polcy-aware traffc engneerng n SDNs, by assumng that the traffc has to pass a gven sequence of network functons. Lukovszk et al. [21] recently consdered mddlebox placements n a n-node network so that each source-destnaton par n a gven set has a path of length at most L wth one mddlebox n t, and each mddlebox can be used by at most k pars. They devsed an approxmaton algorthm wth an approxmaton rato of O(log mn{n, k}) for the problem, under the assumpton that only one VM (or a network functon) s assocated wth each request, and each server can accommodate no more than k VMs. Clearly, ths assumpton s over-smplfed as the length of a servce chan of each request may be far greater than one. Lukovszk and Schmd [22] acheved several mportant theoretcal results under deal assumptons that each server can accommodate only one VM and dfferent VMs for dfferent network functons consume the

3 HUANG et al.: EFFICIENT ALGORITHMS FOR THROUGHPUT MAXIMIZATION IN SDNs WITH CONSOLIDATED MIDDLEBOXES 633 same amount of computng resource. Xu et al. [32] devsed the very frst approxmaton algorthm for the NFV-enabled multcastng problem and onlne algorthm wth a compettve rato for dynamc admssons of NFV-enabled multcast requests wthout the knowledge of future request arrvals. Followng the same assumpton n [5] and [12], n ths paper we assume that mddleboxes are mplemented as software applcatons runnng as VMs n servers or data centers. Meanwhle, there are many possbltes for resource sharng, one of whch s to use a dedcated VM for each NFV. However, consderng a chan of network functons s often made up of several functons [28], ths approach wll clearly not be feasble as physcal resources wll easly be depleted, and wll be wasteful of resources snce most functons are lght-weght and can therefore be processed by a sngle VM, e.g., by contaners wthn the VM [9]. Therefore, we further adopt the dea of consoldated mddleboxes [12], where every flow obtans all ts requred functonal treatment at a sngle PM, because the consoldated mddlebox model smplfes traffc routng, helps reduce the number of routng rules n the swtches, and removes the topology dependence between dfferent mddleboxes. On the other hand, n contrast to [5], we do not allow routng traffc va multple paths from ts source to ts destnaton, because most network functonaltes wll be appled to the entre packet flow, e.g., encrypton and decrypton should only be appled to an entre message. III. PRELIMINARIES A. System Model We consder a software-defned network represented by a drected graph G = (V, E), where V s the node set and E s the edge set. Each node v V represents an SDN-enabled swtch, whle each drected edge u, v E represents a lnk from swtch u to swtch v. Each swtch v V s equpped wth a Ternary Content-Addressable Memory (TCAM) forwardng table that can accommodate at most L v forwardng rules. A subset of swtches n V s connected to physcal machnes (PMs) to mplement mddleboxes as vrtual machnes. As such a swtch and ts attached PM usually are connected by a hgh-speed optcal lnk, the latency between them s neglgble. In the rest of ths paper, the swtches and ther attached PMs wll be used nterchangeably. Denote by V pm ( V) the set of swtches that have attached PMs. Wthout loss of generalty, we assume that each PM attached to a swtch v V pm has lmted computng resource capacty, denoted by C v. If swtch v V \ V pm, then C v = 0. Smlarly, each lnk e E has a bandwdth capacty B e. We assume that there s a logcally centralzed SDN controller for network G that collects and processes user requests, by nstallng forwardng rules nto the forwardng tables n swtches, assgnng the mddleboxes for the requests to PMs, and allocatng bandwdth on lnks. B. User Requests We assume that tme s slotted nto equal tme slots. User requests are scheduled by the centralzed SDN controller n the begnnng of each tme slot. Let S(t) be the set of arrved user requests n tme slot t. Each user request has a certan amount of bandwdth demand to route ts traffc n G from a source swtch to a destnaton swtch that passes through a sequence of mddleboxes, and the request also has the end-to-end delay requrement. Let r S(t) be a user request, represented by a quntuple r = s, t, b, SC, d, where s, t V are, respectvely, ts source and destnaton swtches, b s ts bandwdth demand, SC s ts servce chan, and d R + s ts end-to-end delay constrant. Admsson of request r therefore nvolves routng the traffc from the source swtch s to the destnaton swtch t va a routng path P = s,...,t subject to the specfed constrants. Followng the same assumpton as n [12], [23], [26], and [27], we assume that servces n SC are run n a sngle VM and dfferent VMs servng dfferent requests can be consoldated to a sngle Physcal machne (PM). Specfcally, when the traffc of request r arrves at the PM hostng the VM for ts servce chan SC, the traffc wll be drected to the VM and the servces n SC are appled n the specfed order. Performng the servces n SC for r thus wll consume the computng resource of a PM. Denote by C(, j) the amount of computng resource needed by SC n a PM attached to swtch v j V pm. Notce that some servces n SC may alter the volume of the traffc of request r. For nstance, the volume of traffc ncreases f encrypton s appled to the traffc, whle the volume of traffc decreases f compresson s appled to the traffc. We here defne λ R + as the rato between the volumes of the traffc of request r before and after processng at a PM. Snce request r requres an amount b of bandwdth to route ts traffc before processng, t needs an amount λ b of bandwdth to route the processed traffc. The value of λ for each request r s gven and can be derved from hstorcal traces of smlar requests [6]. In addton, each request r has a tolerant end-to-end delay requrement d. Suppose that request r s admtted wth a routng path P from ts source s to ts destnaton t, and ts servce chan SC s mplemented on a PM-attached swtch v V pm on P.Letd(P ) and d(, v) be the network delay experenced by r va path P and the processng delay of r at PM v, respectvely. The network delay d(p ) s proportonal to the number of swtches on P, and the average processng delay d(, v) depends on the complexty of the servce chan SC whch usually s gven as apror. Then, the end-to-end delay D of r va path P s the sum of the network delay of P and the processng delay of SC,.e., d(p )+d(, v). It has to be guaranteed that d(p )+d(, v) d for every admtted request r. C. Problem Defnton Gven an SDN G = (V, E), a subset of swtches V pm ( V) wth each attachng a PM wth computng capacty C v,the forwardng table capacty L v for each swtch v V, the bandwdth capacty B e for each lnk e E, and a set of user requests S(t) at tme slot t, thenetwork throughput maxmzaton problem n G s to admt as many user requests n S(t) as possble such that the number of requests admtted s maxmzed whle the end-to-end delay d, bandwdth demand b, and computng demand C(, j) of the servce chan SC of each

4 634 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER 2017 Fg. 1. An example of the SDN G constructed from an nstance of the GAP wth four tems and four bns. We then generate a set of requests S: For each tem n I, we add to S a request r = s, t, b, SC, d, where s s set to the swtch n V, t s set to the vrtual snk t, b = 0, the computng resource demand C(n, m) to process ts servce chan at m V pm s sze(n, m), and d =. Therefore, routng the set of requests S nto network G s an nstance of the network throughput maxmzaton problem. We fnsh by notng that the network throughput maxmzaton problem has a soluton of admttng K requests f and only f the GAP wth dentcal profts has a soluton of proft K. admtted request r S(t) s met, subject to resource capacty constrants n G. GvenanSDNG = (V, E), a subset of swtches V pm ( V) wth each attachng a PM wth computng capacty C v, the forwardng table capacty L v for each swtch v V, the bandwdth capacty B e for each lnk e E, and a gven tme horzon T that conssts of T equal tme slots, assume that the set of requests arrved at tme slot t s S(t) and the duraton of each request r = s, t, b, SC, d,τ S(t) n the system s τ tme slots wth 1 τ τ max,theonlne network throughput maxmzaton problem n G s to admt as many user requests as possble durng tme horzon T such that the number of requests admtted s maxmzed whle the end-to-end delay d, bandwdth demand b, and computng demand C(, j) of the servce chan SC of each admtted request r S(t) s met, subject to resource capacty constrants n G. D. NP-Hardness We show that the network throughput maxmzaton problem s NP-hard by the followng lemma. Lemma 1: The network throughput maxmzaton problem n a software-defned network G = (V, E) s NP-hard. Proof: We show that the network throughput maxmzaton problem n G = (V, E) s NP-hard, by a polynomal reducton from the generalzed assgnment problem (GAP) whch s a well-known NP-hard problem [7]. Gven an nstance of the GAP n the form of a set of bns B ={b 1,...,b n },aset of tems I ={ 1,..., m }, bn capactes cap: B R + and sze : B I R +. For each tem j wth 1 j m and bn b k wth 1 k n, we are gven a sze sze(j, k) and a proft proft(j, k). The problem s to pack a subset U I of tems to the bns n B such that the total proft by these tems s maxmzed. The GAP problem s a well-studed problem. We frst construct an SDN G = (V, E), through addng a stand-alone swtch for each tem n I, a PM-attached swtch b for each bn b n B, a vrtual snk v 0 that s servng as the common destnaton for all requests, a lnk from each standalone swtch to each PM-attached swtch, and a lnk from each PM-attached swtch to the vrtual snk v 0. That s, V = I B {v 0 } and E ={, b I, b B} { b, v 0 b B}. Fg. 1 shows an example of the constructed SDN G = (V, E). The forwardng table sze at each node n V and the bandwdth resource capacty of each lnk n E are set to nfnty. Moreover, V pm = B and the computng capacty of each node m n V pm s set to cap(m), the capacty of bn m. IV. INTEGER LINEAR PROGRAM In ths secton, we formulate the network throughput maxmzaton problem as an Integer Lnear Program (ILP), where x s a decson varable wth value 1 f request r s admtted and 0 otherwse. z v s a decson varable wth value 1 f and only f the traffc of r s processed by the PM attached to swtch v V pm. For brevty, denote by δ + (v) and δ (v) the sets of leavng and enterng edges of a swtch v V, respectvely. In addton, to dstngush between traffc before and after beng processed at a PM, we ntroduce two decson varables w pre (e) and w post (e) wth value 1 f and only f lnk e carres the unprocessed and processed traffc, respectvely. The detaled descrpton s gven n Fg. 2, Constrant (2) ensures that f and only f a request r S(t) s admtted, t wll be processed n exactly one PM. The volume of the traffc may change after the processng at v, whle the volume s conserved at other non-termnal swtches except the swtch v V pm where t s processed. Constrants (3) and (4) capture traffc changng at PMattached swtches that process traffc of user requests and traffc conservaton at non-termnal swtches. Specfcally, f request r s processed at v V pm, then () exactly one ncomng edge of v carres the unprocessed traffc and none of the outgong edges of v carres the unprocessed traffc; and () exactly one of the outgong edges of v carres the processed traffc, and none of the ncomng edges of v carres the processed traffc. Otherwse, f the traffc of r s not processed by the PM attached to swtch v V pm but goes through v, ether () exactly one ncomng edge and one outgong edge of v carry the unprocessed traffc, or () exactly one ncomng edge and one outgong edge of v carry the processed traffc. Constrants (5) and (8) ensure that no unprocessed traffc enters any source swtch s and no processed traffc leaves the termnal swtch t. Constrants (6) and (7) handle the cases where the traffc of a request v s processed at the source swtch s or the termnal swtch t. Constrant (9) enforces that the end-to-end delay requrement, whch s the sum of the network delay D n (P ) and the processng delay D p (, v), of every admtted request s met, where the network delay D n (P ) s calculated by e E (wpre (e) + w post (e)) and the processng delay D p (, v) s v V zv D p(, v). Snce z v s 1 only for node v that mplements the consoldated mddleboxes for request r, only the processng delay at the node v s ncurred.

5 HUANG et al.: EFFICIENT ALGORITHMS FOR THROUGHPUT MAXIMIZATION IN SDNs WITH CONSOLIDATED MIDDLEBOXES 635 Fg. 2. An ILP formulaton of the network throughput maxmzaton problem. Constrant (10) enforces the bandwdth capacty constrant for each lnk e E. Constrant (11) mposes the forwardng table capacty constrant for each swtch v V, and Constrant (12) models the computng capacty constrant of PMs attached to each swtch v V pm. Constrants (13), (14), and (15) restrct the range of decson varables to 0 and 1 nclusvely. Constrant (16) ndcates that f there s no PM at a swtch v V \ V pm, then t cannot process any request. Snce the ILP soluton s tme-consumng, t s only applcable when the problem sze s small. The rest of ths paper wll develop effcent, scalable solutons to the problem. V. A HEURISTIC ALGORITHM In ths secton, we focus on devsng an effcent heurstc for the problem. We frst propose a cost model to capture the dynamc resource usages n G, and then devse the algorthm through a reducton that reduces the problem nto shortest path fndngs n a seres of auxlary graphs derved from G. A. A Novel Cost Model of Resource Usages and the Constructon of An Auxlary Graph Gven an SDN G, t contans dfferent types of resources such as computng resources at servers, TCAM szes at swtches, and bandwdth resources at lnks. Desgnng an effcent algorthm for the network throughput maxmzaton problem needs to utlze these resources judcously, through the gudance of an effcent cost metrc that can accurately capture the usages and utlzatons of dfferent resources. In the followng, we frst propose a cost model of resource usages. We then reduce the problem of concern n G nto another problem of fndng shortest paths n a seres of auxlary graphs G that are derved by mplementng the servce chan at dfferent servers n G.

6 636 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER 2017 Fg. 3. The auxlary graph constructon of G from G for the th request: (a) the orgnal SDN G = (V, E); and (b) the correspondng auxlary graph G = (V, E ) of G. Fg. 4. Augmentng auxlary graph G on the left to G,v on the rght for swtch v V pm. Gven an SDN G = (V, E) and a request r = s, t, b, SC, d, the auxlary graph G = (V, E ; ω ) for request s constructed as follows. For each swtch v n V, two vertces v and v are added to V, and a drected edge v, v s added to E. For each lnk u, v n E, an edge u, v s added to E,.e., V = {v, v v V} and E ={ v, v v V} { u, v u, v E}. Intutvely, each edge v, v n G represents swtch node v and an edge u, v represents lnk u, v n G. An example of such an auxlary graph s gven n Fg. 3. The cost model of resource usages n G s proposed as follows. For a gven type of resource, the margnal cost of ts usage dramatcally nflates wth the ncrease of ts utlzaton rato, snce the larger proporton of the resource s occuped, the hgher rsk the resource capacty wll be volated. We therefore use an exponental functon to model the cost of resource usage. Denote by RL v, the resdual capacty of the forwardng table at v V and RB e, the resdual bandwdth of lnk e = v, u E when request r arrves. Then, the weghts of ther correspondng edge e E n G are ω (e ) = α 1 RL v, Lv f e = v, v E, β 1 RB v,u, B v,u f e = v, u E, (1) where α and β are constants wth α, β > 1. The larger the values of α and β, the more the resources wth hgh utlzatons wll be dscouraged to use, snce ther margnal costs wll ncrease wth the ncrease of ther utlzaton ratos. Notce that the usage cost of computng resource n PMs has not been ncorporated nto the auxlary graph G, because admttng a request r va a PM-attached swtch v V pm does not necessarly consume the computng resource of the PM. Only f the servce chan SC of r s realzed n t wll the computng resource of ts attached PM be consumed. B. Algorthm The basc dea behnd the proposed algorthm s to reduce the problem n G nto fndng the shortest paths n a seres of graphs G wth 1 S. In the followng we frst consder a sngle request admsson. We then extend the soluton to the admssons of a set of requests. The detaled algorthm s descrbed as follows. We frst consder admttng a request r S(t) where r = s, t, b, SC, d. We fnd a shortest path n G = (V, E ) from s to t such that ts correspondng routng path n G meets both ts bandwdth demand b and ts end-to-end delay d and there s a swtch v V pm attached a PM n the path wth suffcent computng resource to process ts servce chan SC. Specfcally, we frst remove the edges wthout adequate resources from G, and then construct G = (V, E ) from the resultng graph G. To nclude computng resource n PMs for the admsson of request r, we then augment G for each PM-attached swtch v V pm and denote by G,v = (V,v, E,v ) the graph obtaned by augmentng G for a PM-attached swtch v V pm.the only dfference between G,v and G s that the drected edge v, v s removed, and a new node v and edges v, v and v, v are added to V,v and E,v, respectvely, as shown n Fg. 4 (b). Moreover, the weght of edge v, v s dentcal to the weght of v, v n G whle the weght of v, v s RCv 1 γ Cv, where γ > 1 s a tunng parameter whch usually s a constant, RC v s the resdual computng capacty, and C v s the capacty of v. Therefore, f v V pm s consdered to process servce chan SC of request r, routng the traffc of r s to fnd a path P (v) n G,v that s the concatenaton of a shortest path n G,v from s to v and a shortest path n G,v from v to t.letl(p (v)) and d(p (v)) be the length and delay of P (v),.e., l(p (v)) = e P (v) ω (e) and d(p (v)) = e P (v) d(e)+d(, v), where d(, v) s the processng duraton of SC of r at the PM attached to swtch v. The problem of admttng request r n G s then reduced to the problem of fndng a shortest path P (v) from one of the augmented auxlary graphs G,v derved from node v wth the mnmum length mn{l(p (v)) v V pm } that meets the end-to-end delay d. The detaled descrpton of the algorthm s gven n Procedure 1. We say that the derved routng path P for request r at step 6 n Procedure 1 s a pseudo-routng path or a walk,.e., the nodes and lnks on P may appear multple tmes, t can even contan cycles. Ths s unavodable for a certan type of network topologes. In the followng, we show the exstence of a smple shortest path P n G for request r f G meets certan condtons. Lemma 2: Gven a drected weghted graph G = (V, E), a specfc node v, and a request r wth source s and destnaton t, there s a smple shortest path n G from s to t that passes through node v f any path n another graph H from nodes v 0 to v does not contan any artculaton ponts, where v 0 s a vrtual node and edges v 0, s and t, v 0 are two vrtual edges wth weghts of zeros, and they are added to graph G,.e., H = (V {v 0 }, E { v 0, s, t, v 0 }) s then obtaned.

7 HUANG et al.: EFFICIENT ALGORITHMS FOR THROUGHPUT MAXIMIZATION IN SDNs WITH CONSOLIDATED MIDDLEBOXES 637 Procedure 1 Admttng a Sngle Request r Input: an SDN G = (V, E) and the current consderng request r = s, t, b, SC, d Output: fnd a routng path P = s,...,v V pm,...,t that satsfes b, SC, and d f t exsts. 1: Construct the auxlary graph G = (V, E ; ω ) for G; 2: P sel ; /* a path n an augmented auxlary graph wth the mnmum sum of edge weghts */ 3: l mn ; /* the mnmum length of routng paths */ 4: for each PM-attached swtch v V pm do 5: Construct G,v by augmentng G ; 6: Let P (v) be the concatenaton of a shortest path n G,v from s to v and a shortest path n G,v from v to t ; 7: f (d(p (v)) + d(, v) d )&(l(p (v)) l mn ) then P (v); 9: l mn l(p (v)); 10: v mn v; /* whch PM wll be used */ 11: end f 12: end for 13: The correspondng pseudo-routng path (walk) P n G s 8: P sel then derved from P sel va PM v mn f t exsts; Proof: It s known that v 0 s only connected wth nodes s and t n H, f there s an artculaton pont u n any path from v 0 to v, ths mples that any path between s (or t) and node v must pass through u, thus, f a path n G from s to t must contan u, then t appears n the path at least twce. Lemma 2 provdes a necessary condton of the exstence of a smple path n G from s to t that passes through v. That s, such a smple path exsts f any path n H from v 0 to v does not contan artculaton ponts. If for any request r and a specfed node v, the condton n Lemma 2 holds, a smple shortest path n G from s to t va v can be found as follows. We start wth the mnmum-cost two edge-dsjont path problem: Gven two nodes s and t n G(V, E), the problem s to fnd two edge-dsjont paths between s and t such that the sum of weghted edges n these two paths s mnmum. There s an effcent algorthm for ths problem due to Suurballe [30], and an mproved algorthm later s proposed by Suurballe and Tarjan [31]. To fnd two edge-dsjont paths n graph G from s to t such that the cost sum of the two paths s mnmum, Suurballe s algorthm proceeds as follows. It frst fnds a shortest path n G from s to t. It then reverses the drecton of the edges n the shortest path, and fnds a shortest path n the resultng graph from s to t. As a result, two edge-dsjont paths between s and t are then found through the exclusve unon of the two found paths, and the cost sum of the two paths s the mnmum one [30]. Clearly, ths algorthm can be modfed to fnd two node-dsjont paths between a par of nodes so that the cost sum of the two paths s mnmum, by adoptng the node splttng technque [30]. We now consder the smple shortest path problem n G between a par of nodes s and t that passes through a specfed node v. We reduce ths problem n G to the problem of fndng two node-dsjont paths n another graph H such Algorthm 1 A Heurstc for Admttng a Set of Requests S(t) Input: an SDN G = (V, E) and a set of requests S(t) Output: Determne whch request r S(t) to be admtted and ts routng path P sel 1: S S(t); /* the set of requests to be admtted */ 2: whle S = do 3: for each request r S do 4: Fnd a routng path P sel for request r, by nvokng Procedure 1; 5: f path P sel does not exst then 6: S S \{r }; /* remove r from S */ 7: end f 8: end for 9: Let r 0 be the request wth l(p sel 0 ) = mn r S(t){l(P sel )}; 10: Admt request r 0 usng the routng path P sel 0, and update the resource avalabltes of G by deductng the resources for accommodatng P sel 0 ; 11: S S \{r 0 }. 12: end whle that the cost sum of the two paths s mnmum. We frst construct a drected auxlary graph H = (V H, E H ) where V H = {v, v v V} {v 0 } and E H ={ u, v, v, u (u, v) E} { v 0, s, v 0, t }, by addng a vrtual node v 0 and vrtual edges nto H and assgned both newly added vrtual edges v 0, s and v 0, t wth weghts of zeros. We then fnd two edge-dsjont paths n H between v 0 and v such that the weghted sum of the paths s mnmum. We fnally have a smple path n G from s to t va v that s derved from the found two node-dsjont paths, by removng the vrtual node v 0 and ts ncdent two edges. The resultng path between s and t s a smple path va v and the sum of ts weghted edges s the mnmum one. Havng consdered a sngle request admsson, n the followng we deal wth the admssons of a set of requests S(t) at tme slot t, by admttng the requests one by one untl no more requests can be admtted. A non-admtted request n S(t) s admtted mmedately f t has the mnmum mplementaton cost at that moment. Specfcally, gven a to-be-admtted request r S(t), Procedure 1 s employed to fnd a routng path for r wthout commttng the admsson whch means that the SDN controller does not allocate resources to meet the demands by ths request. A found path P sel wth the mnmum cost among all remanng requests n S(t) wll be admtted and ts demanded resources wll be allocated to t, the resdual resource avalabltes n G are updated accordngly. Meanwhle, f Procedure 1 fals to fnd a path P sel for r, request r wll be rejected at tme slot t. Ths procedure repeats untl every request n S(t) s ether rejected or admtted. The detaled descrpton s gven by Algorthm 1. C. Algorthm Analyss In the followng, we frst show that the soluton delvered by Algorthm 1 s feasble, and then analyze ts tme complexty.

8 638 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER 2017 Lemma 3: Gven the augmented auxlary graph G,v = (V,v, E,v ) derved from G = (V, E) and a swtch v V pm for request r = s, t, b, SC, d S(t), the concatenaton of a shortest path from s to v and another shortest path n G,v from v to t wll result n a vald pseudo-routng path n G from s to t wth PM-attached swtch v n the path. Alternatvely, a smple shortest path from s to t through v delvered by applyng Suurballe s algorthm s also a feasble soluton f G meets the condton n Lemma 2. Proof: Let P (v) be the concatenaton of a shortest path from s to v and a shortest path from v to t n G,v.For smplcty, we use a lnk-derved edge to represent an edge u, v n E,v that s derved from edge u, v n E, and a swtch-derved edge to denote an edge v, v n E,v that s derved from swtch v V. We clam that () path P (v) conssts of lnk-derved and swtch-derved edges alternatvely; and () path P (v) can satsfy the requrements of request r,.e., the bandwdth demand b, the forwardng table demand, the computng resource demand for ts servce chan SC, and the end-to-end requrement d. Clam () s obvous because there s only an outgong edge for each swtch v,.e., v, v. Clam () holds because the augmented auxlary graph G,v s the result of removng the edges and swtches n G that cannot meet resource requrements of request r, and path P (v) s feasble only when ts end-to-end delay s no greater than d. The feasblty of the smple shortest path va a data center f t does exst can be proven smlarly, omtted. Theorem 1: Gven an SDN G = (V, E) wth a set V of swtches and a set E of lnks, a subset V pm V of swtches wth attached PMs, a set of user requests S(t) at tme slot t, there s an algorthm, Algorthm 1, for the network throughput maxmzaton problem, whch delvers a feasble soluton n O( S(t) 2 V 4 ) tme. Proof: The soluton delvered by Algorthm 1 s feasble because each auxlary graph s constructed from the subgraph of G that only ncludes the resources wth suffcent resdual capactes for the request. Consequently, the routng path n G derved from the found path n G s feasble. The tme complexty of Algorthm 1 s analyzed as follows. In Procedure 1, the constructon and augmentaton of the auxlary graph G take O( V + E ) tme, whle fndng a shortest path n each of the V pm augmented auxlary graphs G,v takes O( V 3 ) tme. Procedure 1 thus takes O( V 3 + V + E ) = O( V 3 ) tme. For each request r S(t), Procedure 1 s nvoked at most V pm tmes. The number of requests s O( S(t) ). If we make use of Suurballe s algorthm to fnd a smple shortest path from s to t n G,v through v, t takes O( E + V log V ) = O( E + V log V ) tme as the constructon of the auxlary graph H and fndng shortest paths n H take no more than that amount of tme. The tme complexty of Algorthm 1 thus s O( S(t) 2 V pm V 3 ) = O( S(t) 2 V 4 ). The theorem holds. VI. A FASTER HEURISTIC ALGORITHM Although Algorthm 1 delvers a near optmal soluton emprcally, whch can be seen n later expermental evaluatons, ts runnng tme s qute hgh and may fal to respond to user requests on tme, consderng user requests arrve one by one wthout the knowledge of future request arrvals. In ths secton we devse a faster heurstc that strves for the non-trval trade-off between the accuracy of a soluton and the runnng tme of obtanng the soluton. A. Overvew A key ngredent of ths faster heurstc s that a canddate soluton to admt a subset of S(t) of requests s based on the resdual resource capactes of G n the begnnng of tme slot t, and there s no updatng to these resdual capactes when all requests n S(t) are beng consdered. Thus, a canddate soluton s dentfed frst. It then further refnes the canddate soluton teratvely untl no resource capacty volaton occurs. B. Algorthm We frst fnd a set of canddate routng paths P n G for each request r = s, t, b, SC, d S(t), wthout consderng the resource capacty constrants of G, where a shortest path from s to t s treated as a canddate path of r as long as t has one PM-attached swtch n V pm that satsfes b, SC, and d.as the servce chan SC of r must be served at one of V pm PMattached swtches, we can fnd at most V pm canddate shortest paths for each r. Notce that we fnd canddate routng paths for requests n S(t) on the augmented auxlary graphs based on the resource avalablty of G as of the begnnng of tme slot t, through fndng a shortest path from s to v and a shortest path from v to t n G,v for each request r S(t) and v V pm.letp (v j ) = s,...,v j,...,t be a found path n G,vj for request r, whereas v j ( V pm )saswtchthat fulflls the servce chan SC and P (v j ) meets the resource and end-to-end delay constrants of r. Denote by P be the set of canddate paths for request r, we then have, P ={P (v j ) v j V pm }. (2) Havng the set of canddate paths P for each request r, we then pck only one canddate path P (v j ) n P for request r n a way such that the cost sum of the selected paths for all requests s mnmzed, whle ensurng that the computng capacty of each PM s not volated. In essence, selectng a path P (v j ) P to route request r S(t) s equvalent to selectng a PM attached to a swtch v V pm to mplement SC for r. As dfferent PMs may have dfferent computng capactes, ths means that the processng SC of r at dfferent PMs wll ncur dfferent computng resource demands. We thus reduce ths problem to the GAP, whch s defned as follows. Gven a set of tems I and a set of bns B, where each bn b B has a capacty cap(b), each tem I has a sze sze(, b), and a proft proft(, b) f tem s placed n bn b, the problem s to place a subset of tems U ( I )nbnsb such that the sum of the profts of tems n U s maxmzed and the sum of szes of tems placed n every bn s no more than the capacty of the bn. We now treat each PM-attached swtch v j V pm as a bn and each request r n S(t) as an tem, whereas the capacty of each bn v j s ts resdual computng capacty,.e., cap(v j ) = LC vj,

9 HUANG et al.: EFFICIENT ALGORITHMS FOR THROUGHPUT MAXIMIZATION IN SDNs WITH CONSOLIDATED MIDDLEBOXES 639 Fg. 5. An example of a bpartte graph G b. the sze of an tem r nabnv j s the computng demand of the servce chan SC n the PM attached to v j,.e., sze(r, v j ) = C(, j), and the proft of placng an tem r n a bn v j s the recprocal of the length of the canddate path that fulflls r on v j,.e., proft(, j) = l(p 1 (v j )). Havng reduced the network throughput maxmzaton problem to the GAP, we now solve the GAP and each soluton to the GAP yelds a soluton to the orgnal problem. Specfcally, we use the algorthm proposed by Cohen et al. [7] that guarantees a (2 ɛ)-approxmaton rato, where ɛ s a constant wth 0 <ɛ 1, to solve the GAP. Denote by U a soluton found by ths algorthm as a placement of a subset of tems n bns. U yelds a potental admsson of requests n S(t): for every request r treated as an tem, f t s placed n a bn representng v j V pm, then t s admtted wth the routng path P (v j ); otherwse, r s rejected. Due to the constructon of the GAP, admttng requests n S based on the soluton U to the GAP ensures that the sum of computng demands of requests of whch the servce chans are fulflled n the same PM wll not exceed the computng capacty of the PM. However, the bandwdth and forwardng table capactes may be volated, as routng paths may have overlappng resources. Now, for each request allocated to a bn, ts computng demand can be met wthout volatng the computng capacty of the bn. Some requests however may volate the bandwdth and forwardng table sze capactes of some lnks and nodes whle routng ther traffc. We thus perform adjustments to elmnate such potental resource volatons by selectvely rejectng some requests. Let = P (v j ) = s,...,v j,...,t be the path to route the traffc of request r accordng to U, where v j V pm. The basc dea behnd the adjustment here s to carefully fnd such a path wth resource capacty volatons teratvely and remove ts request from admsson. Ths procedure contnues untl there s no volaton of resource capacty. To ths end, a bpartte graph G b = (U b, V b, E b ) s constructed, where U b s the set of selected routng paths for all potentally admtted requests, P sel V b s the set of edges n r S(t)E(P sel ) whch each corresponds to a resource n G. There s an edge between a node P sel U b and a node e V b f e s n P sel. The weght of edge (P sel, e) E b s the rato of the demand of r on that resource to the sum of those of all requests on that resource, whch represents the contrbuton of r to the resource capacty volaton of e. An example of such a bpartte graph G b s shown n Fg. 5. To elmnate resource capacty volatons, we teratvely remove one node P sel and ts ncdent edges n G b wth the maxmum weghted sum of the ncdent edges, and update Algorthm 2 A Faster Heurstc for Routng a Set of Requests S(t) Into a G Input: an SDN G = (V, E) and a set of user requests S(t) Output: Routng decsons for each request r S(t) 1: Buld an auxlary graph G = (V, E ) for G; 2: Intalze P, the set of canddate routng paths n G for all requests n S(t), to ; 3: for each user request r S(t) do 4: P ; /* the set of canddate paths for request r */ 5: for each PM-attached swtch v j V pm do 6: Fnd a path P (v j ) for r va node v j, by nvokng Procedure 1; 7: f P (v j ) exsts then 8: P P {P (v j )}; 9: end f 10: end for 11: f P s empty then 12: Reject request r ; 13: else 14: P P {P }; 15: end f 16: end for 17: Construct an nstance of the GAP by representng each request as an tem and each node n V pm as a bn; 18: Solve the GAP nstance by nvokng the algorthm n [7]; 19: Construct a bpartte graph G b = (U b, V b, E b ) that reflects potental capacty volatons; 20: whle there are edges n E b do 21: Update G b by the removal of such a node n U b that has the maxmum weghted sum of ts ncdent edges and ts ncdent edges from E b. 22: end whle G b by removng nodes n V b that ther resource overloadngs are avoded due to the removal of node P sel. For example, n Fg. 5, both P sel 1 and P sel 2 volate the computng capacty constrants of e 1, e 2, and e 3. Snce P sel 2 results n more volatons of resource capacty constrants than P sel 1 does, t wll be removed frst. Ths procedure contnues untl no edge s left n E b, a feasble soluton wll be obtaned ultmately. The detaled descrpton s gven n Algorthm 2. C. Algorthm Analyss In the followng, we show that the soluton delvered by Algorthm 2 s a feasble soluton. We then analyze the tme complexty of the proposed algorthm. Theorem 2: GvenanSDNG = (V, E) wthasetv of swtches and a set E of lnks, a subset V pm V of swtches wth each attachng wth a PM, a set of user requests S(t), there s an algorthm for the network throughput maxmzaton problem, Algorthm 2, whch delvers a feasble soluton n O( S(t) V 3 + V S(t) 3 ɛ ) tme, where ɛ s a gven constant wth 0 <ɛ 1. Proof: Recall that Algorthm 2 conssts of three phases: () fnd a set of canddate routng paths for each request; () select only one routng path for each request to meet

10 640 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER 2017 Algorthm 3 Onlne Algorthm Wthn a Fnte Tme Horzon T Input: an SDN G = (V, E) and tme horzon T Output: determne whch request r S(t) to be admtted and ts routng path P sel at each tme slot t wth 1 t T. 1: for t 1toT do 2: Release all resources occuped by the requests that left n the end of tme slot (t 1), and recalculate the resdual resources n G; 3: Let S(t) be the set of arrved requests n the begnnng of tme slot t; 4: f S(t) = then 5: Fnd a subset S (t) S(t) of requests that are admssble at tme slot t, by nvokng ether Algorthm 1 or Algorthm 2 based on the avalable resources n G. 6: end f 7: end for computng capactes of nodes n V pm ; and () elmnate the requests that volate bandwdth or forwardng table capactes. The feasblty of the soluton delvered by Algorthm 2 mmedately follows from Phase (). The rest s to analyze the tme complexty of Algorthm 2. Phase () takes O( S(t) V 3 ) tme, because O( V pm ) = O( V ) shortest paths are found for each request r S(t) n augmented auxlary graphs and each shortest path takes O( V 2 ) tme. The runnng tme of Phase () s domnated by the tme requred to solve the GAP, whch s ) [7]. Phase () takes O( S(t) ( V + E )) tme, there are O( S(t) ( V + E )) edges n the bpartte graph G b, followng the constructon of the bpartte graph. In the worst scenaro, each request volates the resource capactes on all swtches and lnks. The theorem thus holds. O( V S(t) 3 ɛ VII. ONLINE ALGORITHM In ths secton, we study the onlne network throughput maxmzaton problem, by consderng dynamc admssons of user requests wthn a fnte tme horzon T. We wll make use of the proposed algorthms n the prevous secton to solve ths problem. We assume that the system evolves over tme. The tme s parttoned nto equal tme slots, and the user request admsson schedulng proceeds n the begnnng of each tme slot. Some mplementng requests may also leave the system, and the resources occuped by them wll be released back to the system n the end of the current tme slot. The released resources wll be avalable n the begnnng of the next tme slot. The detaled onlne algorthm for dynamc request admssons s gven n Algorthm 3. Theorem 3: Gven an SDN G = (V, E) wth a set V of swtches and a set E of lnks, a subset V pm V of swtches wth each attachng wth a PM, and a fnte tme horzon T, there s an algorthm for the onlne network throughput maxmzaton problem, Algorthm 3, whch delvers a feasble soluton n O( T t=1 ( S(t) V 3 + V S(t) 3 ɛ )) tme f Algorthm 2 s used as ts subroutne, where ɛ s a gven constant wth 0 <ɛ 1. Proof: The tme complexty of Algorthm 3 per tme slot s the dentcal to the one for Algorthm 2, omtted. VIII. PERFORMANCE EVALUATION In ths secton, we evaluate the performance of the proposed algorthms through expermental smulatons, usng real and synthetc SDNs. We start wth the expermental envronments, we then evaluate the performance of the proposed heurstc algorthms for the network throughput maxmzaton problem. We also evaluate the performance of the onlne algorthms for the onlne network throughput maxmzaton problem. We fnally nvestgate the mpact of parameters on the performance of the proposed algorthms. A. Expermental Envronment We adopt commonly used, real network topologes ncludng GÉANT [23] and several ISP networks from [29] nthe smulatons, where GÉANT [23] s a European network consstng of 40 nodes and 122 lnks. The sze of the forwardng table of each swtch s set from 1,000 to 8,000 randomly [18]. The bandwdth of each Internet lnk vares from 1,000 Mbps to 10,000 Mbps [17]. There are nne PMs for the GÉANT topology as set n [12] and the number of PMs n ISP networks are provded by [26]. The computng capacty of each PM s from 4,000 to 8,000 MHz [13]. The delay of a lnk s between 2 mllseconds (ms) and 5 ms [17], [18]. We consder fve types of mddleboxes: Frewall, Proxy, NAT, IDS, and Load Balancng, and ther computng demands are adopted from [12] and [23]. The runnng tme s obtaned based on a machne wth a 3.40GHz Intel 7 Quad-core CPU and 16 GB RAM. The default accuracy parameter ɛ n solvng GAP s set to 0.1. Unless otherwse specfed, these parameters wll be adopted n the default settng. Each request r = s, t, b, SC, d S(t) s generated as follows. Gven a network G = (V, E), two nodes n V are randomly drawn as the source s and the destnaton t of request r. The bandwdth demand b s randomly drawn from 10 to 120 Mbps [1] and the end-to-end delay d s set from 40 ms to 400 ms randomly [24]. We evaluate Algorthm 1 and Algorthm 2 aganst a baselne heurstc whch s descrbed as follows. Sort all requests n S(t) n ncreasng order of ther computng resource demands, and then, for each request r = s, t, b, SC, d n S(t), fnd a shortest path n G from s to a PM-attached swtch v ( V pm ) wth the mnmum number of hops from s and a shortest path from v to t. We refer to ths mnmum-hop-based baselne as algorthm MH, and algorthm ILP, Algorthm 1 and Algorthm 2 as ILP, ALG-1 and ALG-2, respectvely. Each value n fgures s the mean of the results of 30 trals. B. Performance of Dfferent Algorthms Wthn One Tme Slot In the followng, we nvestgate the performance of the proposed algorthms ILP, ALG-1, ALG-2, and MH n the GÉANT topology wthn a sngle tme slot.

11 HUANG et al.: EFFICIENT ALGORITHMS FOR THROUGHPUT MAXIMIZATION IN SDNs WITH CONSOLIDATED MIDDLEBOXES 641 Fg. 6. Performance of dfferent algorthms on the GÉANT wthn one tme slot. Fg. 7. Performance of dfferent algorthms n the GÉANT by varyng the number of swtches from 100 to 600, whle the number of requests s fxed at 160 per tme slot. Fg. 6 (a) shows the number of requests admtted by dfferent algorthms, when the number of requests arrved at a tme slot s n the range from 40 to 160. It can be seen that both algorthms ALG-1 and MH can admt as many requests as ILP does f there are less than 100 requests. Otherwse, only algorthm ALG-1 can acheve a comparable throughput as ILP. Ths means that the network throughput of algorthm ALG-2 s nferor to algorthm ALG-1, and the gap between ther performance enlarges from nearly zero at S(t) =40 to 21 at S(t) =160. The reason s that algorthm ALG-2 wll reject more requests wth the ncrease on the number of requests, as the lkelhood of routng paths that algorthm ALG-2 fnds for dfferent requests beng overlappng and resource volaton soars. Meanwhle, t can be seen that algorthm MH outperforms algorthm ALG-2 only when the number of requests s small. Specfcally, when there are 160 requests, the number of requests admtted by algorthm MH s only 60% of that by algorthm ALG-2 but runs much faster. The reason behnd s that algorthm MH does not guarantee that the routng path of request r from ts source s to ts destnaton t

12 642 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER 2017 Fg. 8. Performance of dfferent onlne algorthms n the GÉNT wthn a tme horzon of 200 tme slots, where the number of requests arrves at each tme slot follows a Posson dstrbuton wth a mean of 30. has the mnmum weght, snce t fnds shortest paths from s to a PM-attached swtch and from that PM-attached swtch to t separately. Fg. 6 (b) llustrates the amounts of tme spent by dfferent algorthms, from whch t can be seen that the runnng tme of algorthm ILP s several orders of magntude of the other mentoned algorthms, whle algorthm MH s the fastest one, and ALG-2 s faster than ALG-1 sgnfcantly. In addton, the runnng tme of algorthm ILP starts rsng when there are more than 80 requests. The reason s that when the number of requests s small and ther resource demands are relatvely small compared to the network capacty, many feasble solutons that acheves the best performance exst, yet as the resource demands become ncreasngly consderable relatve to the network capacty, fewer optmal solutons exst, and thus ILP spends a sgnfcant amount of tme on searchng for such an optmal soluton. Fg. 6 also demonstrates that algorthm ILP suffers poor scalablty and cannot fnsh wthn a reasonable amount of tme when the problem sze s large. We now evaluate the performance of dfferent algorthms f the number of requests arrved follows a Posson dstrbuton wth the mean between 40 and 160. The results of the four mentoned algorthms are summarzed n Fg. 6 (c)-(d), from whch t can be seen that the smlar behavor patterns are present n Fg. 6 (a)-(b). For nstance, t can be seen that the number of requests admtted by algorthm ALG-1 s dentcal to that by algorthm ILP when the number of requests s no more than 70. Meanwhle, the number of admtted requests by algorthm ALG-2 s on a par wth that of algorthm ALG-1, and t outperforms algorthm MH by 40% when there are more than 120 requests n the set. Smlarly, algorthm ILP s the slowest one whle algorthm MH s the fastest one. In partcular, when the mean of the number of requests s set at 140, the runnng tme of algorthm ILP s more than 1,000 tmes of that of algorthm ALG-1, whlst algorthm ALG-1 s more than sx tmes slower than algorthm ALG-2. Although the runnng tme of algorthm MH s the fastest, the number of requests admtted by t s the smallest one. We fnally evaluate the performance of dfferent algorthms by varyng the network sze. As publcly avalable topologes such as [17] and [29] have lmted szes, we adopt the wdely used Barabás-Albert model [3] to generate networks of dfferent szes. Namely, we vary the number of swtches n an SDN from 100 to 600 whle fxng the number of requests at 160. The results are depcted n Fg. 7. It can be seen that from Fg. 7 (a) that algorthms ALG-1 and ALG-2 acheve the smlar throughput, whle algorthm MH admts only no more than half the requests admtted by ether of the two heurstcs. Fg. 7 (b) reveals that algorthm ALG-2 runs much faster than algorthm ALG-1. In contrast to hgh admsson ratos delvered by algorthms ALG-1 and ALG-2, algorthm MH admts less than one half as many as requests as the other two algorthms, neutralzng ts advantage of havng the lowest runnng tme among all three algorthms ndcated n Fg. 7 (b). Fg. 7 (b) also reveals that algorthm ALG-2 runs much faster than algorthm ALG-1, e.g., algorthm ALG-1 spends 61,208 ms on admttng 160 requests to a network wth 600 swtches, whle algorthm ALG-2 takes only 2,106 ms. C. Performance of Onlne Algorthms We now consder a tme horzon that conssts of 200 tme slots, under whch we evaluate the performance of the onlne versons of the proposed algorthms, assumng that the number of requests at each tme slot follows a Posson dstrbuton wth a mean of 30, and each admtted request spans from 1 to 10 tme slots randomly. The results are summarzed n Fg. 8. It can be seen from Fg. 8 (a) that algorthm MH has the lowest network throughput among the mentoned algorthms. On the other hand, algorthms ALG-1 and ALG-2 utlze resources more effcently, and hence admt much more requests than that of algorthm MH by 150% and 50%, respectvely. It can also be seen from Fg. 8 (b) that the runnng tme of algorthm MH s neglgble compared wth those of algorthms ALG-1 and ALG-2. It must be notced that ths runnng tme comes at the cost of admttng much fewer requests. Although the runnng tme of algorthm ALG-2 s less than that of algorthm ALG-1, the gap between them becomes smaller. Specfcally, algorthm ALG-2 s only half of the runnng tme of algorthm ALG-1 whle t has only 10% of the runnng tme of algorthm ILP. The man reason s that the addtonal tme ncurred from constructng an nstance of the GAP cannot be gnored when the

13 HUANG et al.: EFFICIENT ALGORITHMS FOR THROUGHPUT MAXIMIZATION IN SDNs WITH CONSOLIDATED MIDDLEBOXES 643 Fg. 9. The accumulatve number of admtted requests and the runnng tme of dfferent onlne algorthms based on algorthms ALG-1, ALG-2 and MH for a tme horzon of 200 tme slots n ISP networks. Fg. 10. The accumulatve number of admtted requests and runnng tme of dfferent onlne algorthms based on algorthms ALG-1, ALG-2, andmh wth dfferent maxmum duratons of requests when the network s of the GÉANT. number of requests s relatvely small, whch wll be offset when the number of requests s greater. The rest s to evaluate algorthms ALG-1, ALG-2, and MH n three network topologes from [29]: AS-4755 s a network wth 121 swtches and 296 lnks, AS-1755 wth 172 swtches and 762 lnks, and AS-3967 wth 212 swtches and 886 lnks. The results are llustrated n Fg. 9, from whch t can be seen that n terms of the performance, algorthm ALG-1 s the best whle algorthm MH s the worst. The performance gap between algorthm ALG-1 and algorthm ALG-2 s small compared to that n the GÉANT, snce the sze of these three networks s larger than that of the GÉANT, and routng paths delvered by algorthm ALG-2 for dfferent requests at each tme slot are less lkely to overlap. In AS-4755, the dfference on requests admtted by algorthms ALG-1 and ALG-2 s less than 500, whle algorthm MH admts no more than 40% the number of requests by the other two algorthms. On the other hand, the performance of algorthm MH n both AS-1755 and AS-3967 mproves due to the larger resource capacty of the network. The accumulatve runnng tme of these three algorthms when admttng requests nto dfferent networks are

14 644 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 14, NO. 3, SEPTEMBER 2017 shown n Fg. 9 (d)-(f). The results n both Fg. 9 (d) and (e) are smlar: the runnng tme of algorthm MH s slghtly smaller than that of algorthm ALG-2, and algorthm ALG-1 s much slower than both algorthms ALG-1 and MH. However, we notce from Fg. 9 (f) that algorthm ALG-2 s faster than algorthm MH, snce algorthm ALG-2 only needs to fnd shortest paths n a graph once. D. Impact of Request Duratons on the Performance of Dfferent Onlne Algorthms We fnally evaluate the mpact of the maxmum duraton of requests on the performance of dfferent onlne algorthms based on algorthms ALG-1, ALG-2, and MH, by varyng the maxmum duraton from 5 tme slots to 25 tme slots. The results are presented n Fg. 10. FromFg. 10 (a)-(b) we can see that the longer the maxmum duraton, the fewer requests admtted by all mentoned algorthms, because longer duratons result n less avalable resources n the network. Fg. 10 (a)-(c) show that when the maxmum duraton of requests s fve tme slots, algorthm ALG-2 admts as many requests as algorthm ALG-1, yet algorthm MH only admts half the number of requests as the two heurstcs. We also see that an ncrease n the request duratons has a greater mpact on algorthms ALG-2 and MH than that on algorthm ALG-1, as algorthm ALG-1 s more exhaustve. In addton, t can also be seen from Fg. 10 (d)-(f) that longer duratons are assocated wth less runnng tme, because f other parameters keep unchanged and request duratons become longer, more resources wll be fully occuped and accordngly, the auxlary graphs wll have fewer vertces and edges, shortenng operatons on the auxlary graphs and reducng the runnng tme of all algorthms. IX. CONCLUSION In ths paper, we studed the admssons of user requests wth each havng a sequence of network functons n an SDN so that the network throughput can be maxmzed, subject to the constrants of forwardng table capacty, network bandwdth capacty, and computng resource capacty at PMs. We frst formulated an ILP soluton when the problem sze s small. We then devsed two heurstc algorthms that strve for a fne tradeoff between the soluton accuracy and the runnng tme to obtan the solutons. We also nvestgated the dynamc admssons of requests wthn a fnte tme horzon by extendng the proposed algorthms to solve dynamc request admssons. We fnally evaluated the performance of the proposed algorthms through smulatons, usng real and synthetc network topologes. Expermental results demonstrated that both proposed algorthms admt more requests than a baselne algorthm, and the qualty of solutons delvered s on a par wth that of optmal solutons yet the proposed algorthms are sgnfcantly faster. 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15 HUANG et al.: EFFICIENT ALGORITHMS FOR THROUGHPUT MAXIMIZATION IN SDNs WITH CONSOLIDATED MIDDLEBOXES 645 [26] Z. A. Qaz et al., SIMPLE-fyng mddlebox polcy enforcement usng SDN, n Proc. ACM SIGCOMM, Hong Kong, 2013, pp [27] V. Sekar, N. Eg, S. Ratnasamy, M. K. Reter, and G. Sh, Desgn and mplementaton of a consoldated mddlebox archtecture, n Proc. USENIX NSDI, San Jose, CA, USA, 2012, p. 24. [28] J. Sherry et al., Makng mddleboxes someone else s problem: Network processng as a cloud servce, n Proc. ACM SIGCOMM, Helsnk, Fnland, 2012, pp [29] N. Sprng, R. Mahajan, and D. Wetherall, Measurng ISP topologes wth rocketfuel, n Proc. ACM SIGCOMM, 2002, pp [30] J. W. Suurballe, Dsjont paths n a network, Networks, vol. 4, no. 2, pp , [31] J. W. Suurballe and R. E. Tarjan, A quck method for fndng shortest pars of dsjont paths, Networks, vol. 14, no. 2, pp , [32] Z. Xu et al., Approxmaton and onlne algorthms for NFV-enabled multcastng n SDNs, n Proc. IEEE ICDCS, 2017, pp [33] S. Q. Zhang et al., Sector: TCAM space aware routng on SDN, n Proc. IEEE Int. Teletraffc Congr., 2015, pp Zchuan Xu (M 17) receved the B.Sc. and M.E. degrees from the Dalan Unversty of Technology n Chna n 2008 and 2011, and the Ph.D. degree from the Australan Natonal Unversty n 2016, all n computer scence. He was a Research Assocate wth the Department of Electronc and Electrcal Engneerng, Unversty College London, U.K. He s currently an Assocate Professor wth the School of Software, Dalan Unversty of Technology, Chna. Hs research nterests nclude cloud computng, software-defned networkng, network functon vrtualzaton, wreless sensor networks, algorthmc game theory, and optmzaton problems. Metan Huang receved the B.Sc. (wth Frst Class Hons.) degree n computer scence wth the Australan Natonal Unversty n He s currently pursung the Ph.D. degree wth the Research School of Computer Scence, Australan Natonal Unversty. Hs research nterests nclude softwaredefned networkng, algorthm desgn and analyss, and cloud computng. Wefa Lang (M 99 SM 01) receved the B.Sc. degree from Wuhan Unversty, Chna, n 1984, the M.E. degree from the Unversty of Scence and Technology of Chna n 1989, and the Ph.D. degree from the Australan Natonal Unversty n 1998, all n computer scence. He s currently a Professor wth the Research School of Computer Scence, Australan Natonal Unversty. Hs research nterests nclude desgn and analyss of energy effcent routng protocols for wreless ad hoc and sensor networks, cloud computng, software-defned networkng, desgn and analyss of parallel and dstrbuted algorthms, approxmaton algorthms, combnatoral optmzaton, and graph theory. Song Guo (M 02 SM 11) receved the Ph.D. degree n computer scence from the Unversty of Ottawa, Canada, n He s currently a Full Professor wth the Department of Computng, Hong Kong Polytechnc Unversty, Hong Kong. Hs research nterests are manly n the areas of protocol desgn and performance analyss for wreless networks and dstrbuted systems. He has publshed over 250 papers n refereed journals and conferences n these areas and was a recpent of three IEEE/ACM Best Paper Awards. He currently serves as an Assocate Edtor of the IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, an Assocate Edtor of the IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTING for the track of Computatonal Networks, and on edtoral boards of many others. He has also been n organzng and techncal commttees of numerous nternatonal conferences. He s a Senor Member of the ACM.