Distributed Resource Allocation in Full-duplex Relaying ... - IEEE Xplore

Globecom 2014 - Wireless Networking Symposium

Distributed Resource Allocation in Full-duplex Relaying Networks with Wireless Virtualization ∗ Key

Gang Liu∗‡ , F. Richard Yu† , Hong Ji∗ , and Victor C.M. Leung‡ Lab. of Universal Wireless Comm., Ministry of Education, Beijing University of Posts and Telecom., P.R. China † Depart. of Systems and Computer Eng., Carleton University, Ottawa, ON, Canada ‡ Depart. of Electrical and Computer Eng., The University of British Columbia, Vancouver, BC, Canada E-mail: [email protected]; richard [email protected]; [email protected]; [email protected];

Abstract—Wireless network virtualization has attracted a lot of interests recently. Furthermore, recent advances of selfinterference cancellation techniques enable full-duplex relaying (FDR) systems, which transmit and receive simultaneously in the same frequency band with high spectrum efficiency. In this paper, we propose a resource allocation scheme for virtualized FDR networks. The resource allocation problem is formulated as a mixed combinatorial and non-convex optimization problem, which involves high computational complexity. Therefore, we transfer the original problem to a convex optimization problem, which can be solved efficiently. In addition, with recent advances in alternating direction method of multipliers (ADMM), we develop an efficient ADMM-based distributed resource allocation algorithm. Extensive simulations are presented to verify the effectiveness of the proposed scheme. Index Terms—Wireless network virtualization, full-duplex relaying, resource allocation, ADMM.

I. I NTRODUCTION AND M OTIVATIONS Recently, network virtualization has attracted a lot of interests. The basic idea for network virtualization [1] is to split the role of the traditional internet service providers (ISPs) into two entities: infrastructure providers (InPs), which create and manage the physical infrastructure, and service providers (SPs), which provide various services to end users by aggregating the virtualized resources from multiple InPs. With the ever-increasing wireless traffic and services, it is natural to extend virtualization from wired networks to wireless networks [1]–[3]. Several works have been done for wireless network virtualization. A research report from GENI [4] described some proposals and issues of wireless network virtualization. Hoffmann et al. discussed the generalized control and management frameworks in [5]. The authors of [2] investigated virtual resource sharing mechanisms, where the dynamic interactions among SPs and InPs are modeled as a stochastic game. Zaki et al. [6] proposed a virtualization framework for long term evolution (LTE) systems, in which a hypervisor is used to virtualize the eNB and manage the physical resources. Scheduling schemes for virtualized WiMAX networks were studied in [7]. For WLANs virtualization, a SplitAP architecture and a resource sharing algorithm were proposed in [8]. Although some excellent works have been done for wireless network virtualization, most existing works do not consider relaying. However, relaying has been regarded as one of the key components of the future wireless networks. Furthermore,

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recent advances of self-interference cancellation techniques [9], [10] enable full-duplex relaying (FDR) systems, which transmit and receive simultaneously in the same frequency band with high spectrum efficiency. While FDR can improve the performance significantly, resource allocation is a complicated problem in FDR networks since the residual self-interference couples the transmission and reception of FDR [9], [11]. Nevertheless, resource allocation plays a very important role in energy efficiency, spectrum efficiency and quality of service (QoS) provisioning in wireless networks [12], [13]. To the best of our knowledge, resource allocation for virtualized FDR networks has not been well studied. In this paper, we propose a resource allocation scheme for virtualized FDR networks, where multiple InPs and multiple SPs coexist. In the proposed scheme, the radio spectrum, base stations (BSs) and full-duplex relay stations (RSs) are virtualized as virtual resources, which can be dynamically allocated to different users of SPs. We formulate the resource allocation as an optimization problem, which maximizes the total utility, considering not only the revenue earned by serving users but also the cost of power. The formulated problem is a mixed combinatorial and non-convex optimization problem, which involves high computational complexity. Therefore, we transfer the original problem to a convex problem, which can be solved efficiently. With recent advances in alternating direction method of multipliers (ADMM) [14], we develop an efficient ADMM-based distributed resource allocation algorithm. Extensive simulations show that the proposed scheme can substantially improve the performance of virtualized FDR networks. In addition, the InPs, SPs and users can benefit from the proposed virtual resource allocation scheme. Bold-face lower-case letters and bold-face upper-case letters are used for vectors and matrices. Let (·)T , (·)H and (·)−1 denote transpose, conjugate transpose and inverse of a matrix respectively. diag{·} is the diagonal element vector of a square matrix. E{·} denotes expectation operator. Cm×n and IK stand for space of m × n matrix and K × K identity matrix. N is the set of natural number and Rp×q denotes p × q real matrix. II. S YSTEM D ESCRIPTION The virtualized FDR network with multiple InPs and multiple SPs is shown in Fig. 1. We assume there are M InPs. Each InP manages a cellular network with one BS and several fullduplex RSs. There are I SPs, which provide various services

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to their subscribers through the same substrate network. Based on the virtualization frameworks in [1], [2], a virtual resource manager (VRM) is introduced to abstract the virtual physical resources (e.g., BSs, RSs, subcarriers, power, etc.) of different InPs and it dynamically allocates the virtual resources to users from different SPs. There are K subcarriers with bandwidth W , M BSs and L RSs. Each BS, RS and user are equipped with N antennas. The set of users for SP i is denoted as Ui . Then, let U = ∪Ii=1 Ui be the set of all the users and |U | be the total number of users. 1) One-hop Users: To maximize the information rate between the BSs and one-hop users, the optimal precoding and post-processing matrices in [15] are adopted. Assume m ,R0 HB ∈ CN ×N is the channel matrix of one-hop user u Tu ,k m ,R0 from BS m on subcarrier k and nB ∈ CN ×1 is additive Tu ,k Gaussian noise vector at the receiver of one-hop user u with Bm ,R0 H m ,R0 covariance matrix Rnn = E{nB Tu ,k (nTu ,k ) } = IN . Let k,1 Bm ,R0 Bm ,R0 Bm ,R0 T Λi = [λTu ,k,1 , λTu ,k,2 , ..., λTu ,k,N ] be the eigenvalue −1 Bm ,R0 m ,R0 H vector of (HB Tu ,k ) Rnn HTu ,k . According to [15], the transmission rate of one-hop user u on  subcarrier k via BS N Bm ,R0 Bm ,R0 m ,R0 m is n=1 W log2 1 + pTu ,k,n λTu ,k,n , where pB Tu ,k,n is the allocated transmission power for user u on subcarrier k via BS m in spatial channel n. Thus, the total rate for onehop user u on all subcarriers at all the BSs is expressed as R1u =

K  M  N  m=1 k=1 n=1

  Bm ,R0 Bm ,R0 (1) am,0 u,k W log2 1 + pTu ,k,n λTu ,k,n

am,0 u,k

is the subcarrier allocation indicator. If subcarrier where k is allocated to user u via BS m, then am,0 u,k = 1; otherwise, Bm ,R0 Bm ,R0 am,0 = 0. Note that we use index 0 in H u,k Tu ,k , pTu ,k,n and m,0 au,k to indicate that one-hop user is served by BSs directly. 2) Two-hop Users: Two-hop users access to BSs via the help of RSs. The decode and forward full-duplex relaying scheme [16] is used at the RSs. Linear MMSE receivers are implemented at RSs and end users, and the optimal linear precoding matrices [10] are adopted at the BSs and RSs to diagonalize the BS-RS-User channels on each subm ,Rl l ,Tu carrier. Let HB ∈ CN ×N and HR ∈ CN ×N be k k channel matrices from BS m to RS l and that from RS l to user u on subcarrier k, respectively. By singular value m ,Rl decomposition (SVD), they can be rewritten as HB = k Bm ,Rl Bm ,Rl Bm ,Rl Rl ,Tu Rl ,Tu Rl ,Tu Rl ,Tu Λk Vk and Hk = Uk Λk Vk . Uk Hence, the equivalent signal-to-noise ratio (SNR) for each spatial channel n on subcarrier k of the BS m-to-RS l link and the RS l-to-user u link can be obtained from the diagonal Rl ,Tu m ,Rl matrices ΛB and with diagonal element vectors k  Λk  m ,Rl m ,Rl m ,Rl m ,Rl T ) = [ λB , λB , ..., λB ] and diag(ΛB k k,2 k,N  k,1   Rl ,Tu Rl ,Tu Rl ,Tu Rl ,Tu T ) = [ λk,1 , λk,2 , ..., λk,N ] , respecdiag(Λk tively. According to [10], the capacity between NBS m and user u via RS l on subcarrier k can be given as n=1 W log2 (1 +

min{

Bm ,Rl Bm ,Rl pk,n λk,n Rl Rl ,Tu δk,n pk,n +1

Fig. 1.

The architecture of the virtualized full-duplex relaying networks.

rate for a given two-hop user u is given as N M  L  K   m,l 2 Ru = au,k W log2 (1+

m=1 l=1 k=1 n=1  m ,Rl Bm ,Rl λ pB k,n k,n Rl ,Tu Rl ,Tu , pk,n λk,n }) min{ R Rl ,Tu l δk,n pk,n + 1

(2)

where am,l u,k is resource allocation indicator. When user u is served by BS m via RS l on subcarrier k, am,l u,k = 1; otherwise, Rl am,l = 0. δ is the residual self-interference gain on spatial u,k k,n channel n of subcarrier k at RS l and different self-interference Rl cancellation technologies may result in different δk,n . Moreover, the residual self-interference couples the links from BSs to RSs and the links from RSs to two-hop users, which makes the resource allocation in FDR networks more complicated and different from that in half-duplex relay networks. III. P ROBLEM F ORMULATION In our virtualized FDR network, each user can either be served by any BS directly or indirectly with the help of any RS. Hence, for a given user u, the utility function can be defined as its total service rate, which is given as R(u) = Ru1 + Ru2 , where Ru1 and Ru1 are defined in (1) and (2) respectively. This utility function for users is widely used in existing literature (e.g., [10]). For SP i with user set Ui , the weighted sum of user utility is used as its  utility function, which can be expressed as FSP (Ui ) = βi u∈Ui ωu R(u), where βi is the price for SP i charged by the VRM, ωu is a positive weight for user u specified by each SP based on different scheduling criteria. This weighted sum utility function provides SPs with certain flexibility to design their scheduling policy. For the VRM, we consider an energy-aware utility function, which is defined as the payoff of all the InPs earned by serving the SPs taking into account of the energy consumption cost. M I  L    FV RM (U ) = βi ωu R(u)−c pBm + p Rl

Bm ,Rl l ,Tu Rl ,Tu l ,Tu , pR and pR k,n λk,n }), where pk,n k,n

are the allocated power at BS m and RS l on the spatial channel n of subcarrier k, respectively. Then the transmission

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m=1

i=1 u∈Ui

+

M L K I     i=1 u∈Ui m=1 l=1 k=1

+

N  m,l

au,k

l=1

m ,Rl l ,Tu (pB + pR k,n k,n )

n=1

K M   m=1 k=1

am,0 u,k

N  n=1

 m ,R0 pB Tu ,k,n

(3)

Globecom 2014 - Wireless Networking Symposium

where the first term is the total amount of money that the SPs pay to the InPs, the second term is the total cost for energy consumption of all the InPs, c is the price of per unit energy consumption, pBm and pRl are the circuit energy consumption of BS m and RS l, respectively. In each scheduling cycle, the VRM needs to dynamically allocate the virtual resources to the SPs. The objective of the resource allocation problem is to maximize the utility (3) under the constraints in the following. Firstly, the transmission power on each subcarrier should be kept under certain level to put a limit on the amount of out-of-cell interference. Also, the transmission power is also limited by the of power Hence, we Idynamics the  L amplifiers. M m,l N Bm ,Rl have C1 : [ a (p + i=1 m=1 l=1 u,k n=1 k,n u∈Ui   M N Rl ,Tu m,0 Bm ,R0 pk,n ) + m=1 au,k n=1 pTu ,k,n ] ≤ PT , ∀k, where PT is the maximum power on each subcarrier. Secondly, similar to [10], [17], in order to guarantee the QoS, the SPs will provide each user with a minimum transmission rate based on its service state (e.g., queue length). Let Rumin be the minimum rate threshold for user u, then its transmission rate satisfies C2 : R(u) ≥ Rumin , ∀u. Thirdly, to avoid intra-cell interference, each subcarrier can only be used by one user. Furthermore, for simplicity, each user can only be served either by one BS or by one BS and one RS on one subcarrier. Therefore, resource U the M L allocation m,l indicator should satisfy C3 : u=1 m=1 l=0 au,k ≤ K M L m,l m,l 1, ∀k; C4 : k=1 m=1 l=0 au,k ≤ 1, ∀u; C5 : au,k ∈ {0, 1}, ∀m, l, u, k. Then, the resource allocation problem can be formulated as arg

B

,R

max

B

m,l m m 0 {am,0 u,k ,pTu ,k,n ,au,k ,pk,n

,Rl

R ,Tu

l ,pk,n

}

FV RM (U )

The formulated problem jointly optimizes the access point selection, subcarrier and transmission power allocation. However, this makes it difficult to solve, since it is a mixed combinatorial and non-convex problem. The combinatorial nature comes from the integer constraints C3, C4 and C5. The non-convexity is caused by the objective function and constraint C2, because the residual self-interference in (2) couples the links from BSs to RSs and the links from RSs to two-hop users. IV. D ISTRIBUTED R ESOURCE A LLOCATION VIA ADMM A. Problem Transformation To decouple the links from BSs to RSs and the links from m ,Rl RSs to two-hop users, we introduce a new variable pB Tu ,k,n Bm ,Rl Bm ,Rl Rl ,Tu such that pTu ,k,n = pk,n + pk,n , which represents the total transmission power on the spatial channel n of subcarrier k for two-hop user u via BS m and RS l. Then, we have: Theorem 1: When the utility function in (3) is maximized, Rl Rl ,Tu m ,Rl Bm ,Rl l ,Tu Rl ,Tu we have pB λk,n /(δk,n pk,n + 1) = pR k,n k,n λk,n . Lemma 1: When the utility function (3) is maximized, the transmission rate Ru2 for two-hop user u can be rewritten as N L  M  K   l ,Tu Rl ,Tu ˜ u2 = R am,l W log2 (1 + pR u,k k,n λk,n ) (5) n=1

l ,Tu pR k,n

=

 B ,R B ,R B ,R B ,R (λTum,k,nl )2 +4ΨTum,k,nl pTum,k,nl −λTum,k,nl R

R ,T

,

l λ l u 2δk,n k,n Bm ,Rl Rl Bm ,Rl Rl ,Tu ΨTu ,k,n = δk,n λk,n λk,n .

Bm ,Rl m ,Rl l ,Tu +λR and λB Tu ,k,n = λk,n k,n With Theorem 1 and Lemma 1, the optimization problem (4) can be transformed to a concave problem with respect m ,Rl to (w.r.t) power allocation variables pB Tu ,k,n , l ∈ {0, 1, ..., L} Bm ,Rl Bm ,Rl l ,Tu by substituting pTu ,k,n = pk,n + pR k,n , (5) and l ,Tu pR k,n

=

 B ,R B ,R B ,R B ,R (λTum,k,nl )2 +4ΨTum,k,nl pTum,k,nl −λTum,k,nl R

into it.

R ,Tu

l λ l 2δk,n k,n

Unfortunately, the combinatorial nature still exists. Similar to [18], we relax am,l u,k , l ∈ {0, 1, ..., L} in C5 to be a real value between zero and one instead of a Boolean. Then, am,l u,k can be interpreted as a time sharing factor for user u on subcarrier k via BS m and RS l. It has been shown in [10] that the relaxation is optimal for the system with a large number of subcarriers. We further introduce the following m,l Bm ,Rl m ,Rl auxiliary variables p˜B Tu ,k,n  = au,k pTu ,k,n , l ∈ {0, 1, ..., L} and  B ,Rl , F˜V RM (U ) = FV RM (U ) p ˜ m B

,R

pTum,k,nl =

Tu ,k,n m,l a u,k

B

,R

B

m ,pTum,k,nl =pk,n

,Rl

R ,Tu

l +pk,n

so the transformed problem is rewritten as arg

min B

,R

{am,l ˜Tum,k,nl } u,k ,p

−F˜V RM (U )

(6)

s.t. M L N

I   M  N    B ,R  Bm ,R0 m l ˜ p˜Tu ,k,n + p˜Tu ,k,n ≤ PT , ∀k C1 : i=1 u∈Ui m=1 l=1 n=1

m=1n=1

˜ : R(u) ˜ C2 ≥ Rumin , ∀u C3, C4

(4)

s.t. C1, C2, C3, C4, C5

m=1 l=1 k=1

where

˜ : 0 ≤ am,l ≤ 1, ∀m, l, u, k C5 u,k   ˜ B ,Rl where R(u) = R(u) p ˜ m B ,R B ,R B pTum,k,nl =

Tu ,k,n m,l a u,k

m ,pTum,k,nl =pk,n

,Rl

R ,Tu

l +pk,n

.

Property 1: If the problem (6) is feasible, it is convex w.r.t m ,Rl all optimization variables {am,l ˜B u,k , p Tu ,k,n , ∀m, l, u, k, n}. B. Problem Solving via ADMM ADMM [14] is a simple but powerful algorithm that is well suited to distributed convex optimization. One of the important properties of ADMM is its quick convergence to a modest accuracy of the optimal solution. Generally, there are two basic forms for ADMM, namely unscaled form and scaled form. Since the scaled form of ADMM is more convenient than the unscaled form, the scaled form of ADMM will be used. Let x ∈ R(M (L+1)|U |K(N +1))×1 be the variable vector, which consists of one of the permutation m ,Rl of variables {am,l ˜B and z ∈ Tu ,k,n , ∀m, l, u, k, n}, u,k , p (M (L+1)|U |K(N +1))×1 be an auxiliary vector, which consists R m,l Bm ,Rl of the same permutation of {zu,k , zTu ,k,n , ∀m, l, u, k, n}. Also, we define Φs as the set of variable vectors, which satisfy constraint Cs, s = 1, 2, ..., S, where S = 5 is the number of constraints. Then, the feasible set of problem (6) can be written as Φ = ∩Ss=1 Φs . Inspired by [19], we introduce an indicator function g(z) such that g(z) = 0 when z ∈ Φ; otherwise, g(z) = +∞. With these notations, problem

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(6) of minimizing −F˜V RM (U ) on set Φ is equivalent to arg min{−F˜V RM (U ) + g(z)} x,z

(7)

a,b

s.t. x − z = 0 which is essentially a general form consensus problem with regularization [14], because we will find that the function −F˜V RM (U ) and the constraint x − z = 0 are separable across each BS m. Hence, the auxiliary variable z can be viewed as the global consensus variable, the constraint x − z = 0 is the consensus constraint, and the indicator function g(z) can be regarded as the regularization function, which will be handled by a global variable node (i.e., the VRM in our case). Since the scaled form ADMM is adopted, the augmented Lagrangian can be given as Lρ (x, z, μ) = −F˜V RM (U ) + g(z) − (ρ/2) μ 22 +(ρ/2) x − z + μ 22 , where μ is the scaled dual variable vector, and it consists of scaled Bm ,Rl m,l dual variables μm,l u,k and μTu ,k,n corresponding to au,k and m ,Rl p˜B Tu ,k,n , respectively. Based on the iterations of ADMM [14], the iterations of the considered problem is written as xt+1 := arg min −F˜V RM (U )+ x

L K I M ρ      m,l m,l t t 2 (au,k − [zu,k ] + [μm,l u,k ] ) 2 i=1 u∈Ui m=1 l=0 k=1

 N  Bm ,Rl Bm ,Rl t Bm ,Rl t 2 + (˜ pTu ,k,n − [zTu ,k,n ] + [μTu ,k,n ] )

(8)

n=1

I   M  K L   t+1 t zt+1 := arg min{ [([am,l +[μm,l u,k ] u,k ] − z∈Φ

i=1 u∈Ui m=1 l=0 k=1

N 

m,l 2 zu,k ) +

,Rl m ,Rl t+1 m ,Rl t 2 ([˜ pB −zTBum,k,n +[μB Tu ,k,n ] Tu ,k,n ] ) ]} (9)

n=1

μt+1 := μt + xt+1 − zt+1

(10)

where t stands for the iteration index. 1) x-Update: It can be found that x-update is separable across each BS m after substituting (3), (5) into (8). Therefore, x-update can be decomposed into M subproblems, which can be solved locally at each BS. Let xm be the optimization variable vector associated with BS m. More specifically, xm m ,Rl consists of am,l ˜B u,k and p Tu ,k,n with a given m. Hence, the subproblem to be solved at BS m can be given as (11). Obviously, it is an unconstraint convex problem because of the convexity of problem (6). Since it involves solving non-linear logarithmic equations to compute the closed form solution, steepest descent method [19] is adopted at each BS to search for the numerical solution in xm -update. The convergence is guaranteed by its convexity. Recall that we have relaxed the resource allocation indicator to be a real value between zero and one, hence we have to rem ,Rl ,∗ cover it to a Boolean. Assume am,l,∗ ˜B u,k and p Tu ,k,n are the optimal solutions obtained directly from steepest descent method. Inspired by [18], we first computethe marginal benefit for each ∂Lρ (xm )  m,l am,l , u,k as follows Qu,k = ∂am,l  u,k

B

where Lρ (xm ) is the objective function of problem (11). Then, the indicator can be recovered to a Boolean as follows: m,b m,b am,l,∗ = 1 if Qm,l u,k u,k = max Qu,a and Qu,a ≥ 0; otherwise,

,R

B

,R ,∗

m,l,∗ p˜Tum,k,nl =p˜Tum,k,nl ,am,l u,k =au,k

am,l,∗ u,k = 0. This is a common method to deal with the resource allocation indicators in existing literature (e.g., [10], [18]). The distributed implementation of x-update can be realized as follows. At first, the subproblem (11) is solved in parallel using only local CSI at each BS via steepest descent method to update xm in each iteration, and then the results of xm are reported to the VRM. At last, the process of x-update is completed by combining the values of xm into xt+1 at the VRM. 2) z-Update: In fact, z-update is to find the Euclidean projection [19] of xt+1 + μt onto the intersection of S closed convex sets Φs , which is a classical mathematical problem investigated in [20]. A simple iterative projection method [20] can be adopted by iteratively applying the projection operators onto the individual sets. We denote Π(r|C) as the projection of a point r onto a set C. Let zτs and θ τs , τ ∈ N, s = 1, 2, ..., S, be two sequences, which will be generated in the iterative projection procedure. Then, the iterative projection procedure can be given as zτs = Π(zτs−1 − θ sτ −1 |Φs ) and θ τs = zτs − (zτs−1 − θ τs −1 ), where τ ∈ N is the iteration index, zτ0 = zSτ −1 , z0s = xt+1 + μt and θ 0s = 0, s = 1, 2, ..., S. According to [20], the sequence zτs converges strongly to the solution of (9) because of the convexity of the constraint ˜ C3, C4, C5 ˜ of problem (6) are sets. Since the constraints C1, linear and CSI is not involved in these four constraints, their corresponding projection operations can be realized directly at the VRM without cooperation with any BSs by the standard convex optimization method [19], e.g., interior point method. However, the projection operation corresponding to Φ2 is in ˜ is related to the CSI of another case because constraint C2 different users, and exchanging them will result in enormous signaling overhead. Next, we will discuss the distributed implementation of this projection operation. To find the projection of a given point r onto Φ2 is to solve ˜ arg min ˆz − r 2 , s.t. R(u) ≥ Rumin , ∀u, where ˆz and r ˆ z have the similar structure with z. It can be shown that this problem is convex since its objective function and constraint are convex. Hence, the duality gap is equal to zero and solving the dual problem is equivalent to solving its original problem [19]. Dual decomposition [17], [19] is adopted to solve it in a distributed manner. written as  TheLagrangian can be ˜ L(ˆz, ξ) = ˆz − r 2 + Ii=1 u∈Ui ξu (Rumin − R(u)), where ξ = [ξ1 , ξ2 , ..., ξ|U | ]T is the Lagrange multiplier vector. Then, the dual problem can be given as max min L(ˆz, ξ). By dual ξ ≥0 ˆz decomposition, it can be decomposed into a master problem and several subproblems. Substituting (1) and (5) into the dual problem and rearranging terms, we can find that the subproblem to be solved at BS m takes the similar form with problem (11). Hence, a similar steepest descent method can be applied to solve the subproblems. After the subproblems are solved at each BS, the service rates of all users at each BS are reported to the VRM. Then, the master problem is solved by

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 xt+1 m

:= arg min xm

I  L  K  N  i=1 u∈Ui l=0 k=1n=1

m ,Rl c˜ pB Tu ,k,n

−

I K  N  i=1 u∈Ui k=1 n=1

 βi ωu am,0 u,k W

log2

1+

m ,R0 Bm ,R0 p˜B Tu ,k,n λTu ,k,n

 +

am,0 u,k

 ⎤ Bm ,Rl Bm ,Rl m,l Bm ,Rl m ,Rl 2 (λB ) + 4Ψ p ˜ /a − λ Tu ,k,n Tu ,k,n Tu ,k,n Tu ,k,n u,k βi ωu am,l )⎦ u,k W log2 (1+ Rl 2δ k,n i=1 u∈Ui l=1 k=1 n=1 I

 L K I   L  K  N  ρ     m,l m,l t m,l t 2 Bm ,Rl Bm ,Rl t Bm ,Rl t 2 (au,k − [zu,k ] + [μu,k ] ) + (˜ pTu ,k,n − [zTu ,k,n ] + [μTu ,k,n ] ) + 2 i=1 n=1 i=1 I   L  K  N 

u∈Ui l=0 k=1

(11)

u∈Ui l=0 k=1

Algorithm 1 Distributed resource allocation algorithm via ADMM in virtualized full-duplex relaying networks 1: Initialization a) At each BS m, collect CSI of users within its coverage; b) Initialize z0 ∈ Φ, μ0 > 0 and a stop criterion threshold ζ > 0 at the VRM; 2: for t = 0, 1, 2, ..., do 3: a) Broadcast zt and μt to each BS; 4: b) At BS m, update xt+1 m via steepest descent method; 5: c) At the VRM, update xt+1 by combining the results of xt+1 m from each BS; 6: d) Update zt+1 via iterative projection method; 7: e) Update μt+1 via (10) at VRM; 8: if xt+1 − xt 2 ≤ ζ, then go to Step 10; 9: end for 10: Output the optimal resource allocation policy xt+1 .

the gradient method, which iteratively updatesthe Lagrange

+ κ+1 κ ˜ , ∀u, where multipliers by ξu = ξu − u (Rumin − R(u)) κ is the iteration index and u is a positive step size. The convergence is guaranteed if the chosen step sizes satisfy the general conditions in [21]. 3) μ-Update: Compared with x-update and z-update, the process of μ-update is quite simple. After collecting the updated xt+1 and zt+1 , μ-update can be performed directly via (10) at the VRM. Hence, the distributed resource allocation algorithm via ADMM is summarized as Algorithm 1. It is able to converge to the optimal resource allocation policy of problem (6). The convergence is guaranteed by [14]. V. S IMULATION R ESULTS AND D ISCUSSIONS In this section, we compare the performance of the proposed scheme with several schemes. We consider a round geographical area with three concentric ring-shaped discs. The outer, middle and inner boundaries have radii of 1 km, 0.66 km and 0.33 km, respectively. The considered area is covered by several ISPs simultaneously. When virtualization is used, each ISP is split into one InP and one SP. Each InP has one BS, three RSs and two subcarriers with bandwidth W = 10 kHz each. The BSs and RSs are randomly deployed in the inner and outer areas, respectively. Each BS, RS or user terminal is equipped with two antennas. The circuit power of each BS/RS and the maximum transmission power on each

The proposed virtualized FDR network Variation of the proposed scheme Existing virtualized cellular network without FDR Existing celluar network with FDR Traditional cellular network without FDR and virtualization

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subcarrier are set to 25mW and 100mW, respectively. The minimum transmission rate for each user is 100 kbit/s. The price for power c is 100 per Watt. All the results are averaged from 200 simulation runs. We compare the following schemes: (a) A traditional cellular network without FDR and virtualization in [17]; (b) A cellular network with FDR but without virtualization in [10]; (c) A virtualized cellular network without FDR in [6]; (d) A variation of the proposed scheme with only spectrum virtualization; (e) The proposed virtualized FDR network with Rl spectrum and infrastructure virtualization. The average of δ¯k,n is 0.001 and the augmented Lagrangian parameter ρ is set to 0.01. We consider two InPs and two SPs. The price charged by VRM for SP1 and SP2 are set to β1 = 300 per Mbit/s and β2 = 250 per Mbit/s, respectively. The total utility of SPs and InPs are illustrated in Fig. 2 and Fig. 3 respectively. The average utility of users with different schemes is also given in Fig. 4 when U = 4. From Fig. 2 to Fig. 4, it can be observed that the proposed scheme and its variation always outperform the virtualized network without FDR. This is because, the proposed scheme with FDR is able to further enhance the service rate of cell edge users with higher spectrum efficiency and lower power consumption, which results in larger utility for both SPs and InPs. Also, it can be found that the variation of the proposed scheme is superior to the existing cellular network with FDR. Moreover, there is an another appreciable performance gain of our proposed scheme compared to its variation. The reason

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Globecom 2014 - Wireless Networking Symposium

and users can benefit from the virtualized resource allocation scheme.

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ACKNOWLEDGMENT This work is jointly supported by the National 863 Project of China (Grant No. 2014AA01A705) and the National Natural Science Foundation (Grant No. 61271182).

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Fig. 4. The average utility of users with M = 2, L = 6, K = 4, U = 4. The labels (a), (b), (c), (d) and (e) denote the various schemes described above.

is that, with spectrum and infrastructure virtualization, there is more degree of freedom for resource allocation and hence a user is able to connect to a better access point via a better subcarrier with better channel conditions. That is to say, access point selection gain and spectrum selection gain can be obtained from our proposed scheme. In addition, it can be seen from Fig. 2 and Fig. 3 that when the number of users is larger than that of the available subcarrier (four in total), there is also a small performance gain for every scheme. This means that the resource manager is able to serve the users with better channel conditions chosen from a larger set of user and hence multi-user diversity gain can be obtained. VI. C ONCLUSIONS In this paper, the problem of resource allocation in virtualized FDR networks has been investigated. We first introduced the idea of virtualization into FDR networks. Radio spectrum, BSs, and full-duplex RSs are virtualized as virtual resources, which can be dynamically allocated to different users. Then, the problem of resource allocation was formulated as a combinatorial and non-convex problem. To solve it efficiently, the original problem was converted into a convex problem and a distributed resource allocation algorithm based on ADMM was developed. Simulation results showed that all the InPs, SPs

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