Distributed Channel Assignment in Cognitive Radio Networks: Stable Matching and Walrasian Equilibrium
aa r X i v : . [ c s . I T ] J a n Distributed Channel Assignment in Cognitive RadioNetworks: Stable Matching and WalrasianEquilibrium
Rami Mochaourab, Bernd Holfeld, and Thomas Wirth
Abstract —We consider a set of secondary transmitter-receiverpairs in a cognitive radio setting. Based on channel sensingand access performances, we consider the problem of assigningchannels orthogonally to secondary users through distributedcoordination and cooperation algorithms. Two economic modelsare applied for this purpose: matching markets and competitivemarkets. In the matching market model, secondary users andchannels build two agent sets. We implement a stable matchingalgorithm in which each secondary user, based on his achievablerate, proposes to the coordinator to be matched with desirablechannels. The coordinator accepts or rejects the proposals basedon the channel preferences which depend on interference from thesecondary user. The coordination algorithm is of low complexityand can adapt to network dynamics. In the competitive marketmodel, channels are associated with prices and secondary usersare endowed with monetary budget. Each secondary user, basedon his utility function and current channel prices, demands a setof channels. A Walrasian equilibrium maximizes the sum utilityand equates the channel demand to their supply. We prove theexistence of Walrasian equilibrium and propose a cooperativemechanism to reach it. The performance and complexity of theproposed solutions are illustrated by numerical simulations.
Index Terms —cognitive radio; spectrum sensing; resourceallocation; distributed algorithms; stable matching; Walrasianequilibrium; English auction; combinatorial auctions
I. I
NTRODUCTION
In cognitive radio settings, secondary users (SUs) are capa-ble of adapting their transmissions intelligently [1]. Throughthe detection of spectrum holes, the SUs can use the unoccu-pied channels licensed to the primary users for communication.This mechanism is called opportunistic spectrum access [2]and corresponds to the interweave paradigm described in [3].Generally, there exists a tradeoff between the optimizationof the secondary systems’ performance and the primary sys-tems’ performance [4]. Our objective is to find an assignmentof the primary channels to the SUs taking into account bothsecondary and primary user performances. For a survey onchannel assignment mechanisms in cognitive radio networksplease refer to [5]. Since a cognitive radio network is a
This work has been submitted to the IEEE for possible publication.Copyright may be transferred without notice, after which this version mayno longer be accessible.Rami Mochaourab was with Fraunhofer Heinrich Hertz Institute, Berlin,Germany. Now he is with ACCESS Linnaeus Centre, Signal ProcessingDepartment, School of Electrical Engineering, KTH Royal Institute of Tech-nology, 100 44 Stockholm, Sweden. Phone: +4687908434. Fax: +4687907260.E-mail: [email protected]. Bernd Holfeld and Thomas Wirthare with Fraunhofer Heinrich Hertz Institute, Berlin, Germany. E-mail:[email protected], [email protected]. distributed and less regulated system, we are interested inchannel assignment mechanisms which are implemented in adistributed way. We study such mechanisms using matchingmarkets and competitive markets with indivisible goods.Although the applications and solutions of the two marketmodels are conceptually different, there exist similarities be-tween the two market models [6]. First, both solutions of themarket models lead to an assignment, which is in our case, anorthogonal assignment of the channels to the SUs. Moreover,both models assume autonomous and rational agents who areable to decide locally between different alternatives. Theseproperties are favorable for distributed operation of the SUs.Nevertheless, both models, rely on communication based onbinary decisions reflecting a proposal in stable matching ora demand in competitive markets. Hence, the application ofthe two models has practical implementation in cognitiveradio networks in which coordination can be achieved withlow communication overhead. In addition to their distributedand low communication overhead properties, optimality of thesolutions of both frameworks within specified performanceregions make the application of these models attractive forresource allocation in communication networks. We relate tosome of these works, after discussing the differences betweenthe two frameworks.The differences between the two models are as follows:Competitive markets use prices as means to coordinate thedemands (decisions) of the consumers to buy goods and areupdated by an auction mechanism to reach a solution. Instable matching, on the other hand, no prices are involvedbut the two sets of agents, i.e. SUs and channels, exchangeproposals based on preference relations of each agent withinthe two sets. Through sequences of acceptances and rejections,a stable matching is reached. In the competitive market model,only the consumers’ preferences (utility functions) are needed.In Section III, we further discuss the differences of the twosolutions for our cognitive radio scenario.
A. Application of Matching and Competitive Market Models
In two-sided matching markets [7], two sets of agents areto be matched, corresponding to the SUs and the primarychannels. Each agent in one set has preferences over the agentsin the other set. A matching of the agents in the two sets isstable when no pairs of agents prefer each other compared totheir current matching.Matching market models for resource allocation in wirelessnetworks have been recently applied in several works. In [8], the framework of two-sided stable matching is applied forresource allocation in wireless networks and its merits revealedregarding distributed implementation and efficiency. Stablematching for channel assignment in cognitive radio settingshas been applied in [9]–[12]. In [9] and [10], one-to-one stablematching is considered where the utility of the secondary andprimary users are chosen to be identical due to the fact that theSUs cannot obtain the performance measures of the primaryusers. In this case, the stable matching of SUs to the primarychannels is proven to be unique. In addition, in [10] stablematching is successfully implemented through opportunisticCSMA techniques. Reference [11] applies the model in [10]to interweave cognitive radio settings with identical utility forsecondary and primary users. While in [10], [11], the utilityof both types of agents are the same, in [12] the utility ofthe primary users depend on the interference leakage fromSUs and the utility of the SUs are their achievable rates in theprimary channels. In this context, many-to-one stable matchingis applied.Stable matching for channel assignment in a single radiocell is applied in [13] where two-sided matching takes intoaccount the utilities of the users in the uplink and the downlinktransmissions. In [14], the stable matching framework isapplied for cross-layer scheduling in the downlink of a singlecell where the utility of a user is his sum rate and the utilityof the resources includes the user queue state of the buffer.In the context of physical layer security, stable matching oftransmitter-receiver pairs to friendly jammer is proposed in[15]. In [16], uplink user association in small cell networksis considered using many-to-one stable matching as well ascoalitional games. The user utilities are based on quality ofservice (QoS) and coverage aspects.In competitive markets [17], also referred to as one-sidedmatching markets [18], there exists a set of agents which wantto buy quantities of goods. The prices of the goods regulatethe quantities bought by the consumers and are adapteddepending on the demand and supply of the goods. TheWalrasian equilibrium is a state in which the demand equalsthe goods’ supply. In order to reach a Walrasian equilibrium,a price adjustment process is required. This process is relatedto auction mechanisms and its advantage is the distributedimplementation aspect and the limited amount of informationexchange required between the users and the coordinator.Competitive market models have found a few applicationsfor resource allocation problems in communication networks.Please refer to [19], [20] for a discussion on these applications.Also, for a recent survey on auction mechanisms for resourceallocation in wireless networks, see [21]. In cognitive radiosettings, auctions have been applied for distributed channelassignment in [22]–[24]. In [22], repeated auctions in theuplink of a secondary cell are proposed for the allocation ofthe primary channel resources to the SUs. Distributed auctionsare studied in [23] for energy efficient channel assignmentin cognitive radios. Moreover, in [24], a distributed auctionmechanism is proposed to find optimal one-to-one channelassignment to the SUs where CSMA mechanisms are utilizedto implement the solution.
B. Contributions and Outline
In this work, we consider a set of transmitter-receiver pairsas SUs and each user seeks the assignment of a set ofprimary channels. Our objective, formulated in Section III,is to optimize both the secondary and primary users’ perfor-mance through coordinated and cooperative distributed chan-nel assignment. We assume that each primary channel can beassigned to one SU while an SU can be assigned to multiplechannels. However, an SU is restricted to use a maximumnumber of channels called quota which improves the fairnessin the channel assignment.We propose a coordinated channel assignment (Section IV )which exploits many-to-one stable matching. Here, we assumethat a coordinator exists which can communicate with the SUs.We characterise in worst case the number of bits each SU hasto exchange with the coordinator in order to reach a stablematching. In addition, we provide conditions under which thestable matching is unique and primary user optimal. Our modeldiffers from the models used in [10], [11], by the followingtwo aspects: multiple channels are assigned per SU, and theutility functions of the primary channels are different from theutility functions of the SU. One main difference to [12] is ourapplication of stable matching in interweave cognitive radio.For cooperative channel assignment, we study a competitivemarket model with indivisible goods [25] in Section V. Theutility function of an SU is the weighted sum of his achievablerate and the utility of the primary users whose channelshe is assigned to. We prove the existence of a Walrasianequilibrium which maximizes the weighted sum-performanceof the secondary and primary systems. To reach the Walrasianequilibrium through a cooperative mechanism, we exploit anEnglish auction algorithm from [26]. The cooperative mecha-nism requires the exchange of L bit information between theSUs. In comparison to auction algorithm studied in [24], ourmechanism is able to assign multiple channels to each user.Numerical simulations are provided in Section VI beforewe draw the conclusions in Section VII. Notations:
Vectors are written in boldface letters. Sets arewritten in calligraphic font. |S| is the cardinality of the set S . | c | is the absolute value of c ∈ C . The Q-function isgiven as Q ( x ) = √ π R ∞ x exp( − u / du . The inverse ofthe Q-function is Q − ( x ) . x ∼ CN (0 , a ) is a circularly-symmetric Gaussian complex random variable with zero meanand variance a . Pr ( x ) is the probability of an event x . R + isthe set of nonnegative real numbers.II. S YSTEM M ODEL
Consider a set K = { , . . . , K } of secondary transmitters-receiver pairs and a set of orthogonal channels L = { , . . . , L } licensed to primary users. Each secondary user (SU) wants touse a set of channels for communication. The system modelis illustrated in Fig. 1.We assume that the distributed assignment of the channelsto the SU can be done either using a coordinator or throughdirect communication between the SUs. In the stable matchingmodel studied in Section IV, we assume the existence of acoordinator which is connected to the SUs through low-rate primary transmitter l ∈ L primary receiver l ∈ L secondary transmitter k ∈ K secondary receiver k ∈ K g [ l ] h [ l ] k ˜ g [ l ] ˜ h [ l ] k z [ l ] k Fig. 1: Illustration of the system model.links. In Section V, we do not assume the existence of acoordinator, but require that the SUs can directly communicatewith each other.
A. Secondary System Performance
An SU is allowed to access a set of channels if these aredetected to be idle. We assume a primary user (PU) operates ina time-slotted fashion and starts transmission at the beginningand for the duration of a time-slot. Each SU at the beginningof the time slot is assumed to make a number N of sensingobservations in each channel l . The sensing problem of SU k is the decision between two hypothesis on whether PU l isactive ( H [ l ] k, ) or not ( H [ l ] k, ). The two hyposesis correspond to: H [ l ] k, : x l,k ( t ) = w k ( t ) , t = 1 , . . . , N, (1) H [ l ] k, : x l,k ( t ) = p P l z [ l ] k s l ( t ) + w k ( t ) , t = 1 , . . . , N, (2)where s l ( n ) ∼ CN (0 , is the transmitted signal of PU l , P l isthe average primary transmission power, w k ( n ) ∼ CN (0 , σ ) is additive white Gaussian noise, and z [ l ] k ∼ CN (0 , isthe quasi-static block flat-fading channel from PU l to SU k assumed constant during the time-slot.Let f [ l ] k = Pr ( H [ l ] k, | H [ l ] k, ) and d [ l ] k = Pr ( H [ l ] k, | H [ l ] k, ) bethe false alarm and detection probability of the detector at SU k , respectively. The access probability of SU k in channel l isgiven as θ [ l ] k = (1 − ϑ [ l ] )(1 − f [ l ] k ) + ϑ [ l ] (1 − d [ l ] k ) , where ϑ [ l ] is PU l transmission probability.After spectrum sensing, an SU can be assigned a channel l if he detects that PU l is idle. The signal from secondarytransmitter k received at secondary receiver k on channel l is y [ l ] k = ( h [ l ] k √ P k s k + ˜ g [ l ] k √ P l s l + w k , PU l is active; h [ l ] k √ P k s k + w k , otherwise, (3)where s k ∼ CN (0 , is the transmitted signal, P k is thetransmission power assumed to be the same in all channels, h [ l ] k is the channel from secondary transmitter k to its receiver, and w k ∼ CN (0 , σ ) is additive white Gaussian noise. We assumethat P k is fixed and equal for all SUs and define the signal-to-noise ratio (SNR), used in the simulations, as SNR := P k /σ .The average achievable rate in bits/s/Hz of SU k in channel l can be formulated as u su k ( l ) = (1 − ϑ [ l ] )(1 − f [ l ] k ) log P k | h [ l ] k | σ ! + ϑ [ l ] (1 − d [ l ] k ) log P k | h [ l ] k | σ + P l | ˜ g [ l ] k | ! . (4)The first term in the summation above is the average achiev-able rate when the PU is idle (also called opportunistic rate[27]) and the second term corresponds to the achievable rateon transmission simultaneously with the PU.If an SU k is assigned the set B ⊆ L of channels, hisaverage sum-rate is then u su-sum k ( B ) = X l ∈B u su k ( l ) , (5)where u su k ( l ) is defined in (4) and u su k ( ∅ ) = 0 . In this work,we introduce the following channel assignment constraint: Themaximum number of channels an SU k can be assigned tois restricted to a maximum of q k ∈ N , called quota , and isassumed to be fixed for each SU. B. Primary System Performance
If channel l is assigned to SU k , then the performance of PU l decreases in both probability of misdetection (1 − d [ l ] k ) andinterference P k | ˜ h [ l ] k | , where ˜ h [ l ] k is the channel from secondarytransmitter k to primary receiver l . Accordingly, we formulatethe utility function of a PU l as: u pu l ( k ) = φ l (1 − d [ l ] k , P k | ˜ h [ l ] k | ) , (6)where φ l ( x, y ) ≤ φ l ( x ′ , y ′ ) for x ≥ x ′ and y ≥ y ′ . If noSU is active in channel l , the interference-free utility of PU l is u pu l ( ∅ ) = φ l (1 , . We additionally define self-matching ofchannel l as u pu l ( l ) = u pu l ≤ u pu l ( ∅ ) , l ∈ L , (7)where the value u pu l reflects a threshold for a QoS requirementof PU l . This QoS requirement will be incorporated later inthe stable matching framework in Section IV.Since the utility of a PU is largest without interference fromSUs, the region R pu = { ( r , . . . , r L ) ∈ R L + | r l ≤ u pu ( ∅ ) , l ∈ L} , (8)contains all jointly achievable performances for the PUs. Asubset of R pu , specified as e R pu = { ( u pu ( a ) , . . . , u pu L ( a L )) ∈ R pu | a l ∈ K ∪ { l } ,a l = ∅ , X a l ′ = kl ′ ∈L ≤ q k , k ∈ K , l ∈ L} , (9)does not contain the performance tuples in which a PUoperates alone, i.e., a l = ∅ for all l , but only the performancetuples of the PUs when SUs are assigned to them or when thePUs are self-matched as specified in (7). Also, the region in(9) takes into account the quota restrictions on the SUs. Later,we utilize the definition of e R pu to relate to existing efficiencyresults for stable matching. III. P
ROBLEM D ESCRIPTION
Our objective is to find an assignment of primary channelsto the SUs through distributed mechanisms. Define the assign-ment variable x ( B , k ) = (cid:26) , B ⊆ L is assigned to SU k ∈ K ; , otherwise. (10)In addition, define the following set of assignment constraints: x ( B , k ) ∈ { , } , ∀B ⊆ L , k ∈ K , (C1) X B∋ l X k ∈K x ( B , k ) ≤ , ∀ l ∈ L , (C2) X B⊆L x ( B , k ) ≤ , ∀ k ∈ K , (C3) |B| x ( B , k ) ≤ q k , ∀B ⊆ L , k ∈ K , (C4) u pu l ( k ) x ( B , k ) ≥ u pu l x ( B , k ) , ∀ l ∈ B , ∀B ⊆ L , k ∈ K . (C5)Constraint (C2) ensures that only one SU is allocated per chan-nel, and constraint (C3) ensures that each SU is associatedwith one subset of L . The user quota constraint is in (C4) andconstraint (C5) specifies a QoS threshold for each PU.We will utilize the definition of the constraints (C1) − (C5)to describe the coordination and cooperation mechanisms wepropose in this work. A. Coordination Mechanism
Our first objective is to propose a low complexity coordi-nation algorithm which matches the SUs to the PU channelsexploiting the existence of a coordinator. The SUs and PUchannels form two agent sets and each agent has a preferenceover each agent in the other set. These preferences are accord-ing to the utility functions defined in Section II-A and SectionII-B, respectively. For this purpose, we use many-to-one stablematching (Section IV) as an assignment of the channels to theSUs.Generally, distributed implementation of stable matchingrequires communication between one agent set and the otherin order to exchange proposals. In this work, we assumethat the coordinator receives proposals from the SUs andaccepts or rejects them on behalf of the PU channels. In orderto implement the coordinated stable matching algorithm, theinformation which should be available at SU k is u su k ( l ) in(4) for all l ∈ L and his quota q k , while the coordinatorneeds the information of u pu l ( k ) in (6) for all l ∈ L and all k ∈ K . The information u pu l ( k ) = φ l (1 − d [ l ] k , P k | ˜ h [ l ] k | ) at SU k requires the knowledge of the probability of misdetection − d [ l ] k , which is known at the SU, and the interference atthe primary receiver l . Assuming time division duplex (TDD)systems, the channel gain | ˜ h [ l ] k | from secondary transmitter k to primary receiver l is almost identical to the channel gain We impose the orthogonality constraint on the channel assignment becausethe frameworks we exploit from stable matching and competitive markets donot take externalities [28] into account which would exist in nonorthogonalassignments. In our context, externalities are the interdependencies of theallocation of channels to some users on a given channel assignment to aspecific user. Settings with externalities are generally much more complex toanalyze, especially regarding the stability of distributed resource allocationalgorithms. Recent application of stable matching with externalities for userassociation in small cell networks can be found in [29]. from primary receiver l to secondary transmitter k and canbe made available at the SU during channel estimation tocalculate the interference P k | ˜ h [ l ] k | . Having this information,we assume that each SU k forwards u pu l ( k ) for all l ∈ L tothe coordinator in an initialization phase.If the coordinator also knows the utilities of the SUs, u su k ( l ) for all k ∈ K and l ∈ L , then the Hungarian method can beapplied at the coordinator to find an assignment which satisfies(C1) − (C5). The advantage of stable matching, however, is itsflexibility to adapt to network dynamics and also to complexityrequirements. We discuss these issues in Section IV-B. B. Cooperative Mechanism
The cooperative mechanism relies on direct communicationbetween the SUs and does not require the existence of acoordinator as in the coordination mechanism. We consider theoptimization of both the secondary and primary performancemeasures which is a multi-objective optimization problem.One method for solving multi-objective optimization problemsis by optimizing the weighted sum of the objectives [31] whichcan be formulated for user k utilizing resource set B as: W ( B , k ) = λu su-sum k ( B ) + (1 − λ ) X l ∈B u pu l ( k ) , (11)with λ ∈ [0 , . Here, λ is a parameter which can be usedto increase the priority of one objective to the other. If λ isclose to one, the secondary system performance is given moreimportance in the optimization above than the primary userperformance, while if λ is close to zero, the primary systemperformance is prioritized. The value of λ must be defined inregard of the network specifications.The integer optimization problem we are interested to solveis stated as follows: maximize X k ∈K X B⊆L W ( B , k ) x ( B , k ) s.t. (C1) − (C4) . (12)The solution of (12) is a Walrasian equilibrium of an asso-ciated competitive market with indivisible goods studied inSection V. The Walrasian equilibrium can be reached througha distributed English auction which we exploit to providea decentralized and optimal cooperative channel assignmentmechanism.In order to implement the cooperative mechanism to solve(12), each SU k must know W ( { l } , k ) for all l ∈ L , i.e., SU k must know u pu l ( k ) in (6) for all l ∈ L , his utility function u su k ( l ) in (4) for all l ∈ L , the weight λ , and also his quota q k . Moreover, all SUs must have knowledge of a commonparameter α > which will be used as a price incrementingfactor. IV. M ANY - TO -O NE S TABLE M ATCHING
We propose assigning SUs to the channels associated withthe PUs by a framework for which stability serves as solution The Hungarian method [30] is an algorithm that solves the fundamentalone-to-one assignment problem in combinatorial optimization. The many-to-one channel assignment with quotas in (12) is a form of the generalizedassignment problem for which the Hungarian method can be applied when q k -many virtual SUs with a quota of one are introduced for each SU k ∈ K . concept instead of optimality. In cognitive radios, where SUaccess on channels is opportunistic and complex regulationentities are commonly absent, a stable distributed assignmentprocess is favored. The applied framework involves a two-sided matching market where a coordinator is acting on behalfof the PU side to support the decision-making. We assumethat one primary channel l ∈ L is matched to one SU k ∈ K while the latter can be assigned to up to q k primary channels,where q k ∈ N is the maximum matching quota , see SectionII. This resource allocation is named college admission [32]or hospitals/residents problem [33] in the stable matchingliterature.A stable matching is produced by a distributed processthat matches together preference relations of the primariesand the secondaries over the other agents each. The orderof preferences is given by the strictly ranked rate utilitiesin (4) and (6). In some cases, matching one agent withhimself (denoted as being unmatched) might be preferred to amatching with other agents in the stable matching framework.Therefore, we define the matching of SU k ∈ K to himselfas u su k ( k ) = u su l ( ∅ ) = 0 which means that the device doesnot transmit on any primary channel. On the other hand,matching a PU with himself should intuitively lead to theutility u pu l ( ∅ ) = φ l (1 , as in Section II-B, since the PUoccupies its channel l alone and no SU transmits in l . However,we specify the utility of a self-matched primary channel to avalue u pu l , see (7), reflecting a threshold for a QoS requirementof the PU. In doing so, a PU prefers being matched to a SUif its own utility remains higher than the threshold and prefersself-matching if matching with a SU does not guarantee QoS.The coordinator is in charge of monitoring the QoS issues ofthe PU in the stable matching framework. A. Stable Matching Model
The stable matching problem is described by the tuple [8] hL , K , { u pu l } l ∈L , { u su k } k ∈K , { q k } k ∈K i , where L is the set ofprimary channels, K is the set of SUs, u pu l and u su k are theutility functions of the PUs and SUs given in (6) and (5). Thequotas q k are associated with the SUs. Definition 1:
A matching M is from the set K ∪ L into theset of unordered family of elements of
K ∪ L such that | M ( l ) | = 1 for every PU l ∈ L where M ( l ) = l if M ( l ) / ∈ K ,2) ≤ | M ( k ) | ≤ q k for every SU k ∈ K where M ( k ) = k if M ( k )
6⊂ L ,3) M ( l ) = k if and only if l ∈ M ( k ) .In Definition 1, M ( l ) denotes the matched SU of PU l or self-matching and M ( k ) denotes the subset of PUs matched to SU k or self-matching, respectively. Definition 2:
The matching M is individually rational ifthere exists no PU l ∈ L for which u pu l ( l ) > u pu l ( M ( l )) andno SU k ∈ K for which u su k ( k ) > u su k ( j ) , j ∈ M ( k ) [7].Individually rational matching ensures that no user, primary orsecondary, would prefer being matched to himself than withthe current matching. Definition 1 is adopted from [7] despite the fact that our framework doesnot fill an under-subscribed matching set with multiple copies of the self-matched agent.
Definition 3:
The matching M is blocked by the pair ( k, l ) ∈ K×L if (i) u pu l ( k ) > u pu l ( M ( l )) and (ii) | M ( k ) | < q k and u su k ( l ) > or u su k ( l ) > u su k ( l ′ ) for some l ′ ∈ M ( k ) .Accordingly, a matching is blocked by ( k, l ) if these prefereach other to their current matching. Definition 4:
A matching M is stable if it is individuallyrational and not blocked by any pair ( k, l ) ∈ K × L .There may exist several stable matchings. Let all stablematchings lead respectively to the SU and PU performanceregions R su SM and R pu SM ⊆ e R pu , where e R pu is in (9). Next,we provide an algorithm which reaches a stable matching andreveal its performance in these regions. B. Distributed Implementation of Stable Matching
Algorithm 1 implements a distributed coordination mecha-nism proposed in [34] to deliver a stable matching. Here, theSUs start proposing to be matched with their preferred PUchannels (Line 2) and a low-complex coordinator responds onbehalf of the PUs (starting Line 3). The information neededat the SUs and the coordinator prior to the execution ofAlgorithm 1 is stated in Section III-A. In the given protocol,SU k proposes to be matched to a channel l by sendinga message Ψ k to the coordinator if k has not reached itsquota and prefers this channel it is not already matched with(condition in Line 1). The message Ψ k can be of ⌈ log ( l ) ⌉ bits which is the length of the base-2 equivalent of l . Thecoordinator reacts to the proposal by sending a one bit messageto an SU k to indicate acceptance or rejection. The coordinatoraccepts SU k on a channel l (Line 7 and 8) only if the QoSrequirement of the PU is fulfilled (Line 4). Otherwise, thecoordinator rejects SU k (Lines 5 and 6). Also, in order toreduce the total number of iterations of the algorithm, thecoordinator triggers messages to exclude selected SUs fromproposing to certain channels (Lines 9 and 10). Including therejection information in Line 6, these messages are of L bitsand indicate for each SU which channels he need not proposeto.In Algorithm 1, each SU proposes at most once to bematched with a specific resource. Thus, the worst case totalnumber of proposals by an SU to the coordinator is L . Proposition 1:
The worst case number of bits that is ex-changed between one SU and the coordinator during Algo-rithm 1 is L + L + P Ll =1 ⌈ log ( l ) ⌉ . Proof:
The term P Ll =1 ⌈ log ( l ) ⌉ is the total number of bitsneeded to indicate the channel indexes in the L proposals fromthe SU. L bits are needed in total to indicate the acceptance orrejection from the coordinator to the L proposals from the SU.The term L is due to L bit messages sent from the coordinatorto the SUs in Lines 9 and 10 of Algorithm 1.The actual number of proposals by a single SU depends on hisquota and also on the matching of the channels to the otherSUs. If the quota of an SUs is small, then a few proposalscan be sufficient to reach the matching quota limit and stopthe SU from further proposals. Also, when several SUs arealready accepted on some channels, the number of channelswhich an SU can propose to may decrease due to Lines 9and 10 in Algorithm 1. In Section VI, we provide extensivesimulations on the average number of proposals from an SU. Algorithm 1
Distributed SU-proposing stable matching. while some SU k ∈ K is under-subscribed ( | M ( k ) | < q k or M ( k ) = k ) and max l ∈L ,l/ ∈ M ( k ) u su k ( l ) > u su k ( k ) do Proposal by SU k : send out index l ⋆ =argmax l ∈L ,l/ ∈ M ( k ) u su k ( l ) of most preferred PU Coordinator Response: if QoS is ensured, i.e. u pu l ⋆ ( k ) > u pu l ⋆ then if PU l ⋆ is engaged to any SU k ⋆ = k then inform k ⋆ on releasing engagement with l ⋆ ,giving M ( k ⋆ ) = { k ⋆ } if it was | M ( k ⋆ ) | = 1 and M ( k ⋆ ) = M ( k ⋆ ) \ { l ⋆ } otherwise accept engagement temporarily; set M ( l ⋆ ) = { k } inform SU k on approving the engagement, giving M ( k ) = { l ⋆ } if k was unmatched ( M ( k ) = { k } )and M ( k ) = M ( k ) ∪ { l ⋆ } otherwise for all i ∈ K such that u pu l ⋆ ( i ) < max { u pu l ⋆ ( k ) , u pu l ⋆ } do eliminate preference on i ( u pu l ⋆ ( i ) = 0 ) and disqual-ify SU i from proposing to l ⋆ ( u su i ( l ⋆ ) = 0 )Observe in the implementation of stable matching thatif new secondary users arrive to the network and proposeto the coordinator to be matched to a set of channels, thecoordinator can use Algorithm 1 with the initialization of thecurrent stable matching. In contrast, the application of theHungarian method necessitates the network wide optimizationproblem to be solved again. Nevertheless, the stable matchingalgorithm can be terminated at any time instance associated toa desirable complexity level to retrieve an orthogonal matchingof resources to the SUs. Such properties of the algorithm makeit adaptable to changes in the network and also to specifiedcomplexity or implementation requirements.Initializing Algorithm 1 with unmatched SUs and channelsas M ( l ) = { l } ∀ l , M ( k ) = { k } ∀ k , the terminating state is anSU-optimal stable matching which is weak Pareto optimal inthe set R su SM according to [7, Corollary 5.9] but is the worststable matching for the PUs [7, Corollary 5.30]. Next, weprovide conditions under which our stable matching is uniqueand also sum-performance optimal for the PUs. Theorem 1:
For q k ≥ L for all k , the stable matching is unique and leads to the maximum sum performance point in e R pu defined in (9). Proof:
The proof is provided in Appendix A.The result above generalizes the uniqueness result of one-to-one matching in [9, Proposition III.1] to the case of many-to-one matching. V. W
ALRASIAN E QUILIBRIUM
In the previous section, we studied two-sided matchingwhere the SUs on one side are matched to primary channelson the other. In this section, we study a market model whereonly one entity is represented (SUs) but its utility is theweighted combination of the utilities of the secondary andprimary users given in (11) in Section III. Contrary to the The set of all weak Pareto optimal points in a performance region R aredefined as [35, p. 14] W ( R ) = { x ∈ R | there is no y ∈ R with y > x } . previous section, we now do not assume the existence of acoordinator. However, we require that the SUs are able tocommunicate with each other in order to find an assignmentof the channels which solves Problem (12). The mechanismexploits the market model studied next. A. Competitive Market Model
A competitive market with indivisible goods [25] is com-posed of a set of consumers and a set of goods. The consumersin our setting are the SUs in K and the goods correspond to theprimary channels in L . Here, we define the unit-less utilityfunction of consumer k using the utility function in (11) withan additional restriction on his quota q k as follows U k ( A ) = maximize B⊆A λu su-sum k ( B ) + (1 − λ ) X l ∈B u pu l ( k ) s.t. |B| ≤ q k . (13)The function above is called the q k - satiation of the weightedsum-perfromance [25, Section 2].Each good l has a price p l ≥ which is in monetaryunits and we assume that each SU is endowed with sufficientamount of monetary budget which enables him to buy bundlesof goods. The unit-less net utility of SU k is v k ( B , p ) := U k ( B ) − X l ∈B p l . (14)Given the prices of the goods (primary channels) p =( p , . . . , p L ) , the demand correspondence of SU k is the setof goods which maximizes his net utility: D k ( p ) = {A ⊆ L | v k ( A , p ) ≥ v k ( B , p ) ∀B ⊆ L} . (15)Later in Algorithm 2 in Section V-B, we specify a method toefficiently calculate the demand for each SU. The outcome of acompetitive market is a Walrasian equilibrium which specifiesthe prices of the channels at which each SU buys the channelshe demands and no channel is bought by more than one SU. Definition 5: [25, Section 2] A
Walrasian equilibrium is atuple ( p , X , X , . . . , X K ) , where p ∈ R L + is a price vector,and ( X , . . . , X K ) is a partition of L , i.e., X k ∩ X j = ∅ forall k = j , and S Kk =0 X k = L , such that (i) for each k ∈ K , v k ( X k , p ) ≥ v k ( A , p ) for all A ⊆ L , and (ii) the price of anyobject in X is zero.A Walrasian equilibrium exists if and only if the utilityfunction U k in (13) satisfies [25]:1) monotonicity : for all A ⊂ B ⊂ L , U k ( A ) ≤ U k ( B ) ,2) gross substitutes condition : for any two price vectors p ′ and p such that p ′ ≥ p (the inequality is componen-twise), and any A ∈ D k ( p ) , there exists B ∈ D k ( p ′ ) such that { i ∈ A | p ′ i = p i } ⊆ B .The gross substitute condition implies that if an SU demands aset of channels, and prices of some channels increase, the SUwould still demand the channels whose prices did not change. Theorem 2:
A Walrasian equilibrium exists in our setting. The utility function can be made unit-less by dividing the terms with theirassociated unit of measure.
Algorithm 2
Calculate demand A k of SU k . Input : prices p = ( p , . . . , p L ) ; quota q k Init : Π = { π , . . . , π L } with π l = v k ( { l } , p ) ; A k = ∅ sort Π in descending order to obtain Π sorted set A k as the first q k elements in Π sorted which are strictlylarger than zero. Proof:
The proof is provided in Appendix B.There is a direct relation between the solution of (12)and the Walrasian equilibrium of the associated competitivemarket model with indivisible goods [36]. The existence of aWalrasian equilibrium ensures that the solution of (12) is iden-tical to the solution of its linear programming relaxation [36,Theorem 1.13] in which the integer constraint (C1) is replacedby the convex constraint x ( B , k ) ≥ , ∀B ⊆ L , k ∈ K .Next, we will describe the English auction which is theprice adjustment mechanism needed to reach the Walrasianequilibrium. B. English Auction
In the English auction [26, Section 5], if a channel issimultaneously demanded by more than one SU then its priceis increased. The auction terminates when each channel isdemanded by at most one SU. This auction mechanism iswithin the combinatorial auctions frameworks classified in[21] and has been rarely applied in the context of wirelesscommunication due to their complexity. In the following, weshow that the steps required during the English auction in ourmodel can be calculated efficiently.In order to perform the English auction, we first need toefficiently calculate the consumer demand in (15). Afterwards,we need to calculate the set of channels which are simultane-ously demanded by more than one SU. This set is called theaggregate excess demand. These issues are addressed in thesame order next.The consumer demand in (15) seems at first sight hard tosolve since a search over all L subsets of L is needed. Notethat in [26] no method is provided to calculate the demand, butis only assumed that the demand can be calculated efficiently.This assumption is known under the existence of a demandoracle . We show that the consumer demand in our case canbe solved in polynomial time with the number of channelsusing a greedy approach. First, we need the following result. Lemma 1: If p l > for all l ∈ L , then A k ∈ D k ( p ) satisfies |A k | ≤ q k for all k ∈ K . Proof:
The proof is provided in Appendix C.From Lemma 1, if the prices of each good are strictly largerthan zero, then a user demands at most as many resources ashis quota. In order to calculate the demand of an SU, we usethe following assumption to ensure that the best q k channelsof a user k are unique. Assumption 1:
For any price vector p > , and for allconsumers k , the net utilities satisfy v k ( { l } , p ) = v k ( { l ′ } , p ) for any two goods l, l ′ ∈ L . Theorem 3:
For given prices p > , Algorithm 2 finds theconsumer demand set which is the smallest subset of all setsin D k ( p ) defined in (15). Algorithm 3
Calculate excess demand Z . Input : demand A , . . . , A K Init : Z = ∅ for l = 1 , . . . , L do for all k, j = 1 , . . . , K and j = k do if l ∈ A k ∩ A j then Z = Z ∪ { l } Proof:
The proof is provided in Appendix D.The complexity of calculating the demand of SU k inAlgorithm 2 requires a sorting algorithm such as Quick Sortwhich requires on average O ( L log L ) comparisons.Later, in the distributed implementation of the Walrasianequilibrium in Section V-C, it is required that each SU reportshis demand set to the other SUs. If an SU knows all other SUs’demands, he can calculate the excess demand set which iscomposed of the channels simultaneously demanded by morethan one SU. First, we need the following definitions beforedefining the excess demand set.The requirement function of consumer k is defined as K k ( B , p ) := min A∈ D k ( p ) |A ∩ B| , (16)and reveals the smallest number of elements in commonbetween B and the demanded channels by SU k at given prices p . From [26, Theorem 2], we have K k ( B , p ) ≤ K k ( B , q ) for p ≥ q (componentwise inequality) and p l = q l for l ∈ L \ B .This means that K k ( B , p ) decreases when the prices of theobjects inside B increase.Since the demand from Algorithm 2 is the smallest subset ofall demand sets following Theorem 3, we have the followingresult: Corollary 1:
The requirement function can be calculated as K k ( B , p ) = |A k ( p ) ∩ B| , where A k ( p ) is the demand setcalculated using Algorithm 2.Define the function which counts the number of times eachchannel in B is demanded as K K ( B , p ) := X Kk =1 K k ( B , p ) . (17)From [26, Corollary of Theorem 3], a necessary condition thatany channel in set B is not demanded by more than one SUat the same time is K K ( B , p ) − |B| ≤ . Hence, in Walrasianequilibrium with prices p ∗ , it must hold K K ( B , p ∗ ) − |B| ≤ for all B ⊆ L . Define O ( p ) := {A ⊆ L | K K ( A , p ) − |A| ≥ K K ( B , p ) − |B| , for all B ⊆ L} , (18)which collects the set of channels B ∈ O ( p ) that maximize K K ( B , p ) − |B| . The excess demand set Z ( p ) is the smallestelement of O ( p ) and can be calculated using Algorithm 3 bychecking whether each channel is simultaneously demandedby more than one SU. Algorithm 3 requires in worst case LK calculations. Theorem 4:
For given prices p , Algorithm 3 finds theexcess demand set Z ( p ) . Proof:
The proof is provided in Appendix E.
Algorithm 4
Implementation of Walrasian Equilibrium bymodified English Auction. Input : price incrementing factor α > Init : p (0) l = ǫ, l ∈ L ; t = 0 repeat Each SU k calculates A ( t ) k using Algorithm 2 andbroadcasts it to all SUs Each SU calculates Z ( p ( t ) ) using Algorithm 3 Each SU updates the prices as p ( t +1) = p ( t ) + α δ ( p ( t ) ) (19)with δ i ( p ( t ) ) = (cid:26) , i ∈ Z ( p ( t ) ) ; , otherwise t = t + 1 until Z ( p ( t − ) = ∅ C. Distributed Implementation of Walrasian Equilibrium
The English auction, proposed in [26] and proven to reacha Walrasian equilibrium, can be implemented by an auctioneer(coordinator) which, upon collecting the demands from all theusers, updates the prices and broadcasts them to the SUs. How-ever, since we do not assume the existence of a coordinatorin this section, we formulate a cooperative implementationbased on the SUs exchanging the channel demands betweenthemselves. Instead of updating the prices at the auctioneer,each SU can update the prices locally knowing the demandsof all SUs. In Algorithm 4, we provide an implementationof this mechanism. As in the stable matching coordination inSection IV, the SUs need only communicate the indices of thechannels they demand. Thus, given L channels, each SU needsto send L bits of information to the other SUs to reveal hisdemand. Specifically, SU k sends the L bit message Ψ k to theother SUs with [Ψ k ] l = 1 if l ∈ A k and [Ψ k ] l = 0 otherwise.Given the demand sets from all SUs, each SU calculates theexcess demand set and updates the prices by incrementing theprices of the channels in excess demand by a factor α . Notethat only in case the demand set of an SU has changed it isnecessary that the SU broadcasts this update to the other SUs.The choice of the price incrementing factor α for the pricesinfluences the speed of convergence of the algorithm. Forsufficiently small α , i.e., α → , Algorithm 4 converges tothe Walrasian equilibrium (Definition 5). For relatively large α ,some channels may not be demanded by any SUs. The reasonfor this is that for a channel which has been demanded by morethan one SU, the price update of this channel does not take intoaccount the SUs’ utilities such that a high price incrementingfactor can make the channel suddenly unattractive to all SUs.Algorithm 4 is guaranteed to converge since prices of thechannels can only be incremented and the SU utilities arefinite valued. VI. N UMERICAL R ESULTS
We assume an energy detector is used at each SU k with a detection threshold γ [ l ] k . The false alarm probability P U s u m r a t e [ b it s / s / H z ] (a) q k = 1 , ∀ k ∈ K P U s u m r a t e [ b it s / s / H z ] (b) q k = 2 , ∀ k ∈ K P U s u m r a t e [ b it/ s / H z ] stable matchingrandom matchingmaximum PU sum rateWalrasian equilibriummaximum rate boundquoted rate bound (c) q k = L, ∀ k ∈ K Fig. 2: Comparison of stable matching and Walrasian equilib-rium in the PU and SU sum rate regions for different quotasat 0 dB SNR.and probability of detection of the energy detector are re-spectively approximated by f [ l ] k = Q (cid:18) γ [ l ] k − Nσ σ √ N (cid:19) and d [ l ] k = Q γ [ l ] k − N ( σ + P l | z [ l ] k | ) q Nσ ( σ +2 P l | z [ l ] k | ) ! [37]. We fix f [ l ] k = 0 . and choose N = 20 sensing observations. By calculating the threshold γ [ l ] k , we can determine the detection probability. The probabil-ity of primary transmission of all PUs is set to ϑ [ l ] = 0 . .For simulations, we adopt the primary utility function φ l tobe the average achievable rate: φ l (1 − d [ l ] k , P k | ˜ h [ l ] k | ) = ϑ [ l ] d [ l ] k log (cid:18) P l | g [ l ] | σ (cid:19) + ϑ [ l ] (1 − d [ l ] k ) log P l | g [ l ] | σ + P k | ˜ h [ l ] k | ! , (20)and assume no specific QoS requirements of the PUs.In Fig. 2, we plot the average performance region of theprimary (y-axis) and secondary systems (x-axis), where thenumber of SUs is K = 10 and number of primary channelsis L = 20 . All channels are independently and identicallyRayleigh distributed and the simulations are averaged over random instances.In Fig. 2, the region inside the quoted rate bound includesonly the performance of the SUs and PUs in the channelsin which the SUs are assigned to. This region is included inthe region marked as maximum rate bound which includes allchannel assignment possibilities to the SUs without quota re-strictions. The quoted rate bound includes the stable matchingand Walrasian equilibrium channel assignments. We generateboth boundaries using the Hungarian optimization method [30] −10 −5 0 5 10 15 20010203040506070 signal−to−noise ratio [dB] P U s u m r a t e [ b it s / s / H z ] Fig. 3: Average sum-rate of the PUs for increasing SNR andquotas q k = 2 for all k . −20 −10 0 10 20 3001020304050 signal−to−noise ratio [dB] S U s u m r a t e [ b it s / s / H z ] stable matchingrandom matchingmaximum SU sum rate Fig. 4: Average sum-rate of the SUs for increasing SNR andquotas q k = 2 for all k .which finds the optimal channel assignment by maximizing theweighted sum rate of SUs and PUs in a centralized way, seeSection III.In Fig. 2(a), we plot the performance of stable matchingfollowing Algorithm 1 for q k = 1 , for all k ∈ K . The outcomefavours the SUs and is near the quoted boundary showingthat the sum performance of the SUs from stable matching isnear optimal. Note, that in this setting not all channels can beassigned to the SUs due to the quota restriction. In Fig. 2(b),we set q k = 2 , for all k ∈ K . In this setting, the outcomes ofstable matching do not reach the boundary but are closer toit than the random matching scheme discussed further below.Note, that our proposed quick terminating algorithm leads tothe best stable matching for the SUs but there may exist otherstable matchings. Nevertheless, our stable matching outcomeshows a fair trade-off in terms of giving an acceptable PU per-formance. In Fig. 2(b), we also plot the Walrasian equilibriumusing Algorithm 4 with price incrementing factor α = 0 . and weight λ = 0 . . By choosing λ = 0 . , we achieve inWalrasian equilibrium the maximum sum performance of thePUs and SUs in the quoted region. Note, that all points on theboundary of the quoted region can be obtained as Walrasianequilibria for different values of λ . In Fig. 2(c), the quotasare specified such that any SU can be assigned all primarychannels. Following Theorem 1, the stable matching is unique −20 −10 0 10 20 301357 signal−to−noise ratio [dB] a vg . nu m b e r o f p r opo s a l s p e r S U q = L = 20q = 2q = 1 Fig. 5: Average number of proposals from each SU to thecoordinator for different quotas and increasing SNR occurringduring Algorithm 1.and is sum-rate optimal in the quoted region.In comparison to stable matching for different quota val-ues in Fig. 2(a-c), random matching chosen to satisfy theassociated quota constraints does not show preference inperformance to neither PUs nor SUs. In our random matchingscheme, we first introduce q k -many virtual SUs with a quotaof one for each SU k ∈ K . Then, we apply random one-to-onematching achieving a number of min( L, P k ∈K q k ) matchingpairs. A. Performance of Stable Matching
In Fig. 3 and Fig. 4, the average sum rate of the PUsand SUs are plotted respectively for increasing SNR and for q k = 2 , for all k ∈ K . The number of SUs is K = 10 and number of primary channels is L = 20 . All channelsare independently and identically Rayleigh distributed andwe generate again random instances for averaging in thesimulations. In Fig. 3, it is shown that the performance lossof the PUs is very low in stable matching compared to thesetting without the operation of SUs. Hence, the coexistencewith the SUs does not lead to much performance degradationsto the PUs. In Fig. 4, the average sum rate of the SUs isshown to be always larger with Algorithm 1 than with therandom matching scheme but does show a significant gap tothe maximum possible sum rate of the SUs. Note, that themaximum SU sum rate is obtained at the point where the PUsum rate is lowest which is an unsatisfying operating point fora cognitive radio network, where primary user communicationis prioritized. Hence, the SU sum rate reached by stablematching comes at an acceptable level. Since Algorithm 1is SU-optimal, outcomes of other stable matching schemeswould perform worse for the SUs. At high SNR, the sum rateperformance of the SUs grows linearly. Note that in this SNRregime the detection probability of each SU approaches zerosince the noise is much smaller compared to the primary signalpower. Accordingly, the achievable rate of an SU is not limitedby the interference from a PU.In Fig. 5, the complexity of Algorithm 1 is revealed bycounting the number of matching proposals per secondary userover SNR in the simulation scenario described above. Theaverage number of SU proposals increases for larger quotas a vg . nu m b e r o f p r opo s a l s p e r S U q = 1q = 2q = L Fig. 6: Average number of proposals by the SUs duringAlgorithm 1 at dB SNR. The number of PUs is double thenumber of SUs, and the quotas of the SUs are equal to q . −3 −2 −1 α a vg . nu m b e r o f d e m a nd s / b i d s p e r S U English auctionDistributed auction
Fig. 7: Comparison of average number of demands per userduring Algorithm 4 and the average number of bids from thedistributed auction algorithm in [24] for different values of α with L = K = 10 and q k = 1 for all k . SNR is 0 dB. q k = q , for all k ∈ K . For q k = 1 , for all k ∈ K , where the SUsrequire only one channel, the stable matching algorithm tendsto match each SU with its first preference of the channels. Onlyfew re-matchings occur. Hence, a very low average number ofSU proposals (slightly above one) is needed for terminatingthe protocol. Re-matchings occur when a user, engaged to achannel, is released due to a proposal from another user whichgives the channel a higher utility. For large q k requirements,frequent re-matchings lead to a much higher proposal numberin the presented stable matching algorithm. Generally however,it is shown that only a few number of proposals are requiredto reach a stable matching. In Fig. 6, the average number ofproposals during Algorithm 1 is plotted for increasing numberof SUs. The number of PU channels is set to be doublethe number of SUs. It is shown that the average number ofproposals during the stable matching algorithm increases withincreasing quotas. B. Performance of English Auction
In Fig. 7 and Fig. 8, we compare the average number ofdemands/bids and the performance loss from optimal channelassignment by Hungarian method approaches, respectively,with respect to the price incrementing factor α used inAlgorithm 4 and also required in the distributed auctionmechanism in [24, Section IV]. Note that in [24, Section V], −3 −2 −1 α p e rf o r m a n ce l o ss [ b it s / s / H z ] English auctionDistributed auction
Fig. 8: Comparison of performance loss in outcome of Al-gorithm 4 and the distributed auction algorithm in [24] fordifferent values of α with L = K = 10 and q k = 1 for all k .SNR is 0 dB. −20 −10 0 10 20 300200400600800 signal−to−noise ratio [dB] a vg . nu m b e r o f d e m a nd s / b i d s p e r S U English auctionDistributed auction
Fig. 9: Comparison of average number of demands per userrequired for the auction algorithms for different SNR valueswith α = 0 . and L = K = 10 and q k = 1 for all k .an implementation of the distributed auction mechanism isprovided using opportunistic CSMA which needs no exchangeof information between the users. In order to compare theperformance to the algorithm in [24] which assigns a singlechannel per user, we set q k = 1 , for all k ∈ K and choose L = K = 10 . We average the performance at SNR = 0 dBover channel realizations. In Fig. 7, it can be seen that theauction algorithm in [24] requires on average larger numbersof bids per SU compared to the number of demands per user inthe English auction. However, according to Fig. 8, the Englishauction is shown to be more sensitive over the choice of α . Alarger α increases the convergence rate of the English auction,but leads to high performance loss. The low sensitivity of thedistributed auction in [24] to the price incrementing factor isdue to the fact that this auction method makes use of the pricesof both the best and second best object a user would demandto determine his bid. This cannot be exploited in the Englishauction since the demand of a user is a set of channels.In Fig. 9 and Fig. 10, we plot the average number ofdemands and the SU sum rate for different SNR values. Thechosen price increment factor for the auction algorithms is α = 0 . . It can be seen that the distributed auction requiresmore average number of bids for larger SNR values than theEnglish auction requires number of demands. Both algorithms −20 −10 0 10 20 30051015202530 signal−to−noise ratio [dB] S U s u m r a t e [ b it s / s / H z ] English auctionDistributed auctionCentralized optimal assignment
Fig. 10: Comparison of average SU sum rate for different SNRvalues. The price incrementing factor α is . and L = K =10 and q k = 1 for all k .achieve very close performance to the optimum as is shownin Fig. 10. The optimum is reached centralized by means ofthe Hungarian method. Note that the performance loss of bothalgorithms is due to the non-infinitesimal chosen value of theprice incrementing factor.VII. C ONCLUSION
We considered the problem of assigning primary channelsto SUs for communication in a cognitive radio. For thisproblem, we proposed two solution concepts, stable matchingand Walrasian equilibrium, and provided coordination andcooperation algorithms to reach them in a distributed manner.Both concepts lead to different performances for the secondaryand primary users. We relate both solutions utilizing theachievable sum performance regions. While stable matchingrelies on stability of the assigned secondary and primaryusers, the Walrasian equilibrium maximizes the weighted sumutilities. In contrast to the stable matching framework, pricesare required in the competitive market model in order to definethe Walrasian equilibrium.The complexity of SU-optimal stable matching algorithmsin terms of the average number of proposals is shown tobe very low by extensive simulation. The complexity of theEnglish auction to reach a Walrasian equilibrium depends,however, on the choice of the price incrementing parameter.Future works may devise a mechanism which adapts the priceincrementing parameter intelligently.A
PPENDIX
A. Proof of Theorem 1
The proof is based on the result in [38] (also given in[7, Theorem 5.13]) which we restate here in relation to ourcognitive radio setting.
Theorem 5 ( [38]):
Let k be a secondary user with quota q k , and let M be a stable matching such that | M ( k ) ∩ L| < q k .Then for any stable matching M ′ , M ′ ( k ) = M ( k ) .Thus, if a stable matching does not strictly satisfy the quotaof a secondary user with equality, then this user is assignedthe same set of primary channels in any other stable matching.Since q k = L for all k ∈ K , and the existence of at least one stable matching is guaranteed, then we can do the followingtwo-case study: 1) If | M ( k ) | < q k for all k ∈ K , then thestable matching is unique Theorem 5 holds for all users. 2) If | M ( k ) | = q k for some user k ∈ K then | M ( j ) | = 0 for all j = k . Following Theorem 5 all users j = k are unmatchedin all other stable matchings. What is left is to considerstable matchings in which the user k is assigned strictly lesschannels than his quota. However, since a matching M ′ , with | M ′ ( k ) | < q k is not individually rational for user k accordingto Definition 2, then M ′ is not a stable matching. Hence, M is unique.Considering Algorithm 1, each SU k proposes to everychannel since his quota q k ≥ L . The coordinator then acceptsthe proposal of an SU k in a channel l if he gives thehighest performance for PU l . Accordingly, the unique stablematching is sum performance optimal for the PUs within theperformance region e R pu defined in (9). B. Proof of Theorem 2
We have to show that the utility function in (13) satisfiesthe monotonicity property and the gross substitutes condition.The monotonicity property is obviously satisfied because ifadditional channels are provided to an SU k , the utility U k in (13) does not decrease. In order to prove that the utilityfunction in (13) satisfies the gross substitutes condition, wemust first prove that φ k ( B ) = λu su-sum k ( B ) + (1 − λ ) X l ∈B u pu l ( k ) , (21)satisfies the gross substitute property. Afterwards, the op-eration in (13) on φ k ( B ) is called the q k - satiation of φ k and preserves the gross substitutes property according to [25,Section 2]. The function φ k ( B ) satisfies the gross substitutesproperty because it is additively separable [25, Section 2], i.e., φ k ( B ) in (21) can be expressed as φ k ( B ) = P l ∈B φ k ( { l } ) . C. Proof of Lemma 1
The proof is by contradiction. Given p l > for all l ∈ L ,assume for some SU k a demand set A ∈ D k ( p ) satisfies |A| > q k . Then, according to his utility function in (13), a setof channels R ⊂ A with |R| ≥ |A| − q k give no additionalperformance to SU k since the maximization in (13) is overat most q k channels and we can write U k ( A ) = U k ( A \ R ) .The net utility of SU k in (14) satisfies v k ( A , p ) = U k ( A ) − X l ∈A p l (22a) = U k ( A \ R ) − X l ∈A\R p l − X l ′ ∈R p l ′ (22b) = v k ( A \ R , p ) − X l ′ ∈R p l ′ . (22c)Hence, v k ( A , p ) < v k ( A \ R , p ) which contradicts that A isa demand for SU k . D. Proof of Theorem 3
The consumer utility function in (13) is proven in Theorem2 to satisfy the gross substitutes condition. We will use anequivalent property to the gross substitutes condition calledthe single improvement property [25] in the proof.
Definition 6:
The utility function U k ( A ) satisfies the singleimprovement property if for any price vector p and set ofchannels A 6 = D k ( p ) , i.e., A is not a demand set for SU k ,there exists another set B such that v k ( A , p ) < v k ( B , p ) witheither |A \ B| ≤ , or |B \ A| ≤ .Let A k be the output of Algorithm 2 and assume A k is not a demand set, i.e., A k / ∈ D k ( p ) . Then in order tostrictly improve v k ( A k , p ) we can, according to the singleimprovement property in Definition 6, do either of threepossibilities:(i) adding an element to A k , i.e., find a ∈ L \ A k s.t. v k ( A k ∪ { a } , p ) > v k ( A k , p ) (ii) removing one element from A k , i.e., find b ∈ L \ A k s.t. v k ( A k \ { b } , p ) > v k ( A k , p ) (iii) or do both (i) and (ii).Case (i): If |A k | = q k , then adding an element a ∈ L \ A k requires removing another element b ∈ A k following Lemma1. Since the elements in A k are the first q k best channels, thenremoving one element to add the element a does not lead tostrict performance improvement having that the net utilitiesin the resources are distinct according to Assumption 1. If |A k | < q k , then the net utility with channel a ∈ L \ A k isnonpositive, i.e., v k ( a, p a ) ≤ because otherwise it wouldbe a member of A k from Algorithm 2. Hence, no strictperformance improvement can be made by adding an elementto A k .Case (ii): Since v k ( b, p b ) > for all b ∈ A k , then removingone element from A k leads to strict performance degradation.Since both (i) and (ii) lead to no strict performance im-provement, then (iii) cannot be satisfied. Accordingly, A k is ademand set, i.e., belongs to the set D k ( p ) . From (ii), the A k is the smallest demand set in D k ( p ) and is contained in otherdemand sets in D k ( p ) . Specifically, another demand set canbe constructed from A k when |A k | < q k by adding a resource l ∈ L \ A k to A k which satisfies v k ( { l } , p ) = 0 . E. Proof of Theorem 4
Following Theorem 3, the demand is the smallest elementin D k ( p ) in (15). Then, for an SU k , the intersection of hisdemand set A k from Algorithm 2 with the aggregate excessdemand Z from Algorithm 3 is smallest compared to otherdemand sets in D k ( p ) .First, we prove that Z belongs to the set O ( p ) in (18). Thatis, we must prove:(i) K K ( Z , p ) − |Z| = K K ( Z ∪ { a } , p ) − |Z ∪ { a }| , a ∈ L \ Z (ii) K K ( Z , p ) −|Z| > K K ( Z\{ b } , p ) −|Z \ { b }| , b ∈ Z .In order to prove (i), let the element a ∈ L\Z be demanded bySU j only, i.e., a ∈ A j . Note, that if channel a is demandedby more than two SUs, then it would belong to the set Z according to Algorithm 3. Then we can write K K ( Z ∪ { a } , p ) − |Z ∪ { a }| (23a) = X Kk =1 |A k ∩ Z ∪ { a }| − |Z ∪ { a }| (23b) = X Kk =1 ,k = j |A k ∩ Z| + |A j ∩ Z ∪ { a }| − |Z| − (23c) = X Kk =1 ,k = j |A k ∩ Z| + |A j ∩ Z| + 1 − |Z| − (23d) = X Kk =1 |A k ∩ Z| − |Z| . (23e)In order to prove (ii), let b ∈ Z be demanded by SUs j and ℓ , i.e., b ∈ A j and b ∈ A ℓ . Then, K K ( Z , p ) − |Z| = K X k =1 |A k ∩ Z| − |Z| (24a) = K X k =1 k / ∈{ j,ℓ } |A k ∩ Z| + |A j ∩ Z| + |A ℓ ∩ Z| − |Z| (24b) = X Kk =1 ,k / ∈{ j,ℓ } |A k ∩ Z \ { b }| + |A j ∩ Z \ { b }| + |A ℓ ∩ Z \ { b }| + 2 − |Z \ { b }| − (24c) = X Kk =1 |A k ∩ Z \ { b }| − |Z \ { b }| + 1 (24d) > K K ( Z \ { b } , p ) − |Z \ { b }| . (24e)Thus, Z ∈ O ( p ) and from (ii), Z is the smallest element in O ( p ) . R EFERENCES[1] S. Haykin, “Cognitive radio: brain-empowered wireless communica-tions,”
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