On the number of active links in random wireless networks

On the number of active links in random wireless networks Hengameh Keshavarz1 , Ravi R. MAZUMDAR2 , Rahul Roy3 and Farshid Zoghalchi4 1 Department of Communications Engineering, University of Sistan and Baluchestan, Zahedan, Iran (E-mail: [email protected]) 2 Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (E-mail: [email protected]) 3 Indian Statistical Institute, New Delhi, India (E-mail: [email protected]). 4 Department of Mathematics, University of Toronto, Canada (E-mail:[email protected]) arXiv:1412.3098v1 [cs.IT] 9 Dec 2014 May 3, 2022 Abstract This paper presents results on the typical number of simultaneous point-to-point transmissions above a minimum rate that can be sustained in a network with n transmitter-receiver node pairs when all transmitting nodes can potentially interfere with all receivers. In particular we obtain a scaling law when the fading gains are independent Rayleigh distributed random variables and the transmitters over different realizations are located at the points of a stationary Poisson field in the plane. We show that asymptotically with probability approaching 1, the number of simultaneous 1 transmissions (links that can transmit at greater than a minimum rate) is of the order of O(n 4 ). These asymptotic results are confirmed from simulations. Keywords: Wireless networks; Rayleigh fading; path-loss; heavy-tailed distributions; rate constrained links. 1 Introduction Consider the situation where there are n transmitter-receiver pairs that are randomly distributed over the spatial domain. A transmitter will transmit to its designated receiver only if it can deliver a rate greater than a certain minimum rate. Otherwise it will choose not to transmit. A natural question to ask is what is the number of simultaneous transmitter-links that can exist? Of course, in a particular situation, the numbers are dictated by the geometry of the transmitter placements. Nevertheless, over all possible random configurations we can obtain some insights on the simultaneous number of links when the number n is large and the area is finite. This is the question that we address in this paper. In particular we show a concentration of distribution type of result when the transmitters have a uniform distribution over the area. The problem is motivated by networks of base stations that wish to communicate to users nearby such that a minimum rate can be guaranteed otherwise it does not transmit to reduce the interference in the network. Over all realizations, this number is random but there is typicality in behavior. The model can be thought of an instance where cellular towers are located in a given area that transmit to users in their vicinity with a given power and will do so only if they can provide a minimum rate to the user. If they choose to transmit they will cause interference at receivers of other transmitting towers and the aim is to estimate the number of such two-way communications possible as a function of the number of towers distributed uniformly over the area over different realizations. The pioneering work of Gupta and Kumar [1] was the first concrete approach based on a simple communication model of exclusion called a node exclusive model and exploited a uniform random geometric structure of node placement. In their model, interference only affects the size and geometry of exclusion regions. Since then, many researchers have tried to consider more realistic situations (i.e. the communication model, link loss model) and present tighter throughput bounds. As shown in [9] the 1 assumption of a simple communication model as in [1] can lead to overly optimistic results. Assuming a power-law path-loss model for each source-destination pair channel, analysis based on the models considered in [1] and [2] shows that the per-node throughput scales with Θ( √1n ), where n denotes the total number of nodes in the network. Introducing multi-path fading effects, in [3] the authors assume that the channel gains are drawn independently and identically distributed (iid) from a given probability density function (pdf). As a particular example, [3] shows that the throughput scaling law of the Rayleigh fading channel is logarithmic. In [11, 12], a rate-constrained single-hop wireless network with Rayleigh fading channels is considered. An upper bound and a lower bound of the order ln(n) on the number of active links supporting a minimum rate are obtained. The result is based on a threshold activation policy and the idea is to choose a threshold such that the given rate can be achieved. In [4], each channel gain is a product of a path-loss term and a non-negative random variable modeling multi-path fading and having an exponentially-decaying tail. In this case, the achievable   per-node throughput scales with Ω √ 1 n(ln n)3 . In [5], the same channel model is considered and it is shown that for a path-loss exponent α > 2 and any absorption modeled by exponential attenuation, a per-node throughput of the order Ω( √1n ) is achievable. We consider a wireless network of n transmitter-receiver node pairs where any transmitting user can potentially cause interference at a receiver node.The aim is to estimate the number of transmitterreceiver pairs that can simultaneously exist such that they can transmit at a rate of at least Rmin over random realizations of the transmitter-receiver pairs. We assume that the channel gains are due to two components, a fading gain that is Rayleigh distributed that we assume is i.i.d. over all channels and a distance based attenuation, the path-loss, that decreases as d−α , α > 2 where d is the distance between an interfering transmitter and receiver. The value of α is typically 3. We assume that the transmitters are uniformly distributed over the domain (made precise later) and the fading gains are independent of the location. We show that the number of simultaneous transmissions between 1 transmitters and their receivers is of the order O(n 4 ). Our results differ from earlier ones reported in [12] in that they only estimate the number of links that have rates above a minimum when all transmitter-receiver pairs are activated. Moreover, the geometric aspects were not directly addressed. The paper is organized as follows: In Section II, the network model is introduced. Section III presents the main results where we show that the combination of multipath fading and random distance attenuation induces a Pareto type of distribution for the interfering gains. In Section IV we conclude with some simulation results that confirm the principal result. We use the following notation: we say fn f (n) is O(gn ) if lim supn fgnn < ∞ and f (n) ∼ g(n) means lim = 1. Similarly for a sequence of random n→∞ g(n) n variables {Xn } and a deterministic function fn we say that Xn is O(fn ) if lim supn→∞ X fn < ∞ a.s. n Similarly we say Xn ∼ o(fn ) a.s. if lim X fn = 0 a.s.. We also use the terminology a.a.s to refer to a property holding asymptotically almost surely. 2 Network Model Consider a wireless network with n transmitter and receiver pairs as shown in Figure 2. It is assumed that a transmitter i transmits to its receiver i through a stationary i.i.d fading channel denoted by hii . Transmitting nodes j 6= i can interfere with receiver i and the channel gain between interfering transmitters at a receiver i is denoted by hji , j 6= i. It is assumed that the dominant factor affecting the gain between a transmitter and receiver is only due to multipath fading, i.e., distance is ignored, for example at a fixed distance from the transmitter. The scenario is one of a receiver being in the vicinity of a base station. The rate it receives is only affected by the interference from other transmitters that 2 are transmitting to their receivers at the same time. Destination i Inactive node h Source i ii Destination Source h ji Interference Source j Active link Figure 1: A wireless network with active links (−) and interference channels (−−) Throughout the paper, by sources and destinations, we mean transmitting nodes and receiving nodes respectively. Destinations are conventional receivers without multi-user detectors; in other words, no broadcast or multiple-access channel is embedded in the network. Nodes transmit signals with maximum power of P or remain silent during each time slot. Let (ti , ri ) ∈ R2 × R2 , i ≥ 0 denote the location of the ith transmitter-receiver pair. The random model we consider is as follows. Let r1 , r2 , . . . be a marked Poisson point process of intensity n on the plane R2 with the receiver i located at ri having a mark ti ∈ R2 . We assume that ti depends on ri however in such a way that (a) the process t1 , t2 , . . . is a Poisson point process on R2 of intensity n, and, (b) ri and tj are independent whenever i 6= j. This occurs, for example, when ti = ri + Wi , where W1 , W2 , . . . is a sequence of i.i.d. bounded random vectors. Let S1 be the disc of unit area centered at the origin. From the Palm theory of Poisson point processes (see Daley and Vere-Jones (1988)[6, Ch. 12]) we know that (a) the number of transmitters lying in a disc of unit area centered at the location rj of the ith receiver, i.e., Nj := #{ti : i 6= j, 1 ≤ i < ∞} ∩ (rj + S1 ), has a Poisson distribution with mean n, and, (b) given Nj = k, the k points {ti : i 6= j, 1 ≤ i < ∞} ∩ (rj + S1 ) are uniformly distributed in S1 . At the steady state of the system, the signal, Yi , received at receiver i, is given by X Yi = hii Xi + hji Xj +Zi (1) j∈Ai ,j6=i where hii denotes the link fading channel between transmitter i and receiver i, Ai denotes the set of active transmitter-receiver pairs in the unit area neighbourhood of the ith receiver at ri and Zi ∼ CN (0, σ 2 ) represents background noise at node i during a time-slot. Note Xi = P if i is transmitting and 0 otherwise. We assume that the channel activation from slot to slot is independent and each node uses a threshold based strategy for activation, .i.e., Xi = P if |hii |2 > h0 where h0 is a threshold. This 3 is referred to as a TBLAS (Threshold Based Activation Strategy) in [12]. Only nodes that can sustain a given rate of transmission are activated and thus such a strategy is not fully decentralized. However, this allows us to obtain an estimate of the largest number of concurrent activated links. Let Nn = {j : 1 ≤ j ≤ Ni } denote the set of possible links in the unit area region ri + Si centered at ri , where Ni is the Poisson number of transmitters in this region, as described earlier. The achievable rate bits of link i can be thus be written as     P |hii |2   X Ri ≤ B ln 1 +  2  σ2 + P |hji |  (2) j∈Ai ,j6=i where B is the spectrum bandwidth. The above is an equality if the noise and channel gains are gaussian which is the case here. For convenience we take B = 1 throughout the paper. In the remainder of the paper our goal is to estimate m = #|Ai |, the cardinality of Ai with the maximum number of simultaneous transmitting links that can exist at a time when we require that all transmitters in A must be able to transmit at a rate greater than Rmin . Note Ai ⊂ Nn and hence m depends on n , i.e., m = m(n), is random and depends on the channel gain realization. We show that when n is large the distribution of m(n) sharply concentrates around a given value modulo constants. Let us denote by γi (n):     P |hii |2   X γi (n) = B ln 1 +  2 2  σ + P |hji | (3) j∈Ai ,j6=i 3 Main results In delay-sensitive applications, each active link needs to support a minimum rate. Due to limited transmitted power and interference from other active source-destination pairs, it is not always possible for all nodes to keep this minimum rate. Hence, only nodes with good channel conditions should be active while others remain silent during each time slot. Consider the received signal model given by (1). Let Am denote the set of active transmitters. Define the stochastic rate γi,m of link i as   ∆  γi,m = ln 1 +  σ2 +   P |hii |2 1[|hii |2 >h0 ]  X  2 P |hji | 1[|hjj |2 ≥h0 ]  (4) j∈Am ,j6=i Then the maximum number of active links supporting the minimum rate is given by the following optimization problem.  Mn = max |Am |, m ≤ Nn γi,m ≥ Rmin , 4 i ∈ Am , (5) (6) Clearly, with fixed P and Rmin , the maximum number of active links is a random variable which depends on the Poisson point process, the channel gains |hij |2 ; i, j = 1, . . . , n and the interference caused by nodes transmitting at the same time. In wide area networks attenuation due to path-loss plays a significant role in determining the quality of the link, i.e., the rate at which a transmitter-receiver pair can communicate. In our model the channel gain between a transmitter-receiver pair i denoted by hii is due to fading only with Gaussian channel conditions. This leads to channel gains between T-R pairs to be Rayleigh distributed. In the sequel we use hij to denote h2ij , i, j ∈ {1, 2, . . . , n}. Without loss of generality we assume that hii has an exponential distribution with mean 1. Accounting for both fading and path-loss between a transmitter i and receiver j for i 6= j is characterized by a channel gain of the form: −α hij = gij Dij 1{Dij ≤1} , α ≥ 2 (7) where gij is the fading gain which we assume to be an exponential random variable with mean 1 (Chisquared with two degrees of freedom and mean 1), Dij is the distance from transmitter i to receiver j and α is the path loss exponent which is typically 3. In the expression (7), the indicator function guarantees that there is no effect on a receiver from a transmitter at a distance 1 or more from it. From the scaling properties of the Poisson process we may deduce that this restriction is minor. Indeed, the transformation x 7→ ax applied to a Poisson process of intensity n, keeps its Poissonness intact and just changes the intensity to n/(a)2 . Thus a bound on the radius of influence of a receiver may be adjusted with a corresponding change in intensity of the Poisson process. Over different realizations the distances Dij are assumed to be random and denote the distances from transmitter i to receiver j noting that the transmitters (and receivers) form a stationary Poisson field with intensity n. A similar model has also been considered Baccelli and Singh [7] in the context of spatial random access schemes. In the following we denote by g the density gij of the exponential random variable with mean 1 representing the fading gain between a transmitter i and receiver j ( i 6= j), and by D the random variable with the same distribution as each of the i.i.d. distance random variables of Dij ( i 6= j) from transmitter i to receiver j whose distribution is obtained from the spatial distribution of the transmitter-receiver pairs. We assume that Dij is independent of the random variable whose density is gij . Lemma 3.1 Fix b > 0. For any receiver j and transmitter i and hij as above, we have c1 z −β ≥ P(hij > z|ti ∈ rj + S1 ) ≥ c2 z −β for all z ≥ b, (8) where β = α2 and c1 = c1 (α) > 0, c2 = c2 (α, b) > 0 are constants that depend on α but are bounded, i.e. hij is heavy-tailed. Proof: Without loss of generality, suppose the receiver i is located at the origin, so that there are Ni transmitters in the region S1 which have an effect on the receiver i, where Ni is a Poisson random variable of mean n as discussed in (b) of the description of the model. Given ti ∈ S1 , we know that the transmiter i is uniformly located in the ball S1 , so that P(Dij ≤ 2 u) = u2 = πsπ for 0 ≤ u ≤ 1.with the density of Dij being pd (u) = 2u1{0≤u≤1} Therefore, −α α P(hij > z|ti ∈ rj + S1 ) = P(gij Dij > z|ti ∈ rj + S1 ) = P(gij > zDij |ti ∈ rj + S1 ) Z 1 P(gij > zuα |ti ∈ rj + S1 )pd (u)du = 0 Z 1 α ue−zu du = 2 0 5 where we have used fact that P(gij > x) = e−x for x > 0. Now, Z 1 Z z 2 2 −zuα −α ue du = z v α −1 e−v dv 0 0 2 2 2 = Γ( )P ( , z)z − α α α whereΓ(a) is the Gamma function and P (a, x) is the Incomplete Gamma function for a, x > 0. Note that by definition P (a, 0) < P (a, x) ≤ P (a, ∞) = 1. This completes the proof. Now, from the fact the the points are independent, the above shows that the channel gains from the interferers are i.i.d. distributed as (3.1). From (3.1) we see that the distribution of the channel gains is a generalized Pareto distribution. Thus the classical Strong Law of Large Numbers (SLLN) does not apply due to the infinite mean since α ≥ 2 and typically is 3 for the far field model in wireless communications. However, a suitable normalization of the partial sums of heavy-tailed random variables can be associated with a SLLN due to Marcinkiewicz and Zygmund [17, Theorem 2.1.5] that we state below. Theorem 3.1 Let p ∈ (0, 2) and Sn = Pn n i=1 Xi − p1 where {Xi } are i.i.d. Then the following SLLN holds: (Sn − an) → 0, a.s. (9) for some real constant a if and only if E|X0 |p < ∞. If {Xi } satisfy (9) and p < 1 we can choose a = 0 while if p ∈ (1, 2) then a = E[X1 ] In our context we need a slight generalization of the above result. First we note that Lemma 3.2 Let X be a non-negative r.v whose tail distribution is given by (8) with α ≥ 2. Then for any 0 < p < α2 , the r.v. X p is integrable, i.e. E[|X|p ] < ∞ Proof: The proof readily follows from the fact that for 0 < p < (10) 2 α <1 1 P(X p > z) = P(X ≥ z p ) ≥ c2 z 2 − αp = c2 z −(1+δ) , for some δ > 0. Hence X p has a finite mean. Corollary 3.1 Let X1 , X2 , . . . be i.i.d. non-negative random variables, each having a probability density function whose tail distribution is given by (8) with α > 2 and Sn as in Theorem 3.1. Let Nn be a Poisson random variable with mean n, independent of the random variables X1 , X2 , . . .. Then we have −1 2 (11) Nn p SNn → 0, a.s., ∀ 0 < p < < 1 α Proof: Note that SN n 1/p Nn = Sn Sn SNn − ( 1/p − 1/p ). n1/p n Nn 6 Now writing Sn n1/p − (i) by Theorem 3.1, ( S Nn 1/p Nn n − as ( nS1/p S Nn 1/p Nn n − )1{|Nn −n|≤√n ln n} + ( nS1/p Sn S Nn √ ≤2 )1 − n1/p Nn1/p {|Nn −n|≤ n ln n} Pn+√n ln n √ X j=n− n ln n i n1/p S Nn 1/p Nn )1{|Nn −n|>√n ln n} we note that → 0 almost surely as n → ∞, (ii) by Chebychev’s inequality P {|Nn − n| > √ n ln n} ≤ 1 Var(Nn ) = → 0 as n → ∞ 2 n(ln n) (ln n)2 thus an application of Slutsky’s theorem (see Grimmett and Stirzaker[10, p 318]) completes the proof of the corollary. We now state the main result of this paper. Proposition 3.1 (Main Result) Consider a dipole random SINR graph whose channel gains between transmitters i and j with i 6= j are i.i.d. and whose tail distribution is given by (8) and the direct channel gain hii is exp(1) distributed arising from a Rayleigh fading model. Let ηn denote the number of simultaneous transmitter-receiver pairs that can transmit at a rate of at least Rmin . Then, 1 ηn ∼ n 4 a.a.s. We prove the result through showing several intermediate results. Let h0 > 0 be a threshold and define the Bernoulli random variables:  1 if hii > h0 ξj = 0 if hii < h0 (12) (13) P i and let: Mn = N i=1 ξi denote the number of good or potentially active channels. Let p0 = P(ξi = 1) = P(hii > h0 ) and choose h0 = γ ln n for 0 < γ < 1. Then we can show the following result: P i −h0 Lemma 3.3 Let Mn = N i=1 ξi where {ξi } are i.i.d. {0, 1}, random variables with E[Xi ] = p0 = e where h0 = γ ln n for 0 < γ < 1/2. Then as n → ∞ P(Mn = O(n1−γ )) → 1. (14) Proof: ξi =  1, 0, with probability p0 with probability 1 − p0 for i = 1, 2, . . . , n. Then, the number of “good” links has the same distribution as Mn = which satisfies the Binomial distribution B(Nn , p0 ). p0 = P(hii > h0 ) = exp (−h0 ) 1 = nγ Hence np0 = n1−γ . 7 (15) P Nn i=1 ξi , (16) Now from the fact that ξ ∈ {0, 1} , using Hoeffding’s inequality, see [10, Example 8, p 477] : ∞ X P(|Mn − Nn p0 | > εn ) = P(|Mn − kp0 | > εn ) k=0 ∞ X ≤ e−n nk k! exp(−ε2n /(2k(1 − n−γ ))) k=0 e−n nk . k! Now let εn = na for some a ∈ (1/2, 1 − γ) and, for a given η > 0, let n0 be large such that for all P e−n nk < η. Thus, for n ≥ n0 , n ≥ n0 (i) exp(−ε2n /(2n log n(1 − n−γ ))) < η and (ii) ∞ k=n log n k! nX log n k=0 exp(−ε2n /(2k(1 − n−γ ))) e−n nk + k! ∞ X k=n log n ∞ X e−n nk e−n nk ≤η + η = 2η k! k! 0 where we used the fact that exp(−ε2n /(2k(1 − n−γ ))) ≤ exp(−n2a /(2n log n(1 − n−γ ))) for k < n log n. Therefore we have P(|Mn − np0 | > εn ) ≤ 2η for n ≥ n0 . Since η > 0 is arbitrarily small, we have P(|Mn − Nn p0 | > εn ) → 0 as n → ∞. The result is established by Slutksy’s theorem and on noting εn → 0. that as n → ∞, by the strong law of large numbers, Nn /n → 1 and n1−γ Next we show that the minimum rate constraint is satisfied for at least nδ transmitter-receiver pairs in A for any 0 < δ < γ. Lemma 3.4 Consider a dipole SINR random graph with n transmitter-receiver pairs. Suppose the channel gains are direct channel gains hii are exp(1) distributed and the cross transmitter-receiver channel gains denoted by hij , i 6= j are i.i.d with distribution given by (8). Let Am ⊂ Nn denote the set of m active transmitter-receiver pairs. Then, asymptotically almost surely, every set Am of cardinality m = nδ with 0 < δ < γ < 21 can support a mimimum rate Rmin . Define the set: Uε,m = {ω : 1 m 1 p X j∈Am ,j6=i hji 1[hjj >h0 ] ≤ ε} (17) 2 α by Theorem 3.1 P(Uε,m ) → 1 as m → ∞. However we need the following Clearly for p < estimate of probability of the complement of Uε,m . First note that the r.v.’s hji 1[hjj >h0 ] are i.i.d. for j 6= i and moreover c1 P(hji 1[hjj >h0 ] > z) = P(hji > z)P(hjj > h0 ) ∼ 2 (18) z α nγ by independence of hji and hjj for j 6= i. This shows that the random variables hji 1[hjj >h0 ] are also heavy tailed with the same exponent − α2 . Now we use the principle of the single large jump for heavy tailed random variables [18, Chapter 3] Theorem 3.2 Let {Xi }ni=1 be a collection of n i.i.d. sub-exponential distributions with common distribution F (x). Then: P(X1 + X2 + · · · + Xn > x) ∼ P( max Xi > x) ∼ 1 − F (x)n ∼ n(1 − F (x)) as x → ∞ 1≤i≤n 8 (19) Applying Theorem 3.2 to we obtain: P for every fixed ε > 0, and m(n) → ∞ as n → ∞ j∈Am ,j6=i hji 1[hjj >h0 ] 1 P(U c ε,m ) = P(Ω/Uε,m ) ∼ mP(hji 1[hjj >h0 ] > m p ε) c1 ∼ m 1 2 (m p ε) α nγ 2 2 1− pα −α ∼ c1 m ε n−γ → 0 as n → ∞ (20) (21) 2 <0 Note that pα < 2 and hence 1 − pα Let us show that if i ∈ Am when m ∼ nδ , δ < 1 then the minimum rate constraint is met when ω ∈ Uε,n . Let Xi,m be the (random) rate as defined before in (4). Now, choose ε = γe−Rmin n − pδ ln n. Then, δ p since n ε → ∞ the conditions of Theorem 3.2 and (20) are met. Without loss of generality let us take the transmit power P = 1     hii 1[hii >h0 ] ∆   X Xi,m 1[Uε,n ] = ln 1 +  1[Uε,n ] 2  σ + hji 1[hjj ≥h0 ]  j∈An ,j6=i h0 ≥ ln 1 + ! 1[Uε,n ] 1 σ 2 + (m − 1) p ε   γ ln n − a.s. n → ∞ ∼ ln 1 + −R γe min ln n ∼ ln(1 + eRmin ) ≥ Rmin a.s. n→ ∞ since 1[Uε,n ] → 1 a.s. n → ∞ by the SLLN given in Theorem 3.2. Let us now show that indeed nδ transmitter -receiver pairs can simultaneously transmit above the rate Rmin provided δ ≤ γ2 thus completing the proof of the main result. First of all, in light of the above result, it follows that: c {ω : Xi,m < Rmin } ⊂ Uε,m where Ac denotes Ω/A. Therefore noting: c P (Xi,m < Rmin ) ≤ P Uε,m from the union bound with ε = γe−Rmin n P [ i∈Am − pδ ln n ! {Xi,m < Rmin } ≤ X
P(Xi,m < Rmin ) i∈Am c ≤ mP Uε,m 1 p
≤ c1 m2 (m ε) (22) 2 −α ≤ const.n2δ−γ 9 n−γ 1 2 (ln n) α (23) →0 (24) ′ s are identically distributed, (23) where (22) follows from the union bound and fact that the Xi,m follows from (20) and (24) follows by our choice of ε. Since nδ denotes the cardinality of the set of “good” transmitters and it implies that δ ≤ γ2 , γ < 12 and therefore δ < 14 and we can make it as close to 41 as needed. The proof of the upper-bound can be obtained by noting that when the direct fading gains are Rayleigh, max1≤i≤n hii ∼ ln n. Therefore if γ > 1 the cardinality of the ”good” set of probable links goes to zero. From Lemma 14, γ < 12 , Then, it can be seen that our estimate of nδ , δ ≤ γ2 with 0 < γ < 12 is maximal in that if the cardinality is higher then asymptotically the rate constraint cannot be met. This completes the proof of the result. Remark 3.1 The results rely on the independence hypothesis of the channel gains. If we consider a simplified model with i.i.d. Rayleigh fading ignoring the geometric aspects of the problem (i.e. ignoring path loss) the it can be shown the typical number of rate constrained links is ∼ (log n)2 which is much lower than the reported result. Thus spatial aspects help improve the total communication rates due to path loss effects making interference from more distant transmitters be negligible. This scaling law gives an idea of typical behavior over many realizations of the wireless system due to placement of n transmitters in a bounded region. 4 Simulation Results We simulated a dipole random model presented in section 2 and assumed that the T-R channel, i.e. the hii gains are i.i.d. exp(1) and the interfering channel gains, hij , i 6= j are i.i.d, Pareto with α = 3. The maximum transmitted power was taken as P = 0.032 watt (i.e. 15 dBm which is typical power for WiFi). The spectrum bandwidth is B = 22 MHz (typical for WiFi) and the background noise variance is σ 2 = 0.01. Numerical results on each figure were generated by Monte-Carlo simulations. Figure 2(a) shows the number of links supporting a minimum rate of 100 Kbps versus the total number of possible T-R pairs. Both simulation results (in blue) and the theoretical estimate (in red) shifted by an additive constant given by Proposition 3.1 are indicated on this figure. It can clearly be seen that there is a constant gap between the numerical and theoretical results as seen from the 1 simulation results that are centered around the line C1 + n 4 where C1 is a constant. Likewise, Figure 2(b) shows the number of links supporting a minimum rate versus the total number 1 of users for Rmin = 150 Kbps .Once again we see that the asymptotic C1 (Rmin ) + n 4 provides a very good estimate of the number of simultaneous T-R pairs when there are more than 100 T-R pairs. For the case of Rmin = 100Kbps the additive constant is C1 = 192 while for the case Rmin = 150Kbps, the constant is given by C1 = 145. It is not difficult to see that the constant C1 should be inversely proportional to Rmin . 5 Acknowledgment This work was supported by Natural Sciences and Engineering Research Council (NSERC) of Canada. RM would like to acknowledge the support and hospitality of LINCS (Laboratory of Information, Networks and Communication Sciences) and INRIA-ENS, Paris. References [1] P. Gupta and P. R. Kumar, “The capacity of wireless networks”, IEEE Trans. Information Theory, vol. 46, no. 2, pp. 388-404, March 2000. 10 250 180 160 200 Number of active links Number of active links 140 150 100 120 100 80 60 50 40 20 0 0 2000 4000 6000 Total number of links 8000 0 10000 (a) Active links vs. total for Rmin = 100kbps 2000 4000 6000 Total number of links 8000 10000 (b) Active links vs. total for Rmin = 150kbps Figure 2: Number of active links vs. total number for different minimum rates [2] M. Franceschetti, O. Dousse, D. N. C. Tse, and P. Thiran, “Closing the gap in the capacity of wireless networks via percolation theory”, IEEE Trans. Information Theory, vol. 53, no. 3, pp. 1009-1018, March 2007. [3] R. Gowaikar, B. Hochwald, and B. Hassibi, “Communication over a wireless network with random connections”, IEEE Trans. Information Theory, vol. 52, no. 7, pp. 2857-2871, July 2006. [4] S. Toumpis and A. J. Goldsmith, “Large wireless networks under fading, mobility, and delay constraints”, IEEE Conf. Computer Communications (INFOCOM), pp. 609-619, Hong Kong, China, March 2004. [5] Y. Nebat, “A lower bound for the achievable throughput in large random wireless networks under fixed multipath fading”, Intern. Symp. Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), pp. 1-10, Boston, USA, April 2006. [6] D.J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Series in Statistics, Springer-Verlag, New York,1988. [7] F. Baccelli and C. Singh, “Adaptive Spatial Aloha, Fairness and Stochastic Geometry”, CoRR, 2013. Available from: http://arxiv.org/abs/1303.1354. [8] A. Gut, Probability: A Graduate Course, Springer, 2005. [9] V. P. Mhatre, C. P. Rosenberg, and R. R. Mazumdar,“ On the capacity of ad hoc networks under random packet losses”, IEEE Trans. on Information Theory, Vol 55 (6), 2009, pp 2494-2498. [10] G.R. Grimmett and D. R. Strizaker, Probability and Random Processes, 3rd. Ed, Oxford Science Publ., 2001. [11] M. Ebrahimi, M. A. Maddah-Ali, and A. K. Khandani, ”Throughput scaling laws for wireless networks with fading channels,” IEEE Trans. Information Theory, vol. 53, no. 11, pp. 4250 4254, November 2007. 11 [12] M. Ebrahimi and A. K. Khandani, ”Rate-constrained wireless networks with fading channels: Interference-limited and noise-limited regimes,” IEEE Trans. Information Theory, Vol 57 (12), 2011, pp. 7714-7731. [13] M. Haenggi, “On distances in uniformly random networks”, IEEE Trans. Information Theory, vol. 51, no. 10, pp. 3584-3586, October 2005. [14] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Second Edition, Reading, Massachusetts: Addison-Wesley, 1994. [15] V. K. Rohatgi, An Introduction to Probability Theory and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1976. [16] M. Franceschetti, M. D. Migliore, and P. Minero, “The capacity of wireless networks: informationtheoretic and physical limits”, IEEE Trans. on Information Theory, 55(8), pp. 3413-3424, August 2009 [17] P. Embrechts, C. Kluppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Risk, Springer-Verlag, Berlin, 1997. [18] S. Foss, D. Korshunov, and S. Zachary; An introduction to heavy-tailed and subexponential distributions. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2011. 12