Distributed Admission and Power Control for Cognitive Radios in Spectrum Underlay Networks John George Ahmed Sultan Mohammed Nafie Wireless Intelligent Networks Center (WINC) Nile University, Cairo, Egypt john.tadrous@nileu.edu.eg Wireless Intelligent Networks Center (WINC) Nile University, Cairo, Egypt asultan@nileuniversity.edu.eg Wireless Intelligent Networks Center (WINC) Nile University, Cairo, Egypt mnafie@nileuniversity.edu.eg Abstract— In this paper we investigate admission control and power allocation for cognitive radios in an underlay network We consider the problem of maximizing the number of supported secondary links under their minimum QoS requirements without violating the maximum tolerable interference on primary receivers in a cellular network An optimal solution to our problem is shown in previous works to be NP-hard We propose an efficient distributed algorithm with reasonable complexity that provides results close to the optimum solution without requiring neither a large amount of signaling nor a wide range of information about the system parameters Our algorithm is compared with previously proposed algorithms to demonstrate its relative efficiency.1 I I NTRODUCTION Recent studies by the FCC show that the current utilization of some spectrum bands is as low as 15% [1] On the other hand, it is commonly believed that there is a crisis of spectrum availability at frequencies that can be economically used for wireless communications via fixed spectrum allocation This misconception is strengthened by a look at the FCC frequency chart [2] that indicates multiple allocations over all of the frequency bands In other words, there is fierce competition for the use of spectra, especially in the bands below GHz while in the same time most of these bands suffer from underutilization Therefore, there is an increasing interest in developing new efficient techniques for spectrum management and sharing as in [3] which consequently motivates the concept of cognitive radios [4] and dynamic spectrum access Cognitive radios are frequency agile devices that can sense the spectrum, identify vacant frequency chunks in licensed bands, and make use of them as long as no harmful interference is produced on primary users Moreover, cognitive radios have to evacuate these frequency chunks upon the arrival of primary users or when the interference produced by cognitive radios on primary users exceeds certain limit A challenging question arises as how to provide Quality of Service (QoS) for cognitive radios while preventing them from violating the maximum tolerable interference on primary receivers at the same time The operation of cognitive radios in dynamic spectrum access environment is divided into two paradigms, spectrum overlay and spectrum underlay [5] In spectrum overlay, cognitive radios are allowed only to access the spectrum chunks This work was partially supported by the Egyptian National Telecommunications Regulatory Authority that are completely empty of any primary operation In this case, cognitive radios have to sense the spectrum first to ensure that there is a vacant spectrum chunk to be accessed by them In spectrum underlay systems, cognitive radios are allowed to share underutilized frequency bands besides the primary users, such that they not violate the interference temperature declared by primary receivers Hence interference constraints are placed on cognitive radios in the spectrum underlay model A lot of work has been done to solve the problem of admission control and power allocation for cognitive radios with QoS and interference constraints [6]-[9] In [6], a centralized user removal algorithm based on treepruning technique was proposed, the proposed algorithm leads to optimal supported set of secondary links but with extensive computations Moreover, a distributed game theoretic approach based on sequential play was introduced, but the sequential play converges to local optimal solutions In [7], a distributed algorithm was introduced that aims at minimization of the total transmit power by primary and secondary links However, the primary users were allowed to increase their transmission power without bounds In order to maximize the number of admitted secondary links under QoS and interference constraints, the authors of [8] developed two centralized removal algorithms namely I-SMIRA and I-SMART(R) However, the removal criteria in both algorithms require the knowledge of instantaneous channel gains between system nodes to be kept at a central controller which is practically difficult In [9], joint admission control and rate/power allocation for secondary links was investigated while only mean channel gains are available However, the authors not show how far their algorithm from the optimal solution In this paper we propose a distributed dynamic algorithm for admission and power allocation for cognitive radios that aims at maximizing the number of admitted secondary links with their target QoS, without violating the interference constraints allowable by primary receivers in an underlay network Our proposed algorithm can be implemented by secondary users alone without the need of a dedicated controller Moreover, the secondary users not have to keep track of the instantaneous channel gains of the whole system if they can measure their own signal-to-interference-and-noise-ratio (SINR) locally The algorithm is performed online by the secondary links so that it 978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings where, P G(p) = B/R is the processing gain in case that primary transmitters are employing spread spectrum technique, B and R are the primary system bandwidth and transmission rate respectively, f is the other-cell interference factor, and N0 is the background noise power We assume that the received power, Pr , from a primary transmitter to BS j is fixed at this BS The transmitted power from a primary user i to its BS j is given by, (u) Pi = Pr (u) gj,i , i = 1, 2, · · · , Kj (2) (u) where, gj,i is the uplink channel gain between the primary transmitter i and its intended BS j The interference, ηj , produced by secondary transmitters on BS j is, N Fig 7-cell system model i=1 can account for any dynamic changes of the system parameters in an adequate time (like interference constraints on primary receivers, number of secondary links currently in the system, variation of channel conditions, etc.) Our simulations show that the proposed algorithm for admission of secondary links yields results that are close to the NP-hard optimum solution, but with reduced complexity The rest of this paper is organized as follows The system model is described in Section II We formulate the admission control and power allocation problem in Section III Our proposed solution and related implementation issues are given in Section IV Section V presents the numerical results Our work is concluded in Section VI We consider a spectrum underlay model where secondary users can share the same frequency band with primary users In this model we focus on the uplink transmission from primary users toward their base stations (BS), whereas secondary users are communicating with each other in an ad-hoc fashion The decisions for admission and power control for secondary links are applied in a distributed manner in which only one secondary link is allowed to take a decision about its transmit power at a time, while other secondary links remain in their previous states That is, decisions are taken in a round robin fashion (RR) A system with seven cells is shown in Fig A Primary Network Model: Our system model is very close to those in [8] and [9] so we use almost the same notations for the system description For the given system model there are M cells of primary network, each cell has one BS in its center Each cell j has Kj , j = 1, 2, · · · , M , primary users transmitting in the uplink direction The SINR of the primary link i is measured at BS j as follows: (p) gj,i Pi , j = 1, 2, · · · , M (3) Where N is the number of secondary transmitters, Pi is the (p) transmission power of transmitter of secondary link i, and gj,i is the channel gain from the transmitter of secondary link i to primary BS j B Secondary Network Model: We assume that there are N secondary link pairs, each comprising a transmitter and one intended receiver The secondary links are distributed uniformly over the primary network area of coverage and are communicating in an ad-hoc mode We denote the interference produced by all primary transmitters on the secondary link i by Ni , K Ni = j=1 II S YSTEM M ODEL μj (p) ηj = (su) (u) gi,j Pj , i = 1, 2, · · · , N (4) M where K = j=1 Kj is the total number of primary transmit(su) ters in the system, and gi,j is the channel gain from primary transmitter j to the receiver of secondary link i Thus, the SINR at the secondary link i is (s) μi = P Gi gi,i Pi (s) N j=1,j=i gi,j Pj + Ni + N0 , i = 1, 2, · · · , N (5) where P Gi = RBi is the processing gain of secondary link i, (s) Ri is the transmission rate of secondary link i and gj,i is the channel gain between the transmitter of secondary link j to the receiver of secondary link i In underlay models, spread spectrum techniques are sometimes applied by secondary links so that their transmission power can be regarded as noise at primary receivers [5] However, our proposed algorithm is applicable to any system where users share the same operating band, i.e., spread spectrum techniques are not necessary III P ROBLEM F ORMULATION The admission control and power allocation technique Pr = P G(p) , j = 1, 2, · · · , M should admit and allocate power for a set of secondary links (1 + f )(Kj − 1)Pr + ηj + N0 (1) that fulfills the following conditions First, the interference 978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings produced by secondary links on primary receivers should not violate a certain limit The SINR of any primary transmitter is required to be higher than certain threshold γ (p) , i.e, (p) μj ≥ γ (p) , j = 1, 2, · · · , M (6) Pi (t + Δt) = Pimax , Inequality (6) can be rewritten as, Pr ≥ γ (p) (1 + f )(Kj − 1)Pr + ηj + N0 (7) P G(p) Pr − (Pr (Kj − 1)(1 + f ) + N0 ) ≥ ηj γ (p) (8) P G(p) Therefore, Ij The interference constraints can then be written as, ηj ≤ Ij , j = 1, 2, · · · , M (9) where Ij is the maximum tolerable interference by primary receiver (BS) j in order to keep the constraint of (6) satisfied Second, since SINR reflects the amount of QoS gained by each secondary link, the SINR at the receiver of each secondary link i is required to be greater than a certain target SINR That is, μi ≥ γi , i = 1, 2, · · · , N (10) where γi is the target SINR for secondary link i Note that each secondary link cannot increase its transmission power beyond a certain level, Pi ≤ Pimax , i = 1, 2, · · · , N (11) The set of the admitted secondary links is called the active set and denoted by ℵ Our problem can be defined as the following optimization problem, maximize subject to all links can be supported at their QoS requirements DCPC iteratively allocates power to secondary link i synchronously or asynchronously with other links according to the following formula, |ℵ| , j = 1, 2, · · · , M ηj ≤ Ij , i = 1, 2, · · · , N μi ≥ γi Pi ≤ Pimax , i = 1, 2, · · · , N (12) where |ℵ| denotes the cardinality of the active set This optimization problem is proved in [8] to be NP-hard when all of the requesting secondary links cannot be supported to operate concurrently under the given constraints IV O NLINE D ISTRIBUTED A LGORITHM A Distributed Power Control: Most of the relevant proposals use the distributed constrained power control (DCPC) algorithm [10] as an efficient power control algorithm for spectrum underlay models [6][8],[11] In fact, DCPC can be applied distributively or in a central controller that has complete knowledge of all instantaneous channel gains and QoS requirements of all users The DCPC algorithm aims at allocating power to links such that the required SINR for each secondary link is achieved at the minimum possible transmission power, provided that Pi (t)γi μi (t) , i = 1, 2, , N (13) where, Pi (t) is the power of secondary link i at interval t, and Pi (t + Δt) is the power of the secondary link i at the next time interval As proved in [10] the power updates of all of secondary links will converge to a fixed power vector P(s) , regardless of the values of the initial powers Each element in P(s) denotes the steady state power of one of the secondary transmitters If the fixed power vector P(s) is found to contain any elements with value P max , then that means the current set of links cannot be supported at its QoS requirements Alternatively, if P(s) does not contain any P max value, then it means the current set of requesting links can be supported at exactly their target QoS requirements Note that, the fixed power vector does not include any information about the interference on primary BSs B Generic Description of the Algorithm Our algorithm uses asynchronous DCPC, specifically the power updates are performed in a round robin fashion (RRDCPC), asynchronous DCPC converges to a fixed power vector faster than synchronous DCPC [11] We divide the current set of requesting secondary links into two sets: the first one is called the active set that contains all the links with transmission power greater than 0, and denoted by ℵ Whereas the second set is called the inactive set that contains all the links with transmission power set to 0, and denoted by , where at any instant |ℵ| + | | = N We describe our algorithm as follows, 1) The secondary links start with certain initial power vector Pinit = (P1init , P2init , , PNinit ), such that Pinit Pmax , and all of them are kept in the active set ℵ while the set is initially empty, i.e = {} and ℵ = {1, 2, · · · , N } 2) In a round robin (RR) fashion one secondary link updates its transmission power according to a formula that resembles (13), that is, Pjmax , 0, γj Pj (LN +j−1) μj (LN +j−1) ,j ∈ ℵ j∈ (14) where L is an integer and j is the current link performing its power update and the term LN + j is the index of the current power update, while other secondary links maintain their transmission power unchanged Secondary links are assumed to perform the power updates based on the local measurements of their SINR i.e., they are not required to keep track of channel gains 3) The power updates of step will continue till one of the following three events occurs: Pj (LN +j) = 978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings a) Event 1: The system with the active set ℵ converges to a fixed power vector P(s) , that does not violate any of constraints in (12) with the secondary links in ℵ admitted b) Event 2: During the power updates the interference constraint on a primary BS is violated Consequently, the BS broadcasts a warning signal that it has interference larger than the tolerable level c) Event 3: When a secondary link j is updating its transmission power according to (14), it finds that Pj (LN +j) = Pjmax implying that this link would not reach it’s target QoS 4) The system should handle the occurrence of any of events or efficiently and continues from step until convergence at event with the admitted links being in set ℵ Our distributed algorithm should take an action once any violation of constraints is detected by either of events or even if RR-DCPC has not completely converged as it might be not efficient to wait for the convergence of RRDCPC while interference constraints on primary receivers are being violated Hence, after any violation of constraints we deactivate one secondary link as will be described below of either of events or as an indicator for its infeasibility in order to quickly remove any possible violation of (9) The following two steps are performed to handle events or 3: 1) Selecting one secondary link to join : We use a simple criterion to select a secondary link for joining the set in order to decrease the amount of signaling exchanged among secondary links After the power update of secondary transmitter j, if events or occur, link j is the link to join the set That is, the user joining set is the one whose power update is followed by a constraint violation 2) Re-activating a secondary link: For the link j to be deactivated and join the set , it should first check the possibility of replacing a link i, where i ∈ , such that i may become re-activated and joins ℵ If replacing i results in an active set that was not tested before, link j joins while the link i joins ℵ Otherwise, when the link j cannot replace any of the inactive links in (because the resulting active set, with i and without j, was estimated before as infeasible), link j joins the inactive set without activating any of the inactive links Hence, the size of is then incremented by while the size of ℵ is then reduced by C Handling Events and 3: D Implementation Issues : Upon the detection of the infeasibility of the current inactive set (after the convergence of the DCPC), [8] and [11] proposed removal of at least one link at a time followed by DCPC again The efficiency of this kind of handling the infeasibility events basically depends on the removal criterion used and the number of links removed at a time Removal criteria like those in I-SMIRA and I-SMART(R) require information about all of system parameters be kept at a central controller e.g., instantaneous channel gains between all nodes, QoS requirements of secondary links, current allowable level of interference on primary BSs Instead, we suggest use of a simple criterion for selecting one secondary link to be deactivated (to join the set ) However, the secondary links in are not completely removed from the system, i.e., they can be reactivated Reactivating links from the set can compensate the inefficiency of the selection criterion by allowing different sets of requesting secondary links to check the possibility of being admitted Since it is proved in [10] that the asynchronous power updates like those used in (14) always converge to a fixed power vector regardless of the values of the initial power vector, then fixed power vector P(s) violating (9) or (10) must violate it had the power vector started with another initialization Consequently, we can conclude the following fact: • Fact : Under stationary channel conditions, an inactive secondary link in set cannot be activated if it will result in an active set ℵ that was infeasible before Although, the infeasibility of the active set ℵ is checked after the convergence of the DCPC, we take the occurrence For our algorithm to be distributed, a control channel should exist so that the secondary links can exchange control information (about the recent inactive set and interference constraint violation) during execution of the proposed algorithm Each secondary link should keep the last version of the current inactive set Upon each join process the link that is supposed to join should inform the other links that it would leave the active set, and which secondary link, if any, would be re-activated Each link would then update its version of accordingly There is no need for the secondary links to know the maximum tolerable interference on primary receivers nor what is the current value of interference they produce on each receiver Instead, the primary BS broadcasts a warning signal over the control channel if the interference produced by secondary links exceeds the tolerable allowable level The secondary links not have to keep track of channel gains between themselves and primary receivers Once a warning signal from a primary BS is heard, the secondary link that performed the last power update has to join the set As mentioned in the previous subsection, when a secondary link j joins the set , it first checks the possibilities of replacing any inactive link i, where i ∈ , producing a new active set ℵ This, of course, requires the availability of the history of either the active or inactive sets progression Since our proposed algorithm is distributed, it is enough for each secondary link to keep a list of only the past inactive sets in which it was included, in addition to the current inactive set Thus, if j finds that a new inactive set is produced when it replaces i in , i joins ℵ and j joins If several members in qualify to be activated, for simplicity, j replaces the first 978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings V S IMULATION R ESULTS In this section we present simulation results for of our proposed algorithm In order to compare the performance of the algorithm with that in [8], we use the same system model and parameters Primary users are communicating with a single BS in the uplink direction The secondary transmitters are located in an area of size 2000m × 2000m with BS of the primary network located at the center The receiving node of each secondary link is placed randomly in a 1000m × 1000m square with its transmitting node at the center The channel gain between any transmitter j and receiver i is modeled as gi,j = K0 × 10βi,j /10 × d−4 i,j , where di,j is the corresponding distance, βi,j is a random Gaussian variable with zero mean and a standard deviation of dB to account for shadowing effects, and K0 = 103 is a factor that includes some system parameters such as antenna gain and carrier frequency The total noise and interference at the receiving node of Fig Simulation Model 0.4 Optimum I−SMIRA Proposed with lists Proposed with 1/0 vectors 0.35 0.3 Outage Probability qualified one If all possible replacements are to produce an active set previously estimated to be infeasible, then j joins and the size of expands by one All secondary links then clear their stored histories and keep only the current We preferred keeping the history of progression of inactive sets instead of the active sets at each secondary link, because the inactive set is initially empty and then grows one by one, that in turn would facilitate the search process rather than searching in history of previous active sets Sometimes when the network conditions are severe with large number of requesting secondary links, the number of removed secondary links is large, which in turn may result in a large history of inactive sets maintained by secondary links As a result the join process might consume much time in searching as well as memory for storing previous inactive sets For this reason, we propose a simple technique for handling join processes This technique states that, instead of maintaining a list of previous inactive sets in which a secondary link was included at that secondary link, each secondary link would maintain a vector of size N with components set to zero, then after each replacement between any two secondary links i and j each link of them will set the vector component corresponding to the other link to However, if a link j is supposed to join the inactive set j will first check if it could replace any link i in or not via the components of its maintained vector A replacement occurs only if j finds the component of index i in the vector of j is zero Moreover, for the case in which the inactive set size is increased by then all secondary links should clear their maintained vectors (setting their components to zeros) The simulation results showed that both ways (using lists of previous inactive sets, or vectors with or components) produce almost same number of admitted secondary links Finally, the proposed algorithm is assumed to converge within a time period over which the system is stationary We are currently investigating the effects of mobility of users and variations of their arrivals and departures so that they are accounted for in our future work 0.25 0.2 0.15 0.1 0.05 10 15 20 Target SINR dB Fig Outage probability vs target SINR of secondary links I = 5(Ni + N0 ) and N = all secondary links is Ni + N0 = 10−10 W, the maximum transmission power on secondary links is Pimax = 0.1 W, the bandwidth is B = 5.12 MHz, and the minimum transmission rate on secondary links is Ri = 64 kbps For each simulation run, the locations of secondary links are generated randomly Fig gives the outage probability calculated as the ratio of the average number of removed links after convergence to the average number of requesting links, versus target SINR of secondary links We averaging over 1000 simulation runs using N = and a maximum tolerable interference at the primary receiver, I, equal to 5(Ni + N0 ) We can see from the figure that the proposed algorithm produces outage probability that is close to the optimum solution obtained by exhaustive search for the smallest possible set of removed links This is the same for both cases of the secondary links storing the past inactive sets or storing a vector tracking replacements Fig shows the number of admitted secondary links versus the number of requesting secondary links for the proposed algorithm and the optimal algorithm The result shows that the number of admitted secondary links is close to the optimum admitted number This figure is produced for I = 5(Ni + N0 ) and SINR=15dB To simulate the complexity of our proposed algorithm, we take the number of comparisons in our proposed second 978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings VI C ONCLUSION 11 Optimum Using lists Using 1/0 vectors 10 Number of admitted links 5 10 11 Number of requesting links 12 13 14 15 Fig Number of admitted secondary links vs number of requesting secondary links I = 5(Ni + N0 ) and SINR=15dB 9000 Using 1/0 vectors I−SMIRA Average number of basic operations 8000 We address the problem of admission control and power allocation for cognitive radios in spectrum underlay networks Our design objective is to maximize the number of admitted secondary links under their QoS requirements without violating the interference temperature on primary receivers Since an optimal solution to our problem is proved to be NP-hard, we propose a suboptimal distributed algorithm Our algorithm, in contrast with previous works, allows the removed links to be re-activated to compensate for the non optimality of the used simple removal criterion In order to maintain reduced complexity, we use a simple criterion to control the number of reactivation of inactive secondary links Our distributed algorithm is supposed to be applied online based on local information in each secondary link and small amount of signaling that could be exchanged over a control channel Simulation results show that the proposed algorithm produces close results to the optimum solution with reasonable complexity 7000 R EFERENCES 6000 5000 4000 3000 2000 1000 0 10 15 20 Number of requesting secondary links 25 30 Fig Average number of basic operations vs number of requesting links I = 5(Ni + N0 ) and SINR=15 dB storage scheme (where secondary links maintain vectors of or components) as a measure of basic operations We run the algorithm for different number of requesting secondary links At each run we calculate the number of performed comparisons and average the results over 100 simulations Moreover, we run I-SMIRA algorithm under the same simulation setup and estimated the number of basic operations performed by it R−1 as i=0 (N − i)2 , where R is the number of removed links by I-SMIRA Note that the complexity of I-SMIRA is reported to be O(N ) [8],[11] Afterwards, we plot the average number of basic operations of both algorithms in Fig It is clear from the figure that the proposed second storage scheme has a smaller number of basic operations (comparisons) than I-SMIRA The basic operation of I-SMIRA can be considered as an addition or multiplication operation Here, we note that the first storage scheme (where secondary links maintain lists of previous inactive sets that they were included in) requires large number of comparisons in searching for previously formed inactive sets Wee can see from the results of Figs 3,4,5, the second storage technique is more efficient than both I-SMIRA and our first proposed technique in terms of complexity and outage probability [1] FCC Spectrum policy task force report, FCC 02-155 Nov 2002 [2] http://www.fcc.gov/oet/info/database/spectrum/ [3] FCC Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies, notice of proposed rule making and order, FCC 03-322 Dec 2003 [4] J Mitola III,“Cognitive Radio: An Integrated Agent Architecture for Software Defined Radio” Doctor of Technology Dissertation, Royal Institute of Technology (KTH), Sweden, May, 2000 [5] Ian F Akyildiz, Won-Yeol Lee, Mehmet C Vuran, and Shantidev Mohanty, “NeXt generation/dynamic spectrum access/cognitive radio wireless networks: A survey,” Computer Networks Journal(Elsevier), September 2006 [6] Y Xing, C N Mathur, M A Haleem, R Chandramouli, and K P Subbalakshmi, “Dynamic spectrum access with QoS and interference temperature constraints,” IEEE Trans Mobile Comp., vol 6, no 4, pp 423433, April 2007 [7] M H Islam, Y.-C Liang, and A T Hoang, “Distributed power and admission control for cognitive radio networks using antenna arrays,” in Proc IEEE DySPAN07, Dublin, Ireland, pp 250-253, 17-20 Apr 2007 [8] L Le and E Hossain, “Resource allocation for spectrum underlay in cognitive radio networks,” IEEE Transactions on Wireless Communications, to appear [9] D I Kim, L Le, and E Hossain, “Joint rate and power allocation for cognitive radios in dynamic spectrum access environment,” IEEE Transactions on Wireless Communications, submitted [10] S A Grandhi and J Zander, “Constrained power control,” Wireless Personal Commun., vol 1, no 4, 1995 [11] SM Andersin, Z Rosberg, and J Zander, “Gradual removals in cellular PCS with constrained power control and noise,” ACM/Baltzer Wireless Networks J., vol 2, no 1, pp 27-43, 1996 978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings ... ISTRIBUTED A LGORITHM A Distributed Power Control: Most of the relevant proposals use the distributed constrained power control (DCPC) algorithm [10] as an efficient power control algorithm for... Liang, and A T Hoang, ? ?Distributed power and admission control for cognitive radio networks using antenna arrays,” in Proc IEEE DySPAN07, Dublin, Ireland, pp 250-253, 17-20 Apr 2007 [8] L Le and. .. “Constrained power control, ” Wireless Personal Commun., vol 1, no 4, 1995 [11] SM Andersin, Z Rosberg, and J Zander, “Gradual removals in cellular PCS with constrained power control and noise,”