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IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. X, NO. X, XX 2018 1

Profit Maximization for Admitting Requests withNetwork Function Services in Distributed CloudsYu Ma, Weifa Liang, Senior Member, IEEE, Zichuan Xu, Member, IEEE, and Song Guo, Senior Member,

IEEE

Abstract—Traditional networks employ expensive dedicated hardware devices as middleboxes to implement Service Function Chainsof user requests by steering data traffic along middleboxes in the service function chains before reaching their destinations. NetworkFunction Virtualization (NFV) is a promising virtualization technique that implements network functions as pieces of software in serversor data centers. The integration of NFV and Software Defined Networking (SDN) further simplifies service function chain provisioning,making its implementation simpler and cheaper. In this paper, we consider dynamic admissions of delay-aware requests with servicefunction chain requirements in a distributed cloud with the objective to maximize the profit collected by the service provider, assumingthat the distributed cloud is an SDN that consists of data centers located at different geographical locations and electricity prices atdifferent data centers are different. We first formulate this novel optimization problem as a dynamic profit maximization problem. Wethen show that the offline version of the problem is NP-hard and formulate an integer linear programming solution to it. We thirdlypropose an online heuristic for the problem. We also devise an online algorithm with a provable competitive ratio for a special case ofthe problem where the end-to-end delay requirement of each request is negligible. We finally evaluate the performance of the proposedalgorithms through experimental simulations. The simulation results demonstrate that the proposed algorithms are promising.

Index Terms—Network Function Virtualization; Software Defined Networking; Distributed Data Centers; Online Algorithms; ServiceFunction Chain Consolidation; Profit Maximization; Request Admission Scheduling.

F

1 INTRODUCTION

Traditional networks are built upon proprietary softwareplatforms tied onto proprietary hardware devices thatevolved slowly [4]. Provisioning network services in tra-ditional networks takes a great deal of time and effort,because different network hardware devices need to beacquired and configured in an appropriate manner. Also,user requests in traditional networks are implemented bydedicated hardware such as routers and associated (vendor-specified) protocols, which lack of flexibility, manageability,and efficiency [27]. Profits thus are being eroded by esca-lating operational expenses (OPEX) and capital expenses(CAPEX), and flat or declining revenue [20]. Network Func-tion Virtualization (NFV) and Software Defined Networking(SDN) have been envisaged as next-generation networkingparadigms to enable fast service deployment, inexpensiveand flexible service provisioning in future communicationnetworks [4]. Specifically, SDN introduces the concept ofseparation between the data plane and the control plane,thus provides more flexibility in steering network flow,while NFV enables highly optimized packet-processing net-work functions to run on commodity servers. Typically, eachflow goes through a sequence of network functions in a spec-ified order to perform necessary packet processing, such asequence is termed as the service function chain (SFC) of the

• Y. Ma and W. Liang are with the Research School of Computer Science,The Australian National University, Canberra, ACT 2601, Australia. E-mails: [email protected], [email protected]

• Z. Xu is with School of Software, Dalian University of Technology, Dalian,116024, China. E-mail: [email protected]

• S. Guo is with the Department of Computing, The Hong Kong PolytechnicUniversity, Hong Kong. Email: [email protected]

flow [22]. The integration of SDN and NFV enables flexibleresource allocations to service function chains and optimiz-ing resource utilization for user request admissions [27].It thus can reduce network service providers’ operationalexpenses and increase their profits significantly [5], [20],[28].

The adoption of SDN and NFV technologies poses sev-eral important challenges for network service providers. Thefirst challenge is how to maximize the profit of networkwork service providers by admitting as many as requestswhile reducing their operational costs. This can be achievedby creating instances of service function chains in datacenters and making use of the instances to implementNFVs of user requests. However, different user requestsnot only have different amounts of resource demands butalso have different service function chains and end-to-enddelay requirements. The second challenge is that, differentdata centers may have different energy consumption ratesdue to various specifications of servers. Also, the electric-ity prices of different geo-graphical data centers may bequite different. The last challenge is how to dynamicallyadmit user requests and find routing paths that includedata centers to process their data traffic, assuming that therequests arrive one by one without the knowledge of futurerequest arrivals. Tackling the aforementioned challengesrequires novel frameworks, algorithms, and technologies,as conventional routing algorithms and routing protocolsdesigned for traditional networks are not applicable to NFV-enabled SDNs.

The novelties of work in this paper include: a noveloptimization framework for dynamic admissions of NFV-enabled requests in an SDN that jointly considers computingand bandwidth resources provisioning at data centers and




links to meet delay requirements of user requests, with theaim to maximize the profit of the network service provider;efficient online algorithms for the dynamic profit maxi-mization problem are developed. Particularly, an onlinealgorithm with a provable competitive ratio for a specialcase of the problem is also proposed.

The main contributions of this paper are summarizedas follows. We formulate a dynamic profit maximizationproblem in a distributed cloud, by dynamically admittingdelay-aware requests with service function chain require-ments, assuming that the admission of each request is basedon as pay-as-you-go, subject to various network resourcecapacities. We first show the offline version of the problemis NP-hard and formulate an integer linear programming(ILP) solution to it. We then propose an efficient heuris-tic for the dynamic profit maximization problem. We alsodevise an online algorithm with a provable competitiveratio for a special case of the problem where the end-to-end delay requirement is negligible. We finally conductempirical evaluation on the performance of the proposedalgorithms through experimental simulations. Experimentalresults demonstrate that the proposed algorithms are verypromising.

The rest of the paper is organized as follows. Section 2reviews related work. Section 3 introduces notions, nota-tions, and the problem definitions. Section 4 formulates anILP solution to an offline version of the problem. Section 5develops a heuristic for the problem, and Section 6 devisesan online algorithm with a provable competitive ratio fora special case of the problem where the end-to-end delayrequirement of each request can be negligent. Section 7evaluates the performance of the proposed algorithms em-pirically, and Section 8 concludes the paper.

2 RELATED WORK

There have been several studies on NFV-enabled user re-quest routing in various networks [16], [17], [18], [29], [31],where a user request with a service function chain require-ment is referred to as the NFV-enabled request. For exam-ple, Martins et al. [18] introduced a virtualization platformto improve network performance, by extending existingvirtualization technologies to support the deployment ofmodular, virtual middleboxes on lightweight VMs. Wanget al. [29] investigated the problem of dynamic networkfunction composition by proposing a distributed algorithmfor the problem. Jia et al. [13] studied the dynamic pro-visioning of VNF service function chains across differentdata centers to minimize the operational cost of networkservice providers. They proposed an efficient online algo-rithm with a provable competitive ratio for the problem,using regularization techniques. Li et al. [16] investigatedQoS-aware NFV-enabled user request routing problem byimplementing the VNFs of the service function chain of eachrequest in a single data center. Huang et al. [11] studiednetwork throughput maximization problem by consolidat-ing all VNFs in a service chain into a single server. Theyproposed two heuristic algorithms for the problem throughstriving for non-trivial tradeoffs between the accuracy ofa solution and the running time to obtain the solution.Jia et al. [12] considered the operational cost minimizationproblem by utilizing pre-installed VNF instances. They also

provided a prediction mechanism to dynamically create andremove VNF instances in data centers. Xu et al. [31] inves-tigated the throughput maximization problem by makinguse of existing VNF instances, assuming that all requestsare given in advance. Most aforementioned studies focusedon developing either exact solutions for the problems byformulating Integer Linear Programming (ILP) solutions, orfast, scalable heuristic solutions for the problems.

There are several recent studies focusing on theprofit maximization by admitting NFV-enabled requests inSDNs [7], [9], [23], [32], [33]. For example, Zhang et al. [33]devised a novel online stochastic auction mechanism forthe network service chain provisioning and pricing for NFVservice providers. They adopt the primal-dual optimizationframework with a learning-based strategy for obtaining re-source prices. They developed an online auction mechanismwith a provable competitive ratio that is able to maximizethe expected social welfare of the NFV ecosystem includingthe NFV provider, customers and resource suppliers. Gu etal. [9] proposed an online auction mechanism for dynamicVNF service chain provisioning and pricing in a single datacenter, through incorporating the primal-and-dual approxi-mation technique and Myersons characterization of truthfulmechanism together. Racheg et al. [23] explored servicefunction chain provisioning for distributed data centers withthe aim to maximize the profit of network service providersby incorporating the variability of energy prices in differentlocations. They proposed a heuristic for the profit maximiza-tion problem by enumerating all possible embedding pathsfor user requests which makes their algorithms unscalable,thereby restricting the applicability of their algorithm in thereal world. Ghribi et al. [7] studied a profit maximizationproblem in SDNs. They devised an algorithm by joint VNFplacement and traffic steering. However, they considereddeploying service function chains in a multi-stage approach,which is to deploy each VNF instance one by one, whichmay lead to sub-optimal solutions. Yuan et al. [32] pro-posed a workload-aware revenue maximization approachby jointly considering VNF instance deployment and userrequest routing paths for each request admission.

In addition, the problem of minimizing energy consump-tion of distributed cloud networks has been extensivelyexplored in literature [2], [21], [34]. For example, Zhang etal. [34] observed that a large fraction of energy is wasted onmaintaining excessive service capacity during low workloadperiods of data centers. They devised an online algorithmfor minimizing the power consumption of data centers bydynamically switching on/off servers, according to resourcedemands of user requests. On the other hand, cloud ser-vice providers usually deploy their data centers in geo-graphically different locations to meet ever-growing servicedemands of users from different regions. The electricityprices for data centers at different geographical locations aredifferent, thus, the costs of energy consumption of these datacenters are different too. Exploiting time-varying electricityprices in different regions has been shown a promisingtechnique to minimize the operational costs of data centers,which has been explored in previous studies [23], [24], [25],[26], [30].

Unlike these aforementioned studies, in this paper wewill study a dynamic profit maximization problem that




admits NFV-enabled requests in SDNs through VNF in-stance placement while meeting delay requirements of userrequests, assuming that requests arrive one by one with theknowledge of future request arrivals. We aim to maximizethe profit of the service provider through reducing its opera-tional cost. This is achieved by incorporating varying energyconsumption rates and electricity prices among data centers,and performing smart resource allocations to fully utilizingthe resources in SDNs.

3 PRELIMINARIES

In this section, we first introduce the system model, notionsand notations, and then define the problems precisely.

3.1 System model

We consider a Software Defined Network (SDN) modelledby a directed graph G = (S ∪ V,E), where S is a set ofSDN-enabled switch nodes and V is a set of data centerswith rich computing resources that are geographically co-located with some of the switch nodes, and E is a set oflinks between SDN-enabled switches and between SDN-enabled switches and data centers. Denote by Qv the setof active homogeneous servers in each data center v ∈ V ,and for each server s ∈ Qv , let Cs be its computingcapacity. Thus, the computing capacity Cv of data centerv is Cv =

∑s∈Qv Cs. Notice that different data centers

may deploy different types of servers. Furthermore, the datatraffic of each user request consumes link bandwidth andincurs data transmission delays on links in the routing path.Denote by Be and de the bandwidth capacity and the delayon link e ∈ E.

3.2 Virtualized network functions and service functionchains

In conventional networks, network functions such as Fire-walls, Intrusion Detection Systems (IDSes) and Load Bal-ancers (LBs) are implemented by expensive, dedicatedhardware middleboxes installed at network routers andswitches [4]. Network function virtualization (NFV) thatmakes use of common x86 architecture to implement net-work functions as software components in data centers, hasbecome a promising technology to significantly reduce theoperational expenses (OPEX) and capital expenses (CAPEX)of network service providers, which introduces a new di-mension for cost savings on dedicated hardware middle-boxes and flexible deployment of network functions. Theimplementation of virtualized network functions (VNFs)in data centers consumes their computing resource. Also,different VNFs may have different processing delays de-pending on the resources that are allocated to them.

Since different requests may have different types ofservice function chains, establishing a routing path for eachof the requests needs steering its flow to pass through thespecified VNFs in its service function chain in a particularorder. An ordered sequence of VNFs is defined as a servicefunction chain (SFC). In this paper we assume that the VNFsof the SFC of each request is consolidated into a single VMin a data center for processing. This assumption has beenadopted in [3], [10], [11].

3.3 User requests with service function chain require-mentsConsider a user request rj = (sj , tj ;SFC

(k)j , bj , φj , Dj) that

requires to transfer its data traffic from a source sj to adestination tj with a given packet rate φj > 0, and the datatraffic must pass through the instances of VNFs in its servicefunction chain SFC(k)

j . We assume that the service functionchains among all requests can be classified into K types.We also assume that each instance of a VNF can process abasic packet rate [31], and a request rj with φj packet rateimplies it needs φj instances of its required type of servicefunction chain and the amount bj (= φj · b(k)) of bandwidthresource, where b(k) is the bandwidth requirement of a basicdata packet rate of a request with SFC(k). Dj is the end-to-end delay requirement of request rj . We also assume thatthe data traffic of each request will be processed in a singledata center.

Denote by SFC(k) the service function chain of type kand C(SFC(k)) the computing resource consumption of aninstance of the service function chain of type k with 1 ≤k ≤ K. Assume that for each type of service function chainSFC(k), C(SFC(k)) > 1. Each instance of service functionchain SFC(k)

v in data center v has data processing rate µ(k)v ,

which is also measured by the amount of basic packet rate.Figure 1 is an illustrative example of a request admission inG.

data center

switch

request routing

request generators

SFC instances

sv

t

Fig. 1: An illustrative example of the admission of a userrequest in a Software Defined Network (SDN), where therequest is issued at a source node s and its data traffic isprocessed at a data center v that contains the SFC instancesof the request, and the processed data traffic then is routedto its destination t.

3.4 End-to-end delay requirements of user requestsEach request rj has an end-to-end delay requirement Dj

that specifies the maximum duration of per data packetfrom its source to its destination, which includes both theprocessing and queuing delay in a data center and thetransmission delay along links.

Each data packet of an admitted request rj will bequeued in its service function chain instances in a datacenter before being processed, which will incur both thequeuing and processing delays when each of the packetspasses through the service function chain instances. Each




user request rj with packet rate φj has φj instances of itsrequired type of service function chain, we thus assumethere is an M/M/n queue for φj SFC instances of requestrj . The average queuing delay is 1

φj ·µ(k)v −φj

= 1

φj ·(µ(k)v −1)

.

Considering that the data processing rate of service func-tion chain SFC(k)

v is µ(k)v , the processing delay of SFC(k)

v indata center v is 1

µ(k)v

. Denote by dj(SFC(k)v ) the processing

and queuing delay of a packet at a data center v, i.e.,

dj(SFC(k)v ) =

1

φj · (µ(k)v − 1)

+1

µ(k)v

. (1)

Furthermore, traffic route for request rj along a linke ∈ E incurs a data transmission delay de. Denote byP (sj , v) the first segment of the routing path for the datatraffic of request rj from node sj to data center v ∈ V .The data transmission delay along this segment P (sj , v) isdj(P (sj , v)) =

∑e∈P (sj ,v)

de. Similarly, the data transmis-sion delay along the second segment P (v, tj) of the routingpath is dj(P (v, tj)) =

∑e∈P (v,tj)

de. The end-to-end delayof data transmission of request rj along the routing path viadata center v thus is

d(rj) = dj(P (sj , v)) + dj(SFC(k)v ) + dj(P (v, tj)). (2)

To meet the end-to-end delay requirement of rj , we haved(rj) ≤ Dj .

3.5 Energy consumption and profit by admitted re-quests

Denote by Ev the energy consumption of data center v ∈ V ,which is determined by the number of homogeneous servers|Qv| in v. We adopt a linear energy consumption model tocapture the energy consumption of each server [23], [26],[34], that is, for a server s ∈ Qv in data center v ∈ V , theamount of energy Es consumed by s per time unit is

Es = E idlev + ηv · Us, (3)

where Us = Us/Cs is the utilization of computing resourcein server s while Us is the workload (the amount of com-puting resource being occupied at the moment) of servers and Cs is its computing capacity, E idlev is the amount ofenergy consumed per time unit when server s is idle, whichusually is fixed. The energy consumption of a server s isproportional to its workload, and ηv is constant with ηv > 0,and the amount of energy Ev consumed in data center vper time unit thus is the sum of the amounts of energyconsumed at all servers in Qv , i.e.,

Ev =∑s∈Qv

Es =∑s∈Qv

(E idlev + ηv · Us)

= |Qv| · E idlev + ηv∑s∈Qv

Us,

= |Qv| · E idlev + ηv · |Qv| ·UvCv

, (4)

where Uv =∑s∈Qv Us is the workload and Cv is the

computing capacity of data center v ∈ V .A network service provider typically charges its users by

admitting their requests on a pay-as-you-go basis throughadopting a common revenue collection model [23]. The rev-

enue collected RVj by admitting a request rj is proportionalto its computing and bandwidth resource demands, that is,

RVj = λ1 · φj · C(SFC(k)) + λ2 · bj , (5)

where λ1 and λ2 are constant weights for computing andbandwidth selling prices, and φj is the packet rate of requestrj .

The profit p(A) collected by admitting a set A of requeststhus is the difference of the sum of the revenues collectedto the cost of the amount of energy consumed at all datacenters for the services [23].

p(A) =∑rj∈A

RVj −∑v∈V

ρv · Ev, (6)

where ρv is the electricity price per unit energy at thelocation of data center v ∈ V , andRVj and Ev are defined byEquations (5) and (4) respectively. Notice that the electricityprices at different locations and different time slots may besignificantly different [30].

3.6 Problem statements

Given an SDN G = (S ∪ V,E) and a set of NFV-enabledrequests R, the profit maximization problem is to admit asmany requests in R as possible so that the profit collectedby the network service provider is maximized, subject tonetwork resource constraints on G.

We now define a dynamic version of the problem. Givenan SDN G = (S ∪ V,E), let r1, r2, . . . rj be a sequenceof requests that arrive one by one without the knowledgeof future arrivals, the dynamic profit maximization problem inG is to maximize the total profit collected by admitting asmany as requests in the sequence while meeting the end-to-end delay requirement of each admitted request, subject toresource capacity constraints on both data centers and links.

The profit maximization problem is NP-hard, which canbe shown by a reduction from the knapsack problem. Thedetailed reduction is omitted, due to space limitation.

3.7 Approximation and competitive ratios of algo-rithms

The approximation ratio of an approximation algorithm isdefined as follows. A γ-approximation algorithm for a mini-mization problem P1 is a polynomial time algorithm A thatdelivers an approximate solution whose value is no morethan γ times the optimal one for any instance of P1 withγ > 1, where γ is termed as the approximation ratio ofalgorithm A.

Let OPT and S be an optimal solution of the offlineversion of a problem and the solution delivered by an onlinealgorithm A′ for a maximization problem P2 of dynamicrequest admissions, where requests arrives one by one with-out the knowledge of future arrivals. The competitive ratio ofan online algorithm A′ is ξ if S

OPT ≥ ξ for any instance I ofproblem P2 with 0 < ξ < 1.

4 ILP FOR THE PROFIT MAXIMIZATION PROBLEM

In this section, we formulate an integer linear programming(ILP) solution to the profit maximization problem.




For each request rj ∈ R, a binary constant akj = 1 if rjis the type k request; akj = 0 otherwise. A binary variablexj = 1 if rj is admitted; xj = 0 otherwise. Recall that RVj isthe amount of revenue collected if request rj is admitted. Abinary variable yj,v is used to determine which data centerwill accommodate the instances of the service function chainof rj , where yj,v = 1 if it is implemented by the servicefunction chain instances in data center v; yj,v = 0 otherwise.For each request implementation in the network, we treatthe data traffic of the request as a flow inG from its source toits destination. For clarity, denote by ψ−(u) and ψ+(u) thesets of incoming and outgoing edges of a node u ∈ S ∪ V ,respectively. The proposed ILP is described as follows.

maximize∑rj∈R

RVj · xj −∑v∈V

ρv · |Qv| · E idlev −

∑v∈V

ρv · |Qv| ·ηv(∑rj∈R

∑Kk=1 a

kj · φjC(SFC(k)) · yj,v)Cv

,

(7)subject to:∑v∈V

yj,v = xj , ∀rj ∈ R (8)∑e∈ψ−(u)

zsvj (e)−∑

e∈ψ+(u)

zsvj (e) = 0,

∀u ∈ S \ {sj}, rj ∈ R (9)∑e∈ψ+(u)

zvtj (e)−∑

e∈ψ−(u)

zvtj (e) = 0,

∀u ∈ S \ {tj}, rj ∈ R (10)∑e∈ψ+(sj)

zsvj (e) = xj , ∀rj ∈ R (11)

∑e∈ψ−(tj)

zvtj (e) = xj , ∀rj ∈ R (12)

∑e∈ψ−(v)

zsvj (e)−∑

e∈ψ+(v)

zsvj (e) = yj,v,∀v ∈ V, rj ∈ R

(13)∑e∈ψ+(v)

zvtj (e)−∑

e∈ψ−(v)

zvtj (e) = yj,v,∀v ∈ V, rj ∈ R (14)

∑e∈ψ−(sj)

zsvj (e) = 0, ∀rj ∈ R (15)

∑e∈ψ+(tj)

zvtj (e) = 0, ∀rj ∈ R (16)

∑e∈E

de(zsvj (e) + zvtj (e)) +

∑v∈V

dj(SFC(k)v ) · yj,v ≤ Dj ,

∀rj ∈ R (17)∑rj∈R

bj · (zsvj (e) + zvtj (e)) ≤ Be, ∀e ∈ E (18)

∑rj∈R

∑1≤k≤K

akj · φjC(SFC(k)) · yj,v ≤ Cv, ∀v ∈ V (19)

zsvj (e), zvtj (e) ∈ {0, 1}, ∀e ∈ E, rj ∈ R (20)

xj , yj,v ∈ {0, 1}, ∀v ∈ V, rj ∈ R (21)

Constraints (9) and (10) ensure the flow conservation atany switch node of a flow except the source and sink nodesof the flow. Constraints (11) and (12) enforce data traffic flow

entering and leaving the network in accordance with theadmission of rj . As an edge e ∈ E can be used in both thefirst segment P (sj , v) and the second segment P (v, tj) ofthe routing path of request rj via data center v, two decisionvariables zsvj (e) and zvtj (e) are adopted to distinguish theusage of the edge in these two segments, respectively, wherezsvj (e) = 1 if edge e is in P (sj , v); zsvj (e) = 0 otherwise.Similarly, zvtj (e) = 1 if edge e is in the second segmentP (v, tj); zvtj (e) = 0 otherwise. Constraints (13) and (14)specify whether the service function chain of rj is imple-mented in a data center v ∈ V . Constraints (15) and (16)ensure that no any fractional flow of rj before it is processedby its SFC instance will enter its source node sj and no anyfractional flow of rj processed by its SFC instance will leavefrom its destination tj . Constraint (17) enforces the end-to-end delay requirement. Constraints (18) and (19) imposebandwidth and computing resource capacity constraints forall links and data centers in the network, respectively.

5 ONLINE ALGORITHM FOR THE DYNAMIC PROFITMAXIMIZATION PROBLEM

In this section, we deal with the dynamic profit maxi-mization problem. We first investigate the delay-constrainedshortest path problem via a specified node, which will be usedas a subroutine for the dynamic profit maximization prob-lem. We then propose an efficient online algorithm for theproblem.

5.1 A delay-constrained shortest path via a specifiednode

Given a graph G = (S ∪ V,E), a request rj from a sourcesj to a destination tj with an end-to-end delay constraintDj , and a specified node v ∈ V , the delay-constrained shortestpath via a specified node problem inG is to find a minimum costpath (or route) from sj to tj that passes through the specifiednode v, while the end-to-end delay of the route is no greaterthan Dj . This problem is NP-hard as its special case thedelay-constrained shortest path problem is NP-hard [14].

Assume that the implementation of the service functionchain of a request rj = (sj , tj ;SFC

(k)j , bj , φj , Dj) by in-

stances of network functions in a data center v, and theprocessing and queuing delay for the data traffic of rj atv is dj(SFC

(k)v ). To meet its end-to-end delay constraint

Dj , its routing delay dj(P (sj , v)) + dj(P (v, tj)) from itssource sj to its destination tj must be no greater thanDj − dj(SFC(k)

v ). However, finding such a minimum costrouting path for request rj from sj to tj via v is chal-lenging, and existing algorithms for the delay-constrainedshortest path problem is not applicable. We instead developa novel algorithm for it. That is, for request rj and datacenter v ∈ V , we construct an auxiliary graph G′j,v fromG, we reduce the problem in G to the delay-constrainedshortest path problem in G′j,v = (V ′, E′), where V ′ ={u′, u | ∀u ∈ S ∪ V }, and E′ = {〈u′, w′〉, 〈u,w〉 | ∀〈u,w〉 ∈E}. We then merge the specified data center node v andits copy v′ into a single node v, i.e., V ′ = V ′ \ {v′},E′ = E′ \ (

⋃〈u′,v′〉∈E′{〈u′, v′〉} ∪

⋃〈v′,u′〉∈E′{〈v′, u′〉}) ∪

(⋃〈u′,v′〉∈E′{〈u′, v〉} ∪

⋃〈v′,u′〉∈E′{〈v, u′〉}) for any u′ ∈ V ′.

Figure 2 shows the construction of G′j,v .




u

sj

w

tj

v

u’

v’

tj’

w’

sj’

x x’

Fig. 2: An example of the construction of an auxiliary graphG′j,v for finding a delay-constrained shortest path via aspecified node v for request rj .

Having the constructed auxiliary graph G′j,v , a delay-constrained shortest path P ′(sj , t′j) in G′j,v from node sj tot′j with a delay no greater than Dj − dj(SFC(k)

v ) can thenbe found if it does exist, by applying the approximationalgorithm due to Juttner et al. [14]. The pseudo-routing path(route) P (sj , tj ; v) in G for request rj via data center v canthen be derived from P ′(sj , t

′j), where a pseudo path (route)

is a path in which nodes and links can appear multipletimes. The detailed description of the algorithm for finding adelay-constrained shortest path via a specified node is givenin Algorithm 1.

Algorithm 1 Finding a delay-constrained shortest path in Gvia node v for request rj

Input: An SDN G = (S ∪ V,E), a data center v ∈ V , arequest rj = (sj , tj ;SFC

(k)j , bj , φj , Dj).

Output: A delay-constrained shortest path via data centerv; if it does not exist, data center v cannot be usedfor implementing service function chain for rj withoutviolating its delay constraint.

1: Construct the auxiliary graph G′j,v from G;2: Find an approximate delay-constrained shortest pathP ′(sj , t

′j) in G′j,v from sj to t′j subject to the delay

constraint on its routing path such that dj(P (sj , v)) +dj(P (v, tj)) ≤ Dj − dj(SFC(k)

v ), by applying the algo-rithm due to Juttner et al. [14];

3: if P ′(sj , t′j) in G′j,v exists then4: A routing path P (sj , tj ; v) in G is derived from

P ′(sj , t′j), by replacing each edge in P ′(sj , t

′j) with

its corresponding edge in G;5: else6: Request rj cannot make use of data center v for

service function chain implementation.

Lemma 1. (i) If there is a directed path in G′j,v from sj tot′j , it must pass through node v, i.e., v is an articulationpoint in G′j,v . (ii) Any delay-constrained shortest pathP ′(sj , t

′j) in G′j,v from sj to t′j cannot pass through node

v twice. (iii) Each edge or node in G appears at mosttwice in the route P (sj , tj ; v) in G which is derived fromP ′(sj , t

′j).

Proof: We first show claim (i) by contradiction. As-sume that there is a directed path from sj to t′j that does notpass through node v. We remove v and its incident edges

from G′j,v , this will not impact the directed path since v isnot on it. However, the removal of v from G′j,v will lead tothe resulting graph disconnected and nodes sj and tj′ willnot be in the same connected component. Then, there is notany directed path from sj to t′j , which forms a contradiction.

We then prove claim (ii). Assume that there is such adelay-constrained shortest path P ′(sj , t′j) from sj to t′j thatpasses through v twice. Then, P ′(sj , t′j) must contain adirected cycle with v in it. If we break the cycle by removingsome of its edges and nodes except v while keeping t′jreachable from sj . The length of the resulting path from sjto t′j thus is shorter than that of P ′(sj , t′j), which contradictsthe fact that P ′(sj , t′j) is a shortest delay-constrained pathfrom sj to t′j .

We finally show claim (iii). It can be seen that P ′(sj , t′j) isa simple path in G′j,v . Each edge 〈u,w〉 ∈ E is contained inthe route P (sj , tj ; v) in G derived from P ′(sj , t

′j) if 〈u,w〉 ∈

P ′(sj , t′j) or 〈u′, w′〉 ∈ P ′(sj , t′j). Thus, each edge or node

in G appears at most twice in P (sj , tj ; v).

Theorem 1. Given a graph G = (S ∪ V,E), a request rj ,and a specified node v ∈ V , there is an approximationalgorithm with an approximation ratio of (1 + ε) for thedelay-constrained shortest path problem via a specifiednode, which takes O(|E|2 log4 |E|) time, where ε is aconstant with 0 < ε ≤ 1.

Proof: For a request rj and any specified node v ∈ V ,the construction of auxiliary graphG′j,v fromG takesO(|S∪V | + |E|) time. Finding an approximate delay-constrainedshortest path in G′j,v takes time O(|E′j,v|2 log

4 |E′j,v|), byapplying the (1 + ε)-approximation algorithm due to Jut-tner et al. [14], where |E′j,v|(= 2|E|) is the number ofedges in G′j,v . Thus, the running time of Algorithm 1 isO(|E′j,v|2 log

4 |E′j,v|+O(|S ∪V |+ |E|)) = O(|E|2 log4 |E|).

5.2 Online algorithm

We now propose an online algorithm for the dynamic profitmaximization problem.

We start by introducing a usage weight model to mea-sure various network resource consumptions. Specifically,given the dynamics of user requests and networks, there isa need of a cost model to capture dynamic consumption ofvarious resources in the network in order to better guidethe admissions of future requests and utilize the resources.Intuitively, overloaded resources usually have higher prob-abilities of violating the resource demands of admitted re-quests due to the high dynamics of resource consumptions.This will affect the admissions of future requests eventually.Thus an under-loaded resources should be encouraged tobe used while overloaded resources should be restricted(discouraged) to be used when conducting resource allo-cations to meet resource demands of admitting requests.If a specific resource has been highly utilized, it shouldbe less used in the future by assigning it a higher usageweight; otherwise (a specific resource has rarely been used),it should be encouraged to be used by assigning it a lowerusage weight. The usage weight model is given as follows.




When request rj arrives, for each data center node v ∈V , the usage weight cv(j) of using the computing resourceat data center v by request rj is defined as

cv(j) = ρv · Cv(α1−Cv(j)Cv − 1) ∀v ∈ V, (22)

where α is a tuning parameter with α > 1, Cv(j) is theresidual computing resource at data center v when requestrj arrives with Cv(j) = Cv(j − 1) − φj · C(SFC(k)) ifrequest rj admitted and Cv(0) = Cv , φj is the packet rateof request rj , and ρv is the electricity price per unit energyat the location of data center v.

For each link e ∈ E, the usage weight ce(j) of using thebandwidth resource at link e in the route for request rj canbe defined similarly, that is,

ce(j) = Be(β1−Be(j)Be − 1) ∀e ∈ E, (23)

where β is a tuning parameter with β > 1, and Be(j) is theresidual bandwidth at link e when request rj arrives withBe(j) = Be(j − 1)− bj if rj admitted and Be(0) = Be.

Denote by ρmax and ρmin the maximum and minimumelectricity prices among the locations of all data centers,i.e., ρmax = maxv∈V {ρv} and ρmin = minv∈V {ρv}. Thenormalized usage weights of each data center v ∈ V and eachlink e ∈ E of G are defined as follows when request rjarrives.

ωv(j) =cv(j)

ρmax · Cv=

ρvρmax

· (α1−Cv(j)Cv − 1) ∀v ∈ V,

(24)

ωe(j) =ce(j)

Be= β1−Be(j)Be − 1 ∀e ∈ E. (25)

For the admission of a request rj , denote by Gj =(S ∪ V,E;ωv(j), ωe(j)) a weighted version of a subgraphof G by removing those nodes and links that do not haveenough residual resources to accommodate rj , in which itsdata center nodes and edges are assigned with the definednormalized usage weights. Then, for each data center v ∈ V ,a delay-constrained shortest path (route) P (sj , tj ; v) in Gjfrom sj to tj via v is found by applying Algorithm 1. Onedelay-constrained shortest path in G via a data center v′ isidentified if P (sj , tj ; v′) = minv∈V {P (sj , tj ; v)}.

To avoid admitting request rj that demands too muchresources (leading to a higher implementation usage weight,thereby undermining the performance of the SDN), we thenadopt the following admission control policy: request rj willbe rejected if either (i) ωv′(j) > σ1 for the chosen datacenter v′ ∈ V or (ii)

∑e∈P (sj ,tj ;v′)

ωe(j) > σ2, where σ1and σ2 are admission control thresholds of computing andbandwidth resources, respectively, where σ1 = (n − 1) ·ρmin/ρmax, σ2 = n− 1, and n = |S ∪ V |.

The detailed algorithm for the dynamic profit maximiza-tion problem is given in Algorithm 2.

The time complexity of the proposed online algorithm,Algorithm 2, is stated in the following theorem.

Theorem 2. Given an SDN G = (S ∪ V,E) with aset V of data centers, there is an online algorithm,Algorithm 2, for the dynamic profit maximizationproblem, which delivers a feasible solution for eachrequest in O(|E|2 log4 |E| · |V |) time.

Algorithm 2 Maximizing the profit for the dynamic profitmaximization problem

Input: An SDN G = (S∪V,E) with a set V of data centers,a sequence of requests rj = (sj , tj ;SFC

(k)j , bj , φj , Dj)

∈ R, assuming that requests arrive one by one withoutthe knowledge of future arrivals.

Output: A solution to maximize the profit. If a request rjis admitted, a data center v′ to implement its servicefunction chain and its routing path (route) from sj to tjvia v′ will be delivered.

1: A ← ∅;2: while request rj arrives do3: Construct the auxiliary graph Gj for request rj ;4: Find a delay-constrained shortest path (route)

P (sj , tj ; v′) via data center v′ among all delay-

constrained shortest paths via all different data cen-ters in V , by invoking Algorithm 1 for rj via eachdata center v ∈ V such that the one via node v′ is theminimum one; if no such path exists, reject rj ;

5: Determine whether rj to be accepted or rejected bythe defined admission control policy;

6: if rj is accepted then7: A ← A∪ {rj};8: return A.

Proof: We analyze the time complexity ofAlgorithm 2. The assignment of weights on nodes andedges of Gj takes O(|S ∪V |+ |E|) time. For each request rjand each data center v, finding a delay-constrained shortestpath via a node v takes O(|E|2 log4 |E|) time. Finding onedelay-constrained shortest path which is the minimum oneamong all such delay-constrained shortest paths thus takesO(|E|2 log4 |E| · |V |) time. Algorithm 2 therefore takestime O(|E|2 log4 |E| · |V |) to determine whether a requestshould be admitted.

In practice, almost all SDNs are planar, and the degreeof each node is a small constant, i.e., |E| = O(|V |), thus, thetime complexity of the algorithm can be further reduced toO(|V |3 log4 |V |).

6 ONLINE ALGORITHM FOR A SPECIAL DYNAMICPROFIT MAXIMIZATION PROBLEM

In this section we consider a special case of the dynamicprofit maximization problem where the end-to-end delayconstraint can be neglected, for which we devise an onlinealgorithm with a provable competitive ratio.

6.1 Online Algorithm with a provable competitive ratioThe algorithm is to perform a minor modification to Step4 in Algorithm 2. That is, a shortest path P (sj , tj ; v

′)instead of a delay-constrained shortest path, in Gj will befound. The detailed description of Algorithm 3 throughthe modification to Algorithm 2 is given as follows.

6.2 The competitive ratio and time complexity analysisof the online algorithmThe rest is to analyze the performance of Algorithm 3. Wefirst show the upper bound on the total usage weight of




Algorithm 3 Maximizing the profit for a special dynamicprofit maximization problem without end-to-end delay con-straintsInput: An SDN G = (S∪V,E) with a set V of data centers,

a sequence of requests rj = (sj , tj ;SFC(k)j , bj , φj) ∈ R,

assuming that requests arrive one by one without theknowledge of future arrivals.

Output: A solution to maximize the profit. If a request rjis admitted, a data center v′ to implement its servicefunction chain and its routing path (route) from sj to tjvia v′ will be delivered.

1: A ← ∅;2: while request rj arrives do3: Construct the auxiliary graph Gj for request rj ;4: Find a shortest path (route) P (sj , tj ; v′) via data cen-

ter v′ which is the minimum one among all shortestpaths via all different data centers in V , by invokingthe modified version of Algorithm 1 (i.e., set delayrequirement Dj of rj as +∞) for rj via each datacenter v ∈ V such that the path length via node v′ isthe minimum one; if no such path exists for all datacenters, reject rj ;

5: Determine whether rj to be accepted or rejected bythe defined admission control policy;

6: if rj is accepted then7: A ← A∪ {rj};8: return A.

admitted requests when request rj arrives in terms of thedefined usage weight model by Equations (22) and (23).We then provide a lower bound on the usage weight ofa rejected request by Algorithm 3 but admitted by anoptimal offline algorithm. We finally derive the competitiveratio of Algorithm 3.

We start by showing the upper bound on the total usageweight of all admitted requests on the arrival of request rjby Algorithm 3 in the following lemma.

Lemma 2. Given an SDN G = (S∪V,E) with computing ca-pacity Cv at each data center v ∈ V and link bandwidthcapacity Be at each link e ∈ E, denote by A(j) the set ofrequests admitted by Algorithm 3 prior to the arrivalof request rj . The sum of the usage weights of all datacenters and links in G defined in Equations (22) and (23)prior to the arrival of rj thus are∑

v∈Vcv(j) ≤ ρmax · C(j)(σ1 + n− 1) logα, (26)

and ∑e∈E

ce(j) ≤ B(j)(σ2 + n− 1) log β, (27)

respectively, provided that φj′ · C(SFC(k′)) ≤minv∈V {Cv}

logα and bj′ ≤ mine∈E{Be}log β with 1 ≤ j′ ≤ j,

where C(j) (=∑rj′∈A(j) φj′ · C(SFC(k′))) and B(j)

(=∑rj′∈A(j) bj′ ) are the accumulated amounts of com-

puting and bandwidth resources being occupied by theadmitted requests in A(j) at the moment, respectively.

Proof: Consider a request rj′ ∈ A(j) admitted by

the online algorithm Algorithm 3 and its service functionchain is implemented in data center v ∈ V , the usage weightof data center v in G by admitting request rj′ is

cv(j′ + 1)− cv(j′)

= ρv · Cv(α1−Cv(j′+1)Cv − 1)− ρv · Cv(α1−Cv(j′)

Cv − 1)

= ρv · Cv(α1−Cv(j′+1)Cv − α1−Cv(j′)

Cv )

= ρv · Cvα1−Cv(j′)Cv (α

Cv(j′)−Cv(j′+1)Cv − 1)

= ρv · Cvα1−Cv(j′)Cv (α

φj′ ·C(SFC(k′))

Cv − 1)

= ρv · Cvα1−Cv(j′)Cv (2

φj′ ·C(SFC(k′))

Cvlogα − 1)

≤ ρv · α1−Cv(j′)Cv · φj′ · C(SFC(k′)) · logα. (28)

Notice that Inequality (28) holds due to that 2x − 1 ≤ x for0 ≤ x ≤ 1.

If an edge e ∈ E is in the routing path P (sj′ , tj′ ; v) ofrj′ for request rj′ admission by Algorithm 3, the usageweight of edge e in the admission of request rj′ is

ce(j′ + 1)− ce(j′)

= Be(β1−Be(j

′+1)Be − 1)−Be(β1−Be(j

′)Be − 1)

= Be(β1−Be(j

′+1)Be − β1−Be(j

′)Be )

= Beβ1−Be(j

′)Be (β

Be(j′)−Be(j′+1)Be − 1)

= Beβ1−Be(j

′)Be (β

bj′Be − 1),

= Beβ1−Be(j

′)Be (2

bj′Be

log β − 1)

≤ β1−Be(j′)

Be · bj′ · log β. (29)

We then calculate the sum of the usage weights of alldata centers and all edges in G when admitting request rj′ .If a data center was not chosen for implementing the servicefunction chain of rj′ , its usage weight does not change afterthe admission of rj′ ; similarly, if an edge is not includedin P (sj′ , tj′ ; v), its usage weight remains unchanged afterthe admission of rj′ . The difference of the sum of the usageweights of data centers before and after admitting rj′ is∑

v∈Vcv(j

′ + 1)− cv(j′)

= cv(j′ + 1)− cv(j′)

≤ ρv · α1−Cv(j′)Cv · φj′ · C(SFC(k′)) · logα, by (28)

= ρv · φj′ · C(SFC(k′)) logα((α1−Cv(j′)Cv − 1) + 1)

= ρv · φj′ · C(SFC(k′)) logα(ωv(j′) · ρmax

ρv+ 1)

= φj′ · C(SFC(k′)) logα(ωv(j′) · ρmax + ρv)

≤ ρmax · φj′ · C(SFC(k′)) · (σ1 + 1) logα (30)

≤ ρmax · φj′ · C(SFC(k′)) · (σ1 + n− 1) logα. (31)

Inequality (30) holds due to the fact that if request rj′ isadmitted, then Condition (i) of the admission control policyis satisfied, i.e., ωv(j′) ≤ σ1.

The difference of the sum of the usage weights of all




edges in G before and after admitting rj′ is∑e∈E

(ce(j′ + 1)− ce(j′))

=∑

e∈P (sj′ ,tj′ ;v)

(ce(j′ + 1)− ce(j′))

≤∑


(β1−Be(j′)

Be · bj′ · log β), by (29)

= bj′ · log β∑


((β1−Be(j′)

Be − 1) + 1)

= bj′ · log β

∑e∈P (sj′ ,tj′ ;v)

(β1−Be(j′)

Be − 1) +∑


1

= bj′ · log β

∑e∈P (sj′ ,tj′ ;v)

ωe(j′) +

∑e∈P (sj′ ,tj′ ;v)

1

≤ bj′ · (σ2 + n− 1) log β. (32)

The sum of the usage weights of all data centers prior tothe arrival of request rj thus is

∑v∈V

cv(j) =

j−1∑j′=1

∑v∈V

cv(j′ + 1)− cv(j′)

=∑

rj′∈A(j)

∑v∈V

cv(j′ + 1)− cv(j′)

≤∑

rj′∈A(j)

(ρmax · φj′ · C(SFC(k′))(σ1 + n− 1) logα),

by (31)

= ρmax · (σ1 + n− 1) · logα∑

rj′∈A(j)

φj′ · C(SFC(k′))

= ρmax · C(j) · (σ1 + n− 1) logα, (33)

and the sum of the usage weights of all edges in G prior tothe arrival of request rj is

∑e∈E

ce(j) =

j−1∑j′=1

∑e∈E

ce(j′ + 1)− ce(j′)

=∑

rj′∈S(j)

∑e∈E

ce(j′ + 1)− ce(j′)

≤∑

rj′∈S(j)

(bj′ · (σ2 + n− 1) log β), by (32)

= (σ2 + n− 1) · log β∑

rj′∈A(j)

bj′

= B(j) · (σ2 + n− 1) log β. (34)

We now provide a lower bound on the usage weightof a rejected request by Algorithm 3 but admitted by anoptimal offline algorithm OPT . Before we proceed, we chooseappropriate values for α and β in Equations (22) and (23)prior to the arrival of any request rj with its SFC(k) asfollows.

2n ≤ α ≤ minv∈V

{2

Cv

φj ·C(SFC(k))}, (35)

2n ≤ β ≤ mine∈E

{2Bebj}. (36)

By Inequality (35), for any v′ ∈ V , any request rj ∈ R,any edge e′ ∈ E, and any k′ with 1 ≤ k′ ≤ K, we have

2

Cv′

φj′ ·C(SFC(k′)) ≥ min

v∈V

{2

Cv

φj ·C(SFC(k))}≥ α, (37)

2Be′bj′ ≥ min

e∈E

{2Bebj}≥ β. (38)

Lemma 3. Let T (j) be the set of requests rejected byAlgorithm 3 but admitted by algorithm OPT priorto the arrival of request rj . Let POPT (sj′ , tj′ ; v∗) be therouting path in Gj′ by algorithm OPT for any requestrj′ ∈ T (j) with 1 ≤ j′ ≤ j, and assuming that thedata center v∗ is identified for the implementation of theservice function chain SFC(k) of request rj′ . Then,∑e∈POPT (sj′ ,tj′ ;v∗)

ωe(j′) + ωv(j

′) ≥ min{σ1, σ2} = σ1.

(39)

Proof: Consider a request r′j , which is rejected byAlgorithm 3 by one of the three cases: (i) there is nosufficient computing resource in any data center for imple-menting the service function chain of the request; (ii) there isno sufficient bandwidth in G for routing its data traffic; and(iii) the normalized usage weight (defined by Eq. (24)) of theselected data center to implement its service function chainis too high, or the sum of the normalized usage weights ofedges (defined by Eq. (25)) in its routing path is too high.

Case (i). Let v∗ be the data center selected to imple-ment the service function chain of request r′j of type k′.As the request is rejected by Algorithm 3 due to in-sufficient available computing resource, this implies thatCv∗(j

′) < φj′ · C(SFC(k′)). We thus have

ωv(j′) = ωv∗(j

′)

=ρvρmax

· (α1−Cv∗ (j′)

Cv∗ − 1)

>ρminρmax

· (α1−φj′ ·C(SFC(k′))

Cv∗ − 1)

≥ ρminρmax

· (α1− 1logα − 1), by (37)

=ρminρmax

· (α2− 1) ≥ σ1.

Case (ii). If request rj′ is rejected, then there is an edgee′ ∈ P (sj′ , tj′ ; v

∗) that does not have sufficient residualbandwidth to accommodate the request. This implies thatBe′(j

′) < bj′ . Therefore, the length of P (sj′ , tj′ ; v∗) isgreater than σ2.∑

e∈P (sj′ ,tj′ ;v∗)we(j

′) ≥ we′(j′)

= β1−Be′ (j

′)Be′ − 1

> β1−

bj′Be′ − 1, since Be′(j′) < bj′

≥ β1− 1log β − 1, by (38)

=β

2− 1 ≥ σ2.




Case (iii). Although Algorithm 3 is able to de-liver a shortest path P (sj′ , tj′ ; v) for request rj′ , it re-jects rj′ due to the fact that the implementation us-age weight of rj′ is too expensive, violating condition(i) or (ii) of the admission control policy, i.e., eitherωv(j

′) > σ1 or∑e∈P (sj′ ,tj′ ;v)

ωe(j′) > σ2. Consequently,∑

e∈POPT (sj′ ,tj′ ;v∗) ωe(j′)+ωv(j

′) ≥ min{σ1, σ2}, Lemma 3thus follows.

We finally analyze the competitive ratio of Algorithm 3in the following theorem.

Theorem 3. Given an SDN G = (S ∪ V,E) with com-puting capacity Cv of each data center v ∈ V andbandwidth capacity Be for each link e ∈ E, assumethat there is a sequence of requests with each requestrj = (sj , tj ;SFC

(k)j , bj , φj) arriving one by one without

the knowledge of future arrivals, there is an onlinealgorithm, Algorithm 3, with a competitive ratio ofO(log n) for the special case of the dynamic profit max-imization problem without the end-to-end delay con-straint, and the algorithm takes O(|S ∪ V |2 · |V |) timeto admit each request when α = 2n and β = 2n, wheren = |S ∪ V |.

Proof: Let COPT (j) and BOPT (j) be the total amountsof computing and bandwidth resources being occupied bythe admitted requests by the optimal offline algorithm OPTprior to the arrival of request rj , and let POPT (sj′ , tj′ ; v∗)be the optimal routing path for request rj′ ∈ T (j) andv∗ ∈ V the data center to implement the service func-tion chain of rj′ . Recall that B(j) =

∑rj′∈A(j) bj′ and

C(j) =∑rj′∈A(j) φj′ · C(SFC(k′)) are the accumulated

bandwidth and computing resources being occupied byadmitted requests inA(j) at the moment, respectively. Then,

(n− 1)(BOPT (j)− B(j)) ≤ (n− 1)∑

rj′∈T (j)

bj′

=∑

rj′∈T (j)

bj′(n− 1)

≤∑

rj′∈T (j)

bj′ ·ρmaxρmin

(ωv∗(j

′) +∑

e∈POPT (sj′ ,tj′ ;v∗)

ωe(j′))

=∑

rj′∈T (j)

bj′ ·ρmaxρmin

( ∑e∈POPT (sj′ ,tj′ ;v∗)

ce(j′)

Be+

cv∗(j′)

ρmax · Cv∗)

≤ ρmaxρmin

∑rj′∈T (j)

bj′( ∑e∈POPT (sj′ ,tj′ ;v∗)

ce(j)

Be+

cv∗(j)

ρmax · Cv∗)

(40)

=ρmaxρmin

∑rj′∈T (j)


ce(j) · bj′Be

+cv∗(j) · bj′ρmax · Cv∗

)

=ρmaxρmin


ce(j)

∑rj′∈T (j) bj′

Be

+ cv∗(j)

∑rj′∈T (j) bj′

ρmaxCv∗

)≤ ρmaxρmin

∑e∈POPT (sj′ ,tj′ ;v∗)

ce(j) +γcv∗(j)

ρminlet γ =

jmaxj′=1{bj′},

(41)

≤ ρmaxρmin

∑e∈E

ce(j) +1

ρmin

∑v∈V

cv(j) · γ

≤ 2(n− 1) · ρmaxρmin

(γ · C(j) logα+ B(j) log β). (42)

Inequality (40) holds since the utilization ratio of bothcomputing and bandwidth resources do not decrease andthus the usage weight of each node or edge in Gj does notdecrease with more and more request admissions.

Let γ be the maximum bandwidth demanded by allrequests, i.e., γ = max{bj′ | 1 ≤ j′ ≤ j}, and let δ be themaximum computing resource demanded by any request,i.e., δ = max{φj′ ·C(SFC(k′)) | 1 ≤ j′ ≤ j}. Inequality (41)holds since the accumulated bandwidth consumption ofall admitted requests whose service function chains areimplemented in any data center v ∈ V is no greater thanCv ·γ by the assumption that C(SFC(k)) > 1 for any type kof service function chain, and data center v is able to admitat most Cv requests, i.e.,

∑rj′∈T (j)

∑e∈POPT (sj′ ,tj′ ;v∗) bj

′ ≤γ ·Cv∗ . Meanwhile, all algorithms for the problem, includingthe optimal offline algorithm OPT , the accumulated usageof bandwidth in any link is no greater than the link band-width capacity, i.e.,

∑rj′∈T (j)

∑e∈POPT (sj′ ,tj′ ;v∗) bj

′ ≤ Be.By Inequality (42), we have

BOPT (j)− B(j) ≤ 2 · ρmaxρmin

(γ · C(j) logα+ B(j) log β).

(43)

The following inequality can be shown by adopting a simi-lar method as above.

COPT (j)− C(j) ≤ 2 · ρmaxρmin

(C(j) logα+ δ · B(j) log β).(44)

Recall thatA(j) is the set of requests admitted by the on-line algorithm, Algorithm 3, and T (j) is the set of requestsrejected by Algorithm 3 but admitted by algorithm OPTprior to the arrival of request rj .

Let p(A(j)) and p(T (j)) be the total profit for the ad-missions of requests inA(j) and T (j), respectively. We hereabuse the notation OPT to denote it as both the optimaloffline algorithm and the solution delivered by the optimalalgorithm. It is clear that the profit collected by admitting arequest is always non-negative as this is guaranteed by theadmission control policy. Without loss of generality, for anyserver s at a data center v, we assume that the amount ofits energy consumption when it is in idle status to its totalenergy consumption is no greater than a constant ξ with0 < ξ < 1, i.e., Eidlev

Eidlev +ηv·Us ≤ ξ. Thus, for each data centerv ∈ V , we have ∑

s∈Qv Eidlev∑

s∈Qv E idlev +∑s∈Qv ηv · Us

≤ ξ. (45)

The competitive ratio of Algorithm 3 then is

p(A(j))OPT

≥ p(A(j))p(A(j)) + p(T (j))

=1

p(T (j))p(A(j)) + 1




=1

λ1·∑rj′∈T (j) φj′ ·C(SFC(k′))+λ2·

∑rj′∈T (j) bj′−

∑v∈V ρvEv

λ1C(j)+λ2B(j)−∑v∈V ρvEv

+ 1

, by Eq. (6)

≥ 1λ1·(COPT (j)−C(j))+λ2·(BOPT (j)−B(j))−

∑v∈V ρvEv


+ 1

≥ 12· ρmaxρmin

·λ1(C(j) logα+δB(j) log β)+2· ρmaxρmin·λ2(γC(j) logα+B(j) log β)−

∑v∈V ρvEv


+ 1, by Ineqs. (43) and (44)

=1

2· ρmaxρmin·λ1(C(j) logα+δB(j) log β)+2· ρmaxρmin

·λ2(γC(j) logα+B(j) log β)−∑v∈V ρvEv

λ1C(j)+λ2B(j)−∑v∈V ρv|Qv|Eidlev −

∑v∈V ρv|Qv|ηv·Uv

+ 1, by Eq. (4)

≥ 12· ρmaxρmin

·λ1(C(j) logα+δB(j) log β)+2· ρmaxρmin·λ2(γC(j) logα+B(j) log β)

λ1C(j)+λ2B(j)−∑v∈V ρv|Qv|Eidlev −

∑v∈V ρv|Qv|ηv·Uv

+ 1

≥ 12· ρmaxρmin

·λ1(C(j) logα+δB(j) log β)+2· ρmaxρmin·λ2(γC(j) logα+B(j) log β)

λ1C(j)+λ2B(j)−( ξ1−ξ+1)·a1C(j)

+ 1, by Ineq. (45) and let a1 = max

v∈V

{ρv · |Qv| · ηvCv

}=

1

2· ρmaxρmin(λ1(

C(j)B(j) logα+δ log β)+λ2(γ

C(j)B(j) logα+log β))

λ1C(j)B(j)+λ2− a1

1−ξC(j)B(j)

+ 1

=1

2· ρmaxρmin(λ1(

C(j)B(j) logα+δ log β)+λ2(γ

C(j)B(j) logα+log β))

λ1C(j)B(j)+λ2−a2 C(j)

B(j)+ 1

, let a2 =a1

1− ξ

=1

2 log 2n · ρmaxρmin

(λ1+λ2γ)x+λ2+λ1δ(λ1−a2)x+λ2

+ 1, when α = β = 2n and let x =

C(j)B(j)

=1

2 log 2n · ρmaxρmin(1 + (λ2γ+a2)x+λ1δ

(λ1−a2)x+λ2) + 1

, (46)

where x = C(j)B(j) , α = β = 2n, a1 = maxv∈V

{ρv·|Qv|·ηvCv

},

a2 = a11−ξ . Notice that a1, a2, and ξ are constants.

Since the profit collected is always non-negative, theterm (λ1 − a2)x + λ2 in Inequality (46) is always non-negative too. The rest is to estimate the upper bound on term2 log 2n · ρmaxρmin

(1+ (λ2γ+a2)x+λ1δ(λ1−a2)x+λ2

)+1 in the denominator of

Inequality (46). Let f(x) = (λ2γ+a2)x+λ1δ(λ1−a2)x+λ2

, x ∈ (0,+∞). Thefirst derivative of f(x) is

f ′(x) =(λ2γ + a2)λ2 − (λ1 − a2)λ1δ

((λ1 − a2)x+ λ2)2, (47)

and the term 1((λ1−a2)x+λ2)2

in f ′(x) always is non-negative.In the following we show that function f(x) is upperbounded. We distinguish our discussion into the followingthree cases. Case 1. If λ2γ+a2

λ1−a2 −λ1δλ2

> 0, then f ′(x) > 0 andlimx→+∞ f ′(x) = 0. Function f(x) thus is an increasingfunction and limx→+∞ f(x) = λ2γ+a2

λ1−a2 . Case 2. If λ2γ+a2λ1−a2 −

λ1δλ2

< 0, then f ′(x) < 0 and limx→+∞ f ′(x) = 0. Thus, f(x)is a decreasing function and limx→0 f(x) = λ1δ

λ2. Case 3. If

λ2γ+a2λ1−a2 −

λ1δλ2

= 0, then f ′(x) = 0 and f(x) = λ2γ+a2λ1−a2 = λ1δ

λ2.

In summary, the upper bound of f(x) is λ2γ+a2λ1−a2 if

λ2γ+a2λ1−a2 −

λ1δλ2≥ 0; λ1δ

λ2otherwise. Therefore, we have

p(A(j))OPT

≥

1

2 log 2n· ρmaxρmin(1+

λ2γ+a2λ1−a2

)+1if λ2γ+a2

λ1−a2 −λ1δλ2≥ 0,

1

2 log 2n· ρmaxρmin(1+

λ1δλ2

)+1otherwise.

(48)Notice that λ1, λ2, a2, ρmax, ρmin, δ, and γ are constants.

Thus, p(A(j)) ≥ OPTθ1·logn+θ2 , where θ1 and θ2 are non-

negative constants, when α = β = 2n.The time complexity analysis of Algorithm 3 is omit-

ted, due to space limitation.

7 PERFORMANCE EVALUATION

In this section, we evaluate the performance of the pro-posed algorithms through experimental simulations. Wealso investigate the impact of important parameters on theperformance of the proposed algorithms.

7.1 Experiment settingsWe consider an SDN G = (S∪V,E) consisting of from 10 to250 nodes. We adopt a commonly used tool GT-ITM [8] togenerate network topologies. The number of data centers ineach network is set at 10% of network size. The computingcapacity of each data center varies from 4,000 GHz to8,000 GHz [31]. The number of servers in each data centeris randomly drawn between 1,000 and 2,000. The energy




consumption of an idle server per second is drawn between100 watt and 200 watt [25] which is a constant; otherwise,its energy consumption is determined by its energy con-sumption rate η that is drawn within [0.5, 1.5] [19]. For eachdata center, the unit energy price per hour in its locationis drawn from [15, 50] per MWh [24], [30]. The bandwidthcapacity of each link varies between 2, 000 Mbps and 20, 000Mbps [15], and the transmission delay on a link varies from2ms to 5ms [15]. The revenue accumulation weights λ1 andλ2 are two constants randomly drawn within [0.05, 0.12]and [0.15, 0.22] respectively, following typical charges inAmazon EC2 [1]. The number of different types of servicefunction chainsK is set at 10. The bandwidth requirement ofeach type of service function chain instance is set between10 Mbps and 20 Mbps [6], and their computing demandis set from 2 GHz to 4 GHz [6], [18]. The processing rateof each service function chain instance is randomly drawnfrom 2 to 20 basic data packets per millisecond [18]. For eachgenerated request rj , two switch nodes in S are randomlyselected as its source sj and destination tj . The packet rateφj of each request rj is randomly drawn from 1 to 10packets/second [16]. Its end-to-end packet delay constraintDj varies from 10 ms to 100 ms [18], and its type of servicefunction chain SFC(k) is randomly chosen from the Ktypes. The value in each figure is the mean of the results outof 50 SDN instances of the same size. The running time ofan algorithm is obtained on a machine with 3.4 GHz Intel i7Quad-core CPU and 16GB RAM. Unless otherwise specified,these parameters will be adopted in the default setting.

7.2 Performance evaluation of online algorithmsWe refer to online algorithms Algorithm 2 andAlgorithm 3 for the problem with and without end-to-enddelays as algorithms OLPM_Delay and OLPM_NDelay, re-spectively. We first evaluate the performance of these two al-gorithms against the optimal ILP solutions with and withoutdelay constraints that are termed as algorithms ILP_Delayand ILP_NDelay, respectively. We then evaluate the onlinealgorithms against two baseline heuristics: Linear_Delayand Linear_NDelay which represent with and withoutthe end-to-end delay constraints, where for each requestrj , algorithms Linear_Delay and Linear_NDelay firstremove all nodes and links from G that do not have enoughresidual resources to admit rj , and then assign each datacenter node and each link in the resulting subgraph witha weight of one. They finally find a (delay-constrained)shortest path in the resulting subgraph from sj to tj viaa data center v ∈ V without (with) the end-to-end delayconstraint on the request.

We now evaluate the performance of algorithmsOLPM_Delay and OLPM_NDelay against the optimal so-lutions of algorithms ILP_Delay and ILP_NDelay, byvarying the number of requests from 800 to 2,400 whilefixing the number of data centers at 3. Fig. 3 depicts thetotal profits and running times of different algorithms. Wecan see from Fig. 3 (a) that algorithm OLPM_Delay achievesa near optimal profit, i.e., it achieves at least 88.3% ofthe performance compared with the optimal solution de-livered by algorithm ILP_Delay. For example, algorithmOLPM_Delay can achieve 93.1% of the optimal profit deliv-ered by algorithm ILP_Delay when the number of requests

800 1200 1600 2000 2400number of requests

6e+03

7e+03

8e+03

accu

mu

late

d p

rofi

t

ILP_DelayOLPM_Delay

(a) The accumulated profit


1e+04

1e+05

1e+06

1e+07

runnin

g t

ime

(ms)

ILP_DelayOLPM_Delay

(b) The running time

Fig. 3: Performance of different algorithms with 3 datacenters and with delay constraints.

is no more than 1,200; and it achieves 88.3% of the optimalprofit for 2,400 requests. It can be seen from Fig. 3 (b) thatalgorithm ILP_Delay takes a prohibitively long time, whilealgorithm OLPM_Delay only takes a small fraction of therunning time of algorithm ILP_Delay. Notice that with theincrease on the number of requests, algorithm ILP_Delayfails to deliver a solution within a reasonable amount oftime when the number of requests reaches 2,400. The similarperformance behaviors can be observed between algorithmsOLPM_NDelay and ILP_NDelay too, which are illustratedin Fig. 4 (a) and (b), respectively.


7e+03

8e+03

9e+03ac

cum

ula

ted

pro

fit

ILP_NDelayOLPM_NDelay



1e+03

1e+04

1e+05

1e+06

1e+07

runnin

g t

ime

(ms)

ILP_NDelayOLPM_NDelay


Fig. 4: Performance of different algorithms with 3 datacenters and without delay constraints.

The rest is to evaluate the performance of algo-rithms OLPM_Delay and OLPM_NDelay against theircorresponding benchmark heuristics Linear_Delay andLinear_NDelay, respectively, for a set of 10,000 requests,by varying the number of nodes from 10 to 250 whilekeeping the other parameters fixed, i.e., α = β = 2n,σ1 = (n− 1) · ρmin/ρmax, and σ2 = n− 1.

Fig. 5 demonstrates the performance behaviors ofthe mentioned algorithms. From Fig. 5 (a) and 6 (a),we can see that both algorithms OLPM_Delay andOLPM_NDelay outperform their benchmark counterparts:algorithms Linear_Delay and Linear_NDelay. Specif-ically, algorithm OLPM_Delay outperforms algorithmLinear_Delay, and the performance gap between thembecomes larger and larger with the increase in network size.As shown in Fig. 5 (a), algorithm OLPM_Delay delivers48.2% more profits than that by algorithm Linear_Delaywhen the network size is set at 250. The similar perfor-mance can be observed by algorithm OLPM_NDelay againstalgorithm Linear_NDelay too. In particular, as shownin Fig. 6 (a), the accumulated profit delivered by algo-rithm OLPM_NDelay is 45.4% more than that by algorithmLinear_NDelay when n = 250.

The rationale is that both algorithms OLPM_Delay andOLPM_NDelay assign higher usage weights to over-utilizedresources while assigning lower usage weights to under-utilized resources, thus the resources in the network can be




50 100 150 200 25010network size n

0

1e+04

2e+04

3e+04

4e+04ac

cum

ula

ted p

rofi

t OLPM_DelayLinear_Delay


50 100 150 200 25010network size n

0

1e+05

2e+05

3e+05

run

nin

g t

ime

(ms) OLPM_Delay

Linear_Delay


Fig. 5: Performance of different online algorithms by varyingnetwork size from 10 to 250 and with delay constraints.

fully utilized to meet the demands of user requests, therebyachieving higher profits. In contrast, both algorithmsLinear_Delay and Linear_NDelay do not take into ac-count the utilization of resources, thus the resource usage onsome data centers and links are overloaded. Fig. 5 (b) depictsthe running time curves of algorithms OLPM_Delay andLinear_Delay, where algorithm Linear_Delay takesless time than algorithm OLPM_Delay in all network sizes.The similar results on the running time of algorithmsOLPM_NDelay and Linear_NDelay can also be seen inFig. 6 (b).

50 100 150 200 25010network size n

0

1e+04

2e+04

3e+04

4e+04

5e+04

accu

mula

ted p

rofi

t OLPM_NDelayLinear_NDelay


50 100 150 200 25010network size n

0

4e+04

8e+04

1.2e+05

1.6e+05

run

nin

g t

ime

(ms) OLPM_NDelay

Linear_NDelay


Fig. 6: Performance of different online algorithms by varyingnetwork size from 10 to 250 without delay constraints.

7.3 Parameter impact on the algorithm performanceIn the following, we first evaluate the impact of the admis-sion control threshold parameters σ1 and σ2 on the per-formance of algorithms OLPM_Delay and OLPM_NDelay.Fig. 7 plots their performance curves with and without theadmission control thresholds, from which it can be seen thatboth of them deliver less profits if no admission controlpolicy is applied. With the increase in network size n,the performance gap of algorithm OLPM_Delay with andwithout the admission control thresholds becomes largeras shown in Fig. 7 (a). For instance, the ratio of the profitdelivered by algorithm OLPM_Delay with and without theadmission control grows from 1.18 to 1.59 when n = 10 and250 respectively. This is due to that the distance betweenthe source and the destination of a request can be very longin a large network, thus consuming much more bandwidthresource for routing its data traffic. Under the admissioncontrol policy, algorithm OLPM_Delay is able to reject thoserequests beyond the given threshold, thereby enabling toadmit future requests and achieving higher profits. The per-formance gap of algorithm OLPM_NDelay with and withoutthe admission control is similar to the one of algorithmOLPM_Delay with and without the admission control, asshown in Fig. 7 (b).

We then evaluate the impact of the maximum durationTmax of request implementation on the performance of

50 100 150 200 25010network size n

0

1e+04

2e+04

3e+04

accu

mula

ted p

rofi

t σ1 = (n-1)*ρ

min/ρ

maxσ

2 = n - 1

σ1 = σ

2 = ∞

(a) The accumulated profit byOLPM_Delay

50 100 150 200 25010network size n

0

1e+04

2e+04

3e+04

4e+04

accu

mula

ted p

rofi

t σ1 = (n-1)*ρ

min/ρ

maxσ

2 = n - 1

σ1 = σ

2 = ∞

(b) The accumulated profit byOLPM_NDelay

Fig. 7: The accumulated profit delivered by OLPM_Delayand OLPM_NDelay with thresholds σ1 = (n − 1) ·ρmin/ρmax, σ2 = n − 1, and without the thresholds σ1 =σ2 =∞ when α = β = 2n.

algorithms OLPM_Delay and OLPM_NDelay in a networkwith 200 nodes, by varying Tmax from 2 to 32. Assume thatthe finite time horizon is slotted into equal time slots. Therequest admissions and resource releases are performed inthe beginning of each time slot. The duration of each requestimplementation is randomly drawn in [1, Tmax]. The profitcollected by admitting a request rj is the duration of requestimplementation in a data center times its profit p(rj) in asingle time slot. We consider a time horizon consisting of200 time slots with up to 1,000 requests per time slot. FromFig. 8 (a), it can be seen that the longer the maximum dura-tion, the more profits delivered by algorithm OLPM_Delay.The similar performance of algorithm OLPM_NDelay can beobserved in Fig. 8 (b).

0 50 100 150 200timeslot

0

2e+06

4e+06

6e+06

accu

mula

ted p

rofi

t T.max = 32T.max = 16T.max = 8T.max = 4T.max = 2

(a) The accumulated profit byalgorithm OLPM_Delay

0 50 100 150 200timeslot

0

2e+06

4e+06

6e+06

8e+06

accu

mula

ted p

rofi

t T.max = 32T.max = 16T.max = 8T.max = 4T.max = 2

(b) The accumulated profit byalgorithm OLPM_NDelay

Fig. 8: The accumulated profit of online algorithmsOLPM_Delay, and OLPM_NDelay with different maximumdurations of requests when the network size is 200.

8 CONCLUSION

In this paper, we investigated the dynamic profit maximiza-tion problem in distributed clouds by dynamically admit-ting delay-aware requests with network function servicerequirements, assuming that each request is admitted ona pay-as-you-go basis, subject to various network resourcecapacity constraints. We first proposed an ILP solution forthe offline version of the problem. We then developed anonline heuristic for the problem. We also devised an onlinealgorithm with a provable competitive ratio for a specialcase of the problem where the end-to-end delay constraintof each request can be negligible. We finally evaluated theperformance of the proposed algorithms through experi-mental simulations. Experimental results demonstrate thatthe proposed algorithms are promising.

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Yu Ma received his BSc degree with the firstclass Honours in Computer Science at the Aus-tralian National University in 2015. He is cur-rently a PhD candidate in the Research Schoolof Computer Science at the Australian NationalUniversity. His research interests include Soft-ware Defined Networking, Internet of Things(IoT), and Social Networking.

Weifa Liang (M’99–SM’01) received the PhDdegree from the Australian National Universityin 1998, the ME degree from the University ofScience and Technology of China in 1989, andthe BSc degree from Wuhan University, Chinain 1984, all in Computer Science. He is cur-rently a Full Professor in the Research Schoolof Computer Science at the Australian NationalUniversity. His research interests include designand analysis of energy efficient routing protocolsfor wireless ad hoc and sensor networks, cloud

computing, Software-Defined Networking, design and analysis of paral-lel and distributed algorithms, approximation algorithms, combinatorialoptimization, and graph theory. He is a senior member of the IEEE anda member of the ACM.

Zichuan Xu (M’17) received his PhD degreefrom the Australian National University in 2016,ME degree and BSc degree from Dalian Univer-sity of Technology in China in 2011 and 2008,all in Computer Science. He was a ResearchAssociate at Department of Electronic and Elec-trical Engineering, University College London,UK. He currently is an Associate Professor inSchool of Software at Dalian University of Tech-nology, China. His research interests includecloud computing, software-defined networking,

network function virtualization, wireless sensor networks, routing pro-tocol design for wireless networks, algorithmic game theory, and opti-mization problems.

Song Guo (M’02-SM’11) received the PhD de-gree in computer science from the University ofOttawa, Canada in 2005. He is currently a FullProfessor at the Department of Computing, theHong Kong Polytechnic University, Hong Kong.His research interests are mainly in the areasof protocol design and performance analysis forwireless networks and distributed systems. Hehas published over 250 papers in refereed jour-nals and conferences in these areas and re-ceived three IEEE/ACM best paper awards. Dr.

Guo currently serves as Associate Editor of IEEE Transactions on Par-allel and Distributed Systems, Associate Editor of IEEE Transactions onEmerging Topics in Computing for the track of Computational Networks,and on editorial boards of many others. He has also been in organizingand technical committees of numerous international conferences. Dr.Guo is a senior member of the IEEE and the ACM.

Documents

Profit Maximization for Admitting Requests with Network ...users.cecs.anu.edu.au/~Weifa.Liang/papers/MLXG18.pdf · algorithm with a provable competitive ratio for a special case of