price-based spectrum management in cognitive radio networks 2008

74 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 1, FEBRUARY 2008

Price-Based Spectrum Managementin Cognitive Radio NetworksFan Wang, Marwan Krunz, and Shuguang Cui, Member, IEEE

AbstractCognitive radios (CRs) have a great potential to im-prove spectrum utilization by enabling users to access the spectrumdynamically without disturbing licensed primary radios (PRs). Akey challenge in operating these radios as a network is how to im-plement an efficient medium access control (MAC) mechanism thatcan adaptively and efficiently allocate transmission powers andspectrum among CRs according to the surrounding environment.Most existing works address this issue via suboptimal heuristicapproaches or centralized solutions. In this paper, we propose anovel joint power/channel allocation scheme that improves the per-formance through a distributed pricing approach. In this scheme,the spectrum allocation problem is modeled as a noncooperativegame, with each CR pair acting as a player. A price-based itera-tive water-filling (PIWF) algorithm is proposed, which enables CRusers to reach a good Nash equilibrium (NE). This PIWF algorithmcan be implemented distributively with CRs repeatedly negotiatingtheir best transmission powers and spectrum. Simulation resultsshow that the social optimality of the NE solution is dramaticallyimproved through pricing. Depending on the different orders ac-cording to which CRs take actions, we study sequential and parallelversions of the PIWF algorithm. We show that the parallel versionconverges faster than the sequential version. We then propose acorresponding MAC protocol to implement our resource manage-ment schemes. The proposed MAC allows multiple CR pairs to befirst involved in an admission phase, then iteratively negotiate theirtransmission powers and spectrum via control-packet exchanges.Following the negotiation phase, CRs proceed concurrently withtheir data transmissions. Simulations are used to study the perfor-mance of our protocol and demonstrate its effectiveness in termsof improving the overall network throughput and reducing the av-erage power consumption.

Index TermsCognitive radio networks (CRNs), iterativewater-filling (IWF), medium access control (MAC) protocol de-sign, power/rate control, spectrum management.

I. INTRODUCTION

THE concept of a cognitive radio (CR) has recently trig-gered great interest within the research community (see [1]for a comprehensive survey). The term cognitive radio wasfirst coined by Mitola [2] as the point in which wireless per-sonal digital assistants (PDAs) and the related networks are suf-ficiently computationally intelligent about radio resources and

Manuscript received April 20, 2007; revised October 20, 2007. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Randall Berry.

F. Wang and M. Krunz are with the Department of Electrical and Com-puter Engineering, University of Arizona, Tucson, AZ 85721 USA (e-mail:[email protected]; [email protected]).

S. Cui is with the Department of Electrical and Computer Engineering, TexasA & M University, College Station, TX 77843 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTSP.2007.914877

related computer-to-computer communications to a) detect usercommunications needs as a function of use context and b) toprovide radio resources and wireless services most appropriateto those needs. Mitolas definition, however, does not specifythe radio architecture for the physical and link layers. More re-cently, the FCC [3] suggested that any radio with adaptive spec-trum awareness is to be referred to as a CR. Specifically, a CRshould be able to adapt its transmission parameters to the neigh-borhood environment. CRs are expected to be deployed in bothmilitary and commercial applications.

Several scenarios can be found for operating a cognitive radionetwork (CRN). In this paper, we focus on an opportunisticCRN where the CRs are secondary users that coexist with pri-mary radios (PRs). The PRs are licensed to operate over cer-tain frequency bands. They do not cooperate with or even pro-vide feedback to the CRs. CRs continuously sense the channeland exploit spectrum holes for their transmissions. One ofthe main challenges in an opportunistic CRN is how to de-sign an efficient and adaptive channel access scheme that sup-ports dynamic channel selection and power/rate allocation ina distributed (ad hoc) CRN environment. An efficient designis one that tries to maximize the CRNs performance withoutdisturbing PR transmissions. A typical measure of efficiency isthe achievable sum-rate across all CR pairs. Unfortunately, theproblem of maximizing the sum-rate over a multi-user, interfer-ence channel subject to individual power constraints is a non-convex optimization problem [4]. Such a problem becomes evenmore intractable when we allow multiple CRs to share the samechannel, as we now have to consider CR-to-CR interferences inaddition to PR-to-CR and CR-to-PR interferences.

Several attempts have been made to solve the aforementionedinterference channel problem. One well-known resource alloca-tion scheme, called iterative water-filling (IWF), was first pro-posed in [5], where a noncooperative game was used to modelthe spectrum management problem with each user iterativelymaximizing its own rate. This per-user optimization problem isconvex and leads to a water-filling solution. For the two-usercase, it was proven that the Nash Equilibrium (NE) exists andthe IWF algorithm converges to the NE under certain conditions.However, this NE is generally not Pareto optimal [6] and may bequite inefficient in term of sum-rate [7]. This is because in a non-cooperative game, each user only has the incentive to maximizeits own utility function without considering the overall systemperformance. A centralized spectrum management scheme wasproposed in [7], which greatly improves the system performanceover the IWF scheme by utilizing a centralized spectrum man-agement center (SMC). However, such an approach cannot beimplemented in an ad hoc opportunistic CRN, where none of

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WANG et al.: PRICE-BASED SPECTRUM MANAGEMENT IN COGNITIVE RADIO NETWORKS 75

the CRs has global knowledge of the entire CRN to function asthe SMC.

Given the above, we are motivated to design a channel/power/rate allocation scheme that overcomes the inefficiency of IWFand yet can be implemented in a distributed fashion. Specif-ically, we provide incentives to CR users such that they canreach a more socially efficient NE. A commonly used incentivetechnique in game theory is pricing (a thorough review is pro-vided in [8]). Previously, pricing techniques have been imple-mented in various wireless networks such as cellular networks,ad hoc networks, and peer-to-peer networks (e.g., [9][13]). Inthis paper, we apply pricing techniques to CRNs. We propose aprice-based iterative water-filling (PIWF) algorithm, and showthat this algorithm maintains the simplicity and distributed oper-ation of the IWF algorithm while achieving better bandwidth ef-ficiency (i.e., higher sum-rate). The effectiveness of the pricingtechnique depends on the selection of the pricing functions,which is a challenging problem by itself. Although there mayexist an optimal pricing function that allows the NE to con-verge to a Pareto-optimum solution, the search for such a pricingfunction generally requires a central controller and is hard to beimplemented in a distributed manner. Some suboptimal pricingfunctions have been proposed in the literature. For example, theauthors in [10] proposed an auction-like pricing scheme for mo-bile ad hoc networks (MANETs). The unit price in this scheme(uniform across all users) is gradually increased until the systemsettles down at a feasible NE. A similar approach was taken in[9], where the users of a wireless data network keep increasingtheir prices in a uniform fashion until one user begins to receivea decreasing utility. Both of the previously mentioned pricingschemes achieve feasible NEs and improve the system perfor-mance. However, the achieved NEs are not guaranteed to beglobally optimal, which is partially caused by the fact that bothof the two approaches take a uniform unit price for all players inthe game. In our work, we determine a user-dependent pricingfunction, which not only improves the NE, but also achievesglobally or locally optimal NE after a few iterations. Such apricing function can be determined by allowing each CR userto distributively acquire its neighborhood information via con-trol-packet exchanges. Note that a similar pricing analysis wasrecently conducted in [13] in the context of interference man-agement. Much of the emphasis and convergence results wererelated to a different utility function and network settings thanthe ones in our paper.

Another problem of applying the IWF algorithm in [5] toCRNs is that this algorithm only considers a total power con-straint for each user. In a CRN, PRs impose strict power con-straints over each frequency band, so CR users have to abide byfrequency-dependent power constraints. Such constraints willaffect the response of each CR user and thus the achieved NE. Inthis paper, we incorporate a frequency-dependent power maskconstraint into the optimization problem.

In our proposed algorithm, each user maximizes its ownutility function (which includes a pricing function) by per-forming a single-user price-based water-filling, while treatingthe interference from other CR users at each subband as ad-ditive white Gaussian noise (AWGN). The same procedureiterates sequentially, eventually converging to the NE. If the

number of users in the network is large, sequential updatingmay suffer from slow convergence. Therefore, we also discussa parallel PIWF algorithm (the parallel concept was intro-duced in [14]), which is an instance of the Jacobi scheme:At each iteration, CRs update their strategies simultaneously,based on the interference measured in the previous iteration.Simulations indicate that this parallel version converges fasterthan the sequential PIWF algorithm. Both the sequential andparallel PIWF algorithms require CRs to be synchronized andthe system parameters to be correctly estimated for each CR.These conditions may not be satisfied in practical systems.To overcome this problem, a relaxed update scheme canbe used (as in [15][17]) and is studied in our work. For therelaxed version of the PIWF algorithm, each CR is requiredto remember its most recent policy choices together with thechoices of other users. The relaxed algorithm is more robust toinaccurate estimates and channel oscillations, but it may impactthe convergence speed.

Our PIWF algorithms are then integrated into the design ofa distributed medium access (MAC) protocol for CRNs. Thisprotocol allows CRs to dynamically select channels and adaptto different transmission powers and rates. We discuss how thevarious versions of PIWF impact the MAC design. Simulationsare conducted to compare the performance of the proposed pro-tocol against other adaptive protocols.

The rest of this paper is organized as follows. The systemmodel is described in Section II. Section III formulates thenoncooperative game and introduces the pricing techniques.We then discuss the PIWF algorithms for solving the NE inSection IV and design the corresponding MAC protocol inSection V. In Section VI, we provide simulation results ofthe PIWF algorithms and compare them with the classic IWFalgorithm. Finally, we draw conclusions and discuss futureextensions in Section VII.

II. SYSTEM MODEL

We consider a hybrid network consisting of several PRNs andone CRN. The CRN contains CR pairs. The total spectrumconsists of orthogonal frequency channels with central fre-quencies , where . Each PR in a PRN mayoperate over one or multiple channels. Letand denote the sets of CR links and chan-nels, respectively.

Each CR may simultaneously transmit over multiple chan-nels. It can also receive over multiple channels (from thesame transmitter) at the same time. However, we require thateach CR operates in a half-duplex manner, meaning that itcannot receive while transmitting, and vice versa. Letdenote the total noise-plus-interference level measured by CRuser over channel . This quantity includes the PR-to-CRinterference, the CR-to-CR interference, and the thermalnoise. We assume that when not transmitting, CR is ca-pable of measuring over all channels . Let

, which is used by CRto perform channel selection, power control, and rate allocation,as described later.



Fig. 1. Example of channel allocation for four CR links.

The motivation of using CR technology is to enhance thespectrum utilization by allowing CR users to share the spectrumwith PRs. Some previous work [18] assumed that CR transmis-sions do not interfere with each other, i.e., only one CR user canoperate over a given channel in a given neighborhood (alongwith the PRs). In this way, there is no spectrum sharing amongCR users. Such schemes limit the number of admitted CR links,especially when the number of channels is small. In our work,we allow multiple CR users to share a particular channel. Fig. 1depicts a channel allocation example for a CRN with and

. The dark square indicates that a channel is utilized by aCR. For example, link 1 uses channels 1 and 2, while link 4 useschannel 1 only. We denote the set of utilized channels for CRlink as . In the above example, and .The transmission power vector of CR link over all channels isdenoted by , whereis the transmission power of CR on channel . If channelbelongs to , ; otherwise, .

To ensure feasible spectrum sharing, we impose the followingconstraints.

1) Maximum transmission power constraint: The total trans-mission power of a CR over the selected channels shouldnot exceed , i.e., . Here, weassume that the total power constraint is the same for allusers. It is easy to extend the treatment to the case where

is user-dependent.2) CR-to-PR power mask constraint: The transmis-

sion power of CR on channel is constrained by, where is the power mask

on channel . Such a per-device power mask is easier toverify at the design stage from a practical point of view. Forexample, the power mask is often specified by FCC regu-lations. In this case, CR vendors need to design the radiowhile ensuring its RF transmission power meets the FCCpower mask. Such a philosophy is often used in variouswireless technologies (e.g., UWB, Wi-Fi, Walkie-Talkies,etc.). Note that because the number of active CR links thatshare a given frequency band varies in time and space,it is impractical to design the hardware to account fora neighborhood-dependent power mask. We use thevectorto denote the power mask on all channels. In the followinganalysis, we assume that is given a priori.

3) Minimum signal-to-interference-and-noise ratio (SINR)constraint: If the received SINR over a given channel isbelow the SINR threshold , the CR will not usethat channel.

We assume that the CRs are either static or are moving slowlycompared to the convergence time of the resource assignment al-gorithms. This assumption is generally acceptable because ouriterative algorithms operate on the time scale of few millisec-onds, whereas pedestrian and vehicular mobility impacts thenetwork topology on the time scale of seconds. In addition, CRsare homogeneous, meaning that they follow the same operationrules and have the same system constraints.

III. PROBLEM FORMULATIONIn a noncooperative CRN, each CR user is interested in

maximizing its own achievable rate. Such a greedy behaviorcan be modeled using game theory. Game theory analyzes theinteractions of players in decision-making processes. It can beused to identify distributed optimal strategies for the players[19]. A normal game is expressed as: ,where is a finite set of rational players;

is the action space with being the ac-tion set for player ; and is the utility (payoff) func-tion of player , which depends on the strategies of all players.We can model the channel/power allocation problem in a CRNas a noncooperative game, in which the players are the CR users;their actions are the transmission power vector (i.e., the actionfor user is given by ); andtheir utility functions are associated with their actions and thequality of the channels. Note that a CR user in the game denotesa CR link consisting of a pair of CRs.

A. Utility FunctionIn our game, the utility function of user can be con-

sidered as the reward received by this user from the net-work. This reward should depend on the users actionand the union set of all other users actions , where

. While the selectionof the utility function is not unique, the selected utility functionmust have physical meaning for the particular application. Anatural selection of the utility function for CR link (also usedin [5], [17], and [20]) is its transmission rate, given by (1),shown at the bottom of the next page, where denotesthe channel gain between the transmitter of link and thereceiver of link over channel , denotes thePR-to-CR interference at the receiver of CR link over channel

, and denotes the received thermal noise power onchannel . In the paper, this relationship is taken as Shannonscapacity formula. In a practical multirate wireless system, thepower-rate relationship takes the form of a staircase, and theuser sets the transmission rate to the maximum possible rate(among a finite set of rates) that satisfies the SNR thresholdat the given transmission power value. It is straightforwardto extend our design to accommodate such a power-rate re-lationship. Given the utility function in (1), users select theirtransmission powers to maximize their own utility functions,and under certain conditions, they eventually reach at a NEafter several iterations. As discussed in Section I, because of



the noncooperative nature of the game, each CR user behavesselfishly. Thus, the resulting NE may be far from the Paretooptimum [6]. In practice, we are interested in maximizing aweighted sum of the utilities of all users, defined as

(2)where denotes the weight assigned to CR user , which maybe interpreted in different ways (e.g., priority factor of user ).Note that the power assignment that solves (1) is a Pareto-op-timum solution.

To drive the NE towards the Pareto optimum boundary, weuse pricing as an incentive for each selfish CR user to work in acooperative manner. A new utility function with pricing is thendefined as follows:

(3)

with (4), shown at the bottom of the page, where repre-sents the pricing function for user on channel . As discussedin Section I, our goal is to choose a user-dependent pricingfunction that can drive the CR users to converge to an efficientNE. How to define this pricing function will be discussed inSection III-C.

B. Game FormulationGiven the price-based utility function in (3), each CR user

iteratively selects its power vector to maximizesubject to the constraints listed in Section II. This results in thefollowing noncooperative game :

(5)

If there is a solution to the above game, then it would be theone that achieves the NE. Note that the above game differs fromthe game studied in [5] in the form of the utility function and theadditional power mask constraint. Thus, the existence proofs in[5] and [20] cannot be directly applied.

The following proposition show that a NE solution alwaysexists for the above game.

Proposition 1: For any given and values, thereis at least one NE for the game in (4).

Proof: The game in our setup can be shown to be a concavegame if the following two properties are satisfied:

1) the action space is a closed and bounded convex set;2) the utility function is concave over its strategy

set.It is straightforward to show that the two properties are satisfiedby the game . Because a concave game always admits at leastone NE [21], the proposition follows immediately.

Given the existence of a NE solution, we need to design analgorithm for CR users to reach the NE. However, before we dothat, we first investigate the form of the optimal pricing function.

C. Optimal Pricing Function

Pricing is an idea that originated from economics (e.g., [8]). Itdenotes the cost of commodities for individual decision makers.In the power control context (e.g., [9] and [11]), pricing is usedas an incentive mechanism to improve the efficiency of the NE.To illustrate, in Fig. 2, we depict an example of the Pareto-op-timal frontier and the NE for a two-user game. In general, theNE is not Pareto optimal. Previous pricing techniques usuallyimprove the achieved NE (i.e., moving it closer to the Pareto-op-timal frontier) using heuristic pricing functions, since an optimalpricing function generally requires global information and couldhardly be deployed in a distributed manner. For example, in [9]and [11], the pricing function is a suboptimal linear functionwith a fixed linear pricing factor for all players.

The pricing function can take various forms. A linear pricingfunction is commonly used because of its implementation sim-plicity. In this paper, we use a user-dependent linear pricing

(1)

(4)



Fig. 2. Nash equilibrium and Pareto-optimal Frontier.

function that drives the NE close to the Pareto optimal fron-tier with each player having only local and certain neighbor-hood information. The neighborhood information is acquiredvia control packets that are exchanged during the channel accessprocess (see Section V for details). Note that a similar pricinganalysis was recently conducted in [13] in the context of inter-ference management.

Proposition 2: If there exists a NE for the game and if thisNE is Pareto optimal, then the linear pricing function factor foruser should be

(6)

where denotes the set of neighbors for user .Proof: By definition, a NE is the solution to the individual

utility optimization problem for each user given all otherusers actions. In our formulation, each individual optimizationproblem is a convex problem with the linear constraints C1C3in (4). So the Lagrangian function for user can be written as

(7)

where , , and are the Lagrangian multipliers (nonneg-ative real numbers). The K.K.T. conditions [22] are given by

(8)

In contrast, to solve the social optimization problem (1) withconstraints C1C3, the Lagrangian function can be written as

(9)

The K.K.T. conditions for the optimization problem in (1) aregiven by

(10)



By comparing K.K.T. conditions in (8) and (10), to obtain thesame solution, we must have

(11)

By substituting into (9), we have

(12)If the transmitter of link and the receiver of link are not

neighbors, i.e., the transmission of link at the maximum powercannot reach the receiver of link , the channel gain isset to zero. Thus, the optimal pricing factor for link only de-pends on its neighborhood information. We then have the resultin Proposition 2.

Intuitively, a higher pricing factor will prevent userfrom using a large transmission power on channel . In viewof (5), for link to determine its optimal pricing factor, thefollowing procedure is needed: If a neighbor is to transmitover channel , it needs to broadcast its transmission power

, the measured total noise and interference , andthe channel gain between the transmitter and the re-ceiver of link . The above information can be incorporated intothe control packets of the MAC protocol (details in Section V).In addition, can be measured from the received signalpower of the control packet.

IV. ITERATIVE ALGORITHMS

From the propositions in the previous section, we can use thefollowing iterative procedure to reach the NE. Each individualCR user, say , first adjusts its linear pricing factor overall channels according to (5), and then determines its best action[6], i.e., the optimal channel/power/rate combination, by mea-suring the total noise-plus-interference level over all chan-nels. The best response of user is to maximize its individualutility function (3) subject to the constraints C1C3. The sameprocedure is repeated for all users in the network. If such a pro-cedure converges, then by definition, it has to converge to a NEof the game in (4).

Note that the utility function in (1) is monotonically in-creasing with given that the other users powers arefixed, and the only condition that prevents user from choosinginfinitely large transmission power is the total power constraintC2. In our work, after adding the linear pricing function, theutility function (3) now leads to finite optimal power settingseven without the constraint C2.

Proposition 3: Treating other users transmissions as inter-ference, the best response of user is given by

(13)with

(14)

Fig. 3. IWF versus PIWF.

where , with , denotes the Euclidean projection ofonto the interval , i.e., if , if

, and if . The water level is chosento satisfy the total power constraint C2.

A similar result for the IWF algorithm is provided in [17]. Al-though we have an additional pricing function, a similar analysiscan be used to reach the result in Proposition 3. We also providean alternative proof in the Appendix using the sequential opti-mization technique as discussed in [23] and [24].

Note that without the power mask constraint and without thepricing function (i.e., for all and ), (11) becomesthe classical water-filling solution. Fig. 3 graphically illustratesthe difference between the traditional water-filling [5] and theprice-based water-filling solution (11). The variable water levelin the right-hand side of Fig. 3 is because of the addition of thepricing factor in (12).

Several approaches are available for CR users to converge tothe NE according to the best response function defined in (11).Naturally, CR users may make their decisions one after anotheror in parallel, which corresponds to sequential and parallel up-date procedures. The specific algorithms will be described nextalong with their convergence properties.

A. Sequential Price-Based Iterative Water-Filling

If CR users are to make their best-response decisions sequen-tially according to a fixed order, we have a sequential PIWF al-gorithm, and the algorithm can be generalized as follows.

Algorithm 1 Sequential PIWF

0: Initialize , and ; initializeiteration count .

1: Repeat iterations:

2: ;

3: for to users do

4: for to channels do

5: Estimate the total interference plus noise level ;

6: Compute the pricing factor using (5);7: Estimate the channel gain using the received signalpower of the control packet.

8: end for



9: ;

10: Transmit on selected channels using .

11: end for

12: until or forall .

The condition is thestopping criteria for the PIWF algorithm. Normally, is set toa small value, such as 5%. If that condition is not satisfied after

iterations, the algorithm terminates. The above algorithmis akin to the Gauss-Seidel procedure [25], where the playerstake their turns sequentially and act on the most recent policyinformation obtained from the other players. In a two-userscenario, the th iteration for user 1 can be expressed as

(15)

The Nash equilibrium is thus a fixed point [26] under the map-ping . For the -user case, the expression is more compli-cated, but we will keep the notation as the mapping betweenthe previous power vector and the current power vector. For theIWF algorithm, to ensure convergence to the NE, several suf-ficient conditions were proposed in the literature. The conver-gence condition was first provided in [5] for two-user cases andin [20] for -user cases. More recently, the convergence condi-tions have been further relaxed in [17] and [27].

Since the utility function (3) in our formulation possessesan additional pricing function, the previous convergence proofsmay not be applicable any more. In fact, since we are using atime-varying pricing factor in Algorithm 1, the mapping func-tion is also time-varying. Thus, the fixed point theoremthat underlies the proofs in [17] and [27] can no longer be used.The convergence proof with a time-varying mapping functionis challenging and will be left for a future work. However, con-vergence is always observed in our simulations for various net-work conditions. Fig. 4 depicts the convergence behavior overseveral iterations with and . In the test net-work, ten CR pairs are randomly placed in a square area. Thefigure shows the average sum-rate improvement of the sequen-tial PIWF over the classic IWF algorithm for 1000 runs, withthe starting sum-rate of the IWF algorithm normalized to one.The two algorithms converge at comparable speeds, but the NEsolution for the sequential PIWF algorithm is much better thanthe NE of the classic IWF algorithm.

Although the convergence proof for the time-varying pricingfunction is difficult to establish, if the pricing factor remainsfixed over a few iterations, the convergence proof in [17] is stillapplicable. This is because adding a linear pricing function witha fixed pricing factor to the utility function in (1) has no impacton the convergence proof in [17]. If we take the result in [17] andapply it to our CRN setting, we have the following proposition.

Proposition 4: Given a linear pricing function with a fixedpricing factor over a few iterations, the sequential update pro-

Fig. 4. Normalized system sum-rate versus iterations.

Fig. 5. Example network with three CR links.

cedure converges to the unique NE if one of the two followingsets of conditions is satisfied:

(16)

(17)

From (14) and (15), the convergence and the uniqueness ofthe NE are ensured if the CRs that share the same channel arefar apart from each other, and thus cause weak interference onone another. Fig. 5 graphically illustrates these two conditionsin a CRN that consists of three CR links. Each link is allo-cated two channels (e.g., nodes and are allocated channels1 and 3). Condition (14) indicates that for each CR receiver,the summation of the normalized channel gains between thatreceiver and all interfering CR transmitters that share the samechannel (normalized by the channel gain of the intended packet)is less than 1. For example, for node , this condition reducesto . Similarly,this condition is applied to receivers and . If this condition issatisfied at all CR receivers, then according to Proposition 4 thesequential update procedure converges to the unique NE. Thesecond condition (15) indicates that for each CR transmitter, thesummation of the normalized channel gains between that trans-mitter and unintended CR receivers sharing the same channelis less than 1. For example, for transmitter , (15) reduces to

. Similarly,we can derive this condition for transmitters and . If this



condition is satisfied at all CR transmitters, it is also sufficientto assert that the sequential update procedure converges to theunique NE.

If the number of users in the network is large, the sequentialupdate procedure may suffer from slow convergence. Therefore,we now discuss a parallel version of the PIWF algorithm.

B. Parallel Price-Based Iterative Water-Filling

Algorithm 2 describes a parallel update version of the PIWFalgorithm (the parallel IWF concept was first introduced in[14]). The stopping criteria for the parallel PIWF are the sameas those of the sequential PIWF. The parallel PIWF algorithmis related to the Jacobi computational procedure [28], wherein each iteration CR users simultaneously perform price-basedwater-filling based on the interference generated by the otherusers in the previous iterations. In a two-user case, the counter-part of (13) is

(18)

Algorithm 2 Parallel PIWF

0: Initialize , and ; initializeiteration count .

1: Repeat iterations:

2: ;

3: for to users do

4: for to channels do

5: Estimate the total interference plus noise level ;

6: Compute the pricing factor using (5);7: Estimate the channel gain using the received signalpower of the control packet.

8: end for

9: ;

10: end for

11: for to users do

12: Transmit using .

13: end for

14: until or forall .

In [17], it was proved that the convergence conditions for theparallel IWF and the sequential IWF are the same. For a time-varying PIWF, the proof is not applicable. But if the pricingfactor of the linear pricing function remains fixed over a fewiterations, we can apply the corresponding proof and arrive atthe following corollary of Proposition 4.

Fig. 6. Convergence speed of the sequential/parallel PIWF.

Corollary 1: If the conditions of Proposition 4 are satisfied,the parallel update procedure converges to the unique NE of thegame.

Corollary 1 shows that stability under the Gauss-Seidel pro-cedure coincides with stability under the Jacobi iteration. Fur-thermore, following the argument in [17], one can prove that anyasynchronous computation in which the players act at randomand use the most recent available policy from other players con-verges to a NE, as long as no players remain idle for an infiniteduration. Hence, the achieved NE based on asynchronous up-dates coincides with the NE achieved with parallel or sequentialupdates.

The parallel and sequential PIWF procedures are distributedalgorithms that maximize the total achievable data rate. Bothhave the same implementation complexity of the traditionalIWF. As shown in Fig. 4, these two algorithms greatly out-perform IWF. In Fig. 6, we can see that the parallel PIWFconverges faster than the sequential PIWF, especially for alarge number of users. In this simulation, we assume that CRsare randomly located in a square area and five channels areavailable for their transmissions. Whether the players act se-quentially or in parallel makes a difference in the MAC design.In Section V, we discuss the impact of the update procedure onthe MAC design.

C. Relaxation Algorithms

Both the sequential and parallel PIWF algorithms require thesystem parameters to be correctly estimated for each CR. Thiscondition may not be satisfied in practical systems. To over-come this problem, a relaxed update scheme can be used (as in[15][17]), and will be discussed here for completeness. In sucha relaxed version, each CR is required to remember its mostrecent policy choices together with the choices of other users.The relaxed algorithms are more robust to occasional estimationerrors and channel oscillations, but lead to certain degradationin the convergence speed.



More specifically, we can achieve a relaxed version of the se-quential PIWF algorithm if the best response function in Algo-rithm 1 is replaced by

(19)

where the factor can be interpreted as the memoryfactor. The larger the value of , the longer the memory of thealgorithm. With a larger , the algorithm is more robust to esti-mation errors at the cost of slower convergence.

Similarly, we can arrive at a relaxed version of the parallelPIWF algorithm if the best response in Algorithm 2 is replacedby

(20)

As proved in [17], the relaxation algorithms converge to theunique NE of the game for any under the conditionsin Proposition 4.

All the above proposed iterative algorithms are rate-adaptive(RA), where the data rates of users are maximized under trans-mission power constraints. Similarly, we can design margin-adaptive (MA) algorithms where users attempt to minimizetheir transmission powers while satisfying a target data rate.Both RA and MA algorithms follow similar mechanisms. Dueto space limitations, we do not discuss MA algorithms in thispaper.

V. MAC PROTOCOL DESIGNIn this section, we describe a MAC protocol that allows

CR users to operate efficiently in an opportunistic CRN. Thisprotocol implements the distributed channel/power allocationstrategies discussed in the previous sections. It should be notedthat a number of multichannel MAC protocols have beenproposed in the context of CRNs (e.g., [18] and [29][31]).Most of them do not allow multiple CR transmissions withinthe same neighborhood to overlap in frequency channels, sothere is no interference among CR users. Such a restrictionsimplifies the MAC design, but limits its spectrum efficiency.A natural extension (analogous to the improvement offeredby the POWMAC protocol [32] over the classic CSMA/CA)is to allow CR users to overlap in spectrum, provided thattheir mutual interference does not lead to collisions. The IWFalgorithm [5] and the no-regret algorithm [33] were proposedas two possible enabling techniques. However, the works in [5]and [33] provide only channel/power allocation algorithms anddo not offer a practical MAC design.

In this section, we incorporate our price-based channel/powerallocation algorithms into an operational MAC protocol. Sincethe IWF algorithm is a special case of the proposed PIWF algo-rithm, our MAC protocol can be simplified to accommodate theclassic IWF algorithm, thus complementing the work in [5].

A. AssumptionsWe consider a CRN setting with the following features.

A dedicated control channel or a coordinated controlchannel [31] is used to support a community of CR users.Control packets are transmitted over the control channelusing a pre-assigned power value .

Channel gains between any two terminals are symmetric. The channel gain is static for the duration of several control

packets and a flow of data packets.

B. Protocol Overview

Our MAC protocol uses three types of control packets for thehandshaking between a CR transmitter and a CR receiver: Re-quest-to-Send (RTS), Clear-to-Send (CTS), and Decide-to-Send(DTS). Unlike in the classic CSMA/CA scheme and other mul-tichannel MAC protocols for CRNs, these control packets arenot used to exclusively reserve channels (i.e., prevent neigh-boring CRs from accessing the reserved channels), but rather toexchange some information within the neighborhood. Such in-formation is used by terminals to determine their transmissionparameters.

The control packets are exchanged within a certain duration,referred to as the contention window (CW). A CW can be initi-ated asynchronously by any CR user that has packets to transmitand that is not aware of any active CWs in its neighborhood.Such a user is referred to as a master user. Other CR users thatfollow the schedule of an ongoing CW are called slave users.Note that the master/slave designation of a user is dynamic, i.e.,it changes with traffic and mobility conditions. The objective ofthe CW is to allow several pairs of CR nodes to repeatedly nego-tiate their transmission channels and powers. As shown in Fig. 7,the CW is divided into two parts. The first part, referred to as theaccess window (AW), is used by CR nodes to compete for ad-mission to the CW and initialize their transmission policies. Thesecond part, referred to as the training window (TW), is usedby the CR nodes to repeatedly negotiate their channel/powerpolicies (as explained later). Note that the AW can be consid-ered as the first iteration of the training process. CR nodes thathave been successfully admitted during the CW transmit a flowof data packets over one or multiple data channels (as deter-mined during the CW) within a data window (DW). The dura-tions for the AW and DW are changed adaptively, similar to thesingle-channel POWMAC protocol [32]. As for TW, its size (inslots) is dictated by the convergence speed of the iterative re-source allocation algorithm. In general, an unnecessarily largevalue increases the overhead, but does not necessarily improvethe throughput (as shown in Fig. 4). On the other hand, a smallvalue may give suboptimal results. In Section VI, we study theperformance of the MAC protocol under various TW sizes.

C. Operation Details

1) Access Window: When a CR node intends to commu-nicate with another node , it first needs to contend during theAW. If node is not aware of any ongoing AW in its neigh-borhood, it initiates a new AW (i.e., it becomes a master user).Otherwise, node contends during one of the slots of the on-going AW. In either case, node first backs off by a randomamount of time, selected from , before accessingthe channel.



Fig. 7. Overview of the MAC operation with two CR transmissions (A ! B and C ! D).

The AW consists of a number of fixed-size slots. The size ofeach slot is plus the durations of the RTS, CTS, and DTSpackets, plus three SIFS durations (SIFS denotes the short inter-frame spacing between successive control packets). In each slot,CR nodes compete for admission following a standard CSMAapproach.

If CR successfully receives the RTS packet from , it needsto decide the initial channel/power policy for the link .This is done as follows.

First, node estimates the channel gain between itself andnode (denoted by ). This is facilitated by knowl-edge of the RTSs transmission power and the re-ceived power of the RTS. From , CR computes

for all . The determination offrom is made possible by knowing the carrier fre-quencies and by assuming a certain path-loss model. Forexample, under the two-ray model [34] and for a giventransmission power, ,where is the carrier frequency of the control channel.

Next, node measuresover all data

channels. Note that for the sequential PIWF algorithm,if there are previous CTS/DTS packets that have beenreceived in the same AW, is computed as the sum ofthe current and the predicted CR-to-CR interference,which is obtained by assuming that the neighboring linkstransmit using the channels/powers specified in theirCTS/DTS.

Then, node determines the pricing factor forall data channels . For the sequential PIWF algorithm,

is computed using (5), where the neighborhood in-formation is obtained from previously received CTS/DTSpackets in the same AW. For the parallel PIWF algorithm,

is initialized to 0. Finally, based on the above information, node decides its

best-response transmission policy according to Proposition3.

After the above procedures have been executed, node willsend a CTS, announcing its channel/power allocation. The CTSincludes and for all , which areused by neighboring CRs to update their best responses. Note

that even if the set of selected channels is empty (i.e., thecomputed transmission power is zero for all channels), the link

will still be admitted in the AW. This is because thedata transmission may later be allowed to proceed afterseveral iterations in the TW.

If node receives the CTS from , it will respond with aDTS packet, repeating the information included in the CTS.This DTS is used to alleviate the hidden terminal problem asin [32].

The above steps are repeated by CR pairs in every AW slot.2) Training Window: CR nodes that are admitted in the AW

iteratively negotiate their transmission parameters in the TW,following the same order of their admissions in the AW. In con-trast to the AW, the TW is accessed in a TDMA manner. TheTW consists of a number of slots (the TW size), where each slotis used to conduct one iteration of the channel/power allocationalgorithm, using CTS and DTS packets. Note that there is noneed for the RTS during the TW, since new admissions are notallowed.

In each iteration, the receiver of a CR link updates the trans-mission policies based on the policies of its neighbors. The up-dates are made based on either the sequential or the parallelscheme. Specifically, if the sequential PIWF algorithm is ap-plied, the transmission policy of each CR user is made based onthe policies of all previous users in the same iteration (obtainedfrom CTS/DTS packets) and those of the other users in the pre-vious iteration, as described in Algorithm 1. If the parallel PIWFalgorithm is applied, the policy of each CR user is made basedon the policies of other CR users in the previous iteration, asdescribed in Algorithm 2. Note that the AW is regarded as theinitial iteration of the training process. After each computation,the receiver sends a CTS, announcing its transmission policy.Upon receiving the CTS, the transmitter will send a DTS, re-peating the information in the CTS.

3) Data Window: The last negotiated transmission policiesin the TW are used by the CR nodes for data transmissions inthe DW. In the DW, a flow of data packets is transmitted fromeach CR transmitter. The length of the flow is selected suchthat the channel conditions remain static over the entire flow.Obviously, the DW size needs to be selected according to thechannels coherence time.



D. Simplified Packet-Based MAC DesignThe above MAC design can be used for flow-based channel

access, where a sequence of data packets are transmitted usingconverged channel/power policies agreed upon during the TW.Thus, the sum-rate of all competing CRs is likely to be max-imized if the channels remain static over the duration of thedata flow. However, if sum-rate optimality is not critical, we cansimplify the protocol by removing the TW and only allow for asingle data-packet transmission in the DW. This design then be-comes packet-based, and the convergence is now achieved afterseveral sessions of CW and DW (provided that channel condi-tions remain static within this period).

Note that in the previous section, all CR nodes contend inthe AW with equal probability. In contrast, in the packet-basedMAC design, the admitted users in the previous AW have pri-orities in accessing the control channel over other CR users.Specifically, the admitted links in the previous AW will contendin the current AW without backoff, according to their order inthe previous AW, as long as they still have packets to transmit.After these links have been admitted, other links compete forthe remaining slots, following the backoff mechanism that wasdiscussed in the previous section. Such a design is meant tofacilitate the convergence behavior. To ensure fairness amongusers, we set a limit on the maximum number of continu-ously prioritized packets. Specifically, if one CR user acquiresthe channel in one session, it will have priorities in accessing thecontrol channel for the next packet transmission durations(as long as the user has packets to transmit). The parameter isselected to be larger than the convergence time of the algorithm.

The channel/power policies are updated in the AW followingsimilar procedures to the flow-based MAC. The only differenceis that the interference-plus-noise level is now estimated fromthe previous DW, instead of the previous iteration in the TW.In Section VI, we compare the performance of this design withthat of the flow-based MAC.

E. Implementation of Relaxed AlgorithmsThe implementation of the relaxed version algorithms to

the MAC design is straightforward. Each CR receiver memo-rizes its most recent policy, and makes decisions based on (17)for the sequential algorithm [or (18) for the parallel algorithm].Since the convergence speed of the relaxed PIWF is slower,a larger TW size is needed for the flow-based MAC protocol.Similarly, for the packet-based MAC, a larger number of datapackets are transmitted before the convergence can be achieved.

VI. PERFORMANCE EVALUATIONTo evaluate the effectiveness of the proposed MAC, we con-

duct MATLAB-based simulations of a hybrid network with onePRN and one CRN. Nodes in these networks are uniformly dis-tributed over a square area of length 100 m. The PRN operatesin the 300-MHz band, occupying five nonoverlapping 1-MHzchannels, with ten PRs in each channel. The time is divided intoslots, each of length 10 ms. In each slot, each PR attempts totransmit with a probability of , the PRs activity factor. Thetransmission power of each PR is 1 Watt when it is on, and theantenna length is 5 cm.

We use the following signal propagation model to simulatethe PR-to-CR and CR-to-CR interference over channel [34]:

(21)

where is the received power over channel , is thedistance between the transmitter and the receiver, is thereference distance, is the reference received power atdistance over channel , and is the path loss exponent.We set meter, , and we compute asfollows [34]:

(22)

where is the transmission power on channel ,and are the transmitter and receiver antenna gains onchannel , and is the wavelength of the carrier frequencyof channel . For simplicity, we set for allchannels.

We simulate ten pairs of CRs. The maximum transmissionpower for CR is 1 Watt, which ensures that CRs are within themaximum transmission range of each other. The AWGN noiselevel is set to dBm over all channels. Each CR trans-mitter generates burst of packets according to a Poisson processwith parameter burst/s. Each burst has an exponentially dis-tributed duration with mean s. The traffic rate for CR is de-fined as . We set the CR-to-PR power mask to 0.5 Watt forall channels.

We compare the performance of the proposed flow-basedPIWF-MAC protocol with the packet-based PIWF-MAC pro-tocol, against the flow-based IWF-MAC protocol. Since theIWF algorithm is a special case of the PIWF algorithm, the pro-posed MAC protocols are also applicable to the IWF algorithm.The DW for the flow-based MAC protocol allows ten datapackets to be transmitted in a row. The comparison is in termsof the system throughput and the average power consumption.The system throughput is defined as the average volume of CRtraffic bits that are transmitted in 1 s, and the power consump-tion is calculated as average power consumption by all CRs.

The resulting performance is depicted in Figs. 810. Fig. 8(a)shows the system throughput versus the traffic rate. As ex-pected, the flow-based PIWF-MAC protocol gives the highestthroughput. This throughput improvement over IWF-MACbecomes more significant at higher traffic rates. It is interestingto see that the simplified packet-based PIWF-MAC protocolexhibits comparable system throughput with the flow-basedPIWF-MAC protocol. Besides achieving a higher systemthroughput, the PIWF-MAC protocols also save transmissionpower, as shown in Fig. 8(b). This is because in IWF, usersgreedily maximize their own rates using the maximum trans-mission power, while such greedy behaviors are overcomeby the pricing technique used in PIWF. Note that althoughthe packet-based PIWF-MAC consumes less energy in con-trol overhead than the flow-based PIWF-MAC, it consumesmore energy in data transmissions. This is because in thepacket-based PIWF-MAC, the optimal power assignment isachieved after several packet transmissions (due to the absenceof a control-based training phase). The confluence of the



Fig. 8. Performance when = 0:1. (a) System throughput versus traffic rate. (b) Average power consumption versus traffic rate.

Fig. 9. Performance under traffic rate = = 0:5. (a) System throughput versus PR activity factor. (b) Average power consumption versus PR activity factor.

two energy-consumption factors still favors the flow-basedPIWF-MAC.

Fig. 9(a) depicts the network throughput versus the PRsactivity factor . As expected, a higher results in a higherPR-to-CR interference, which negatively affects the throughput.Fig. 9(b) shows the corresponding average power consumption.In all cases, PIWF-MAC protocols consume less power.

Finally, Fig. 10(a) shows the throughput versus the trainingwindow size. Since the simplified packet-based PIWF-MACdoes not employ a TW, we only compare the flow-basedPIWF-MAC and flow-based IWF-MAC. Intuitively, a largerTW size will ensure that CR users will converge to the NE.However, as seen in Fig. 4, 23 iterations are normally suffi-cient to reach a near-optimal sum-rate. The same behavior isobserved from the MAC simulations. As seen in Fig. 10(a),taking a TW of size 2 is enough to achieve 95% of the max-

imum system throughput in the simulation setup. Fig. 10(b)shows the corresponding average power consumption.

VII. CONCLUSIONIn this paper, we proposed a PIWF algorithm for spectrum

sharing in cognitive radio networks. Our PIWF algorithm canbe implemented distributively with CRs repeatedly negotiatingtheir transmission powers and spectrum. Simulation resultsshowed that the proposed algorithm greatly improves the NEcompared with the one achieved using the IWF approach.Based on the order by which CR nodes make their resourceallocation decisions, we studied sequential and parallel versionsof the PIWF algorithm. The parallel update scheme was shownto converge faster than the sequential update scheme, espe-cially for a large number of users. We also presented relaxedversions of the PIWF algorithms, which are more robust to



Fig. 10. Performance under traffic rate = = 0:7 and = 0:1. (a) System throughput versus TW size. (b) Average power consumption versus TW size.

estimation errors and channel oscillations at the cost of slowerconvergence. Based on the PIWF algorithms, flow-based andpacket-based MAC protocols were designed. Our simulationresults showed that the PIWF-MAC protocol achieves consider-ably higher system throughput compared with the IWF-MAC,with less energy consumption.

APPENDIX

Proof of Proposition 3:Proof: We first solve the optimization problem without the

power mask constraint C3, using the method of Lagrange mul-tipliers. This leads to a water-filling solution of the form [5]

(23)

If is the optimal power allocation over channel , thenthe slope of the utility function must be positive atthe point . Otherwise, a power vector with a smaller

could reach a higher utility , with all the con-straints satisfied. Thus, the utility function is mono-tonically increasing between 0 and . Accordingly, if anyof the in (21) violates the upper bound C3, then the cor-responding bounded optimal solution must be the upper bound

itself (a similar approach was also adopted in [23]).After bounding by , the remaining power willbe further water-filled over other channels, thus reaching the re-sult in (11).

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Fan Wang received the B.S. degree from TsinghuaUniversity, China, in 2001, and the M.S. degree fromthe University of Arizona, Tucson, in 2004. He is cur-rently pursuing the Ph.D. degree in the Departmentof Electrical and Computer Engineering, Universityof Arizona.

His current research interests are in system archi-tecture and communication protocol designs for cel-lular, ad hoc, and cognitive radio wireless networkswith emphasis on spectrum, power and rate control.He has several journal articles and refereed confer-

ence papers in this area. During the summer of 2006, he was a member of theWiMAX Development Group, FREESCALE, Inc.

Mr. Wang won the Best Student Paper Award in the Second InternationalConference on Cognitive Radio Oriented Wireless Networks and Communica-tions (CrownCom 2007). He serves as a reviewer for many IEEE conferencesand journals.

Marwan Krunz received the Ph.D. degree in elec-trical engineering from Michigan State University,East Lansing, in 1995.

He is a Professor of electrical and computer en-gineering at the University of Arizona, Tucson, andthe Director of the Advanced Networking And Wire-less Communications Group. He joined the Univer-sity of Arizona in January 1997, following a briefpostdoctoral stint at the University of Maryland, Col-lege Park. He previously held visiting research posi-tions at INRIA (Sophia Antipolis, France), HP Labs

(Palo Alto, CA), Paris VI, and US West (now Qwest) Advanced Technologies.His research interests lie in the fields of computer networking and wirelesscommunications. His recent interests include power/rate control in wireless andsensor networks, channel access and routing protocols, media streaming, qualityof service routing, and optical networking. He previously worked on traffic anal-ysis, modeling and performance evaluation, packet video modeling, and QoSprovisioning in high-speed networks.

Dr. Krunz is a recipient of the National Science Foundation CAREER Award(19982002). He currently serves on the editorial board for the IEEE/ACMTRANSACTIONS ON NETWORKING, the IEEE TRANSACTIONS ON MOBILECOMPUTING, and the Computer Communications Journal. He was a guestco-editor for special issues in IEEE Micro Magazine and IEEE CommunicationsMagazine. He served as a technical program chair for the IEEE WoWMoM2006, the IEEE International Conference on Sensor and Ad hoc Communica-tions and Networks (SECON 2005); the IEEE INFOCOM 2004 Conference;and the 9th Hot Interconnects Symposium, August 2001. He has served on,and continues to serve on, the executive and technical program committees ofmany international conferences and on the panels of several NSF directorates.He has given several tutorials at premier wireless networking conferences.

Shuguang Cui (S99M05) received the Ph.D. de-gree in electrical engineering from Stanford Univer-sity, Stanford, CA, the M.Eng. degree in electrical en-gineering from McMaster University, Hamilton, ON,Canada, in 2000, and the B.Eng. in radio engineeringwith the highest distinction from Beijing Universityof Posts and Telecommunications, Beijing, China, in1997.

He is currently an Assistant Professor in electricaland computer engineering at Texas A&M University,College Station. From 1997 to 1998, he was with

Hewlett-Packard, Beijing, China, as a System Engineer. In the summer of 2003,he worked at National Semiconductor, Santa Clara, CA, on the ZigBee project.From 2005 to 2007, he worked as an Assistant Professor in the Department ofElectrical and Computer Engineering, University of Arizona, Tucson. His cur-rent research interests include cross-layer energy minimization for low-powersensor networks, hardware and system synergies for high-performance wirelessradios, statistical signal processing, and general communication theories.

Dr. Cui was a recipient of the NSERC Graduate Fellowship from the Na-tional Science and Engineering Research Council of Canada and the CanadianWireless Telecommunications Association (CWTA) graduate scholarship. Hehas been serving as the TPC co-chair for the 2007 IEEE Communication TheoryWorkshop and the ICC08 Communication Theory Symposium. He is currentlyserving as an associate editor for the IEEE COMMUNICATION LETTERS and theIEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.


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