Wireless Network Slicing: Generalized Kelly Mechanism Based Resource Allocation
Yan Kyaw Tun, Nguyen H. Tran, Duy Trong Ngo, Shashi Raj Pandey, Zhu Han, Choong Seon Hong
aa r X i v : . [ c s . N I] J u l Wireless Network Slicing: Generalized KellyMechanism Based Resource Allocation
Yan Kyaw Tun, Nguyen H. Tran,
Senior Member, IEEE,
Duy Trong Ngo,
Member, IEEE,
Shashi Raj Pandey, Zhu Han,
Fellow, IEEE, and Choong Seon Hong,
Senior Member, IEEE
Abstract —Wireless network slicing (i.e., network virtualiza-tion) is one of the potential technologies for addressing the issueof rapidly growing demand in mobile data services related to5G cellular networks. It logically decouples the current cellularnetworks into two entities; infrastructure providers (InPs) andmobile virtual network operators (MVNOs). The resources ofbase stations (e.g., resource blocks, transmission power, antennas)which are owned by the InP are shared to multiple MVNOs whoneed resources for their mobile users. Specifically, the physicalresources of an InP are abstracted into multiple isolated networkslices, which are then allocated to MVNO’s mobile users. Inthis paper, two-level allocation problem in network slicing isexamined, whilst enabling efficient resource utilization, inter-slice isolation (i.e., no interference amongst slices), and intra-slice isolation (i.e., no interference between users in the sameslice). A generalized Kelly mechanism (GKM) is also designed,based on which the upper level of the resource allocation issue(i.e., between the InP and MVNOs) is addressed. The benefitof using such a resource bidding and allocation framework isthat the seller (InP) does not need to know the true valuationof the bidders (MVNOs). For solving the lower level of resourceallocation issue (i.e., between MVNOs and their mobile users),the optimal resource allocation is derived from each MVNOto its mobile users by using KKT conditions. Then, bandwidthresources are allocated to the users of MVNOs. Finally, the resultsof simulation are presented to verify the theoretical analysis ofour proposed two-level resource allocation problem in wirelessnetwork slicing.
Index Terms —Generalized Kelly Mechanism, resource alloca-tion, wireless network virtualization, wireless network slicing.
I. I
NTRODUCTION N OWADAYS, wireless networks have faced with an explo-sive growth of mobile data traffic because of the dramaticincrease in the use of mobile devices, and consequently, datagreedy applications. To address the ever growing network traf-fic, in recent years, wireless network slicing has been a central
Manuscript received October 10, 2018; revised June 16, 2019; acceptedJune 29, 2019; data of current version . .; date of publication ... This workwas supported by the National Research Foundation of Korea (NRF) grantfunded by the Korea government (MSIT) (NRF-2017R1A2A2A05000995).*Dr. CS Hong is the corresponding author.Yan Kyaw Tun, Shashi Raj Pandey, and Choong Seon Hong are with theDepartment of Computer Science and Engineering, Kyung Hee University,Yongin-si, Gyeonggi-do 17104, Rep. of Korea, e-mail: { ykyawtun7, shashiraj,cshong } @khu.ac.kr.Nguyen H. Tran is with the School of Information Technologies, The Uni-versity of Sydney, NSW 2006, Australia, email { nguyen.tran } @sydney.edu.au.Duy Trong Ngo is with the School of Electrical Engineering and Com-puting, The University of Newcastle, Callaghan, NSW 2308, Australia,email { duy.ngo } @newcastle.edu.au.Zhu Han is with the Electrical and Computer Engineering Department,University of Houston, Houston, TX 77004, and the Department of ComputerScience and Engineering, Kyung Hee University, Yongin-si, Gyeonggi-do17104, Rep. of Korea, email { zhan2 } @uh.edu. topic of research. Wireless network slicing decouples mobilenetwork operators (MNOs) in the current wireless network intotwo bodies: InPs and MVNOs. The physical wireless networkincluding physical infrastructure such as base stations, cellsites, radio towers, antennas, physical resource blocks (RBs),backhaul, core network, transmission networks, transmissionpower, etc., are owned and operated by an InP. Physicalresources from multiple InPs are leased by the MVNOs tocreate their own virtual networks for delivering particularservices such as VoIP, live streaming, video conferencing,and video telephony, to their network users. By enabling thesharing of physical resources, wireless network slicing enableseffective reduction in operational expenditures (OPEX) andcapital expenditures (CAPEX) of mobile network operators(MNOs). It also enables a flexible network operation byfacilitating the coexistence of multiple MVNOs on a sharedinfrastructure [1].Though network slicing is the potential technology forfuture mobile networks, there remains several challengingissues to address. Among them, one important issue is howto efficiently slice and split radio resources (i.e., bandwidthor physical RBs) into multiple slices for MVNOs who mustmeet the dynamic demands of their mobile end users, whilstensuring the key requirements of inter-slice and intra-slice isolation [2], [3]. In this regards, as defined in 5G architectureproposed by [4], virtualization of network functions relies onnetwork function virtualization (NFV) and software definednetwork (SDN) technologies. Specifically, NFV enables theabstraction of the resources and facilitates in sharing themamong multiple tenants for future network services [5]. Here,the virtualization layer, referred to as a hypervisor, enablesan agile network environment which is managed by theSDN-based open standard application programming interface(API). A number of SDN controllers have been developedto enable a flexible and programmable radio access network(RAN), namely the SD-RAN platform [6], [7], [8]. In [2], theauthors designed a virtualization substrate, in particular a flowscheduler, and implemented it to meet the key requirementsof efficient resource utilization, customization, and isolation inwireless resource virtualization. In [7], the authors designedthe controller, namely FlexRAN that uses an agent API whichtransparently communicates to UEs. The control protocolsof such controller can make scheduling decisions such asresource block allocation. With the software enabler, theeNodeB only has to handle the data plane and the operationssuch as obtaining and setting the configurations, applyingthe scheduling decisions, maintaining the flow and so on are abstracted via the control plane with the help functionsprovided by the APIs. Similar to these described virtualizedwireless network architecture based on SDN in [9], [10],[11], [12], network slicing functionality can be considered oneinstance of the virtual machine (VM) in our proposed systemmodel.Efficient resource allocation helps to improve resource uti-lization, ensures the quality of services for the end users andfurthermore, provides energy efficiency. The resource alloca-tion problem in wireless network slicing is more challengingwhen selfish agents (i.e., MVNOs) are involved. Therefore,under such scenario where the agents act greedily, it is impor-tant to design an appropriate incentive plan in order to achievesocial efficiency. In this regard, to address the challenges inthe efficient resource allocation in wireless network slicing,two prominent frameworks are implemented in the wirelessnetwork slicing. In the first approach, the InP acts as a centralplayer and can directly allocate resources to mobile usersof MVNOs as per the predetermined resource requirement.In the second approach, MVNOs take part in the resourcescheduling to their users instead of the InP. Firstly, the InPinteracts with MVNOs and allocates resources to them. Then,the MVNOs will manage the individual resource allocation(i.e., scheduling) to their own mobile users. Therefore, withthe involvement of MVNOs, the resource allocation designcorresponds as a two-level problem. Most of the existingresearch works investigated the first resource allocation designwhere they ignored the role of MVNOs [13], [14], [15],[16]. Unlike existing works that only focus on maximizingnetwork utilization, our problem formulation considers thenetwork economics issue in wireless network slicing. It in-cludes monetary profit to the InP in terms of efficient resourceallocation strategy for multiple associated MVNOs, and thecorresponding economic interactions between MVNOs and itsusers. In this work under wireless network slicing, we focuson the two-level resource allocation problem to maximize theindividual and the aggregate valuation of MVNOs. Here, themost important challenge is the resource allocation amongMVNOs with fairness guarantee.Under the aforementioned challenges, we design a general-ized Kelly mechanism (GKM) [17] to address the upper-levelproblem and make use of the Karush-Kuhn-Tucker (KKT) con-ditions in addressing the lower-level of the resource allocationproblem. The GKM belongs to one of the auction algorithmswhere each agent (i.e., MVNO) can submit an individual bidfor resources to the seller (i.e, InP), while an InP receives bidsfrom different MVNOs and then allocates resources to eachbidding agent (i.e., MVNO) proportionally to their bidding val-ues [18]. The Kelly mechanism (KM) [19] is suitable for price-taking agents, i.e., agents who have no power to influencethe market price of the available resources with their biddingvalue. That is only possible when there are a large numberof agents (i.e., MVNOs) in the resource allocation auction.However, the GKM is suitable for both price-anticipating andprice-taking agents. Here, the price-anticipating agent meansan agent’s bidding value can influence the market price of theresources. Such price anticipating agents’ bidding values maylead to loss in efficiency, and the social welfare (i.e., sum of all MVNO’s valuation). At that time, GKM can reduce theloss of efficiency. Note that there are several effective auctionmechanisms such as the Vickrey-Clarke-Groves (VCG) [20]which focuses on the scenario where the agents bid truthfully,i.e., every agent has to submit its true valuation as a bid.However, as valuation is the private information of agents,they will not submit it to the seller. In the GKM, even if theagents do not submit their true valuations, the seller can stillinduce the marginal valuations of the agents. A. Research Contributions
In order to address the challenges and issues of resourceallocation in wireless network slicing as mentioned above, wepropose an efficient resource allocation framework by usingthe GKM. Summary to our main contributions is: • Firstly, a two-level resource allocation problem in wire-less network slicing is proposed. Then, the GKM isdesigned to address the upper-level of the proposedresource allocation problem. In the GKM, MVNOs willsubmit their individual bidding values to the InP inorder to request wireless resources. The InP will furtherallocate its physical resources to MVNOs according totheir bidding values. Then, each MVNO will use thatwireless resources allocated by the InP to serve its mobileusers. The most important challenges of the resourceallocation in the network slicing such as isolation andfairness between MVNOs are handled by the proposedproblem formulation. • We next perform the theoretical analysis of GKM proper-ties such as the existence of a unique Nash equilibrium,and the optimal resource allocation to MVNOs under theNash equilibrium. Then, we analyze the influence powerof each bidder (i.e., MVNO) in the market which is theability of the MVNO to change the market price of theresources. To control the market influence power of thebidders (i.e., MVNOs), the seller (i.e., InP) introduces thepenalty value that is attached with the cost for each bidderin GKM. We further analyze the effect of this penaltyvalue for each MVNO under the Nash equilibrium. • Finally, we use KKT conditions to address the lower-level of the proposed problem (i.e., between MVNOsand their mobile end users), and provide the closed-formsolution to this problem. Moreover, we also consider anincomplete information scenario in which each MVNOdoes not know the channel condition of its mobile usersdue to estimation error, or wireless channel delay. Wefurther extend our work into multiple resources scenariowhere each MVNO requests multiple resources (e.g.,bandwidth, power) from an InP. • In simulation section, we first present the resource al-located to MVNOs under the GKM. Then, we com-pare the achieved valuation of each MVNOs under pro-posed GKM with others: Equal Sharing, traditional Kellymechanism, and Optimal solution. The proposed schemeachieves a significant performance gain: up to , and in comparison to Equal Sharing, and traditional Kellymechanism with our proposed algorithm, respectively. Physical MBS
Virtual BS Virtual BS
Virtual BS
VirtualizationMobile Virtual Network Operator(MVNO (cid:282)
1) Mobile Virtual Network Operator (MVNO (cid:282) Mobile Virtual Network Operator(MVNO (cid:282)
M)Infrastructure Provider (InP)
Fig. 1: A model of wireless network slicing.We also observe that our proposed solution frameworkachieves near Optimal solution. Further, we also demon-strate the allocated bandwidth to each user of MVNOunder KKT conditions.The rest of this paper is organized as follows: Section IIsummarizes related works. The system model and wirelessnetwork slicing framework are introduced in Section III. Sec-tion IV presents the two-level resource allocation problem inwireless network slicing and proposes the solution mechanism.The extension of our proposed multiple resource allocationproblem in wireless network slicing is presented in SectionV. Section VI discusses about the simulation results. Finally,Section VII concludes the paper.II. R
ELATED W ORKS
Resource layer, along with network slice instance layer, andservice instance layer is one of the integral parts of networkslicing in 5G architecture proposed by [4]. A network slicesupports at least one type of service, and should be mutu-ally isolated, manageable and support multi-tenants, multi-services [21], [22]. In this regard, the recent 3GPP R15 [23]specifications and standards define tailored services such asmassive machine type communication (mMTC), ultra reliableand low latency communications (URLLC), and enhancedmobile broadband (eMBB). Therefore, a proper design ofthe resource allocation solution that is flexible, scalable anddemand-oriented in wireless virtualization has to be done,which is the scope of this paper.Proportional allocation (i.e., Kelly mechanism) in one-sidedresource allocation auction was investigated in [24], [25]. Theyshowed that under the assumption of price-taking agents, Kellymechanism achieves maximum value of social welfare. Inorder to reduce the loss efficiency gap and the market influencepower of the price-anticipating agents, [26] studied a GKMby setting a penalty value for each price-participating agentaccording to their bid. Moreover, the theoretical limitations ofboth GKM and Kelly mechanism were presented in [27]. In [28], the authors have proposed a stochastic game-basedspectrum allocation in virtualized wireless networks. Althoughthe proposed resource allocation scheme achieved higher re-source utilization, MVNOs are not considered in resourceallocation design. Moreover, as InP manages resources andallocates it directly to the mobile users of MVNOs in acentralized manner, the computation complexity is high.The work of [29], [30] introduced a joint resource allocationand admission control strategy for an orthogonal frequencydivision multiple access (OFDMA) based virtualized wirelessnetwork. Here, both resource-based and rate-based MVNOswere considered, and a joint optimization problem for powerand resource allocation was formulated for maximizing theoverall sum rate of the corresponding MVNOs. But thesignificance of MVNOs was ignored in [29], [30] and theuser scheduling was instead performed by the base stationsof the InP. The dynamic resource management in wirelessvirtualized networks was proposed in [31]. The developed dy-namic resource sharing approach can result in higher resourceutilization and better system efficiency.In [32], the authors introduced an auction game modelfor the users to bid for radio resources. Moreover, auctionmechanism based power allocation in the LTE air interface vir-tualization was proposed in [33]. The authors of [34] proposedan LTE framework with an added entity called “hypervisor” ata base station. The hypervisor enables sharing of RBs amongthe MVNOs without interfering with each other. In [35], acombination of wireless network virtualization and massiveMIMO was considered. Then, the authors formulated a re-source (i.e., bandwidth, power, antennas) allocation problem asa hierarchical structure and implemented a combinatorial VCGauction mechanism for solution. In most of the existing works,the responsibility of MVNOs was missing and they did notconsider economic models of wireless network slicing. In ourwork, we consider both two-level resource allocation problemand economic model of wireless network slicing. Moreover,we highlight the responsibility of MVNOs in the resourceallocation in wireless network slicing.III. S
YSTEM M ODEL
A wireless network slicing where a single InP having aphysical macro base station (MBS), and a set of mobilevirtual network operators, M = { , , . . . , M } that providesparticular mobile services to their users is shown in Fig. 1.Specifically, in this work, we consider 4G architecture forgeneral virtualization, similar with the works in [15], [2].The MBS is operating on the total bandwidth of R andeach MVNO m ∈ M provides services to the users S m = { , , . . . , S m } . A fraction of the total bandwidth R allocatedto each MVNO m ∈ M is defined as r m . Here, a hypervisoris deployed by the InP at the MBS to slice its physicalresources for leasing among multiple MVNOs. A centralquestion is how the InP will schedule its wireless bandwidth Considering the 3GPP specification and standards for 4G architecture[36], the resource allocation problem considers resource block (RB) as theminimum allocation unit. However, to make the problem tractable, we usethe continuous form of resource as ‘bandwidth’, similar to the works in [37],[38], [39], to solve the problem.
TABLE I: Summary of Notations.
Notation Definition M Set of MVNOs, |M| = M S m Set of mobile users of MVNO m ∈ M , |S m | = S m R Total bandwidth capacity of an InP r m Bandwidth allocated to MVNO m ∈ M r m ( b ) Bandwidth allocated to MVNO m ∈ M depends onthe bidding vector b of MVNOs b The vector of bidding values of MVNOs b m Bidding value of MVNO m ∈ M B Sum of bidding values of all MVNOs v m ( r m ( b )) Valuation of MVNO m depends on theallocated resource r m ( b ) c m ( b ) Cost function of MVNO m ∈ M q m Penalty value of MVNO m ∈ M q The vector of penalty values of MVNOs v ′ ( r m ( b )) Marginal valuation of MVNO m ∈ M µ m Market influence power of the MVNO m ∈ M β The virtual price of bandwidth (per Hertz) x ms The allocated resource to the user s ∈ S m of MVNO m ∈ M E The resource competition matrix e m The vector of the resource allocated to MVNO m ∈ M Q The penalty matrix B The bidding matrix amongst multiple MVNOs to give services to their mobileusers. Because the InP cannot access the information of userssuch as QoS requirements and channel conditions. Therefore,a possible solution is to allocate bandwidth to MVNOs first,and afterwards each MVNO allocates the wireless bandwidthto its users. This approach is regarded as a two-level solutionapproach.In this work, the resource (i.e., bandwidth) allocation prob-lem in wireless network slicing is decomposed into two levels.In the upper level, as shown in Fig. 2, the InP decides how toefficiently allocate bandwidth to multiple MVNOs and whichaims to maximize the social welfare (i.e., aggregate valuationof MVNOs). In the lower level, each MVNO manages resourcescheduling to its mobile users by considering its own utility.We formulate the upper-level problem as an auction-basedresource allocation problem for which the GKM is proposedfor solution. Each MVNO m ∈ M will report its own biddingvalue b m (0 ≤ b m < ∞ ) to the InP in each resourceallocation round. Depending on the bidding values of allMVNOs, they will receive a proper allocation of bandwidthform the InP. The bandwidth allocation among MVNOs willbe straightforward when the InP knows the characteristics (i.e.,valuation) of the MVNOs. However, the valuation function isthe private information of each MVNO and it is related withthe dynamic channel conditions of its users. After that, eachMVNO will assign the bandwidth to its users as per their QoSrequirements.IV. T WO -L EVEL R ESOURCE A LLOCATION IN W IRELESS N ETWORK S LICING
In this work, the resource allocation problem in the wirelessnetwork slicing can be decoupled into two levels: 1) resourceallocation between InP and MVNOs, and 2) resource alloca-tion from MVNO to its mobile users.
Infrastructure Provider (InP)MVNO - 1 MVNO - M
MVNO - 2 . . . . .
Mobile Virtual Network Operators(MVNOs)
Fig. 2: Generalized Kelly Mechanism.
A. Upper-level Problem
Depending on the number of users and their QoSrequirement, each MVNO decides the required wirelessbandwidth. Let us define the valuation function v m ( r m ( b )) , b = { b , b , . . . , b M } is the vector of thebidding value, as the satisfaction of the MVNO m ∈ M . Assumption 1 : The valuation function v m ( r m ( b )) isstrictly increasing, concave and continuous over the domain r m > .This assumption is widely used for utility or valuationfunctions in communication networks [17], [40], [41]. Here,the InP will allocate its fraction of resource (bandwidth)to each MVNO according to the reported bidding value ofMVNOs. It means that InP will allocate the largest ratioof bandwidth to the MVNO with the highest bidding value.Thus, the GKM framework [17] can be used to express theinteraction of InP and MVNOs, where the objective is tomaximize the aggregate valuation of MVNOs. Therefore, thewireless bandwidth allocation to the MVNOs from the InP ina virtualized network is formulated as follows: max X m ∈M v m ( r m ( b )) (1)s.t. r m ( b ) ∩ r n ( b ) = ∅ , for m = n, and m, n ∈ M , (2) M X m =1 r m ( b ) ≤ R, (3)var. r m ( b ) ≥ , ∀ m ∈ M , (4)which considers the optimal division of the total bandwidth R of the InP under the GKM. Constraint (2) ensures the isolationbetween different MVNOs. As the MBS has limited amount ofbandwidth, constraint (3) guarantees the allocated bandwidthof all MVNOs not exceed the total bandwidth of the MBS and(4) ensures that the resource allocated to each MVNO mustbe positive value.Solving problem (1)-(4) is possible once the valuation ofMVNOs is known at the InP. However, the valuation is theprivate value of each MVNO. Therefore, the MVNOs will not share these information to the InP so as to maximize theirown utilities with allocation of bandwidth. Upon submissionof the bidding value b m , each MVNO will receive the fractionof the total bandwidth of MBS r m ( b ) accordingly. Let r = { r , r , . . . , r M } be the resource allocation vector which isdetermined by the proportional allocation [17] as follows: r m ( b ) = b m P Mm =1 b m R, ∀ m ∈ M , (5)where P Mm =1 b m = B is the total bidding value at the InP.Here, the proposed resource allocation scheme guarantees thefairness among MVNOs [see Remark 1]. Each MVNO m ∈M has the cost function c m ( b ) = q m b m which depends onthe bidding value b m and q m , the penalty parameter whichis varying according to the bidding value. Then, the payofffunction of the MVNO m is defined as u m ( r m ( b )) = v m ( r m ( b )) − c m ( b )= v m ( r m ( b )) − q m b m , ∀ m ∈ M , (6)where v m ( r m ( b )) is the valuation of MVNO m with allocatedresource r m based on bidding value b m and the penalty vectorof all MVNOs with the bidding value is q = { q , q , . . . , q M } . Remark 1.
The proportional allocation scheme canmaintain fairness amongst competing MVNOs. This isbecause the allocation of resources is based upon theproportion of bidding values of each MVNO, i.e., biddinghigher means getting more resource.
Proposition 1 : The optimal bidding of each MVNO m ∈M is b m = 1 q m r m ( b ) v ′ ( r m ( b ))(1 − µ m ) , ∀ m ∈ M , (7)where µ m is the market influence power of the MVNO m ∈M to be explained later in this section. Proof:
See Appendix A.
Proposition 2 : The unit penalty parameter for each MVNOis q m = 1 β v ′ m ( r m ( b )) (cid:18) − r m ( b ) R (cid:19) , ∀ m ∈ M . (8) Proof:
See Appendix B.However, the penalty parameter for each MVNO m ∈ M depends on its valuation. As the valuation of each MVNOis its private information, it will not reveal it to the InP. Insuch a case, one can employ an iterative algorithm that allowsthe InP to approximate the penalty for each MVNO from theinformation of the previous iteration. Therefore, the penaltyfor MVNO m ∈ M at the k th iteration is as follows [18]: q km = q k − m + R − ( r m ( b )) k − M − − Rq k − m P Mm =1 q k − m ! , ∀ m ∈ M . (9) Proof:
See Appendix C.Without the penalty parameter q m , ∀ m ∈ M , each MVNOwill try to obtain a large proportion of wireless bandwidthresource from the InP by bidding higher, in fact, as much aspossible. For this reason, the InP imposes a control parameter, defined as the penalty vector q to obtain true valuations ofthe MVNOs and balance the bandwidth allocation betweenthe competing MVNOs. In this regards, the InP interacts withthe MVNOs as follows: 1) InP informs the penalty parameter q m to each MVNO m , 2) considering the penalty parameter,each MVNO submits its bidding value b m to get resourcesfrom the InP, and 3) the InP broadcasts the virtual priceof the bandwidth (per Hertz). Therefore, the virtual price ofbandwidth (per Hertz) is β = P Mm =1 b m R . (10)After knowing the virtual price for the bandwidth, eachMVNO can derive the fraction of the bandwidth it receives as r m = b m β , ∀ m ∈ M . To this end, when there are few MVNOsin the resource allocation, the bidding value of each MVNOwill largely influence the virtual price. Therefore, we canobserve that each MVNO is capable enough to manipulate theoutcome of the resource allocation game. However, an increasein the number of MVNO will eventually eliminate suchinfluences, i.e., the market influence power of each MVNOis low. In this regards, with an infinite number of MVNO inthe resource allocation, the individual market influence abilityof the MVNOs approaches to zero. Note that, in the real world,an infinite number of MVNOs is not possible. Therefore, in ourformulation, we have considered the market influence powerof each MVNO m ∈ M as µ m = b m P Mm =1 b m , ∀ m ∈ M . (11)From (11), we observe that the market influence power ofMVNO m ∈ M is coupled with the bids of other competingMVNOs, however, these bids are their private information. Insuch a case, one can employ an iterative algorithm that allowsthe MVNO to know the approximate market influence powerfrom the information related with the previous iterations. Thus,for the MVNO m ∈ M , its market influence power at the k th iteration can be defined as µ km = 1 − b ( k − m q ( k − m ( r m ( b )) ( k − v ′ m (cid:0) ( r m ( b )) ( k − (cid:1) . (12)In our bandwidth allocation (i.e., bandwidth competitionamong MVNOs) game, each MVNO will adopt a strategy b m to maximize its utility u m ( b ) as u m ( b m ; b − m , q ) = v m ( r m ( b )) − q m b m , m ∈ M , (13)where b − m = [ b , . . . , b m − , b m +1 , . . . , b M ] denotes the strat-egy profiles of all the other MVNOs except m . Then, foreach MVNO, with the strategy profile b ∗ m , ∀ m ∈ M , thereexists a unique Nash equilibrium in the formulated resourcecompetition game if the following relation is satisfied: u m ( b ∗ m ; b ∗− m , q ) ≥ u m ( b m ; b ∗− m , q ) , ∀ b m ≥ . (14) Theorem 1 (Uniqueness of Nash equilibrium): When
M > , at least two components of b m > and Assumption 1holds. For any q m ∈ q , there is a unique Nash equilibriumfor the resource competition game with the strategy profile b m > , ∀ m ∈ M . Proof:
See Appendix D.In order to distinguish the equilibrium conditions ofthis bandwidth allocation (i.e., resource competition amongMVNOs), the function ˆ v ( r m ) is introduced as ˆ v m ( r m ( b )) = 1 q m (cid:18) − r m ( b ) R (cid:19) v m ( r m ( b ))+ 1 q m R Z r m v m ( z ) dz. (15)The efficient bandwidth allocation to MVNOs in this resourcecompetition among MVNOs can be explored according to thefollowing optimization problem max X m ∈M ˆ v m ( r m ( b )) (16)s.t. M X m =1 r m ( b ) ≤ R, (17)var. r m ( b ) ≥ , ∀ m ∈ M . (18)Let r ∗ m be the solution to the above optimization problem. Proposition 3 : There exists a unique Nash equilibriumof the resource allocation (i.e., resource competition amongMVNOs) shown in Theorem 1. Under that unique Nash equi-librium, the allocated bandwidth r m to each MVNO m ∈ M is the solution to the above optimization problem shown in(16) with constraints (17) and (18). Proof:
See Appendix E.
B. Lower-level Problem
In the lower-level, each MVNO aims at maximizing itsvaluation by allocating the obtained bandwidth from the InP.The valuation of each MVNO m ∈ M is the sum of thelogarithmic function of data rate of its users. Therefore, thevaluation of an MVNO m ∈ M can be defined as: v m ( r m ) = max S m X s =1 log (cid:18) x ms r m log (cid:18) p s h s N (cid:19) + 1 (cid:19) (19)s.t. S m X s =1 x ms ≤ , ∀ m ∈ M , (20)var. x ms ∈ [0 , , ∀ s ∈ S m , ∀ m ∈ M , (21)where p s is the downlink transmitted power of the BS to amobile user s . Note that we assume fixed power allocation perbandwidth (Hertz) in this paper. Moreover, h s is the channelgain of user s , N is the noise power, and x ms representsa fraction of bandwidth of MVNO m assigned to s where x ms ∈ r m . (20) and (21) are the constraints for the fraction ofwireless bandwidth allocated to each subscriber of the MVNO m ∈ M . As constraints (20) and (21) are linear, the constraintset is affine and objective function (19) is concave. Therefore,the valuation function v m ( r m ( b )) of each MVNO m ∈ M satisfies Assumption 1. C. Optimal Bandwidth Allocation
The bandwidth allocation problem in (19) is a convexproblem. Thus, the optimal solutions for (19) can be obtainedvia Lagrangian duality [42]. Here, the Lagrangian of (19) is L ( x ms , λ, ν ) = S m X s =1 log (cid:18) x ms r m log (cid:18) p s h s N (cid:19) + 1 (cid:19) + λ (cid:18) − S m X s =1 x ms (cid:19) , (22)where λ ≥ is the Lagrangian multiplier defined for constraint(21). By using the KKT conditions, we get the optimalbandwidth allocated to each user s ∈ S m as x m ∗ s = 1 r m | S m | " r m + S m X s =1 α ∗ − α ∗ ! , ∀ s ∈ S m . (23) Proof:
See Appendix F.
D. Lower-level Problem with Incomplete Information
In a practical scenario, it is hard for MVNOs to getprecise information of the channels (bandwidth) because of theestimation errors, and the wireless channel delay. To addressthe uncertainty of the wireless channel, in this work weconsider that the wireless bandwidth follows Rayleigh fading[43]. Since there is no complete information at the MVNO,we need to introduce an outage probability constraint in (19)as v m ( r m ) = max S m X s =1 log (cid:18) x ms r m log (cid:18) p ms H ms N (cid:19) + 1 (cid:19) (24)s.t. (20) − (21) , Prob (cid:18) ρ min s > log (1 + p ms h ms N ) (cid:19) ≤ ǫ, ∀ s ∈ S m , (25)where ρ min s = log (1 + p ms H ms N ) , and ǫ is a predeterminedthreshold on outage probability. In this work, without loss ofgenerality, we consider that the threshold value is the same forall mobile users. Then, we can rewrite the outage probabilityfor the QoS constraint (25) as Prob (cid:26) ρ min s > log (1 + p ms h ms N ) (cid:27) ≤ ǫ, ⇔ Prob ( γ ms ≤ ρ min s − p ms ) ≤ ǫ, ⇔ ρ min s ≤ log (1 + p ms F − γ ms ( ǫ )) , (26)where F γ ms ( . ) is the cumulative distribution function (CDF) of γ ms for the user s of the MVNO m and F − γ ms ( . ) is its inverse.With Rayleigh fading, we get F γ ms as: F γ ms ( a ) = Z a aσ e − ( a )22 σ d ( a )= 1 − e − ( a )22 σ , (27) Algorithm 1
GKM Algorithm for Optimal Bandwidth Allo-cation in a Wireless Virtualized Network Initialization:
Initialize q (0) m , b (0) m , µ m ← , ∀ m ∈ M ; Set k ← Each MVNO m ∈ M estimates the market influencepower µ ( k ) m according to (12); The InP calculates the penalty for each MVNO q ( k ) m by(9) and then informs each MVNO m ∈ M Each MVNO m ∈ M updates the bidding value b ( k ) m by(7) and then submits to the InP who then sends the virtualprice β ( k ) in (10) to all MVNOs; for m = 1 to M do Based on (5), the InP calculates the amount of band-width r ( k ) m ( b ) to be allocated to MVNO m ∈ M ; end for The InP distributes r ( k ) m ( b ) to each MVNO m ∈ M ; for m = 1 to M do for s = 1 to S m do Each MVNO m allocates optimal bandwidth x ms toits mobile users based on (27); end for end for Each MVNO m ∈ M calculates v ( k ) m ( r m ) ; Increment: k ← k + 1; Repeat lines 3 to 15 until convergence.where σ is the scale parameter, and a = ρ min s − p ms .From (24), we can remove the outage probability of theQoS constraint in (26) and rewrite (19) as v m ( r m ) = max S m X s =1 log (cid:16) x ms r m log (cid:16) p ms F − γ ms ( ǫ ) (cid:17) + 1 (cid:17) (28)s.t. (20) and (21) . From the KKT conditions, we get the optimal bandwidthallocation to each user s ∈ S m as follows x m ∗ s = 1 r m | S m | " r m + S m X s =1 ν ∗ − ν ∗ ! , ∀ s ∈ S m , (29)where ν ∗ = log (1 + p ms F − γ ms ( ǫ )) .V. W IRELESS N ETWORK V IRTUALIZATION WITH M ULTIPLE R ESOURCES
In this section, we consider that each MVNO m ∈ M needs C divisible (e.g., power, wireless bandwidth, antennas,computation capacity, storage capacity, etc.,) at the same timeto service mobile users. We can model C × M resourcecompetition matrix E as E = ( e , e , . . . , e M ) = e . . . e M ... . . . ... e C . . . e CM (30)where ( e c . . . e cM ) is the row vector that indicates theallocation of the resource c ∈ C of the InP among M MVNOs, and e m shows the allocation of C resources to an MVNO m ∈ M . Assumption 2 : The valuation function v m ( e m ) is concave,strictly increasing, and continuous over the domain e m > [17].Here, we define the social welfare maximization problem inmultiple divisible resources as max X m ∈M v m ( e m ) (31)s.t. e cm ∩ e cn = ∅ , for m = n, m, n ∈ M , ∀ c ∈ C , (32) M X m =1 e cm ≤ R c , ∀ c ∈ C , (33)var. e cm ≥ , ∀ m ∈ M , ∀ c ∈ C , (34)where R c is the maximum capacity of resource c ∈ C . (32)ensures the intra-isolation among different MVNOs for allresources c ∈ C of the InP. As the resources provided bythe InP are limited, (33) guarantees that the allocated resource e cm to all MVNOs do not exceed the total resource capacity R c .Similar to the resource competition matrix E , we define thepenalty matrix Q and the bidding values matrix B of MVNOsas Q = q . . . q M ... . .. ... q C . . . q CM , B = b . . . b M ... . . . ... b C . . . b CM , (35)where q cm represents the penalty for MVNO m to bid forresource e cm at the InP, and b cm denotes the bidding valueof MVNO m for the divisible resource c at the InP. Let us,respectively, denote by q m and b m the penalty and biddingvalue with regard to MVNO m ∈ M . Also denote by Q c and B c the penalty and bid with regard to the resource c ∈C , respectively. The resource c ∈ C is allocated to MVNOsaccording to e cm ( B c ) = b cm P Mm =1 b cm , ∀ c ∈ C . (36)The utility function of MVNO m ∈ M is defined as u m ( B , Q ) = v m ( e m ( B )) − q T m b m , ∀ m ∈ M , (37)where each MVNO chooses its bidding strategy to maximizeits utility defined as u m ( b m ; b -m , Q ) = v m ( e m ( B )) − q T m b m , ∀ m ∈ M . (38)The bidding profile matrix B ∗ is a Nash equilibrium for anyMVNO m ∈ M if the following equation is satisfied: u m ( b ∗ m ; b ∗ -m , Q ) ≥ u m ( b m ; b ∗ -m , Q ) , ∀ b m ≥ . (39)The proof of the existence of a unique Nash equilibrium,optimal resource allocation and optimal bidding value for eachMVNO are already shown in Section IV. Fig. 3: Bandwidth allocation to each MVNO under theproposed (GKM) algorithm.VI. S
IMULATION R ESULTS
A. Simulation setting and performance metrics
In this section, we evaluate the performance of our proposedGKM algorithm in a wireless virtualized network for optimalbandwidth allocation. The network scenario of our simulationincludes a single InP with one MBS and 4 MVNOs with10, 5, 4, and 3 mobile users who are positioned randomlywithin the coverage area of the MBS, respectively. The radiusof the macrocell is set as 500m. At the MBS, the maximumavailable bandwidth is 10MHz, the maximum transmit poweris 43dBm, and the thermal noise density is considered as -174dBm/Hz. The path loss model is
P L = 40 log ( d ) −
10 log ( Gh t h r ) + 10 λ log ( dd ) + X g , where d is the actualand d is the reference distance between the transmitter and thereceiver, respectively, h t and h r are respective heights of thetransmitter and the receiver, and a Gaussian random variable X g . The small-scale fading model is Rayleigh fading.In this work, we maximize both aggregate valuation ofMVNOs, and the individual valuation of each MVNO byoptimizing the performance metrics of both bandwidth andpower. B. Detailed Numerical Results
In this section, we will discuss the detailed numerical resultsto show the efficacy of our proposed mechanism in a wirelessvirtualized network.Fig. 3 shows the bandwidth resource allocated to theindividual MVNOs under the proposed GKM algorithm.Here, MVNO-1 is allocated 4.5455MHz, MVNO-2 is allo-cated 2.273MHz, MVNO-3 is 1.812MHz and MVNO-4 is1.364MHz, respectively. From the above results, we observethat the MVNO who has more mobile users receives a largerfraction of bandwidth resources owned by the InP. For thisreason, in Fig. 3 MVNO-1 is allocated more bandwidthresources compared with the others. It is also clear that theMVNO with more mobile users will invest or bid much morethan other MVNOs to get more bandwidth to fulfill the servicerequirement of its mobile users. Fig. 4 compares the achieved valuation for each MVNO asthe function of allocated bandwidth resource under different al-gorithms: our proposed algorithm, the traditional Kelly Mecha-nism [19], the Equal Sharing and the Optimal solutions. Underthe Equal Sharing mechanism, the InP allocates an equalamount of bandwidth to all MVNOs. The Optimal solutionis achieved under zero market influence power of MVNOs,i.e., no MVNO can alter the market price of the resource. Thisscenario accounts for the price-taking buyers (MVNOs) whichis of our particular interest. As an example, from Fig. 4, weobserve that the median of the achieved valuation of MVNO-1 is 119 (Proposed), 117 (Kelly Mechanism), 106 (EqualSharing), and 120 (Optimal). Moreover, we can also see thelowest and the highest valuation of MVNO-1 is 106 - 133.5(Proposed), 105.4 - 130 (Kelly Mechanism), 93 - 129.3 (EqualSharing), 109 - 133.6 (Optimal). Therefore, our proposedalgorithm achieves a higher valuation than doing the traditionalKelly Mechanism and Equal Sharing for all MVNOs. Also, itis comparatively close to the Optimal solution, demonstratingits efficacy. It is clear that our solution approach is better thanthe traditional Kelly Mechanism and the Equal Sharing.Furthermore, Fig. 5 demonstrates the convergence of theachievable valuation of MVNOs in the proposed GeneralizedKelly Mechanism (GKM). At the beginning of the algorithm,as the fraction of bandwidth is randomly allocated to eachMVNO, the MVNOs evaluate the valuation randomly. In thesubsequent iterations, each MVNO chooses its best strategy,i.e., bidding value, to achieve the highest valuation. From Fig.5, we observe that our proposed GKM algorithm convergesto the equilibrium point in just 5 iterations. Here, MVNO-1 achieves the highest valuation compared with the otherMVNOs. This is because it has more associated users, andgets more fraction of bandwidth from the InP.Fig. 6 demonstrates the solution for the lower-level problem.The bandwidth resource for individual users in each MVNOis assigned as per (23). From Fig. 6, we can notice thatthe amount of bandwidth that MVNOs allocated to each oftheir users. As an example, for MVNO-1, the median of theallocated bandwidth to its mobile users are around (0.44, 0.46,0.47, 0.46, 0.49, 0.43, 0.48, 0.48, 0.48, 0.49) MHz. Fromresults, the amount of bandwidth allocation among users ofthe same MVNO are different.In Fig. 7, we show the adversity of the number of users inthe network. We observe that our proposed algorithm obtainsthe Optimal solution for the larger network. This is due to theincrease in the bidding value of the corresponding MVNOsto the users for obtaining resources. Consequently, with theincrease in the number of buyers in the network, the marketprice is less likely to be affected as discussed before. Similarly,with the increase in the number of MVNOs, the achievableaggregate valuation of each MVNO will be the same as theoptimal social welfare, as observed in Fig. 8.Fig. 9 depicts the achieved data rate for each MVNOunder incomplete information. It is evident that the data rateincreases with a more relaxed outage constraint threshold. Fora sufficiently large outage threshold, a significant gain in theachieved data rate is observed.Fig. 10 represents the allocated power to each MVNO under
Fig. 4: Comparison of the achieved valuation for (a) MVNO-1, (b) MVNO-2, (c) MVNO-3, (d) MVNO-4.Fig. 5: Convergence of valuation of MVNOs in the proposedGKM algorithm.our proposed algorithm where the InP allocates 19.625dBm toMVNO-1, 9.717dBm to MVNO-2, 7.83dBm to MVNO-3 and5.827dBm to MVNO-4, respectively. This result is similar tothe bandwidth allocation because the MVNO who has moremobile users gets a larger share of the resources. We also showthe power allocation from each MVNO to its respective usersin Fig. 11, which is the solution of the lower-level problem.As we have discussed in the lower-level of the bandwidthallocation problem, the power allocation to each user dependson the channel condition which is assigned to that user.Finally, Fig. 12 shows the achieved valuation for eachMVNO as the function of allocated bandwidth and powerresources by the InP under different algorithms. Similar to theindividual resource allocation, our proposed algorithm resultsin a higher valuation than do the traditional Kelly Mechanismand Equal Sharing scheme. Further, we observe that the valua-tion is comparatively near to the Optimal solution. In Fig. 13,we present the convergence of valuations of all MVNOs undermultiple resources (i.e., power, and bandwidth) allocation. We observe the convergence of our proposed algorithm in lesserthan 8 iterations. VII. C
ONCLUSION
In this paper, we have formulated a two-level optimalbandwidth allocation problem for wireless network slicing. Inthe upper-level, the GKM is introduced to model MVNOsas bidders who compete for the bandwidth from the InPin order to serve their mobile users. The InP, who is theseller of the resources, then executes the bandwidth allocationprocess under the GKM to fulfill these requests. In the lower-level, each MVNO allocates the optimal bandwidth resourceto its users. Morever, we consider the incomplete informationscenario where the MVNO does not know the exact channelstate information for its users. Finally, we consider the multipleresource scenario where MVNOs compete with each otherfor power and bandwidth resources from the InP. Simulationresults have reflected that the aggregated valuation of theMVNOs following our proposed algorithm outperforms thatby the traditional Kelly Mechanism, Equal Sharing, and isnearly close to the Optimal.A
PPENDIX AP ROOF OF P ROPOSITION b m , the stationarycondition of (6) can be obtained as v ′ ( r m ( b )) ∂r m ( b ) ∂b m − q m = 0 . (40)MVNO m ∈ M in this resource competition is a price-anticipating agent, and the resource r m ( b ) allocated to MVNO m is dependent on its bidding value b m . Therefore, using (10), r m ( b ) = b m β , and applying the first-order derivative, we have ∂r m ( b ) ∂b m = 1 β (cid:18) − b m β ∂β∂b m (cid:19) . (41) Fig. 6: Bandwidth allocation to each user of (a) MVNO-1, (b) MVNO-2, (c) MVNO-3, (d) MVNO-4.Fig. 7: Aggregate valuation of MVNOs for different numberof users.Fig. 8: Aggregate valuation of MVNOs under differentnumber of MVNOs.By using (10), the above (40) and (41) are rewritten as v ′ ( r m ( b )) (cid:18) − b m β (cid:19) = βq m . (42) According to (10), the optimal bidding strategy of MVNO m is b m = 1 q m r m ( b ) v ′ ( r m ( b ))(1 − µ m ) , ∀ m ∈ M . (43)A PPENDIX BP ROOF OF P ROPOSITION m ∈M depends on its bidding value. ∂u m ∂b m = v ′ m ( r ( b )) (cid:18) ( P Mm =1 b m − b m ) R ( P Mm =1 b m ) (cid:19) − q m = 0 , (44) ⇔ β v ′ m ( r m ( b ))(1 − r m ( b ) R ) − q m = 0 , (45) ⇔ β v ′ m ( r m ( b ))(1 − r m ( b ) R ) = q m . (46)A PPENDIX C PROOF OF UNIQUE PENALTY FOR EACH
MVNOFrom (43) and (10), q ∗ m v ′ m ( r ∗ m ( b ))( R − r ∗ m ( b )) = M X m =1 b m , ∀ m ∈ M . (47)Therefore, q ∗ m : q ∗ n = v ′ m ( r ∗ m ( b ))( R − r m ( b )) : v ′ n ( r ∗ n ( b ))( R − r ∗ n ( b )) , ∀ m, n ∈ M . (48)Suppose the penalty vector q induces the optimal bandwidthallocation vector r . The optimality condition for the optimalbandwidth allocation is v ′ m ( r ∗ m ( b )) = v ′ n ( r ∗ n ( b )) , ∀ m, n ∈ M [18]. Therefore, (48) becomes R − r ∗ m ( b ) q ∗ m = R − r ∗ n ( b ) q ∗ n = M R − R P Mm =1 q ∗ m , ∀ m, n ∈ M . (49) . . . . ε505560657075 R a t e ( bp s ) +4.5454e6MVNO1 . . . . ε262830323436 R a t e ( bp s ) +2.2727e6MVNO2 . . . . ε808284868890 R a t e ( bp s ) +1.8181e6MVNO3 . . . . ε343638404244 R a t e ( bp s ) +1.3636e6MVNO4 Fig. 9: Rate achieved by each MVNO with respect to different outage thresholds.
MVNO-1 MVNO-2 MVNO-3 MVNO-4
Index of MVNOs P o w e r ( d B m ) Fig. 10: Power allocation at MVNOs.From (42), R − r ∗ m ( b ) M − q ∗ m P Mm =1 q ∗ m R. (50)Inspired by the optimal condition in (50), we iteratively updatethe penalty of each MVNO m ∈ M using the informationof the previous iteration. Therefore, at each iteration, the InPupdates the penalty of each MVNO according to q km = q k − m + R − ( r m ( b )) k − M − − Rq k − m P Mm =1 q k − m ! , ∀ m ∈ M . (51)A PPENDIX DP ROOF OF T HEOREM b m > , ∀ m ∈ M . First, we have ∂u m ∂b m = v ′ m ( r ( b )) (cid:18) b − m R ( P Mm =1 b m ) (cid:19) − q m = 0 (52) ⇔ q m v ′ m b m R P Mm =1 b m ! b − m R ( P Mm =1 b m ) ! = 1 (53) ⇔ q m v ′ m b m R P Mm =1 b m ! × R P Mm =1 b m − b m R ( P Mm =1 b m ) ! = 1 . (54)When b m > , it is true that q m v ′ m b m R P Mm =1 b m ! R P Mm =1 b m − b m R ( P Mm =1 b m ) ! = 1 . (55)When b m = 0 , we also have q m v ′ m (0) ≤ . (56)There exists a unique Nash equilibrium [17] if the above twoconditions are satisfied. A PPENDIX EP ROOF OF P ROPOSITION r m ( b ) of (15) is ∂ ˆ v m ( r m ( b )) ∂r m ( b ) = 1 q m (1 − µ m ) v ′ m ( r m ( b )) . (57)When µ m and r m ( b ) are greater than zero, ∂ ˆ v m ( r m ( b )) ∂r m ( b ) > in (57). From (57), (1 − µ m ) is strictly decreasing in b m .Moreover, the allocated bandwidth r m depends on the biddingvalue b m of MVNO. Therefore, r m ( b ) is also decreasing.Thus, ∂ ˆ v m ( r m ( b )) ∂r m ( b ) is monotonically decreasing. For this reason, ˆ v m ( r m ( b )) is a concave function, and hence, the optimizationproblem (16) has a unique maximum value. Fig. 11: Power allocation to each user of (a) MVNO-1, (b) MVNO-2, (c) MVNO-3, (d) MVNO-4.Fig. 12: Comparison of achieved valuation under multiple resources for (a) MVNO-1, (b) MVNO-2, (c) MVNO-3, (d)MVNO-4.Fig. 13: Convergence of valuation of MVNOs for multipleresources in the proposed GKM algorithm. Here, we define the Lagrangian of (16) as L ( r m , ρ ) = X m ∈M ˆ v m ( r m ( b )) + ρ " R − M X m =1 r m ( b ) , (58)where ρ ≥ is the Lagrangian multiplier for the constraint(16). Therefore, the KKT conditions can be expressed by usingthe first-order derivative of (58) w.r.t r m and ρ as ∂L ( r m , ρ ) ∂r m ( b ) = 1 q m [(1 − µ m ) v ′ m ( r m ( b ))] − ρ ≤ , if r m ≥ , ∀ m ∈ M , (59) ∂L ( r m , ρ ) ∂ρ = R − M X m =1 r m ( b ) ≥ , if ρ ≥ , (60)When ρ > , q m v ′ m ( r m ( b ))(1 − µ m ) = ρ. (61) From (43) and (61), ρ = β . By using (10), we can clearlyobserve that the optimal bidding strategy in (7) is satisfied.So, the optimal resource allocation to each MVNO at theequilibrium is defined as follows: r ∗ m ( b ) = b m q m v ′ ( r m ( b ))(1 − µ m ) , ∀ m ∈ M . (62)To sum up, ρ = β,ρ > , when P Mm =1 r m ( b ) = R,ρ = 0 , when P Mm =1 r m ( b ) < R, (63)A PPENDIX FP ROOF OF O PTIMAL B ANDWIDTH A LLOCATION
The KKT conditions can be expressed with the first-orderderivative of (22) w.r.t x ms and λ as ∂L∂x ms = log (cid:16) p s h s N (cid:17) x ms r m log (cid:16) p s h s N (cid:17) + 1 − λ ≤ , if x ms ≥ , ∀ s ∈ S m , (64) ∂L∂λ = 1 − S m X s =1 x ms ≥ , if λ ≥ , (65)Solving (64) gives the bandwidth allocated to each user s ∈S m as x m ∗ s = 1 λ ∗ r m − r m log (cid:16) p s h s N (cid:17) , ∀ s ∈ S m , (66)where λ ∗ = 1 | S m | r m + S m X s =1 (cid:16) p s h s N (cid:17) + . (67)Thus, the optimal bandwidth allocated to each user s ∈ S m is x m ∗ s = 1 r m | S m | " r m + S m X s =1 α ∗ − α ∗ ! , ∀ s ∈ S m (68)where α ∗ = log (cid:16) p s h s N (cid:17) .R EFERENCES[1] C. Liang and F. R. Yu, “Wireless network virtualization: A survey,some research issues and challenges,”
IEEE Communications Surveys& Tutorials , vol. 17, no. 1, pp. 358–380, First Quarter 2015.[2] R. Kokku, R. Mahindra, H. Zhang, and S. Rangarajan, “NVS: Asubstrate for virtualizing wireless resources in cellular networks,”
IEEE/ACM Transactions on Networking , vol. 20, no. 5, pp. 1333–1346,Oct 2012.[3] A. Haider, R. Potter, and A. Nakao, “Challenges in resource allocationin network virtualization,” in , vol. 18, 20May 2009.[4] N. Alliance, “5G white paper,”
Next generation mobile networks, whitepaper , pp. 1–125, 2015.[5] J. Ordonez-Lucena, P. Ameigeiras, D. Lopez, J. J. Ramos-Munoz,J. Lorca, and J. Folgueira, “Network slicing for 5G with sdn/nfv: Con-cepts, architectures, and challenges,”
IEEE Communications Magazine ,vol. 55, no. 5, pp. 80–87, 2017.[6] T. Chen, H. Zhang, X. Chen, and O. Tirkkonen, “Softmobile: Controlevolution for future heterogeneous mobile networks,”
IEEE WirelessCommunications , vol. 21, no. 6, pp. 70–78, 2014. [7] X. Foukas, N. Nikaein, M. M. Kassem, M. K. Marina, and K. Konto-vasilis, “Flexran: A flexible and programmable platform for software-defined radio access networks,” in
Proceedings of the 12th Internationalon Conference on emerging Networking EXperiments and Technologies .ACM, 2016, pp. 427–441.[8] W. Wu, L. E. Li, A. Panda, and S. Shenker, “Pran: Programmable radioaccess networks,” in
Proceedings of the 13th ACM Workshop on HotTopics in Networks . ACM, 2014, p. 6.[9] E. J. Kitindi, S. Fu, Y. Jia, A. Kabir, and Y. Wang, “Wireless networkvirtualization with sdn and c-ran for 5G networks: Requirements,opportunities, and challenges,”
IEEE Access , vol. 5, pp. 19 099–19 115,2017.[10] N. F. V. NFV and U. Cases, “Etsi gs nfv 001 v1. 1.1 (2013-10),” 2013.[11] R. A. Addad, T. Taleb, M. Bagaa, D. L. C. Dutra, and H. Flinck,“Towards modeling cross-domain network slices for 5G,” in . IEEE, 2018, pp.1–7.[12] X. Zhang and Q. Zhu, “Scalable virtualization and offloading-basedsoftware-defined architecture for heterogeneous statistical qos provision-ing over 5G multimedia mobile wireless networks,”
IEEE Journal onSelected Areas in Communications , vol. 36, no. 12, pp. 2787–2804,2018.[13] I. Malanchini, S. Valentin, and O. Aydin, “Generalized resource sharingfor multiple operators in cellular wireless networks,” in
Proc. IEEE Inter-national Wireless Communications and Mobile Computing Conference(IWCMC) , Nicosia, Cyprus, 4-8 Aug. 2014, pp. 803–808.[14] J. Van De Belt, H. Ahmadi, and L. E. Doyle, “A dynamic embeddingalgorithm for wireless network virtualization,” in
Proc. IEEE 80thVehicular Technology Conference (VTC Fall) , Vancouver, BC, Canada,14-17 Sept. 2014, pp. 1–6.[15] M. I. Kamel, L. B. Le, and A. Girard, “LTE wireless network virtual-ization: Dynamic slicing via flexible scheduling,” in
Proc. IEEE 80thVehicular Technology Conference (VTC Fall) , Vancouver, BC, Canada,14-17 Sept. 2014, pp. 1–5.[16] X. Lu, K. Yang, and H. Zhang, “An elastic sub-carrier and powerallocation algorithm enabling wireless network virtualization,”
WirelessPersonal Communications , vol. 75, no. 4, pp. 1827–1849, April 2014.[17] R. T. Ma, “Efficient resource allocation and consolidation with selfishagents: An adaptive auction approach,” in
Proc. IEEE InternationalConference on Distributed Computing System , Nara, Japan, 27-30 June2016, pp. 497–508.[18] Y. Yang, R. T. Ma, and J. C. Lui, “Price differentiation and controlin the kelly mechanism,”
Performance Evaluation , vol. 70, no. 10, pp.792–805, Oct 2013.[19] F. Kelly, “Charging and rate control for elastic traffic,”
Transactions onEmerging Telecommunications Technologies , vol. 8, no. 1, pp. 33–37,Feb 1997.[20] N. Nisan and A. Ronen, “Computationally feasible VCG mechanisms,”
Journal of Artificial Intelligence Research , vol. 29, pp. 19–47, May 2007.[21] A. Galis and K. Makhijani, “Network slicing landscape: A holisticarchitectural approach, orchestration and management with applicabilityin mobile and fixed networks and clouds,” 2018.[22] A. Nakao, P. Du, Y. Kiriha, F. Granelli, A. A. Gebremariam, T. Taleb,and M. Bagaa, “End-to-end network slicing for 5G mobile networks,”
Journal of Information Processing
Mathematics of Operations Research , vol. 29, no. 3,pp. 407–435, Nov 2004.[25] J. R. Correa, A. S. Schulz, and N. E. Stier-Moses, “The price ofanarchy of the proportional allocation mechanism revisited,” in
Proc.International Conference on Web and Internet Economics . Springer,Dec 2013, pp. 109–120.[26] R. Ma, D. M. Chiu, J. C. Lui, V. Misra, and D. Rubenstein, “Onresource management for cloud users: A Generalized Kelly mechanismapproach,”
Electrical Engineering, Tech. Rep , 2010.[27] V. Syrgkanis and E. Tardos, “Composable and efficient mechanisms,”in
Proceedings of the forty-fifth annual ACM symposium on Theory ofcomputing , Palo Alto, California, USA, 1-4 June. 2013, pp. 211–220.[28] F. Fu and U. C. Kozat, “Stochastic game for wireless network virtual-ization,”
IEEE/ACM Transactions on Networking , vol. 21, pp. 84–97,February 2013.[29] Y. K. Tun, C. W. Zaw, and C. S. Hong, “Downlink power allocationin virtualized wireless networks,” in
Proc. Network Operations andManagement Symposium (APNOMS) , Seoul, South Korea, 27-29 Sept.2017, pp. 346–349. [30] S. Parsaeefard, V. Jumba, M. Derakhshani, and T. Le-Ngoc, “Jointresource provisioning and admission control in wireless virtualizednetworks,” in Proc. IEEE Wireless Communications and NetworkingConference , LA, USA, 9-12 March. 2015, pp. 2020–2025.[31] Y. Zaki, L. Zhao, C. Goerg, and A. Timm-Giel, “LTE wireless virtual-ization and spectrum management,” in
Proc. IEEE Wireless and MobileNetworking Conference (WMNC) , Budapest, Hungary, Oct 2010.[32] S. Bashar and Z. Ding, “Admission control and resource allocationin a heterogeneous OFDMA wireless network,”
IEEE Transactions onWireless Communications , vol. 8, no. 8, pp. 4200–4210, August 2009.[33] B. Fan, H. Tian, and B. Liu, “Game theory based power allocation inLTE air interface virtualization,” in
Proc. IEEE Wireless Communica-tions and Networking Conference , New Orleans, LA, USA, 9-12 March2015, pp. 972–976.[34] Y. Zaki, L. Zhao, C. Goerg, and A. Timm-Giel, “A novel LTE wirelessvirtualization framework,” in
Proc. International Conference on MobileNetworks and Management , Santander, Spain, Sept 2010, pp. 245–257.[35] K. Zhu and E. Hossain, “Virtualization of 5G cellular networks asa hierarchical combinatorial auction,”
IEEE Transactions on MobileComputing
Optical Fiber Communication Conference . Optical Society of America,2019, pp. W3J–2.[38] L. Chen, Y. Feng, B. Li, and B. Li, “Efficient performance-centricbandwidth allocation with fairness tradeoff,”
IEEE Transactions onParallel and Distributed Systems , vol. 29, no. 8, pp. 1693–1706, 2018.[39] Q. Wu and R. Zhang, “Common throughput maximization in uav-enabled ofdma systems with delay consideration,”
IEEE Transactionson Communications , vol. 66, no. 12, pp. 6614–6627, 2018.[40] N. C. Luong, P. Wang, D. Niyato, Y.-C. Liang, Z. Han, and F. Hou,“Applications of economic and pricing models for resource managementin 5G wireless networks: A survey,”
IEEE Communications Surveys &Tutorials , 2018.[41] A.-L. Jin, W. Song, and W. Zhuang, “Auction-based resource allocationfor sharing cloudlets in mobile cloud computing,”
IEEE Transactionson Emerging Topics in Computing , vol. 6, no. 1, pp. 45–57, 2018.[42] S. Boyd and L. Vandenberghe,
Convex optimization . CambridgeUniversity press, 2004.[43] X. Tang, P. Ren, Y. Wang, and Z. Han, “Combating full-duplex activeeavesdropper: A hierarchical game perspective,”