Cross-Layer Scheduling for OFDMA-based Cognitive Radio Systems with Delay and Security Constraints
Xingzheng Zhu, Bo Yang, Cailian Chen, Liang Xue, Xinping Guan, Fan Wu
aa r X i v : . [ c s . N I] D ec Cross-Layer Scheduling for OFDMA-basedCognitive Radio Systems with Delay and SecurityConstraints
Xingzheng Zhu, Bo Yang, Cailian Chen, Liang Xue, Xinping Guan, Fan Wu
Abstract —This paper considers the resource allocation prob-lem in an Orthogonal Frequency Division Multiple Access(OFDMA) based cognitive radio (CR) network, where the CRbase station adopts full overlay scheme to transmit both privateand open information to multiple users with average delayand power constraints. A stochastic optimization problem isformulated to develop flow control and radio resource allocationin order to maximize the long-term system throughput of openand private information in CR system and ensure the stabilityof primary system. The corresponding optimal condition foremploying full overlay is derived in the context of concurrenttransmission of open and private information. An online resourceallocation scheme is designed to adapt the transmission of openand private information based on monitoring the status ofprimary system as well as the channel and queue states in theCR network. The scheme is proven to be asymptotically optimalin solving the stochastic optimization problem without knowingany statistical information. Simulations are provided to verifythe analytical results and efficiency of the scheme.
Index Terms —Cognitive radio, physical-layer security, delay-aware network, full overlay, cross-layer scheduling.
I. I
NTRODUCTION T HE emergency of high-speed wireless applications andincreasing scarcity of available spectrum remind re-searchers of spectrum utilizing efficiency. The concept ofCR provides the potential technology in increasing spectrumutilizing efficiency [1], [2] because CR allows unlicensedusers (also known as secondary users (SUs)) to access somespectrum which is already allocated to primary user (PU) orlicensed user who has the authority to access the spectrum byspectrum sensing [3], [4]. As another promising technologyof high speed wireless communication system, OFDMA is acandidate for CR systems [1] due to its flexibility in allocatingspectrum among SUs [5]. Hence, OFDMA-based CR networksare catching great attention [6], [7]. This paper focuses on anOFDMA-based CR network without loss of generality.
Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected]. Zhu, B. Yang, C. Chen and X. Guan are with Department of Automation,Shanghai Jiao Tong University, and Key Laboratory of System Control andInformation Processing, Ministry of Education of China, Shanghai 200240,China (Emails: { wendyzhu, bo.yang, cailianchen, xpguan } @sjtu.edu.cn). B.Yang, C. Chen and X. Guan are also with the Cyber Joint InnovationCenter, Hangzhou, China. L. Xue is with School of Information and ElectricalEngineering, Hebei University of Engineering, Handan 056038, China (Email:[email protected]). F. Wu is with the Department of Computer Scienceand Engineering, Shanghai Key Laboratory of Scalable Computing andSystems, Shanghai Jiao Tong University, Shanghai 200240, China (E-mail:[email protected]). B. Yang is the corresponding author of this paper. In order to exploit the capacity of the whole OFDMA-bsed CR system, this paper aims at maximizing the sec-ondary network capacity in consideration of the whole systemtransmission efficiency. Thus, the following three main issuesshould be considered.Firstly, an efficient spectrum sharing scheme is essentialfor exploiting the unused spectrum in OFDMA-based CRnetwork. When a SU wants to access some spectrum, it mustensure that the spectrum is not accessed by any PU or adaptits parameter to limit the interference to PU. Both of these twomentioned spectrum utilization manners, known as overlay andunderlay schemes, are conservative in some ways, since theyignore the PU’s ability to tolerate some inference.Secondly, due to that CR networks as well as many otherkinds of wireless communication systems have a nature ofbroadcast, security issues at physical layer have always beenunavoidable in designing CR systems. Furthermore, to SUs,it is obviously practical that there exist both private and opentransmission requirements. Then, the scheduling among thesetwo different kinds of transmission should be considered.In addtion, delay performance is an indispensable quality ofservice (QoS) index in scheduling different transmissions.Last but not least, the dynamic nature of OFDMA-basedCR communication system brings another big challenge. Therandom arrival of user requests (from both PU and SU)and time-varying channel states renders dynamic resourceallocation instead of fixed ones in exploiting the OFDMAsecondary network capacity.Aiming at the above issues, the contributions of this paperare threehold: • First, this paper adopts a novel full overlay spectrumaccessing scheme by exploiting PU’s tolerance to inter-ference. Besides, the theoretic proof of full overlay’soptimality is given in the presence of both open andprivate transmissions. • Second, a joint encoding model is introduced to allowboth private and open transmissions towards SUs withthe full overlay spectrum sharing scheme. A dynamicresource allocation scheme consisting of flow controland radio resource allocation is developed by solvinga formulated stochastic optimization problem under thedelay and power constraints. • Third, the proposed dynamic resource allocation schemeis proven to be close to optimality although its imple-mentation only depending on instantaneous information.This paper is organized as follows. Section II presents the related work. In Section III, we introduce the systemmodel and relevant constraints in detail. Section IV formulatesthe problem. In Section V, we introduce our cross-layeroptimization algorithm. We give the performance bound andstability results in Section VI. Two different implementationsare proposed in Section VII. In Section VIII, some simulationresults are shown. Finally, we conclude this paper in SectionIX. II. R
ELATED W ORK
There have been many works on spectrum sharing inOFDMA-based CR networks [8]–[10]. According to [11],[12], the access technology of the SUs can be divided in twocategories: spectrum underlay and spectrum overlay. The firstcategory means that SUs can access licensed spectrum duringPUs’ transmission, while as is mentioned in [12], this approachimposes severe constraints on the transmission power of SUssuch that they can operate below the noise floor of PUs, e.g, in[8], [13], [14]. The second category means that SUs can onlyaccess licensed spectrum when the PU is idle, e.g, in [9], [10],[15]–[17]. Considering both these two strategies suffer fromsome drawbacks, the authors in [18] propose a new cognitiveoverlay scheme requiring SUs to assess and control theirinterference impacts on PUs. In general, the cognitive basestation (CBS) controls the aggregate interference to primarytransmission by allowing SUs to monitor channel qualityindicators (CQIs), power-control notifications and ACK/NAKof primary transmission. In this paper, this novel thought isextended into an OFDMA-based CR system.On the other hand, dynamic resource allocation plays a crit-ical role in exploiting OFDMA network capacity. The overallperformance as well as the multiuser diversity of the systemcan be improved by proper dynamic resource allocation [17],[19]–[25]. Thus, dynamic resource allocation in OFDMA-based CR system has been attracting more attention recently.The corresponding spectrum sharing schemes in [8]–[10] areall realized by dynamic resource allocation.Besides the interference constraints, the works of delayaware transmission are also quite relative to this paper. Huangand Fang in [26] investigate both reliability and delay con-straints in routing design for wireless sensor network. Cui etal. in [27] summarize three approaches to deal with delay-aware resource allocation in wireless networks. A constrainedpredictive control strategy is proposed in [28] to compensatefor network-induced delays with stability guarantee. Thosethree methods are based on large deviation theory, Markovdecision theory and Lyapunov optimization techniques. Asto the first two methods, they have to know some statisticalinformation on channel state and random arrival data rate todesign algorithm, while these prior knowledge is expensiveto get, even unavailable. To overcome this problem, manyauthors pay attention to Lyapunov optimization techniques.References [29] and [30] investigate scheduling in multi-hop wireless networks and resource allocation in cooperativecommunications, respectively as two typical applications ofLyapunov optimization in delay-limited system. In this paper,we utilize this tool to dispose the resource allocation problemin OFDMA-based CR networks. As for secure transmission, Shannon’s information theorylaid the foundation for information-theoretic security [31] andthe concept of wire-tap channel was proposed in [32]. Therehas been some research on exploiting security capacity inOFDMA network by dynamic resource allocation, such as in[33] and [34]. In CR area, the study of secure transmissionfrom information-theoretic aspect is very limited. Pei et al. in[35] first investigate secrecy capacity of the secure multiple-input single-output (MISO) CR channel. Kwon et al. in [36]utilize the concept of security capacity to explore MISO CRsystems where the secondary system secures the primary com-munication in return for permission to use the spectrum. Boththese two works focus on only private message transmission.The security and common capacity of cognitive interferencechannels is analyzed in [37]. The entire capacity of a MIMObroadcast channel with common and confidential messages isobtained in [38]. The paper [39] considers the problem ofoptimizing the security and common capacity of an OFDMAdownlink system by dynamic resource allocation. This paperfurther considers the transmissions of private and open flowsin CR networks with delay constraints.III. S
YSTEM M ODEL
The system model consists of multiple primary links andmultiple secondary links as Fig. 1 shows. The total bandwidth B is divided into M subcarriers equally using Orthogonal Fre-quency Division Multiplexing (OFDM). Assume that M = B holds for simplicity of expression. The subcarrier set of thenetwork is denoted as M = { , , · · · , M } and m ∈ M denotes subcarrier index. The downlink case is considered.The primary link is from a single primary base station (PBS)to K PUs. Secondary links are from a common CBS to N SUs. We denote k ∈ { , , · · · , K } and n ∈ { , , · · · , N } asthe indexes of PU and SU respectively. The system operatesin slotted time, and T is the length of a time slot. Hereafter, [ tT, ( t + 1) T ) is just denoted by t for brevity.The set of subcarriers occupied by PU k on timeslot t is denoted as Γ P Uk ( t ) = { τ k ( t ) , τ k ( t ) , · · · , τ km k ( t ) ( t ) } where m k ( t ) is the number of subcarriers occupied by PU k and Γ P Uk ( t ) ⊆ { , , · · · , M } . The power set P P Uk ( t ) = { P mk ( t ) | m ∈ M } is the set of transmission power from PBSto PU k , where for m ∈ Γ P Uk ( t ) , P mk ( t ) > , else P mk ( t ) = 0 .For brevity, we will omit the time index ( t ) somewherein further discussion. P SU = { p mn |∀ n, ∀ m } denotes theoverall SUs power allocation policy set and p mn represents thepower allocated by CBS to user n in subcarrier m . Denote Γ SUn = { ̟ mn |∀ m } as the subcarrier assignment policy of SU n , where ̟ mn is either 1 representing subcarrier m is assignedto SU n , or 0 otherwise. Then let Γ SU = { Γ SUn ( t ) , ∀ n } be theoverall subcarrier assignment policy of secondary network.Due to the orthogonal properties of OFDMA technology,there exists no mutual influence between every two SUs.However, there exists mutual interference between the primaryand secondary networks when PU and SU access in the samesubcarrier.The channel gains include the one of secondary user n on subcarrier m , h mn and the one of primary user k on subcarrier m , H mk . The additive white gaussian noise(AWGN) is σ . The corresponding subcarrier gain-to-noise-ratio (C/I) in slot t are thus defined as a mn ( t ) = h mn ( t ) σ and A mk = H mk ( t ) σ respectively as illustrated in Fig.1. Theset a ( t ) = { A mk ( t ) , a mn ( t ) , ∀ n, ∀ m, ∀ k } represents the systemchannel state information (CSI). All channels are assumed tobe slow fading, and thus a ( t ) remains fixed during one slotand changes between two [40]. In this work, there exists anreasonable assumption that the system CSI is known to BS.As in [41], BS can get full-CSI by utilizing pilot symbolsand CSI feedback process. Besides, at the beginning of everyslot, PU reports P mk A mk to PBS. For example, the PU reports areceived-signal-strength index to PBS in packets such as RSSIreports. We assume the CBS will listen to the information toderive P mk A mk before accessing subcarrier m [18], [41].Denote h mkS as the cross-link interference channel gain fromCBS to PU k on subcarrier m and let a mkS = h mkS σ . Similarly,denote h mnP as the cross-link interference channel gain fromPBS to SU n on subcarrier m and let a mnP = h mnP σ . It isassumed that a mkS and a mnP can be got by the CBS. a mkS canbe estimated by CBS from the PU feedback signal based onreciprocity. a mnP can be estimated by SUs through trainingand sensing and the estimation results are sent to CBS [41].Beyond that, information about cross-link channel state couldalso be measured periodically by a band manager either [8],[42].Compared to pervious work, this paper considers a morecomplicated and practical situation of SU transmission. TheCBS transmits both private and open data to each SU as Fig.2shows. The private data has security requirement and open datahas long-term time-average delay constraint. Instead of thatboth open and private data have delay constraint, only delayconstraints on open transmissions are considered in this paperfor simplifying the mathematic expressions, since the handlingof delay constraint in secure transmission is totally the same asopen transmission. Actually, in real wireless communicationsystems, there exists some private transmission having no strictdelay constraint, e,g. updating contact information in mobiledevices. At the beginning of every time slot, random datapackets arrive at CBS. CBS decides whether to admit it intothe system or not. Besides, CBS is also in charge of resourceallocation to assign power and subcarriers among SUs. CBSutilizes the information of data queue and CSI to allocateresources. The system performance can be optimized and thequeuing delay of open data can be ensured to fulfill by flowcontrol and resource allocation.In the side of CBS, the amount of open data packet of SU n , D on ( t ) , and private data, D pn ( t ) that arrive at CBS duringslot t are independent identically distributed (i.i.d) stochasticprocesses, e,g. Bernoulli processes, with the long-term averagearrival rates λ on and λ pn , and their upper bounds are µ max and D max , respectively. These packets can not be transmitted totarget users instantaneously due to the time-varying channelconditions and they are enqueued at the CBS. However, onlyparts of these packets are admitted into each queue towards Also called gain-to-noise-plus-interference-ratio when SU and PU accessin the same subcarrier.
Primary Network
Secondary Network M u t u a l I n t e r f e r e n ce SU 1 SU NSU n
Subcarrier 1...
Subcarrier m ...Subcarrier M
Primary & Secondary Channel to Noise RatioCross-link Channel to Noise Ratio … …
Fig. 1. General network model each user for stability reason to be specified later. The amountsof open and private data admitted by respect queues are T on ( t ) and T pn ( t ) and CBS is in charge of determining T on ( t ) and T pn ( t ) according to a certain principle which would bespecified in Section V. A. Capacity model
In OFDMA-based CR networks, SU and PU can access inthe same subcarrier with mutual interference. However, due tothe characteristic of OFDMA networks, each subcarrier cannot be assigned to more than solitary user in any secondaryor primary network. Thus the following formulation is set toensure the limitation in CBS: ≤ N X n =1 ̟ mn ≤ , ∀ m (1)CBS will realize the occupied subcarrier set Γ P Uk = { m | P mk > , ∀ m } , and we denote Γ SU = { , , · · · , M } − S Kk =1 Γ P Uk . Thus the transmission rates of PU and SUs canbe analysed by dividing M subcarriers into two parts: oneis m ∈ S Kk =1 Γ P Uk where there exists interference betweenPU and SUs; another is m ∈ Γ SU which means SUs canaccess these subcarriers without influencing primary link. Thusaccording to information theory the transmission rate of PU k on subcarrier m is: R mk = ( log (1 + P mk A mk a mkS p mn ′ ) m ∈ Γ P Uk , n ′ ∈ ˜Γ m m ∈ Γ SU where ˜Γ m is the set of SUs accessing subcarrier m . Further-more, since in secondary network, only one SU can accessone subcarrier, n ′ is the only one element in set ˜Γ m .It should be noticed that the total transmission rate in anOFDMA network equals to the sum rates on all subcarriers.So the transmission rate of PU is: R P Uk = X m ∈ Γ PUk R mk (2)The channel capacities of SU n on subcarrier m can beexpressed as: C mn = ( log (1 + p mn a mn P mk ′ a mnP ) m ∈ S Kk =1 Γ P Uk , k ′ ∈ ˆΓ m log (1 + p mn a mn ) m ∈ Γ SU where ˆΓ m is the set of PUs accessing subcarrier m . Fur-thermore, since only one PU can access one subcarrier, k ′ is the only one element in set ˆΓ m . Denote R SUn = P m C mn as the sum transmission rate of SU n without considerationof security.By introducing the joint transmission model, open and pri-vate data of one SU can be transmitted simultaneously. Openmessage is jointly encoded with security message as randomcodes. In this way, although open message may be decodedby eavesdroppers, security message would be perfectly secureif the channel fading is properly utilized [43]. According tothe theory of physical-layer security [34], if the transmissionrate of private data is less than security capacity , the pro-posed joint-encoding model can at least realize physical-layersecurity in theory. [44], [45] propose physical-layer securityrealization applications using error correcting codes and pre-processor, which lays the foundation of realizing physical-layer security of the joint encoding model. For each SU, CBSmakes decision if his secure data could be transmitted in thisslot and this decision is expressed as the secure transmissioncontrol vector ζ = ( ζ , ζ , · · · , ζ N ) . The indicator variable ζ n = 1 implies that private and open messages are encoded atrate ˆ R pn and R SUn − ˆ R pn respectively in timeslot t and ζ n = 0 means that only open messages can be transmitted at rate R SUn .When CBS is transmitting private messages to SU n , all theother SUs except SU n are treated as potential eavesdroppers[34]. According to [46], subject to perfect private of SU n ,the instantaneous private rate of SU n on subcarrier m is theachievable channel capacity minus the highest eavesdroppercapacity if there is no cooperation among eavesdroppers. Foreach SU n , we define the most potential eavesdropper onsubcarrier m as SU ˜ n and ˜ n = arg max n ′ ,n ′ = n a mn ′ . So the securitycapacity of SU n on subcarrier m is: ˆ R mpn = ( [ C mn − log (1 + p mn b mn P mk ′ b mnP )] + m ∈ Γ P Uk , k ′ ∈ ˆΓ m [ C mn − log (1 + p mn b mn )] + m ∈ Γ SU (3)where [ · ] + = max {· , } , b mn = a m ˜ n and b mnP is the cross-linkCSI from PBS to SU ˜ n on subcarrier m . Obviously, ˆ R pn = P m ∈ M ˆ R mpn . Thus the achievable private rate of user n is: R pn = ζ n ˆ R pn and the open rate of user n is: R on = R SUn − R pn . B. Queuing model
There exist data queues in both PBS and CBS. Althoughwe want to maximize the weighted throughput of SUs, PU
Primary Network
Private data flowOpen data flow Secondary Network SU 1SU NSU n ......
Cognitive Base station
Valve
Flow control Resource allocation P r i v a t e & O p e n t r a n s m i s s i o n r p a n d r o P r i va t e & O p e n t r a n s m i ss i on P r i va t e & O p e n t r a n s m i ss i on D e l ay r pn and r on r p N a n d r o N Interference
CSI
Queuing Delay Open data queuePrivate data queue Delivery Delay PBS … Fig. 2. Transmission model of secondary network queue stability is a constraint in ensuring that PU’s long-term throughput is not affected by SU’s transmission. It isassumed that the transmission rate of PBS without interferenceis sufficient to serve PU’s demand. However, the primarynetwork and the secondary network will be influenced by eachother if they work on the same channel. The transmissionrate decrease of PU is due to the interference brought bySU transmission, while the CBS can adjust its schedule tolimit interference in order to ensure that PU’s time-varyingrate demands can be satisfied. Later, the notation of queuestability will be used to measure whether PU’s demand can befulfilled. In [18], the interference is limited by that PU queueis kept stable under the influence caused by the only one SUaccess. We continue to utilize this technique in scheduling ourmulti-SU access system.First, it is necessary to introduce the concept of strongstability . As a discrete time process, Q ( t + 1) = [ Q ( t ) − S ( t )] + + D ( t ) is strongly stable if: lim sup t →∞ t t − X τ =0 E { Q ( τ ) } < ∞ (4)In particular, a multi-queue network is stable when all queuesof the network are strongly stable . According to Strong Stabil-ity Theorem in [47], for finite variable S ( t ) and D ( t ) , strongstability implies rate stability of Q ( t ) . The definition of ratestability can be found in [47] and omitted here.Furthermore, according to Rate Stability Theorem in [47], Q ( t ) is rate stable if and only if d ≤ s holds where d =lim t →∞ t P t − τ =0 D ( τ ) and s = lim t →∞ t P t − τ =0 S ( τ ) .Since the data can not be delivered instantly to PUs orSUs, there are data backlogs in the PBS and CBS waitingfor transmitting to respective users.
1) PU queue:
In PBS, the data queue of PU k is updatedas following: Q k ( t + 1) = [ Q k ( t ) − R P Uk ( t )] + + D P Uk ( t ) (5)where D P Uk ( t ) is the amount of data packets randomly arrivingat PBS during slot t with the destination of PU k . Weassume D P Uk ( t ) is an i.i.d stochastic process with its upper bound of D P Umax and its long-term average arrival rates λ k =lim t →∞ t P t − τ =0 D P Uk ( τ ) . As it has been mentioned before, Q k should be kept stable by limiting SUs’ interference to primarylink. As Rate Stability Theorem shows, Q k is rate stable ifand only if r P Uk ≥ λ k where r P Uk , lim t →∞ t P t − τ =0 R P Uk ( τ ) .Therefore, if PU system is strongly stable, its long-termtransmission is not affected by SUs.
2) SU data queues:
In CBS, there exist actual data queuesof open and private data which are represented by Q on and Q pn respectively for all n ∈ { , · · · , N } . These queues are updatedas follows: Q on ( t + 1) = [ Q on ( t ) − R on ( t )] + + T on ( t ) (6) Q pn ( t + 1) = [ Q pn ( t ) − R pn ( t )] + + T pn ( t ) (7)All Q k , Q on and Q pn have initial values of zero. Wedefine t on , lim t →∞ t P t − τ =0 T on ( τ ) , t pn , lim t →∞ t P t − τ =0 T pn ( τ ) as the long-term time-average admission rates of open dataand private data respectively. The long-term time-averageservice rates of Q on and Q pn are also defined as: r on , lim t →∞ t P t − τ =0 R on ( τ ) and r pn , lim t →∞ t P t − τ =0 R pn ( τ ) . Q on and Q pn should be kept strongly stable in order to ensure the raterequirements of open and private date can be supported by theCR system, which means t on ≤ r on and t pn ≤ r pn hold.Virtual queues of open data, X on ( t ) , and private data X pn ( t ) are introduced in (8) and (9) to assist in developing ouralgorithms, which would guarantee that the actual queues Q on and Q pn are bounded deterministically in the worst case. X on ( t + 1) = [ X on ( t ) − T on ( t )] + + µ on ( t ) (8) X pn ( t + 1) = [ X pn ( t ) − T on ( t )] + + µ pn ( t ) (9)Denote µ on and µ pn as the virtual admission rates of open dataand private data, which are upper bounded by D on and D pn respectively. Notice that X on , X pn , µ on and µ pn do not stand forany actual queue and data. They are only generated by theproposed algorithms. According to queuing theory, when X on and X pn are stable, the long-term time-average value of µ on and µ pn would satisfy: ν on = lim t →∞ t t − X τ =0 µ on ( τ ) ≤ t on (10) ν pn = lim t →∞ t t − X τ =0 µ pn ( τ ) ≤ t pn (11)To summarise, as shown in Fig. 2 the control space χ of the system can be expressed as χ = { P SU , Γ SU , ζ, T } ,which includes admission control T = { T on , T pn |∀ n } , powercontrol decision P SU , subcarrier assignment Γ SU and securitytransmission control ζ . C. Basic constraints1) Power consumption constraint:
Let E , P ∀ n, ∀ m p mn as total power consumption of the whole system in one timeslot. There exists a physical peak power limitation P max that E cannot exceed at any time: ≤ E ≤ P max (12) The long-term time-average power consumption also has anupper bound P avg , which is proposed for energy conservation: e ≤ P avg (13)where e = lim t →∞ t P t − τ =0 E { E ( τ ) }
2) Delay-limited model:
The queuing delay is defined asthe time a packet waits in a queue until it can be transmitted.Each SU has a long-term time-average queuing delay ρ on forits open data transmission. To each SU, it proposes a delayconstraint ρ n as in (14) for its open transmission. ρ on ≤ ρ n (14)IV. P ROBLEM F ORMULATION
Considering the simplicity and understandability of math-ematic analysis, a special case of one single primary link isconsidered in the following. In the single PU case, the onlyone PU is indexed with number . In part C of Section V, thegeneral results of multi-PU case are listed for completeness. A. Optimization objective and constraints
Following above descriptions, the objective of this paper isto improve throughput of secondary network while ensuringstability of primary network. So the problem is formulated as:Maximize the sum weighted admission rates of all SUs andstabilize the PU data queue Q at the same time. Let θ n and ϕ n for all n be the nonnegative weights for private and opendata throughput. Then the optimal problem can be formulatedas: Maximize N X n =1 { θ n t pn + ϕ n t on } (15)Subject to: ≤ t pn ≤ λ pn , ∀ n ≤ t on ≤ λ on , ∀ n t = ( t pn , t on ) ∈ Υ lim sup t →∞ t t − X τ =0 E { Q ( τ ) } < ∞ (13) , (14) where Υ is the network capacity region of secondary links.Define the service rate vector as υ = ( r on , r pn ) . The definitionof network capacity region Υ is the region of all non-negativeservice rate vectors υ for any possible control actions [47].When the CBS takes a kind of control policy under a certainchannel condition, the secondary links will have a decidednetwork capacity and the network capacity region is the setof network capacities under all possible control policies andall channel conditions. In the proposed system, the controlpolicy of CBS should fulfill subcarrier assignment rule (1),peak power constraint (12) and stabilize all queues includingactual queues and virtual queues. So actually, the controlpolicy that can achieve the network capacity region should satisfy the following constraints: (1) , (12)lim sup t →∞ t P t − τ =0 E { Q on ( τ ) } < ∞ lim sup t →∞ t P t − τ =0 E { Q pn ( τ ) } < ∞ lim sup t →∞ t P t − τ =0 E { X on ( τ ) } < ∞ lim sup t →∞ t P t − τ =0 E { X pn ( τ ) } < ∞ Theoretically, we can get the optimal solution to (15) ifwe get the distribution of the system CSI and external dataarrival rate beforehand. However, this information can not beobtained accurately. In this paper an online algorithm requiringonly current information of queue state and channel state isproposed and will be described in detail then.
B. Optimality of SU overlay
Before detailing the control algorithm, it should be specifiedthe conditions that make SU overlay play a positive role inthis cognitive transmission model other than traditional accessmethods. We focus on presenting a sufficient condition onoverlay for constant channel conditions here, then we willextend it to time-varying situation.In the case of static network condition, the optimal problemof SUs’ weighted throughput is simplified asMaximize: P Nn =1 { θ n r pn + ϕ n r on } (16)Subject to r P U = λ where we only consider the optimal case when r P U = λ .Notice here, the system maximal weighted sum data rate underfull overlay scheme must be greater than or at least no worsethan that when SU can only access the subcarrier which is notoccupied by PU. It is easy to understand that full overlay is amore general access scheme than spectrum overlay which isa special access situation. We can get an intuition that whenall subcarriers are assumed to be accessed by PU, SU datarate would be positive under full overlay scheme instead ofzero under traditional overlay scheme. Thus what we want toprove is the sufficient condition of that SUs perform betterin consideration of PU transmission other than accessing thelicensed subcarrier roughly. Let κ be the fraction of time thatPU is actively transmitting, thus: r P U = κ X m ∈ Γ PU log (1 + A m P m a m S p mn ) (17) r pn = { X m ∈ Γ PU { (1 − κ )[log (1 + a mn p mn ) − log (1 + b mn p mn )] + + κ [log (1 + p mn a mn P m a mnP ) − log (1 + p mn b mn P m b mnP )] + } + X m ∈ Γ SU [log (1 + a mn p mn ) − log (1 + b mn p mn )] + } ζ n (18) r on = X m ∈ Γ SU log (1 + a mn p mn ) + X m ∈ Γ PU [ κ log (1+ p mn a mn P m a mnP ) + (1 − κ ) log (1 + a mn p mn )] − r pn (19)We have the following lemma: Lemma 1:
In high SINR region, a sufficient condition forfull overlay to be optimum in SU n accessing subcarrier m (both security and open transmission) is: a m S ≤ min { C nm , C nm } , ∀ m (20)where C nm = b mn / [(1 + P m b mnP + b mn P max ) log (1 + b mn P max )] , C nm = a mn / [(1 + P m a mnP + a mn P max ) log (1 + a mn P max )] .We can have an intuitive explanation on Lemma 1 , for SU n ’s accessing subcarrier m . If the cross link (from CBS toprimary link) condition is bad enough (worse than weightedCBS-to-SU channel condition C nm and weighted CBS-to-eavesdropper channel condition C nm ), the full overlay schemewould be the optimal scheme when both security and opentransmission happen. The proof of Lemma 1 can be found inAppendix C.It would be obvious to derive the following lemma onsufficient condition of optimality of the whole system overlay.Thus we get:
Lemma 2:
In high SINR region, a sufficient condition forfull overlay to be optimum in the whole OFDMA-based CRsystem is: a m S ≤ min n { C nm , C nm } , ∀ m (21)Notice that, the sufficient condition does not mean thatsubcarrier m ∈ Γ P U would provide a greater data rate than m ′ ∈ Γ SU under the same power allocation scheme. It meansthat for m ∈ Γ P U , full overlay would achieve the optimalresult other than any other access policy such as partial overlayor underlay. We assume the sufficient condition of Lemma 2 isfulfilled in this paper and we proceed considering time-varyingchannels then.V. O
NLINE C ONTROL A LGORITHM
It is worth noticing that problem (15) has long-term time-average limitations on power consumption and queuing delay.Using the technique similar to [47], we construct power virtualqueue Y and delay virtual queue Z n to track the powerconsumption and queuing delay respectively. These virtualqueues do not exist in practice, and they are just generatedby the iterations of (22) and (23): Y ( t + 1) = [ Y ( t ) − P avg ] + + E ( t ) (22) Z n ( t + 1) = [ Z n ( t ) − ρ n µ on ] + + Q on ( t ) (23)Similar to actual queues, Y and Z n have initial values ofzero. According to Necessary Condition for Rate Stability in[47], if Y is stable, constraint (13) is satisfied. In addition, if Z n is stable, q on = lim t →∞ t P t − τ =0 E { Q on ( τ ) } ≤ ρ n ν n ≤ ρ n t on holds. According to Little’s Theorem, q on /t on = ρ on , when Z n is stable, the delay constraint (14) would be achieved. It willbe proven that the proposed optimal control algorithm canstabilize these queues in section VI, that is to say the long-term time-average constraints are fulfilled.Using virtual queues X n , Z n and Y , we decouple problem(15) into two parts: one is flow control algorithm whichdecides the admission of data, and another is resource allo-cation algorithm in charge of subcarrier assignment, power allocation and secure transmission control in every slot. Allthese control actions aim at secondary links and happen inCBS. The whole algorithm is named CBS-side online controlalgorithm (COCA). A. Flow control algorithm
When external data arrives at CBS, CBS will decide whetherto admit it according to queue lengthes. Let V be a fixed non-negative control parameter. Let q omax ≥ µ max and q pmax ≥ D max hold. They are actually the deterministic worst caseupper bounds of relative queue length to be proven later. Theflow control rules of open data and private data are obtainedby solving (24) and (25) respectively:Minimize T on [ Q on − q omax + µ max ] (24)Subject to: ≤ T on ≤ D on Minimize T pn [ Q pn − q pmax + D max ] (25)Subject to: ≤ T pn ≤ D pn The corresponding solutions to (24) and (25) are easy to get: T on = (cid:26) if Q on − q omax + µ max ≥ D on otherwise (26) T pn = (cid:26) if Q pn − q pmax + D max ≥ D pn otherwise (27)Here we can have an intuitive explanation on flow controlrules. They work like valves. When any actual data queueexceeds some threshold, the corresponding valve would turnoff and no data would be admitted.As to virtual variable µ on and µ pn , there are also theirrespective virtual flow control algorithms (28) and (29) soas to update virtual queues X on and X pn which will play animportant role in resource allocation:Minimize µ on [ q omax − µ max q omax X on − ρ n Z n − V ϕ n ] (28)Subject to: ≤ µ on ≤ D on Minimize µ pn [ q pmax − D max q pmax X pn − V θ n ] (29)Subject to: ≤ µ pn ≤ D pn Solutions to (28) and (29) are (30) and (31) respectively: µ on = ( if ( q omax − µ max q omax X on − ρ n Z n − V ϕ n ) ≥ D on otherwise (30) µ pn = ( if ( q pmax − D max q pmax X pn − V θ n ) ≥ D pn otherwise (31) B. Resource allocation algorithm
The resource allocation policy can be found in solving thefollowing optimization problem.Maximize U ( P SU , ζ ) (32)Subject to: (1) , (12) where U ( P SU , ζ ) = P Nn =1 ( X on Q on q omax R on + X pn Q pn q pmax R pn ) + Q R P U − Y E .At the beginning of every slot, all X on , X pn , Q on , Q pn and Y can be regarded as constants because they all have beendecided in the previous slot. Q can be estimated by CBSby overhearing PBS feedback. In section VII we propose animperfect estimation scheme of Q and compare the perfor-mances of perfect and imperfect estimations in simulations.Notice that, the resource allocation is determined at thebeginning of every slot and all queues are updated at the endof every slot.Firstly, we can easily decide the vector ζ maximizing U by assuming that all elements of ζ are continuous variablesbetween 0 and 1 and in further discussion, we can get adiscrete implementation of ζ n .We take partial derivative in U ( P SU , ζ ) with respect to ζ n : ∂U ( P SU , ζ ) ∂ζ n = ( X pn Q pn q pmax − X on Q on q omax ) M X m =1 ˆ R mpn (33)Observing (33), P Mm =1 ˆ R mpn is no-negative and U is mono-tonic in ζ n , and thus the optimality condition of securetransmission control is: ζ ∗ n = ( if ( X pn Q pn q pmax − X on Q on q omax ) ≥ otherwise (34)Then we use ζ ∗ to assign subcarrier and power which isthe solution to the following optimization problem PS,PS: Maximize e U ( P SU ) Subject to: (1) , (12) where e U ( P SU ) = U ( P SU , ζ ∗ ) . PS is a typical Weighted SumRate (WSR) maximization problem, and it is difficult to find aglobal optimum since e U ( P SU ) is neither convex nor concaveof P SU . Obviously, PS has a typical D.C. structure whichcan be optimally solved by D.C. programming [48]. In [49]there lists a dual decomposition iterative suboptimal algorithmsolving this kind of constrained nonconvex problem instead ofD.C. programming. In addition, because of the characteristicsof OFDMA networks, the duality gap is equal to zero evenif PS is nonconvex when the number of subcarriers is closeto infinity [50]. So we take a more computationally effectivedual method to solve PS and due to space limitation, we givethe key steps here only.We define R mpn = ζ ∗ n ˆ R mpn and R mon = C mn − R mpn . Thenthe Lagrange function of PS is expressed as: J ( δ, P SU ) = M X m =1 { N X n =1 [ X on Q on q omax R mon + X pn Q pn q pmax R mpn − Y p mn ]+ Q R m } + δ ( P max − E ) (35)where δ is the non-negative Lagrange multiplier for the peakpower constraint in problem PS. The dual problem of PS is: min δ ≥ H ( δ ) , where H ( δ ) = max P SU ≥ { J ( δ, P SU ) } . When δ is fixed, we can decide the parameters P SU maximizing the objective of H ( δ ) . Observing H ( δ ) , we findthat it can be decoupled into M subproblem as: H ( δ ) = M X m =1 max P mSU J m ( δ, P mSU ) + δP max = M X m =1 max P mSU N X n =1 J mn ( δ, p mn ) + δP max (36)where P mSU = { p mn | ≤ n ≤ N } , J m ( δ, P mSU ) = P Mm =1 J mn ( δ, p mn ) , J mn ( δ, p mn ) = X on Q on q omax R mon + X pn Q pn q pmax R mpn − ( Y + δ ) p mn + Q R m ( n ) and R m ( n ) = (cid:26) m ∈ Γ SU R m m ∈ Γ P U , ̟ mn = 1 (37)For m ∈ Γ SU , we can get p m ∗ n by taking partial derivativeof J ( δ, P SU ) with respect to p mn and making (38) equal tozero:for m ∈ Γ SU : ∂ ( J ( δ, P SU )) ∂p mn = X on Q on q omax { a mn p mn a mn − ζ n [ a mn p mn a mn − b mn p mn b mn ] } + X pn Q pn q pmax ζ n [ a mn p mn a mn − b mn p mn b mn ] − ( Y + δ ) (38)However, for m ∈ Γ P U , a global optimal solution p m ∗ n maximizing J mn can be got easily by an exhaustive searchsuch as clustering methods or enumerative methods [51] andit is computationally tractable [50], [52].Substituting (34) and p m ∗ n into J mn ( δ, P SU ) , the results aredenoted as J m ∗ n . For any subcarrier m , it will be assigned tothe user who has the biggest J m ∗ n ( δ, P SU ) . Let n ∗ m be theresult of subcarrier m ’s assignment which is given by: n ∗ m = arg max n J mn , ∀ n and ̟ m ∗ n = (cid:26) if n = n ∗ m otherwise (39)Let E ∗ = P Nn =1 P Mm =1 p m ∗ n ̟ m ∗ n . As to the value of δ , weuse subgradient method to update it as in (40), δ ( i + 1) = [ δ ( i ) − ς △ δ ( i )] + (40)where △ δ ( i ) = P max − E ∗ ( t, i ) . △ δ ( i ) is the subgradient of H ( δ ) at δ and ς is the step size which should be a smallpositive constant. In addition, index i stands for iterationnumber. When the subgradient method converges, the resourceallocation is completed.From the above description, we can find some principles ofresource allocation. Remark 1 : In (34), both virtual and actual queues of open aswell as private data reflect the gap between the correspondinguser’s demand on data rate and the data rate that the systemcan provide. Thus, X on Q on q omax and X pn Q pn q pmax can be regarded as thetransmission urgency of open data and private data. Onlywhen the transmission urgency of private data exceeds opendata, CBS would allocate some resource to transmit privatedata. Otherwise, CBS would use the user’s entire resource to TABLE IA
LGORITHM D ESCRIPTIONS
Proposed online control algorithm in timeslot t Flow control:Use (26), (27), (30), (31) to calculate T on , T pn , µ on and µ pn respectively. Resource allocation: a) Set the Lagrange multiplier δ = δ ini , ( δ ini : An initial value of δ ). b) For each ( n, m ) i) Use (34) to calculate ζ ∗ n .ii) Use (38) or exhaustive search to find p m ∗ n .iii) Use (39) to calculate ̟ m ∗ n . c) Use (40) to update δ and calculate △ δ ( i ) . d) If |△ δ ( i ) | > △ δ c , goto b) , else proceed.( △ δ c : converge condition of △ δ ) Update the queues:Use (6), (7), (8), (9), (22) and (23) to update all queues including Q on ,Q pn , X on , X pn , Y , Z n . transmit open data due to delay constraint. In PS, it is easy tofind that a bigger Y results in less power allocated to everyuser, which will reduce the system power consumption. Alsowe let Q to be the weights of R P U in PS. It means thatif the transmission pressure of PU is high, CBS will allocateless power in subcarrier set Γ P U to avoid causing too muchinterference on primary link. Remark 2 : In the sub-problem of PS, the transmission powerof PBS is assumed to be external variables. Even for theworst case that PBS does not control its transmission poweractively, the proposed resource allocation algorithm aims tomaximize Q R P U in PS by adjusting the interference fromthe secondary networks to primary networks. Thus, it can befound that the proposed algorithm actually does not affect theenergy consumption of primary networks too much. C. Control algorithm of Multi-PU case
Flow control algorithm is the same as (26), (27), (30) and(31).Resource allocation of multi-PU implementation is thesolution to problem MPS:MPS:Maximize: N X n =1 X on Q on q omax R on + N X n =1 X pn Q pn q pmax R pn + K X k =1 Q k R P Uk − Y E (41)Subject to: (1) , (12) In next section, the algorithm performance with single PUis analysed. It is easy for readers to prove that multi-PUimplementation ensures primary data queue stability and fur-thermore enjoys a similar performance as single PU situation.VI. A
LGORITHM P ERFORMANCE
Before the analysis it is necessary to introduce some aux-iliary variables. Let t ∗ = ( t p, ∗ n , t o, ∗ n ) be the solution to the following problem: max t : t ∈ Υ P Nn =1 θ n t pn + ϕ n t on , Subject to: e ≤ P avg And t ∗ ( ǫ ) = ( t p, ∗ n ( ǫ ) , t o, ∗ n ( ǫ )) denotes the solution of: max t : t + ǫ ∈ Υ P Nn =1 θ n t pn + ϕ n t on Subject to: e ≤ P avg According to [53], it is true that: lim ǫ → N X n =1 { θ n t p, ∗ n ( ǫ )+ ϕ n t o, ∗ n ( ǫ ) } = N X n =1 { θ n t p, ∗ n + ϕ n t o, ∗ n } (42)The algorithm performance will be listed in Theorem 1 and
Theorem 2 . Theorem 1:
Employing the proposed algorithm, both actualqueues of open data Q on ( t ) and private data Q pn ( t ) in CBS havedeterministic worst-case bounds: Q on ( t ) ≤ q omax , Q pn ( t ) ≤ q pmax , ∀ t, ∀ n (43) Theorem 2:
Given q omax > µ max + C omax + µ max ǫ , (44) q pmax ≥ D max + C pmax + D max ǫ , (45) ρ n > q omax ν o, ∗ n ( ǫ ) , ∀ n (46)where ǫ is positive and can be chosen arbitrarily close to zero.The proposed algorithm performance is bounded by: lim inf t →∞ t t − X τ =0 N X n =1 { θ n T pn ( τ ) + ϕ n T on ( τ ) }≥ N X n =1 { ϕ n t o, ∗ n ( ǫ ) + θ n t p, ∗ n ( ǫ ) } − BV (47)where B is a positive constant independent of V and itsexpression can be found in appendix B.In addition, the algorithm also ensures that the long-termtime-average sum of PU queue Q and virtual queues X on , X pn , Z n , Y has an upper bound: lim sup t →∞ t t − X τ =0 { N X n =1 ( X on + X pn + Z n ) + Y + Q }≤ B + V N P n =1 { [ θ n t p, ∗ n + ϕ n t o, ∗ n ] } σ (48)where o ≤ σ ≤ ǫ . The proof of Theorem 1 is in appendix A.Theorem 2 and the definition of σ can be found in appendixB. Remark 3 (Network stability) : According to the definitionof strongly stability as shown in (4), (43) and (48) indicatethe stabilities of all queues in the network system. As aresult, the network system is stabilized and the long-term time-average constraints of delay and power are satisfied. Notice here that Q ’s stability is proved means the PU queue stabilityconstraint is fulfilled. Q ’s stability means that the long-termthroughput performance is uninfluenced. In addition, if PU’sarrival rates are within the stability region of PU networks, Q ’s stability can be ensured by the proposed schedulingalgorithm for any transmission power of PU base station.Therefore, the transmission power of PU network is notaffected in this situation. Furthermore, (43) states that all theactual queues of open data and private data have deterministicupper bounds, and this characteristic means that the CBS canaccommodate the random arrival packets with finite buffer. Remark 4 (Optimal throughput performance) : (47) states alower-bound on the weighted throughput that our algorithmcan achieve. Since B is a constant independent of V , ouralgorithm would achieve a weighted throughput arbitrarilyclose to P Nn =1 { ϕ n t o, ∗ n ( ǫ ) + θ n t p, ∗ n ( ǫ ) } for some ǫ ≥ .Furthermore, given any ǫ ≥ , we can get a better algorithmperformance by choosing a larger V without improving thebuffer sizes. In addition, as it is shown in (42), when ǫ tendsto zero, our algorithm would achieve a weighted throughputarbitrarily close to P Nn =1 { ϕ n t o, ∗ n + θ n t p, ∗ n } with a tradeoffin queue length bounds and long-term time-average delayconstraints as shown in (44)-(46). Thus we can see that withsome certain finite buffer sizes, the proposed algorithm canprovide arbitrarily-close-to-optimal performance by choosing V , and V ’s influence on queue length is shifted from actualqueues to virtual queues.VII. I MPLEMENTATION WITH I MPERFECT E STIMATION
CBS needs the information of queue length from primarynetworks to decide the resource allocation among SUs. [17]considers a situation that queue length information is sharedamong all the nodes, but in CR environment it is impossible toknow the non-cooperative PU’s queue information precisely.Compared with getting perfect information about Q k , it ismore realistic to know the time-average packet arrival rateof PUs. Considering this, in this section, we propose animperfect estimation of Q k by CBS. And the performanceof this estimation will be showed in simulation section. If thePU k is busy, the estimated queue length in CBS is: ˆ Q k ( t + 1) = [ ˆ Q k ( t ) − R P Uk ( t )] + + ( λ k + ι ) (49)where ι is an over-estimated slack variable to promise primarylink stability. CBS can get the precise information when PU isidle by listening to primary link ACK to find that no power isused to transmit PU k ’s data packets. In this situation, ˆ Q k = Q k = 0 perfectly holds.As to the control algorithm, we use ˆ Q k to substitute Q k in resource allocation algorithm. For simplicity, we name thisimplementation COCA-E (CBS-side online control algorithmwith estimated PU queue).VIII. S IMULATION
In this section, we firstly simulate COCA performance inan examplary CR system with a single primary link andsecondary network consisting of one CBS, eight SUs and64 subcarriers. All weights of open data and private data Q p , Q o , Q a nd Z Time slot (A) Actual &Virtual Queues of SU 8 and PU Queue,V=50 Q Q Q Z X o , X p a nd Y Time slot (B) Virtual queues of SU 8,V=50 X X Y Fig. 3. Queue evolutions over 4500 slots are set to be 0.8 and 1 respectively. The main algorithmparameters of secondary network are set as: P avg = 0 . W , P max = 1 W , ρ n = 60 , ∀ n , q omax = 200 , q pmax = 1000 , µ max = 50 , D max = 20 and λ on = n ∗ . ∗ D max , λ pn = n ∗ . ∗ µ max , for n ∈ { , , · · · , } . The long-term time-average arrival rate of PU λ is set to be and D P Umax = 200 .We simulate the multipath channel of primary and secondarynetworks as Rayleigh fading channels and the shadowingeffect variances are 10 dB. The cross-link channels betweenPBS to SUs and CBS to PU are simulated as long-scale fading.All parameters in the following parts are set the same as thesementioned here, except for other specification.In Fig. 3 and Fig. 4, we set average value of a m S , a S =0 . , and average value of a mnP , a nP = 21 , V = 50 and weshow both primary and secondary networks’ queue evolutionover 4500 slots. Because all SUs’ data queues ( Q on , Q pn ) andvirtual queues ( X on , X pn , Z n ) enjoy similar trends, we take SU8 as an example. Fig. 3 (A) shows the dynamics of SU 8’sdata queues Q o , Q p , virtual delay queue Z and PU queue Q . It is observed that both actual data queues are strictlylower than their own deterministic worst case upper bound,which verifies Theorem 1 . That Q is stable in Fig. 3 (A)illustrates that our algorithm can ensure PU queues stabilityfrom simulation aspect. Besides, in Fig. 3(B), we can alsosee that virtual queues X o , X p and Y are bounded. So Fig.3 shows that all queues are bounded, which means that thenetwork system is stabilized and the long-term time-averageconstraints of delay and power are satisfied.Fig. 4 directly shows eight SUs’ long-term time-averageadmitted rates and service rates of open data and private data,respectively. Notice that, every user’s admitted rate is smallerthan service rate and this promises the stabilities of actual dataqueues.Fig. 5 shows the relationship between the weighting param-eters and long-term time-average service rates. To show theeffects more clearly, we consider the scenario consisting ofonly one SU and one PU with fixed ϕ = 450 and variational θ ∈ { , . , , , } . The long-term time-averagearrival rate of SU is set as: λ p = 8 and λ o = 260 . The control T i m e - a v e r g e d a t a r a t e ( b i t s / s l o t ) (B)Time-average admitted & service rate of private data of different SU Admitted rate of private rateService rate of private rate1 2 3 4 5 6 7 801020304050 SU index T i m e - a v e r g e d a t a r a t e ( b i t s / s l o t ) (A)Time-average admitted & service rate of open data of different SU Admitted rate of open dataService rate of open data
Fig. 4. Long-term time-average admitted and service rate of all SUs
Weights of private data throughput θ S e r v i ce r a t e o f op e n d a t a r o Weights of private data throughput θ S e r v i ce r a t e o f p r i va t e d a t a r p Fig. 5. The time average service rate ( r p and r o ) versus the weights ofprivate data θ parameter V is set to be . Each value in Fig. 5 is obtainedby averaging the converged results of 5000 times. Fig. 5 showswith the increase of θ the long-term time-average service rateof private data increases while the one of open data decreases,which illustrates the effect of throughput weights on long-termtime-average service rates.Fig. 6 demonstrates the relationship between different long-term time-average network performance versus control pa-rameter V . In order to compare PU and SU performance,the similar scenario including one PU and one SU is alsoconsidered here. The average data arrival rates of SU areset as: λ o = 250 and λ p = 10 . In general, the bigger V results in the higher SU open and private transmission ratesas Fig. 6 (B) and Fig. 6 (C) respectively show. Fig. 6 (A)demonstrates PU transmission rate decreases as V increases.Notice here, although r P U decreases, even when V = 380 , r P U approximates and is greater than λ = 140 , whichpreserves PU queue stability. Fig. 6 (D) shows the queuingdelay performance also improves as V increases.The implementation of COCA-E with imperfect estimated Q is simulated. We set the over-estimated slack variable
30 80 180 280 380144146148150152154156158 r P U Parameter V(A) PU time-average rate versus different V
30 80 180 280 380280300320340360380 r o Parameter V(B) SU time-average open transmission rate versus different V
30 80 180 280 38014.414.614.81515.215.4 r p Parameter V(C) SU time-average private transmission rate versus different V
30 80 180 280 38011.011.021.031.041.051.061.07 ρ o Parameter V(D) SU time-average queuing delay versus different V
Fig. 6. COCA performances (long-term time-average PU rate, SU rates andSU queuing delay) versus control parameter V . R C O C A - R C O C A - E Time slot(C) The difference of PU rate under under two implementations R o , C O C A - R o , C O C A - E Time slot(A) The difference of SUs sum open transmission rate between two implementations R p , C O C A - R p , C O C A - E Time slot(B) The difference of SUs sum private transmission rate between two implementations
Fig. 7. The rate difference of COCA and COCA-E implementation during4500 slots. ι to be 0.01. We show the differences of the sum servicerate of SUs and R P U between COCA and COCA-E in Fig.7 (A), Fig. 7 (B) and Fig. 7 (C), respectively. We can seethat all the differences are around zero, and SU sum rateis more effected than R P U by the imperfect estimation ofPU queue information. More directly, the influence of ι onthe long-term time-average rate difference between COCAand COCA-E is simulated in Fig. 8, where each record isan averaged result of 1000 converged results. Fig. 8 (C)shows that r COCA − r COCA − E becomes more negative as ι increases, which means that the rate decline of PU causedby SU transmissions decreases as ι increases. More directly,if we want to make sure PU transmission is less influenced,we should choose a larger ι . While a larger ι inevitably makesSUs’ transmission rates decrease as Fig. 8 (A) and Fig. 8 (B)show. IX. C ONCLUSIONS
In this paper, we propose a cross-layer scheduling anddynamic spectrum access algorithm for maximizing the long- r o , C O C A - r o , C O C A - E Slack variable ι (A) Time-average SU open rate difference versus ι r p , C O C A - r p , C O C A - E Slack variable ι (B) Time-averge SU private rate differences versus ι r C O C A - r C O C A - E Slack variable ι (C) Time-averge PU rate difference betwenn COCA and COCA-E Fig. 8. The long-term time-average rate differences of COCA and COCA-Eimplementation versus different over-estimated slack variable ι . term average throughput of open and private information in anOFDMA-based CR network. We derive the sufficient conditionto guarantee that full overlay is optimal in this system. Theproposed algorithm can provide a flexible scheduling imple-mentation of open and private information while ensuring thestability of primary networks as well as performance require-ments in CR systems with finite buffer size. Furthermore, theproposed algorithm is proved to be close to optimality withcurrent network states in time-varying environments.A PPENDIX AP ROOF OF T HEOREM t satisfying Q on ( t ) ≤ q omax , itis obviously true for all queues initialized to zero. We provethat for t +1 the same holds. Obviously, there exists two cases.Firstly, we suppose Q on ( t ) ≤ q omax − µ max and we can easilyget Q on ( t + 1) ≤ q omax . Else, if Q on ( t ) > q omax − µ max , thenaccording to (26), T on ( t ) = 0 . Then Q on ( t + 1) ≤ Q on ( t ) ≤ q omax . The proof of Q pn ≤ q pmax is similar and omitted here.A PPENDIX BP ROOF OF T HEOREM Q = { Q , Q on , Q pn , X on , X pn , Y, Z n } . We define Lya-punov function L ( Q ) as: L ( Q ) = 12 { N X n =1 [ q omax − µ max q omax X on + Z n + 1 q omax Q on ( t ) X on + q pmax − D max q pmax X pn + 1 q pmax Q pn X pn ] + Y + Q } (50)According to [47], △ L ( Q ) is defined as the conditionalLyapunov drift for slot t : △ L ( Q ) , E { L ( Q ( t + 1)) − L ( Q ( t )) | Q ( t ) } (51)According to ( | x − y | + z ) ≤ x + y + z − x ( y − z ) , wecan get the results below: q omax − µ max q omax [ X on ( t + 1) − X on ( t )] ≤ q omax − µ max q omax { µ max − X on ( t )[ T on ( t ) − µ on ( t )] } (52) [ Q o n ( t + 1) X on ( t + 1) − Q o n ( t ) X on ( t )] q omax ≤ q omax µ max +( µ max + C o max ) − Q on ( t )[ R on ( t ) − T on ( t )] q omax X on ( t ) (53)The queues of private data have similar inequalities above.Furthermore, we can derive that: △ L ( Q ) − V N X n =1 E { θ n µ pn + ϕ n µ on | Q } ≤ B − Q E { R P U − D P U | Q } − Y E { P avg − E | Q } + N X n =1 { X on ( C o max + µ max )2 q omax + X pn ( C p max + D max )2 q pmax − X pn Q pn q pmax E { R pn − T pn | Q }− X on Q on q omax E { R on − T on | Q } − (1 − D max q pmax ) X pn E { T pn − µ pn | Q }− (1 − µ max q omax ) X on E { T on − µ on | Q } − Z n E { ρ n µ on − Q on | Q }− V E { θ n µ pn + ϕ n µ on | Q }} (54)where B = ( D P U max + R max + P max + P avg ) + N [ q omax µ max + (1 − µ max q omax ) µ max + (1 − D max q pmax ) D max + q pmax D max ] + N P n =1 ( ρ n µ max + q o max ) and C pmax =max n { R pn } , C omax = max n { R on } , R max = max { R P U } . Herewe can find that our algorithm minimizes the right hand side(RHS) of (54).In order to prove Theorem 2 , we introduce
Lemma 3 . Lemma 3:
For any feasible rate vector t ∈ Υ , there existsa a -only policy SR which stabilizes the network with the dataadmitted rate vector, ( µ pn,SR ( t ) , t pn,SR ( t ) , µ on,SR ( t ) , t on,SR ( t )) ,and the service vector, ( R pn,SR ( t ) , R on,SR ( t )) , independent ofdata queues. For all t and all n ∈ { , , ..., N } , the flowconstraints are satisfied: E { µ on,SR ( t ) } = E { T on,SR ( t ) } = E { R on,SR } E { µ pn,SR ( t ) } = E { T pn,SR ( t ) } = E { R pn,SR } Notice that, the stationary randomized policy SR makes de-cisions only depending on channel condition and independentof queue backlogs. Furthermore it may not fulfill the delayconstraints. Similar proof of a -only policy is given in [17]and the proof of Lemma 3 is omitted here.We can control the admitted rate of t ranging from t ∗ ( ǫ ) to t ∗ ( ǫ )+ ǫ arbitrarily and resulting in that both t ∗ ( ǫ ) and t ∗ ( ǫ )+ ǫ are within Υ . It is assumed that the sufficient conditionof full overlay optimum (21) is satisfied in our system, soaccording to Lemma 2 , full overlay can achieve the optimalresult. Besides, according to
Lemma 3 , it is true that there existtwo different a -only policies SR and SR which satisfy: E { T on,SR } = E { R on,SR } = E { µ on,SR } = t o, ∗ n ( ǫ ) (55) E { T pn,SR } = E { R pn,SR } = E { µ pn,SR } = t p, ∗ n ( ǫ ) (56) E { T on,SR } = E { R on,SR } = E { µ on,SR } = t o, ∗ n ( ǫ ) + ǫ (57) E { T pn,SR } = E { R pn,SR } = E { µ pn,SR } = t p, ∗ n ( ǫ ) + ǫ (58)In addition, for policy SR and SR , it is easy to provethat: E { R P U ,SR } ≥ λ + ǫ (59) E { E SR } ≤ P avg − ǫ (60)Our algorithm minimizes RHS of (54) among all possiblepolicies including SR policy, thus we can get : △ L ( Q ) − V N X n =1 E { θ n µ pn + ϕ n µ on } ≤ B + Y { E { E SR } − P avg } − Q { E { R P U ,SR } − λ } + N X n =1 { Z n Q on + C o max + µ max q omax X on + C p max + D max q pmax X pn +[ q omax − µ max q omax X on − Z n ρ n − V ϕ n ] X on E { µ on,SR } + E { T on,SR } X on q omax [ Q on + µ max − q omax ] − X on Q on q omax E { R on,SR } + E { T pn,SR } X pn q pmax [ Q pn − q pmax + D max ] − X pn Q pn q pmax E { R pn,SR } + E { µ pn,SR } [ q pmax − D max q pmax X pn − V θ n ] (cid:9) (61)After substituting (55)-(58) , (59) and (60) into the RHS of(61) and transforming it, we can derive that: △ L ( Q ) − V N X n =1 E { θ n µ pn + ϕ n µ on } ≤ B − ǫ ( Y + Q ) − X { t o, ∗ n ( ǫ ) ρ − q max } Z − V N X n =1 { ϕ n t o, ∗ n ( ǫ ) + θ n t p, ∗ n ( ǫ ) }− N X n =1 X on q omax { ǫ ( q omax − µ max ) − C o max + µ max }− N X n =1 X pn q pmax { ǫ ( q pmax − D max ) − C p max + D max } (62)So when (44)-(46) hold, we can find ǫ > that ǫ ≤ ǫ ( q omax − µ max ) − Co max + µ max q omax , ǫ ≤ t o, ∗ n ( ǫ ) ρ n − q max and ǫ ≤ ǫ ( q pmax − D max ) − Cp max + D max q pmax . Thus: △ L ( Q ) − V N X n =1 E { θ n µ pn + ϕ n µ on } ≤ B − V N X n =1 { ϕt o, ∗ n ( ǫ )+ θ n t p, ∗ n ( ǫ ) } − σ ( N X n =1 { X on + X pn + Z n } + Y + Q ) (63)where σ = min { ǫ, ǫ } . It can be got that when (44), (45) and (46) hold, (48) and lim inf t →∞ t t − X τ =0 N X n =1 { θ n µ pn ( τ ) + ϕ n µ on ( τ ) } ≥ N X n =1 { ϕ n t o, ∗ n ( ǫ ) + θ n t p, ∗ n ( ǫ ) } − BV (64)are satisfied by applying the theorem of Lyapunov Optimiza-tion, Theorem 4.2 in [47], on (63) directly. Furthermore, (48)implies that (10) and (11) hold since X on and X pn are keptstable. So after substituting (10) and (11) into (64), (47) holds.Hence the proof of Theorem 2 is completed.A
PPENDIX CP ROOF OF L EMMA r P U = λ , we can represent κ as afunction of p mn and substitute it into (16). Then, the optimumsolution can be found by solving: max p mn λ f ( P SU ) + N X n =1 M X m =1 ζ n { ( θ n − ϕ n )[log (1 + a mn p mn ) − log (1 + b mn p mn )] + + ϕ n W log (1 + a mn p mn ) } (65)s.t. ≤ X ∀ n, ∀ m p mn ≤ P max where f ( P SU ) = P Nn =1 { ( θ n − ϕ n ) ζ n { P m ∈ Γ PU [log (1 + p mn a mn P m a mnP ) − log (1 + p mn b mn P m b mnP )] + − P m ∈ Γ PU [log (1 + a mn p mn ) − log (1 + b mn p mn )] + } + ϕ n P m ∈ Γ PU [log (1 + a mn p mn P m a mnP ) − log (1 + a mn p mn )] + } / P m ∈ Γ PU { log (1 + A m P m a m S p mn ) } and we denote the denominator of f ( p ) as ∆ d .It is reasonable to make an approximation of f ( P SU ) underthe assumption that PU is in a high SINR region such that log (1 + a m S p mn + A m P m ) ≈ log (1 + A m P m ) .One sufficient condition of full overlay scheme achievingthe optimal solution of (65) is that κ = 1 makes the objectiveof (66) greater than any other κ ≥ . max
b mn .In this situation, security transmission happens with positiveprivate transmission rate and [log(1 + p mn a nn ) − log(1 + p mn b nn )] + = log(1 + p mn a nn ) − log(1 + p mn b nn ) . We derive thefirst derivative of the objective of (66): ( θ n − ϕ n )( a mn a mn p mn − b mn b mn p mn )+ ϕ n W a mn a mn p mn + λ ∂f ( P SU ) ∂p mn (67)Then for any m ∈ Γ P U , we derive that: ∂f ( P SU ) ∂p mn = a m S ∆ d (1 + a m S p mn ) { θ n { log (1 + b mn p mn )(1 + a mn p mn ) +[log (1 + p mn a mn P m a mnP ) − log (1 + p mn b mn P m b mnP )] } + ϕ n [log (1 + p mn a mn P m a mnP ) − log (1 + p mn b mn P m b mnP )] } +1∆ d { ϕ n [ b mn P m b mnP + b mn p mn − b mn b mn p mn ]+ θ n [( b mn b mn p mn − a mn a mn p mn ) + ( a mn P m a mnP + a mn p mn − b mn P m b mnP + b mn p mn )] } (68)Substituting (68) into (67), we can get the numeration ofthe first derivative of the objective of (66): ∆ d [ θ n (∆ d − λ )( a mn a mn p mn − b mn b mn p mn )+ θ n λ ( a mn P m a mnP + a mn p mn − b mn P m b mnP + b mn p mn )+ ϕ n (∆ d − λ ) b mn b mn p mn + ϕ n λ b mn P m b mnP + b mn p mn ]+ ϕ n [log (1 + b mn p mn P m b mnP ) − log (1 + b mn p mn )]+ a m S a m S p mn λ { θ n { [log (1 + p mn a mn P m a mnP ) − log (1 + p mn b mn P m b mnP )] }} − log (1 + a mn p mn )(1 + b mn p mn ) ≥ ∆ d [ θ n λ ( a mn P m a mnP + a mn p mn − b mn P m b mnP + b mn p mn )+ ϕ n λ b mn P m b mnP + b mn p mn ] − a m S λ { θ n [log (1 + a mn p mn ) − log (1 + b mn p mn )] + ϕ n log (1 + b mn p mn ) }≥ θ n λ ( a mn P m a mnP + a mn p mn − b mn P m b mnP + b mn p mn )+ ϕ n λ b mn P m b mnP + b mn p mn − a m S λ ϕ n log (1 + b mn p mn ) − a m S λ θ n [log (1 + a mn p mn ) − log (1 + b mn p mn )] (69)The last inequality in (69) holds under the assumption that A m ≫ a m S .For any pair of ( θ n , ϕ n ) , the RHS of the last inequality of(69) is greater than zero if and only if ( a mn P m a mnP + a mn p mn − b mn P m b mnP + b mn p mn ) − a m S [log (1 + a mn p mn ) − log (1 + b mn p mn )] ≥ (70) b mn P m b mnP + b mn p mn − a m S log (1 + b mn p mn ) ≥ (71)Sufficient conditions for full overlay are: a m S ≤ ¯ C mn = a mn P m a mnP + a mn p mn − b mn P m b mnP + b mn p mn log (1 + a mn p mn ) − log (1 + b mn p mn ) (72) a m S ≤ ¯ C mn = b mn P m a mnP + b mn p mn log (1 + b mn p mn ) (73)So the sufficient condition of full overlay when ζ n = 1 and a mn > b mn is: a m S ≤ min { ¯ C nm , ¯ C nm } (74)And for ζ n = 0 or a mn ≤ b mn we can get similar conclusionand omit the process here. The sufficient condition is: a m S ≤ ¯ C nm = a mn P m a mnP + a mn p mn log (1 + a mn p mn ) (75)It is easy to find that for any a m S ≤ min { ¯ C nm , ¯ C nm } , a m S is definitely no greater than ¯ C nm . So for any m , the sufficientcondition for the optimum of user n accessing this subcarrier m in full overlay mode is a m S ≤ min { ¯ C nm , ¯ C nm } For P max > p mn , C nm ≤ ¯ C nm and C nm ≤ ¯ C nm hold.Hence the results in Lemma 1 are proved.R
EFERENCES[1] T. A. Weiss and F. K. Jondral, “Spectrum pooling: an innovative strategyfor the enhancement of spectrum efficiency,”
Communications Magazine,IEEE , vol. 42, no. 3, pp. S8–14, 2004.[2] X. Huang, D. Lu, P. Li, and Y. Fang, “Coolest path: spectrum mobilityaware routing metrics in cognitive ad hoc networks,” in
DistributedComputing Systems (ICDCS), 2011 31st International Conference on .IEEE, 2011, pp. 182–191.[3] R. Deng, J. Chen, X. Cao, Y. Zhang, S. Maharjan, and S. Gjessing,“Sensing-performance tradeoff in cognitive radio enabled smart grid,”
Smart Grid, IEEE Transactions on , vol. 4, no. 1, pp. 302–310, March2013.[4] R. Deng, J. Chen, C. Yuen, P. Cheng, and Y. Sun, “Energy-efficientcooperative spectrum sensing by optimal scheduling in sensor-aidedcognitive radio networks,”
Vehicular Technology, IEEE Transactions on ,vol. 61, no. 2, pp. 716–725, Feb 2012.[5] E. Lawrey, “Multiuser ofdm,” in
Signal Processing and Its Applications,1999. ISSPA’99. Proceedings of the Fifth International Symposium on ,vol. 2. IEEE, 1999, pp. 761–764.[6] X. Zhou, G. Y. Li, and G. Sun, “Multiuser spectral precoding forofdm-based cognitive radios,” in
Global Telecommunications Conference(GLOBECOM 2011), 2011 IEEE . IEEE, 2011, pp. 1–5.[7] ——, “Low-complexity spectrum shaping for ofdm-based cognitiveradios,” in
Wireless Communications and Networking Conference(WCNC), 2011 IEEE . IEEE, 2011, pp. 1471–1475.[8] S. M. Almalfouh and G. L. Stuber, “Interference-aware radio resourceallocation in ofdma-based cognitive radio networks,”
Vehicular Technol-ogy, IEEE Transactions on , vol. 60, no. 4, pp. 1699–1713, 2011.[9] Y. Zhang and C. Leung, “Resource allocation for non-real-time servicesin ofdm-based cognitive radio systems,”
Communications Letters, IEEE ,vol. 13, no. 1, pp. 16–18, 2009.[10] ——, “Cross-layer resource allocation for mixed services in multiuserofdm-based cognitive radio systems,”
Vehicular Technology, IEEE Trans-actions on , vol. 58, no. 8, pp. 4605–4619, 2009.[11] B. Wang and K. Liu, “Advances in cognitive radio networks: A survey,”
Selected Topics in Signal Processing, IEEE Journal of , vol. 5, no. 1, pp.5–23, 2011.[12] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,”
Signal Processing Magazine, IEEE , vol. 24, no. 3, pp. 79–89, 2007.[13] J. Huang, R. A. Berry, and M. L. Honig, “Spectrum sharing withdistributed interference compensation,” in
New Frontiers in DynamicSpectrum Access Networks, 2005. DySPAN 2005. 2005 First IEEEInternational Symposium on . IEEE, 2005, pp. 88–93.[14] L. Le and E. Hossain, “Qos-aware spectrum sharing in cognitivewireless networks,” in
Global Telecommunications Conference, 2007.GLOBECOM’07. IEEE . IEEE, 2007, pp. 3563–3567. [15] M. Levorato, U. Mitra, and M. Zorzi, “Cognitive interference manage-ment in retransmission-based wireless networks,”
Information Theory,IEEE Transactions on , vol. 58, no. 5, pp. 3023–3046, 2012.[16] S. Huang, X. Liu, and Z. Ding, “Distributed power control for cognitiveuser access based on primary link control feedback,” in
INFOCOM,2010 Proceedings IEEE . IEEE, 2010, pp. 1–9.[17] L. Georgiadis, M. J. Neely, and L. Tassiulas, “Resource allocation andcross-layer control in wireless networks,”
Foundations and Trends R (cid:13) inNetworking , vol. 1, no. 1, pp. 1–144, 2006.[18] F. E. Lapiccirella, X. Liu, and Z. Ding, “Distributed control of multiplecognitive radio overlay for primary queue stability,” IEEE transactionson wireless communications , vol. 12, no. 1, pp. 112–122, 2013.[19] J. Jang and K. Lee, “Transmit power adaptation for multiuser ofdmsystems,”
Selected Areas in Communications, IEEE Journal on , vol. 21,no. 2, pp. 171–178, 2003.[20] S. W. Kim, B.-S. Kim, and Y. Fang, “Downlink and uplink resourceallocation in ieee 802.11 wireless lans,”
Vehicular Technology, IEEETransactions on , vol. 54, no. 1, pp. 320–327, 2005.[21] K. Seong, M. Mohseni, and J. Cioffi, “Optimal resource allocation forofdma downlink systems,” in
Information Theory, 2006 IEEE Interna-tional Symposium on . IEEE, 2006, pp. 1394–1398.[22] Y. Zou, T. Chen, and S. Li, “Network-based predictive control ofmultirate systems,”
IET control theory & applications , vol. 4, no. 7,pp. 1145–1156, 2010.[23] X. Zhu, J. Yue, B. Yang, and X. Guan, “Flow rate control and resourceallocation policy with security requirements in ofdma networks,” in
Intelligent Control and Automation (WCICA), 2012 10th World Congresson . IEEE, 2012, pp. 1020–1025.[24] Z. Shen, J. Andrews, and B. Evans, “Adaptive resource allocation inmultiuser ofdm systems with proportional rate constraints,”
WirelessCommunications, IEEE Transactions on , vol. 4, no. 6, pp. 2726–2737,2005.[25] G. Li and H. Liu, “Dynamic resource allocation with finite bufferconstraint in broadband ofdma networks,” in
Wireless Communicationsand Networking, 2003. WCNC 2003. 2003 IEEE , vol. 2. IEEE, 2003,pp. 1037–1042.[26] X. Huang and Y. Fang, “Multiconstrained qos multipath routing inwireless sensor networks,”
Wireless Networks , vol. 14, no. 4, pp. 465–478, 2008.[27] Y. Cui, V. Lau, R. Wang, H. Huang, and S. Zhang, “A survey ondelay-aware resource control for wireless systemsłlarge deviation theory,stochastic lyapunov drift, and distributed stochastic learning,”
Informa-tion Theory, IEEE Transactions on , vol. 58, no. 3, pp. 1677–1701, 2012.[28] Z. Yuanyuan, L. Shaoyuan, and N. Yugang, “Networked predictivecontrol of constrained linear systems with stability guarantee,” in
ControlConference (CCC), 2010 29th Chinese , July 2010, pp. 4355–4360.[29] D. Xue and E. Ekici, “Delay-guaranteed cross-layer scheduling in multi-hop wireless networks,” arXiv preprint arXiv:1009.4954 , 2010.[30] R. Urgaonkar and M. Neely, “Delay-limited cooperative communicationwith reliability constraints in wireless networks,” in
INFOCOM 2009,IEEE . IEEE, 2009, pp. 2561–2565.[31] C. Shannon, “Communication theory of secrecy systems,”
Bell systemtechnical journal , vol. 28, no. 4, pp. 656–715, 1949.[32] L. Ozarow and A. Wyner, “Wire-tap channel ii,” in
Advances inCryptology . Springer, 1985, pp. 33–50.[33] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resourceallocation for secure ofdma systems,”
Vehicular Technology, IEEETransactions on , vol. 61, no. 6, pp. 2572–2585, 2012.[34] X. Wang, M. Tao, J. Mo, and Y. Xu, “Power and subcarrier allocation forphysical-layer security in ofdma-based broadband wireless networks,”
Information Forensics and Security, IEEE Transactions on , vol. 6, no. 3,pp. 693–702, 2011.[35] Y. Pei, Y.-C. Liang, L. Zhang, K. C. Teh, and K. H. Li, “Secure commu-nication over miso cognitive radio channels,”
Wireless Communications,IEEE Transactions on , vol. 9, no. 4, pp. 1494–1502, 2010.[36] T. Kwon, V. W. Wong, and R. Schober, “Secure miso cognitive radiosystem with perfect and imperfect csi,” in
Global CommunicationsConference (GLOBECOM), 2012 IEEE . IEEE, 2012, pp. 1236–1241.[37] Y. Liang, A. Somekh-Baruch, H. V. Poor, S. Shamai, and S. Verd´u,“Capacity of cognitive interference channels with and without secrecy,”
Information Theory, IEEE Transactions on , vol. 55, no. 2, pp. 604–619,2009.[38] E. Ekrem and S. Ulukus, “Capacity region of gaussian mimo broadcastchannels with common and confidential messages,”
Information Theory,IEEE Transactions on , vol. 58, no. 9, pp. 5669–5680, 2012. [39] X. Zhu, B. Yang, and X. Guan, “Cross-layer scheduling with secrecydemands in delay-aware ofdma network.” in Wireless Communicationsand Networking Conference (WCNC), 2013 IEEE.
IEEE, 2013, pp.1339–1344.[40] D. Tse and P. Viswanath,
Fundamentals of wireless communication .Cambridge university press, 2005.[41] M. Wallace, J. R. Walton, and A. Jalali, “Method and apparatus formeasuring reporting channel state information in a high efficiency,high performance communications system,” Oct. 29 2002, uS Patent6,473,467.[42] H. A. Suraweera, P. J. Smith, and M. Shafi, “Capacity limits and perfor-mance analysis of cognitive radio with imperfect channel knowledge,”
Vehicular Technology, IEEE Transactions on
Signals, Systems and Computers (ASILOMAR), 2010Conference Record of the Forty Fourth Asilomar Conference on . IEEE,2010, pp. 47–51.[47] M. Neely, “Stochastic network optimization with application to commu-nication and queueing systems,”
Synthesis Lectures on CommunicationNetworks , vol. 3, no. 1, pp. 1–211, 2010.[48] Y. Xu, T. Le-Ngoc, and S. Panigrahi, “Global concave minimizationfor optimal spectrum balancing in multi-user dsl networks,”
SignalProcessing, IEEE Transactions on , vol. 56, no. 7, pp. 2875–2885, 2008.[49] L. Venturino, N. Prasad, and X. Wang, “Coordinated scheduling andpower allocation in downlink multicell ofdma networks,”
VehicularTechnology, IEEE Transactions on , vol. 58, no. 6, pp. 2835–2848, 2009.[50] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimiza-tion of multicarrier systems,”
Communications, IEEE Transactions on ,vol. 54, no. 7, pp. 1310–1322, 2006.[51] R. Horst and H. Tuy,
Global optimization: Deterministic approaches .Springer, 1996.[52] R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, and T. Bostoen,“Optimal multiuser spectrum balancing for digital subscriber lines,”
Communications, IEEE Transactions on , vol. 54, no. 5, pp. 922–933,2006.[53] A. Stolyar, “Maximizing queueing network utility subject to stability:Greedy primal-dual algorithm,”