具有QoS约束的频谱共享的分布式功率控制和随机接入.pdf

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Distributed power control and random access for spectrum sharing with QoS constraint

Introduction

Related work

System model

Problem formulation

Distributed algorithm design via layered decomposition method

Equivalent convex formulation

Primal-dual decomposition

Joint random access and power control under interference temperature constraint

Simulation results

Simple network scenario

General network scenario

Conclusions and future work

References

Computer Communications 31 (2008) 4089–4097 Contents lists available at ScienceDirect Computer Communications j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m c o m Distributed power control and random access for spectrum sharing with QoS constraint q Bo Yang a,b,*, Yanyan Shen b, Gang Feng a, Chengnian Long b, Zhong-Ping Jiang d, Xinping Guan b,c a Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong SAR, PR China b Center for Networking Control and Bioinformatics, The Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, PR China c Department of Automation, School of Electronic, Information, and Electrical Engineering, Shanghai Jiaotong University, Shanghai 200240, PR China d Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA a r t i c l e i n f o a b s t r a c t Article history: Received 24 December 2007 Received in revised form 20 August 2008 Accepted 20 August 2008 Available online 29 August 2008 Keywords: Wireless communication Spectrum sharing Dynamic spectrum access Constrained optimization 1. Introduction Distributed spectrum sharing with minimum quality of service (QoS) requirement and interference tem- perature (IT) constraint is studied in this paper. This problem can be formulated as a non-convex optimi- zation problem with conﬂicting constraints. To make solutions to this problem feasible, random access and power control are jointly considered. The challenges in solving this problem arise from the coupling in utility functions, the conﬂicting constraint sets, and coupled control variables. Moreover, there is no centralized controller or base station in networks to coordinate unlicensed users’ transmission and pro- tect active users’ QoS under IT constraint. By introducing variable substitution and transformation, we derive a distributed random access and power control algorithm that can achieve global optimal solution to the original problem. Convergence of the algorithm is proven theoretically. Simulation results demon- strate that both QoS guarantee and interference avoidance can be achieved even with channel gain variations. Ó 2008 Elsevier B.V. All rights reserved. The frequency spectrum for radio signal transmission is cur- rently allocated in a static manner, where bands are licensed to users by government agencies and thus licensed users have the exclusive rights to access the allocated band. At the same time, studies show that the spectrum has been used inefﬁciently in li- censed bands [1]. In contrast, the spectrum left by licensed users cannot meet the increasing demands for radio resources. To solve this demand and utilization dilemma, dynamic spectrum sharing approaches appear to be a good alternative. One of the approaches, called the underlay sharing approach exploits the spread spectrum techniques in cellular networks. Once a spectrum allocation map has been acquired, unlicensed devices can coexist with licensed users on the same frequency, provided that unlicensed users can limit the interference to licensed users. To quantify the interfer- q This work was supported in part by the Research Grants Council of Hong Kong SAR, P.R. China, under Grant Cityu 112907, the China National Outstanding Youth Foundation under Grant 60525303, NSFC under Grant 60604012, 60804030. This material was also based upon work supported by the National Science Foundation under Grant OISE-0430145. Part of this work was accomplished when the ﬁrst author was visiting the Polytechnic Institute of New York University under a WICAT Fellowship. * Corresponding author. Tel.: +852 2784 4601. E-mail addresses: bo.yang@ieee.org (B. Yang), megfeng@cityu.edu.hk (G. Feng), dylcn@ysu.edu.cn (C. Long), zjiang@control.poly.edu, zjiang@poly.edu (Z.-P. Jiang), xpguan@ysu.edu.cn (X. Guan). 0140-3664/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2008.08.016 ence, the Federal Communication Commission (FCC) proposed a new metric named the interference temperature (IT) [2]. The IT is deﬁned to be the radio frequency power measured at a receiving antenna per unit bandwidth. In this paper, we focus on the underlay spectrum sharing in ad hoc networks where all unlicensed users spread their transmission powers across the entire available band under the IT constraint. In this context, power control is used to limit interference among unlicensed users and to licensed receivers and, hence, maximize spectrum usage. The signal-to-interference-and-noise ratio (SINR) at the unlicensed receiver determines the successful transmission at physical layer. However, to satisfy the minimum SINR require- ment that characterizes the quality of service (QoS) constraint on the bit-error rate at individual receivers, the transmitting power of an unlicensed user may exceed its maximum value or even vio- lates the IT constraint. To balance the minimum SINR requirement and IT constraint, and further to efﬁciently and fairly utilize spec- trum, transmission power and channel access must be determined by coordination among unlicensed users [3]. In ad hoc networks without infrastructure, this coordination should be implemented distributively. Taking these factors into consideration, we formulate a joint random access and power control problem under IT and QoS con- straints. To protect the primary transmission, some nodes called measurement points are deployed to monitor the real-time IT of the concerned band at their locations. Due to the interference rela- tionship among wireless links and coupling between transmission

4090 B. Yang et al. / Computer Communications 31 (2008) 4089–4097 power and channel access, the formulated problem is non-convex and non-separable in control variables. After variable transforma- tion and introduction of an auxiliary variable, the optimization problem can be turned into a convex one. Then a distributed ran- dom access and power control algorithm is proposed within the framework of layering as optimization decomposition [4]. The transmission power of a link is updated not only to maintain its own QoS but also to limit interference to peer unlicensed users as well as licensed users. The random access of a transmission is aimed at satisfying its own channel access requirement and sup- porting other active links’ QoS requirement. The only involvement of a measurement point in the implementation of this algorithm is that when the aggregate IT violates the given upper bound the measurement point will broadcast the current IT to all the unli- censed transmitters. Furthermore, we prove that the proposed algorithm can converge to the global optimum. Simulation demon- strates that convergence of this algorithm can be ensured even with channel gain variations. The rest of this paper is organized as follows. In Section 2, we review some related work on spectrum sharing and non-convex optimization. The system model is presented in Section 3. The spectrum sharing problem is formulated in Section 4. In Section 5, we propose our joint random access and power control algo- rithm for spectrum sharing. Convergence results of the algorithm are also given in Section 5. Section 6 includes performance evalu- ation of the proposed algorithms. We conclude the paper and give future research direction in Section 7. 2. Related work Spectrum sharing between licensed users and unlicensed users can be classiﬁed as overlay and underlay schemes. In the overlay spectrum sharing unlicensed users access the network dynamically by using a portion of spectrum that has not been used by licensed users. For more discussions on the overlay spectrum sharing, see [5,2]. We focus on the underlay spectrum sharing, especially the scheme based on IT model. In [6], Huang et al. proposed the uplink spectrum sharing algorithm based on auction theory. They charac- terized the selﬁshness of unlicensed users without considering the maximum transmission power and QoS constraints of individual users. The base station, which is co-located with IT measurement point, allocates the uplink transmission powers proportionally to unlicensed users’ bids. Wang et al. [7] investigated the individual users’ throughput maximization under IT constraint. To penalize the individual users’ selﬁshness, they relied on the pricing scheme in [8] to limit strong interference in power control. However, the pricing scheme with the participation of a centralized base station (BS) cannot be implemented in ad hoc networks. It is noted that all these literatures consider how to utilize spectrum without QoS constraint. Xing et al. [3] formulated the spectrum sharing under IT and QoS constraint as a geometric programming problem in cel- lular networks. To make it possible for each active user to maintain minimum QoS requirement under IT constraint, some active unli- censed links will be switched off temporarily by solving a central- ized operator problem at BS. To mitigate the implementation overhead, a learning automata algorithm is proposed in [3] and BS needs to collect each unlicensed links’ SINR and aggregate IT for updating uplink access probability. A non-cooperative power control game with IT constraint was formulated by Jia and Zhang in [9] for ad hoc networks, where the IT measurement point in- forms the unlicensed link that generates the maximum received power to back off when current measured IT exceeds the given upper bound. However, the back-off scheme in [9] did not take into account QoS requirement of unlicensed users. The key difference between those works mentioned and ours is that this paper studies a distributed spectrum sharing scheme to satisfy the minimum QoS and IT constraints simultaneously in ad hoc networks. Similar log transformation approach was proposed in [10] to derive power control in cellular networks by solving a non-convex optimization problem satisfying QoS constraints. In our formulation, there is also the coupling between different kinds of control variables, i.e., transmission power and channel access probability. This coupling together with coupled utility and con- straint sets have to be tackled when searching for an efﬁcient algo- rithm which can be easily implemented. 3. System model The wireless network considered here is formulated as a set L of radio links with each link corresponding to an unlicensed trans- mitter and a receiver. Each transmitter is assumed to have a ﬁxed channel gain to its intended receiver as well as ﬁxed gains to all other receivers in the network. The quality of each link is deter- mined by the SINR at the intended receiver. In a network with j L j interfering links we denote the SINR for the ith user as ci ¼ P Giipi j–i;j2LGijpjÞ=S ; g2 þ ð where Gij > 0 is the channel gain from the transmitter of the jth link to the receiver of the ith link, pi is the power of the ith transmitter, g2 is the background noise power that is assumed to be the same for all users, and S is the normalized spreading sequence. The IT is a measurement of the power and bandwidth occupied by interference. Interference temperature TI is speciﬁed in Kelvin and is deﬁned as [11] ; kB TIðfc; BÞ ¼ PIðfc; BÞ where PIðfc; BÞ is the average interference power in Watts centered at fc; covering bandwidth B measured in Hertz. Boltzmann’s constant k is 1:38 1023 J/K [11]. For a given bandwidth, FCC imposes an IT limit to restrict the interference to licensed receiver at a given frequency in a particular location. In this paper, we assume that there are measur- ing points monitoring the real-time IT at particular band in their loca- tions. Thus, if the aggregate IT at measurement points is lower than a given threshold, the protection to the licensed receiver can be achieved. Then the constraint that the total received power does not exceed that threshold at a measurement point is given by ð1Þ i¼1 where G0i is the channel gain from user i0s transmitter to the measure- ment point, and T > 0 is a pre-deﬁned threshold. The model (1) con- siders that only a single measurement point is deployed. Under the scenario that multiple measurement points are deployed, all IT con- straints should be satisﬁed simultaneously. Our method proposed la- ter can be extended to the multiple measurement points case. XjLj piG0i 6 T; 4. Problem formulation The demand for QoS from unlicensed transmission is denoted by a utility of SINR, ulðclÞ; 8l; where ulðÞ is a twice differentiable, increasing function of cl: The minimum QoS requirement of link l is cl0: The social utility maximization problem with QoS and IT con- straints can be formulated as follows: P PjLj i¼1 l2L ulðclÞ maxp2P subject to cl P cl0 8l; piG0i 6 T; ðP1Þ

B. Yang et al. / Computer Communications 31 (2008) 4089–4097 4091 l where P ¼ fpl; l 2 L j 0 6 pl 6 pmax g: To introduce the criterion of ﬁnding a feasible solution p 2 P that solves (P1), a normalized link ( gain matrix G is deﬁned as Gjl ¼ cl0 0 for l–j; for l ¼ j: Gjl=S Gll " The normalized noise vector is # n ¼ c10g2 G11 c20g2 G22 ; ; . . . ; cjLj0g2 GjLjjLj T ; and g ¼ ½G01; . . . ; G0jLjT; pmax ¼ ½pmax With the similar analysis as in [3], the criterion of ﬁnding a feasible solution p 2 P that solves (P1) is given as qðGÞ < 1, ðI GÞ1 n pmax; and ððI GÞ1nÞTg 6 T: ; . . . ; pmax jLj T: 1 However, the feasible solution to (P1) may not exist given a set of active links. One alternative method is to let some transmission pairs be switched off temporarily to support other active links’ QoS requirement (minimum SINR) under IT constraint (1) as in [3]. If the lth link is active, its SINR cl is given by cl ¼ P Gllpl j–lGljpjrjÞ=S ; ð2Þ g2 þ ð where rj is an indicator variable deﬁned as rj ¼ 1; jth link is active; 0; otherwise: ð3Þ Given the expression of SINR in (2), problem (P1) turns into Due to the fact that rl; 8l are binary values, it is difﬁcult to design distributed algorithm solving (P1)0 efﬁciently. Thus, we relax the binary valued constraint on the variable rj in (3) and replace the discrete variable rj 2 f0; 1g by a continuous variable qj 2 ½0; 1: The variable qj can be thought of as the proba- bility of the transmitter of the jth link being active in a transmis- sion slot. Thus, by the relaxation on the binary valued variable rj; the problem of joint power control and scheduling is trans- formed into one of power controlled random access, where the objective is to determine for each link the probability of transmis- sion and the transmission power in order to share the spectrum efﬁciently and fairly with the minimum SINR and IT constraints. We now express the problem of power controlled random access as a constrained optimization problem ðulðclÞ þ xvlðqlÞÞ max p2P;q2Q subject to cl P cl0 8l 2 L; P P ðP2Þ l2L plqlG0l 6 T; l2L where P ¼ fpl; l 2 L j 0 6 pl 6 pmax x > 0 is a constant. We use the average SINR l g; Q ¼ fql; l 2 L j 0 6 ql 6 1g; P cl ¼ g2 þ Gllpl j–lGljpjqj =S to quantify the QoS. The second item vlðqlÞ in the objective function is an increasing, differentiable, and concave function of ql; which max p 2 P r 2 f0; 1g subject to ðP1Þ0 P l2L ulðclÞ PjLj cl P cl0 8l; plG0lrl 6 T: l¼1 captures the delay requirement for the lth link. xvlðqlÞ also serves as a regulation term that enables (P2) to be transformed into a con- vex optimization problem shown later. The minimum acceptable SINR criterion for error free packet reception is now replaced by a minimum expected SINR criterion, i.e., cl P cl0 8l 2 L: In general, problem (P2) is non-convex and non-separable, thus extremely difﬁcult to solve distributively for the global optimum. We show next that under a readily veriﬁable sufﬁcient condition on curvatures of the utility functions, the problem (P2) can be turned into a convex and separable optimization problem. 5. Distributed algorithm design via layered decomposition method 5.1. Equivalent convex formulation First, we give the following notations. Let yl ¼ cl and logarithmic change of variables: ^ql ¼ log ql; ^pl ¼ log pl 8l. Correspondingly, the domains of these variables by excluding the boundary region can g; ^Q ¼ f:^ql; l 2 L j be denoted as ^P ¼ f^pl; l 2 L j M 6 ^pl 6 log pmax P M 6 ^ql 6 0g; ^Y ¼ f^yl; l 2 L j M 6 ^yl 6 log ymax g; where M is a 6 pmax l Gll=g2: Let sufﬁciently large positive constant. Note that ymax ^ulð^ylÞ ¼ ulðe^ylÞ; ^vlð^qlÞ ¼ vlðe^qlÞ; ^Ilð^pl; ^qlÞ ¼ g2 þ j–leðlog Glj=Sþ^pjþ^qjÞ and rewrite the constraint sets in (P2) with respect to the logarithmic change of variables as follows: l l l ð ð ð ^Cl ^yl; ^p; ^ql ^Dl ^p; ^ql ^El ^p; ^qð Þ ¼ ^yl þ log ^Il ^pl; ^ql Þ ¼ log cl0 þ log ^Il ^pl; ^ql Þ ¼ log G0le^plþ^ql X ! ð log T 6 0: Þ ^pl log Gll 6 0; Þ ^pl log Gll 6 0; l ð4Þ ð5Þ ð6Þ Using geometric programming [12] and variable substitute tech- nique, we can convert the optimization problem (P2) into a convex programming problem (P3) ð ^ul ^ylð Þ þ xvl ^qlð P Þ Þ ðP3Þ max subject to ð4Þ—ð6Þ l2L : The main result is given in the following Theorem. Theorem 1. Given that v00 lðylÞ=yl for all l 2 L; the non-convex programming problem (P2) can be equiv- alently converted to the convex programming problem (P3). lðqlÞ=ql; u00 l ðylÞ P u0 l ðqlÞ P v0 Proof. Without loss of generality, we can replace the equality yl ¼ cl for all l; by an inequality in the constraint (4) and rewrite lulðylÞ: We also rewrite one of the objective functions the constraints in (P2) with standard geometric programming form P lulðclÞ as P P ðP3Þ0 l2L max subject to yl Il pl; ql ðulðylÞ þ xvlðqlÞÞ ð Þ Gllpl ð P 6 1; cl0c1 T1 G0lqlpl 6 1: l l2L Þ1 6 1; ð7Þ Note that the ﬁrst inequality in (7) will always be achieved with an equality at the optimum. The next step of this problem reformula- tion is to take log of both sides of the constraint in (7) and a log change of variables: ^yl ¼ log yl; ^ulð^ylÞ ¼ ulðe^ylÞ; ^vlð^qlÞ ¼ vlðe^qlÞ; ^ql ¼ log ql; ^pl ¼ log pl 8l 2 L: This reformulation turns the problem (P3)0 into (P3). Based on the constraints for the curvatures of ulðÞ; vlðÞ in Theorem 1, it is easy to verify that the utility functions ^ulð^ylÞ, ^vlð^qlÞ are concave with respect to ^yl and ^ql [13]. To show prob- P P lem (P3) is a convex programming problem, one only needs to prove the convexity of constraints set in (4)–(6). We can see that l2LG0le^qlþ^plÞ are, respectively, convex j–lGlje^qjþ^pj =S þ g2Þ; logð logð

4092 B. Yang et al. / Computer Communications 31 (2008) 4089–4097 with respect to q; p due to the convexity of log of a sum of exponen- tial of linear functions [14]. h 5.2. Primal-dual decomposition In this section, we present a primal-dual decomposition method to solve the convex programming problem (P3). Consider the dual to the primal problem (P3) min k0;l0 Dðk; lÞ ð8Þ with partial dual function Dðk; lÞ ¼ max ~P; ^y2 ^Y;^q2^Q;^p2 ^P ð9Þ Since the function Gð^y; kÞ ¼ P lð^ulð^ylÞ kl^ylÞ is concave with re- spect to ^y; we can solve the subproblem in (10) by considering the ﬁrst-order necessary optimality condition oGð^y; kÞ=o^yl ¼ ^u0 lð^ylÞ kl ¼ 0: The auxiliary variable is adjusted by ylðt þ 1Þ ¼ e^u01 ðklðtÞÞ h i : l ymax l eM We assume that there exists an optimal solution to the subproblem (11). The detailed solution method is deferred to the next subsec- tion. Now we come to solve the dual problem (8). Note that the dual function Dðk; lÞ is not differentiable. Therefore, we cannot use the usual gradient methods and we will instead solve the dual problem using subgradient methods [15]. Let ð^y; ^p; ^qÞ be the associated opti- mal solutions to the subproblems (10), (11), respectively. Note that ð D k ; l D k; lð l P l2L X ^pl k ^ul ^ylð Þ k X l ^yl l þ l k ^pl k Þ ¼ max ^y2 ^Y l2L X þ max ^q2^Q;^p2 ^P ^ul ^ylð Þ k X l ^yl l2L X þ l þ l l þ l k l2L Þ ^ul ^ylð Þ kl^yl ð X l2L X X þ ^pl kl þ ll Þ kl k þ P lkl ^Clð^yl; ^p; ^qlÞ ^Cl ^yl; ^p; ^ql kl þ ll Þ ¼ l2L l2L l l l log log l l þ l X log j–l X j–l ^Dl ^p; ^ql ð P ð D k ; l ð l2L Þ P Þ D k; lð P lð^ulð^ylÞ þ x^vlð^qlÞÞ lll where ~P ¼ ^Dlð^p; ^qlÞ: Noting that we relax only the ﬁrst and second constraints in (P3) by introducing Lagrangian multipliers k; l; which are also known as consistency price and link price, respectively. (P3) and (8), If respectively, we have D P ~P with the weak duality theorem. Since (P3) is a convex programming, there is no duality gap, i.e., D ¼ ~P: ~P and D denote the optimal value of The maximization in (9) can be decomposed into the following two subproblems: X D1ðkÞ ¼ max ^y2 ^Y l D2 k; lð Þ ¼ max ^q2^Q;^p2 ^P s:t: ð6Þ Þ 0 ^ulð^ylÞ kl^yl ð X B@ kl þ ll l ^pl kl þ ll log þx^vl ^qlð Þ ! 1 ð10Þ CA Gljpjqj=S þ g2 P j–l ð11Þ Remark 1. With the dual decomposition method, the system optimization problem (P3) can be decomposed by the auxiliary variable into the maximization problem in (10), which maximizes the corresponding SINR, and the resource allocation problem in (11), which is used to determine the link’s access probability and power level. These two subproblems interact through consistency prices k and link prices l: X j–l ! Gljpjqj=S þ g2 ! þ x^vl ^qlð ! log Gll ! l þ l l Þ þ k l l þ l log Gll ; ! þ x^vl ^qlð Þ þ k þ x^vl ^qlð Þ þ kl þ ll log Gll ; Gljpjqj=S þ g2 ! Gljpjqj=S þ g2 Þ ll l : l Thus, it is veriﬁed that ^C ¼ ð ^Clð^yl; ^p; ^qlÞ 8l 2 LÞ; ^D ¼ ð ^Dlð^p; ^qlÞ 8l 2 LÞ are the subgradient of dual function Dðk; lÞ at point k and l; respectively [15]. By the subgradient method, we obtain the following algorithm for price adjustment: ð12Þ klðt þ 1Þ ¼ klðtÞ þ aðtÞ ^Cl ^yl; ^p; ^ql Þ þ llðt þ 1Þ ¼ llðtÞ þ aðtÞ ^Dl ^p; ^ql ð13Þ Þ where t is the iteration number, aðtÞ is the positive scalar stepsize, and ‘‘½þ” denotes the projection onto the set Rþ of non-negative real numbers. ð ð þ ; : 5.3. Joint random access and power control under interference temperature constraint In the following, we introduce the penalty function method to solve the joint random access and power control problem (11). We use a non-smooth penalty function method as follows: max ^q2^Q;^p2 ^P where Þ; W ^p; ^qð ( X W ^p; ^qð Þ ¼ kl þ ll ^pl kl þ ll Þg j max ^El ^p; ^qð l2L þx^vl ^qlð ð14Þ ! X j–l Þ; 0 log Gljpjqj=S þ g2 and j is a positive penalty constant. Since the objective function of problem (9) is concave, problem (14) is a convex optimization prob-

B. Yang et al. / Computer Communications 31 (2008) 4089–4097 4093 ¼ kl þ ll lem with convex constraints which can be solved using a subgradi- ent projection algorithm. It can be shown that oW o^pl oW o^ql X GklplðtÞqlðtÞmkðtÞ=S j plqlG0l X j2LpjqjG0j GklplðtÞqlðtÞmkðtÞ=S j plqlG0l j2LpjqjG0j ¼ xv0 P P ð15Þ ð16Þ l ^qlð Þ k–l k–l ; ; 8< : where mkðtÞ ¼ kkðtÞ þ lkðtÞ P ckðtÞ GkkpkðtÞ ; pjqjG0j 6 T; 0 if j2L ¼ 1 otherwise: ð17Þ ð18Þ Then the subgradient projection algorithm that solves (14) is ob- tained as follows. On each logical link l; transmission is decided to take place with probability " " ( log qlðtÞ þ bðtÞ ( log plðtÞ þ bðtÞ oW o^ql oW o^pl qlðt þ 1Þ ¼ exp plðt þ 1Þ ¼ exp ) # ) # 1 : ð19Þ ^p¼^pðtÞ;^q¼^qðtÞ eM pmax l eM ; ð20Þ ^p¼^pðtÞ;^q¼^qðtÞ The transmission power is updated by the transmitter of link l as where bðtÞ is the step size.During each time slot, the following algo- rithm is updated until convergence. Algorithm 1. 1. At the transmitter of link l, 8l the consistency price k is updated according to (12) and the link price l is updated according to (13). a message mlðtÞ (17) is broadcasted to all other transmit- ters by a ﬂooding protocol. its transmission power plðtÞ and access probability qlðtÞ are updated according to (20) and (19), respectively. 2. At the measurement point, it broadcasts the aggregate value l2LpjqjG0j to all transmitters, if the aggregate IT violate the the on monitoring keeps it upper-bound. Otherwise, interference. (a) (b) (c) P Remark 2. From the practical implementation point of view, the P complexity of Algorithm 1 comes from the message passing in (15) and (16). The computations of (15) and (16) require the kth link, 8k–l; to transmit k–lGklplðtÞqlðtÞmkðtÞ=S to l. As the number of transmitters increase, the convergence rate of Algorithm 1 will become slower as shown in simulation part. One method to decrease the complexity is to limit the message passing within a small neighborhood of l [16], since the term (17) is multiplied by Gkl; which is inverse proportional to the distance between the lth transmitter and the kth receiver, 8k–l: Another method to decrease the complexity of Algorithm 1 is to design non-cooperative ran- dom access algorithm without message passing. Interested readers are referred to [17]. Remark 3. Another limitation of Algorithm 1 is due to the mea- surement of IT violation in (18). Using IT constraint as a protection of primary transmission does not require the interaction between primary and secondary transmissions. However, the IT measured at the measurement point is different from that at primary receiv- ers. We refer interested readers to [18] for more details on IT measurement. Theorem 2. Given small enough positive constants a; b and large enough positive constant j; the distributed joint power control and random access algorithm (12), (13), (19), (20) converges statistically to the global optimum of problem (P2). Proof. First we show that qðtÞ; pðtÞ are bounded with the subgra- (20). Let wðtÞ ¼ ðqðtÞ; pðtÞÞ; Wðk; lÞ ¼ dient algorithm (19), ðqðk; lÞ; pðk; lÞÞ be the set of optimal solutions to (14) with a given ðk; lÞ: We further deﬁne dðw; WÞ ¼ minw02Wkw w0k and NrðWÞ ¼ fw j dðw; Wðk; lÞÞ 6 rg: We have the following results readily derived based on the convergence properties of subgradient algorithm [19]. Let fwbðtÞg denote the sequence of vectors deﬁned by the iterative procedures stated in (19), (20) with bðtÞ ¼ b; 8t: Then there exist a K < 1 and a function rðbÞ such that limb!0þ rðbÞ ¼ 0; and for all j > K; t!1 dðwðtÞ; NrðbÞðWÞÞ ¼ 0: lim Thus, qðtÞ; pðtÞ are bounded. Next we show that the subgradient algorithms (12), (13) with constant step size a converges statistically to the optimal value ðk; lÞ of the dual function Dðk; lÞ: Recall that in (12), (13) the following iterative procedures are updated: ! # ^Il ^pl; ^ql Þ þ ^yl ^Il ^pl; ^ql Þ þ ; ! # ð21Þ ð22Þ þ : l l Þ ð Gllpl X " " Gllpl ð ð ll t þ 1 klðt þ 1Þ ¼ klðtÞ þ a log Þ ¼ llðtÞ þ a log klðtÞ k klðtÞ þ a log llðtÞ þ a log X Introduce the Lyapunov candidate Þ ¼ 1 ð V kðtÞ; lðtÞ 2a X ð ð V k t þ 1 Þ; lðt þ 1Þ 6 1 X 2a þ 1 X 2a klðtÞ k X ( þ a X 2 llðtÞ l X þ ^yl ð ! ^Il ^pl; ^ql ¼ 1 2a Gllpl ð ( 2 þ Gllpl log log þ ð Þ ð l l l l l l l l þ a 2 ð ^Il ^pl; ^ql Gllpl Þcl0 ; log l þ log c0l X ! l 2 þ 1 2a llðtÞ l l 2 ; ! ! 2 ^Il ^pl; ^ql Þ þ ^yl k l ^Il ^pl; ^ql Þ Gllpl klðtÞ k ! l 2 X ) ! l þ 1 2a Þcl0 ^Il ^pl; ^ql Gllpl 2 þ log c0l l l ð ^Il ^pl; ^ql Þ þ ^yl log Gllpl llðtÞ l l ! 2 ! ) 2 ð23Þ ð24Þ ð25Þ where the ﬁrst inequality is due to ðmax½z; 0Þ2 6 z2: By the sub-gra- ! ) dient property [15], we have ) ! ( X ( l klðtÞ k X D kðtÞ; lðtÞ ð ^Il ^pl; ^ql Þ D k ð þ ^yl ; l Þ 6 log ð Þ l llðtÞ l Gllpl ð log ^Il ^pl; ^ql Gllpl Þcl0 l2L þ l2L Substitute the above into (23)–(25) Vðt þ 1Þ 6 VðtÞ þ D k ð ; l Þ D kðtÞ; lðtÞ ð Þ þ aY 2 ;

4094 2 whereX 4 ! 2 þ log ! 3 5 2 ^Ilð^pl; ^qlÞcl0 log Gllpl ^Ilð^pl; ^qlÞ Gllpl þ ^yl 6 Y; l2L since ^y; ^p; ^q belong to compact set and ^p–0: Applying the inequal- ity recursively, we have V k t þ 1 ð DðkðsÞ; lðsÞÞ Dðk X ; lÞ ð t Þ þ atY 2 s¼1 Þ D k ð ; l Þ Þ 6 V kð1Þ; lð1Þ Þ ð t By Jensen’s inequality, we have Þ; lðt þ 1Þ ð ð Þ 6 V kð1Þ; lð1Þ Þ andP s¼1 D k sð Þ; lðsÞ t P s¼1DðkðsÞ; lðsÞÞ ð ð t t t X t s¼1 where kðtÞ ¼ 1 t kðsÞ; P DðkðsÞ; lðsÞÞ; X lðtÞ ¼ 1 t t s¼1 lðsÞ; þ aY 2 : b y t i l i b a b o r p s s e c c A since DðkðsÞ; lðsÞÞ is a convex function with respect to k; l: Further. we have D k; l Taking limits as t ! 1; yields D 6 Vð1Þ þ aY 2 : t lim sup t!1 D k; l D 6 aY 2 : Thus according to the deﬁnition of statistical stability in [20], the subgradient algorithm (12), (13) converges statistically to within aY=2 of the optimal value ðk ; lÞ: Then, given small enough positive constants a; b and large enough positive constant j; the distributed algorithms (12), (13), (19), (20) converge statistically to the global optimal solution to problem (P2). h 6. Simulation results 6.1. Simple network scenario We ﬁrst present a simple network topology to illustrate the ef- fects of network parameters and control parameters on algorithm performance. Consider an ad hoc network consisting of four trans- mission pairs. The parameters are set as follows except otherwise speciﬁed. The target SINR is c0 ¼ 12 and the background noise is g2 ¼ 5 1013. The low rate data users are considered with a spreading gain of S ¼ 128: The IT bound is 4 108. The maximum power for each transmission pair is 0.5 W. The channel gain matrix is given as follows: 0:010 0:080 0:018 0:0901 0:050 0:020 0:005 0:036 0:003 0:122 0:100 0:297 0:0996 0:016 0:138 0:010 G ¼ 104 3 7775: 2 6664 ð26Þ ½ The path gain from four transmitters to the IT point is G0 ¼ 107 2:3682 1:4784 1:8582 1:1909 ð27Þ Using the criterion in Section 4, it can be checked that the social optimization solution to (P1) is infeasible. Hence, we turn to the proposed joint random access and power control algorithm. We T: B. Yang et al. / Computer Communications 31 (2008) 4089–4097 a ) W ( r e w o p i i n o s s m s n a r T 0.1 0.08 0.06 0.04 0.02 0 0 1 0.8 0.6 0.4 0.2 0 0 R N S I 25 20 15 10 5 0 0 link pair 1 link pair 2 link pair 3 link pair 4 100 200 300 Iteration 400 500 link pair 1 link pair 2 link pair 3 link pair 4 500 100 200 300 Iteration 400 Fig. 1. Transmission power and access probability. link pair 1 link pair 2 link pair 3 link pair 4 100 200 300 Iteration 400 500 Fig. 2. SINR at receivers over time. P l2LððclÞ1 ðqlÞ1Þ: The adopt the system utility function as step-size is chosen as a ¼ 0:04; b ¼ 0:1: The convergence of trans- mission power and access probabilities of four links are shown in Fig. 1. It can be found that four links’ SINR are controlled to c0 ¼ 12 after 200 iterations with j ¼ 100 in Fig. 2. To investigate the effects of j on the aggregate interference at the IT measure- ment point, Fig. 3 plots the measured IT with different j, where the case of j ¼ 0 corresponds to power control and random access without IT constraint. It is shown that the measured IT exceeds 4 108 with j ¼ 0. It can also be observed in Fig. 3 that the sys- tem becomes more sensitive to the violation of IT constraint with bigger j: The inﬂuence of different IT constraints on system utili- ties is demonstrated in Fig. 4, where system utilities become stable with higher power threshold. This is due to the fact that the back- ground noise weights much more in the low power threshold and becomes negligible in higher cases. In this section, we also compare the performance of Algorithm 1 in this paper with that of centralized algorithm in [3]. The four link pairs considered here are endowed with the same priority. With

B. Yang et al. / Computer Communications 31 (2008) 4089–4097 4095 1 x 10−7 0.9 ) W ( r e w o p d e v e c e r l a t o T i 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 κ=0 κ =10 κ =100 a R N S I b ) W link pair 2 link pair 3 link pair 4 50 100 150 200 250 Iteration 300 350 400 450 30 25 20 15 10 5 0 x 10−9 6.5 6 5.5 5 4.5 0 50 100 150 200 250 Iteration 300 350 400 450 Fig. 5. SINR and IT controlled by [3]. ( r e w o p d e v e c e r l a o T t i a R N S I 25 20 15 10 5 0 0 link pair 1 link pair 2 link pair 3 link pair 4 100 200 300 Iteration 400 500 b ) W ( r e w o p i d e v e c e r l a t o T 8 6 4 2 0 x 10−8 κ =10 0 100 200 300 Iteration 400 500 Fig. 6. SINR and total received power over time with channel gain variations. true values. The SINR converges around the same objective value after a larger period as shown in Fig. 6(a) and the aggregate inter- ference temperature is also controlled below the constraint value as shown in Fig. 6(b). 6.2. General network scenario A general ad hoc topology as in [23] is further considered in this paper. The size of network is a 300 m 300 m square one, where 100 nodes are located in the square area in a random placing man- ner. An IT measurement point is located at the center of the consid- ered area. Two nodes can formulate a secondary transmission link if their distance is no more than 50 m. The background noise is g2 ¼ 1:0 1010: The channel gain is given by Gij ¼ Ksijðd0=dijÞ.; 100 200 300 Iteration 400 500 Fig. 3. Total received power at measurement point over time. − 6 −7 −8 −9 −10 −11 −12 −13 y t i l i t u m e t s y S −14 10 −10 −8 10 − 6 10 Interference temperature constraint (dB) −4 10 −2 10 Fig. 4. System utilities with different power threshold. the centralized searching algorithm considering both QoS con- straint and IT constraint [3], only three nodes in this numerical example are allowed to communicate with their respective receiv- ers. The active users’ SINR are shown in Fig. 5(a), which satisfy QoS requirement. The stochastic learning automata algorithm in [3] also results in the similar performance but with longer conver- gence time, which is not simulated here. Fig. 5(b) shows that the aggregate power at the measurement point is far from the con- straint value 4 108 with only three active secondary transmis- sion pairs. Note that the power control algorithm employed by [3] is the standard power control [21], which alone does not con- sider the interference generated to either peer secondary users or IT measurement point. Hence, it does not require the message passing like in Algorithm 1. However, the power control [21] has to relay on searching algorithm in [3] to switch off link 1 to support other links’ QoS, whose implementation requires whole network information. The path gain Gjl; G0l 8j; l 2 L used in (15)–(18) can be mea- sured by the method in [22]. However, due to mobility of nodes and variation of fading process, the estimation of channel gains cannot be very accurate. To evaluate the effect on convergence of the proposed algorithm by the variation of Gjl; G0l in Algorithm 1, Gjl; G0l are generated at random between +10% and 10% of their

4096 B. Yang et al. / Computer Communications 31 (2008) 4089–4097 R N S I 50 40 30 20 10 0 0 link pair 1 link pair 2 link pair 3 link pair 4 link pair 5 link pair 6 link pair 7 link pair 8 link pair 9 link pair 10 1 2 3 4 5 Iteration 6 7 8 9 10 x 105 Fig. 7. SINR at receivers over time. where dij is the distance between transmitter j and receiver i, K and d0 are two constants with K ¼ 106 and d0 ¼ 10 m, the path loss exponent . ¼ 4; and the shadowing factor sij is random, indepen- dent and identically generated from a lognormal distribution with a mean of 0 dB and variance of d ¼ 8 dB. Note that in the following simulation, we generate sij 8sij every 20 time steps according to its distribution. Other parameters are set the same as in Section 6.1. This network scenario is used to simulate that several primary users communicate to a BS in the uplink direction. Transmitting nodes corresponding to the secondary users are randomly located in a rectangular area and the BS of primary network is located at the center of considered area. Thus, the IT measurement point is co-located with BS to monitor the aggregate secondary interfer- ence and protect primary transmission. Fig. 7 plots the average SINR of selected links from the consid- ered area with j ¼ 1000. It is shown that most of those links’ SINR are controlled around the target value of 12. The longer time con- vergence than Fig. 2 is due to a larger number of nodes in concur- rent transmission and time-varying channel gains. Due to the time- varying network environments, we use the IT violation probability to demonstrate the interference avoidance effect. The IT violation probability is deﬁned as the portion of time that the aggregated IT exceeds the constraint T: The measure of violation probability is obtained by counting over 105 iterations. Fig. 8 displays the vio- e c n e r e f r e n t i r o f y t i l i b a b o r p n o l i t a o V i i s t n a r t s n o c e r u t a r e p m e t 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 100 T=η2 T=10* 2 η T=20* 2 η 200 300 400 500 κ 600 700 800 900 1000 Fig. 8. Violation probability for interference temperature constraints versus different j: lation probability versus different j for different IT constraints. The results in Fig. 8 show that even with very stringent IT constraint ðT ¼ g2Þ, a very low violation probability can be achieved with proper selection of j: 7. Conclusions and future work The distributed spectrum sharing among unlicensed users with interference temperature and QoS constraints is studied in this pa- per. To ﬁnd feasible transmission powers for supporting active links’ QoS, unlicensed users are allowed to access networks oppor- tunistically. Then joint random access and power control is formu- lated as a non-convex optimization problem. After variable substitution and log transformation, the problem is then trans- formed to a convex optimization problem, and correspondingly, a distributed algorithm for updating links’ transmission powers and access probabilities is derived. With the proposed algorithm, the transmission of a link not only achieves its own QoS require- ment but also avoids strong interference to other peer unlicensed users and primary transmissions. This algorithm is proved to con- verge to a global optimal solution. Simulation results verify the theoretical results. The joint design of power control and random access is con- sidered under IT constraint, which is known as the underlay scheme. However, this scheme seems conservative when primary transmission is silent. One possible direction of future work is to exploit the combinational advantages of overlay schemes and underlay schemes, where the secondary transmission can occupy the shared spectrum with IT constraint in the presence of pri- mary transmission, and access the spectrum freely, otherwise. Distributed design and implementation are also challenges in this topic. References [1] M. McHenry, Spectrum white space measurements, presented to New America Foundation BroadBand Forum, June 2003. [2] I.F. Akyildiz, Won-Yeol Lee, M.C. Vuran, S. Mohanty, Next generation/dynamic spectrum access/cognitive radio wireless networks: a survey, Comput. Networks J. (Elsevier) 50 (September) (2006) 2127–2159. [3] Y. Xing, C.N. Mathur, M.A. Haleem, R. Chandramouli, K.P. Subbalakshmi, Dynamic spectrum access with QoS and interference temperature constraints, IEEE Trans. Mob. Comput. 6 (4) (2007) 423–433. [4] M. Chiang, S.H. Low, A.R. Calderbank, J.C. Doyle, Layering as optimization decomposition: a mathematical theory of network architectures, Proc. IEEE 95 (1) (2007) 255–312.

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