A Two-Phase Channel and Power Allocation Scheme for Cognitive Radio Networks

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We consider a cognitive radio network in which a set of base stations make opportunistic unlicensed spectrum access to transmit data to their subscribers. As the spectrum of interest is licensed to another (primary) network, power and channel
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  The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06) A TWO-PHASE CHANNEL AND POWER ALLOCATION SCHEME FORCOGNITIVE RADIO NETWORKS Anh Tuan Hoang and Ying-Chang LiangInstitute for Infocomm Research21 Heng Mui Keng Terrace, Singapore 119613 { athoang, ycliang } @i2r.a-star.edu.sgA BSTRACT We consider a cognitive radio network in which a set of basestations make opportunistic unlicensed spectrum access totransmit data to their subscribers. As the spectrum of in-terest is licensed to another (primary) network, power andchannel allocation must be carried out within the cognitiveradio network so that no excessive interference is caused toany primary user. For such a cognitive network, we proposea two-phase channel/power allocation scheme that improvesthe system throughput, defined as the total number of sub-scribers that can be simultaneously served. In the first phaseof our scheme, channels and power are allocated to base sta-tions with the aim of maximizing their total coverage whilekeeping the interferencecaused to each primary user below apredefined threshold. In the second phase, each base stationallocates channelsto their active subscribersbased on a max-imal bipartite matching algorithm. Numerical results showthat our proposed resource allocation scheme yields signifi-cant improvement in the system throughput.I. I NTRODUCTION The traditional approach of fixed spectrum allocation to li-censed networks leads to spectrum underutilization. In re-cent studies by the FCC, it is reportedthat there are vast tem-poral and spatial variations in the usage of allocated spec-trum, which can be as low as 15% [3]. This motivates theconcepts of   opportunistic unlicenced spectrum access  thatallows secondary cognitive radio networks to opportunisti-cally exploit the underulized spectrum. In fact, opportunisticspectrum access has been encouraged by both recent FCCpolicy initiatives and IEEE standadization activities [4,6].On the one hand, by allowing opportunistic spectrum ac-cess, the overall spectrum utilization can be improved. Onthe other hand, transmission from cognitive networks cancause harmful interference to primary users of the spectrum.Therefore, important design criteria for cognitive radio in-clude maximizing the spectrum utilization and minimizingthe interference caused to primary users.In this paper, we consider a cognitive radio network thatconsists of multiple cells. Within each cell, there is a basestation (BS) supporting a set of fixed users called customerpremise equipments (CPEs). We consider the downlink sce-nario. The spectrum of interest is divided into a set of non-overlapping channels. Each CPE can be either  active  or  idle andaBS needsexactlyonechanneltoserveeachactiveCPE.The spectrum is licensed to a set of primary users (PUs). Forthe cognitiveradionetwork,twooperationalconstraintsmustbe met: •  the total amount of interference caused by all oppor-tunistictransmissionsto eachPU mustnotexceeda pre-defined threshold, •  for each CPE, the received signal to interference plusnoise ratio (SINR) must exceed a predefined threshold.We definethe system throughput  as thetotal numberofactiveCPEs that can be simultaneously served.Note that in order to implement the above system, thereshould be a mechanism for secondary users, i.e., BSs andCPEs, to sense the spectrum and detect the presence of pri-mary users. This is a challenging problem and is beyond thescope of this paper. Here, we simply assume that the posi-tions and operating channels of all PUs are known.We propose a  Two-phase Resource Allocation (TPRA) scheme that improves the system throughput and can be im-plementedwith a reasonable complexity. In the first phase of our scheme, channels and power are allocated to BSs withthe aim of maximizing their total coverage while keepingthe total interference caused to each PU below a predefinedthreshold. The coverage of a particular BS is the number of CPEs that can be supported by the BS using at least one of its allocated channels. In the second phase of TPRA, eachBS allocates channels within its cell so that the number of active CPEs served is maximized. This is done by solving arelated  maximal bipartite matching  problem. Numerical re-sults showthatourproposedTPRA schemeyieldssignificantimprovement in the system throughput.Works on channel allocation in cognitive radio networksinclude[10]and[11]. In[10],WangandLiuconsideraprob- lem of opportunistically allocating licensed channels to a setof cognitive base stations so that the total number of channelusages is maximized. In [11], Zheng and Peng consider aproblem similar to [10]. However, they introduce a rewardfunction that is proportional to the coverage areas of basestations and also allow the interference effect to be channelspecific. Both problemsin [10]and [11]are studied based on graph-coloringframeworks.There are two significant differences between our work and [10] and [11]. Firstly, instead of looking at the total number of channel usages or the coverage area of base sta-tions, we are interested in the number of subscribers that areactually served. While doing so, we take into account thefact that subscribers are not always active. Secondly, a majordrawback of the works in [10,11] lies in their oversimpli- fied binary interference model, which is based on whetheror not the coverage areas of two base stations overlap. Thisis unrealistic and does not capture the aggregate interferenceeffects when multiple transmissions simultaneously happenon one channel. Our model overcomes this by consideringthe interference effects based on received SINR.  The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06) 0 100 200 300 400 500 600 700 800 900 100001002003004005006007008009001000 PU BS CPE Figure 1: Deployment of a cognitive radio network.Works on channel-allocation/power-controlproblems thatmodel interference effects based on received SINR include[2] and [7]. The objective of  [2] is to maximize spectrum utilization while that of  [7] is to minimize total transmitpower to satisfy the rate requirements of all links. How-ever,[2] and [7] do not considerthe scenario of opportunistic spectrum access and there is no issue of protecting primaryusers. In a broader context, our work is related to the classof power control problems for interfering transmission linkswith SINR constraints [1,5,9]. In fact, similar to [1,5,9], we use Perron-Frobeniuos theorem to check the feasibility of aparticular channel allocation.The rest of this paper is organized as follows. In SectionII., we introduce our system model and the control prob-lem. In Section III., we present the TPRA scheme. Numeri-cal results showing the performance of our proposed controlscheme will be discussed in Section IV.. Finally, in SectionV., we conclude the paper and outline the future research.II. P ROBLEM  D EFINITION  A. System Model We consider an opportunistic spectrum access scenario de-picted in Fig. 1. The spectrum of interest is divided into  K  channelsthatarelicensedtoaprimarynetworkof  M   primaryusers (PUs). In the same area, a cognitive radio network isdeployed. This cognitivenetworkconsists of   B  cells. Withineach cell, there is a base station (BS) serving a number of fixed customer premise equipments (CPEs) by opportunisti-cally making use of the  K   channels. Channel allocation andpower control must be applied to the cognitive radio network to ensure that each PU experiences an acceptable level of in-terference.Let  N   denote the total number of CPEs. We consider thedownlink scenario in which data are transmitted from BSsto CPEs. Moreover, we assume that each CPE is only ac-tive and requires data transmission with probability  p a ,  0  < p a  ≤  1 . Assuming that a BS needs exactly one channel toserve each active CPE, we define the system throughput asthe total number of active CPEs that can be simultaneouslyserved. Our objective then is to find a channel/power allo-cation scheme that achieves good averagesystem throughputwhile appropriately protecting all primary users.  B. Operational Requirements1) SINR Requirement for CPEs For the sake of brevity, we use the phrase  ”transmission to-ward CPE   i ”  to refer to the downlink transmission from theBS serving CPE  i  toward CPE  i .Let  G cij  be the channel power gain from the BS servingCPE  j  to CPE  i  on channel  c . Let  P  ci  denote the transmitpower for the transmission toward CPE  i  on channel  c ,  0  ≤ P  ci  ≤  P  max . Ifchannel c is notassignedforthetransmissiontoward CPE  i , then  P  ci  = 0 . The SINR at CPE  i  is given by: γ  ci  =  G cii P  ci N  o  +  N j =1 ,j  = i  G cij P  cj ,  ∀ i  ∈ { 1 , 2 ,...N  } ,  (1)where  N  o  is the noise power spectrum density of each CPE.For reliable transmission toward CPE  i , we require that γ  ci  ≥  γ.  (2)In practice,  γ   can be the minimum SINR required to achievea certain bit error rate (BER) performance at each CPE. 2) Protecting Primary Users Let Π c denote the set of all PUs that use channel  c  and let G c pi  be the channel gain from the BS serving CPE  i  to PU  p on channel  c . We require that, for each PU, the total interfer-ence from all opportunistic transmissions does not exceed apredefined threshold  ζ  , i.e., N   i =1 P  ci  G c pi  ≤  ζ,  ∀  p  ∈ Π c ,  ∀ c  ∈ { 1 , 2 ,...K  } .  (3) C. Feasible Assignments Beforemovingon, let us address the question of whetherit isfeasible to assign a particular channel  c  simultaneously to aset of transmissions toward  m  CPEs:  ( i 1 , i 2 ,...i m ) . Here,feasibility means there exists a set of positive transmit powerlevels  P  c = ( P  ci 1 ,P  ci 2 ,...P  ci m ) T  such that all the SINR con-straints ofthe m CPEs are metwhile theinterferencescausedto PUs do not exceed the acceptable threshold.If we define an  m × 1  vector  U  c as: U  c =  γN  o G ci 1 i 1 , γN  o G ci 2 i 2 , ... γN  o G ci m i m  T  (4)and an  m  × m  matrix  F  c as: F  crs  =   0 ,  if   r  =  s γG ciris G cirir ,  if   r   =  s, r,s  ∈ { 1 , 2 ...m }  ,  (5)then the SINR constraints of   m  CPEs  ( i 1 , i 2 ,...i m )  can bewritten compactly as: ( I   − F  c ) P  c ≥  U  c .  (6)From the Perron-Frobenioustheorem [1,5,9], (6) has a posi- tive component-wisesolution P  c if and onlyif the maximumeigenvalue of   F  c is less than one. In that case, the Pareto-optimal transmit power vector is: P  c ∗ = ( I   − F  c ) − 1 U  c .  (7)  The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06) HerePareto-optimalmeansthat if  P  c is a positivepowervec-tor that satisfies (6), then  P  c ≥  P  c ∗ component-wise. Dueto this fact, the following 2-step procedure can be used tocheck the feasibility of assigning a particular channel  c  tothe transmissions toward the set of CPEs  ( i 1 ,i 2 ,...i m ) . Two-step Feasibility Check: •  Step 1: Checkifthe maximumeigenvalueof  F  c definedin (5) is less than one. If not, the assignment is notfeasible, otherwise, continue at Step 2. •  Step 2: Using (7) to calculate the Pareto-optimal trans-mit power vector  P  c ∗ . Then, check if   P  c ∗ satisfies theconstraints for protecting PUs in (3) and the maximumpower constraints, i.e.  P  c ∗ ≤  P  max . If yes, concludethat theassignmentis feasible and P  c ∗ is thepowervec-tor to use. Otherwise, the assignment is not feasible.If it is feasible to assign channel  c  to the transmis-sions toward CPEs  ( i 1 ,i 2 ,...i m ) , we simply say  the set  ( i 1 ,i 2 ,...i m )  is feasible on channel  c .III. T WO -P HASE  R ESOURCE  A LLOCATION  A. Motivations We are interested in channel/power allocation schemes thatcan simultaneously serve a good number of active CPEswhile protectingall PUs fromexcessiveinterference. To pro-tect PUs, all BSs have to coordinate their transmit powers oneach channel. That requires a global control mechanism. Onthe other hand, CPEs in the network can switch between ac-tive and idle states frequently. In that case, it is preferablethat changes in CPEs’ states are dealt with locally, withineach cell. This will reduce the amount of recalculations andsignaling/updates involved. These observations motivate our Two-Phase Resource Allocation (TPRA)  scheme.  B. The Two-Phase Resource Allocation Scheme1) Phase 1 - Global Allocation: In this phase, channels and transmit powers are allocated toBSs so that the interference caused to each PU is below atolerable threshold. At the same time, we aim to cover asmany CPEs as possible. When talking about coverage here,we do not care whether a CPE is active or idle. That will betaken care of in the second phase of the TPRA scheme.Consider a particular channel  c . For each BS, the higherpower it transmits on  c , the more CPEs it can cover. How-ever, the higher the transmit power of the BS, the more in-terference it causes to PUs and other cells. This interferencereduces the number of CPEs that can be covered using chan-nel  c  in other cells.We note that it is extremely hard to fully characterize theabove dual effects of varying base stations transmit powerson the number of CPEs being covered in the whole network.Therefore, we rely on the following intuition for making ourchannel/powerallocationdecisions. ABSthatisnearanyPUusing channel  c  should only transmit at low power to reduceinterference. On the other hand, a BS that is faraway fromall PUs using channel  c  can transmit at higher power. EachBS can use a set of channels on which it can transmit at highpower to cover faraway CPEs. The same BS can use a set of channels on which it can only transmit at low power to covernearby CPEs.Based on the above intuition, we propose the followingprocedure to allocate channels/powers to BSs. We process K   channels one at a time. For channel  c , let  Γ c pb  denote thechannelgainfrombase station b to primaryuser  p anddefine: Γ c ∗ b  = max  p ∈ Π c { Γ c pb } .  (8)We do the following: •  Sort the base stations in the ascending order of   Γ c ∗ b  , i.e.,form  ( b 1 ,b 2 ,...b B )  where  Γ c ∗ b n ≤  Γ c ∗ b m ,  ∀ 1  ≤  n <m  ≤  B . The base stations will be processed one at atime in this order. •  For base station  b n , determine a particular CPE  i n that  b n  should cover. This is done as follows. Giventhe set  ( i 1 ,i 2 ,...i n − 1 )  of CPEs being covered by ( b 1 ,b 2 ,...b n − 1 ) , let  V   cn  be the set of all CPEs in thecell of   b n  such that  ( i 1 ,i 2 ,...i n − 1 ,i )  is feasible onchannel  c  (see the two-step feasibility check in SectionII-C.). Then  i n  is the CPE that has the weakest channelgain from  b n , i.e., i n  = arg min i ∈ V   cn { G cii }  (9)It can happenthat the set V   cn  is empty. Then, with somelittle abuse of notation, we set  i n  = 0  to indicate that noCPE is covered by  b n . •  After processing all BSs in the order  b 1 ,b 2 ,...b B , weobtain a set of CPEs  ( i 1 ,i 2 ,...i B ) . Using (7), deter-mine the transmit power to serve each of these CPEs,i.e.,  ( P  ci 1 ,P  ci 2 ,...P  ci B ) . •  Finally, based on  ( P  ci 1 ,P  ci 2 ,...P  ci B ) , determinethe N  × K   coverage matrix  C  , where  C  ( i,c ) = 1  indicates thatCPE  i  can be served by the corresponding BS on chan-nel  c . This can be checked based on (2).It can happen that when sorting BSs based on  Γ c ∗ b  , there areties among BSs. In that case, the BSs with less number of CPEs covered so far (can be checked from coverage matrix C  ) will be processed first. 2) Phase 2 - Local Allocation Based on the coverage matrix  C   obtained in the first phase,channel allocation can be carried out within each cell, in amanner independent to what happens in the rest. The proce-dure is as follows. •  First, determine all active CPEs in the cell. •  Next,forma bipartitegraphthatrepresentsthecoverageof the cell. This is done by representingthe set of activeCPEs as a set of vertices, which are connected to an-other set of vertices representingthe available channels.Notethat anedgeexists betweenthe vertexrepresentingCPE  i  and the vertex representing channel  c  if and onlyif   C  ( i,c ) = 1 . This is demonstrated in Fig. 2. •  Now, the problem of maximizing the number of activeCPEs served is equivalent to the problem of maximiz-ing the number of disjoint edges in the resulting bipar-tite graph. Two edges in a graph are disjoint if they donot share any end. This is called the maximal bipartitematching problem.  The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06) CPEs Channels 1 3 2 4 1 3 2 Figure 2: Representing the coverage within one cell as a bi-partite graph. Edges with the same color represent the samechannel.There are a host of algorithms for finding the maximalmatching of a bipartite graph. In this paper, we obtain maxi-mal bipartite matching based on Berge’s Theorem of findingalternating augmenting paths [8]. C. Other Channel/Power Allocation Schemes To relatively evaluate the performance of our proposedTPRA scheme, let us consider the following other resourceallocation schemes. 1) Random Allocation In this so called  Random  scheme, the two-phase approachis still followed. However, the decisions in each phase aremade in random manners.In the first phase, for each channel  c , the base stationsare processed one at a time following a random order, e.g., ( b 1 ,b 2 ,...b B ) . For base station  b n , instead of pickinga CPEto cover according to (9), we just arbitrarily choose any CPE i  with  i  ∈  V   cn .In the second phase, for each cell, no maximal bipartitematching is carried out. Instead, active CPEs in the cell areprocessed one at a time according to a randomly chosen or-der. For each active CPE, we assign the first free channel.This continues until all active CPEs of the cell are served orno more channel is available. 2) Non-overlapping Allocation In this scheme, the  K   channels are first partitioned into  B disjoint groups, each consists of   ⌊ K/B ⌋  channels. Each of the base stations is then assigned one group of channels tosupport its active CPEs. Channel groups are formed and as-signedasfollows. Pickanarbitraryorderofthebasestations,e.g.,  ( b 1 ,b 2 ,...b B ) , and let them to choose their  ⌊ K/B ⌋ channels one at a time in this order. This means  b n  can-not pick the channels chosen by  b 1 ,b 2 ,...b n − 1 . Among allchannel left, each of the channel c  chosen by  b n  must satisfy: Γ c ∗ b n <  Γ c ∗ b m ,  ∀ 1  ≤  n < m  ≤  B.  (10)Each BS can assign channels to its active CPEs based onthe simple allocation procedure of the Random scheme dis-cussed above. We call this the  Non-overlapping Channel Al-location (NOCA)  scheme. 3) Allocation Based on An Interference Graph In [2], Behzad and Rubin proposethe powercontrolschedul-ing algorithm (PCSA) that improves the system throughputwhile also guaranteeing the SINR constraints of all trans-mission links. In the PCSA scheme, an interference graph, 10 15 20 25 30 35 40 45 50 55 6010%20%30% 40%50% 60% 70%80%90%100% No. of primary users    %   o   f   C   P   E  s  c  o  v  e  r  e   d TPRANOCARandomPCSA Figure 3: Percentage of CPEs being covered vs no. of PUs.No. of BSs =  4 , no. of CPEs =  300 , no. of channels =  16 .which captures the pairwise interference effects among alltransmissions, is first constructed. After that, the prob-lem of channel allocation to maximize the number of non-interferinglinks can be convertedinto the problemof findinga maximum independent set of the interference graph.In Section IV., we will test the performance of PCSA un-der two scenarios. In the first scenario, we reapply the algo-rithm every time there is a change in state of any CPE. Wecall this PCSA G (PCSA Global). In the second scenario,we apply the algorithm to the whole network once, and afterthat, the changes in CPEs’ states are only dealt with locally.We call this PCSA L (PCSA Local).IV. N UMERICAL  R ESULTS AND  D ISCUSSION  A. Simulation Model We consider a square service area of size  1000  ×  1000 m  inwhich a cognitive radio network is deployed. The servicearea is further divided into  B  = 4  adjacent cells, each is asquare of size  500  ×  500 m . A BS is deployed at the centerof each cell to serve CPEs within the cell. The total numberof CPEs is  N   = 300  and each CPE is active with probability  p a  = 0 . 1 . We vary  M  , the total number of PUs, from  10 to  60 . All CPEs and PUs are randomly deployed across theentire service area with a uniform distribution. A samplenetwork is shown in Fig. 1.The number of channels available is  K   = 16 . We assumea free-space path loss model with the path-loss exponent of  4 . We assume that each PU randomly picks and uses one of the  K   channels. The noise power spectrum density at eachCPE is  N  o  = 100 dBm . The required SINR for each CPE is 15 dB . The maximum tolerable interference for each PU is 90 dBm . For each BS, the maximum transmit power on eachchannel is  P  max = 50 mW  .  B. Performance Analysis As the number of active CPEs served is closely related tohow many CPEs in the network are covered, let us look atthepercentageofCPEs beingcoveredfirst. InFig. 3, we plotthe percentage of CPEs being covered versus the number of PUs when four schemes TPRA, NOCA, Random, and PCSAare employed. As expected, when the number of PUs in-creases, the coverage of each of the four schemes decreases.  The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06) 10 15 20 25 30 35 40 45 50 55 600510152025    N  o .  o   f  a  c   t   i  v  e   C   P   E  s  s  e  r  v  e   d No. of primary usersPCSA_GTPRANOCARandomPCSA_L Figure 4: Number of active CPEs being served versus no. of PUs. No. of BSs =  4 , no. of CPEs =  300 , probability of aCPE being active =  0 . 1 , no. of channels =  16 .The coverage of TPRA is best because in this scheme (firstphase), we deliberately seek to cover faraway CPEs. On theother hand, the coverage of PCSA is worst. This is becausewhen using the interference graph approach, PCSA tends toonly cover nearby CPEs to minimize interference. The cov-erage of NOCA is close to that of TPRA. This is because inNOCA scheme, base stations employ non-overlappingchan-nel groups and therefore, can transmit at high power to reachfaraway CPEs. The coverage of Random scheme is signif-icantly worse than that of TPRA, but still much better thanthat of PCSA.Next, in Fig. 4, we plot the average number of activeCPEs served versus the number of PUs for TPRA, NOCA,Random, and PCSA G and PCSA L. Clearly, as PCSA G isallowed to respond globally to changes in CPEs’ states, itsperformance outperforms the rest. We present the through-put of PCSA G here just to show what can be achieved  if wecan tolerate  the computational and signaling costs of alwayscarrying out global control.As can be seen in Fig. 4, our TPRA scheme consis-tently outperforms NOCA and Random schemes. The gainof TPRA, relative to NOCA, is higher when the number of PUs is small. This is because when the number of PUs issmall, there is more chance for channel reuse but NOCA isnot flexible enough to take advantage of that. The gain of TPRA, relative to Random, is higher when the number of PUs increases. This is because Random scheme does nottake PUs into account when carryingout allocation. It is alsointeresting to see how PCSA performs when it is subject tothe local update constraint, i.e. PCSA L. The throughput of PCSA L is much worst than all the other schemes. This isbecause, as shown in Fig. 3, the coverage of PCSA is verylow compared to that of other schemes.In Fig. 5, the number of channels is increased from  16  to 24 . The performance trends are similar to those of Fig. 4.We have results for other sets of system parameters and thetrends are also similar to what have been discussed.V. C ONCLUSIONS In this paper, we consider the problem of channel-allocation/power-controlto maximize the system throughput 10 15 20 25 30 35 40 45 50 55 608101214161820222426 No. of primary users    N  o .  o   f  a  c   t   i  v  e   C   P   E  s  s  e  r  v  e   d TPRANOCARandom Figure 5: Number of active CPEs being served versus no. of PUs. No. of BSs =  4 , no. of CPEs =  300 , probability of aCPE being active =  0 . 1 , no. of channels =  24 .ofa cognitiveradionetworkthatemploysopportunisticspec-trum access. At the same time, a realistic control framework is formulated to guarantee protection to primary users andreliable communications for cognitive nodes.We propose the TPRA scheme that achieves good sys-tem performance while can be implemented at reasonablecomplexity. Numerical results show that our proposedscheme achieves significant performance gain, relative toother schemes.For future research, we are currently extending this work to consider fairness among CPEs as well as their QoS. Atthesametime,ajointnetwork-admission/resource-allocationframework is being developed based on the system model of this paper.R EFERENCES[1] N. Bambos, S. C. Chen, and G. J. Pottie. Radio link admission algo-rithms for wireless networks with power control and active link qualityprotection. In  Proc. of IEEE INFOCOM  , Boston, MA, Nov. 1995.[2] A. Behzad and I. Rubin. Multiple access protocol for power-controlledwireless access nets.  IEEE Transactions on Mobile Computing ,3(4):307–316, Oct.-Dec. 2004.[3] FCC. Spectrum policy task force report, FCC 02-155. Nov. 2002.[4] FCC. Facilitating opportunities for flexible, efficient, and reliablespectrum use employing cognitive radio technologies, notice of pro-posed rule making and order, FCC 03-322. Dec. 2003.[5] G. J. Foschini and Z. Miljanic. A simple distributed autonomouspower control algorithm and its convergence.  IEEE Transactions onVehicular Technology , 42(4):641–646, Apr. 1993.[6] IEEE 802.22 Wireless RAN. Functional requirements for the 802.22WRAN standard, IEEE 802.22- 05/0007r46. Oct. 2005.[7] G. Kulkarni, S. Adlakha, and M. Srivastava. Subcarrier allocation andbit loading algorithms for OFDMA-based wireless networks.  IEEE Transactions on Mobile Computing , 4(6):652–662, Nov./Dec. 2005.[8] K. Mehlhorn and S. Naher.  The LEDA Platform of Combinatorial and Geometric Computing . Cambridge University Press, 1999.[9] D. Mitra. An asynchronous distributed algorithm for power control incellular radio systems. In  Proceedings of 4th WINLAB Workshop onThird Generation Wireless Information Networks , Rutgers University,New Brunswick, NJ, Oct. 1993.[10] W. Wang and X. Liu. List-coloring based channel allocation for open-spectrum wireless networks. In  Proceedings of IEEE 62nd Vehicular Technology Conference (VTC’05 Fall) , Dallas, Texas, Sep. 2005.[11] H. Zheng and C. Peng. Collaboration and fairness in opportunisticspectrum access. In  Proceedings of IEEE International Conference onCommunications (ICC’05) , Korea, May 2005.
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