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Abstract—In next generation wireless mobile networks, net- work virtualization will become an important key technology. In this paper, we firstly propose a ...

IEEE INFOCOM 2015 Workshop on Mobile Cloud and Virtualization

Distributed Resource Allocation in Virtualized Wireless Cellular Networks based on ADMM Chengchao Liang and F. Richard Yu Depart. of Systems and Computer Eng., Carleton University, Ottawa, ON, Canada Email: [email protected]; [email protected] Abstract—In next generation wireless mobile networks, network virtualization will become an important key technology. In this paper, we firstly propose a resource allocation scheme for enabling efficient resource allocation in wireless network virtualization. Then, we formulate the resource allocation strategy as an optimization problem, considering not only the revenue earned by serving end users of virtual networks, but also the cost of leasing infrastructure from infrastructure providers. In addition, we develop an efficient alternating direction method of multipliers (ADMM)-based distributed virtual resource allocation algorithm in virtualized wireless networks. Simulation results are presented to show the effectiveness of the proposed scheme. Index Terms—Wireless network virtualization, virtual resource allocation, distributed algorithm, alternating direction method of multipliers

I. I NTRODUCTION In the field of information and communication technologies, virtualization is a well-applied concept (e.g., virtual machines and virtual private networks). With the recent advances of wireless mobile services and applications, it is natural to extend virtualization to wireless mobile networks. Using virtualization, wireless resources can be shared among many users, and wireless network infrastructure can be decoupled from the services that it provides, so that differentiated services can share the same infrastructure, maximizing their utilization. Moreover, wireless network virtualization provides easier migration to newer technologies while supporting legacy technologies by isolating part of the network [1]. Compared to the relatively stable medium in wired networks, the medium in wireless mobile networks is timevarying and has the inherent broadcast nature, which makes wireless network virtualization a challenging task. Recently, some works have been done for wireless network virtualization, e.g., wireless local area network (WLAN) virtualization approach [2], wireless network virtualization projects ((GENI) [3] and (VITRO) [4]), and Virtualizing eNodeB [5]. Although some excellent works have been done for wireless network virtualization, most existing works do not consider virtualization with heterogeneous networks (HetNets), where there are heterogeneous cell types, such as macro and small cells. However, HetNets have been regarded as one of the key components of the future wireless networks to im-

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prove network capacity and energy efficiency [6]. Therefore, it is necessary to take HetNets into account in wireless network virtualization. Furthermore, in wireless network virtualization, mobile virtual network operators (MVNOs) lease the network resources from infrastructure providers, create virtual resources, and operate the virtual resources. As MVNOs will play key roles in the future telecommunication market, the benefits of (MVNOs) should be carefully guaranteed, which are not well considered in most existing works. Meanwhile, resource allocation plays a very important role in energy efficiency, spectrum efficiency and quality of service (QoS) provisioning in wireless networks [7], [8]. When virtualization and HetNets are jointly considered, the problem of resource allocation becomes even more challenging. In this paper, we propose a wireless network virtualization framework for enabling efficient wireless network virtualization in next generation wireless mobile HetNets. We formulate the virtual resource allocation as an optimization problem, which maximizes the utility of mobile virtual network operators (MVNOs), considering not only the revenue earned by serving end users but also the cost of leasing infrastructure. With recent advances in distributed convex optimization, we develop an efficient alternating direction method of multipliers [9] (ADMM)-based distributed virtual resource allocation and in-network caching scheme. Simulation results are presented to show the effectiveness of the proposed scheme. The rest of this paper is organized as follows. Section II introduce wireless network virtualization and system model. The optimization problem of wireless network virtualization is presented in Section III. Section IV discusses our proposed ADMM-based virtual resource allocation algorithm. Simulation results are discussed in Section V. Finally, we conclude this study in Section VI. II. W IRELESS V IRTUAL R ESOURCE A LLOCATION An important requirement in wireless network virtualization is an efficient virtual resource allocation. In this section, we formulate the wireless virtual resource allocation strategy as an optimization problem. A feasible and efficient algorithm to enable the mapping process is the key of successfully implementing mobile network virtualization. MVNOs can benefit from this algorithm not only because it provides

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the solution of virtualization, but also it should maximize MVNOs’ revenue through allocating physical resource. This optimization problem will be formed and solved in section III and section IV. Before these, we will present the system model of our network. A. System Model In this section, business model of mobile network virtualization, the scheme of network virtualizing and communication model for a cellular network are presented. A general single-input-single-output (SISO) downlink cellular network is considered in our system with assuming perfect frequency and time synchronization as well as perfect channel estimation. 1) Virtualization Model: The business model used in our paper is given in [1]. Briefly explaining, infrastructure providers (InPs) provide (through leasing) the physical substrate infrastructure (e.g., data center, radio access networks (RANs) and backhaul networks) to MVNOs while service providers (SPs) lease virtual network from MVNOs to provide specific services to end users. Obviously, with mobile network virtualization, MVNOs will play a key role in future mobile network exist as a ”connector” in the network. Each SP provides certain services (e.g., video, voice or game) to its users through the virtualized resources. To realize virtualization, resources of substrate mobile networks (e.g., base stations (BSs), radio resource, computing resource, storage resource) should be allocated to each slice (virtual resource allocated to one SP) dynamically by MVNO based on the contract with SPs. The virtualization and management of slices is executed by wireless virtualization controller (or called hypervisor in some researches) where the requirements of virtualization (e.g., isolation, customization and utilization) can be realized. The wireless virtualization controller can be equipped in a control center centralized or distributed at each BS. With virtualization, each SP can schedule next serving users and allocate necessary bandwidth to users based its own QoS requirements. Assuming the pre-agreed bandwidth of ¯ i , SP can allocate any data rate r¯ki slice allocated to SP i is R  ¯i, to its serving user ki under the constrains ki ∈Ki r¯ki ≤ R where Ki is the set of the scheduled users of SP i. Thus, when MVNO conducts the allocation of substrate resource to user ki , r¯ki requested by SP i should be guaranteed; otherwise the SP will not pay for this user since the agreement is not satisfied. Obviously, MVNO needs to give payment to InPs based on some pre-agreements policies (the usage of resources) where InPs will charge MVNO as its actual usage. Specifically, MVNO should dynamically pay the usage of radio resource (e.g., spectrum) to InPs of RANs and bandwidth (e.g., data rate) consumption to InPs of backhaul. In practical networks, unlike big BSs, femtocells are using the Internet (e.g., digital subscriber line or Fiber to the x) as their backhaul [6]. Thus, without losing generality, we assume that the InPs of RANs and backhaul are different. The unit price of radio resource

units/MHz set by RANs’ InPs corresponding to BS j is μ ran j while the unit price of backhaul set by backhaul InPs is μ tn j units/Mbps, where bn represents backhaul network. Similarly, multiple SPs are considered in our model and the charging policies between SPs and MVNOs where the payment given by SP i for user ki is νisp . 2) Wireless Mobile Cellular Network Model: We consider a mobile cellular network comprising J BSs. The set of BSs is denoted by J , and j is used to indicate one of the BSs. In this paper, the cellular network is limited in a finite area where there are only one macro BS and several small cell BSs. These BSs belong to different InPs with different leasing prices. Even we only consider a single macrocell system in our paper, it can be easily extended to the multi-cell case. Let I denote the set of SPs. For each SP i, each allocated user is denoted by k i , and Ki is the set of users of SP i. The licensed spectrum used by different InPs is orthogonal, which means there is no interference between InPs. The spectrum used within in one InP is overlaid, which means downlink interference between macrocell and femtocells as well as among femtocells are both considered. The spectrum bandwidth allocated to BS j is W j Hz and backhaul capacity of BS j is Bj bps. Fixed equal power allocation mechanism is used, where the normalized transmit power on BS j is p j watts/Hz. Therefore, by using Shannon bound, the spectrum efficiency of user k i who associate with BS j is (1) bki j = log2 (1 + γki j )   where γki j = gki j pj / is the signal to l,l=j gki l pl + σ interference-plus-noise ratio (SINR) between user k i and BS j. gki j is the large-scale channel gain that includes pathloss and shadowing. σ is the power spectrum density of additive white Gaussian noise. Let aki j denote the association indicator, where a ki j = 1 means that user ki associates to BS j; otherwise aki j = 0. Practically, each user only associates to only one BS; thus  aki j = 1, ∀i, k (2) j∈J

αki j ∈ [0, 1] is used to denote the percentage of radio resource allocated by BS j to user k i . Certainly,  aki j αki j ≤ 1, ∀j (3) i∈I,ki ∈Ki

The expected instantaneous data rate of user k i is  aki j αki j Wj bki j Rki j =

(4)

j∈J

In our model, we assume the backhaul bandwidth usage of user ki is the same as instantaneous data rateR ki j [10]. Thus, the total required bandwidth of BS j is i∈I,ki ∈Ki Rki j . Since the capacity of backhaul is limited,  Rki j ≤ Bj , ∀j (5) i∈I,ki ∈Ki

must be hold.

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3) Utility Function: Let us consider that only one user k is associated with BS j, the revenue, radio resource cost bn and backhaul cost are ν isp Wj bki j , μran j Wj and μj Wj bki j , sp ran bn respectively, where ν i ,μj ,μj are defined in Subsection II-A1. Thus, the net revenue of allocating radio resource to user k is defined as. bn ωki j = νisp Wj bki j − μran j Wj − μj Wj bki j

(6)

Since the proportion of radio resource allocated to user k is controlled by y ki j , we can formulate the total utility of MVNO by  u (ωki j αki j aki j ) (7) UMV N O = I,K,J

where u(·) is a utility function that is a nondecreasing and convex function normally. In this paper, we adopt the wellknown logarithmic function u(x) = log x, x > 0 to our utility function. The logarithmic function has been widely used in where utility estimation is calculation [11]. Another constraint that represents the data rate allocated by SP i to user k i is given by  aki j αki j Rki j ≥ r¯ki , ∀i, k (8) J

III. P ROBLEM F ORMULATION In this section, we will formulate the optimization problem to maximize the aggregate utility of MVNO. A. Formulation The aggregate utility maximization problem is shown as follows:  max + aki j u (ωki j αki j ) aki j ,αki j ∈R I,K,J (9) s.t. (2), (3), (5), (8) It is equivalent to take a ki j outside utility function without loss any optimality. If a = 1, we have au(α) = u(a, α); if a = 0 that means user is not served by BS so that no resource will be allocated, u(a, α) = 0 and au(α) = 0. Note that aki j ≤ 1 and αki j ≤ 1 are eliminated by giving (2) and (3). aki j αki j appeared in (3), (5) and (8) are equivalent to α ki j . We use aki j αki j instead of αki j that is equivalent, because it is more convenient to transfer it to a convex problem. Problem (9) is difficult to solve based on the following observations: • •



The feasible set of (9) is nonconvex as a result of the binary variables {a ki j }. The objective function is not convex due to the product relationship between {a ki j } and convex function of {αki j }. The size of the problem is very large.

B. Problem Transformation Based on the research in [12] and [13], {a ki j } in (9) can be relaxed to be real value variables that 0 ≤ a ki j ≤ 1. From a long-term view to interpret relaxed a ki j , this is a time sharing factor that represents the ratio of time when user k i serving by BS j [13]. Nonetheless, the problem is still not tractable due to the nonconvex objective function. Thus, to make the problem (9) tractable and solvable, we introduce an equivalent problem of (9) following the approach in [14]. If ˜ki j ) /aki j ] = 0 we define α ˜ ki j = αki j aki j and aki j u [(ωki j α for aki j = 0, it can be proven that the following problem:    ωki j α ˜ ki j aki j u max aki j (10) I,K,J ˜ ˜ ˜ s.t. (2), (3), (5), (8) is equivalent to problem (9). The proof of 10 is similar to [14]. The relaxed problem (9) can be recovered by substitution of variable α ˜ ki j = αki j aki j into problem (10) except a ki j = 0. However, if xki j = 0, yki j = 0 is certainly hold because of the optimality. Thus, problem (10) and problem (9) are equivalent. C. Convexity Theorem 3.1: If problem (10) is feasible, it is jointly convex with respect to all optimization variables a ki j , αki j ∀i, j, k. Proof The proof of the convexity is motivated by similar problem in [14], we describe briefly as follows. Firstly, we prove the continuity of the function f (t, x) = x log(t/x), t ≥ 0, x ≥ 0 at the point of x = 0. Let s = t/x, t log s t = lim log s = t lim =0 s→∞ s x→0 x s→∞ s f (t, x) = x log(t/x), t ≥ 0, x ≥ 0 is the well-known perspective operation [15] of log, where convexity is preserved. The perspective function of a concave function is a concave function. Since aki j log [(ωki j α ˜ ki j ) /aki j ] = 0 for aki j = ˜ ki j ) is only a linear combination of y, 0 and (ωki j α aki j log [(ωki j α ˜ ki j ) /aki j ] is the perspective function of ˜ki j ) that is concave. Our objective function in log (ωki j α problem (10) is a sum of concave function, and all of the constraints are linear constraints. Thus, problem (10) is a concave problem. f (t, 0) = lim x · log

The feasibility of problem (10) can be guaranteed by admission control policy. Since the number of BSs in HetNet is comparable to the number of users [6], it is not hard to guarantee the feasibility. Since the problem (10) is a convex problem, a lot of methods (e.g., interior point method) can be used to solve it. However, as we mentioned above, with the increase of the number of BSs, the size of problem will be very large. Practically, even with a powerful computing center, the overhead of deliver enough local information (e.g., channel status

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information (CSI)) to global center is extremely inefficient. Therefore, for the purpose of implementation, a distributed algorithm running on each BS should be adopted. But, due ˜ (5), ˜ and (8) ˜ of problem (10), the to the constrains (2), (3), problem is not separable with respect to the BSs. Specifically, the user association indicator that represents the connection between users and BSs is a global variable. Thus, in order to achieve a decentralized optimization algorithm, this coupling has to be decoupled appropriately, which is discussed in the next section. D. Decoupling of Association Indicators Following the approach in [16] and [17], we introduce local copies of the global association indicators. Each local variable can be considered as the preference of each BS about the association of users. Let us introduce a set of new variables to represent the local copies of our association indicators. To lighten the notation, from now on, we use k to denote all the users instead of ki . If we define the a as the vector of association indicators {a kl , ∀l, k} (note that we change the index of BS from j to l), the local copy of a at BS j is ˆj . Formally, denoted as a a ˆjkl = akl , ∀j, k, l ˆj and α By means of the local vectors a ˜ j , let feasible local variable set for each BS j ∈ J  ⎧ ˆjkl = 1, ∀j, k ⎪ ⎪ l∈J a ⎨ α ˜ ≤ 1, ∀j ˆj , α χj = a ˜j k∈K kj α ⎪ ⎪ k∈K ˜ ki j Rki j ≤ Bj , ∀j ⎩ ˜ kj Rkj ≥ r¯k , ∀k j∈J α

(11) us define a ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(12)

and an associated local utility function, respectively, as     (ωkj α ˜ kj ) ˆj , α a − Ka ˆjkj u ˜ j ∈ χj j a ˆ kl fj = (13) ∞ otherwise With this notation, the global consensus problem of problem (10) can be written as    ˆj , α max gj a ˜j J (14) s.t. a ˆjkl = akl , ∀j, k, l Obviously, our objective function is separable across the BSs in the network. However, the global association variables is still involved in the consensus constrain. In next chapter, we will apply ADMM for approach the problem (14) in a distributed method.

to a modest accuracy of the optimal solution. It has been successfully used in many cases, such as in statistical learning problems, engineering design, multi-period portfolio optimization, time series analysis, network flow, and scheduling [9], [17], [18]. It takes the form of a decompositioncoordination procedure, in which the solutions to small local subproblems are coordinated to find a solution to a large global problem. Furthermore, ADMM can be viewed as an attempt to blend the benefits of dual decomposition and augmented Lagrangian methods for constrained optimization [9], [18]. Generally, ADMM is able to solve min x,z

f (x) + g(z)

s.t.

(15)

Ax + Bz = c

where x ∈ Rq×1 , z ∈ Rr×1 , A ∈ Rp×q , B ∈ Rp×r and c ∈ Rp×1 . B. Problem Solving via ADMM In this section, the proposed algorithm for virtual resource allocation and in-network caching via ADMM is described. According to [9], our problem (14) is a global consensus problem, since the constrain is that all the local variables should agree. The initial step to apply ADMM to problem (14) is that an augmented Lagranian with corresponding global consensus constrains should be formed. Let λ jkl , ∀j ∈ J , l ∈ J , k ∈ K be the Lagrange multipliers corresponding to the consensus constrain in problem (14). The augmented Lagrangian for problem (14) is  j     j ˆ ,α a ,α ˜ j }j∈J , {a}, {λj } = fj a ˜j + Lρ {ˆ 



j∈J k∈K,l∈J



ˆjkl − akl λjkl a



J

ρ + 2



j∈J k∈K,l∈J

 2 a ˆjkl − akl (16)

j

where λ is the vector of the Lagrange multipliers and ρ ∈ R++ is a positive constant parameter for adjusting the convergence speed of the ADMM [9]. Based on the iteration of AMDD with consensus constrain [9], the ADMM method applied to problem (14) consists of following sequential optimization steps:  j  [t+1] ˆ ,α ˜ j }j∈J := arg min{fj a ˜j {ˆ aj , α   ρ   2  j[t] [t] [t] ˆjkl − akl + a ˆjkl − akl } + λkl a 2 k∈K,l∈J

k∈K,l∈J

IV. V IRTUAL R ESOURCE A LLOCATION VIA A LTERNATING D IRECTION M ETHOD OF M ULTIPLIERS

{a}[t+1] := arg min{

A. Introduction to Alternating Direction Method of Multipliers with Consensus Constrain

+

ρ 2



j∈J k∈K,l∈J

ADMM [9], [18] 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

[t+1]

{λj }j∈J

363





j[t]

λkl

j∈J k∈K,l∈J  2 j[t+1] a ˆkl − akl }



j[t+1]

a ˆkl

  [t] ˆj[t+1] − a[t+1] := λj + ρ a

− akl

(17) 

(18) (19)

IEEE INFOCOM 2015 Workshop on Mobile Cloud and Virtualization

Obviously, the first step (17) and third step (19) can be completely decentralized, which find the local optimum association, radio resource allocation and caching strategies, as well as local Lagrange multipliers. The second step (18) can be optimized by a center controller of MVNO. Each step will be presented in the following part of this section. In step 1, local association, radio resource allocation and caching strategies are separable across each BS j. Therefore, ˜ j }j∈J -update can be decomposed into J subproblems, {ˆ xj , y which can be solved locally at each BS. Thus, each BS j solves the following optimization problem at iteration t:  j   j ˆ ,α ˆ max Ljρ = fj a ˜ j + hj a

2400 2200

Toatal Utility of MVNO

2000

 l∈J

1400

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1000

(20)

a ˆjkl = 1, ∀k, l

1600

1200

{ˆ a j ,α ˜j }

s.t.

1800

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800 0

where  j ˆ = hj a

 k∈K,l∈J

  ρ ˆjkl − akl + λjkl a 2

 k∈K,l∈J



a ˆjkl − akl

2

(21) Apparently, problem (20) is a convex problem but still intractable to compute the closed form solution. Hence, it should be solved by some general numerical methods (e.g., interior-point methods and successive approximation methods) that provide efficient ways. In this paper, we use the primal-dual interior-point method [15] that provides efficient way for convex problems. Recall that we have relaxed the association indicator to a real value between zero and one in stead of a binary variable in Subsection III-B. Thus, we have to recover it to binary after we get the optimum solution of (17). The binary recovery deals with computing the marginal benefit for each user k. Then, the indicator {x} can be recovered by  1 if Qjnl = maxk {Qjkl , k ∈ K} and Qjkl > 0 j anl = 0 otherwise (22) where Qjnl = ∂Ljρ /∂ajnl is the first partial derivation of a jnl . Compared with the updating of local variables, {a}-update and {λj }-update are quite simple since it is only an unconstrained quadratic optimization problem. After the collec˜ j } and {λj }, {a}-update can be done by many tion of {ˆ aj , α efficient ways [15]. In this paper, we use an interior-point method in the simulations. V. S IMULATION R ESULTS AND D ISCUSSIONS In the simulations, we consider two RAN InPs, two backhaul InPs, one MVNO and three SPs. RAN InP 1 owns a twotier HetNet with one macro BS with price of 100 units/MHz and 10 small BSs with price of 90 units/MHz. RAN InP 2 owns only 10 small BSs with 80 units/MHz. The price of backhaul InPs 1 for macro BS and small BSs of RAN InP 1 is 1 units/Mbps, while the price of backhaul InP 2 for small BSs of RAN InP 2 is 1.2 units/Mbps. The average numbers of the scheduled users of 3 SPs are assumed to be equal, and the

100

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400 500 600 Iteration Steps

700

800

900

1000

Fig. 1. Convergence of ADMM and the effect of ρ. The number of users = 20, Required Data Rate = 8Mbps, Macro BS Backhaul Delay = 10ms, Small BSs Backhaul Delay = 20ms.

prices for requesting virtual resources are 15units/Mbps, 20 units/Mbps and 18 units/Mbps, respectively. Transmit power of 49dBm for macro BS and of 20dBm for small BSs are considered in our simulations. The bandwidth is 20MHz. In our simulations, the location of the macro BS is fixed in the center and the locations of 20 small BSs are uniformly distributed in a area where the radius is 250meter representing urban environment [19]. Referred to [12], we user a path loss L(d) = 34 + 40log(d) and L(d) = 37 + (d) to model the macro cell and small cell propagation, respectively. The lognormal shadowing with standard deviation 8dB for macro cell and 4dB for small cell are assumed in our paper. The power density of thermal noise power is -174dBm/Hz. The backhaul delay of small BSs is 10ms longer than macro BS. The backhaul capacity of macro BS is 10Gbps and of small BS is 2Gbps. To compare our proposed algorithm, three benchmarks are also considered. The first baseline is a centralized algorithm based on solving problem (10) by interior methods directly and round up based on (22). The second baseline is a traditional max-SINR association algorithm, all users associate to the BSs who provide the maximum received SINR, and each BS performs proportional fairness resource allocation. The third one is a traditional virtualization scheme called hard slicing that means each base station slices fixed ratio of resource to individual SP. Fig.1 shows the convergence of the proposed ADMM-based algorithm and the effect of parameter ρ in ADMM. The y-axis is the total utility of MVNO and xaxis is the iteration step index. Observing from the figure, the gap between ADMM-based algorithm and the centralized algorithm is narrow. This means the effectiveness of ADMMbased algorithm is equivalent to centralized algorithm considering the overall utility. It can be found that the results with

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Average Total Utility of MVNO (Optimization Object)

ACKNOWLEDGMENT 3500

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and in part by Huawei Technologies Canada CO., LTD.

Baseline (MaxSINR+PF) Hard Slicing Soft Slicing (ADMM-based)

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R EFERENCES 2500

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Number of Access Users (All 3 SPs)

Fig. 2. Comparison of the MVNO utility. ρ = 1.0. Required Data Rate = 8Mbps, Macro BS Backhaul Delay = 10ms, Small BSs Backhaul Delay = 20ms.

different ρ finally converges to almost same utility value with only small gap. However, ρ affects the rate of convergence. ρ = 1.0 gives higher rate than ρ = 0.5 especially before the 100th iteration. From Fig.1, a significant decrease of utility gap between Centralized algorithm and proposed ADMMbased algorithm can be found from 1st iteration to 10th iteration. After 10th iteration, the gain of running iteration is still increasing but with less rate. Thus, a tradeoff is exiting between acceptable utility value and iteration steps. Fig. 2 shows the total utility of MVNO in different schemes. We can see that the utility of MVNO increases with the increase of the number of users for all of these three cases. This is because more payment can be obtained from SPs with the increase of the number of users. From Fig. 2, we can observe that the proposed ADMM-based algorithm gives better performance than the traditional max-SINR and the hard slicing virtualization algorithm. VI. C ONCLUSIONS AND F UTURE W ORK In this paper, we jointly studied wireless network virtualization and heterogeneous networks in next generation cellular networks. We proposed a wireless network virtualization framework for enabling wireless network virtualization. Then, we formulated the virtual resource allocation strategy as an optimization problem, which maximizes the utility of mobile virtual network operators. In addition, we developed an efficient ADMM-based distributed virtual resource allocation. Simulation results were presented to show that the performance of utility and capacity can be substantially improved in the proposed scheme. The InPs, SPs and MVNOs can benefit from the proposed wireless network virtualization algorithm. Future work is in progress to consider admission control and full-duplexing [20] in the proposed framework.

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