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NTRODUCTION tdn ebr IEEE, Member, Student fScnayUsers Secondary of balhShami, Abdallah 30 Mexico 7360, Engineering, ege Primak, Serguei a e-mail: da, formation nstituto d In ed. kers um me ce es in s, d s ebr IEEE Member, h rbe fgmswt nucranpplto,i other in population, uncertain addresses an analysis, with Poisson theory games game for of of problem results random branch the Myerson’s and This on th [19]. heterogenous based solve games provide a SUs, to for we of order access specific, population in multiple more analysis of be and problem extension To interpretation natural DCF different mechanism. a a Enhanced is DCF which the the the [18], the by addresses of 802.11e in paper approached IEEE SUs this in that of included contrast, to number In similar the [17]. problem in [16] randomness [15] t the different system having nor [14] of and users, possibility [13] the of in either consider authors not throughput the do the Notwithstanding, maximize scheme. to in the how described shown of is had algorithm authors backoff The 802.11 [14]. IEEE the in window o h U nodrt aiietetruhu sotie.A obtained. is throughput the maximize to p order chann the in access SUs t to the to attempting for interference of restrict probability optimal and acces An with users PU. the secondary algorithm coordinate the to MAC the order among in distributed proposed a is memory [13] one-slot In equally.) treated esadrciesi laskondrn h nlssand analysis ( probability the access during common known a share always transmit users is all of receivers number the and that ters assumes which author networ ALOHA CR the an in Nonetheless, slotted probability calculates delivery on author successful the based the maximizes probability [12] enforc to In access order players. optimal in the [11] among in Nash fairness proposed with the ALOHA is slotted model, reception on multipacket based this characte game-theoretic is in pricing region a a stability that Furthermore, its [8] and show exist in authors must performance equilibrium hand, The the studied. informatio other perfect evaluate with is the ALOHA to slotted On multipacket order of model DCF. in number the total known the of that is th the assumed of players how is throughput it consider of sum [10] the not in affect does Moreover, can system. game traffic of DCF types out proposed different which pointed the as points However, [9], manner. access t competitive in transmit a or nodes in all stations frames hence channel, data base the to no access game the are In control a there 802.11. [9] IEEE DCF In in Distrib the [9]. mechanism the for (DCF) [8] point Function starting [2] a Coordination as [1] tools presented as theory is model game such theory using literature analyzed previous been in has problem (MAC) pritn rtclt oto h eeto fteconten the of selection the control to protocol -persistent ebr IEEE, Member, aeiKontorovich, Valeri i.e. elw IEEE, Fellow, l sr are users all rized. ypes uted heir tion ks. he el n e e e s 1 - 2 words, when the number of players1 is unknown and can be (not counting himself), who choose each possible action. The modelled as a random variable. number of players for each possible element in C is listed in The paper is organized as follows. Section II summarizes a vector called the action profile. Finally, the last term of the the theoretical basis of Poisson Games and two multiple access tuple is the utility defined as u = (ut)t∈T where ut(a, x) is examples are given. Section III presents our novel Poisson the payoff that a player of type t receives when a pure action game model. We calculate the optimal mixed strategies and the a is chosen and the number of players who choose action b is optimal penalizations used in the game. Section IV extends the x(b), for all b ∈C. Now, if the participants play in accordance aforementioned analysis to the case of two types of SUs and to the strategy σ, we call σt(a) the probability that a player provides an accurate analytical approximation to the Pareto of type t chooses the pure action a. Using the decomposition frontier. In Section V the impact on the PU based on its activity property again, we can establish that the number of players of is considered and the optimal mixed strategies are calculated type t ∈ T who choose the pure action a is Poisson distributed accordingly. Finally some conclusions are drawn in Section with mean λr(t)σt(a). Since the sum of independent Poisson VI. random variables is also a Poisson variable with mean equal to the sum of the means, the total number of players who II. POISSON GAMES take the pure action a is Poisson distributed with mean λτ(a), where In [19] it is established the concept of a Poisson Game as a τ(a)= r(t)σt(a). special case of a more general type of games called Random t∈T Player Games. In games with population uncertainty [20] [21], X It follows that, a player of type t who plays a pure action there is a nonempty finite set of players types T which is a ∈C while the rest of the players are expected to play using known apriori. In the context of communications systems, this strategy σ has a expected utility of set could contain the different types of services offered by the network (voice, video, data, etc). There is also a finite set of Ut(a, σ)= P (x|σ)ut(a, x), (2) available choices or pure actions that a player may take. For x∈Z(C) C X instance, these could be all the different transmission powers where, that the secondary user may utilize [22] or, in the context of (λτ(b))x(b) P (x|σ)= e−λτ(b) , (3) this paper, the decision to transmit or not. The set of possible x(b)! b actions is the same regardless of the type of player. The main Y∈C characteristic of a Poisson Game is that the total number of while the expected utility whether the player chooses action players of certain types, are modelled as random variables. In θ ∈ ∆(C) is this paper we use the definition given by Myerson [20] which Ut(θ, σ)= θ(a)Ut(a, σ). (4) presents a Poisson Game Γ as the five-tuple (λ, T , r, C,u). a∈C Here, the parameter λ corresponds to the mean number of X users described by a Poisson random variable with probability A. in Poisson Games mass function defined as It is very well known that a Nash equilibrium is achieved λk when each strategy played by all players corresponds to the f(k)= e−λ . (1) k! to all other strategies in such equilibrium [3] [24]. Consequently, no player has anything to gain by changing Thus, the number of players in the game is a Poisson random his own strategy unilaterally. The set of best responses for a variable with average number of players2 λ >> 1. Each user player of type t against a strategy σ is the set of actions that from the complete population belongs to one of the types maximizes his expected utility given that the rest of the players t ∈ T . The probability of a user to be of type t is given by (including those whose type is t) play as prescribed by σ. Let r(t) = Prob(type = t). This information is embedded in the us define the set vector r ∈ ∆(T ) where ∆(·) represents the set of probability distributions over T . By applying the decomposition property Bt(σ)= b ∈C : b ∈ arg max Ut(a, σ) (5) a∈C [20] [23] of the Poisson distribution, we can establish that the   number of players in the game of type t is also a Poisson as the set of pure best responses against σ for a player of type random variable with parameter λr(t). On the other hand, it t. Equally, the set for mixed best responses against σ is the ∗ is assumed that the set C of possible actions is common to set of actions Bt(σ) = ∆(Bt(σ)). Therefore, the strategy σ ∗ ∗ all the players regardless of their type. Thus, the set ∆(C) is a Nash equilibrium if σt ∈ Bt(σ )∀t. is the set of mixed actions associated with the players. In Poisson games, the utility of a specific player depends on its B. Examples and Motivation type, the action he chooses, and on the number of players a) Example 1: Let Γ be a Poisson game with λ = 15, only one type of players, set of available choices C = 1 Throughout this paper the terms players and secondary users shall be used {ON,OFF }, and the utility function: interchangeably. 2It is important to remark that even though in the original paper of Myerson, R if x(ON) ≤ K u(ON,x) = max , the value of λ is chosen to be very large, the validity of the results can be 0 otherwise applied here as long as a relatively large λ is used so that the probability of  having zero players in the Poisson game is negligible. u(OFF,x) = 0 ∀x ∈C, 3

K = 0 K = 2 max max 4 16 100

3.5 14 Poisson Game λ = 15 90 3 12 N = 10 Complete Information N = 15 2.5 10 80 Complete Information N = 20 2 8

1.5 6

Average utility (%) Average utility (%) 70 1 4

0.5 2 60

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p Simulation Poisson Game p 50 Theoretical Fixed No. of Players = 20 Fixed No. of Players = 15 K = 5 K = 7 max Fixed No. of Players = 10 max 40 40 60 Average Utility | R|%

35 50 30 30 40 25 20

20 30 10 15

Average utility (%) Average utility (%) 20 10 10 0 5 0 2 4 6 8 10 12 14 K 0 0 max 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p

Fig. 2. Pareto Optimality Utilities for Example 1.

Fig. 1. Average utility for Poisson Game in Example 1 (λ = 15).

where Kmax is the maximum number of players that can transmit at the same time beside one transmitting player with- probability of transmission by a player increases along side out causing a collision and R is the transmission rate payoff resulting in a higher utility. It is possible to deduce that as when the player achieves a successful transmission. This game Kmax tends to the maximum number of players known in a follows the mixed strategies defined as σ(ON) = p and game with complete information, the average utility converges σ(OF F )=1 − p where p is the probability of transmission to 100%. At the same time, in a Poisson game, due to the by any given player. Hence, using (4) we can calculate the uncertainty on the number of players, the average utility does expected utility as not achieve the maximum when Kmax tends to λ. Thus, it is K +1 important to consider the Poisson game in detail. max ∞ n i λi U(p)= R pn(1 − p)i−n e−λ i n i! n=1 i=n     b) Example 2: Consider the Poisson game defined by K X+1 X max pnλn [Γ(n + 1) − nΓ(n, −λ(1 − p))] Γ= {λ, T , r, C,u}, with expected number of players λ = 15, = R . eλn!(−λ(1 − p))n set of types T = {1, 2} with probabilities r1 and r2, r1 +r2 = n=1 1 respectively, set of choices C = {ON,OFF } and the utility X (6) function: The dependence of U(p) as a function of p is shown in Figure 1 for different values of Kmax. The solid line represents the utility of the Poisson game and the dashed lines represent the utility if this was a game with complete information (i.e., fixed number of players known for all). Considering the fact R if x(ON) ≤ K u (ON,x) = 1 max1 , that a Nash equilibrium predicts in a consistent manner the 1 0 otherwise way in which a game will be played, it is evident that in this  u1(OFF,x) = 0 ∀x ∈C, game there exists just one logical . In other words, the R if x(ON) ≤ K Nash equilibrium is strict, which by definition must occur in u (ON,x) = 2 max2 , (7) 2 0 otherwise non degenerate strategies [24]. The equilibrium occurs when  all players transmit all the time (p = 1). As noted in Figure u2(OFF,x) = 0 ∀x ∈C, 1, this conveys to a zero utility for all the users. This game is designed in such way that the players have no motivation to not transmit. It can be seen that the utilities obtained are very far from the Pareto optimal3 which could be achieved by playing a mixed strategy (p < 1). Figure 2 shows the where Kmax1 and Kmax2 are the maximum number of players achievable utilities if the players were motivated to play Pareto who can transmit simultaneously with type 1 and type 2 dominant strategies, and thus transmitting only a fraction of players respectively. Similarly to Example 1, R1 and R2 are time p< 1. Notice that by increasing the value of Kmax , the the achievable rates in case of a successful transmission. We define the mixed strategies σ1(ON)= p1 and σ2(ON)= p2 3A Pareto Optimality is defined as a specific set of strategies in which no player can change their strategy and have a greater utility without making any as the transmission probabilities by type 1 and type 2 players other player utility worse. respectively. The expected utilities can be calculated by means 4

Player I 1

100 80 0.8 ON OFF 0.6

60 1 50 P 0.4 40 0.2 0 20 1 0 ON 1 0 0.2 0.4 0.6 0.8 1 (−, −) (0, ) 0.5 0.5 P 0 2 P 0 0 P 1 2 Player II 1 30 40 0.8 25 30 20 0.6 OFF (, 0) (0,0) 20 1 15 P 10 0.4 10 0 1 0.2 1 5 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 P 0 0 1 P P 2 2 Fig. 4. Strategic form of multiple access game (N = 2).

Fig. 3. Average utility (λ = 15, Kmax = 7, Kmax = 5 and r1 = 0.3). 1 2 ALOHA fashion where each slot of the system is modelled as a one-stage game. At the beginning of each slot, the players have to choose between the two possible actions C∈{ON,OFF } of eqs. (2)-(4) as follows which represent their ability to transmit or to backoff. Every Kmax +1 n ∞ 1 k i time a player decides to transmit, he can either succeed, in U (p ,p )= R pi (1 − p )i−k which case he gains throughput, or fail due to a collision , 1 1 2 1 i k 1 1 n=1 k=1 (i=k resulting in a penalty (negative throughput) associated with X X X   i ∞ (r1λ) j such failure. Moreover, assume that the system is capable of × e−r1λ pj (1 − p )j−n+k i! n − k 2 2 handling multipacket reception (MPR) [26] , i.e. it is possible   j=n−k   to receive several packets simultaneously4. We consider that j X (r2λ) × e−r2λ , the channel has no influence in the loss of any package, j! therefore the only option for a failure transmission is due to   Kmax2 +1 n ∞ collisions with any package over the MPR limit (K +1). k i max U (p ,p )= R pi (1 − p )i−k As it was shown in Example 1 and Figure 1, it is possible to 2 1 2 2 i k 2 2 n=1 k (i k obtain the maximum utility by controlling the probability of X X=1 X=   i ∞ (r2λ) j transmission among the players. This is equivalent to players × e−r1λ pj (1 − p )j−n+k i! n − k 1 1 choosing to play mixed strategies instead of playing the single   j=n−k   pure strategy. However, as discussed above, they do not have j X (r1λ) any incentive to cease transmission to avoid collisions, so, the × e−r1λ . j! expected outcome would be all of them transmitting resulting   (8) in zero throughput. It is shown in [27] that by introducing a penality to the game it is possible to get closer to the Pareto Figure 3 shows the expected utility for this example. Since optimality. Considering the latter, we propose the following again the players have no incentive to use mixed strategies, game models: the same strict Nash equilibrium occurs in which all players transmit all the time (i.e. p1 = p2 =1) resulting in zero utility to all of them. Nevertheless, unlike Example 1, there exist A. Poisson game, single type of players several Pareto dominant mixed strategies in terms of the pairs Let us reformulate Example 1 by adding a penalty in the of probabilities (p1,p2). Such a pairs form a Pareto frontier case of a collision which contains the set of all optimal strategies [24] [25]. In the R if x(ON) ≤ K following section, we will reformulate the proposed Poisson u(ON,x)= max , −αR otherwise (9) games in order to approach to the Pareto optimality.  u(OFF,x)=0 ∀x ∈C, III. SYSTEM MODEL AND CORRESPONDING GAME where α ≥ 0 is a penalization constant. First, we consider Let us assume for a moment that there is a fixed and known a multiple access game with N ≥ 2 transmitters (players) number N of SUs contending stations while the PU is in the and Kmax = 1. In the case of a collision the transmitter is idle state. The transmission queue for all the SUs is assumed penalized by a constant quantity −αR. Figure 4 shows the to be always nonempty i.e. each user always has a packet 4This might be possible by using certain enhancements to the physical ready to be transmitted right after the completion of each layer such as beamforming in MIMO systems, frequency hopping or multiuser transmission. We consider that the system works in a slotted detection for instance. 5 game in strategic form for the case of N =2. An analogous If N =2, the corresponding value of α is game can be found in [9] as an alternative approach to the 1 distributed coordination function (DCF) in the IEEE 802.11 α(2) = =1. 2 − 1 standard. Each player transmits with probability p following a mixed strategy policy. Consequently, when the number of On the other extreme, when N → ∞, one can conclude that users is known, the multiple access could be cast as a game 1 1 α(∞) = lim N−1 = ≈ 0.5 with the following utilities: N→∞ N e − 1 N−1 − 1 U =0, OFF,k   N−1 N−1 Thus, the range of variation of α is 1 ≤ α ≤ 1. Since UON,k = p(1 − p) R − αRp[1 − (1 − p) ] (10) e−1 N−1 N−1 for smaller N, collisions are frequent, they require a higher = (1 − ϑ)[ϑ R − αR(1 − ϑ )], penalty to defer the SUs from continuous transmission. The for k = 1, 2,...N. Here we make use of the notation ϑ = case of MPR Kmax > 1 could be treated in a similar fashion 1 − p. In order for this game to be in equilibrium we need as follows. The probability of transmitting without collision to insure, by choosing a proper penalty α, that UOFF,k = can be expressed as UON,k =0, ∀k. In other words the following should be true K max N − 1 N−1 N−1 P (p) = p pk(1 − p)N−1−k ϑ R = αR(1 − ϑ ), nc k 1 k=0   N−1 α KX ϑ = . max N − 1 (17) 1+ α = pk+1(1 − p)N−1−k   k k=0   As a result, the mixed strategy in equilibrium given as peq is X achieved by = I1−p(N − 1 − Kmax,Kmax + 1), 1 N−1 α where I1−p(a,b) is the incomplete beta function (see [29], peq =1 − . (11) 1+ α chapter 6) and N >K . For moderately large N, the   max It can be seen that binomial distribution of interferers could be considered as a sum of N − 1 binary random variables. Its distribution 1 if α =0 p = . (12) can be very well approximated by a normal random variable eq 0 if α → ∞  ξ ∼ N (µ, σ2), where Notice that for arbitrary N ≥ 2 and α = 0 (i.e. no collision µ = (N − 1)p, penalty), peq = 1. This shows, as explained in previous section, that in general a game without penalty would not σ2 = (N − 1)p(1 − p) = (N − 1)pq. make sense considering that no data can be transmitted at the equilibrium in pure strategies for N>Kmax. The amount of Therefore, data transmitted for a given α> 0 is then

1 Pnc ≈ Prob(ξ

Nash Equilibrium no player has any incentive to change it. Notwithstanding if 0.7 NE K = 0 max the initial probability of transmission is off from the Nash NE K = 2 max NE K = 5 equilibrium, the game described in (9) will converge to it and 0.6 max will be self-sustaining as can be seen in Figure 5.

0.5 We consider now that the number of players is a random variable distributed according to a probability distribution

0.4 PN (N). Conditioned on the number of players N, the pay-off can be rewritten as

0.3 UOF F |N =0, N−1 N−1 Probability of transmission ( P ) UON|N = pϑ R − p(1 − ϑ )αR, (26) 0.2 = pR[(1 − α)ϑN−1 − α]. 0.1 Averaging over the distribution of N (eq. 4), the unconditional utilities can be expressed as 0 0 50 100 150 200 250 300 350 400 450 500 Iterations UOF F = UOF F |N PN (N)=0

UON =X UON|N PN (N)= Fig. 5. Nash equilibrium for different values of Kmax and N = 20. (27) ∞ ∞ N−1 X pR (1 + α) ϑ PN (N) − α PN (N) =0. " N=1 N=1 # This term represents the largest contribution to Pnc given by X X eq.(17). Furthermore, By making use of the following notation ∞ N −1 pKmax+1(1 − p)N−1−Kmax N−1 Kmax N − Kmax p FN (ϑ)= ϑ PN (N), (28) N−1 = pKmax (1 − p)N−Kmax Kmax 1 − p N=1 Kmax−1 X Np 1 eq. (27) can be rewritten as  ≈ >> 1. (22) K 1 − p max (1 + α)FN (ϑ)= α[1 − PN (0)], Subsequently, the optimized term provides the bulk contribu- or tion to Pnc. Taking one more term in eq. (17) and following α ϑ = F −1 (1 − P (0)) , (29) the same reasoning, it is possible to obtain a very accurate N 1+ α N approximation to the optimum probability of transmission as   −1 where FN is the inverse function of FN (ϑ). For instance, if Kmax +1 Popt ≈ . (23) PN (N)= δ(N − N0) N0 > 0, i.e. the game corresponding to Kmax + N a game with complete information (fixed and known number Using this value of Pnc we can reformulate eq. (10) as of players), then

UOFF,k =0 P (0) = 0, (24) N N0−1 UON,k = RPnc(Popt) − αRPc(Popt) FN (ϑ)= ϑ , where Pc = 1 − Pnc stands for the probability of collision. which coincides with eq. (11). For the Poisson distribution (1) Therefore, at the equilibrium N λ −λ PN (N)= e , (30) UOFF,k = UON,k, N! one can easily obtain Pnc = αPc −λ and PN (0) = e , Pnc α = ∞ ϑN−1λN e−λ 1 − P Kmax+1 −λ nc p=Popt≈ Kmax+N FN (ϑ)= e = [exp(ϑλ) − 1] . (31) N! ϑ K N=1 max N−1 (1 − p)N −1−kpk X = k=0 k . (25) Notice that eq. (31) has to be inverted numerically. The N−1 N−1 N−1−i i (1 − p) p Kmax+1 Pi=Kmax i  p=Popt≈ required equation for equilibrium is then Kmax+N

It isP consistently assumed in all modelled e−λ α [exp(ϑλ) − 1] = 1 − e−λ , problems, that the players take decisions based on the most ϑ 1+ α (32) basic notion of rational play where dominated strategies can eϑλ − 1 α  = eλ − 1 . be iteratively eliminated [1]. It is in this sense that the Nash ϑ 1+ α equilibrium predicts the most likely outcome of the game. If α =0,  The SUs in this game transmit with a certain probability. If eϑλ − 1 =0, such probability coincides with the one in equilibrium, then ϑ 7

does not have solution ϑ =0 since 6 Binary Exponential Backoff W = 16 ϑλ 0 Binary Exponential Backoff W = 32 e − 1 0 P Game complete information lim = λ 6=0. opt K = 7 max ϑ→0 P Poisson Game ϑ 5 opt Consider the reduced Poisson distribution where N = 0

is not possible. For the case in which there is a random 4

K = 5 number of transmitters trying to access, the question arises: max which penalization α should be chosen in order to obtain the maximum throughput? We can substitute the optimum 3 Throughput probability of transmission for each player as K = 2 max 2 Kmax N ∞ N λ −λ Kmax +1 λ −λ Popt ≈ 1 · e + e = N! Kmax + N N! 1 N=0 N=Kmax+1 K = 0 X X max (K + 1) [Γ(K + 1) − K Γ(K , −λ)] max max max max . λ Kmax 0 Kmaxe (−λ) 10 20 30 40 50 60 70 80 90 100 (33) N or λ

Now for the case when Kmax << λ, the first term in (33) can Fig. 6. Throughput comparison between binary exponential backoff scheme be neglected and using asymptotic of Γ(x, a) one obtains the (solid line), game modelled with complete information (dashed line), and following approximation game modeled as a Poisson game (dotted line) for different values of Kmax.

Kmax +1 Popt ≈ . (34) λ + Kmax − 1 Accordingly, analogously to eq. (25), we can calculate the where Poptλ is obtained from eq. (34). In order to assess optimal α using the following the performance of the proposed scheme, we compare the N ∞ Kmax N−1 N−1−k k λ −λ throughput obtained using Poisson games and the one obtained (1 − Popt) P e α = N=Kmax+1 k=0 k opt N! . from a binary exponential backoff algorithm implemented in ∞ N−1 N−1 (1 − P )N−1−iP i λN e−λ PN=Kmax+1 Pi=Kmax i  opt opt N! the IEEE 802.11 standard [10]. Standard contention windows (35) W = 16 and W = 32 were used for comparison. The results P P  0 0 are shown in Figure 6. In [15] it is shown that as the number Throughput Analysis of nodes (players) increases, the throughput converges to a Following the analysis of the MAC with exponential backoff nonzero constant in all cases (1/2 ln 2 for the case of the with MPR [15] [16], we calculate the normalized throughput binary exponential backoff). The case for the game with com- as follows. First, we obtain the conditional probability of plete information is also included. Notice that the throughput obtained by means of the Poisson game outperforms the one having k packets transmitted successfully given that there was at least one player transmitting in any slot as obtained with the classic exponential backoff for the case of small number of users. This can be explained by the large size N k N−k (k) k Popt(1 − Popt) of the contention window compared with the number of users. Psucc = , (36)  PTx where PTx is the probability of having at least one player transmitting in one slot of time which can be computed as

N PTx =1 − (1 − Popt) .

Therefore, the normalized throughput for the case of a game IV. A POISSON GAME WITH TWO TYPES OF PLAYERS with complete information is

Kmax Kmax (k) N k N−k T = kP PTx = k P (1 − Popt ) , succ k optN N In this section we extend the Poisson game interpretation k=1 k=1   X X (37) of Multiple Access to the case of two types of SU, defined by different QoS requirements on their rate. For instance, some where PoptN is given by eq. (23). For the case of Poisson game, eq. (37) becomes users might be using voice services while the rest require a video streaming service. Within this framework we consider K −1 max λj e−λ that type 1 and type 2 players transmit with rate R1 and TR = + R respectively. Depending on the QoS for each type of (j − 1)! 2 j=0 X user, we assume that the maximum number of simultaneous ∞ Kmax n −λ transmissions supported by users of first type could be no n k n−k λ e k Popt (1 − Poptλ ) . (38) more than K + 1 and the maximum supported by the k λ n! max1 n=Kmax k=1   second type could be no more than . From here the X X Kmax2 +1 8

Pareto frontier N = 15, N = 10 K = 5 K = 3 1 2 max max corresponding Poisson game can be modelled as 1 2 0.25 Genetic Algorithm 1 + Kmax R1 if x(ON) ≤ Kmax1 U 0, 2 Simulation u1(ON,x) = , µ N2 + Kmax 2 ¶ Pareto frontier −αR1 otherwise approximation  0.2 P = P u1(OFF,x) = 0 ∀x ∈C, 1 2

1 + Kmax¯ R2 if x(ON) ≤ Kmax2 P1 = P2 = ¯ u (ON,x) = ,(39) N1 + N2 + Kmax 2 −βR otherwise  2 0.15 u2(OFF,x) = 0 ∀x ∈C, 2 U where α,β > 0 are the two penalization constants in order 0.1 to guarantee the convergence to a Nash equilibrium in mixed 1 + Kmax U 1 , 0 6 + strategies . Similarly to eq. (24) for the case of a single type µN1 Kmax 1 ¶

of players, it is possible to write the utilities functions in terms 0.05 of the probabilities of non collision and collision as

U (1) =0, OF F 0 (1) 0 0.05 0.1 0.15 0.2 0.25 (1) (1) U U (p1,p2)= R1P (p1,p2) − αR1P (p1,p2), 1 ON nc c (40) (2) UOF F =0, Fig. 7. Approximation of the Pareto frontier for a complete game with two U (2) (p ,p )= R P (2)(p ,p ) − αR P (2)(p ,p ). ON 1 2 2 nc 1 2 2 c 1 2 type of players (N1 = 15,N2 = 10,Kmax1 = 5,Kmax2 = 3).

Here p1 and p2 are the probabilities of transmission (mixed strategies) of type 1 and type 2 players respectively. Further- ¯ more, let us just assume, as in the case of single type of where Kmax is the arithmetic mean of Kmax1 and Kmax2 . players, that this is a game with complete information with N1 Using these three points, one can construct an approximation and N2 players of each type respectively. The probability of to the Pareto frontier by connecting them with straight lines non-collision for both types can be found using the following as shown in Figure 7. Furthermore, analytical approximation expressions: of such frontier as follows.

Kmax j 1+K ¯ 1 m p + max2 0 ≤ p ≤ 1+Kmax (1) N1 − 1 N2 i+1 N1−i−1 1 1 N +K 1 N +N +K¯ P (p ,p )= p (1 − p ) 2 max2 1 2 max nc 1 2 1 1 p2 = 1+K ¯ 1+K i j − i  max1 1+Kmax max1 j=0 i=1    m2 p1 − N +K N +N +K¯ ≤ p1 ≤ N +K X X  1 max1 1 2 max 1 max1 j−i N2−j+i (43) × p2 (1 − p2) ,   wherem1 and m2 are calculated respectively as Kmax2 j N2 − 1 N1 P (2)(p ,p )= pi+1(1 − p )N2−i−1 nc 1 2 2 2 ¯ 1+K i j − i 1+Kmax max2 ¯ j=0 i=1 ¯ − N1 + N2 + Kmax    N1+N2+Kmax N2+Kmax2 j i X X m = , − N1−j+i 1 ¯ × p1 (1 − p1) .  1+ Kmax  (41) 1+ K¯ m = − max . 2 1+K ¯ ¯ max1 1+Kmax As it was shown in Example 2, there is a tradeoff in the N1 + N2 + Kmax − ¯ N1+Kmax1 N1+N2+Kmax choice of p1 and p2. We can see that if p1 is kept fixed and   (44) p2 gets decreased, type 1 players are benefited and viceversa. Therefore, it is desirable to assign penalties α and β in such The accuracy of such approximation achieved by eq. (43) is a way that the system works in the boundaries of the Pareto shown in Figure 7. We compare the approximation calculated frontier, i.e. to let all the choices of mixed strategies be Pareto with the Pareto frontier obtained by a simulation using a efficient. The question is how to find such a frontier. We start genetic algorithm with 30 iterations in Matlab [30]. All from noticing that if N1 =0 or N2 =0 the maximum utility solutions in a Pareto set are equally optimal, so it is up to for each type of users is given respectively by eq. (23) when the wireless designer to select a solution in that set depending on the application or the QoS goal. Furthermore, extending (1) 1+ Kmax1 (2) 1+ Kmax2 Pnc , 0 or Pnc 0, . this to the case of a Poisson game, the probability of non N1 + Kmax N2 + Kmax  1   2  collision for both types can be obtained using the following In a similar way, using eq. (23) we can see that for the case of equations p1 = p2, the maximum probability of non collision for both

utilities is achieved by Kmax1 j (1) P = P1(i)P2(j − i), 1+ K¯max nc P = , (42) j=0 i=0 N1 + N2 + K¯max X X (45) Kmax2 j 6 (2) Notice that x(ON) is the sum of all transmitting players regardless of Pnc = P1(j − i)P2(i), their type. j=0 i=0 X X 9 Pareto frontier N = 15, N = 10, K = 10, K = 2 max max 1 2 0.16 !! Genetic Algorithm Simulation Theoretical Approximation 0.14

GOOD BAD 0.12 !" !! (ON ) (OF F ) !" "! 0.1

2 0.08 U "!

0.06 Fig. 9. Primary User Activity

0.04

0.02 that the PU transmits in a slot by slot basis with probability PT and that the slots are synchronized between the PU and 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 the SUs. In this sense, we consider a PU transmitting to be in U ¯ 1 an ON state. Let NON be the average number of consecutive slots in which the PU is in an ON state. Furthermore, let us Fig. 8. Pareto Frontier (λ = 30,r1 = 0.3,Kmax = 8,Kmax = 5). 1 2 observe the PU over ν >> N¯ON sequential time slots. Then, on average, there will be νPT ON states and therefore the average number of transitions from OFF to ON state is simply where x−1 s ∞ l νPT η → ≈ . (51) (rnλ) −rnλ l x l−x (rnλ) −rnλ OFF ON P (x)= e + p (1−p ) e . N¯ON n s! x n n l! s=0 l=x   We assume that all the SUs have perfect detection of the PU X X (46) activity, hence when the latter is transmitting, all SUs remain The Pareto frontier for the case of Poisson Games can be silent. Let be the probability that there is a transmission calculated as PSU,T from one or more SU when the state of the PU is turned 1+Kmax2 1+K¯ max m1p1 + 0 ≤ p1 ≤ ¯ ON for the first time after being in an OFF state and by r2λ+Kmax2 −1 λ+Kmax p2 = 1+Kmax1 1+K¯ max 1+Kmax1 consequence, creating a collision with the PU. The average  m2 p1 − ≤ p1 ≤ , r1λ+Kmax −1 λ+K¯max r1λ+Kmax −1  1 1 number of collisions N¯col is given by   (47) νP P where m1 and m2 are calculated respectively as ¯ T SU,T N = P · η → = . (52) col SU,T OFF ON ¯ ¯ 1+K NON 1+Kmax max2 ¯ ¯ − r λ+K −1 λ + Kmax λ+Kmax 2 max2 Consequently, the average probability of collision between m1 = ,  1+ K¯  SUs and the PU is max  (48) 1+ K¯ ¯ m = − max , Ncol PSU,T 2 1+K ¯ Pcol,PU = = . (53) ¯ max1 1+Kmax ¯ λ + Kmax − ¯ νPT NON r1λ+Kmax1 −1 λ+Kmax For the game with full information, the aforementioned prob- following the same technique  as in deriving eqs. (43) and ability represents the probability of having at least one (44). The Pareto Frontier for the Poisson game with two types PSU,T SU transmitting at the same time as the PU. This can be of players is shown in Figure 8. The solid line represents calculated as follows the utility obtained by using eq. (47) to calculate the mixed N strategies in eq. (45). Consequently, the necessary condition PSU,T (p)=1 − (1 − p) . (54) for the system to be in equilibrium is As an example of the previous, let us assume that the transmis- (1) (1) U = U (p1,p2)=0 sions from a single PU operate in a channel inversion with cut- OF F ON (49) (2) (2) off mode [31]. The transmissions follow a Gilbert-Elliot (GE) U = U (p1,p2)=0 OF F ON model (particularly one which imposes a correlation ρ in the thus the penalization factor α and β can be respectively time domain) [32] [33] where the “GOOD” state occurs with obtained as probability PT and the “BAD” state occurs with probability (1) (2) Pnc (p ,p ) Pnc (p ,p ) 1 − PT . In this case, the probability of transmission PT is α = 1 2 and β = 1 2 . (50) (1) (2) directly related to the fading channel as 1 − Pnc (p1,p2) 1 − Pnc (p1,p2) ∞ Here p1 and p2 are obtained from the Pareto frontier by means PT = pγ(γ)dγ (55) of eq. (47). Zγ0 where γ0 is an energy threshold above which a transmission V. GAME MODEL CONSIDERING PRIMARY USER is possible and pγ(γ) is the fading distribution of the channel. ACTIVITY Within this context, we consider that the PU will transmit in In this section we consider the existence of a single PU a slot only within a “GOOD” state and remain silent on the transmitting within the same channel as the SUs. We assume other case. As it can be seen in Figure 9, we can model the 10

a restriction frontier under which the SUs can calculate their 0.12 Approximation (γ = 0.3 ) penalization factor α and β by means of equation (50). Such Numerical (γ = 0.3 ) Approximation (γ = 0.6 ) a frontier can be obtained by rewriting eq. (54) as 0.1 Numerical ( γ = 0.6 ) γ Approximation ( = 0.9 ) N N Th Numerical ( γ = 0.9 ) 1 2 ¯ PSU,T (p1,p2)=1−(1−p1) (1−p2) ≤ NON Pcol , (62)

0.08 and noticing that the maximum PSU,T for each type of player that satisfies eq. (62) occurs when either all type 1 SUs 2

P 0.06 ∗ transmit with probability p1(N1) and none of the type 2 SUs (1) ∗ transmit (i.e. PSU,T (p1(N1), 0)), or, all type 2 SUs transmit 0.04 ∗ with probability p2(N2) and none of the type 1 SUs transmit (2) ∗ (i.e. PSU,T (0,p2(N2))). Hence, an accurate approximation 0.02 of the restriction frontier can be formed by connecting the aforementioned two points with a straight line as shown in

0 Figure 10. It can be seen from the previous that the equality 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 P 1 of such an approximation is very good for the selected values of parameters. We omit a detailed analysis due to lack of space. ∗ Fig. 10. Restriction frontier approximation for N1 = 25,N2 = 20. Finally, in order to calculate p for a Poisson game with the average number of SU denoted as λ, ones has to average eq. (60) over the Poisson distribution as dynamic traffic from the PU with r and q defined as follows ∞ λke−λ [33] p∗(λ)= p∗(k) k! k=0 q = PT (1 − ρ), X∞ (63) (56) 1 λke−λ r = (1 − PT )(1 − ρ), = 1 − 1 − N¯ P Th k . ON col k! k=0 where 0 ≤ ρ ≤ 1 is the correlation coefficient. The average X n  o ∗ duration of the PU in the ON state N¯ON can be calculated as Here p (k) can be expanded in terms of a Taylor series with ∞ respect to the number of players k around the average value i−1 1 1 N¯ON = i(1 − r) r = = . (57) λ to produce r (1 − PT )(1 − ρ) i=1 ln(1−N¯ P T h) − ¯ Th X ∗ ON col ln(1 γ) ln(1 − NON Pcol )(k − λ) Th p (k)= e λ − e λ Let Pcol be a pre-established tolerance threshold defined as λ2 the maximum average probability of collision Pcol,PU the PU + O (k − λ)2 . could be able to tolerate. It follows from equation (53) that (64)  ¯ Th PSU,T (p) ≤ NON Pcol . (58) By taking the first two terms of the latter, we can approximate the solution of eq. (63) for large λ (as assumed throughout this Notice that for P Th ≥ 1/N¯ any value of p satisfies eq. col ON paper) as (58). This means that the SUs can transmit with any probability λ Th and just stop transmitting when they detect the presence of a λe − λ − ln 1 − N¯ON P p∗(λ) ≈ 1 − col . (65) Th ¯ ¯ T h transmitting PU. On the other hand, when P < 1/NON , λ−ln(1−NON P ) col λe col  there is a value p∗ such that Therefore Popt in eq. (34) can be obtained simply by ∗ ¯ Th PSU,T (p )= NON Pcol , (59) ∗ Kmax +1 Popt = min p (λ), . (66) which can be rewritten as Kmax + λ − 1   1 ∗ Th N Analogously, for the case of two types of players eq. (62) p (N)=1 − 1 − N¯ON P . (60) col becomes K If p∗ ≥ max+1 in eq. (23), the impact to PU can be once ∞ ∞ Kmax+N K +1 i j again ignored. Nevertheless, if p∗ < max , a different value PSU,T (p1,p2)= 1 − (1 − p1) (1 − p2) × Kmax+N of P needs to be used in order to calculate the penalty α i=0 j=0 opt X X  (67) in eq. (25). Consequently P in eq. (23) will be given as (r λ)i (r λ)j opt eλ 1 2 ≤ N¯ P Th. i! j! ON col ∗ Kmax +1 Popt = min p (N), . (61) Kmax + N Thus the restriction frontier can be formed, as seen in the pre-   vious case simply by connecting the two points corresponding Considering the analyzed example, one can notice that for Th to a fixed tolerance P , the SUs can transmit using the op- (1) (2) col P (p∗(r λ), 0) and P (0,p∗(r λ)) . timal strategy without significantly affecting the PU if the SU,T 1 1 SU,T 2 2 latter has very poor channel conditions and/or the channel is Figure 11 shows the influence of the PU activity on the strategy uncorrelated. By extending this analysis to the case of two choice by SUs in the case of two type of players for different types of players, the analogous effect of eq. (61) is to create values of Kmax. It is possible to distinguish three different 11

QoS requirements. We showed that the Pareto frontier can be accurately approximated by means of connecting a piece- wise function based on the optimal probability of transmission obtained from each type of players in isolation. Finally, we have considered the impact of the dynamic activity of the PU on the SU optimal strategy. It is shown that the optimal probabilities of transmission for the SUs are influenced by the PU pattern only under certain conditions. More specifically, we have derived the conditions for the Pareto frontier under which the SU activities are limited by either the equilibrium strategy in games without a PU, by constrains on the SINR of a PU, or by both in a piece-wise manner. In particular, it was shown that if the PU has low intermittency in the transmitting intervals, its affects on the SU strategy could be neglected. However, for PUs with relatively bursty activities the strategy of SUs is limited by the SINR requirements at the PU. Fig. 11. Restriction Frontier and ParetoFrontier for Poisson Games (λ = T h 25, N¯ON Pcol = 0.9).

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