Dynamic Decentralized Algorithms for Cognitive Radio Relay Networks
Siavash Bayat, Raymond H.Y. Louie, Branka Vucetic, Yonghui Li
DDynamic Decentralized Algorithms for CognitiveRadio Relay Networks
Siavash Bayat, Raymond H. Y. Louie, Branka Vucetic, Yonghui Li
Centre of Excellence in Telecommunications, School of Electrical and Information Engineering, University of Sydney,Australia
Abstract
We propose a distributed spectrum access algorithm for cognitive radio relay networks with multipleprimary users (PU) and multiple secondary users (SU). The key idea behind the proposed algorithm is thatthe PUs negotiate with the SUs on both the amount of monetary compensation, and the amount of time theSUs are either (i) allowed spectrum access, or (ii) cooperatively relaying the PU’s data, such that both thePUs’ and the SUs’ minimum rate requirement are satisfied. The proposed algorithm is shown to be flexible inprioritizing either the primary or the secondary users. We prove that the proposed algorithm will result in thebest possible stable matching and is weak Pareto optimal. Numerical analysis also reveal that the distributedalgorithm can achieve a performance comparable to an optimal centralized solution, but with significantlyless overhead and complexity.
Index Terms
Cognitive radio, stable matchings, cooperative relaying, overlay model.
I. I
NTRODUCTION
Cognitive radio has been proposed as a promising technology to improve the spectral efficiencyof wireless networks. This is achieved by allowing unlicensed secondary users (SU) to coexist withlicensed primary users (PU) in the same spectrum. This coexistence is facilitated by spectrum accesstechniques, such as those involving an agreement between the PUs and SUs on an acceptable spectrumaccess strategy. The key idea is that the PUs are motivated to lease spectrum bands to the SUs inexchange for some form of compensation.Monetary compensation has been well studied (see e.g., [1–8]), with the predominant approach forspectrum access and performance analysis involving the use of tools from game theory. For example,[7–9] considered a non-cooperative game between the PUs and SUs, while [4, 6] considered a two-stage leader-follower game. For these monetary payment schemes, the PUs are assumed to have sufficient spectrum for leasing to the SUs, such that their own performance requirements are notaffected. Authors in [10] also have considered bilateral bargaining among PUs and SUs. In practice,however, the PUs may desire higher data rates than what its current spectrum can provide.To allow for higher data rates, the use of cooperative relaying has emerged as a powerful techniquedue to its ability to exploit user diversity and provide high reliability and capacity in wireless networks[11]. This is achieved by the use of intermediate relay nodes to aid transmission between the sourceand destination nodes. The use of cooperative relaying is particularly advantageous when the directlink between the source and destination is weak, due to, for example, high shadowing.In this paper, we consider a model where the SUs act as cooperative relays to assist the PUs’transmission in exchange for both spectrum access and monetary compensation, and thus the SUsare effectively providing both a monetary and performance compensation to the PUs. When onlyperformance compensation is considered, this is commonly referred to as the overlay model, andvarious schemes have been proposed [12–16]. However, these schemes considered the scenario wherethe increase in PU’s performance does not necessarily translate into a satisfactory performance for theSUs, and in some cases, the SUs have limited spectrum access opportunities if the PUs have regulardata to transmit [12, 17, 18]. This issue was addressed in [19, 20], where a scheme was proposed whichincreased the PU’s performance while simultaneously satisfying the SU’s requirements. However,these papers [12–14, 19–21] considered a simplified scenario with only one
PU, and did not considermonetary compensation. Besides, some works also considered a simplified scenario with multiplePUs and only one SU.In this paper, we propose a spectrum access strategy for a more general cognitive radio relaynetwork with multiple competitive
PUs and multiple competitive
SUs under the overlay model, whichguarantees a minimum rate requirement for all matched PUs and SUs. We define a PU and SU as matched if the SU cooperatively relays the PU’s data, in exchange for spectrum access and monetarycompensation.The majority of current game theory techniques in cognitive radio, which involve multiple PUsand multiple SUs focus on a framework where only users of the same type are the primary decisionmakers, i.e., either the PUs or the SUs are involved in the game to determine access to the spec-trum resources [22]. However, in this approach, the decision makers do not take into account theperformance metric of the non-decision users, and thus will lead to unacceptably low performance for these users. To obtain better performance, it is thus desirable for the spectrum access strategy toget all users involved in the decision making process, and thus the interactions between the PUs andSUs should be taken into account, along with the varying performance requirements of these users.Moreover, what is not actually captured by the normal and basic game and auction models is thefact that different spectrum resources and relaying services that are offered by PUs and SUs causesdifferent performance levels for the SUs and PUs respectively due to random phenomena such as:fading, noise, and shadowing. More specifically, each PU and each SU supplies a spectrum andrelaying service with a specific and unique rate performance for a SU and a PU respectively. Thus tomodel such a complex interaction between multiple spectrum suppliers and multiple relaying servicesuppliers, a more flexible tool is required. Current algorithms in cognitive radio literature addressingthis issue have thus far focused on centralized-approaches, e.g., the double-auction approach in [23].This approach is clearly not desirable in practice due to the significant amount of overhead requiredin centralized coordination.In this paper, we develop a new distributed spectrum access framework based on auction and matching theory . The proposed distributed algorithms will match the PUs with the SUs based ona preference list, which is unique to each user, and based on both monetary and performancecompensations. Matching users belonging to two different groups (e.g., a PU group and a SU group)based on preference lists has its roots in the dynamic matching theory [24, 25], which was used tomatch users based on a preference list, and recently applied to wireless resource allocation problems[26, 27]. However, the algorithms based on the classic matching theory did not consider the possibilityof dynamic negotiation between the PUs and the SUs with time-varying requirements, which isdesirable for users in next generation wireless access networks.To address this issue, we propose a novel dynamic negotiation algorithm based on multi-itemauction theory [25, 28] and matching theory that requires an intelligent negotiation mechanism, wherethe PUs and the SUs choose between either prioritizing monetary or performance compensation. Ouralgorithm determines the matched pairings between PUs and SUs, such that the SU will providemonetary compensation, and relay its paired PU’s data in exchange for spectrum access. The key ideabehind the algorithm is that the PUs negotiate with the SUs on the amount of monetary compensation,in addition to the time the SUs are either (i) allowed access to the spectrum, or (ii) cooperativelyrelaying the PU’s data, such that both the PUs’ and the SUs’ minimum rate requirement are satisfied.
We then analyze the performance of the proposed algorithm, showing that it results in the best pos-sible stable matching , and is weak Pareto optimal . We introduce a utility function, which incorporatesboth the rate and monetary factors. We demonstrate through numerical analysis that the algorithmcan achieve utilities (i) comparable to the utilities achieved by an optimal centralized algorithm, and(ii) significantly greater than the utilities achieved by a random matching algorithm that is mixedwith a basic negotiation, while also being able to accomplish a high number of matchings with lowoverhead and complexity.When only the rate is important, and monetary compensation is not a priority, we show that ouralgorithm is flexible in terms of prioritizing either the PUs or SUs, by a simple manipulation of globalparameter values. This is in contrast to [29], in which this design flexibility was not achievable. Notethat considering this extra monetary parameter is a significant extension to the algorithm presentedin [29], as considerable changes to the algorithm are required. Moreover, in contrast to [29], wealso present additional analyses including (i) proving that our algorithm results in the best stablematching over all possible stable matchings, (ii) providing an explicit upper bound expression for thenumber of iterations required for convergence of the proposed algorithm, (iii) presenting an explicitexpression for the overhead and (iv) analyzing the complexity performance of our proposed schemecompared to a baseline centralized scheme. Finally, we show that the PUs, which utilize the SUs forcooperative relaying achieve a rate greater than what it would achieve without cooperative relaying,i.e, direct transmission, and thus motivates their participation in the proposed algorithm.This paper is organized as follows. In Section II, we first describe our system model. We thenformulate the optimization problem we are trying to solve in Section III, and present a distributedsolution to this problem in Section IV. Finally, we analyze the performance and the implementationaspects of our proposed algorithm in Section V. For convenience, Table I provides a description ofsome of the parameter values we will be utilizing in this paper.II. S
YSTEM M ODEL
We consider an overlay cognitive radio wireless network, comprising of L PU PU transmitter { PT i } L PU i =1 –PU receiver { PR i } L PU i =1 pairs, with the (cid:2) th pair having a rate requirement of R PU (cid:2) , req , andwith each pair occupying a unique spectrum band of constant size. In the same network, there are L SU SU transmitter { ST i } L SU i =1 –SU receiver { SR i } L SU i =1 pairs, with the q th pair having a rate requirementof R SU q , req , and seeking to obtain access to one spectrum band occupied by a (PT , PR) pair. We
Fig. 1. Secondary user and primary user spectrum-access model. The channel and price and time-slot allocation numbers are indicatedfor (PT (cid:2) , PR (cid:2) ) and (ST q , SR q ) . assume that there are T time-slots per transmission frame, and each (ST , SR) pair has access to amonetary value C .Each PT attempts to grant spectrum access to a unique (ST , SR) pair, as determined by the variousmatching algorithms, described in Sections III and IV, in exchange for (i) the ST cooperativelyrelaying the PT ’s data to the corresponding PR , and (ii) monetary compensation. In particular,without loss of generality (w.l.o.g), let us consider (PT (cid:2) , PR (cid:2) ) , whose transmission is relayed by ST q during a fraction β (cid:2),q ( ≤ β (cid:2),q ≤ ) of T , whilst also receiving a fraction ξ (cid:2),q ( ≤ ξ (cid:2),q ≤ ) of C from ST q , as depicted in Figure 1. We will refer to ξ (cid:2),q and β (cid:2),q as the price and time-slot allocationnumbers, respectively, whose exact values will be determined by the matching algorithms describedin Sections III and IV.During the cooperative relaying stage in the initial β (cid:2),q T time-slots, a fraction τ (cid:2),q ( <τ (cid:2),q < ) isfirst allocated for PT (cid:2) to broadcast its signal to ST q and PR (cid:2) , thus occurring in the first β (cid:2),q τ (cid:2),q T time-slots. In the subsequent β (cid:2),q (1 − τ (cid:2),q ) T time-slots, ST q cooperatively relays the signal from PT (cid:2) to PR (cid:2) . PR (cid:2) and then applies maximum ratio combining (MRC) to the signal received from PT (cid:2) in the first β (cid:2),q τ (cid:2),q T time-slots, and the signal received from ST q in the subsequent β (cid:2),q (1 − τ (cid:2),q ) T time-slots. After this cooperative relaying stage, PT (cid:2) ceases transmission, allowing ST q to transmitto SR q over the spectrum occupied by (PT (cid:2) , PR (cid:2) ) in the final (1 − β (cid:2),q ) T time-slots.In this paper, we consider the amplify-and-forward (AF) relaying protocol, due to its simple andpractical operation, and thus set τ (cid:2),q = . We note, however, that the proposed algorithm is applicableto any relaying protocol, such as the decode-and-forward or compress-and-forward protocol. The AFgain at ST q is chosen such that its instantaneous transmission power is constrained to P SU q . A. Utility Functions
To evaluate the performance of each ( PT , PR ) and ( ST , SR ) pair, we consider the utility function,which comprises of both rate and monetary factors. The utility function is a concept, which showsthe level of satisfaction of a user by combining all parameters to a single number. It represents thenet losses and gains of a user [30]. These parameters can be of different types, and by using suitableweights can be combined together.Utility functions have been widely used in wireless literature to solve various radio resourcemanagement problems [31–33]. These papers include combining different parameters, such as price,delay, signal-to-interference and noise ratio, power, rate, error performance, into a single number.Specifically in cognitive radio, the utility function have been used in [1, 4, 9, 34–36].For (PT (cid:2) , PR (cid:2) ) , the achievable instantaneous rate is given by [11] R PU (cid:2),q ( β (cid:2),q )= β (cid:2),q T (cid:2) DirPR (cid:2) + Γ
RelayPR (cid:2),q (cid:3) (1)where Γ DirPR (cid:2) = γ PT (cid:2) || h PT (cid:2) , PR (cid:2) || d α PT (cid:2) , PR (cid:2) (2)is the receive signal-to-noise-ratio (SNR) at PR (cid:2) of the direct signal from PT (cid:2) , Γ RelayPR (cid:2),q = Γ PT (cid:2) − ST q Γ ST q − PR (cid:2) Γ PT (cid:2) − ST q Γ ST q − PR (cid:2) + 1 (3)is the equivalent receive SNR of the relayed signal at PR (cid:2) from ST q , Γ PT (cid:2) − ST q = γ PT (cid:2) || h PT (cid:2), ST q || d α PT (cid:2), ST q , Γ ST q − PR (cid:2) = γ ST q || h ST q, PR (cid:2) || d α ST q, PR (cid:2) , γ PT (cid:2) = P PT (cid:2) σ is the transmit SNR for PT (cid:2) , γ ST q = P ST q σ is the transmit SNRfor ST q , σ is the noise variance, P PT (cid:2) is the transmission power at PT (cid:2) , h PT (cid:2) , PR (cid:2) and d PT (cid:2) , PR (cid:2) denoterespectively the channel and distance from PT (cid:2) to PR (cid:2) , h PT (cid:2) , ST q and d PT (cid:2) , ST q are respectively the channel and distance from PT (cid:2) to ST q , h ST q , PR (cid:2) and d ST q , PR (cid:2) are respectively the channel and distancefrom ST q to PR (cid:2) and α is the path loss exponent. We also consider a slowly varying block fadingchannel model with a sufficiently long coherence time that is constant over the transmission frame T .Similar to the utility functions defined in [9, 12, 34–36] we define U PU (cid:2),q ( β (cid:2),q , ξ (cid:2),q )= R PU (cid:2),q ( β (cid:2),q ) + ¯ cξ (cid:2),q C (4)where ¯ c ∈ R + , is a variable with unit defined as: rate per unit monetary value.For (ST q , SR q ) , the achievable instantaneous rate is given by R SU q,(cid:2) ( β (cid:2),q )=(1 − β (cid:2),q ) T log (cid:4) SR q,(cid:2) (cid:5) (5)where Γ SR q,(cid:2) = γ ST q || h ST q , SR q ,(cid:2) || d α ST q , SR q (6)is the receive SNR at SR q of the direct transmission signal from ST q , h ST q , SR q ,(cid:2) is the channelcoefficient from ST q to SR q in the spectrum band occupied by (PT (cid:2) , PR (cid:2) ) and d ST (cid:2) , SR q is thedistance from ST q to SR q . The utility for (ST q , SR q ) is thus given by U SU q,(cid:2) ( β (cid:2),q , ξ (cid:2),q )= R SU q,(cid:2) ( β (cid:2),q ) − ¯ kξ (cid:2),q C, (7)where ¯ k ∈ R + is a variable with unit defined as: rate per unit monetary value. We consider a generalcase where ¯ c and ¯ k have the same units, but their values may be different. This is to allow forflexibility in the algorithm design, as will be discussed in Section V.III. P ROBLEM F ORMULATION
In this section, we describe the optimization problem we aim to address. To proceed, we in-troduce some notations. We first define the primary and secondary user sets, respectively, as P = { PU (cid:2) =(PT (cid:2) , PR (cid:2) ) } L PU (cid:2) =1 and S = { SU q =(ST q , SR q ) } L SU q =1 . Moreover, we define a L PU × L SU matchingmatrix M , with m i,j =1 if PU i is matched with SU j , and m i,j =0 otherwise, where the notation m i,j denotes the ( i, j ) th entry of matrix M . From this matrix, we introduce an injective function μ :( P ∪ S ) → ( P ∪ S ∪ { ∅ } ) , such that(a) μ (PU (cid:2) ) ∈ ( S ∪ { ∅ } ) , TABLE IP
ARAMETER D ESCRIPTIONS
Notation Description ξ (cid:2),q Price allocation number β (cid:2),q Time-slot allocation number T Length of transmission frame C Total money each SU has access to in each trans-mission frame τ (cid:2),q Time-slot fraction that PT (cid:2) transmits its signal to ST q α Path loss exponent Γ DirPR (cid:2)
Receive SNR at PR (cid:2) of the direct transmissionsignal from PT (cid:2) Γ RelayPR (cid:2),q
Equivalent receive SNR of the relayed signal at PR (cid:2) from ST q Γ SR q,(cid:2) Receive SNR at SR q of the direct transmissionsignal from ST q γ PT (cid:2) Transmit signal-to-noise-ratio for PT (cid:2) γ ST q Transmit signal-to-noise-ratio for ST q ¯ c Rate per unit monetary value for
PT¯ k Rate per unit monetary value for ST U PU (cid:2),q PU (cid:2) ’s utility while cooperating with ST q U SU q,(cid:2) SU q ’s utility while cooperating with PU (cid:2) M Matching matrix B Time-slot allocation matrix G Price allocation matrix
PULIST (cid:2) PT (cid:2) ’s preference list SULIST q ST q ’s preference list δ PT’s price step number (cid:10)
PT’s time-slot step number (b) μ (SU q ) ∈ ( P ∪ { ∅ } ) ,(c) μ (SU q )=PU (cid:2) and μ (PU (cid:2) )=SU q if m (cid:2),q =1 , for (cid:2) =1 , . . . , L PU and q =1 , . . . , L SU ,(d) μ (SU q )= ∅ if m (cid:2),q =0 , for (cid:2) =1 , . . . , L PU ,(e) μ (PU (cid:2) )= ∅ if m (cid:2),q =0 , for q =1 , . . . , L SU .In the above definition, (a) implies that a PU is matched to a single SU or no PU, i.e., μ (PU (cid:2) )= ∅ ,(b) implies that a SU is matched to a single PU or no source, i.e., μ (SU q )= ∅ , (c) implies that if PU (cid:2) is matched to SU q , then SU q is also matched to PU (cid:2) and m (cid:2),q =1 , (d) implies that if m (cid:2),q =0 then SU q is not matched and (e) implies that if m (cid:2),q =0 then PU (cid:2) is not matched .We also define an L PU × L SU price allocation matrix G with g i,j = ξ i,j , and an L PU × L SU time-slotallocation matrix B with b i,j = β i,j , and where g i,j = b i,j =0 if m i,j =0 . We denote the price and time-slot allocation matrices with continuous elements as G cont and B cont respectively. Mathematically,this implies that the elements of G cont and B cont , respectively take values from the sets { g cont i,j = ξ i,j ∈ R :0 ≤ ξ i,j ≤ } and { b cont i,j = β i,j ∈ R :0 ≤ β i,j ≤ } .Now the main goal for each primary and secondary user is to ensure their minimum rate require-ments are satisfied. When this is achieved, the secondary goal is to maximize their utility functions.Note that the secondary goals for the primary and secondary users cannot be achieved simultaneously,as a higher utility for the primary user will result in a lower utility for the matched secondary user, and vice-versa. It is natural, and often considered in literature (see e.g. [34]), to give preference tothe primary users, i.e. focus on maximizing the primary users’ utility. As such, we now present theoptimization problem: { M opt , B opt , G opt } =arg max { M , B cont , G cont } L PU (cid:6) (cid:2) =1 L SU (cid:6) q =1 m (cid:2),q U PU (cid:2),q ( ξ (cid:2),q , β (cid:2),q )s . t . :( a ) R PU (cid:2),μ † ( (cid:2) ) ( β (cid:2),μ † ( (cid:2) ) ) (cid:2) R PU (cid:2), req , ∀ (cid:2) ∈{ , ,...,L PU } ( b ) R SU q,μ † ( q ) ( β μ † ( q ) ,q ) (cid:2) R SU q, req , ∀ q ∈{ , ,...,L SU } ( c ) R SU q,μ † ( q ) ( β μ † ( q ) ,q ) − ξ μ † ( q ) ,q ¯ kC (cid:2) , (8) ∀ q ∈{ , ,...,L SU } ( d ) (cid:2) L PU (cid:2) =1 m (cid:2),q ≤ , ∀ q ∈{ , ,...,L SU } ( e ) (cid:2) L SU q =1 m (cid:2),q ≤ , ∀ (cid:2) ∈{ , ,...,L PU } ( f ) (cid:3) ξ (cid:2),q (cid:3) , ∀ q ∈{ , ,...,L SU } , ∀ (cid:2) ∈{ , ,...,L PU } ( g ) (cid:3) β (cid:2),q (cid:3) , ∀ q ∈{ , ,...,L SU } , ∀ (cid:2) ∈{ , ,...,L PU } where μ † ( p (cid:2) )= q if μ ( p (cid:2) )= s q , and μ † ( s q )= (cid:2) if μ ( s q )= p (cid:2) .Conditions ( a ) and ( b ) ensure that the minimum required rate for the PUs and SUs are satisfied ,respectively. Condition ( c ) ensures that the SUs always receive a positive utility. Conditions ( d ) and ( e ) respectively ensure that each PU will only be matched with one SU, and vice-versa. Finally,conditions ( f ) and ( g ) , respectively, ensure that the price and time-slot allocation values are keptwithin their bounds.In practice, a centralized controller is required to solve the optimization problem in (8). However,there are three key issues regarding this approach: • Overhead:
The centralized controller will require the feedback on channel conditions and mini-mum rate requirements from each primary and secondary user. Moreover, after the optimizationproblem is solved, the resultant matching between PUs and SUs and price and time-slot allocation From the output of the optimization problem, it can be shown that the final time slot allocation numbers are chosen such that theachievable rate for the matched SUs is equivalent to their minimum rate requirement, and the price-time slot allocation numbers arechosen such that the utility for each matched SU is zero. We can thus present an alternate formulation accordingly, however, we leavethe formulation as described to make it clear that the SU’s rate and utility requirements are satisfied. numbers will then have to be transmitted to the corresponding users. The amount of overheadrequired for this increases with the number of users, and can be quite high, rendering itimpractical. • Complexity:
The optimization problem is non-linear, and requires an exhaustive search over allpossible matching, price and time-slot allocation combinations. Such a problem is known to beNP-hard [5]. • Selfish Users:
We assume all the primary and secondary users are selfish , which means theirgoal is to always maximize their own utilities, then the outcome of the optimization problemmay not be in the best interests of at least one of these users. Selfishness is considered as aninherent behavior of distributed and intelligent users. There are also privacy issues for which acentralized approach may not be ideal.To address these issues, we propose a distributed low-complexity algorithm, which accounts forselfish users. As we will demonstrate in Section V, our algorithm can achieve a performance closeto the solution of the optimization problem in (8) for practical system parameters.IV. P ROPOSED D ISTRIBUTED M ATCHING A LGORITHM
In this section, we describe the proposed algorithm, which determines spectrum access for each (PT , PR) and (ST , SR) pair.
A. Received SNR Assumptions
We first describe two scenarios that will be considered in the proposed algorithm, characterizedby different assumptions on the received SNR at the transmitters and receivers.
1) Complete Received SNR:
In the first scenario, PT (cid:2) has perfect knowledge of the instantaneousreceived SNRs in Γ DirPR (cid:2) and (cid:7) Γ PT (cid:2) − ST q , Γ ST q − PR (cid:2) (cid:8) L SU q =1 . Moreover, ST q has perfect knowledge of theinstantaneous received SNRs in the expressions (cid:7) Γ SR q,(cid:2) (cid:8) L PU (cid:2) =1 . As such, PT (cid:2) and ST q are able torespectively calculate their instantaneous rates in (1) and (5).
2) Partial Received SNR:
In the second scenario, PT (cid:2) has knowledge of the average received SNRsin the term (cid:9) γ ST q d α ST q, PR (cid:2) (cid:10) L SU q =1 and the instantaneous received SNRs in the terms Γ DirPR (cid:2) and (cid:7) Γ PT (cid:2) − ST q (cid:8) L SU q =1 .Moreover, ST q has perfect knowledge of the instantaneous received SNRs in the term (cid:7) Γ SR q,(cid:2) (cid:8) L PU (cid:2) =1 .As such, PT (cid:2) is able to calculate its instantaneous conditional rate, given by the expectation of therate in (1), with respect to { h PT (cid:2) , ST q } L SU q =1 , while ST q is able to calculate its instantaneous rate in (5). Selfish users are a common assumption in cognitive radio literature [30]. For both complete and partial received SNR scenarios, note that each PT and ST does not haveknowledge of the instantaneous received SNRs corresponding respectively to the other PT s and ST s.Moreover, the instantaneous received SNRs can be obtained through standard channel estimationtechniques. B. Users Preference Lists
Each PT has a preference list of ST s which can cooperatively relay the PT ’s message such thatit obtains a rate greater than its minimum rate requirement. In particular, the preference list for PT (cid:2) is given by PULIST (cid:2) = { (ST φ (cid:2) ( j ) , SR φ (cid:2) ( j ) ) } K (cid:2) j =1 (9)where φ (cid:2) ( · ) is a function φ (cid:2) : { , . . . , K (cid:2) }→{ , . . . , L SU } satisfying (cid:11) R PU (cid:2),φ(cid:2) ( q ) ( ξ (cid:2),φ (cid:2) ( q ) , β (cid:2),φ (cid:2) ( q ) ) (cid:2) R PU (cid:2) , req (cid:12) K (cid:2) q =1 (10)for the complete received SNR scenario, and the conditions (cid:11) E h ST φ(cid:2) ( q ) , PR φ(cid:2) ( q ) (cid:13) R PU (cid:2),φ(cid:2) ( q ) ( ξ (cid:2),φ (cid:2) ( q ) , β (cid:2),φ (cid:2) ( q ) ) (cid:14) (cid:2) R PU (cid:2) , req (cid:12) K (cid:2) q =1 (11)for the partial received SNR scenario. The function φ (cid:2) ( · ) also satisfies the ordering U PU (cid:2),φ(cid:2) (1) ( ξ (cid:2),φ (cid:2) (1) , β (cid:2),φ (cid:2) (1) ) >. . .>U PU (cid:2),φ(cid:2) ( K(cid:2) ) ( ξ (cid:2),φ (cid:2) ( K (cid:2) ) , β (cid:2),φ (cid:2) ( K (cid:2) ) ) , implying that the first ST in the listprovides the largest utility. Moreover, K (cid:2) is the number of (ST , SR) pairs satisfying these conditions.Similarly, each ST has a preference list of PT s which, if it transmits in the spectrum band occupiedby the (PT , PR) pair in the list, obtains a rate greater than its minimum rate requirement and a utilitygreater or equal to zero. In particular, the preference list for ST q is given by SULIST q = (cid:11) (PT ψ q ( (cid:2) ) , PR ψ q ( (cid:2) ) ) (cid:12) V q (cid:2) =1 (12)where ψ q ( · ) is a function ψ q : { , . . . , V q }→{ , . . . , L PU } satisfying the conditions (cid:11) R SU q,ψq ( (cid:2) ) ( β q,ψ q ( (cid:2) ) ) (cid:2) R SU q , req (cid:12) V q (cid:2) =1 (13)and (cid:11) U SU q,ψq ( (cid:2) ) ( ξ q,ψ q ( V (cid:2) ) , β q,ψ q ( (cid:2) ) ) (cid:2) (cid:12) V q (cid:2) =1 (14) with the ordering U SU q,ψq (1) ( ξ q,ψ q (1) , β q,ψ q (1) ) >. . .>U SU q,ψq ( Vq ) ( ξ q,ψ q ( V q ) , β q,ψ q ( V q ) ) . The ordering thusimplies that the first PT in the list provides the largest utility. Moreover, V (cid:2) is the number of (PT , PR) pairs satisfying these conditions.In practice, the instantaneous channels can be measured by utilizing common channel estimationtechniques [37]. One possibility is for this channel estimation to be sent via control channels, asconsidered in [1, 12].
C. Proposed Algorithm to Determine the Matching, Price and Time-slot Allocation Matrices
The key idea of the proposed algorithm is that each (PT , PR) pair trades with the (ST , SR) pair,which provides the highest utility, through both cooperative relaying and monetary payment. Thistrading will be done by negotiating on the price and time-slot allocation numbers { ξ (cid:2),q , β (cid:2),q } L PU (cid:2) =1 L SU q =1 .We say PT (cid:2) makes an offer of ( ξ (cid:2),q , β (cid:2),q ) to ST q to imply that PT (cid:2) is willing to allow ST q to transmit,in exchange for ST q (i) cooperatively relaying PT (cid:2) ’s message with time slot allocation number β (cid:2),q and, (ii) providing a monetary payment with price allocation number ξ (cid:2),q .The specific details of the main algorithm are given in Table II. Note that the main algorithm callsupon the function ‘Proposal Update Unit (PUU)’, denoted as PUU ( · , · , · ) , and detailed in Table III.Note that the PUU provides an intelligent response for each PU after its offer is rejected by a SU.To summarize the main algorithm (MA), each PT will first make an offer to the ST , which is firstin its preference list (MA-Step 2-1). The ST will then check if the offering PT is in it’s preferencelist (MA-Step 2-2-1). If it is, and the ST is already matched with another PT , the ST has twochoices: (a) if the offering PT can provide a better utility than the ST ’s current matching, then the ST will reject its current matching in favor of the new matching (MA-Step 2-2-1-a-i), or (b) if theoffering PT can not provide a better utility than the ST ’s current matching, the ST will reject the PT ’s offer (MA-Step 2-2-1-a-ii). If the ST is not matched, then the ST will be matched with theoffering PT (MA-Step 2-2-1-b). If the offering PT is not in the ST ’s preference list, the ST willreject the offering PT (MA-Step 2-2-2). The algorithm will then repeat this procedure with each PT until no more matchings are possible.Note that if the ST rejects a PT , then the proposal update unit (PUU) will be activated, and the PT will either (i) decrease its price allocation number by a price step number δ (PUU-3), or (ii) Note that the PUU is required in the algorithm as we are considering both monetary and rate factors in the utility function. Thisis in contrast to our conference paper [29], which did not have a PUU as monetary factors were not considered. TABLE II
Main Algorithm (MA)
Step 1: Initialization. ( Parameter setting and list construction ) (1) Set ( ξ k,q , β k,q )=( ξ init , β init ) , for k =1 , . . . , L PU , eq =1 , . . . , L SU . (2) Set the price step number δ . (3) Set the time-slot step number (cid:5) . (4) Construct PUs’s preference list,
PULIST k , based on ( ξ init , β init ) ,for k =1 , . . . , L PU . (5) Construct SUs’s preference list,
SULIST q , based on ( ξ init , β init ) ,for q =1 , . . . , L SU . (6) Construct the list of all PT s which are not matched, denoted by MATCH= { PT , . . . , PT L PU } . (7) Set the indexing parameter (cid:6) =1 . Step 2: Find a suitable ST for each PT ( PUs and SUs dynamic negotia-tion ) (1) PT (cid:2) makes an offer of ( ξ (cid:2),q , β (cid:2),q ) to the first ST in its preference list PULIST (cid:2) . Let us denote this ST as ST q . If ( ξ (cid:2),q , β (cid:2),q ) (cid:2) =( ξ init , β init ) , then ST q updates its preference list based on ( ξ (cid:2),q , β (cid:2),q ) and reorder its preferencelist based on the new update. (2)
1) If PT (cid:2) is in ST q ’s preference list, thena) If ST q is already matched to a primary transmitter, denoted by PT curr , theni) If PT (cid:2) is higher up than PT curr on ST q ’s preference list SULIST q then- SU q and PU (cid:2) are matched. Remove PU (cid:2) from MATCH .- Update ( ξ curr ,q , β curr ,q , MATCH , PULIST curr ) by ex-ecuting the Proposal Update Unit, ie., ( ξ curr ,q , β curr ,q , MATCH , PULIST curr )=PUU ( ξ curr ,q , β curr ,q , MATCH , PULIST curr ) .ii) Else- Update ( ξ (cid:2),q , β (cid:2),q , MATCH , PULIST (cid:2) ) by executing the Proposal Update Unit, ie., ( ξ (cid:2),q , β (cid:2),q , MATCH , PULIST (cid:2) )=PUU ( ξ (cid:2),q , β (cid:2),q , MATCH , PULIST (cid:2) ) .b) Else- SU q and PU (cid:2) are matched. Remove PU (cid:2) from MATCH .2) Else - Update ξ (cid:2),q , β (cid:2),q , MATCH , PULIST (cid:2) by executing the Pro-posal Update Unit, ie., ( ξ (cid:2),q , β (cid:2),q , MATCH , PULIST (cid:2) )=PUU ( ξ (cid:2),q , β (cid:2),q , MATCH , PULIST (cid:2) ) .3) If PULIST (cid:2) = ∅ , remove PT (cid:2) from MATCH , and go to Step 2-3, elsego to Step 2-1. (3)
If no more matchings are possible, i.e.,
MATCH= ∅ , then go to Step 4.Else go to Step 2-1 for each PT which is not matched, i.e., go to Step 2-1with PT (cid:2) being the PT corresponding to the first entry in MATCH . Step 4: End of the algorithm decrease its time slot allocation number by a time slot-step number (cid:12) (PUU-4), depending on whichoption maximizes the PT ’s utility, and assuming a positive price and time-slot allocation numberand the minimum data rate requirement for the PT is satisfied.We observe that the proposed algorithm produces a matching, price allocation, and time-slotallocation matrix. Note that the price and time-slot allocation matrix have non-zero entries, which TABLE III
Proposal Update Unit (PUU)
Input (1)
Current price allocation number ξ Old (cid:2),q . (2) Current time-slot allocation number β Old (cid:2),q . (3) Current
MATCHLIST
Old (4)
Current PU Old (cid:2) ’s preference list
PULIST
Old (cid:2) . Output(1)
Updated price allocation number ξ New (cid:2),q . (2) Updated time-slot allocation number β New (cid:2),q . (3) Updated
MATCHLIST
New (4)
Updated PU (cid:2) ’s preference list PULIST
New (cid:2) . Procedure(1) If ξ Old (cid:2),q − δ ≤ β New (cid:2),q =max (cid:2) β Old (cid:2),q − (cid:5), (cid:3) .2) ξ New (cid:2),q = ξ Old (cid:2),q . (2) Else if ξ Old (cid:2),q − δ> and R PU (cid:2),q (cid:2) β Old (cid:2),q − (cid:5), ξ (cid:2),q (cid:3) ≤ R PU (cid:2) , req β New (cid:2),q = β Old (cid:2),q .2) ξ New (cid:2),q = ξ Old (cid:2),q − δ . (3) Else if U PU (cid:2),q (cid:2) β Old (cid:2),q , ξ
Old (cid:2),q − δ (cid:3)
Old (cid:2),q − δ (cid:3) ≥ U PU (cid:2),q (cid:2) β Old (cid:2),q − (cid:5), ξ Old (cid:2),q (cid:3)(cid:3) β New (cid:2),q = β Old (cid:2),q .2) ξ New (cid:2),q = ξ Old (cid:2),q − δ . (5) Update the old preference list
PULIST
Old (cid:2) for (PT (cid:2) , PR (cid:2) ) based onthe updated (cid:2) ξ New (cid:2),q , β
New (cid:2),q (cid:3) , to form a new preference list
PULIST
New (cid:2) . (6) Update the old
MATHLIST
Old by putting PU (cid:2) at the end of thelist, to form a new MATCHLIST
New . take values from a discrete set, in contrast to the continuous set considered in the optimizationproblem in (8). This is due to the update procedure, where the price and time-slot allocation numberschange according to the price and time-slot step numbers δ and (cid:12) . We denote the price and time-slot allocation matrices with discrete elements corresponding to the particular ξ init , β init , δ and (cid:12) values as G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) respectively. Mathematically, the elements of G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) respectively take values from the sets { g disc i,j = ξ i,j ∈ ξ init − mδ : m =1 , . . . , (cid:6) ξ init δ (cid:7)} and { b disc i,j = β i,j ∈ β init − m(cid:12) : m =1 , . . . , (cid:6) β init (cid:10) (cid:7)} .V. P ERFORMANCE AND I MPLEMENTATION A NALYSIS
We now analyze the performance of the proposed algorithm, and consider related implementationissues. We first present some assumptions we will be considering in the analysis. To demonstratethat the (PT , PR) pairs are motivated to participate in the proposed algorithm, we set the minimumrate requirement of each (PT , PR) pair to be the rate of the direct PT to PR link. This is given for (PT (cid:2) , PR (cid:2) ) by R PT (cid:2) , PR (cid:2) = T log (cid:4) DirPR (cid:2) (cid:5) . In this paper, we thus set R PU (cid:2) , req = R PT (cid:2) , PR (cid:2) .We assume all channels experience Rayleigh fading, that randomly distributed as ∼CN (0 , and are constant during the duration of the proposed algorithm and subsequent T transmission time-slots. This is a common assumption for auction based and iterative resource allocation algorithms inwireless networks [38–40].Moreover, the PT s and PR s are located on opposite sides of a square of length two, and thus d PT (cid:2) , PR (cid:2) =2 , while the ST s and SR s are randomly located in an internal square of length one, locatedwithin the square of length two. Moreover, the minimum rate requirements for each (ST , SR) pairis given by R SU , req = { R SU q , req } L SU q =1 =0 . and all curves are generated based on averaging over 20,000instances of the algorithm.Finally, to demonstrate the benefits of the proposed algorithm, we also consider, in addition to thecentralized algorithm in (8), a random matching algorithm where the (PT , PR) pairs are randomlymatched with the (ST , SR) pairs, and denoted by μ rnd . When L PU >L SU , L PU − L SU (PT , PR) pairswill be randomly omitted from the matching, and when L SU >L PU , L SU − L PU (ST , SR) pairs will berandomly omitted from the matching. Once the matchings are randomly established, each matchedPT-ST pair start an interactive negotiation process on the values of ξ (cid:2),q and β (cid:2),q . If the PT-ST pairagree on these values, they will cooperate, otherwise there is no cooperation.We observe that such a random matching requires a centralized approach, and is an upper bound to acompletely distributed random matching with no overhead. However, this is sufficient for comparisonpurposes with the proposed algorithm. Note that the centralized and random matching algorithmrepresent the two extremes in the amount of overhead and complexity required for any algorithm.For all figures, unless indicated otherwise the ‘Centralized’ curves are generated by using (8), the‘Proposed (complete SNR)’ curves are generated as described in Section IV-A1, the ‘Proposed (partialSNR)’ curves are generated as described in Section IV-A2, and the ‘RMBN’ curves are generatedbased on the random match basic negotiation methods as previously described in this Section. A. Incentive Analysis for Selfish and Rational Users
A common and realistic assumption in cognitive radio networks is that the primary and secondaryusers are selfish and rational (see e.g., [1] and [30]). Selfishness implies that users compete witheach other to maximize their individual utility function, with no regard for other user’s utility, whilerationality implies that users always make decisions to increase their utility. It is thus important forthe proposed algorithm to provide, incentive for the users to participate in the proposed algorithm,by taking into account the selfish and rational nature of the users.
1) Accounting for Selfish Users Through Stable Matching:
We first prove that the proposedalgorithm takes into consideration the selfish nature of the users, by showing that a stable matchingis produced. To define a stable matching , we will first define a matching, which is blocked by anindividual, and a matching which is blocked by a pair. To do this, we consider p (cid:2) ∈P and s q ∈S ,where μ ( p (cid:2) ) (cid:9) = s q and μ ( s q ) (cid:9) = p (cid:2) . A matching μ is blocked by an individual p (cid:2) ( s q ) if p (cid:2) ( s q ) prefersnot to be matched, than being matched with its current partner under μ . Mathematically, for p (cid:2) , thisimplies that R PU (cid:2),μ † ( p(cid:2) ) ( β (cid:2),μ † ( p (cid:2) ) ) The proposed algorithm in Section IV-C produces a stable matching, under G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) . Proof: See Appendix A. Theorem 1 is important because it states that the proposed algorithm results in a matching, whereany matched (PT , PR) and (ST , SR) pair will not both achieve a higher utility than if they were torespectively partner with any other (ST , SR) and (PT , PR) pair. 2) Accounting for Rational Users Through Utility Maximization: It is also important that theproposed algorithm takes into account the rational nature of the users. This first means that each PUand SU will obtain at least a better utility if they participated in the algorithm compared to if theywere unmatched. This is equivalent to one of the stable matching conditions; that the matching is not We extend the common definition of stable matchings [41] to incorporate the price and time-slot allocation numbers In general, G and A can have continuous elements as considered in Section III, or discrete elements as considered in Section IV.Any analysis involving the matching μ thus has to also consider the domain of the price and time-slot allocation matrices in order tobe meaningful. blocked by an individual, and was proved in Theorem 1 . The second is that at each iteration of thealgorithm, each PU and SU should always choose the user, which provides the highest utility. Thiscan be directly observed in MA-Step 2-1 for each PU, and MA-Step 2-2-1-a-i and MA-Step 2-2-1-bfor each SU. This implies that none of these matched selfish PUs and SUs have any incentive todeviate from the matching produced by the proposed algorithm. B. Utility Performance We now investigate the utility performance of the proposed algorithm. To do this, we first presentthe following lemma: Lemma 1: The utility for every (PT , PR) pair in the stable matching, produced by the proposedalgorithm in Section IV-C, is greater than or equal to the utility obtained through a stable matchingproduced by any algorithm, under G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) . Proof: See Appendix B. Lemma 1 thus indicates that our algorithm produces the best stable matching, out of every possiblestable matchings, under G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) .A natural question now arises as to how our algorithm performs when compared with non-stablematchings. To answer this, we first show that our algorithm produces a weak Pareto optimal matching,denoted by μ Pareto . A weak Pareto optimal matching is defined as a matching where there exists atleast one matched (PT , PR) pair under μ Pareto , which obtains a utility at least greater than any othermatching, stable or non-stable. Given this definition, we present the following theorem: Theorem 2: The proposed algorithm in Section IV-C is weak Pareto optimal, under G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) . Proof: See Appendix C. Theorem 2 indicates there exists at least one matched (PT , PR) pair under the proposed algorithmthat achieves a utility at least greater than the utility achieved under a non-stable centralized optimalalgorithm, under G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) . This optimal algorithm is similar to the algorithmin (8), but under G disc ( ξ init , δ ) and B disc ( β init , (cid:12) ) , not G cont ( ξ init , δ ) and B cont ( β init , (cid:12) ) .In fact, the proposed algorithm can achieve a utility for every matched (PT , PR) pair very closeto the centralized optimal algorithm in (8), even under G cont ( ξ init , δ ) and B cont ( β init , (cid:12) ) . This can beobserved in Figure 2, which plots the average sum-utility of all matched (PT , PR) pairs vs. time-slot step number (cid:12) for the proposed algorithm, the centralized algorithm in (8), and the RMBN ε A v e r a g e s u m − u tilit y o f m a t c h e d p r i m a r y u s e r s , U P U Σ , μ L SU =10 L SU =2 CentralizedProposed (complete SNR)Proposed (partial SNR)Random Fig. 2. Average sum-utility of all matched (PT , PR) pairs vs. time slot step-number (cid:5) , with ξ init =0 . , β init =0 . , δ = (cid:5) , γ SU = . . . γ SU L SU =25 dB, L PU =2 , γ PU = γ PU =5 dB, R SU , req =0 . , { R PU (cid:2) , req = R PT (cid:2) , PR (cid:2) } L PU (cid:2) =1 , ¯ c =1 , ¯ k =1 and α =4 . algorithm. Note that the average sum-utility corresponds to the sum over all utilities achievedby the matched (PT , PR) pairs, averaged over the channel realizations, and given by U PU Σ ,μ = (cid:15) (cid:2) ∈P μ E (cid:16) U (cid:2),μ † ( (cid:2) ) ( ξ (cid:2),μ † ( (cid:2) ) , β (cid:2),μ † ( (cid:2) ) ) (cid:17) , where P μ corresponds to all the (PT , PR) pairs matched under μ . We first observe in Figure 2 that for the proposed algorithm, the complete and partial receivedSNR scenarios achieve very similar performance, despite different channel assumptions. We nextobserve that the proposed algorithm (i) achieves a sum-utility comparable with the sum-utility of thecentralized algorithm for sufficiently small (cid:12) , and (ii) performs significantly better than the RMBNalgorithm. For example, when the time-slot step number (cid:12) =0 . and L SU =10 , we observe that theproposed algorithm with complete and partial received SNR achieves respectively (i) ≈ % of thesum-utility of the centralized algorithm, and (ii) ≈ % sum-utility increase compared to the RMBNalgorithm. PUs‘ rate per monetary value, ¯ c A v e r a g e s u m − r a t e o f m a t c h e d P U s , R P U Σ , μ L SU =2 L SU =10 CentralizedProposed (complete SNR)Proposed (partial SNR)RandomPUs‘ direct rate Fig. 3. Average total sum-rate of all matched (PT , PR) pairs vs. ¯ c , with ξ init =0 . , β init =0 . , (cid:5) =0 . , δ =0 . , γ SU = . . . γ SU L SU =25 dB, L PU =2 , γ PU = γ PU =5 dB, R SU , req =0 . , { R PU (cid:2) , req = R PT (cid:2) , PR (cid:2) } L PU (cid:2) =1 , ¯ k =15 and α =4 . C. Rate Performance As observed in Section IV-C and Figure 2, the proposed algorithm’s primary focus is on maxi-mizing the PU’s sum-utility, while simultaneously satisfying both the PU’s and SU’s minimum raterequirement. However, some networks may require that (i) only the rate is important and monetaryfactors are not a consideration, and (ii) the SUs also prefer a higher rate, greater than their minimumdata rate requirement. For such networks, the proposed algorithm is flexible in adapting to thesedesign needs.In particular, observe from (4) that increasing ¯ c has the effect of increasing the contribution ofthe monetary value in the (PT , PR) ’s utility function. As observed in the Proposal Update Unit, thishas the effect of the PUs choosing to offer a lower time-slot allocation number, rather than a lowerprice allocation number, to the SUs. This will subsequently result in a higher rate for the SUs, atthe expense of a lower rate for the PUs. Thus the parameter ¯ c can be used to control the rates ofthe PUs and SUs, in accordance with specific system requirements. PUs‘ rate per monetary value, ¯ c A v e r a g e s u m − r a t e o f m a t c h e d S U s , R P U Σ , μ L SU =2 L SU =10 L SU =2 L SU =10 L SU =2 CentralizedProposed (complete SNR)Proposed (partial SNR)Random R SU,req Fig. 4. Average total sum-rate of all matched (ST , SR) pairs vs. ¯ c , with ξ init =0 . , β init =0 . , (cid:5) =0 . , δ =0 . , γ SU = . . . γ SU L SU =25 dB, L PU =2 , γ PU = γ PU =5 dB, R SU , req =0 . , { R PU (cid:2) , req = R PT (cid:2) , PR (cid:2) } L PU (cid:2) =1 , ¯ k =15 and α =4 . This can be observed in Figures. 3 and 4, which respectively plots the average sum-rate of allmatched (PT , PR) pairs vs. ¯ c , and the average sum-rate of all matched (ST , SR) pairs vs. ¯ c , forthe complete received SNR scenario. Note that the average sum-rate of all matched (PT , PR) pairscorresponds to the sum over all rates achieved by the matched (PT , PR) pairs, averaged over thechannel realizations, and is given by R PU Σ ,μ = (cid:15) (cid:2) ∈P μ E (cid:16) R (cid:2),μ † ( (cid:2) ) ( β (cid:2),μ † ) (cid:17) , where P μ corresponds toall the (PT , PR) pairs matched under μ . A similar definition can be made for the sum-rate of thematched (ST , SR) pairs, denoted R SU Σ ,μ . For comparison, we also plot the sum-rate achieved by acentralized optimal algorithm. In particular, in Figure 3, the centralized algorithm produces a sum-rate, which maximizes the total sum-rate of all (PT , PR) pairs, and given by substituting ¯ c =0 intothe optimal algorithm produced from (8). In Figure 4, the centralized algorithm produces a sum-ratewhich maximizes the sum-rate of all (ST , SR) pairs, and is formulated in a way similar to (8), butinterchanging the (PT , PR) pairs with the (ST , SR) pairs in the optimization equation. This results γ ST ) A v e r a g e p e r ce n t a g e o f m a t c h e d p r i m a r y ( s ec ond a r y ) u s e r s L SU =2 L SU =15 CentralizedProposed (complete SNR)Proposed (partial SNR)Random Fig. 5. Percentage of matched (PT , PR) pairs vs. the total number of (ST , SR) pairs L SU , with α init =0 . , β init =0 . , (cid:5) =0 . , δ =0 . , γ SU = . . . γ SU L SU =25 dB, L PU =2 , γ PU = γ PU =5 dB, L PU =2 , R SU , req =0 . , { R PU (cid:2) , req = R PT (cid:2) , PR (cid:2) } L PU (cid:2) =1 , ¯ c =1 , ¯ k =2 and ρ =4 . in the optimization arg max { M , B cont , G cont } L PU (cid:6) (cid:2) =1 L SU (cid:6) q =1 m (cid:2),q R SU q,(cid:2) ( ξ (cid:2),q , β (cid:2),q ) subject to the conditions in (8).We observe in Figures. 3 and 4 that R PU Σ ,μ decreases with ¯ c , while R SU Σ ,μ increases with ¯ c , asexpected. This shows the flexibility of our algorithm in adapting to different primary and secondaryuser priority levels. When ¯ c is low, we observe in Figure 3 that the sum-rate of the matched (PT , PR) pairs of our proposed algorithm achieves a high percentage of the optimal algorithm and significantlygreater than the RMBN algorithm. Similarly, when ¯ c is high, we observe in Figure 4 that the sum-rateof the matched (ST , SR) pairs of our proposed algorithm achieves a high percentage of the optimalalgorithm and significantly greater than the RMBN algorithm. D. Total Number of Matchings The total number of matched (PT , PR) and (ST , SR) pairs is also an important consideration ofany matching algorithm, and is proportional to the total number of users which can achieve theirminimum rate requirements. Figure 5 shows the percentage of matched (PT , PR) pairs vs. the SU’sSNR γ ST = γ ST = γ ST L SU , for different number of (ST , SR) pairs L SU . We observe that the percentageof matched (PT , PR) pairs increases with γ SU and L SU , due to the higher achievable rates that thematched (PT , PR) pairs can achieve through cooperative relaying. Remarkably, we observe that theproposed scheme can achieve a very high matching percentage at high SNR even when L SU =2 ,i.e., ≥ % when γ SU ≥ dB. Moreover, the proposed algorithm delivers a percentage of matchedusers comparable with the centralized algorithm, and significantly greater than the RMBN algorithm.Note that as the minimum rate requirement for the PUs is equal to the rate of their correspondingdirect link transmission without cooperative relaying, Figure 5 thus indicates that the PUs are wellmotivated to participate in the trading framework with the SUs.In practice, the unmatched PT s will transmit directly to their corresponding PR s and thus (PT (cid:2) , PR (cid:2) ) will achieve the rate R PT (cid:2) , PR (cid:2) . However, the unmatched ST s will not be able to transmit at all. Toremedy this, various modifications to the proposed algorithm can be made, which are the subjectof future work such as integrating a fairness mechanism into the algorithm so each ST has a turntransmitting, though at different times. E. Convergence We will now analyze the convergence behavior of the proposed algorithm, as shown in the followingtheorem: Theorem 3: The number of iterations required for the proposed algorithm to converge is upperbounded by: I max = ξ init δ + 1 (cid:12) (cid:4) β init − β MIN (cid:5) (16)where β MIN = min (cid:2) =1 ,...,L PU min q =1 ,...,L SU β min (cid:2),q and β min (cid:2),q = R PU (cid:2), req T log (cid:3) DirPR (cid:2) +Γ RelayPR (cid:2),q (cid:4) . Proof: See Appendix D. Theorem 3 implies that the proposed algorithm converges after a finite number of iterations, andthat this number is dependent on different parameters. For example, we clearly see that the numberof iterations for convergence decreases with (cid:12) and δ , and increases with ξ init and β init . F. Overhead The proposed algorithm is distributed, and thus incurs significantly less overhead and complexitycompared to centralized algorithms. An exact analysis of the amount of overhead and complexityis difficult, due to the dependency on a number of system parameters, such as the minimum sum-rate requirements, the price and time-slot step-numbers and the initial price and time-slot allocationnumbers. We can however, find an expression for the upper bound on the maximum number ofcommunication packets between the PTs adn the STs, given in the following theorem: Theorem 4: The number of communication packets between the PT s and the ST s required in theproposed algorithm is upper bounded by N max =( a L PU + a F ) I max (17) =( a L PU + a F ) (cid:18) ξ init δ + 1 (cid:12) (cid:4) β init − β MIN (cid:5)(cid:19) where where a , a ∈ R + and F =max { L PU , L SU } . Proof: See Appendix E.We observe in (17) that the amount of overhead, and thus the number of iterations, decreaseswith (cid:12) and δ . This is confirmed in Figure 6, which plots the total number of communication packetsexchanged between each PT and all the ST s it communicates with, vs. time-slot step number (cid:12) ,with the same parameters used in Figure 2. We see that the total number of communication packetsconverge to a constant at sufficiently high (cid:12) . This is because if (cid:12) is sufficiently large, the time-slotallocation numbers are updated in the algorithm in such a way that the preference lists for each (PT , PR) and (ST , SR) pair remain unchanged.Denoting the random variable Y as the total number of communication packets as a function ofthe user channels, Figure 7 plots the cumulative distribution function (c.d.f.) of Y , PR ( Y ≤ y ) , vs. y for different number of secondary users L SU , where y is a realization of Y . We observe that theproposed algorithm can achieve low overhead with high probability in various practical scenarios.For example, when L SU =6 , we observe that % of the time, a maximum of only communicationpackets are exchanged between the PUs and the SUs.Figures. 2 and 6, and (17) reveal that (cid:12) can be designed to ensure an acceptable amount of overheadand achievable rate. In particular, we observe from (17) and Figure 6 that decreasing the overhead byincreasing (cid:12) or δ will result in both a lower magnitude and number of price and time-slot allocation ε A v e r a g e nu m b e r o f c o mm un i ca ti on p ac k e t s L SU =6 L SU =2 Proposed (complete SNR)Proposed (partial SNR) Fig. 6. Total number of communication packets vs. (cid:5) for the complete instantaneous received SNR scenario, with α init =0 . , β init =0 . , (cid:5) =0 . , δ =0 . , γ SU = . . . γ SU L SU =25 dB, L PU =2 , γ PU = γ PU =5 dB, R SU , req =0 . , { R PU (cid:2) , req = R PT (cid:2) , PR (cid:2) } L PU (cid:2) =1 , ¯ c =1 , ¯ k =1 and ρ =4 . numbers which can be offered to the (ST , SR) pairs from the (PT , PR) pairs. This will result ina lower utility for the (PT , PR) pairs, and increase the chance the ST s will reject any offer made.This can be observed in Figure 2, which illustrates a tradeoff between performance and overhead.A similar argument can be made for decreasing ξ init and β init . In practice, each PU can adaptivelyadjust it’s time-slot step number (cid:12) based on (i) the acceptable sum utility of PUs in Figure 2 and (ii)the tolerable average number of required communication packets in Figure 6.We note that the packet length required for communication between the PT s and the ST s is veryshort. In particular, assuming that ξ init , β init , δ and (cid:12) are initially known to all users, each PT is onlyrequired to send one bit to the first ST in its preference list indicating an offer, and the corresponding ST only needs to send one bit back to the offering PT indicating either acceptance or rejection. Therequired time for the proposed algorithm execution is thus very small. Specifically, if high speedcontrol channels such as 802.11 are used, the average time for execution is in the order of 100-200nsec [42]. As demonstrated in Figure 6, the total number of communication packets for each PT C D F o f p r opo s e d a l go r it h m c onv e r g e n ce L SU =6 L SU =15 γ SU =25 dB γ SU =20 dB Fig. 7. Cumulative distribution function vs. total number of communication packets for the complete instantaneous received SNRscenario, with α init =0 . , β init =0 . , (cid:5) =0 . , δ =0 . , γ SU = . . . γ SU L SU =25 dB, L PU =2 , γ PU = γ PU =5 dB, R SU , req =0 . , { R PU (cid:2) , req = R PT (cid:2) , PR (cid:2) } L PU (cid:2) =1 , ¯ c =1 , ¯ k =1 and ρ =4 . can be designed to be reasonably small, and thus given the short packet lengths, the total runningtime and amount of overhead from the proposed algorithm can be quite small. G. Complexity We now present a lemma for the complexity of the centralized, proposed and RMBN algorithms: Lemma 2: The complexity of the centralized algorithm is given by (cid:20) O (cid:2) L SU !( L SU − L PU )! L SU + L PU (cid:3) , if L SU ≥ L PU ; O (cid:2) L PU !( L PU − L SU )! L SU + L PU (cid:3) , if L SU See Appendix F.We observe that the centralized method has a significantly larger complexity than both the proposedand randomized algorithm. Its complexity is increasing exponentially with the number of primaryand secondary users. In contrast, the proposed algorithm complexity only increases linearly with the number of primary or secondary users. VI. C ONCLUSION We proposed a distributed algorithm for spectrum access, which guarantees that the PUs’ and SUs’rate requirements are satisfied. Our algorithm hinges on a trading framework between the PUs andSUs, where the PUs and SUs negotiate on combined time-slot and monetary compensation. Timeslot allocation numbers determine the amount of transmission time dedicated for SU transmission,and the length of time the SUs will cooperative relay the PUs’ data. The price allocation numbersexpress the amount of monetary compensation that SU provide for PUs in exchange of spectrumusage. The proposed algorithm, was based on a dynamic bilateral negotiation between the PUs andSUs that resulted in a stable outcome. We proved that the proposed algorithm results in the bestpossible stable matching and is weak Pareto optimal. A numerical analysis also revealed that thedistributed algorithm achieves a performance comparable to an optimal centralized algorithm, butwith significantly less overhead and complexity.A PPENDIX A. Proof of Theorem 1 We first define some notations. Without loss of generality, let { (PT (cid:2) PR (cid:2) ) } L μ prop (cid:2) =1 be the set ofmatched (PT , PR) pairs at the conclusion of the algorithm. We first consider p =(PT , PR ) , withthe preference list at the conclusion of the algorithm denoted by PULIST = { s , . . . , s K } . Associatedwith this preference list are the final price and time-slot allocation numbers associating p and { s q } K q =1 , and denoted respectively by { ξ prop1 ,q } K q =1 and { β prop1 ,q } K q =1 . Note that due to the ordering ofthe preference list, the proposed algorithm will match p with s . We now prove that the proposedalgorithm doesn’t produce any blocking pairs by contradiction. Assume that p and s q , for some q =2 , . . . , K , constitute a blocking pair. This implies that there exists a ξ ,q and β ,q such that U PU ,q ( ξ ,q , β ,q ) >U PU , ( ξ prop1 , , β prop1 , ) , implying that either (i) ξ ,q >ξ prop1 ,q , (ii) β ,q >β prop1 ,q or (iii) both ξ ,q >ξ prop1 ,q and β ,q >β prop1 ,q . Due to the ordering of the preference list PULIST , this cannot occur asthere would have been a period during the algorithm when s q initially rejected p at this β ,q and/or ξ ,q value, as s q received a better offer from another (PT , PR) pair. For this scenario, p and s q thus donot form a blocking pair. A similar proof can be made for the other p (cid:2) s, for (cid:2) =2 , . . . , L PU . The stablematching proof follows by noting that there are no blocking individuals due to the preference lists, i.e., no (PT , PR) and (ST , SR) pairs will be matched respectively with a (ST , SR) and (PT , PR) pair not on its preference list. B. Proof of Lemma 1 The proof is by contradiction. Denote μ prop as the stable matching produced by the proposedalgorithm, and μ alt as another stable matching. Then let us assume that for all (PT , PR) pairs, theutility achieved under μ prop is less than the utility achieved under μ alt . W.l.o.g, consider (PT (cid:2) , PR (cid:2) ) ,and thus the previous statement mathematically implies that U PU (cid:2),μ † alt( (cid:2) ) ( ξ alt (cid:2),μ † alt ( (cid:2) ) , β alt (cid:2),μ † alt ( (cid:2) ) ) >U PU (cid:2),μ † prop( (cid:2) ) ( ξ prop (cid:2),μ † prop ( (cid:2) ) , β prop (cid:2),μ † prop ( (cid:2) ) ) , where ( ξ prop · , · , β prop · , · ) and ( ξ alt · , · , β alt · , · ) denote the price and time-slot allocation numbers under matching μ prop and μ alt respectively. Denoting q = μ † alt ( (cid:2) ) , this means that under the proposed matching μ prop , PT (cid:2) offered ( ξ alt (cid:2),μ † alt ( (cid:2) ) , β alt (cid:2),μ † alt ( (cid:2) ) ) to ST q but was rejected. There are two ways for this to happen. The first is if under matching μ prop , PT (cid:2) with proposal ( ξ alt (cid:2),μ † alt ( (cid:2) ) , β alt (cid:2),μ † alt ( (cid:2) ) ) was not in ST q ’s preference list SULIST q . However, thiscontradicts the implicit assumption that under matching μ alt , PT (cid:2) with proposal ( ξ alt (cid:2),μ † alt ( (cid:2) ) , β alt (cid:2),μ † alt ( (cid:2) ) ) is in ST q ’s preference list, and thus should also be in ST q ’s preference list under matching μ prop .The second alternative is if under matching μ prop , ST q rejected PT (cid:2) in favor of another primarytransmitter, denoted as PT ν . W.l.o.g., let us assume that this is the first rejection that has taken placeunder μ prop . This implies that PT ν prefers ST q to every other (ST , SR) pair, and thus the matching μ alt is blocked by pair ((PT ν , PR ν ) , (ST q , SR q )) , which contradicts the assumption that μ alt is astable matching. C. Proof of Theorem 2 We outline the proof for non-stable matchings, and note that the proof for stable matchings followsdirectly from Lemma 1. The proof is by contradiction. Denote μ prop as the matching produced bythe proposed algorithm, and μ opt as an arbitrary non-stable matching. Then let us assume that everymatched (PT , PR) pair in μ prop achieves a utility less than its utility obtained under matching μ opt .Mathematically, this is represented by { U (cid:2),μ † prop ( (cid:2) ) ( ξ prop (cid:2),μ † prop ( (cid:2) ) , β prop (cid:2),μ † prop ( (cid:2) ) )
1) Same matchings: In this scenario, every matched (ST , SR) and (PT , PR) pair under μ prop is also matched under μ opt . Consider the last (ST , SR) pair which is matched in μ prop , and de-note this pair as (ST L μ prop , SR L μ prop ) . Moreover, w.l.o.g. (ST L μ prop , SR L μ prop ) is matched with (i) (PT L μ prop , PR L μ prop ) under matching μ prop , and (ii) (PT , PR ) under matching μ opt . From the initialcontradiction assumption, note that under μ prop , (PT , PR ) prefers (ST L μ prop , SR L μ prop ) with priceand time slot allocation numbers ( ξ alt1 ,L μ prop , β alt1 ,L μ prop ) than its current matching, which mathematicallyimplies that U PU ,Lμ prop ( ξ alt1 ,L μ prop , β alt1 ,L μ prop ) >U PU ,μ † prop(1) ( ξ prop1 ,μ † prop (1) , β prop1 ,μ † prop (1) ) . There was thus a period duringthe proposed algorithm where (PT , PR ) offered a price allocation number ξ alt1 ,L PU ( >ξ prop1 ,L PU ) , and/ora time-slot allocation number β alt1 ,L PU ( >β prop1 ,L PU ) to (ST L μ prop , SR L μ prop ) , and was rejected. However, as (ST L μ prop , SR L μ prop ) is the last (ST , SR) pair to be matched, this is a contradiction as (ST L μ prop , SR L μ prop ) will not reject any offer from (PT , PR ) . 2) More primary user matchings: In this scenario, there are more (PT , PR) pairs which arematched under μ prop , then matched under μ opt . The initial contradiction assumption is thus clearlyviolated, as the un-matched (PT , PR) pairs which are matched under μ prop , but not under μ opt , havea zero utility. 3) Less primary user matchings: In this scenario, there are less (PT , PR) pairs which are matchedunder μ prop , then matched under μ opt . Every (ST , SR) pair which is not matched under μ prop willthus have a zero utility. However, at least one of these (ST , SR) pairs, denoted as (ST q , SR q ) willobtain a positive utility under μ opt . Combined with the initial contradiction assumption, this impliesthat (ST q , SR q ) and (PT μ † opt ( q ) , PR μ † opt ( q ) ) form a blocking pair for μ prop , which is a contradiction. D. Proof of Theorem 3 From MA-Step 2-2-1-a-i and MA-Step 2-2-2 of the proposed algorithm in Table II, if ST q rejectsthe offer of PT (cid:2) , then PT (cid:2) will decrease its price allocation number β (cid:2),q or its time-slot allocationnumber ξ (cid:2),q by executing PUU. The minimum possible price allocation number that PT (cid:2) can offer ST q is ξ (cid:2),q =0 , while the minimum time-slot allocation number that PT (cid:2) can offer ST q is β min (cid:2),q = 2 R PU (cid:2), req T log (cid:2) DirPR (cid:2) + Γ RelayPR (cid:2),q (cid:3) . (19) PT (cid:2) will not decrease its time-slot allocation number to β (cid:2),q = β min (cid:2),q − (cid:12) , since its minimum raterequirement will not be satisfied. Thus the minimum possible time-slot allocation number that PT (cid:2) can offer in the algorithm is β MIN (cid:2) = min q =1 ,...,L SU β min (cid:2),q . (20)So in the worst case scenario, each PT (cid:2) , (cid:2) =1 , . . . , L PU after (cid:12) ξ init δ + (cid:10) (cid:4) β init − β MIN (cid:2) (cid:5) (cid:13) iterationswill not make any offer to any ST , and according to MA Step 2-2-3, PT (cid:2) will be removed fromthe list of unmatched PTs denoted by MATCH in Table II. Thus the maximum number of iterationsthat all the PT s will be removed from MATCH is given by I max = ξ init δ + 1 (cid:12) (cid:18) β init − min (cid:2) =1 ,...,L PU β MIN (cid:2) (cid:19) . (21) E. Proof of Theorem 4 Following the proposed algorithm in Table II, the PT s and ST s need to communicate together asfollows. In MA-Step 2-1, in the worst case scenario, all the PT s make an offer to all the ST s, andthus L PU communication packets are required. After receiving the PTs offer, the ST s need to informthe PTs whether they accepted or rejected the offers. This process happens in MA-Step 2-2-1-a-i,MA-Step 2-2-1-a-ii and MA-Step 2-2-2 and thus F communication packets are required. Thus in thefirst iteration of the algorithm, a maximum of L PU + F communication packets are required.However, since in the next iterations some of the PUs are matched and also the length of thepreference list of the unmatched PTs is decreased, the number of communication packets willdecrease. Thus after the first iteration, the total number of communication packets at each iterationof the algorithm is a linear function of L PU and F . The maximum number of iterations that thealgorithm ends is also known from Theorem 3. Therefore, the maximum number of communicationpackets packets between the PT s and ST s scales as ( a L PU + a F ) I max . F. Proof of Lemma 2 For the centralized method when L PU (cid:3) L SU , the proof follows by noting that the total number ofmatching combinations between the PTs and STs is L SU !( L SU − L PU )! , while the complexity of solving thelinear programming problem for all possible matching combinations is L SU + L PU [43]. A similarargument can be made when L PU >L SU . For the proposed algorithm, the proof follows by notingthat the complexity is proportional to the total number of times the PTs communicates with the STs,given in (17). For the RMBN method, the proof follows by noting that each PT communicates to arandom ST only once, and thus the complexity is proportional to the number of primary users L PU . 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