Price-based Resource Allocation for Virtualized Cognitive Radio Networks

Abstract

We consider a virtualized cognitive radio (CR) network, where multiple virtual network operators (VNOs) who own different virtual cognitive base stations (VCBSs) share the same physical CBS (PCBS) which is owned by an infrastructure provider (InP), sharing the spectrum with the primary user (PU). The uplink scenario is considered where the secondary users (SUs) transmit to the VCBSs. The PU is protected by constraining the interference power from the SUs. Such constraint is applied by the InP through pricing the interference. A Stackelberg game is formulated to jointly maximize the revenue of the InP and the individual utilities of the VNOs, and then the Stackelberg equilibrium is investigated. Specifically, the optimal interference price and channel allocation for the VNOs to maximize the revenue of the InP and the optimal power allocation for the SUs to maximize the individual utilities of the VNOs are derived. In addition, a low‐complexity ±‐optimal solution is also proposed for obtaining the interference price and channel allocation for the VNOs. Simulations are provided to verify the proposed strategies. It is shown that the proposed strategies are effective in resource allocation and the ±‐optimal strategy achieves practically the same performance as the optimal strategy can achieve. It is also shown that the InP will not benefit from a large interference power limit, and selecting VNOs with higher unit rate utility gain to share the resources of the InP is beneficial to both the InP and the VNOs.

1. Introduction

To increase the wireless network resource utilization efficiency, wireless network virtualization has been increasingly popular for resource management [1]. The physical wireless network operated by the infrastructure providers (InPs) can be shared by multiple wireless virtual networks operated by different virtual network operators (VNOs). Since wireless network virtualization enables sharing of physical resources such as infrastructure and radio spectrum between wireless virtual networks, resource utilization efficiency can be improved significantly. Due to diverse resource limitations, wireless channel fadings, and user requirements, resource allocation among wireless virtual networks is challenging and has received much attention. For example, in [2], an efficient heuristic resource allocation scheme was proposed in virtualized multi‐cell long term evolution (LTE) networks to maximize network sum rate. In [3], a two‐stage power allocation scheme in virtualized LTE networks was proposed. In [4], the dynamic interaction among the InP and the VNOs was modeled as a stochastic game where the InP is responsible for spectrum allocation and the VNOs are responsible for satisfying users’ requirements.

On the other hand, coexistence investigation of heterogeneous wireless networks (HWNs), where different wireless access networks such as Wi‐Fi network and LTE network are deployed within the same area, have attracted a lot of attention recently [5]-[7]. In particular, cognitive radio (CR), as a promising technique to improve spectrum efficiency in HWNs, has received much attention [8]. In CR networks, spectrum efficiency is improved by allowing unlicensed users, also known as secondary users (SUs) to use the spectrum allocated to licensed users, also known as primary users (PUs). The PUs are protected by constraining the interference from the SUs to the PUs. Specifically, in the popular CR underlay model, the SUs and the PUs can coexist simultaneously under the condition that the interference power from the SUs to the PUs is below an acceptable level. Resource allocation in CR networks has been investigated in many works. For example, in [9], aiming at minimizing the bandwidth‐power product, a two‐step resource allocation scheme was proposed under the interference power constraint, the transmit power constraint and the minimum rate constraint. In [10], an optimal resource allocation scheme was proposed to maximize the ergodic sum capacity under the long‐term interference power constraint, the long‐term transmit power constraint and the proportional fairness constraint. In [11], power allocation algorithms were proposed for a CR multiple access channel under the transmit power constraint and the PU outage constraint. In [12], the problem of network selection and channel allocation to minimize the cost of the SUs was investigated and schemes based on particle swarm optimization and genetic algorithm were proposed. In [13], a novel cooperative mechanism for the SUs and the PUs with wireless energy harvesting was proposed and the cat swarm optimization was used to maximize the total throughput of the SUs and the PUs. In [14], the problem of resource allocation for the SUs to maximize the SU ergodic transmission rate under the PU secrecy outage constraint and the SU transmit power constraint with perfect or imperfect channel state information was studied.

Therefore, combining wireless network virtualization and CR can bring higher resource efficiency, especially spectrum efficiency, than either one of them can achieve. However, to the best of our knowledge, resource allocation in virtualized CR networks have not been considered in existing literature. In this paper, we investigate the resource allocation problem for an uplink virtualized CR network. Multiple VNOs who own different virtual cognitive base stations (VCBSs) share the same physical CBS (PCBS) which is owned by an InP. The interference power from the SUs within VCBSs are constrained to protect the PU. Such constraint is applied by the InP through pricing the interference. The price‐based joint channel and power allocation problem for the uplink virtualized CR network is investigated. The problem is formulated as a Stackelberg game to jointly maximize the revenue of the InP (the leader) and the individual utilities of the VNOs (the followers). The Stackelberg equilibrium is investigated for the proposed resource allocation game. We derive the optimal interference price, channel allocation for the VNOs and the optimal power allocation for the SUs to maximize the revenue of the InP and the individual utilities of the VNOs, respectively. A low‐complexity ±‐optimal solution is also proposed for obtaining the interference price and channel allocation for the VNOs. Simulations are conducted to verify the effectiveness of the proposed strategies. From the simulation results, it is shown that the optimal strategy and the ±‐optimal strategy achieve virtually the same performance. It is also shown that a medium interference power limit is advantageous to the InP and the InP will not achieve higher revenue from a larger interference power limit. Beside, it is shown that it is beneficial for both the InP and the VNOs to select VNOs with higher unit rate utility gain to share the resources of the InP.

The rest of the paper is organized as follows. The system model and problem formulation are presented in Section 2. Section 3 investigates the Stackelberg equilibrium and the interference prices and resource allocation strategies. Simulation results are provided in Section 4 to verify the proposed strategies. Section 5 concludes the paper.

 

2. System Models and Problem Formulation

Fig. 1.System model.

We consider a CR network with one PCBS, which is virtualized to N VCBSs, sharing the spectrum with the PU. The PCBS is operated by an InP and each VCBS is operated by a VNO. For notational simplicity, VNO i is assumed to operate VCBS i. The available spectrum is divided into N independent and orthogonal channels. It is assumed that each VCBS can be allocated one channel and one channel can be allocated to only one VCBS at a time1. There are multiple SUs served by each VCBS, and in order to avoid inter-SU interference within a VCBS, we assume that each VCBS schedules only one SU to use the allocated channel at one time. The SU scheduling is assumed to be done and not considered in this paper. We consider the uplink transmission scenario where the SUs transmit signals to VCBSs. All the channels involved are assumed to be block-fading, i.e., the channels keep constant in one transmission block, but may change from one transmission block to another. The instantaneous channel power gains from the SU within VCBS i to VCBS i and the SU within VCBS i to the PU on the channel n are denoted by hi,n and gi,n, respectively. The noises power is denoted by σ2.

We formulate the price‐based resource allocation problem using the Stackelberg game. In the Stackelberg game, a leader who moves first and a set of followers who move subsequently compete with each other on certain resources. Here, the InP is the leader and the VNOs are the followers. We assume that the PU and the InP belong to the same operator, where the PU has the priority to use the spectrum than the PCBS and allows the PCBS to share the spectrum on condition that the interference power from the SUs is restricted. The InP complies with this constraint by pricing the interference power from VNOs and allocating proper channels to the VNOs, aiming at maximizing its own revenue. Then, the VNOs maximize their individual utilities based on the allocated channels and assigned interference prices.

Let pi,n denote the transmit power of the scheduled SU within VCBS i on channel n and let αi,n be a binary variable with αi,n = 1 denoting that channel n is allocated to VCBS i and vice versa. To protect the PU, the interference power constraint is applied as

where Qmax is the predefined interference power limit. Given the above Stackelberg game model, the revenue of the InP is expressed as

where λ, α and p are the interference price vector, the channel allocation matrix and the power allocation matrix, respectively, where α(i,n) = αi,n, p(i,n) = pi,n, and λ(i) = λi, with λi denoting the interference price for VNO i. The objective of the InP is to find the optimal interference price vector and the channel allocation matrix that maximize its revenue under the interference power constraint. The revenue maximization problem for the InP is formulated as

The constraints in (5)-(7) imply that one channel can be allocated to only one VCBS, and one VCBS can occupy only one channel at a time.

At the VNOs’ side, the utility of VNO i on its assigned channel n is defined as

where μi denotes the unit rate utility gain for VNO i. The above utility for VNO i consists of two parts. The first part is the profit for successfully transmission, and the second part is the cost for causing interference to the PU. Then, the utility maximization problem for VNO i is formulated as

P1.1 and P2.1 form a Stackelberg game and our objective is to find the Stackelberg Equilibrium (SE) where neither the InP nor the VNOs have incentives to deviate. Since there is only one InP, the best response of the InP can be found by solving P1.1. For this, the best response of the VNOs shall be found first due to the fact that the InP’s strategy is based on those of the VNOs. Since the utility of VNO i does not depend on other VNOs’ power allocation, the best response of each VNO can be found by solving P2.1. Thus, the SE for the formulated game can be found by solving P2.1 for given ¸ and ®, and then solving P1.1 with the obtained best response of the VNOs.

 

3. Stackelberg Equilibrium Solution

At the VNOs’ side, it can be verified that is convex. Thus, the optimal solution for P2.1, can be obtained by setting the derivative of to zero, i.e.

from which we have

where (.)+ = max(.,0). It is seen that if the interference price is too high for VNO i on channel n, i.e., the scheduled SU is not allow to transmit on channel n in VNO i. Inserting (11) into P1.1, the problem at the InP’s side becomes

The above problem is non‐convex due to the constraint in (16). In the following two subsections, we propose the optimal solution based on the exhaustive search method and the δ‐optimal solution based on the dual optimization method, respectively, for P1.2.

3.1 Optimal solution based on the exhaustive search method

In this subsection, we solve P1.2 optimally using the exhaustive search method, i.e., the optimal solution to P1.2 can be found by finding the maximum objective function in (12)among N! possible channel allocation choices. For a given channel allocation choice, P1.2 can be reformulated as

where gi and hi are the channel power gains related to the SU within VNO i on the allocated channel. In Appendix A, we show that P1.3 is still non‐convex. We solve P1.3 by exploring its special structure. Without loss of generality, we assume that all the VNOs are sorted as

Lemma 1: Assuming that for ∀K ∈ {1,..., N}, then the optimal solution to P1.3 is given by

where

Proof: Refer to Appendix B. ■

Based on Lemma 1, we can obtain the following theorem.

Theorem 1: The optimal solution to P1.3 is given by

where

Proof: Refer to Appendix C.

Based on Theorem 1, the optimal channel allocation solution to P1.2 can be found by finding the maximum among N! possible channel allocation choices using the exhaustive search method. Thus, the complexity of the optimal strategy based on the exhaustive search method is roughly O(N!).

3.2 ±‐optimal solution based on the dual optimization method

In this subsection, we solve P1.2 using the dual optimization method [15]. Usually, the solution provided by the dual optimization method is not optimal for non‐convex problem. However, later in this subsection, we will shown that the optimal duality gap of P1.2 is zero and the proposed strategy based on the dual optimization method is δ‐optimal with δ arbitrarily small.

The partial Lagrangian function of P1.2 with respect to the constraint in (13) can be written as

where ν is the nonnegative dual variable associated with the constraint in (13). The above expression can be rewritten as

The Lagrange dual function can be then expressed as

It can be seen that the optimization variables λ and α are not coupled in the above problem. Thus, the above problem can be separated into two levels of optimization. At the lower level, we obtain the optimal λ for a fixed value of α by solving the following problems, one for each pair of VNO i and channel n, as given by

where and λi,n denotes the value of λi for VNO i occupying channel n. It can be seen that, when is equal to zero. In this case, we can set λi,n = ∞ due to the fact that VNO i is forbidden to use channel n as can be seen from (11). When the problem in (27) is re‐expressed as

It can be verified that the above problem is convex. Thus, by setting the derivative of the objective function in (28) with respect to λi,n to zero, we have the optimal solution of the problem in (27) as Inserting the above λi,n into f(λi,n, ν), we have if In the case of we can also set λi,n = ∞ due to the fact that VNO i can not use channel n as can be seen from (11). Combining the above discussions, we can have

with the solution to the problem in (27) as

At the higher level of optimization, we optimize α given fi,n(ν) as

It can be observed that the above problem is a standard assignment problem that can be optimally and efficiently solved by the Hungarian algorithm [16]. Finally, the value of ν can be obtained by the sub‐gradient‐based method that iteratively updates ν until convergence [17].

Our proposed strategy based on the dual optimization method to solve P1.2 is summarized in Algorithm 1. The sub‐gradient‐based method used for obtaining the value of ν typically converges in a small number Δ [17], [18] and the complexity of the Hungarian algorithm is O(N3)[16]. Thus, the total complexity of Algorithm 1 is roughly O(ΔN3), which is much less complex than the optimal strategy based on the exhaustive search method.

Proposition 1: Let ÛInP be the optimal objective value of P1.2 and D = g(ν*) be the optimal dual objective value with º¤ the optimal dual variable. The optimal duality gap of P1.2 is defined as D - ÛInP, which is zero, i.e., D - ÛInP = 0.

Proof: Refer to Appendix D. ■

Remark 1: It is noted that, for N = 1 we can see more clearly that the optimal duality gap of P1.2 is zero. For N = 1 channel allocation is not needed and P1.2 reduces to the following problem as

where we omit the index i, n as there are only one VCBS and one channel. Since it is easy to see that the optimal solution of the above problem must lead to positive objective function, we omit the operation (.)+ for simplicity. Then, the optimal solution in (20) is simplified as

As for the solution obtained from the dual optimization method, the solution in (30) becomes where the value of ν must be determined by the sub‐gradient‐based method such that the constraint in (36) is satisfied at equality, otherwise, λ can be decreased to let and the objective function in (35) will be increased. Then, inserting Thus, we have It is clearly seen that the solutions provided by the optimal strategy and the strategy based on dual optimization method are the same and the duality gap of P1.2 is zero for N = 1.

Remark 2: It is noted that Algorithm 1 yields a δ‐optimal objective function value with δ arbitrarily small. This is because the sub‐gradient‐based is guaranteed to find the dual variable ν that is within some range of the optimal value due to the existence of error tolerance. The value of δ can be arbitrarily small as long as the error tolerance is arbitrarily small.

Proposition 2: The SE for the formulated Stackelberg game in P1.1 and P2.1 is (λ, α, p), where λ, α are obtained by the optimal strategy presented in Section 3.1 or the δ‐optimal strategy in Algorithm 1, and p is obtained as follows: the transmit power of the scheduled SU, pi,n, within VNO i on channel n with αi,n = 1 is obtained from (11), while the other transmit powers are zero.

 

5. Simulation Results

This section provides simulations to evaluate the performance of the proposed price-based resource allocation strategies. The number of VCBSs/VNOs is assumed to be . It is also assumed that the scheduled SUs and the PU are distributed uniformly around the PCBS, and the distances from the PCBS to the scheduled SUs and the PU are distributed uniformly in the interval [10, 20]. The propagation pathloss model PL = 35 log10(d) + 40 + X dB is adopted, where d is the distance and X is the Log-normal shadow fading with variance 3 dB. The multipath fading is assumed to be Nakgami-m fading with parameter m = 1.5. In addition, we set σ2 = 10-8 W and μi = μ, ∀i. For the purpose of comparison, two reference strategies are presented. The reference strategy 1 subsequently allocates the channel to the VCBS whose SU has the maximum channel power gain hi,n while the reference strategy 2 subsequently allocates the channel to the VCBS whose SU has the minimum channel power gain gi,n. The interference pricing and power allocation of the reference strategies follow the same procedures proposed in the optimal strategy. The following results are obtained by averaging over 100 simulation runs.

Fig. 2 shows the revenue of the InP vs. Qmax under different values of μ. It is seen that the curves of the proposed optimal strategy and the δ-optimal strategy overlap. This indicates that the δ-optimal strategy is practically optimal as δ is arbitrarily small. It is also seen that the optimal strategy and the δ-optimal strategy achieve higher revenues than the reference strategies for small and medium values of Qmax. As Qmax increases, the revenues of the InP achieved by different strategies increase and saturate to the same level for a given μ and a large value of Qmax. This can be explained as follows. From (20), it is observed that λi's achieved by the optimal strategy and the reference strategies are very small for a very large Qmax. On the other hand, for the δ-optimal strategy, since ν → 0 as Qmax → ∞, λi from (30) achieved by the δ-optimal strategy is also very small for a very large Qmax. Thus, as Qmax → ∞, from (12), the revenues of the InP achieved by the four strategies all become Fig. 3 shows the sum utility of the VNOs vs. Qmax under different values of μ. It is shown that the sum utility of the VNOs increases with the increase of Qmax. It is also shown that the proposed optimal strategy and the δ-optimal strategy achieve the same sum utility of the VNOs and outperform the reference strategies.

Fig. 2.Revenue of the InP vs. Qmax.

Fig. 3.Sum utility of the VNOs vs. Qmax.

Figs. 4 and 5 show the revenue of the InP and the sum utility of the VNOs, respectively, vs. μ under different values of Qmax. It is shown that both the revenue of the InP and the sum utility of the VNOs are almost linear increasing functions of μ. It is also shown that the proposed optimal strategy and the δ-optimal strategy achieve higher revenue of the InP and sum utility of the VNOs compared to the reference strategies. In addition, it is shown that performance gaps between the proposed optimal/δ-optimal strategy and the reference strategies increase with the increase of μ.

Fig. 4.Revenue of the InP vs. μ.

Fig. 5.Sum utility of the VNOs vs. μ.

 

5. Conclusions

This paper investigates the price-based resource allocation strategies for an uplink virtualized CR network. The interference power constraint is applied to protect the PU transmission. We use Stackelberg game to jointly maximize the revenue of the InP and the individual utilities of the VNOs. The optimal interference price and channel allocation for the VNOs, and the optimal power allocation for the SUs are derived. Additionally, a low-complexity δ-optimal solution is also proposed for obtaining the interference price and channel allocation for the VNOs.

Simulation results show that the revenue of the InP increases as the interference power limit increases and then saturates for a high interference power limit, while the sum utility of the VNOs increases without limit as the interference power limit increases. It is also shown that the revenue of the InP and the sum utility of the VNOs increase almost linearly with the increase of the unit rate utility gain for VNOs. Beside, it is shown that the proposed optimal strategy and the δ-optimal strategy achieve virtually the same revenue of the InP and sum utility of the VNOs. From these observations, we conclude that the InP will not benefit from a large interference power limit, and thus a medium interference power limit is advantageous to the InP considering the fact that a small interference power limit leads to very low revenue. We also conclude that selecting VNOs with higher unit rate utility gain to share the resources of the InP is beneficial to both the InP and the VNOs.