Power Control and Relay Selection in Cognitive

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Power Control and Relay Selection in Cognitive Radio Ad Hoc Networks using Game Theory Onel L. A. L´opez, Samuel M. S´anchez, Samuel B. Mafra, Student Member, IEEE, Evelio M. G. Fernandez, Member, IEEE, Glauber Brante, Member, IEEE, and Richard D. Souza, Senior Member, IEEE

Abstract—This paper employs game theory to optimize the performance of multiple secondary users (SUs) in a cognitive radio ad hoc network where the primary and secondary users have quality of service (QoS) constraints. The proposed algorithm utilizes a distributed non-cooperative game theory mechanism among SUs that compete to maximize their individual capacities. We propose two utility functions, UcLog and UcLin, with logarithmic and linear cost functions of the perceived interference in the primary receiver, respectively. The value of the utility function varies according to the transmit power and the relay selection strategy. Simulation results show that both utility functions obtain similar results in most scenarios. However, UcLin causes less interference to the primary receiver and has better convergence than UcLog. The two consecutive games mechanism used in the proposed algorithm allows to achieve rates above 95% in all simulated scenarios without QoS constraints in secondary links. Index Terms—Cognitive radio, power control, game theory, relay selection.

I. I NTRODUCTION Cognitive radio (CR) is a paradigm of wireless communications that allows unlicensed secondary users (SUs) to adjust their transmission parameters in order to achieve an efficient usage of radio spectrum resources without causing harmful interference to licensed primary users (PUs) [1], [2]. Power control is of paramount importance and has been investigated from different points of view for CR networks, e.g., to increase the concurrent transmission probability [3]–[5], to maximize the system throughput [6], to optimize SUs sum rate [7], [8], or to ensure quality of service (QoS) [9]. Moreover, cooperative communications [10] have emerged as an effective way of taking advantage of the broadcast nature of wireless transmissions. For instance, [11] investigates the power allocation in decode-and-forward (DF) cognitive dual-hop systems and [12] evaluates the outage behavior and power allocation in a two-user network-coded cooperative CR network, both with the goal of improving the power efficiency of the SUs. In addition, when multiple relays are available, This work was supported by CNPq, CAPES and Araucaria Foundation (Brazil) including a scholarship from the Program for Graduate Students from Cooperation Agreements (PEC-PG, of CAPES/CNPq Brazil). Onel L. Alcaraz L´opez, Samuel Baraldi Mafra and Evelio M. G. Fernandez are with Federal University of Paran´a (UFPR), Curitiba-PR, Brazil (e-mails: [email protected], [email protected], [email protected]). Samuel M. Sanchez is with Central University of Las Villas (UCLV), Santa Clara, Cuba (e-mail:[email protected]) Glauber Brante and Richard D. Souza are with Federal University of Technology-Paran´a (UTFPR), Curitiba-PR, Brazil (e-mails: {gbrante,richard}@utfpr.edu.br)

relay selection schemes have been investigated for CR scenarios in [13]–[17]. By employing cooperation at the secondary network, exact expressions for the outage probability of DF relaying, with relay selection, over Rayleigh fading channels, are derived in [13], while outage probability, average symbol error probability and ergodic capacity are obtained in [14] for amplify-and-forward (AF). Combining relay selection with hybrid automatic repeat request mechanisms, [15] investigates the case of SUs that only transmit during the retransmissions of the PUs. On the other hand, relay selection schemes to assist the PUs are assumed in [16], [17], where the SUs in [16] benefit from cooperation by having access to a larger number of idle slots, while a network matching market is used in [17] to model the interaction between PUs and SUs. Out of the many existing algorithms dealing with power control and relay selection, both centralized [7]–[9] or distributed [17]–[20] approaches have been used. Centralized algorithms require global network information and data processing in large scale to obtain optimum results. In contrast, distributed algorithms use only local information influenced by the global network behavior to take sub-optimum decisions, resulting in less complex data processing. In this context, game theory appears as a field of interest, since it analyzes interactive decision situations in the form of games, which have been mainly linked to power control [21]– [25] and relay selection [19], [20]. The above examples deal with non-cooperative games, where players make independent decisions to optimize their individual utility functions. Then, the Nash Equilibrium (NE) is discussed in all these works as the final solution, so that no individual player can benefit from unilateral deviation. For instance, the utility functions are designed in [21], [22] to restrict the interference to the PUs, while in [23]–[25] the goal is to minimize the total power consumption. Moreover, a low-interference relay selection scheme is designed in [19] by taking into consideration the impacts of the interference from SUs and the selected DF relay, while a joint spectrum management and relay selection algorithm is proposed in [20], whose convergence is ensured by means of variational inequality. Furthermore, we notice that game theory has been widely applied in CR networks to solve either power control or relay selection problems separately, while to jointly address both issues in these scenarios has received less attention. One example is the non-cooperative Stackelberg game proposed in [26] for a system where SUs can relay PU traffic employing the AF cooperation protocol. Also, the authors in [27] address a system where only a source-destination pair

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TABLE I N OTATIONS S D {Si , Di } ns Ri gxy Eid Ei PSi L hi W γxy,k Cir I−i,k δi Cp ui u1i γeq Ith u2i µi,k T1 , T2 (t) Ci ςth φPC−RS φAlg.1 Ip min{x, y} x ¯ |X|

Symbols Set of cognitive transmitters (source nodes) Si Set of cognitive receivers (destination nodes) Di Set of SU pairs hSi , Di i Ri Original number of SU pairs. ns = |S| = |D| Relay node selected to assist the i-th SU pair. mi Ri = Rij ∈Ri dxy Gain between nodes x and y Strategy space for the i-th SU pair Eir when transmission is direct Overall strategy ei S space for the i-th SU pair (Ei = Eid Eir ) Transmit power space for Si P Ri Space of slot times li Slot time chosen by Ri to transmit px,k Bandwidth of the links Cid SINR of the link x → y in time slot k σ2 Capacity of the i-th SU link when transm. is cooperative Ii,k Interference caused to the PU link in the time Ci slot k by all SU links but the i-th link player QoS threshold for Di δp Capacity of the primary link G e−i Function utility for the i-th player UcLog utility function Equivalent SINR for both time slots νi,k Interference threshold at primary receiver β UcLin utility function Cost weight of the i-th link in the time slot k t1 , t2 when using UcLin Maximum values allowed for t1 and t2 Pi (t) Capacity of the i-th SU link at iteration t ςi Threshold for the capacity variation measure Convergence of the PC-RS algorithm φPC Convergence of Algorithm 1 nr Interference utilization rate Operators Minimum value between x and y x|ba ¯ denotes time slot L\{k} Complement of x. k x b Number of elements of the set X

coexists with multiple potential relay nodes. Then, unlike the aforementioned works, we focus on power control and relay selection problems in CR ad hoc networks, where relay nodes are willing to assist SUs. In this sense, the main contribution of this paper lies on the proposal of a two consecutive games mechanism, designed to maximize the individual capacities of the secondary links subject to QoS constraints in primary and secondary receivers. Also, it may employ one of two utility functions, namely UcLog and UcLin, with a logarithmic and a linear cost function of the perceived interference at the PUs, respectively. Then, the average number of concurrent secondary links and the convergence of the algorithm are analyzed through simulations. Our results show that both utility functions obtain similar results in most scenarios, but with UcLin causing less interference to the primary receiver and converging faster than UcLog. Moreover, when the number of available relay nodes increases, UcLin has a steeper growth in the number of concurrent secondary links. Finally, we also show that the implementation of the two consecutive games mechanism allows improving the convergence rates in all simulated scenarios. The remainder of this paper is organized as follows: Section II presents the system model and problem formulation,

i-th cognitive transmitter i-th cognitive receiver i-th SU pair Set of potential relay nodes for the i-th SU pair Number of potential relay nodes for the i-th SU Distance between nodes x and y (x = 0 → PTx and y = 0 → BS) Strategy space for the i-th SU pair when transmission is cooperative Strategy adopted by the i-th player from Ei . edi if communication is direct, otherwise eri Transmit power space for relay Ri Slot time chosen by Si to transmit Transmit power of node x in time slot k Capacity of the i-th SU link when transm. is direct Noise power Interference caused to the primary link by the i-th SU link in the time slot k Capacity of the i-th SU link QoS threshold for primary receiver Game formulation Vector containing the strategies adopted by all players but ei Cost weight of the i-th link in the time slot k when using UcLog Factor to minimize the significance of the middle term of UcLin expression Iteration counters for PC-RS and PC respectively Probability of i-th player leaves the game Measure of the capacity variation for the i-th secondary link between iterations t and t − 1 Convergence of the PC algorithm Number of relay nodes distributed in the effective area of each SU link in simulations If x < a then x → a. If x > b then x → b Best response x for current iteration

while Section III describes the proposed games for power control and relay selection. Section IV discusses some results and, finally, Section V concludes the paper. Notation: Table I summarizes the main symbols and operators used throughout this paper, ordered according to their appearance in the text. II. S YSTEM MODEL We consider the uplink of a primary network composed by a primary transmitter (PTx) and a base station (BS). Then, within the coverage area of the BS, which is πR2 , where R is the radius of the cell, ns cognitive transmitters (source nodes) from the set S = {S1 , S2 , · · · , Sns } are trying to establish communication with their destination nodes, belonging to the set D = {D1 , D2 , · · · , Dns }. Each hSi , Di i pair, i = 1, 2, . . . , ns , can be assisted by a set Ri = {Ri|1 , Ri|2 , ..., Ri|mi } of mi potential relay nodes placed in its effective relay area, from which the most suitable can be selected to cooperate. Let the effective relay area for the i-th SU be understood as the area from where the i-th SU could benefit from cooperation of relay nodes located there. Fig. 1 shows an example with ns = 2 SU pairs, where R1 = {R1|1 , R1|2 , R1|3 } and R2 = {R2|1 , R2|2 , R2|3 , R2|4 }

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represent the sets of possible relay nodes for each SU pair. Notice that there are m1 = 3 and m2 = 4 available relays in each set, respectively. Moreover, the effective relay area for each SU pair is surrounded by a dashed line for R1 and a solid line for R2 . Let us remark that, in this particular scenario, R1|3 and R2|1 refer to a same relay node that is able to assist any of the two secondary links.

node PSi . The set PRi = {pRi : pRi ∈ [pRi min , pRi max ]} includes all the possible power levels for each relay node, Ri , selected to assist the i-th cognitive pair. In addition, li ∈ L = {1, 2} defines the strategy of time slot allocation for the i-th source node, while the complementary time slot hi = L\{li } is used by the selected relay node. The capacity of the i-th link considering direct transmission is given by W X log2 (1 + γSi Di ,k ), (2) Cid = 2 ∀k∈{1,2}

where W is the link bandwidth and γSi Di ,k is the signal-tointerference plus noise ratio (SINR) perceived at Di due to Si transmission in time slot k, and can be written as p g PSi ,k Si Di γSi Di ,k = P . (3) pSj ,k gSj Di + pRj ,k gRj Di +p0 g0Di +σ 2 j6=i

j6=i

Here p0 is the transmit power of PTx and σ 2 is the noise power. On the other hand, when cooperation is employed the capacity of the i-th link is given by [29] Cir = Fig. 1. System model composed by a primary transmitter (PTx) and a base station (BS), in coexistence with two SU pairs. The sets of possible relays are R1 = {R1|1 , R1|2 , R1|3 } and R2 = {R2|1 , R2|2 , R2|3 , R2|4 } with m1 = 3 and m2 = 4 available relays, respectively.

where


W log2 1 + min{γSi Ri ,li , γSi Di ,li + γRi Di ,hi } , (4) 2

γSi Ri ,li = P

j6=i

We also consider a log-distance path loss model as [28] gxy

K = α , dxy

(1)

where dxy is the distance between nodes x and y (with x = 0 denoting the PTx and y = 0 denoting the BS), α is the pathloss exponent and K accounts for other factors as the carrier frequency, heights and gains of the antennas [28]. Moreover, we assume very slow fading channels, so that gxy remains constant during long periods of time. Cooperation occurs in two phases of equal-size time slots, denoted by k ∈ {1, 2}. The source node Si transmits in the slot k = 1, which is received by the destination node Di and by the set of relays Ri . Then, one relay Ri = Ri|j ∈ Ri , for some integer j : 1 ≤ j ≤ mi , is selected to cooperate in the slot k = 2. We assume that the relays operate with the DF protocol, and that retransmission only occurs if the message from the source could be correctly decoded. Moreover, we consider that each secondary link hSi , Di i adopts a strategy ei (ei = edi when direct communication is used or ei = eri when cooperation is employed) from the strategy space Ei composed of two vectorial sets of strategies r Eid = hPS Si ,1 , PSi ,2 i and Ei = hPSi , PRi , Ri , Li such that r d Ei = Ei Ei . Regarding Eid , PSi ,1 and PSi ,2 , these are the strategy sets of transmit power that can be used by the i-th source node in slot 1 and slot 2, respectively, and they are defined indistinctly by the set {pSi : pSi ∈ [pSi min , pSi max ]}, which is also used in the cooperative communication case to denote the levels of possible transmit powers of the i-th source

p g PSi ,li Si Ri (5) pSj ,li gSj Ri + pRj ,li gRj Ri +p0 g0Ri +σ 2 j6=i

is the SINR perceived at relay node Ri due to the transmission of Si in time slot li , γSi Di ,li is the SINR perceived at Di due to the transmission of Si in the time slot li and can be computed according to (3) with k = li , while p g PRi ,hi Ri Di γRi Di ,hi = P (6) pSj ,hi gSj Di + pRj ,hi gRj Di +p0 g0Di +σ 2 j6=i

j6=i

is the SINR perceived at Di due to the transmission of the relay node Ri in the time slot hi . III. G AME D ESCRIPTION A. Optimization Problem

Due to the presence of the primary network, the strategies of the cognitive pairs must be limited by the amount of interference caused in the PUs. The interference generated by the i-th SU on the PUs, in each time slot k, is given by  g p , for k = li if ei = eri    Si 0 Si ,k or ∀k ∈ {1, 2} if ei = edi , (7) Ii,k =    gRi 0 pRi ,k , for k = hi if ei = eri

with the overall interference caused by the remaining SU P nodes given by I−i,k = j6=i Ij,k . In addition, each cognitive pair must satisfy its own QoS requirements, so that our main goal is to maximize the individual capacities of the SUs guaranteeing the QoS requirements

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of the primary (denoted by δp ) and secondary links (denoted by δi ). Therefore, the problem can be formulated as follows: max

ei ∈Ei

subject to

Ci Ci ≥ δi Cp ≥ δp ,

(8)

where Ci is the capacity of the i-th secondary link (given by Ci = Cid as in (2) for direct transmission and Ci = Cir as in (4) for cooperation), and Cp is the capacity of the primary link as follows
W X (9) log2 1 + γ00,k . Cp = 2 ∀k∈{1,2}

Here γ00,k is the SINR of the primary link, computed as γ00,k = P

∀i

pSi ,k gSi 0

p0 g00 P . + ∀i pRi ,k gRi 0 + σ 2

Note that maximizing secondary capacity is carried out in (11) by maximizing the equivalent SINR. Moreover, no player can adopt a strategy that increases the interference beyond Ith , once Ii,k ≥ Ith − I−i,k is not valid due to the logarithm Ii,k function. Also, squaring the rate Ith −I in (11) prevents −i,k from obtaining possible negative values for Ii,k . By its turn, UcLin is expressed as X X Ci µi,k Ii,k , (14) ln(Ii,k )− u2i (ei ,e−i ) = +β W ∀k∈{1,2}

∀k∈{1,2}

where β ≪ 1 is used to prevent obtaining negative solutions, µi,k is a weight associated to the i-th link in slot k, which penalizes the growth of Ii,k , and Ci is the capacity of the i-th link as follows (see Appendix A) W log2 (1 + γeq ). (15) 2 Notice from (11) and (14) that both utility functions are bandwidth independent. Moreover, UcLog maximizes the SUs capacity by exploiting the equivalent SINR, while UcLin focuses on the spectral efficiency to increase the capacity. In addition, it is also worth noting that the interference caused to the PUs with UcLin in (14) is not limited in the same way as with UcLog in (11). Hence, we need some constraints in order to reject the solutions that could jeopardize the QoS of the PUs, which is addressed in the following subsections. Finally, the fact that all SUs in the game must achieve their own QoS requirements also imposes other constraints to the algorithm, which is also handled in the following. Ci =

(10)

Then, to solve the optimization problem in (8), an analytic model in terms of game theory can be stated in its general form as G = h{Si , Di }, {Ei }, {ui }i, where {Si , Di } is the set of players, {Ei } is the set of strategies of the players, and {ui } represent their utility functions. When S players choose their strategies along the full set Ei = Eid Eir , a Power Control and Relay Selection (PC-RS) mode is established. However, the nodes can also play under a Power Control (PC) mode, allowing them to adapt only the transmit power while keeping the rest of the strategy set fixed (e.g., employing the same selected relay node). B. Utility Functions The selection of the utility function depends on the strategy adopted by the i-th player (ei ), and on the strategies adopted by the other players (e−i ). We propose two utility functions to improve the secondary channel capacity for each player, namely UcLog and UcLin. UcLog gives a relevant weight to the equivalent SINR of the link while the increases of the interference caused to the PUs are slightly penalized by a logarithmic cost function. On the other hand, UcLin aims directly at increasing the capacity of the SUs, while it limits the interference caused to the PUs more strictly, by employing a linear cost function. Consequently, the interference caused to the rest of secondary receiver nodes is also strictly limited. The UcLog function is expressed as 2!  X I i,k , (11) νi,k ln 1− u1i (ei ,e−i ) = γeq + Ith −I−i,k ∀k∈{1,2}

where Ith is the allowable interference threshold at the primary receiver, which is given by p g Ith = δp0 00 , (12) 2W − 1 νi,k is the weight associated to the cost of the i-th link in slot k and γeq is the equivalent SINR (see Appendix A), given by  γSi Di ,1 +γSi Di ,2 +γSi Di ,1 γSi Di ,2 , if ei ∈ Eid γeq = (13) min(γSi Ri ,li , γSi Di ,li +γRi Di ,hi ), if ei ∈ Eir .

C. Game Solution To solve non-cooperative games, usually a Nash Equilibrium situation is exploited. A NE is a strategy vector that corresponds to the mutual best response: for each player i, the action selected is the best response to the actions of all other players. Equivalently, a NE is an action tuple where no individual player can benefit from unilateral deviation. Theorem 1 A strategic game h{Si , Di }, {Ei }, {ui }i has at least one NE if, ∀hSi , Di i, the action set Ei is a non-empty compact convex subset of an Euclidean space, and the utility function ui is continuous and quasi-concave on Ei [30]. Lemma 1 The strategic game h{Si , Di }, {Eid }, {ui }i operating only with the direct communication mode (without relaying) has at least one NE point. The proof of Lemma 1 can be found in Appendix B. 1) Direct communication only: Assuming only direct communication, the best response function pbSi ,k for each player hSi , Di i in time slot k is obtained by solving ∂p∂ui = 0. Si ,k Using the UcLog function in (11), the game solution yields q pS −νi,k Yk + (νi,k )2 Yk2 +Xk2(1+γSi Di ,k¯ )2 imax , pbSi ,k = Xk Yk (1+γSiDi ,k¯ ) pS

imin

(16) where the notation x |ba indicates that: if x > b then x ← b (b is the upper bound of x); if x < a then x → a (a is the lower

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bound of x); k¯ denotes time slot L \ {k}; and Xk and Yk are computed as g P Si Di , (17) Xk = P pSj ,k gSj Di + pRj ,k gRj Di +p0 g0Di +σ 2 j6=i

j6=i

Yk =

gSi 0 P P . Ith − ( j6=i pSj ,k gSj 0 + j6=i pRj ,k gRj 0 )

(18)

On the other hand, when UcLin in (14) is used, the best response function is given by " 2 β ln−1 (2) 1 1 pbSi ,k = + − + + 2µi,k gSi 0 4µi,k gSi 0 2Xk 2Xk ∗ !2 ! 12 pSimax ,k −1 −1 ln (2)+2β 2β −ln (2)  + , (19) + 4µi,k gSi 0 4µi,k gSi 0 Xk pS

where p∗S

imax ,k

p∗S

imin

is computed as

imax ,k

!

Ith − I−i,k = min pSi , max gSi 0

.

(20)

Please see Appendix C for the derivation of (16) and (19). Moreover, note that the upper bound of (19) is not fixed but depends on the strategies used by the other players to avoid exceeding Ith . The optimum direct transmission strategy (b edi ) for one player is found by computing only these best responses strategies for its source node in both time slots. 2) Cooperative communication: When cooperation is employed, the best responses are functions of γeq . Then, for UcLog in (11), if γeq = γSi Di ,li + γRi Di ,hi we have q pSi −νi,li Yli + (νi,li Yli )2 + Xl2i max pbSi ,li = , (21) Xli Yli pS

imin

where Xli is computed according to (17) with k = li , while if γeq = γSi Ri ,li , then q pSi −νi,li Yli + (νi,li Yli )2 + Zl2i max pbSi ,li = , (22) Zli Yli pS

imin

where Zli is computed as Z li = P

j6=i

pSj ,li gSj Ri +

P

j6=i

g S i Ri . (23) pRj ,li gRj Ri + p0 g0Ri + σ 2

In (21) and (22), Yli is calculated according to (18) with k = li . In both cases, the transmit power used by the selected relay node can be expressed as ! gSi Di νi,hi pbRi ,hi = − + gRi Di Xhi v !2 !2 pRimax u u g gSi 0 ν Si Di i,hi + , (24) +t gRi Di Xhi gRi 0 Yhi pR

imin

where Xhi and Yhi come from (17) and (18) with k = hi . Similarly, for UcLin if γeq = γSi Di ,li + γRi Di ,hi we have !2 " 2β + ln12 1+γRiDi ,hi 2β + ln12 pbSi ,li = − + + 4µi,li gSi 0 2Xli 4µi,li gSi 0 # 1 p∗Si
  1+γRiDi,hi 2 2β − ln12 (1+γRiDi,hi ) 2 max + , (25) + 2Xli 4µi,li gSi 0 Xli pS

2β + ln12 − pbRi ,hi = 4µi,hi gRi 0 +

1+γSi Di ,li 2Xhi gR gS

i Di

i Di

1+γSi Di ,li  + g 2Xhi gRi Di

"

imin

2β + ln12 4µi,hi gRi 0

!2

+

Si Di

p∗ # 12 Rimax 2β − ln12 (1+γSiDi ,li )   , (26) + gR D 4µi,hi gRi 0 Xhi g i i S D

!2




i

i

pR

imin

where Xli and Xhi come from (17) with k = li and k = hi , respectively, p∗S comes from (20) with k = li , and imax ,li ! Ith − I−i,hi ∗ pRi = min pRimax , . (27) max g Ri 0

On the other hand, if γeq = γSi Ri ,li , !2 " 1 2β + ln−1 2 2β + ln−1 2 − + + pbSi ,li = 4µi,li gSi 0 2Zli 4µi,li gSi 0 # 12 p∗Simax −1 1 2β − ln 2 + , (28) + 2 4Zli 4µi,li gSi 0 Zli pS

imin

and pbRi ,hi is computed according to (26) after using pbSi ,li obtained from (28) to determine the corresponding γSi Di ,li . The derivation of (21), (22), (24)-(26) and (28) are similar to (16) and (19), which we omit here. Moreover, it is worth noting that our algorithm requires complete information as in [18], [21]. Nevertheless, since channel variations are assumed to be very slow, the CSI obtained by players remains valid for long periods. Finally, we observe that there exists a lack of continuity of (11) and (14) when ei ∈ Eir . Therefore, unlike when only direct communication is available, we can not ensure the NE existence for the cooperative communication mode, since we only adopt pure strategies throughout this paper. Moreover, related issues are also discussed in the numerical examples of Section IV. D. Description of the Proposed Algorithm Algorithm 1 shows the Optimization Algorithm for the individual capacities of the secondary links. First, each secondary transmitter starts with the minimum power. Then, the largest possible player set is formed in lines 2-3, discarding the nodes that most interfere at the PUs in order to satisfy the Ith constraint. This step maximizes the number of SUs taking part in the game, thus increasing the number of concurrent

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Algorithm 1. Optimization Algorithm 1: pS ,k = pS i

2: 3: 4: 5: 6: 7: 8: 9: 10:

11: 12: 13: 14: 15: 16: 17: 18: 19: 20:

imin

Sort {Si , Di } in ascendingP order according to Ii . Form {Si , Di }′ , such that ∀hSi ,Di i Ii ≤ Ith and |{Si , Di }′ | is maximum. run the PC-RS algorithm if t1 < T1 then jump to line 9 end if run the PC algorithm if Ci < δi for some hSi , Di i then Each hSi , Di i leaves randomly the competition (up′ ′ date {Si , Di }′ ← {S
i , Di } \hSi , Di i ) with probability Ci Pi = 1 − min δi , 1 . else jump to line 20. end if run the PC-RS algorithm if t1 < T1 then jump to line 9 end if run the PC algorithm jump to line 9 End

Algorithm 2. PC-RS Algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25:

links. Notice that lines 1-3 run only once to form a starting point to the algorithm. Then, when the game ends, the SUs that have been discarded (or even new players) can compete in a second step to find their best strategies. However, lines 1-3 are not needed in these cases. The line 4 of the Optimization Algorithm calls PC-RS in order to find the optimum strategy ebi ∈ Ei for each player, which is described in Algorithm 2. To that end, we select the relay node that maximizes the utility function in that iteration (t1 ) and we compare this strategy (b eri ) with the optimum strategy using direct communication only (b edi ). Then, if cooperation is more advantageous, the selected relay is marked, Ri = Rij , and no other player can use it. Moreover, aiming to balance the interference caused to PUs, li is chosen such that I−i,li > I−i,hi if gSi 0 ≤ gRij 0 , otherwise li must satisfy I−i,li < I−i,hi . Then, γeq is calculated using (13) and selected as part of the cooperative strategy for that player. The PC-RS algorithm ends after T1 iterations or when the value of the capacity variation C (t) − C (t−1) i (t) i (29) ςi = < ςth , (t−1) Ci (t)

where Ci is the capacity of the i-th link at iteration t, with (0) Ci = 0, and ςth is a variation threshold. If the number of iterations of the PC-RS algorithm exceeds T1 , we then run the PC algorithm in line 8 of Algorithm 1, where the strategy space is limited. The goal is to provide a valid solution even when PC-RS does not converge. Such mechanism with two consecutive games improves the convergence of the optimization algorithm even when cooperative conditions do not allow to prove the existence or uniqueness

26: 27: 28: 29: 30: 31:

t1 = 1 while t1 < T1 and ς (t1 ) > ςth do for i = 1 to |{Si , Di }′ | do if ei = eri then pRi ,hi = 0 end if Calculate pbSi ,1 ,b pSi ,2 and form b edi = hb pSi ,1 , pbSi ,2 i for j = 1 to mi do if Rij is already selected then continue end if Select li and hi pRij ,hi = pRij and pSi ,hi = 0 min if Ii,hi > Ith − I−i,hi then pRij ,hi = 0, pSi ,hi = pSi , uij = −∞ min continue end if Calculate pSi ,li , pRij ,hi Form b erij = hpSi ,li , pRij ,hi , Rij , li i Calculate uij = ui (b erij ) end for Find Rij with the larger utility ebri = ebrij edi ) then if ui (b eri ) ≥ ui (b r ebi =b ei and Ri = Rij else ebi = ebdi end if end for t1 ← t1 + 1 end while

of the NE. In fact, if PC-RS and PC converge in φPC−RS % and φPC % of the scenarios, respectively, then Algorithm 1 converges in φAlg.1 = [100 − (100 − φPC−RS)(100 − φPC )]%. The PC mode, by its turn, is described in Algorithm 3, whose maximum number of iterations is denoted by T2 . Here, the communication strategy (direct or cooperative) previously adopted remains unchanged, and only the transmit powers are recalculated at each iteration until ςi < ςth or t2 > T2 . Let us remark that the lack of guarantee for the NE existence in the cooperative communications scenario forces the careful selection of T1 and T2 . If T1 and T2 are too small, the number of possible convergent scenarios decreases. On the other hand, increasing T1 and T2 also increases the computational burden of the algorithm. Furthermore, when the SUs have QoS requirements (δi 6= 0, for some hSi , Di i), it may become necessary to remove some of the players from the game when their QoS targets are far from being met, so that the remaining players can still achieve their requirements. To do that, we employ a probabilistic method where players decide to leave the game randomly, as depicted by lines 9-10 of Algorithm 1. The probability Pi depends on how far δi is to be achieved. Moreover, while there are players with capacities below the target (Ci < δi )

7

was considered in the simulations, so the values of Ci match the spectral efficiency of the system. Moreover, we assume α = 4 and that all mobile stations are equipped with unity-gain antennas at a height of 1 m. In the simulations, the weights associated to the cost factors are fixed and independent of the time slot and the session, and were set at νi,k = 10−1 and µi,k = 2 × 107 , ∀hSi , Di i and k ∈ {1, 2}. These values were chosen in order to obtain an appropriate system performance according to the parameters of Table II.

Algorithm 3. PC Algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

t2 = 1 while t2 < T2 and ς (T1 +t2 ) > eth do for i = 1 to |{Si , Di }′ | do if ebi ∈ Eid then Calculate pbSi ,1 , pbSi ,2 and update ebi else Calculate pbSi ,li , pbRi ,hi and update ebi end if end for t2 ← t2 + 1 end while End

TABLE II S YSTEM PARAMETERS

still playing, the PC-RS and PC algorithms are run repetitively. Otherwise, the optimization algorithm ends. Let us remark that eliminating players without chances of satisfying their QoS requirements allows a better distribution of the resources for the remaining players. Finally, in order to quantify the amount of interference caused at the PUs (which must be below Ith ), we employ the interference utilization rate defined as Ip =

nS X 1 X Ii,k . 2Ith i=1

(30)

∀k

IV. N UMERICAL R ESULTS We evaluate the proposed algorithm by averaging 5000 randomly generated topologies. For each topology, the locations of the ns cognitive pairs (sessions) are uniformly distributed in a circular area of radius R, served by a primary network. Moreover, nr relay nodes are uniformly distributed in the area constrained by dSi Di ≥ dSi Rij and dSi Di ≥ dRij Di , ∀Rij ∈ Ri for each pair hSi , Di i (defining the effective relay area, such that j = 1, 2, . . . , |Ri |), as shown in Fig. 2. As an example, suppose that there are five sessions (ns = 5) and that six relay nodes are uniformly distributed in the effective area of each secondary link (nr = 6). Then, the number of potential relay nodes for the secondary network is 5 × 6 = 30. However, this does not mean that |Ri | = 6, ∀hSi , Di i, since some relays may belong to more than one session, and thus |Ri | ≥ 6.

Parameter ns R d00 W σ2 p0 δp pSi , pRij ∀Si , Rij min min pSi , pRij ∀Si , Rij max max T1 ‡ T2 ‡ ςth β

Value 5 [8] 100 m [8] 25 m [8] 1 Hz [8] 10−15 W [31] 0.1000 W [31] 3 bps [8] 0.0125 W [8] 0.8000 W [8] 40 30 1% 10−8

Moreover, we compare the proposed game theory algorithm, using both UcLog and UcLin utility functions, with the Transmitter Selection with Power Control (TS-PC) algorithm proposed in [8], which employs a centralized power control approach. TS-PC assigns a power level to each secondary transmitter, among m feasible different levels, such that the overall capacity of the secondary network is maximized and the interference caused to PU remains below an established threshold. It is interesting to compare these algorithms in terms of capacity and number of concurrent links of the secondary network, and to highlight the main differences in the results taking into account that the proposed game theory algorithm is distributed while TS-PC is centralized. In the simulations, TS-PC was configured with m = 6 discrete power levels available for the SUs, chosen from the set [−∞ 11 15.5 20 24.5 29] dBm.

Ri∣1

A. Non-Cooperative Scenarios Ri∣3

Ri∣2 d S i Di

Si

Di

Ri∣4 Ri∣mi

Fig. 2. Effective area for relay nodes.

Unless otherwise specified, the system parameters are used according to Table II. Note that a bandwidth of W = 1 Hz

Figs. 3 and 4 compare UcLog, UcLin and the TS-PC algorithm in a non-cooperative scenario, in terms of the overall capacity of the secondary network and of the number of concurrently active secondary links, respectively. Note that, when there are no QoS constraints for the SUs (δi = 0), TSPC performs best in terms of overall capacity of the secondary network as the number of SUs increases (Fig. 3). However, the number of concurrent links is quite smaller than that achieved by the game theory algorithm (Fig. 4). This is due to the very ‡ Experimental tests showed that these are appropriate values for the maximum number of iterations in each case.

8

8

8

6

5

4 TS-PC with δ i=0 bps UcLin with δ =0 bps i

3

UcLog with δ =0 bps i

TS-PC with δ =0.1 bps i

2

UcLin with δ i=0.1 bps

Average overall secondary capacity (bps)

Average overall secondary capacity (bps)

TS-PC with δ i=0 bps

7

UcLin with δ =0 bps

7

i

UcLog with δ i=0 bps TS-PC with δ i=0.1 bps

6

UcLin with δ =0.1 bps i

5

UcLog with δ =0.1 bps i

4 3 2 1

UcLog with δ =0.1 bps i

1

0 1

2

3

4

5

6

7

nS

0

10

20

30

40

50

60

70

80

90

100

d 00 (m)

Fig. 3. Average overall secondary capacity as a function of ns .

Fig. 5. Average overall secondary capacity in function of d00 .

Average number of concurrent secondary links

7 TS-PC with δ i=0 bps UcLin and UcLog with δ =0 bps

6

i

TS-PC with δ i=0.1 bps UcLin with δ i=0.1 bps

5

UcLog with δ =0.1 bps i

4

3

2

1

0 1

2

3

4

5

6

7

nS

Fig. 4. Average number of concurrent secondary links as a function of ns .

nature of these algorithms, since TS-PC allocates power to maximize the overall secondary capacity, even if some SUs are assigned with 0 W (−∞ dBm) for that end. On the other hand, the algorithm proposed here allocates power to maximize the individual capacities of the active players. When QoS constraints are imposed to the secondary links (e.g., δi = 0.1 bps), UcLin and UcLog perform very close to TS-PC in terms of overall capacity of the secondary network, since there are better spectrum sharing opportunities for the sessions that remain in the game when some players have been eliminated. It can also be noted that, for small ns , UcLog and UcLin configurations show similar and even superior performance than TS-PC. This is because TS-PC uses a discrete transmit power set, situation that becomes more critical when there are few SUs. In terms of the number of concurrent secondary links, UcLog and UcLin achieve the same performance when there are no QoS constraints, since only SUs that cause harmful interference to the primary receiver are eliminated (which only depends on their location in the topology), while the difference is still insignificant when QoS constraints are imposed.

Next, Figs. 5 and 6 compare UcLog, UcLin and TS-PC in terms of the overall secondary capacity and the number of concurrent links, respectively, as a function of the distance between PTx and the BS. Increasing d00 reduces the interference threshold at the BS; hence, the SUs have to make a less extensive use of the spectrum resources. The overall secondary capacity (Fig. 5) of UcLog configuration is maximum for d00 = 25 m and has decreasing values for lower distances, because it is a design parameter considered during the cost definitions for the game theory configurations. In spite of that, UcLin has a stable performance when d00 decreases from 25 m to zero, while TS-PC shows a steeper growth in that interval. When d00 > 25 m, both utility functions have decreasing values with increasing distance as can be expected, but UcLin shows the higher decreasing rate until d00 = 70 m. Hence, the performance of UcLog is more sensitive to the variation of system characteristics when the cost factors are fixed. Once again, TS-PC exhibits the best performance in terms of the overall secondary capacity when δi = 0 bps. The existence of less concurrent links in comparison with UcLog and UcLin allows a more rigorous use of the spectrum resources. On the other hand, an opposite behavior is observed in Fig. 6, which shows that there is a trade-off between the overall secondary capacity and the number of concurrent secondary links for all schemes. Therefore, selecting the cost factors for both utility functions is crucial to achieve certain behavior of the system, once increasing or decreasing their values translates into a bigger/smaller overall secondary capacity while the number of concurrent links decreases/increases. Changing the distance between the primary nodes has a similar effect than the variation of the QoS constraint for the primary link (δp ). Variation of some of these parameters translate into changes in the availability of spectrum resources. Thus, increasing (or decreasing) d00 and δp has a somehow similar effect in the performance of the secondary network with the cost factors as previously defined.

9

2 TS-PC with δ i=0 bps

4.5

UcLin and UcLog with δ =0 bps i

TS-PC with δ i=0.1 bps

4

UcLin with δ i=0.1 bps UcLog with δ =0.1 bps

3.5

i

3 2.5 2 1.5 1 0.5

Average number of concurrent secondary links

Average number of concurrent secondary links

5

0

1.95 1.9 1.85 1.8 1.75 1.7 1.65 1.6

UcLin with δ i=0.1 bps UcLog with δ =0.1 bps i

1.55 1.5

0

10

20

30

40

50

60

70

80

90

100

0

1

2

3

4

5

6

7

d 00 (m)

9

10

11

12

13

14

15

Fig. 8. Average number of concurrent secondary links as a function of nr .

Fig. 6. Average number of concurrent links in function of d00 .

6.2

100

6

95

5.8 90 5.6

UcLin with δ =0 bps i

85

UcLog with δ i=0 bps

5.4

UcLin with δ =0.1 bps i

UcLog with δ =0.1 bps

5.2

i

¯Ip (%)

Average overall secondary capacity (bps)

8

nr

80 75

5

UcLin with δ i=0 bps

4.8

70

4.6

65

UcLog with δ =0 bps i

UcLin with δ =0.1 bps i

4.4 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

nr

Fig. 7. Average overall secondary capacity as a function of nr .

UcLog with δ i=0.1 bps

60 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

nr

Fig. 9. Mean utilization rate of the primary interference threshold as a function of nr .

B. Cooperative Scenarios To extend the previous analyzes to a cooperative communications scenario, we present some simulation results as a function of the number of available relay nodes (nr ), while we set ns = 5 and d00 = 25 m. Figs. 7 and 8 show the performance of the game theorybased schemes in terms of the overall secondary capacity and of the concurrent secondary links, respectively. The number of concurrent secondary links when there are no QoS constraints is not shown because it remains unchanged spite the use of cooperative scenarios. In both figures, a positive impact of cooperative communications in the performance metrics is observed. As we can notice, UcLin has a more steady growth than UcLog in terms of the number of concurrent secondary links, and even outperforms UcLog when nr ≥ 8. Nevertheless, when we look at overall capacity, UcLog outperforms UcLin for almost all the cases. The mean interference utilization rate (I¯p ) is shown in Fig. 9 in scenarios with and without secondary QoS constraints. Note that using cooperative communications increases the usage of spectrum resources, which translates in higher interference to

the PUs. Furthermore, when δi = 0, more cognitive users transmit simultaneously, yielding an increased use of the available spectrum resources. On the other hand, UcLog makes a more extensive use of spectrum resources, but in the worst case (nr = 15 with δi = 0 bps) it does not exceed 96% of utilization. As a results, UcLin uses less spectrum resources than UcLog in all scenarios. The convergence of the proposed optimization algorithm is shown in Fig. 10, for both UcLog and UcLin configurations in scenarios without QoS secondary constraints. As previously discussed, the absolute convergence of the proposed algorithm is not always guaranteed since the existence and uniqueness of the NE point cannot be proved for all possible scenarios. However, a two consecutive games mechanism has been established to significantly improve the convergence conditions of the algorithm. The individual convergences of the PC-RS and PC algorithms are shown in Fig. 10, as well as the global convergence rate of the proposed algorithm. As we can notice from Fig. 10, UcLin converges in noncooperative scenarios always reaching a NE point. Comple-

10

100

Convergence of the Algorithms

95

90

85

80

φ

PC-RS

, UcLin

φ PC-RS , UcLog

75

φ φ

70

PC PC

, UcLin , UcLog

φ Alg.1 , UcLin φ

Alg.1

, UcLog

65 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

nr

Fig. 10. Convergence of the optimization algorithm for δi = 0 bps.

mentarily, a marked mean-decreasing tendency for UcLog is observed in the PC-RS mode when nr increases (i.e., in cooperative scenarios), while UcLin has a more steady behavior beyond nr = 2, outperforming UcLog ∀nr . Moreover, since the PC mode is only executed when the PC-RS algorithm does not converge, the number of scenarios where it must be used is significantly smaller and variable as a function of nr . Therefore, the PC mode exhibits a more disperse behavior, specially when UcLog is considered. In addition, Fig. 10 clearly shows that the real convergence of UcLin is close to 100% of the time. By its turn, UcLog has a mean-decreasing behavior, converging within T1 + T2 iterations more than 95% of the time when δi = 0, percentage that is even increased when δi 6= 0, since the players that do not satisfy their QoS requirements exit from the game. V. C ONCLUSIONS This paper proposes a fully distributed game theory-based algorithm for power control and relay selection in a cognitive radio network, aiming at maximizing the individual capacities of the secondary links while guaranteeing both primary and secondary QoS requirements. Two utility functions (UcLog and UcLin) are evaluated and compared with the centralized TS-PC algorithm [8]. Our results show that, when the primary nodes are closer to each other (d00 < 25 m), the spectrum availability is high and UcLin outperforms UcLog in terms of the overall secondary capacity while UcLog outperforms UcLin in terms of concurrent secondary links. On the other hand, when d00 > 25 m, UcLog maximizes the overall secondary capacity while UcLin maximizes the number of concurrent secondary links. Moreover, when only non-cooperative transmissions are employed, the existence of NEs has been proven. In cooperative scenarios, despite the fact that the existence and uniqueness of the NE cannot be proven, the convergence rate is still very high due to the mechanism of employing two consecutive games. In this cases, UcLin converges almost 100% of the time, while UcLog converges more than 95% of the time.

A PPENDIX A If the transmission is direct ei ∈ Eid , so that the capacity of the i-th link is given by (2). With some algebra we get W W log2 (1 + γSi Di ,1 ) + log2 (1 + γSi Di ,2 ) Cid = 2 2 W = log2 [(1 + γSi Di ,1 )(1 + γSi Di ,2 )] 2 W = log2 [1 + γSi Di ,1 + γSi Di ,2 + γSi Di ,1 γSi Di ,2 ] 2 W = log2 [1 + γeq ]. (31) 2 On the other hand, with a cooperative transmission ei ∈ Eir and the capacity of the i-th link is given by (4). Then:
W log2 1 + min{γSi Ri ,li , γSi Di ,li + γRi Di ,hi } Cir = 2 W = log2 [1 + γeq ]. (32) 2 A PPENDIX B The strategy space of each player {Si , Di } is non-empty, closed, compact and convex. Next we are going to prove the quasi-concavity of both utility functions. Expression for u1i given in (11) can be stated as follows using (7), (13), (17) and (18): Eid

u1i =pSi ,k Xk + γSi Di ,k¯ + pSi ,k Xk γSi Di ,k¯ + + νi,k ln(1 − p2S

i ,k

Yk2 ) + νi,k¯ ln(1 − p2S

¯ i ,k

Yk¯2 ).

(33)

Then, 2Yk2 νi,k pSi ,k ∂u1i = Xk (1 + γSi Di ,k¯ ) − , ∂pSi ,k 1 − p2S ,k Yk2

(34)

−2νi,k (Yk )2 (1+(Yk pSi ,k )2 ) ∂ 2 u1i < 0,∀νi,k > 0. = ∂p2S ,k (1−(Yk pSi ,k )2 )2

(35)

i

i

Similarly for u2i given in (14), and using (7), (13), (15) and (17), we have: 1 u2i = log2 (1 + pSi ,k Xk + γSi Di ,k¯ + pSi ,k Xk γSi Di ,k¯ )+ 2 + β ln(pSi ,k gSi 0 ) + β ln(pSi ,k¯ gSi 0 )+ − µi,k pSi ,k gSi 0 − µi,k¯ pSi ,k¯ gSi 0 .

(36)

Then, Xk (1+γSi Di ,k¯ ) ∂u2i 1 = + ∂pSi ,k 2 ln(2) 1+pSi,k Xk +γSi Di ,k¯ +pSi ,k Xk γSi Di ,k¯ β − µi,k gSi 0 + pSi ,k Xk β 1 + − µi,k gSi 0 , (37) = 2 ln(2) 1 + pSi ,k Xk pSi ,k ∂ 2 u2i β −(Xk )2 − 2 = < 0, ∀β ≥ 0. (38) 2 2 ∂pSi ,k 2 ln(2)(1 + Xk pSi ,k ) pSi ,k Therefore, the utility functions u1i and u2i are quasi-concaves since they are continuous; and their second-order derived with respect to pSi ,k , ∀k ∈ {1, 2}, are smaller than zero. Then, according to Theorem 1, the game h{Si , Di }, {Eid }, {ui }i has at least one NE pure strategy, which completes the proof.

11

A PPENDIX C Finding best response means to find the value that satisfies = 0. For UcLog and using (34), the solution yields

∂ui ∂pS ,k i

0 = Xk (1 + γSi Di ,k¯ ) −

2Yk2 νi,k pSi ,k 1 − p2S ,k Yk2 i

0 = Yk2 Xk (1+γSi Di ,k¯ )p2S ,k +2Yk2 νi,k pSi ,k −Xk (1+γSiDi ,k¯ ) i q 2 2 Yk νi,k ± Yk νi,k + Xk2 (1 + γSi Di ,k¯ )2 pSi ,k = . (39) Yk Xk (1 + γSi Di ,k¯ ) Since pSi ,k > 0 and it also must be limited by the transmit power range, [pSi pSi ], then (39) transforms into (16). max min For UcLin and using (37), we have Xk β 1 + − µi,k gSi 0 2 ln(2) 1 + pSi ,k Xk pSi ,k
Xk 0 = µi,k gSi 0 Xk p2S ,k + µi,k gSi 0 − −βXk pSi ,k − β i 2ln 2  1 1 β β 2 2 ln 2 0 = pS ,k + p − − − . (40) i Xk µi,k gSi 0 µi,k gSi 0 Si,k Xk µi,k gSi 0 0=

Let A = p2S

1 Xk ,

B=

ln(2)−1 2µi,k gS 0

and C =

i

β µi,k gS

in (40), then i0

+ (A − B − C)pSi ,k − AC = 0 r B C A (B + C − A)2 pSi ,k = + − ± + AC 2 2 2s 4 2  2  B C A A A B +C pSi ,k = + − ± + + (C −B). (41) 2 2 2 2 2 2 i ,k

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