Dynamic Resource Allocation for Heterogeneous Services in ...

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Dynamic Resource Allocation for Heterogeneous Services in Cognitive Radio Networks with Imperfect Channel Sensing Renchao Xie, F. Richard Yu, Senior Member, IEEE, and Hong Ji, Senior Member, IEEE

Abstract—Resources in cognitive radio networks (CRNs) should be dynamically allocated according to the sensed radio environment. Although some works have been done for dynamic resource allocation in CRNs, many works assume that the radio environment can be sensed perfectly. However, in practice, it is difficult for the secondary network to have the perfect knowledge of a dynamic radio environment in CRNs. In this paper, we study the dynamic resource allocation problem for heterogeneous services in CRNs with imperfect channel sensing. We formulate the power and channel allocation problem as a mixed integer programming problem under constraints. The computation complexity is enormous to solve the problem. To reduce the computation complexity, we tackle this problem in two steps. First, we solve the optimal power allocation problem using Lagrangian dual method under the assumption of known channel allocation. Next, we solve the joint power and channel allocation problem using discrete stochastic optimization method, which has low computation complexity and fast convergence to approximate to the optimal solution. Another advantage of this method is that it can track the changing radio environment to allocate the resources dynamically. Simulation results are presented to demonstrate the effectiveness of the proposed scheme. Index Terms—Cognitive Radio, Imperfect Channel Sensing, Heterogeneous Services, Mixed Integer Programming, Discrete Stochastic Optimization.

I. I NTRODUCTION Cognitive radio is a promising technique to improve the utilization of radio spectrum. The main idea of cognitive radio is that the secondary users (SUs) can sense the radio environment, dynamically adapt communication parameters, and opportunistically utilize the temporarily unused spectrum resource licensed to primary users (PUs) [1], [2]. In cognitive radio networks (CRNs), resources in the secondary network should be dynamically allocated according to the sensed radio environment to maximize the utilization of radio spectrum, and at the same time, limit the interference to Copyright (c) 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was jointly supported by State Key Program of National Natural Science of China (Grant No. 60832009), the National Natural Science Foundation for Distinguished Young Scholar (Grant No. 61001115), Natural Science Foundation of Beijing, China (Grant No. 4102044), and the Natural Sciences and Engineering Research Council of Canada. R. Xie is with the Key Lab. of Universal Wireless Communication, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, P.R. China, and also with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail: [email protected]) F. R. Yu is with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail: richard [email protected]) H. Ji is with the Key Lab. of Universal Wireless Communication, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, P.R. China (e-mail: [email protected])

PUs. Dynamic resource allocation in CRNs has drawn a lot of attention recently. Authors of [3] study the resource allocation for underlay spectrum sharing technology and propose a power control scheme under the assumption that each SU has a minimum QoS requirement. The distributed multi-channel power allocation with QoS guarantee is studied in [4], and the authors propose a distributed power allocation algorithm by applying the Lagrangian dual decomposition to guarantee the QoS of SUs. Authors of [5] study the distributed price-based spectrum management in CRNs, where the resource allocation problem is modeled as a noncooperative game, and a pricebased iterative water-filling algorithm is designed to maximize the SU’s utility function. In [6]–[8], resource allocation is studied from a cross-layer perspective. Although some works [3]–[5], [9] have been done for dynamic resource allocation in CRNs, the authors mainly focus on the resource allocation on the assumption that the radio environment can be sensed perfectly by SUs. However, in practice, it is difficult for SUs to have the perfect knowledge of a dynamic radio environment due to hardware limitation, short sensing time and network connectivity issues in CRNs, where inaccurate channel state information as well as missing detections and false alarms of PUs can occur [10]. In addition, most existing works focus on only one type of service carried by SUs. However, with recent rapid developments of new wireless applications, future CRNs need to support heterogeneous services with diverse quality of service (QoS) requirements. To the best of our knowledge, dynamic resource allocation for heterogeneous services in CRNs with imperfect channel sensing has not been studied in previous works. In this paper, we extend our previous work [11] and study the resource allocation problem in CRNs with imperfect channel sensing. We assume that only the estimate of channel information can be obtained. Based on the imperfect channel information, the secondary base station makes resource allocation for SUs with heterogeneous services. We classify heterogeneous services by QoS requirements, e.g., SUs with minimum rate guarantee and SUs with best-effort service. Furthermore, we introduce the minimum rate constraint condition for SUs with minimum rate guarantee; For SUs with best-effort service, each SU may obtain different resource due to the discrepancy of channel quality. In order to solve the possible unfairness problem, we introduce a proportional fairness constraint. Based on these constraints and imperfect channel information, we formulate the resource allocation problem as a mixed integer programming problem. The computation complexity is enormous to solve the mixed integer programming problem with imperfect channel information. To reduce the computation complexity, we tackle

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the secondary base station. Next, the secondary base station makes decisions to determine whether or not the channels are idle. As we mainly focus on power and channel resource allocation in this paper, we do not describe the cooperatively spectrum sensing scheme in detail, which can be found in tot is the overall test statistic for [17]. If we assume that E m channel m collected by secondary base station, thus we can determine the channel state for channel m by comparing the tot with a decision threshold ε m , which total test statistic Em f can be selected by finding an optimal balance between P m d f d and Pm . Here, Pm and Pm denote the probability of false Fig. 1. Cognitive system model. The spectrum licensed to primary network alarm and the probability of detection, which are expressed as d tot tot is divided into M channels. ”Busy” means that the channels are occupied by f = Pr Em > εm |H0 , Pm = Pr Em > εm |H1 , Pm primary network, and ”Idle” means that secondary network could opportunistically utilize these channels.

this problem in two steps. First, we solve the optimal power allocation problem using Lagrangian dual method [12] under the assumption of known channel allocation. Next, we solve the joint power and channel allocation problem using discrete stochastic optimization method [13], which has been successfully used in operations research [14]. The discrete stochastic optimization method has low computation complexity and fast convergence to approximate to the optimal solution under imperfect channel information. The remainder of the paper is organized as follows. In Section II, the cognitive radio system model is presented. In Section III, the joint power and channel allocation scheme is proposed. In Section IV, the simulation results are illustrated. Finally, we conclude this study in Section V. II. S YSTEM D ESCRIPTION In this section, the system model for a cognitive radio network with heterogeneous services and imperfect channel sensing is presented. Then the resource allocation problem in this system is formulated. A. System Model We consider a cognitive radio system including a primary network and a secondary network as shown in Fig. 1, which is operated in the form of time slotted manner. There is a secondary base station and K tot SUs with heterogeneous services requesting in the secondary network. We assume that the primary network and secondary network take the orthogonal frequency-division multiple access (OFDMA) technology. There are M channels owned by the primary base station in the primary network. In each time slot, the secondary network can sense the M channels and opportunistically utilize the idle channels for heterogeneous services. The time slot for the secondary network consists of three parts: sensing time, resource allocation time and data transmission time. In the sensing time, the secondary network senses the M channels licensed to the primary network and determines the available idle channels. Channel sensing techniques have been extensively studied in the literature [15], [16]. Here we adopt the cooperatively centralized sensing method [15], where each SU senses the channels and sends the sensing information to

where H0 and H1 are the hypotheses that the channels are idle and busy, respectively. Based on the probability of false alarm and the probability of detection, for the sensing in channel m, we have the following four possible results. 1). Channel m is idle, and the decision for channel m at the secondary base station is idle; 2). Channel m is busy, and the decision for channel m at the secondary base station is idle; 3). Channel m is idle, and the decision for channel m at the secondary base station is busy; 4). Channel m is busy, and the decision for channel m at the secondary base station is busy. For the first and fourth cases, the channel state is sensed accurately. The second case is the miss-detection, and the third case is the false alarm detection. Since in this paper we assume that only the idle channels sensed can be used by the secondary network, thus we only consider the first two cases. As we mainly focus on the problem of resource allocation for SUs, without loss generality, we assume that the secondary base station senses N, N ≤ M , idle channels in a time slot and there is a spectrum access scheme (e.g. [18], [19]) that meets the collision constraints of PUs, based on which we design the power and channel resource allocation schemes. Here, when the sensed idle channels includes the second case, we can view the second case as a sensing error and has an interference for SUs during resource allocation, which will cause performance degradation due to sensing errors. Then during the resource allocation time, there are only K ≤ N, K ≤ K tot SUs with heterogeneous services could be accessed, the number of admitted SUs refers to the admission control and users scheduling, which will be studied in the future. In addition, we further assume that these K SUs have heterogeneous service requirements and can be classified into two classes: K 1 SUs with minimum rate guarantee and K 2 SUs with best-effort service. The corresponding sets of these two classes SUs can be denoted as KA and KB , respectively. Moreover, we assume that the secondary base station does not know the perfect channel information for these K SUs on N channels during the resource allocation. Only the estimate of channel information can be obtained. Following the resource allocation time, data is transmitted between SUs and the secondary base station in the data transmission time. B. Problem Formulation When the secondary base station obtains the estimate of channel information, the secondary base station will do the

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optimal resource allocation for SUs with heterogeneous service requirements. To solve the resource allocation problem, we have the following assumptions: • Let each channel be assigned to one SU. We use the binary index ρ k,n ∈ {0, 1} to represent the channel allocation. ρk,n = 1 denotes channel n is allocated to SU k; otherwise ρk,n = 0. • Total power constraint [20], [21]: Let P total denotes the total power budget at the secondary base station for the SUs in each time slot, and p k,n denotes the transmit power for SU k on channel n. Thus, we have K N 1 +K2

ρk,n pk,n ≤ Ptotal .

(1)

n=1

k=1

• Minimum rate guarantee: For SU k in K A , there is a minimum rate threshold to guarantee its transmission performance. Let Rkmin be the minimum rate threshold. The transmission rate for SU k should satisfy the following condition. Rk ≥ Rkmin , ∀k ∈ KA .

(2)

• Proportional fairness constraint: To guarantee the fairness for SU k in KB , we introduce the normalized proportional fairness factor γk . Therefore, we have R k = γk , ∀k ∈ KB , Ri

(3)

i∈KB

where γk , ∀k ∈ KB , is a predetermined value. Based on the above constraints, resources are allocated for SUs. Bandwidth capacity is used as the performance metric in resource allocation. The bandwidth capacity for SU k can be expressed as [22] N 1.5pk,n Gk,n ρk,n W log2 1 + Rk = , ∀k ∈ KA ∪KB , ln (0.2/BERtar ) n=1 (4) where W is the transmission bandwidth on each channel, and |h |2 is the BERtar denotes the target bit error rate, G k,n = k,n n0 signal-to-noise ratio, h k,n denotes the channel gain for SU k on channel n, and n 0 is the additive Gaussian white noise with zero mean and variance σ 2 . Our objective is to maximize the total capacity of CRNs under the constraints. The optimization problem can be formulated as max

pk,n ,ρk,n

K 1 +K 2

k=1

n=1

Rk

k=1

ρk,n pk,n ≤ Ptotal

pk,n ≥ 0, ∀k, ∀n ρk,n = {0, 1} , ∀k, ∀n K1 +K2 ρk,n = 1, ∀n k=1

Rk ≥ Rkmin , ∀k ∈ KA Rk Ri = γk ,∀k ∈ KB . i∈KB

III. J OINT P OWER AND C HANNEL A LLOCATION A LGORITHM The mixed integer programming formulation in (5) introduces both discrete and continuous variables, which will result in enormous computation complexity to solve it. To reduce the computation complexity, we tackle this problem in two steps. First, we present an optimal power allocation scheme using Lagrangian dual method [12] under the assumption of known channel allocation in Subsection III-A. Then, in Subsection III-B, we solve the joint power and channel allocation problem using discrete stochastic optimization method [13], [14], [23]. Finally, the computation complexity and the performance are analyzed in Subsection III-C. A. Optimal Power Allocation for SUs Under the Assumption of Known Channel Allocation Assume that the channel allocation is known, the secondary base station just needs to do the optimal power allocation for SUs to support heterogeneous services under the constraints. We can rewrite the optimization problem in (5) as K 1 +K2 1.5pk,n Gk,n W log2 1 + (6) max pk,n ln (0.2/BERtar ) n∈Ωk

k=1

Subject to:

K1 +K2 k=1

(5)

pk,n ≤ Ptotal

n∈Ωk

pk,n ≥ 0, ∀k, ∀n Ωk ∩ Ωi = ∅, ∀k = i, and ∀k, i ∈ KA ∪ KB Ω 1 ∪ Ω2 ∪ ... ∪ ΩK1 +K2 = {1, 2, ..., N } Rk,n ≥ Rkmin ,∀k ∈ KA n∈Ω k

n∈Ωk K 2

Rk,n

k=1 n∈Ωk

Subject to:

K1 +K2 N

The first inequality denotes that the power assigned to the SUs on different channels should satisfy the total power constraint. The fourth equality constraint represents that each channel can only be assigned to one SU. The last two constraints denote that the SUs with minimum rate guarantee and the SUs with best-effort service have the minimum rate constraints and the proportional fairness constraints, respectively.

= γk ,∀k ∈ KB ,

Rk,n

where Ωk denotes the set of channels assigned to SU k. The third constraint denotes that each channel is allocated to one SU. The optimization problem in (6) is still complex due to the last constraint. Fortunately, inspired by the [24], we can rewrite the last constraint equivalently as follows. R1 : R2 : ... : RK2 = γ1 : γ2 : ... : γK2 , ∀k ∈ KB .

(7)

Now substituting (7) into the last constraint in (6), the optimization problem in (6) is equivalent to solve the Lagrangian dual problem [25] at the top of next page where λ, β k and μk are the Lagrangian multiplier factors, and Γ k,n = 1.5Gk,n . ln(0.2/BERtar )

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K1 +K2

K1 +K2

L(λ, βk , μk , pk,n ) = W log2 (1+ pk,n Γk,n ) + λ pk,n − Ptotal + k=1 n∈Ωk k=1 n∈Ωk K K 1 2 γ1 min + βk W log2 (1 + pk,n Γk,n ) − Rk μk W log2 (1+p1,nΓ1,n ) − γk W log2 (1+pk,nΓk,n ) , k=1

n∈Ωk

k=2

n∈Ω1

n∈Ωk

(8)

For ∀k ∈ KA , we can differentiate (8) and get ∂L ∂pk,n

=

Γk,n W ln 2 1+Γk,n pk,n

Γk,n W +λ+βk ln 2 1+Γk,n pk,n , ∀n∈Ωk .

(9)

We can differentiate (8) for k = 1, k ∈ K B as ∂L ∂p1,n

Γ

1,n = lnW2 1+Γ1,n p1,n +λ+

K2

Γ

1,n μk lnW2 1+Γ1,n p1,n ,

(10)

k=2

K

and for k = {2, . . . , K 2 } , ∀k ∈ KB as ∂L ∂pk,n

Γ

γ1 k,n = lnW2 1+Γk,n pk,n +λ−μk γk

Γk,n W ln 2 1+Γk,n pk,n .

(11)

Let (9), (10) and (11) be equal to 0. For a given particular SU k and the corresponding channel set Ω k , we have Γk,n Γk,m = , 1 + Γk,n pk,n 1 + Γk,m pk,m

(12)

where m, n ∈ Ωk and k ∈{1, 2, . . . , K1 + K2 }. If we assume that the number of channels allocated to SU k is N k and let Γk,1 ≤ Γk,2 ≤ ... ≤ Γk,Nk , without loss generality, we can get the following equation from (12). pk,n = pk,1 +

Γk,n − Γk,1 , ∀n = {1, 2, . . . , Nk }, ∀k. (13) Γk,n Γk,1

So, let the total power allocated to SU k be P k,total , according to (13), we have Pk,total =

Nk

pk,n

k=1

Nk Γk,n − Γk,1 = Nk pk,1 + , ∀k. (14) Γk,n Γk,1 n=2

Therefore when the power allocated to SU k is known, we can realize the channels power allocation for SU k as (13) and (14). Hence, the important thing is how to allocate the power between all SUs. If we let Pk,total , ∀k ∈ KA ∪ KB , be the power allocated to SU k and transfer the inequality constraint condition of minimum rate guarantee to the equality constraint, according to (13), we can get

N k Γ W log2 Γk,n (1+Γ p ) = Rkmin , ∀k ∈ KA , (15) k,1 k,1 k,1 n=1

+ Γ1,1 p1,1 ) = n=1

Nk Γk,n 1 W log (1 + Γ p ) , ∀k ∈ KB . k,1 k,1 2 Γk,1 γk 1 γ1

N1

W log2

Γ1,n Γ1,1 (1

(16)

n=1

Then let Qk =

Nk Γk,n − Γk,1 , ∀k ∈ KA ∪ KB , Γk,n Γk,1 n=2

N N1 k k Γk,n , ∀k ∈ KA ∪ KB . Zk = Γ n=2 k,1

Substituting (14), (17) and (18) into (16), we have

P1,total −Q1 N1 W log (1 + Γ ) + log Z 1,1 2 2 1 = γ1

N1 P −Q Nk k,total k ) + log2 Zk , ∀k ∈ KB . γk W log2 (1 + Γk,1 Nk (19) Then we add the following total power constraint

(17)

(18)

Pk,total = Ptotal .

(20)

k=1

There are (K 1 + K2 ) equations in (15), (19) and (20). We can solve these (K1 +K2 ) equations to find the optimal power allocation for SUs. Alternatively, we can use the NewtonRaphson iterative method [24] or [26] to find the power allocation for SUs. After the power allocation for SUs, each SU can realize the optimal channels power allocation as (13) and (14). B. Joint Power and Channel Allocation Algorithm Based on the discussion in Subsection III-A, we see that the power allocation can be realized under the assumption of known channel allocation. In this subsection, we relax this assumption, and propose a dynamic joint power and channel allocation scheme based on the discrete stochastic optimization algorithm. First, let’s consider the channel allocation. Let the channel allocation indicator matrix be X = {x kn }K×N , where xkn = 1 denotes channel n is allocated to SU k; otherwise xkn = 0. The set of all possible P = K N channel combination subsets is denoted as Φ = {X 1 , X2 , . . . , XP }. Let the mapping from channel allocation indicator matrix X to channel gain matrix subset be H [X]. Then, the channel allocation process is described as follows. The secondary base station selects a channel allocation indicator matrix X p ∈ Φ K Rk , where Rk to optimize the system total capacity R = k=1

is defined in (4). Therefore, the problem of channel allocation can be formulated as a discrete stochastic optimization problem. (21) X∗ = arg max R (H [X]) , X∈Φ

where X∗ denotes the optimal channel allocation combination. The secondary base station can only obtain the estimate of channel information due to the imperfect channel state information estimation. Therefore, in each time slot, it’s necessary to compute the capacity for each channel allocation combination relying on a new channel estimation information, then choose an optimal channel allocation combination with maximum capacity. Suppose that at an iteration time l, the

Here, we can use the training estimation of the channel is H.

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sequence estimate method [27] to realize the channel estimation. And many proposed schemes, which are widely used in traditional OFDMA networks, may be used to increase the channel estimation efficiency, such as take advantage of the parametric channel estimation [28] to reduce the channel estimation error, use the limited feedback schemes (e.g. quantized CSI feedback, CQI feedback, CQRI feedback) [29] to reduce the feedback information of channel estimation. Then the sec [l, X] and ondary base station selects a channel gain subset H computes the relative noisy estimate of the capacity function R (H [X]) denoted as r (l, H [X]). If each r (l, H [X]) is an unbiased estimate of R (H [X]), r (l, H [X]) , l = 1, 2, . . . is a sequence of i.i.d. random variables. The problem of channel allocation can be rewritten as the following discrete stochastic optimization problem. X∗ = arg max R (H (Xp )) = arg max E {r (l, H [Xp ])} . Xp ∈Φ

Xp ∈Φ

(22) To solve (22), there are several methods that can be used. An inefficient method is to compute L estimates of the objective for each of the channel combination and compute the empirical average which approximates the exact value of the objective function. That is, we compute for each X p ∈ Φ, 1 r(l, H[Xp ]) , L L

r L (H[Xp ]) =

(23)

l=1

and then exhaustive search to find X ∗ = max r L (H [Xp ]). Xp ∈Φ

Since for any fixed X p ∈ Φ, r (l, H [Xp ]) is an i.i.d. sequence of random variables, by the strong law of large numbers, r L (H [Xp ]) → E {r (l, H [Xp ])} almost surely as L → ∞. Therefore, based on the channel allocation in (23) and the power allocation proposed in Subsection III-A, now we can give a joint power and channel allocation based on the exhaustive search algorithm as follows. Algorithm 1: Joint Power and Channel Allocation Based on Exhaustive Search Algorithm 1) Initialization: Determine the number of idle channels N and SUs K1 and K2 . 2) Each SU estimates the channel state information and feedbacks it to the secondary base station. 3) For each channel allocation combination, do the power L r (l, H [Xp ]). allocation and get r L (H [Xp ]) = L1 l=1

4) Sort r L (H [Xp ]) as the descending order, i.e. let r L (H [X1 ]) > r L (H [X2 ]) > ... > r L (H [XP ]). 5) Choose the channel allocation X ∗ = X1 . 6) Output the channel allocation and corresponding optimal power allocation. Although the algorithm presented above in principle can realize the optimal joint power and channel allocation, it is highly inefficient and induces enormous computation complexity. Therefore, we propose another algorithm based on the aggressive discrete stochastic approximation algorithm, which has been successfully used in operations research [14] and antenna selection [23]. Define ξ = {e 1 , e2 , ..., eP }, where ep denotes the (P × 1) vector with a one in the pth position and zeros elsewhere. At each iteration, the (P × 1)

probability vector π [l] = {π [l, 1] , π [l, 2] , ..., π [l, P ]} is updated, which represents the state occupation probabilities with element π [l, p] ∈ [0, 1] and p π [l, p] = 1. Let X(l) be the channel allocation indicator matrix chosen at the iteration l. For notational simplicity, we map the sequence of subsets X(l) to the sequence {D [l]} ∈ ξ, where D [l] = e p if X(l) = Xp , p ∈ 1, . . . , P . Therefore, the sub-optimal joint power and channel allocation based on the aggressive discrete stochastic approximation algorithm is as follows. Algorithm 2: Joint Power and Channel Allocation Using Aggressive Discrete Stochastic Approximation Algorithm 1) Initialization: Determine the number of idle channels N and SUs K1 and K2 . 2) Each SU estimates the channel state information and feedbacks it to the secondary base station. 3) The secondary base station selects a channel allocation indicator matrix X(1) ∈ Φ, and set π 1, X(1) = 1, π [1, X] = 0 for all X = X(1) . 4) For l = 1, 2, . . . , do a) Given X(l) combined with another uniformly chosen (l) ∈ Φ\X(l) at iteration time l, the secondary X base station does the power allocation and computes (l) (l) r l, H X . , r l, H X (l) (l) (l) ; , set X(l+1) = X < r l, H X b) If r l, H X otherwise set X(l+1) = X(l) . c) Then, the secondary base station updates all channel allocation subsets’ occupation probabilities: π [l + 1] = π [l] + [l] (D [l + 1] − π [l]) with the decreasing = 1l . step size [l] ∗(l) (l+1) > π l + 1, X , the base station d) If π l + 1, X sets X∗(l+1) = X(l+1) ; otherwise sets X∗(l+1) = X(l) . e) Output the channel allocation combination and corresponding optimal power allocation. f) Set l ← l + 1. end for. In the initialization process, we have K 1 +K2 ≤ N in order to guarantee that each SU can obtain at least one channel. This condition can be satisfied by appropriate admission control. Then, the secondary base station randomly selects the (l) in each iteration channel combination indicator matrix X time l, does the optimal power allocation as Subsection III-A and evaluates the objective function, which can guarantee the convergence and efficiency of the Algorithm 2. Then the algorithm compares the objective function and chooses one with larger objective value. Next, the state occupation probabilities of all the states are updated with the element N umber of chosen Xp . X∗(l) means that the π [l, p] = n most frequently chosen channels combination according to the state probability vector π [l]. To prove the convergence of Algorithm 2 presented above, we firstly present a sufficient condition based on [13] as follows. Lemma 1 (Sufficient convergent conditions) Algorithm 2 converges to the global maximizer of the objective function R (H [X]), if the independent observations r [l, H [X ∗ ]],

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, r [l, H [X]] satisfy the following conditions r l, H X P{r[l, H[X∗ ]] > r[l, H[X]]} > P{r[l, H[X]] > r[l, H[X∗ ]]}. (24) >P r[l, H[X]] > r l, H X . P r[l, H[X∗ ]] > r l, H X (25) It is pointed out in [13] that if the above two conditions are satisfied, the sequence X(n) is a homogeneous irreducible and aperiodic Markov chain with state space Φ, and it spends more time on X ∗ than other states. Condition (24) means that it’s more possible to move into the global optimum X ∗ from any other states than in the other direction. Condition (25) states that it has more chance to go to the optimal state X ∗ than to the other states when the state is not in a optimal state X∗ . Based on the above sufficient condition, we can prove the convergence of Algorithm 2 as follows. Theorem 1 (Global convergence). If the iteration is sufficient. Algorithm 2 converges to the global maximizer. Proof: To prove the convergence of the algorithm, a simplified condition is to check whether r [l, H [X]] − E {r [l, H [X]]} , X ∈ Φ, are i.i.d. symmetric random variables. If it’s satisfied, the algorithm proposed above will satisfy the sufficient convergence condition. The proof is given in [13]. Alternative, we can prove as follows. Let the mean and variance of the objective function for r [l, H [X]] be E {r [l, H [X]]} = μr[H[X]] and V ar {r [l, H [X]]} = 2 σr[H[X]] , respectively. We can get the statistical distribution of r [l, H [X]] through simulations. Let the empirical disn tribution function be F n (y) = n1 I{Yi μr(H[Xq ]) if r (H [Xp ]) > r (H [Xq ]). Therefore, we can rewrite the condition (24) as P{r[l, H[Xp ]]−r[l, H[Xq ]] > 0} > P{r[l, H[Xq ]]−r[l, H[Xp ]] > 0} .

(26)

Here, we notice that the difference of Gaussian variables still keeps the Gaussian property. Hence, the inequality (26) is equivalent to

2 2 +σ P N μr(H[Xp ]) −μr(H[Xq ]) , σr(H[X r(H[Xq ]) > 0 p ])

2 2 >0 . >P N μr(H[Xq ]) −μr(H[Xp ]) , σr(H[Xq ]) +σr(H[X p ]) (27) As μr(H[Xp ]) = max μr(H[X , μ , μ ]) r(H[X ]) r(H[X ]) p q s , we have μr(H[Xp ])−μr(H[Xq ]) > μr(H[Xq ])−μr(H[Xp ]) . Therefore, condition (24) is satisfied due to the same variance. For condition (25), it is equivalent to

2 2 P N μr(H[Xp ]) −μr(H[Xq ]) , σr(H[X >0 +σr(H[X p ]) q ])

2 2 >P N μr(H[Xs ]) −μr(H[Xq ]) , σr(H[X +σ r(H[Xq ]) > 0 . s ]) (28)

Then we can get the following inequality via extensive simulations after obtaining the mean values and variances, μr(H[Xs ]) − μr(H[Xq ]) μr(H[Xp ]) − μr(H[Xq ]) = . (29) 2 2 2 2 σr(H[Xp ]) + σr(H[Xq ]) σr(H[X + σr(H[X s ]) q ]) Therefore, condition (25) is also satisfied. Thus the convergence is proved. C. Computation Complexity and Performance Analysis In this subsection, the computation complexity of exhaustive search algorithm and aggressive discrete stochastic approximation algorithm is analyzed. The exhaustive search algorithm requires to compute the L estimated objective functions for all possible channel combinations for SUs. Therefore, for (K1 + K2 ) SUs and N idle channels, there are (K 1 + K2 )N possible combinations. Moreover, for each channel combination, L estimation objectives for each SU should be computed.

N operations Thus, this requires O L (K1 + K2 ) (K1 + K2 ) to get the optimal resource allocation, which causes enormous computation complexity. By contrast, the Algorithm 2 chooses one of the channels allocation subset in each iteration, computes its objective function, and compares with the objective value corresponding to the channels subset selected in the last iterative time. Therefore, it just needs to have O (2L (K1 + K2 )) operations, which can reduce the computation complexity significantly. The performance of the proposed algorithms will be affected by channel sensing errors. For example, the secondary base station may view channel n as an idle channel even though channel n is occupied by a PU. If a training sequence s n is used in channel n for channel estimation, the received signal by SU k is yk,n = hk,n sn + ψk,n + ϑpk (n) ,

(30)

where ψk,n is a i.i.d. circularly symmetric complex Gaussian variables with unit variance (i.e., ψ k,n ∈ Nc (0, 1)), ϑpk (n) = Pp (n) gpk (n) is the interference from primary base station to SU k on channel n, P p (n) is the transmission power for primary base station on channel n and g pk (n) is the channel gain from primary base station to SU k on channel n. We can get the maximum likelihood estimate [27] of hk,n as

= H + (Ψ + Δ) SH , H

(31)

where S = diag (s) denotes a diagonal matrix with the diagonal elements being the entries of vector s = {s 1 , . . . , sn }, and Δ is the K × N interference matrix. Δ can be described as follows. ⎡ ⎤ ϑp1 (1) ... ϑp1 (N ) ⎦ ... ... , (Δ)K×N = ⎣ ϑpK (1) ... ϑpK (N ) K×N where the element of ϑ pk (n) is the interference from primary base station to SU k on channel n. When there is a sensing error on channel n, we have ϑ pk (n) = Pp (n) gpk (n), otherwise ϑpk (n) = 0, when the secondary base station senses correctly on channel n. Thus, when the optimal training sequence is

7

designed [27], we can get the distribution of channel estimate with sensing errors as

2

hk,n ∼ N hk,n + ϑpk (n) sH . (32) n , 1 + (ϑpk (n))

Total Capacity (bits/s/Hz)

3.7

hk,n (t) = υhk,n (t − 1) + εςk,n (t) ,

(33)

where υ and ε are fixed parameters related through ε = 1/2 , and ςk,n ∼ N (0, 1). Here, we set ν = 0.9. In 1 − ν2 the simulation results, we use “exhaustive search scheme” to represent the exhaustive search scheme in Algorithm 1, “proposed scheme” to represent the aggressive discrete stochastic approximation scheme in Algorithm 2, “perfect CSI” to represent the resource allocation scheme based on perfect channel state information, “existing scheme” to represent the random channel allocation scheme [30] together with optimal power allocation. In Fig. 2, we show the convergence performance of the proposed resource allocation scheme over the number of time slot. From the figure, we can observe that the proposed scheme and exhaustive search scheme converge to the optimal value as the number of time slot increases. Since the estimated objective function is an unbiased estimate of the objective function, as the number of time slot increases, the exhaustive search scheme and proposed scheme approach to the optimal solution with perfect channel information. At the same time,

3.5

3.4 Perfect CSI Exhaustive Search Scheme Proposed Scheme Existing Scheme

3.3

3.1

0

50

100

150 200 Number of Time Slot

250

300

Fig. 2. The convergence in term of total capacity of the cognitive radio network over the number of time slot.


4.2

IV. S IMULATION R ESULTS AND D ISCUSSIONS In this section, we study the performance of the proposed dynamic resource allocation algorithms using computer simulations. In the simulations, we assume that all SUs randomly locate in a rectangular area and communicate with the secondary base station. The size of rectangular area is 100m × 100m. The simulation parameters are set as follows. The spectrum resource licensed to the primary network is divided into 64 channels, and the bandwidth of each channel is 20K Hz. We assume that the total power is 1W , and there are 4 SUs with ratio K1 /K2 = 1 and 6 idle channels. The noise power spectrum density on each channel is N 0 = −100dBm. We also assume that the channel gain from secondary base station to SU k on channel n is slowly time-varying, which is modeled as [23]

3.6

3.2


When the channel estimate follows the Gaussian distribution, we know that the estimated objective function in each iteration time still follow the Gaussian distribution even with sensing error. Similar to the analysis in Subsection III-B, we can get the statistical distribution of the estimated objective function distribution with sensing error through simulations, and use the Kolmogorov-Smirnov test to verify the distribution of estimated objective function. Nevertheless, the mean and standard variance will deviate the optimal estimate of the objective function. The deviation of the estimated objective function will cause the secondary base station to choose other channel allocation combinations rather than the optimal channels combination, which in turn results in degradation of system performance. We will study the performance degradation due to sensing errors through simulations in the next section.

3.8

4

3.8

3.6

3.4

3.2

3

Fig. 3. slot.

0

20

40 60 Number of Time Slot

80

100

The total capacity of the cognitive radio network versus the time

the proposed scheme can reduce the computation complexity significantly compared to the exhaustive search scheme. In Fig. 3, we verify the time-varying channel tracking capability of the proposed scheme from the qualitative perspective. We show that the proposed scheme can adapt to the time-varying radio environment and have a good tracking capability to approximate the optimal solution with perfect channel information. We evaluate the system performance over the number of time slots. The time-varying channel model is based on equation (33). There are K = 3 SUs with K 1 = 1 and K2 = 2, and the number of idle channels N = 5. From the figure, we can see that the obtained total capacity is varying due to the time-varying radio environment in different time slots. And the proposed scheme can track the optimal solution in the time-varying radio environment. In Fig. 4, we evaluate the performance of proposed scheme in term of the total capacity over different total power constraints. In the simulation, there are 4 SUs, 6 idle channels, and the ratio of SUs is K 1 /K2 = 1. We assume that there is no sensing error and no estimation error in the idle channels. In the figure, we can observe that the total capacity achieved by the exhaustive search scheme and the proposed scheme is significant better than that in the existing scheme, and approaches to that in the perfect CSI scheme, which means

8

4.5

4

K1/(K1+K2)=1 4.4

K1/(K1+K2)=1/2

3.5

K1/(K1+K2)=0

3

Capacity (bits/s/Hz)


4.3


2

4.2 4.1 4 3.9

1.5 3.8 1 0.2

Fig. 4. power.

0.4

0.6 0.8 Total Power (W)

1

3.7

1.2

The total capacity of the cognitive radio network versus the total

4

6

8

10 Number of Users

12

14

16

Fig. 6. The total capacity for different SUs’ ratios over the number of SUs. (K1 denotes the number of SUs with minimum rate guarantee, and K2 denotes the number of SUs with best effort service.)

4 K1/K2=2 K1/K2=1 K1/K2=1/2

3

2.5 Total Capacity (bits/s/Hz)


3.5

3

2.5

2

Proposed Scheme 20% of Channel Sensing Error 40% of Channel Sensing Error 60% of Channel Sensing Error

2

1.5

1 1.5 0.2

0.4

0.6 0.8 Total Power (w)

1

1.2

Fig. 5. The total capacity for different SUs’ ratios. (K1 denotes the number of SUs with minimum rate guarantee, and K2 denotes the number of SUs with best effort service.)

that the proposed resource allocation algorithm can obtain a good performance under the condition of imperfect channel information. In Fig. 5, we compare the performance of total network capacity for different ratios of SUs with heterogeneous services. In the simulation, there are 8 idle channels and the total number of SUs is K 1 + K2 = 6, including the SUs with minimum rate guarantee and the SUs with best effort service. The ratios of the number of SUs with minimum rate guarantee and the number of SUs with best effort service are set as K1 /K2 = 2, K1 /K2 = 1, and K1 /K2 = 1/2, respectively. From the figure, we can see that the total capacity increases with the total power increasing. Furthermore, when the ratio of SUs increases, the total capacity also increases. In the optimization problem (5), we introduce the proportional fairness constraint for SUs with best effort service, which has a tradeoff between the total capacity and fairness for SUs with best effort service. Hence, when the ratios of SUs with heterogeneous services increases, the number of SUs with best effort service decreases. In Fig. 6, we evaluate the total network capacity over different numbers of SUs. In the simulation, the total power constraints is 1W , and the available idle channels is 20. From

0.5 0.2

0.4

0.6 0.8 Total Power (W)

1

1.2

Fig. 7. The total capacity over total power with sensing errors. (The total number of channels is 5, and the total number of SUs is 4, with the ratio K1 /K2 = 1.)

the figure, we can observe that the total capacity increases with the total number of SUs increasing. When the ratio of K1 /(K1 + K2 ) increases, the total capacity also increases. This means that the secondary network can obtain better system performance when more SUs with minimum rate guarantee is admitted. This is because that there is a tradeoff between the total capacity and the proportional fairness for SUs when the number of SUs with best-effort service is more than the number of SUs with minimum rate guarantee. In Fig. 7, the total capacity performance is studied with different sensing error situations. In the simulation, the number of idle channels is 5, and the number of SUs is 4 with ratio K1 /K2 = 1. The channel sensing error randomly occurs on all channels. We use the distribution of channel estimate with sensing errors based on (32). In the figure, the percentage of channel sensing error denotes the percentage of the number of channels that are sensed erroneously. From the figure, we can see that the performance significantly decreases as the sensing error increases. This is because that the channel allocation will deviate the optimal channel allocation combination under the sensing errors, which in turn will affect power allocation. In Fig. 8, we analyze the performance degradation over

9

1.4

4 Perfect CSI Exhaustive Search Scheme Proposed Scheme Existing Scheme

1.2 Capacity of Each SU (bits/s/Hz)


3.5

3

2.5

γ(1:1:1:1:1) γ(2:1:1:1:1) γ(4:1:1:1:1)

1

0.8

0.6

0.4

2 0.2

1.5

Fig. 8.

0

0.1

0.2 0.3 0.4 Percentage of Channel Sensing Error

0.5

0.6

The total capacity over the percentage of channel sensing error.

1

0

1

2

3 4 Secondary Users’ Index

5

Fig. 10. Capacity for each SU. (γ is the proportional fairness factor vector.)

the proportional fairness factor.

Minimum Rate of Each SU (bits/s/Hz)

0.9 Reference Minimum Rate Proposed Scheme Existing Scheme Method in [5]

0.8 0.7

V. C ONCLUSIONS AND F UTURE W ORK

0.6 0.5 0.4 0.3 0.2 0.1 0

Fig. 9.

1

2

3 4 Secondary Users’ Index

5

Minimum rate obtained for each SU.

the percentage of the channel sensing error. The simulation parameters are the same as the above. From the figure, we can observe that as the percentage of channel sensing error increases, the performance of total capacity in the secondary network decreases. This is because, when the sensing error happens, the distribution of channel estimation has a deviation, which will affect the channel allocation and power allocation. Therefore, the performance of total capacity decreases. In Fig. 9 and Fig. 10, we evaluate the performance for SUs with only one type of service (i.e., only SUs with minimum rate guarantee or only SUs with best effort service). The total power constraint is 1W and the number of available idle channels is 8. In Fig. 9, we compare the obtained minimum rate for SUs with minimum rate guarantee. In the simulation, we assume that there are 5 SUs with minimum rate guarantee. The reference minimum rate denotes the minimum rate requirement for SUs. We can see that the minimum rate requirement for SUs is guaranteed in the proposed scheme. However, the existing scheme and the method in [9] could not guarantee the minimum rate requirement. In Fig. 10, we also assume that there are 5 SUs with only best effort service, and we evaluate the capacity of each SU over the different proportional fairness factors. From the figure, we can observe that we can guarantee the proportional fairness for SUs and obtain the expectation capacity for particular SU by adjusting

In this paper, we have studied the problem of resource allocation in cognitive radio networks supporting heterogeneous services with imperfect channel sensing. We considered imperfect channel sensing and the QoS requirements for SUs with heterogeneous services. We have formulated the problem of resource allocation as a mixed integer programming problem. To reduce the computation complexity in the formulation with the imperfect channel information, an aggressive discrete stochastic approximate algorithm based joint power and channel allocation has been proposed, which has been proved to be able to fast converge to the optimal solution. Finally, simulation results have been presented to demonstrate the performance of proposed scheme. From the simulation results, we have verified the convergence of proposed scheme and the time-varying tracking capability. Then we have analyzed the performance effect under sensing errors. In our future work, we are mainly consider the issue that how to do joint access control and resource allocation to maximize the total system capacity and minimize the interference to PUs when there are sensing errors in the secondary network. ACKNOWLEDGMENT We thank the reviewers for their detailed reviews and constructive comments, which have helped to improve the quality of this paper. R EFERENCES [1] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [2] F. R. Yu, M. Huang, and H. Tang, “Biologically inspired consensusbased spectrum sensing in mobile ad hoc networks with cognitive radios,” IEEE Networks, pp. 26–30, May 2010. [3] L. B. Le and E. Hossain, “Resource allocation for spectrum underlay in cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5306–5315, Dec. 2008. [4] Y. Wu and D. H. K. Tsang, “Distributed power allcation algorithm for spectrum sharing cognitive radio networks with QoS guarantee,” in Proc. IEEE INFOCOM’09, Rio de Janeiro, Brazil, Apr. 2009.

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Renchao Xie is currently working toward the Ph. D. degree with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications (BUPT), Beijing, China. From November 2010 to November 2011, he visited Carleton University, Ottawa, ON, Canada, as a Visiting Scholar. His current research interests include game theory, resource management in cognitive radio networks, energy efficient resource management in heterogeneous networks and cognitive cooperative communication. Renchao Xie has served on the Technical Program Committee (TPC) of the 2012 IEEE Vehicular Technology Conference (VTC)-Spring. He also served for serval journals and conferences as a reviewer, including ACM/Springer Wireless Networks, EURASIP Journal on Wireless Communications and Networking, (Wiley)Wireless Communications and Mobile Computing, the 2011 IEEE Global Communications Conference and so on. F. Richard Yu (S’00-M’04-SM’08) (S’00-M’04SM’08) received the PhD degree in electrical engineering from the University of British Columbia (UBC) in 2003. From 2002 to 2004, he was with Ericsson (in Lund, Sweden), where he worked on the research and development of 3G cellular networks. From 2005 to 2006, he was with a start-up in California, USA, where he worked on the research and development in the areas of advanced wireless communication technologies and new standards. He joined Carleton School of Information Technology and the Department of Systems and Computer Engineering at Carleton University in 2007, where he is currently an Associate Professor. He received the Ontario Early Researcher Award in 2011, Excellent Contribution Award at IEEE/IFIP TrustCom 2010, the Leadership Opportunity Fund Award from Canada Foundation of Innovation in 2009 and best paper awards at IEEE/IFIP TrustCom 2009 and Int’l Conference on Networking 2005. His research interests include cross-layer design, security and QoS provisioning in wireless networks. Dr. Yu is a senior member of the IEEE. He serves on the editorial boards of several journals, including IEEE Transactions on Vehicular Technology, IEEE Communications Surveys & Tutorials, ACM/Springer Wireless Networks, EURASIP Journal on Wireless Communications Networking, Ad Hoc & Sensor Wireless Networks, Wiley Journal on Security and Communication Networks, and International Journal of Wireless Communications and Networking. He has served on the Technical Program Committee (TPC) of numerous conferences, as the TPC Co-Chair of IEEE INFOCOM-GCN’2012, ICCGCN’2012, VTC’2012S, Globecom’11, INFOCOM-GCN’2011, INFOCOMCWCN’2010, IEEE IWCMC’2009, VTC’2008F and WiN-ITS’2007, as the Publication Chair of ICST QShine 2010, and the Co-Chair of ICUMTCWCN’2009.

Hong Ji received the B.S. degree in communications engineering and the M.S. and Ph. D degrees in information and communications engineering from the Beijing university of Posts and Telecommunications (BUPT), Beijing, China, in 1989, 1992, and 2002, respectively. From June to December 2006, she was a Visiting Scholar with the University of British Columbia, Vancouver, BC, Canada. She is currently a Professor with BUPT. She also works on national science research projects, including the Hi-Tech Research and Development Program of China (863 program), The National Natural Science Foundation of China, etc. Her research interests include heterogeneous networks, peer-to-peer protocols, and cognitive radio.