Resource allocation in a generalized LTE air interface virtualization framework exploiting user behavior
 Bo Fan^{1},
 Hui Tian^{1}Email author and
 Xiao Yan^{2}
https://doi.org/10.1186/s1363801606082
© Fan et al. 2016
Received: 20 October 2015
Accepted: 6 April 2016
Published: 18 April 2016
Abstract
Wireless network virtualization (WNV) is a promising technique to solve the ossification of current networks. In this paper, a generalized Long Term Evolution (LTE) air interface virtualization framework is proposed where virtual operators (VOs) are enabled to share the physical resources owned by the infrastructure provider (InP). This usercentric feature provides VOs with the flexibility to manage their own virtualized networks according to different user traffic demand. Hence, we introduce the “Gini Coefficient” to quantitatively characterize the user traffic behavior among different VOs. In addition, we consider bandwidthpower allocation to optimize system energy efficiency (EE). The resource allocation problem is formulated as a mixed combinatorial and nonconvex optimization problem, which is extremely difficult to solve. To reduce the computational complexity, we decouple the problem into two steps. First, for a given power allocation, we obtain the bandwidth allocation. Adopting bankruptcy game model and the wellknown Shapley value, a heuristic bandwidth allocation algorithm is devised. Second, under the assumption of known bandwidth allocation, we transform the original optimization problem into an equivalent convex optimization problem and obtain the optimal solution via fractional programming. Through simulation, the results of user behavior and resource allocation are jointly analyzed. The user behavior is proved to be effective and the proposed resource allocation outperforms conventional schemes.
Keywords
1 Introduction
Internetdependent lifeway will greatly increase the diversity and density of communication demands in the nextgeneration mobile network (5G) [1]. This will lead to a burst of wireless traffic volume, which is anticipated to increase tenfold for the next decade [2]. However, today’s ossified network operation greatly limits network capacity and efficiency. To cope with the situation, wireless network virtualization (WNV) draws worldwide attention for its potential to enhance flexibility, diversity, manageability, and energy efficiency for current networks [3].
The main idea of WNV is to enable differentiated services to run on common network infrastructure [3]. In WNV, the traditional role of mobile network provider is separated into two parts. One is referred to as infrastructure provider (InP), who owns and supervises the overall physical network infrastructure. The other is virtual operator (VO) whose concentration is paid on offering ondemand services to their customers by purchasing resources from InP.
1.1 Related work
One primary research trend of WNV is Long Term Evolution (LTE) air interface virtualization, where multiple VOs share resource on the air interface [4]. Gudipati et al. [5] proposes logically abstracting multiple LTE eNodeB (evolved NodeB) as one virtual big base station. Hence, all the resource elements can be conceptually thought of as threedimensional resource grid with time, frequency, and space. Li et al. [6] promotes replacing LTE eNodeBs with remote radio units (RRUs) in order to achieve complete virtualization. Thus the distributed control units can be replaced by a central controller, which maintains a global view of the radio access network (RAN). Yang et al. [7] refractors the LTE control plane as softwaredefined to enhance classified user traffic quality of service (QoS). The network performance is therefore improved due to the additional flexibility provided to user service.
Basically, the virtualization of the wireless resource on the air interface can be considered as a resource scheduling problem [8]. Existing researches mainly focus on bandwidth allocation. A contractbased bandwidth allocation algorithm is developed, where resources are allocated based on four types of predefined contracts [9, 10]. CostaPerez et al. [11] proposes virtualizing LTE eNodeB into different bandwidth slices. The virtualized slices are assigned to multiple mobile users according to their differentiated priority of service flows.
Recently, game theory has been applied in the context of WNV. Authors in [12] firstly introduce stochastic game to model the interaction between InP and the VOs. VOs dynamically compete for bandwidth resource from InP, and the allocation is determined when the bidding price is proved to reach a unique Nash equilibrium. To guarantee allocation fairness, bankruptcy game is proposed in [13] to share the limited LTE air interface bandwidth resource. Besides, [14] focuses on power resource allocation in LTE air interface virtualization, where a VickreyClarkeGroves (VCG) auction is used to model InP as auctioneer selling power resource to multiple VOs.
1.2 Main contributions
Brilliant as above researches are, there are still several disadvantages. Firstly, since conventional network architecture no longer suits virtualization, the functionalities in operating virtualized network need to be redefined. And there lacks a systematic virtualization framework for current LTE networks. Secondly, the usercentric feature in WNV is neglected. By purchasing resources from InP, VOs can operate their own virtualized networks. In other words, VOs can expand or shrink their markets and customize different types of services to satisfy whatever their users need to accomplish the communication task. Therefore, we believe that exploiting user behavior can improve the resource allocation efficiency.

A generalized LTE air interface framework is proposed where data plane, control plane, and interface are explicitly defined. To decouple service from infrastructure, conventional LTE eNodeB is split into twofold components: data base station (BS) and signaling controller. Data BS is contained in the data plane, connected through gateway to the Internet. The signaling controller, InP, along with the VO belong to the control plane. InP is in charge of the radio access resource via the interface with the signaling controller. VOs apply for resource from InP to serve the customers through the interface with InP.

A tractable expression to quantitatively characterize user behavior is utilized to indicate the equilibrium/disequilibrium of VOs’ user traffic rate demand, which lays foundation for designing the resource allocation strategies. To the best of our knowledge, this is the first time to introduce user behavior analysis to solve WNV problems.

We consider energy efficiency (EE) optimization from a system perspective with bandwidth and power allocation. The problem is formulated as a mixed combinatorial and nonconvex optimization problem and we decouple it into two steps. A bankruptcy game is adopted to model VOs as players and the bandwidth as the total estate. Using Shapley value, a heuristic bandwidth allocation algorithm is devised to optimize EE. Afterwards, the optimal power allocation is obtained via fractional programming.

Different from existing works, in our simulation, user behavior is analyzed in joint with resource allocation results. In different cases, the change of the user behavior pattern is in line with the resource allocation results. This indicates that user behavior analysis can be a potential technique to be used in WNV to improve its flexibility and efficiency.
The paper is organized as follows. Section 2 and Section 3 present system model and problem formulation, respectively. Detailed resource allocation strategies are investigated in Section 4. Simulation results are analyzed in Section 5. Section 6 concludes the paper.
2 System model

Overall system physical resource block (PRBs, the minimum LTE bandwidth allocation unit, consisting of 12 subcarriers) are ideally orthogonal so there exists no interference problem.Table 1
Basic notations used in the paper
v
VO index
V
total VO number
\(\mathcal {S}\)
cooperative VO set
\(\mathcal {V}\)
total VO set
n
data BS index
N
total data BS number
k
PRB index
K _{ v }
allocated PRBs number of VO#v
m
user index
M _{ v }
user number of VO#v
\({\mathcal {M}}_{v}\)
user set of VO#v
d _{ v }
average user traffic rate demand of VO#v
b _{ v }
minimum PRB number to operate VO#v
c _{ v }
additional claimed PRB number of VO#v
Y
estimated total PRB number to support the whole service area
Φ
Shapley value
y
characteristic function of coalition
\({K_{n}^{\max }}\)
total PRB number of BS n
\(P_{n}^{\max }\)
transmission power budget of BS n
\(R_{{k,n}}^{{m,v}}\)
data rate on PRB k in VO#v of BS n
\(g_{{k,n}}^{{m,v}}\)
user channel condition on PRB k in VO#v of BS n
\(a_{{k,n}}^{{m,v}}\)
PRB allocation indicator
\(p_{{k,n}}^{{m,v}}\)
transmit power on PRB k in VO#v of BS n
P _{ cn }
circuit power of BS n
η
power amplifier inefficiency
R _{tot}
total data rate
P _{tot}
total power assumption

Consider a heavily loaded hotspot area where the available resource is limited to meet the total traffic demand.
2.1 Virtualized air interface model
As in Fig. 1, conventional LTE eNodeB functionalities are split into two components: signaling controller and data BS. Data BS belongs to the dataforwarding plane, connected through gateway (GW) into the Internet. Its main task is to perform baseband processing and realize physical transmission tasks defined by the control plane. The main advantage of such function split is to improve network flexibility and energy efficiency. Energyconsuming data BSs are flexibly deployed and can be activated/deactivated based on different traffic conditions.
Signaling controller hosts control logic of RAN. Its main task is resource management on the air interface. For example, signaling controller can regulate data BSs activation/deactivation through interface X3 [15]. In night time, about 85 % traffic reduction is predicted compared with peakload time [16]. Therefore, in order to optimize network EE, redundant data BSs should be shut down. Also, sleeping BSs can be activated by signaling controller during busy hour.
In addition, signaling controller may also support part of the mobility management functions: when mobile users cross the coverage boundaries of data BSs or signaling controllers, handovers can be realized through the cooperation between signaling controllers and MME (linked via interface S1).
2.2 InPVO model
In Fig. 1, a hotspot area is serviced by several colocated data BSs. Consider one InP and multiple VOs, i.e., VO#1 and VO#2. VOs have distinct service objectives reflecting particular performance targets and constraints.
The task of VO is to ensure a satisfactory quality of experience (QoE) for their users. Since VOs have no direct access to physical network infrastructure, they may wish to buy wireless resource from the InP. Hence, VOs are required to decide how much radio resource to purchase from the InP. After obtaining the corresponding resource, VOs implement their own buffering strategy to satisfy users’ needs (e.g., which user payloads to drop or which payloads to pass to the radio link layer). Therefore, VOs should be equipped with isolated memory space to carry out their own computation task without interfering with each other.
InP undertakes two tasks: (1) supervising the whole RAN and (2) supporting ondemand capacity requests from different VOs. On one hand, InP gathers VOs’ user traffic information through the interface with different VOs. Based on the information, InP analyzes the global user traffic behavior according to a predefined contract or protocol. On the other hand, InP defines the wireless scheduling via management interface towards the signaling controller. The signaling controller, together with data BSs, perform resource allocation and physical transmission.
2.3 User behavior curve
A user behavior model for largescale cellular users is introduced in [17–20]. The concept is based on Gini coefficient, an economic measure of statistical dispersion to evaluate the income distribution of a nation’s residents [21]. It is an efficient index for assessment on regional income or wealth inequality. In this article, we extend the user behavior to more general cases and adopt it as a reference to describe VOs’ traffic volume share in the overall system.
where x _{0}=y _{0}=0.

First, consider the case where users of each VO generate equal traffic rate demand, i.e., d _{1}=d _{2}=⋯=d _{ V }. Thus, Eq. (4) can be transformed as:$$ B = \frac{{2{\sum\nolimits}_{v = 1}^{V} {{\sum\nolimits}_{i = 0}^{v  1} {{M_{i}}{M_{v}}}} + {\sum\nolimits}_{v = 1}^{V} {{M_{v}^{2}}} }}{{2{{\left({{\sum\nolimits}_{i = 0}^{V} {{M_{i}}}} \right)}^{2}}}} $$(6)According to multinomial theorem [22], we have:$${} \begin{aligned} {\left({{\sum\nolimits}_{i = 0}^{V} {{M_{i}}}} \right)^{2}} &= {\sum\nolimits}_{i = 0}^{V} {{M_{i}^{2}}} + 2{\sum\nolimits}_{j = 1}^{V} {{\sum\nolimits}_{i = 0}^{j  1} {{M_{i}}{M_{j}}}} \\ &= {m_{0}^{2}} + {\sum\nolimits}_{i = 1}^{V} {{M_{i}^{2}}} + 2{\sum\nolimits}_{j = 1}^{V} {{\sum\nolimits}_{i = 0}^{j  1} {{M_{i}}{M_{j}}}} \\ &= {\sum\nolimits}_{v = 1}^{V} {{M_{v}^{2}}} + 2{\sum\nolimits}_{v = 1}^{V} {{\sum\nolimits}_{i = 0}^{v  1} {{M_{i}}{M_{v}}} } \end{aligned} $$(7)
Substituting (7) into (5) and (6), we can obtain \(B=\frac {1}{2},h=0\), which means the traffic demands are equivalent over the whole service area across various VOs (equilibrium). This way, the user behavior is completely equality and user behavior curve l _{1} converges to curve l _{0} in Fig. 2.

In the second case, the traffic amounts demanded by multiple VOs are exceedingly disequilibrium. This may correspond to the situation that all the traffic demand are generated by one dominated VO while other VOs generate few amounts of traffic (called as “oligopoly” in economics). In this situation, we have d _{ i }→0,∀i=1,2⋯,V−1. So by observing Eq. (4) we find: \({\sum \nolimits }_{v = 1}^{V} {{\sum \nolimits }_{i = 0}^{v  1} {{M_{i}}{M_{v}}}} \to 0,{\sum \nolimits }_{v = 1}^{V} {{M_{v}^{2}}{d_{v}}} \to {M_{V}^{2}}{d_{V}}\) and \(\left ({{\sum \nolimits }_{i = 0}^{V} {{M_{i}}{d_{i}}}} \right)\left ({{\sum \nolimits }_{i = 0}^{V} {{M_{i}}}} \right) \to {M_{V}}{d_{V}}{\sum \nolimits }_{i = 0}^{V} {{M_{i}}}\). Finally, we have: \(B \to \frac {{{M_{V}}}}{{2{\sum \nolimits }_{i = 0}^{V} {{M_{i}}} }},h \to 1  \frac {{{M_{V}}}}{{{\sum \nolimits }_{i = 0}^{V} {{M_{i}}} }}\). This way, the user behavior is completely inequality and user behavior c urve l _{1} converges to curve l _{2} in Fig. 2. To take a step further, assuming that the user number of VO#V is much less than the total number, i.e., \({M_{V}} < < {\sum \nolimits }_{i = 1}^{V} {{M_{i}}}\), so \(\frac {{{M_{V}}}}{{{\sum \nolimits }_{i = 0}^{V} {{M_{i}}} }} \to 0\). In such condition curve, l _{2} approaches to curve l _{3} and B→0,h→1.
In conclusion, the value of user behavior coefficient generally lies in the interval [0,1]. In practice, however, both extreme values are barely reached. On the one hand, a low user behavior coefficient indicates the traffic demand share across all the VOs follows rather even distribution, with 0 corresponding to complete equilibrium. On the other hand, high user behavior coefficient indicates the traffic demand share of different VOs follows uneven distribution, with one corresponding to complete convergence (i.e., the total traffic demands are requested by few users of the dominated VO).
3 Problem formulation
where A with element \(a_{{k,n}}^{{m,v}}\) and P with element \(p_{{k,n}}^{{m,v}}\) are the PRB allocation vector and the power allocation vector, respectively. C1–C3 are PRB allocation constraints. C2 means that one PRB can be only allocated to one user at most. C3 is used to guarantee the minimum bandwidth (PRB number) requirement of each VO. C4–C5 are the power allocation constraints, and \(P{}_{n}^{\max }\) is the maximum transmit power of BS n.
The resource allocation problem in (10) is a mixed combinatorial and nonconvex optimization problem. The combinatorial nature comes from the PRB allocation constraints C1 and C2. The nonconvexity feature is caused by the fractional form of the objective function. Besides, the variables are mixed integer. Hence, the problem is very difficult to solve. Therefore, we decouple the resource allocation problem into two subproblems: PRB (bandwidth) allocation for a given power allocation and power allocation for a given PRB allocation.
4 Resource allocation strategy
In this section, we first devise a heuristic PRB allocation algorithm using bankruptcy game and Shapley value. Second, power allocation problem is solved by fractional programming.
4.1 PRB allocation algorithm
The problem is a nonlinear integer programming under constraints. Conventional solutions include branchandbound method and intelligent algorithm. For both methods, the computation complexity is rather high, especially when multiple constraints exist. In addition, one major concern of the problem is to guarantee allocation fairness, which is neglected in conventional solutions. Because the rate demands differentiate from user to user, allocation according to the demands efficiently optimizes the resource utilization.
In our work, bankruptcy game is introduced to solve the problem. VOs are modeled as bankrupt players, and total PRBs are modeled as overall estate. The reasons are as follows. First, bankruptcy defines a model where the estate is insufficient to satisfy the demand of all players [24]. Since the PRBs owned by the InP are limited, the total traffic demand should exceed the rate that the PRBs can provide (according to assumption 2). Second, as a cooperative game, bankruptcy game enables VOs to cooperatively share the PRB resource. Third, to guarantee the allocation fairness, Shapley value is adopted as a solution to the bankruptcy game [25].
In order to maintain the virtual network operated, each VO is likely to reserve a minimum number of PRBs b _{ v }, where \({\sum \nolimits }_{v = 1}^{V} {{b_{v}}} \le {\sum \nolimits }_{n = 1}^{N} {K{}_{n}^{\max }} \). In addition, each VO should claim additional PRBs c _{ v } to satisfy its customers’ traffic needs as much as possible. This establishes competition among different VOs. Therefore, to guarantee fairness, the allocation strategy should be centrally handled by the InP based on the instantaneous traffic flow demand reported from VOs.
where symbol [ x] denotes rounding to the nearest integer of x. b _{ v } is the the minimum PRB number to maintain a certain VO network operated and Φ _{ v } is the additional allocated PRB number calculated based on VO’s traffic demand. (15b) means that the total allocated PRB number should meet the maximum PRB number constraint.
The PRB allocation algorithm is presented in Algorithm ??. In each iteration, PRB is allocated to VO user with the aim of maximizing energy efficiency. Operator filter() is defined as: \(\text {filter}({\mathcal {M}_{v}},K_{v})=\emptyset \) if and only if K _{ v }=0; otherwise, \(\text {filter}({\mathcal {M}_{v}},K_{v})={\mathcal {M}_{v}}\). filter() is used to guarantee VO v quitting the allocation when its allocated PRB number exceeds K _{ v }.
4.2 Power allocation
Assuming the PRB allocation is known, the original problem (10) is decoupled to a power allocation problem. To eliminate the nonconvexity of the power allocation problem, we transform it into an equivalent convex optimization. And the optimal power allocation is obtained by Lagrange dual approach.
4.2.1 Problem transformation
where (16) is nonconvex due to the fractional form of the objective function. To make the problem more tractable, several transformations are required to eliminate the nonconvexity. Hence, let \(q = \frac {{{R_{\text {tot}}}}}{{{P_{\text {tot}}}}}\), then we have the following theorem [26].
Theorem 1.
where it is easy to verify that the objective is a concave function with respect to \(p_{k}^{(v,n)}\). Hence, we can first solve (17) and then solve (16) by adopting iterative algorithms as given in Algorithm ??.
Algorithm ?? iteratively approximates the optimal solution of (17) towards a minimum threshold ε (ε>0) and finally achieves optimal EE q with the termination condition satisfied. We give the convergence proof of Algorithm ?? in Appendix 2. The algorithm converges to the optimal EE with a superlinear convergence rate [27].
4.2.2 Power allocation for transformed problem
Therefore, we can solve (19) by decomposing it into two layers. Layer 1: solve the maximization Lagrange problem with respect to \(p_{{k,n}}^{{m,v}}\) under a fix set of Lagrange multipliers. Layer 2: obtain λ _{ n } by minimizing the Lagrange problem.
where \(A_{{k,n}}^{{m,v}} = \frac {{ \sigma ^{2} }}{{g_{{k,n}}^{{m,v}}}}\).
5 Simulation and analysis
Simulation configuration
Parameter  Configuration 

Simulation area  Circular, R=750 [m] 
Coordinate of circle center (0, 0)  
Data BS configuration  3 data BS, 10 MHz bandwidth each 
(corresponds to about 50 PRBs)  
Data BS coordinates  (0,10) (−5,8.66) (5,8.66) 
Data BS power  \({P_{n}^{\max }}=43~\)dBm, P _{circuit}= 25 dBm 
Path loss model  128.1+37.6log10(d) dB, d in km [29] 
Slow fading model  Lognormal with zero mean value 
Standard deviation = 8 dB  
Correlation distance = 50 m  
Fast fading model  Jakes model 
Modulation schemes  QPSK, 16 QAM, 64 QAM 
Link2System interface  Effective exponential SINR mapping [30] 
Video traffic bit rate  242 kbps [13] 
VoIP traffic bit rate  8.4 kbps [13] 
Simulation case
Virtual operators configuration:  VO#1: 30 video users 
Case 1  VO#2: 30 video users 
VO#1: 30 video users  
Virtual operators configuration:  VO#1: 40 VoIP users 
Case 2  VO#2: 20 video users 
VO#2: 30 video+VoIP users  
Virtual operators configuration:  VO#1: 20 VoIP users 
Case 3  VO#2: 50 VoIP users 
VO#3: 20 video users 
In summary, the above three simulation cases reflect different scenarios with different VOs’ service objectives and constraints. Case 1 exhibits ideally identical traffic rate while in case 2 VOs undertake disequilibrium traffic requirements. Case 3 further increases the traffic rate disequilibrium, where the dominated VO#3 generate more than 90 % of the total rate. Correspondingly, user behavior coefficient h takes 0, 0.42, and 0.67. And different h values can effectively reflect different VO traffic condition. Through Figs. 3, 4 and 5, it is observed that the proposed PRB allocation enables multiple VOs to dynamically share the PRBs according to different traffic conditions. By contrast, the static PRB management in legacy LTE network is rather ossified.
From the above analysis, we can conclude that VO#1 is “economical” because of its light traffic burden. By contrast, “greedy” VO#3 claims highrate services and consumes a large amount of resources. While VO#2 is rather “elastic,” for the reason that it expands or shrinks its service level by dynamically obtaining access to mobile resource from the shared pool. This is the socalled flexibility provided by the virtualization, which greatly improves the system efficiency.
6 Conclusions
In this paper, a novel LTE air interface virtualization architecture is proposed based on user behavior analysis. An EE resource allocation problem is studied to support dynamic resource sharing among multiple VOs. We formulate the problem as a mixed combinatorial and nonconvex optimization problem. To reduce the computational complexity, the problem is decoupled into two steps. First, to guarantee fairness, bankruptcy game and Shapley value are introduced to develop a heuristic bandwidth allocation problem. Second, fractional programming and convex optimization are adopted to obtain the optimal power allocation.
In simulation, PRBs are dynamically allocated to multiple VOs based on different user traffic demands. The defined user behavior can effectively reflect the VO traffic condition. The flexibility is proved in the sense that rateconsuming VOs can get more PRB resource while a minimum number of PRBs are reserved for “economical” VOs to maintain the network operated. Besides, energy efficiency is improved in comparison with conventional networks.
7 Appendix 1
7.1 Proof of Theorem 1
From (a), we know that max{R _{tot}(P)−q ^{∗} P _{tot}(P)}=0. From (b), we know that the maximum is taken on at point P ^{∗}. Thus, the first part of our proof is finished.
From (a), we have q ^{∗}≥R _{tot}(P)/P _{tot}(P), i.e., q ^{∗} is the maximum value of problem (16). From (b), we have q ^{∗}=R _{tot}(P ^{∗})/P _{tot}(P ^{∗}), i.e., P ^{∗} is the solution vector of problem (16). Thus, the equivalence of the two problems defined in Theorem 1 is proved.
8 Appendix 2
8.1 Proof of Algorithm 2 convergence
Note that the problem equivalent to (17) is \(\mathrm {F} (q) = \mathop {\max }\limits _{\mathbf {P}} \{{R_{\text {tot}}}({\mathbf {P}})  q{P_{\text {tot}}}({\mathbf {P}})\} \), as defined in Theorem 1. And the proof can be decoupled into three steps as follows:
8.1.1 Step 1
F(q) is a nonnegative function in the definition domain.
Proof.
8.1.2 Step 2
F(q) is a strictly monotonic decreasing function with respect to q.
Proof.
As for any arbitrary pair q _{1}<q _{2}, we have F(q _{1})>F(q _{2}).
8.1.3 Step 3
q _{ l }>q _{ l+1},∀l=1,2,⋯ in Algorithm ??.
Proof.
Since R _{tot}(P _{ l })>0, we have q _{ l+1}>q _{ l }.
Based on steps 1–2, we know that F(q) is a strictly monotonic decreasing function with nonnegative value. According to step 3, q _{ l } increases as iteration number l accumulates, i.e., q _{ l+1}>q _{ l }. Therefore, as long as the iteration time is large enough, this will eventually approach to zero and satisfy the optimality condition of Theorem 1, i.e., l→∞,F(q _{ l })→0.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 61471060) and Funds for Creative Research Groups of China (No. 61421061).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Y Xu, J Wang, Q Wu, Z Du, L Shen, A Anpalagan, A gametheoretic perspective on selforganizing optimization for cognitive small cells. Commun Mag. IEEE. 53(7), 100–108 (2015).View ArticleGoogle Scholar
 Cisco, Global Mobile Data Traffic Forecast Update 2010–2015 White Paper (2011).Google Scholar
 H Wen, PK Tiwary, T LeNgoc, Wireless virtualization (SpringerVerlag, New York, NY, USA, 2013).View ArticleGoogle Scholar
 L Chengchao, FR Yu, Wireless network virtualization: a survey, some research issues and challenges. Commun. Surv. Tutorials IEEE. 17:, 358–380 (2015).View ArticleGoogle Scholar
 A Gudipati, D Perry, LE Li, S Katti, in Proc. ACM SIGCOMM Workshop HotSDN. Softran: Software defined radio access network (ACMHong Kong, China, 2013), pp. 25–30.Google Scholar
 LE Li, ZM Mao, J Rexford, Cellsdn: softwaredefined cellular networks, Princeton Univ.Press, Princeton, NJ, USA. Tech. Rep. TR 92212 Computer Science (2012).Google Scholar
 M Yang, et al, in Proc. ACM SIGCOMM. OpenRAN: A softwaredefined RAN architecture via virtualization (Hong Kong, China, 2013), pp. 549–550.Google Scholar
 Y Zaki, L Zhao, C Goerg, A TimmGiel, LTE mobile network virtualization. Mobile Netw Appl. 16(4), 424–432 (2011).View ArticleGoogle Scholar
 L Zhao, M Li, Y Zaki, A TimmGiel, C Gorg, LTE virtualization: from theoretical gain to practical solution, (San Francisco, CA, USA, 2011).Google Scholar
 Y Zaki, LTE optimization and mobile network virtualization, (SpringerVerlag, New York, NY, USA, 2012).Google Scholar
 X CostaPerez, J Swetina, G Tao, R Mahindra, S Rangarajan, Radio access network virtualization for future mobile carrier networks. Commun Mag. IEEE. 51:, 27–35 (2013).View ArticleGoogle Scholar
 F Fangwen, UC Kozat, Stochastic game for wireless network virtualization. Netw IEEE/ACM Trans. 21:, 84–97 (2013).View ArticleGoogle Scholar
 L Bin, T Hui, A bankruptcy gamebased resource allocation approach among virtual mobile operators. IEEE Commun. Lett. 17(7), 1420–1423 (2013).View ArticleGoogle Scholar
 F Bo, T Hui, L Bin, in IEEE Proceedings of WCNC 2015 IEEE Wireless Communications and Networking Conference. Game theory based power allocation in LTE air interface virtualization (IEEENew Orleans, LA USA, 2015), pp. 972–976.Google Scholar
 Z Zaidi, V Friderikos, MA Imran, Future RAN architecture: SDRAN through a generalpurpose processing platform. Vehicular Technol Mag. IEEE. 10:, 52–60 (2015).View ArticleGoogle Scholar
 A Capone, I Filippini, B Gloss, U Barth, in IEEE Proceedings of the Second IFIP Conference on Sustainable Internet and ICT for Sustainability. Rethinking cellular system architecture for breaking current energy efficiency limits (IEEEPisa, Italy, 2012), pp. 1–5.Google Scholar
 A Osseiran, F Boccardi, V Braun, K Kusume, P Marsch, M Maternia, et al, Scenarios for 5G mobile and wireless communications: the vision of the METIS project. Commun. Mag. IEEE. 52:, 26–35 (2014).View ArticleGoogle Scholar
 Z Xing, Y Rong, Z Yan, G Yue, M Im, L Cuthbert, et al, Energyefficient multimedia transmissions through base station cooperation over heterogeneous cellular networks exploiting user behavior. Wireless Commun. IEEE. 21:, 54–61 (2014).Google Scholar
 Z Xing, Z Yan, Y Rong, W Wenbo, M Guizani, Enhancing spectralenergy efficiency for LTEadvanced heterogeneous networks: a users social pattern perspective. Wireless Commun. IEEE. 21:, 10–17 (2014).View ArticleGoogle Scholar
 H Yu, Z Xing, Z Jiaxin, T Jian, S Zhuowen, W Wenbo, Energyefficient design in heterogeneous cellular networks based on largescale user behavior constraints. Wireless Commun. IEEE Trans. 13:, 4746–4757 (2014).View ArticleGoogle Scholar
 C Gini, “Italian: Variabilità e mutabilità” ‘Variability and Mutability’, C. Cuppini, Bologna, 156 pages. Reprinted in Memorie di metodologica statistica. (E Pizetti, T Salvemini, eds.), (Rome: Libreria Eredi Virgilio Veschi (1955), 1912). https://en.wikipedia.org/wiki/Gini_coefficient.
 M Hazewinkel (ed.), Multinomial coefficient, Encyclopedia of Mathematics (Springer, 2001). http://en.wikipedia.org/wiki/Multinomialtheorem.
 Y Chen, S Zhang, S Xu, G Li, Fundamental tradeoffs on green wireless networks. IEEE Commun. Mag. 49(6), 30–37 (2011).View ArticleGoogle Scholar
 M Pulido, JS Soriano, N Llorca, Game theory techniques for university management: an extended bankruptcy model. Oper Res. 109:, 129–142 (2002).MathSciNetMATHGoogle Scholar
 C Georgios, E Edith, W Michael, Cooperative game theory: basic concepts and computational challenges. IEEE Intell. Syst. 27(3), 86–90 (2012).View ArticleGoogle Scholar
 W Dinkelbach, On nonlinear fractional programming. Manage Sci. 13(7), 492–498 (1967).MathSciNetView ArticleMATHGoogle Scholar
 C Isheden, Z Chong, EA Jorswieck, G Fettweis, Framework for linklevel energy efficiency optimization with informed transmitter. IEEE Trans. Wireless Commun. 11(8), 2946–2957 (2012).Google Scholar
 S Boyd, L Vandenberghe, Convex optimization (Cambridge Univ. Press, Cambridge, UK, 2004).View ArticleMATHGoogle Scholar
 G Piro, L Grieco, G Boggia, F Capozzi, P Camarda, Simulating LTE cellular system: an open source framework. IEEE Trans, Veh. Technol. 60(2), 498–513 (2010).View ArticleGoogle Scholar
 E Westman, Calibration and evaluation of the exponential effective SINR mapping (EESM) in 802.16. Master Thesis, Stockholm, Sweden (2006).Google Scholar