Spectrum Allocation based on Auction in Overlay Cognitive Radio Network

Abstract

In this paper, a mechanism for spectrum allocation in overlay cognitive radio networks is proposed. In overlay cognitive radio networks, the secondary users (SUs) must first sense the activity of primary users (PUs) to identify unoccupied spectrum bands. Based on their different contributions for the spectrum sensing, the SUs get payoffs that are computed by the fusion center (FC). The unoccupied bands will be auctioned and SUs are asked to bid using payoffs they earned or saved. Coalitions are allowed to form among SUs because each SU may only need a portion of the bands. We formulate the coalition forming process as a coalition forming game and analyze it by game theory. In the coalition formation game, debtor-creditor relationship may occur among the SUs because of their limited payoff storage. A debtor asks a creditor for payoff help, and in return provides the creditor with a portion of transmission time to relay data for the creditor. The negotiations between debtors and creditors can be modeled as a Bayesian game because they lack complete information of each other, and the equilibria of the game is investigated. Theoretical analysis and numerical results show that the proposed auction yields data rate improvement and certain fairness among all SUs.

1. Introduction

Cognitive radio (CR) technology is a promising solution to solve the problem of spectrum scarcity and low spectrum utilization associated to classical fixed spectrum assignment schemes [1]. Due to higher priority of primary users (PUs), secondary users (SUs) need to perceive the behavior of PUs in their assigned frequency bands and perform opportunistic spectrum access without or with minimal interference to them. Usually a user or a node trusted by all SUs (or a third party which shares no common interest with any SU) is introduced as an FC to collect the results of SUs’; perception. In this way the FC has a list of vacant spectrum bands which are available for a period of time.

The SUs want to access the vacant bands for transmission. How to allocate vacant bands among the SUs is one common problem in cognitive radios. To improve the spectrum efficiency, many works have been done [2, 3]. In [2] the authors present a game theoretic model for spectrum sharing, where users seek to satisfy their quality of service demands in a distributed fashion. They also extend their model by considering the frequency spatial reuse, and consider the user interactions as a game upon a graph where players only contend with their neighbors. There are a large number of works that attempt to solve the spectrum allocation problem with economical tools such as game theory [4], contract theory [5, 6], auction [7], pricing [4, 8]. Pricing is often used when the seller knows precisely the value of the resource being sold. In [4] the authors formulate an oligopoly market consisting of a few firms and a consumer to investigate the pricing problem. By using a Bertrand game model, they analyze the impacts of several system parameters such as spectrum substitutability and channel quality on the Nash equilibrium and propose a distributed algorithms to obtain the solution for this dynamic game. In cases where the seller only knows limited information of the buyers’ valuations of the resource, contract is more effective because by motivating the buyers truthfully reveal their private valuations, the seller can optimally allocate the resource. In [5] the authors proposed a quality−price contract for the spectrum trading to allocate the spectrum. [6] exploits the incentive effect of cooperative spectrum sharing where SUs relay traffics PUs in exchange for dedicated spectrum access time for SUs’ own communications. The PU-SU interaction is modeled as a labor market and analyzed by contract theory. It must be stressed that these mechanisms should guarantee the spectrum bands are allocated according to SUs true valuations, which is called truthfulness [9]. Truthfulness is important because if SUs can lie to obtain benefits, efficiency and fairness cannot be achieved. Auction mechanism is a suitable approach when the seller has no knowledge about the value of the resource. Many auction mechanisms [7, 10] have been studied because they guarantee truthfulness. However fairness is often ignored or simplified [10]. The authors of this paper believe that SUs’ contribution to the CR network (such as contribution of cooperative sensing) should be rewarded and assigning higher priority of spectrum bands to more contributed SUs is a kind of fairness [11].

In this paper, we propose a spectrum allocation mechanism similar to auction mechanisms. We first quantify the contribution of every SU in the cooperative sensing and reward it with tokens. Then the FC holds an auction for the vacant bands. An SU may use its tokens to bid for spectrum bands in the following period or it may save the tokens for future use. Since the entire vacant bands are considered as one commodity, the SUs may need to cooperate and form coalitions. During the coalition forming process, debtor-creditor relationship may occur among the SUs because of their limited payoff storage. A debtor may ask a creditor for payoff help, and in return provides the creditor with a portion of transmission time to relay data for the creditor. We model the coalition forming process as a coalition formation game and the negotiations between debtors and creditors as a non-cooperative Bayesian game. We further investigate these games with the help of coalitional game theory [12] and non-cooperative game theory, respectively.

The remainder of the paper is organized as follows: Section 2 presents the system model and the proposed auction. Section 3 presents the investigation of coalition formation in the auction through a coalitional game theoretic perspective. Section 4 introduces the potential cooperation among SUs and analyzes the Bayesian equilibria within. In Section 5 we validate our auction mechanism and our theoretical analysis through simulations. And Section 6 concludes the paper.

 

2. System model and proposed auction

2.1 Network model

We consider a CR network consists of N SUs and M potential unoccupied channels. Each SU detects the PU activities on one or more channels according to its different detection performances on different channels. SUs sense the spectrum bands cooperatively and periodically for infinite iterations. An FC is introduced in the network and we assume it could communicate with all CUs via a perfect common control channel (CCC). The FC categorizes the sensing results and obtains global decisions on the channel status. In this work we also assume that the FC is the manager of all vacant channels and it is in charge of spectrum allocation. The FC also adopts some algorithm (such as algorithm in [11], or reputation-based algorithm) to quantify each SU’s contribution to the spectrum sensing, then pays tokens accordingly. If there are any available channels for the SUs in a sensing-transmission round, we address this round as an episode. In an episode each SU either demands some spectrum for transmission or has no data to transmit. The valuations of these spectrum for different SUs may vary due to the fact that even the same bandwidth may produce different transmission performances to different SUs. Here we assume the spectrum valuation of an SU is based on its data rate. We focus on the spectrum allocation problem in one episode. We denote spectrum demand and valuation of SU i by wi and vi respectively, and denote the total available spectrum in this episode by B. We also assume that for any i, wi ≤ B (i.e., the total available spectrum can satisfy at least one SU). A critical problem here is how to allocate the spectrum among the SUs according to their demand and valuation, while satisfying some fairness criterion.

Fig. 1.Transmission time division in cooperation

2.2 Token spectrum auction

We propose a token spectrum auction to address this problem. In this auction, the SUs bid for the spectrum using the tokens they earned in this episode or they saved from previous episodes. Because the currency here is the token which represents the contribution to cooperative sensing, the SUs always have a limited budget and thus they cannot always pay according to their valuations. We denote the token storage of SUi by αi. The token spectrum auction is deployed as follows.

Step 1: The FC announces there is a total bandwidth of B and it will be auctioned as one commodity.

Step 2: Each SU reports its spectrum demand wi and its first bid bi, where bi is calculated according to its valuation.

Step 3: The FC tells each SU the information of all SUs including: wi, bi and token storage αi. Then the FC asks the SUs to form coalitions.

Step 4: The SUs try to form coalitions. There might be negotiations between the SUs.

Step 5: All coalitions report their bid information to the FC. In particular, coalition Sk reports its final bid as well as the exact division of this bid among its members.

Step 6: The FC allocates the available spectrum to a coalition in a statistical manner where coalition Sk obtains the spectrum with probability The FC then charges all members of the winner.

We provide some notes on the proposed auction: In step 2, first bid bi reflects real spectrum valuation of SUi in form of token. bi may exceed its token storage αi and is not necessarily identical to the final payment of SUi. For instance, if bi >αi and SUi failed to get help from other SUs, the final payment of SUi = αi. In step 5, coalition Sk report both its final bid and the division in order to tell the FC how to calculate its winning probability and how to charge its members if Sk wins. Obviously It can be observed that coalitions with higher final bids will have higher probabilities to obtain the spectrum. The potential negotiations in step 4 will be discussed in later sections.

We propose such an auction for the following reasons:

With the token spectrum auction given, we will investigate the behavior of SUs in the auction. We formulate the coalition formation process as a coalition formation game and study it in Section 3. The negotiations and cooperation between SUs are modeled as a non-cooperative game and will be investigated in Section 4.

 

3. Coalition formation game

In this section we formulate and analyze the coalition formation using coalition game theory. We do not consider the potential negotiations between SUs here through these negotiations might introduce some impact to the coalition formation. Before we start, some coalition game theory concepts must be introduced.

3.1 Coalition formation concepts

Researchers have been highly interested in coalition formation game [13-15]. The goal is to find algorithms for characterizing the coalitional structures that form in a network where the grand coalition is not optimal. In [14], the authors present a generic framework for coalition formation where coalitions form and break through two simple merge-and-split rules. Follow their works, we present some definitions.

Definition 1: A coalitional game (N, v) is said to have a transferable utility (TU) if the value v (S) can be arbitrarily apportioned between the coalition’s players. Otherwise, the coalitional game has a non-transferable utility (NTU) and each player will have its own utility within coalition S.

Definition 2: Let N = {1, 2, …, n} be a fixed set of players called the grand coalition. Non-empty subsets of N are called coalitions. A collection is any family C = {C1, C2,…, Cl} of mutually disjoint coalitions, and l is called its size. If additionally Cj= N, the collection C is called a partition of N.

Definition 3: A comparison relation ⊲ is defined to compare two collections R = {R1, R2,…, Rl} and T = {T1, T2,…, Tm} that are partitions of the same set A (same players in R and T). R ⊲ T implies that the way R partitions A is preferred to the way T partitions A based on a criterion.

Many criteria (referred to as orders) can be adopted to compare collections or partitions. In general they are divided into two types: coalition value orders and individual value orders. Coalition value orders compare two collections by using the value of the coalitions inside these collections. For instance, the utilitarian order [16] compares two collections where R ⊲ T indicates On the other hand individual value orders compare two collections using the player’s utility, not the coalition value. An important individual order is the Pareto order. For a collection R, denote the utility of a player i in a coalition Ri ∈ R by ui(R). Pareto order is defined as

with at least one strict inequality for a player j. Pareto order is an adequate comparison relation for NTU games and we will show later the coalition formation here is a NTU game in subsection 3.2 A.

We further introduce some concepts that will later be used to analyze the stability of a partition.

Definition 4: A defection function is a function which associates with each partition R some partitioned subsets of the grand coalition N. A partition P = {P1, P2,…, Pl} of N is -stable if no group of players have an incentive to leave P when they can only form the collections allowed by .

[13] presents some defection functions such as c and hp. c allows formation of all collections in the grand coalition while hp allows formation of all P- homogeneous partitions in the grand coalition. In other words, a partition P is hp-stable, if no players in P have an incentive to leave P through merge-and-split (we will introduce this rule later) to form other partitions, while a partition P is c-stable, if no players in P have an incentive to leave P through any operation to form other collections in the grand coalition.

3.2 Coalition formation in token spectrum auction

A. Game formation

To investigate the behavior of the SUs in the token spectrum auction, we formulate the coalition formation process as a coalition formation game so that we can refer to coalition game theory. The process can be modeled as a (N, v) where N is the set of players and v is the utility function or value of a coalition.

In the proposed auction we aim to encourage players (i.e., SUs) to form coalitions to compete the bandwidth B. Each coalition is supposed to use the spectrum as much as possible, however the total spectrum demand should not exceed available bandwidth B. So we expect ∑i ∈ S wi to be large as possible while ∑i ∈ S wi ≤ B holds, for every coalition S. According to the auction mechanism, a suitable utility function is given by

where p(S) is the probability that coalition S wins the spectrum, R(S) is the total data rate of coalition S and Res(S) is a restriction function. Again we emphasize that in this section we do not consider the potential negotiations and deals in the coalition formation. Therefore

where Ri is the date rate of SUi. If the total demand of a coalition exceeds available bandwidth B, the division among the coalition members cannot be done. To avoid this, we define the restriction function as follows

Obviously v(S) represents the expected total data rate of coalition S in the auction if the bandwidth division can be done among its members. Coalition S will always welcome new members as long as they do not violate the bandwidth demand restriction, since new members increase v(S) by increasing p(S) and R(S). We define utility of SUi in coalition S as

ui(S) represents the expected data rate of SUi when SUi is in coalition S. Utility -∞ indicates that SUi will not have a positive data rate when bandwidth division cannot be done in S. -∞ is used to emphasize this circumstance should never occur and to match with the definition of v(S). In this way utility of each SU in coalition S is a portion of value of S, and v(S) = ∑i ∈ S ui(S) . It can be easily observed that any rational coalition would like to invite an SU to join it as long as this SU does not violate the bandwidth demand restriction, and any SU prefers to accept one of the invitations than compete alone. We also notice that v(S) cannot be arbitrarily apportioned between the members because S cannot alter any member’s bandwidth demand. Therefore the game we formulated is a NTU game and Pareto order is suitable here.

B. Coalition formation algorithm

We propose a coalition formation algorithm based on two simple rules addressed as “merge” and “split” which modify a partition T of SUs set N [14].

Definition 5: Merge Rule: {T1, T2,…, Tk} → where ⊲ {T1, T2,…, Tk}

Definition 6: Split Rule: → {T1, T2,…, Tk} where {T1, T2,…, Tk} ⊲ With these rules, multiple coalitions can merge into one coalition, and one coalition can split into multiple coalitions if merging/splitting operation yields a better collection based on the order ⊲. Here ⊲ is the Pareto order, therefore coalitions will merge (split) only if at least one SU strictly improves its utility through this merging (splitting) without harming other SUs’ utilities.

Remark 1: Suppose ⊲ is a comparison relation. Then every iteration of the merge and split rules terminates.

With the support of Remark 1, an algorithm can be designed based on merge and split rules without considering if it terminates. The coalition formation algorithm works as follows: At the beginning, every SU is a coalition. The FC picks an SU (referred to as the head) to start the merging process. For instance, FC may appoint the SU with the highest “bid per unit bandwidth” (i.e., ) to start so that this SU is given some advantage in competition. The head invites other SUs to join in its coalition Sh in a certain order. If an invited SU rejects the invitation, the head turns to the next SU. The merging ends once Sh reaches bandwidth demand restriction (i.e., ∑i ∈ Sh wi = B) or no SU can join in Sh with a mutual benefit. Then the FC appoints an SU j ∈ N∖Sh as the new head and start the merging among N∖Sh again. The algorithm is repeated until every SU has made its merging decisions, i.e., an SU has either played the role of head or accepted an invitation. A final partition Pfinal of the set of all SUs is formed when the algorithm terminates.

We present some notes on the algorithm. The v(S) and ui(S) we proposed have a significant fall at the edge ∑i ∈ S wi = B, leading to an important property of the coalitions that once two coalitions decide to merge, any of their members will not be split from the large coalition. For instance, if S1 = {SU1, SU2} and S2 = {SU3, SU4} decide to merge intoS3 = {SU1, SU2, SU3, SU4}, S3 will not exclude any of its member. Otherwise S3 would not be formed in the first place. This property implies essentially the proposed algorithm has deployed a complete merge-and-split because even if we allow coalitions to split after Pfinal is formed, no change will occur in Pfinal. So all conclusions of merge-and-split can apply to our algorithm. It should also be noticed that the merging can be simplified because the head can filter the SUs according to bandwidth demand restriction after every acceptance of its invitation. The head only invites qualified SUs and the invited SUs have no incentive to reject in accordance with merge rule. Therefore the time of coalition formation can be significantly reduced.

In practice, every head forms a coalition Sh that maximizes its own utility. Denoting the set of SUs which belong to previous heads’ coalitions by Spre, the current head aims to find

Solving this optimization problem is NP-complete and the complexity increases significantly with the number of SUs N. For small N, (7) may be solved by exhaustive search or iteration method. However for large N there is no practical method to find optimal solutions. We introduce a simple method to help the head with this problem. First the head arranges the available SUs (i.e., ∀i ∈ N∖Spre) decreasingly with regard to payment per unit bandwidth (). Without loss of generality, we assume The head checks if an SU satisfies the bandwidth demand restriction in the above order and invites the first qualified SU (assume it is SUj). Then the head updates its bandwidth demand restriction and continue the search from SUj. This process repeats until the bandwidth demand restriction cannot be satisfied by any SU. The method may not obtain optimal solution for (7), but it significantly reduce time and communication overheads while yielding an acceptable performance in many scenarios. It must be stressed that the method does not take into account the circumstances where SUs negotiate and make deals. Modified method for those scenarios will be introduced in Section 4.

C. Stability analysis

The result of the proposed algorithm is a network partition Pfinal composed of disjoint independent coalitions of SUs. The stability of Pfinal can be investigated using the concept of defection function in Definition 4.

Theorem 1: Every partition obtained from our proposed coalition formation algorithm is hp-stable.

Proof: A hp-stable can be considered as a state where no coalition has an incentive to alter the partition through merge-and-split. The termination of any merge-and-split is evidently hp-stable. And we have explained that essentially our algorithm has deployed a complete merge-and-split in subsection B. Therefore the resulting partition Pfinal is hp-stable.

Before we discuss c-stability, we provide some properties of a c-stable partition proved by the authors of [14].

Theorem 2: If the c-stable partition exists, our proposed coalition formation algorithm will converge to this unique c-stable partition. Otherwise, the proposed algorithm converges to one of the hp-stable partitions.

Proof: It can be proved by Theorem 1 and property 2).

To demonstrate the c-stability and hp-stability explicitly, we introduce two examples.

Example 1: We assume there are 3 SUs. Their bandwidth demands w1=4, w2=3, w3=2. If total available bandwidth is B=4, the c -stable partition exists, i.e., P = {{SU1},{SU2}, {SU3}}. If B=5, the c -stable partition exists, i.e., P = {{SU1},{SU2, SU3}}. Our proposed algorithm guarantees to converge to Pfinal = P.

Example 2: We assume there are 4 SUs. w1=3, w2=3, w3=3, w4=2, and b1=4, b2=3, b3=2, b4=1. We also assume the SUs can afford their bids. If B=5, c-stable partition does not exist and we have three hp-stable partitions. They are P1 = {{SU1, SU4},{SU2}, {SU3}}, P2 = {{SU1},{SU2, SU4}, {SU3}} and P3 = {{SU1},{SU2}, {SU3, SU4}} respectively. Our proposed algorithm converges to Pfinal= P1.

 

4. Potential cooperation between SUs

4.1 Cooperation model

Potential cooperation may exist in coalition formation. Some SUs may have insufficient token storage to afford their spectrum valuation and some SUs may have large token storage but suffer bad channel gains between their transmitters and receivers in this episode. These two types of SUs may cooperate to improve their transmission performance. Assume SUi has extra tokens and suffers a bad channel gain. According to Shannon formula,

where Ri is data rate of SUi, γi is signal-noise-ratio. With a low γi, wi increment leads to very limited data rate improvement. Therefore it is not wise for SUi to invest more tokens for more spectrum. On the other hand, we assume SUj is seeking token support from other SUs to improve its probability of obtaining its demand spectrum. If SUj is willing to share a portion of its transmission time to relay for SUi, then SUi might want to lend SUj some tokens in return. If the relay time slot and lent tokens satisfy both SUi and SUj, this cooperation will bring mutual benefits to both SUs. The cooperation is shown in Fig. 2.

Fig. 2.Transmission time division in cooperation

We assume SUi and SUj transmit in the amplified and forward (AF) mode, SUi achieves a data rate (bits/s/Hz) [17]:

where Pi, Pj are transmission power of SUi and SUj respectively and hi, hj, hij, hji are channel gains shown in Fig. 3. Notice that in the relay time slot SUi and SUj share wi+wj spectrum in their separate transmissions, therefore SUi actually achieves a data rate (bits/s):

In this paper we assume both SUi and SUj know the relay data rate Rr through some mechanism, however SUj does not know direct transmission data rate of SUi (Ri).

Fig. 3.Cooperation between SUi and SUj

Evidently, relay time fraction θ determines the benefit of both SUi and SUj, and a bargaining about θ will be introduced. Under the proposed auction mechanism, the form of this bargaining varies depending on whether SUi and SUj are in the same coalition when bargaining occurs. It is assumed SUi would not negotiate with members of other coalitions since financing them harms its own interest, however SUi tends to finance members of the same coalition (partners) or potential partners. We will discuss the cases where SUi bargains with partners and potential partners in later subsections, respectively.

4.2 Token consumption strategy

We investigate the token consumption strategy before discussing the bargaining cases because the lenders must take into account the value of the tokens they will lend. For different SUs, same tokens may have different values since earning powers are usually different. In the proposed auction mechanism, tokens represent contributions to the network and can only be used to purchase spectrum utilization opportunity. Consequently every SU has a function that maps its achievable data rate in an episode to respective estimated tokens (i.e., first bid). Denote this function of SUk by fk(Rk), where Rk is the data rate. If SUk assumes it has a probability of to win in the auction, fk(Rk) must satisfy the following equations due to the fact that each SU prefers to consume all tokens it earns if the episodes end.

where is the expected data rate of SUk during all episodes and βk is the average tokens SUk earns between two episodes. We assume SUk has knowledge on the distributions of spectrum demand wk and SNR γk, then can be easily obtained through Shannon formula and statistical method. Although there are many functions qualified, we adopt a simple linear function for the “data rate to token” mapping as follows:

In an episode, SUk determines spectrum demand wk, estimates Rk and uses (12) to decide its first bid. Also, when SUk bargain with others, it validates whether data rate improvement introduced by cooperation is worth the tokens lent based on (12).

4.3 Bargain with a partner

If SUj is already a partner of SUi, they share common interest, i.e., spectrum winning probability. In this case, it is appropriate to assume that SUj offers a deal and SUi decides whether to accept. In particular, SUj proposes a relay time slot θ and SUi validates its utility and decides whether to accept. This is a non-cooperative game and non-cooperative game theory can be applied.

Rewrite the notation p(S) as p(T), where T is the total token amount coalition S pays. The utilities of the SUi and SUj based on data rate increment are defined as follows:

where Tm is the total token amount of coalition S will pay if the offer is accepted. Notice that Tm − T = bj − αj. fi-1 (∙) is the inverse function of fi (∙) and fi-1 (Tm − T) can be replaced by The expected utility of SUj can be written

The offer SUj proposes will be accepted if and only if ui(θ) ≥ 0, i.e.,

And SUj attempts to maximize its utility uj(θ), i.e., proposes the optimal θ*

The problem here is SUj does not know SUi direct transmission data rate Ri. Without any prior knowledge, SUj assumes Ri follows a distribution in [K1, K2], where 0 ≤ K1 < K2 < Rr. We have the following theorem:

Theorem 3: Let when A ≥ 0, θ* = 0 and SUi will accept this offer. When A < 0, θ* can be obtained as follows:

where

Proof: Write as

Let ≥ 0, we have

Obviously if ≥ 0, (22) always holds. In that case

Therefore if A ≥ 0, for any , Prob(offer accepted) = 1. Evidently SUj wants θ to be as small as possible. Thus θ* = 0. In essence, this is a circumstance where SUj reminds SUi that increasing SUi’s bid by Tm − T will significantly increase the winning probability of the coalition (as well as its own) and SUi takes this reminding.

If A ˂ 0 ˂ 0. Then

SUj assumes Ri follows a uniform distribution in [K1, K2]. Denote by pacpt (θ) the probability of offer is accepted. We have

It is a piece-wise function. Notice that 0 ≤ K1 < K2 < Rr, therefore

When the utility of SUj is

Differentiating (27) with respect to θ, we have

Evidently θp in (20) is one of the two solutions of (28) and it maximizes uj(θ) (without considering the boundary values). We can prove θp > 0 and uj (θp) > 0 althrough we omit the proof here due to space limitation. However the relationship between , θp and is not constant. Notice that for any θ ∈ [0, θp ), u'j (θ) > 0 and for any > θp, u'j (θ) < 0. Thus we have: if θp≤ , the offer will never be accepted because uj () = 0 and it decreases with θ; if θp ∈ (,), θ*=θp; and if θp ≥ , pacpt (θ) = 1, uj(θ) decreases with θ, SUj checks if P(Tm)(1 − ) Rj − P(T) Rj > 0. If it holds, θ* = ; otherwise θ* does not exist. Therefore Theorem 3 is obtained.

4.4 Bargain with a potential partner

If the first bid of SUj is high but its token storage is relatively low, coalitions will consider whether to invite SUj carefully. If SUj cannot get token fund from others, its contribution to the coalition is negligible. In this circumstance, a coalition S must first figure out if any of its member would fund SUj before it decides whether to invite SUj. Therefore for any coalition member SUi, SUj is a potential partner.

For SUj, its low token storage introduces great disadvantage in its spectrum competition. First, coalition S is not complete when SUj bargains with it, so SUj does not know its final form. Second, SUj does not know if it will be invited by subsequent coalitions as well as their competition power. These uncertainties put SUj in a bad position in the bargaining, therefore it is appropriate to assume in this type of bargaining, SUi will propose an offer and SUj will decide whether to acept. It is natural to assume SUj has a minimum normalized data rate to meet its basic communication demand. If SUi proposes a θ that results in a normalized data rate lower than , SUj will reject this offer. SUi knows this exists, but does not know the value. Define utility of SUj by

where

Notice the coalition is not yet in its complete form, therefore Tm represents the total payment of its current members if SUj is funded. Substituting (30) into (29) we obtain

Utility of SUi remains the same as in (13). SUi attempts to maximize its utility without prior knowledge of . Let (p(Tm) − p(αj)) ⋅ [ Rj (1 − θ) − ] > 0, we obtain

Denote Evidently for any θ ≥ θmax, Rj (1 − θ) − ≤ 0, thereby offer will be rejected. SUi assumes θmax follows a uniform distribution in [K1, K2], where 0 ≤ K1 < K2 < 1. Now we rewrite utility of SUi as

where T represents the total payment of current members (except SUj) of the coalition and

Here we assume Rr > Ri because a potential partner is not yet a partner. If SUj cannot improve data rate of SUi through relaying, SUi will never fund SUj. We have

1) If θ ≤ K1, u'i (θ) = p(Tm)(Rr−Ri) > 0, therefore the optimal value θ* = K1.

2) If K1 < θ < K2,

where A is defined in Theorem 3. Let u'i (θ) = 0, we obtain that maximizes ui (θ). If θp ≤ K1, θ*= K1. If K1 < θp < K2, θ* = θp. However θp ≥ K2 requires both A < 0 and K2 ≤ which results in ui (K2) ≤ 0. Notice ui (θ) increases with θ, so SUi would not assume θmax to be such a value. Therefore the estimation of θmax by SUi must satisfy 0 <

Based on the above discussion, we have the following theorem:

Theorem 4: SUj assumes θmax follows a uniform distribution in [K1, K2]. If A ≥ 0, then 0 ≤ K1 < K2 < 1 ; otherwise Then the optimal value of proposed relay time fraction θ* can be obtained as follows: calculate If θp ≤ K1, θ* = K1; otherwise θ* = θp.

4.5 Modified coalition formation and bidding

In Section 3 we introduced coalition formation without considering potential negotiations and cooperation among SUs. Now we take into account these factors and modify the coalition formation and bidding process. An instance shown in Table 1 is used to illustrate, where total available bandwidth B = 8

Table 1.Instance for modified coalition formation and bidding

Apparently, SU1 is the first head that picks its coalition members. Denote the coalition of SU1 by S1. After SU2 joins S1, S1 needs one of SU5, SU6 and SU7. The member of S1 (SU1 and SU2) negotiates with SU5 respectively. If SU5 makes a deal with either SU1 or SU2, it joins S1; otherwise it remains uninvited. We assume SU5 fails to join S1, so S1 will invite SU6 and its final form will be S1= {SU1, SU2, SU6}. In the next SU3 will be the new head and picks SU4 to join its coalition S2. Again S2 tries to negotiate with SU5. We assume SU4 and SU5 have a deal, hence SU5 joins S2. The coalition formation terminates with S1 = {SU1, SU2, SU6}, S2 = {SU3, SU4, SU5} and S3 = {SU7}. It must be stressed that potential partners will not be chosen as heads. For instance, if SU1, SU2, SU3 and SU4 form coalition S1, then SU5 will not be the head of S2. Instead SU6 will be the head because it has the highest payment per bandwidth even if it is not funded. Before coalitions submit their bids, a negotiation inside S1 (assume it is between SU1 and SU6, and assume the negotiation is successful) is done. Therefore the final payments of coalitions are: S1 pays 24 tokens, S2 pays 21 and S3 pays 3. The payment division inside S1 is: SU1 pays 11, SU2 pays 9 and SU6 pays 4. The payment division inside S2 is: SU3 pays 8, SU4 pays 12 and SU5 pays 1. In the above, SU5 was a potential partner for SU4 therefore discussion in subsection 4.4 applies in their negotiation. SU6 is a partner for SU1 therefore discussion in subsection 4.3 applies.

 

5. Simulation results

In this section we validate the proposed auction mechanism through simulations. We first validate the mutual benefit in the cooperation, then simulate the spectrum allocation.

5.1 Mutual benefit in cooperation

We simulate and illustrate the mutual benefit in cooperation between two partners. We set Ri = 1, Rj = 10, Rr = 10, bi = 10, p(Tm) = 0.56, p(T) = 0.375, Tm = 25, T = 12. Fig. 4 shows the utilities with different relay time fraction θ proposed by SUj. The blue curves are utilities of SUj estimated by itself with different parameters K1 and K2 ([1 1.6] represents K1 = 1 and K2 = 1.6). Different estimations of Ri result in different θ* therefore influence the actual utilities of SUi and SUj. We see θ* of three different estimations are 0.2739, 0.2128 and 0.2490, respectively. The corresponding actual utilities are 0.25, -0.0553 and 0.1258 for SUi, 0.2840, 0.6231 and 0.4220 for SUj. Accurate estimations of Ri may produce relatively large data rate increment of SUj while inaccurate estimations may lead to larger data rate increment of SUi. Notice the cooperation will only be done when both ui > 0 and uj > 0, therefore when SUj assumes Ri ∈ [0.5 1.5], the negotiation will fail. So the cooperation will always introduce mutual benefit for SUi and SUj.

Fig. 4.Utilities of cooperation between partners

To investigate the cooperation between a member of a coalition and its potential partner, we set Ri = 1, Rj = 1, Rr = 10, = 0.8, p(Tm) = 0.55, p(T) = 0.444, p(αj) = 0.022, bi=20, Tm = 50, T = 40. Fig. 5 shows the utilities with different relay time fraction θ proposed by SUi. The blue curves are utilities of SUi estimated by itself with different parameters K1 and K2. Here < K1 < K2. Different estimations of θmax yield different θ* leading to different utilities of SUi and SUj. Three estimations provide three θ* at 0.1494, 0.1744 and 0.2 respectively. The corresponding actual utilities are 0.2278, 0.3403 and 0.4556 for SUi, 0.0242, 0.0122 and 0 for SUj. The offer with θ* = 0.2 will be rejected because uj = 0. Once the negotiation succeeds, the cooperation will lead to a win-win situation.

Fig. 5.Utilities of cooperation between a member and a potential partner

5.2 Token spectrum auction

We investigate the proposed token spectrum auction by studying a specific case with 10 SUs. The parameters of SUs are set as in Table 2. In Table 2, notation [5,7] presents the corresponding variable follows a uniform distribution in [5,7]. In every episode wi, Ri/wi and randomly generate a value in corresponding intervals. We also randomly generate relay data rate shown in Table 3. We run the simulation 10 times and each simulation lasts 100 episodes.

Table 2.Parameters of SUs

Table 3.Data rate for relay transmission

Two contrastive spectrum allocation mechanisms are introduced, named random allocation and token optimal allocation. The random allocation randomly chooses a set of SUs under the condition that the total spectrum demand does not exceed the total available spectrum B in current episode. The token optimal allocation is deployed at the FC. The FC allocates the available spectrum to a set of SUs that maximize its payoff (i.e., the FC aims to collect the maximum tokens in current episode).

Fig. 6 shows the average total data rate in every simulation. In our proposed auction mechanism, due to the randomness in the winner decision, it is possible that an "inefficient" coalition obtains the spectrum resulting in low data rates in some episodes. However this mechanism motivates the SUs with weak competition power. And the significant data rate gain in the relay transmissions in cooperation compensates the aforementioned data rate loss. In fact, averaging the "average total data rate" over 10 simulations we obtain 38.61, 35.85 and 35.39 for three mechanisms respectively. The proposed auction outperforms the other two.

Fig. 6.Average total data rate

Fig. 7 shows spectrum obtaining frequency of each SU. In token optimal allocation, SU1, SU2 and SU3 have very high obtaining frequencies because they earn more tokens every episode (i.e., high βi). With low βi and relatively high spectrum demand wi, SU6 and SU10 show low frequencies (12.4 and 11.9 respectively), and may lose their incentive to participate the spectrum competition. In random allocation, SU8, SU10 and SU7 show high frequencies because w8, w10 and w7 are significantly lower than others. The random allocation neglects the contributions of SUs represented by tokens and obliterates the SUs’ incentive to contribute the network. The proposed token auction shows balanced allocation where SU1 obtains the highest frequency 52.3 and SU10 obtains the lowest frequency 27.4. SUs with low βi improve their frequencies through cooperation. The data indicate all SUs have a fair and acceptable chance to obtain the spectrum in accordance with their contributions.

Fig. 7.Spectrum obtaining frequency of each SU

 

6. Conclusions

This paper proposes a spectrum allocation mechanism based on auction in overlay cognitive radio network. The proposed mechanism quantities contribution of each SU and pays them tokens. The SUs form coalitions and pay tokens to compete the available spectrum in an episode. SUs with low token storage may borrow tokens from other SUs, in return providing relay transmission for the lenders. We model the negotiation between the borrowers and the lenders as Bayesian games because of the incomplete information, and investigate the behaviors and equilibria in these games. Theoretical analysis and simulation results indicate both borrower and lender benefit from cooperation. Simulation results also show our proposed auction yields satisfactory data rate and fairness.