1 Introduction

## 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 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 transmissions (links that can transmit at greater than a minimum rate) is of the order of . These asymptotic results are confirmed from simulations.

On the number of active links in random wireless networks

Hengameh Keshavarz1, Ravi R. MAZUMDAR2, Rahul Roy3and Farshid Zoghalchi4

5

Department of Communications Engineering, University of Sistan and Baluchestan, Zahedan, Iran (E-mail: keshavarz@ece.usb.ac.ir)

6

Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (E-mail: mazum@uwaterloo.ca)

7

Indian Statistical Institute, New Delhi, India (E-mail: rahul.isid@gmail.com).

8

Department of Mathematics, University of Toronto, Canada (E-mail:.zoghalchi@mail.utoronto.ca)

March 3, 2018

## 1 Introduction

Consider the situation where there are 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 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 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 , where 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 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 . In [5], the same channel model is considered and it is shown that for a path-loss exponent and any absorption modeled by exponential attenuation, a per-node throughput of the order is achievable.

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 is if and means . Similarly for a sequence of random variables and a deterministic function we say that is if . Similarly we say a.s. if . We also use the terminology to refer to a property holding asymptotically almost surely.

## 2 Network Model

Consider a wireless network with transmitter and receiver pairs as shown in Figure 1. It is assumed that a transmitter transmits to its receiver through a stationary i.i.d fading channel denoted by . Transmitting nodes can interfere with receiver and the channel gain between interfering transmitters at a receiver is denoted by . 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 are transmitting to their receivers at the same time.

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 or remain silent during each time slot.

Let , denote the location of the th transmitter-receiver pair. The random model we consider is as follows.

Let be a marked Poisson point process of intensity on the plane with the receiver located at having a mark . We assume that depends on however in such a way that (a) the process is a Poisson point process on of intensity , and, (b) and are independent whenever . This occurs, for example, when , where is a sequence of i.i.d. bounded random vectors.

Let 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

• the number of transmitters lying in a disc of unit area centered at the location of the th receiver, i.e., , has a Poisson distribution with mean , and,

• given , the points are uniformly distributed in .

At the steady state of the system, the signal, , received at receiver , is given by

 Yi=hiiXi+∑j∈Ai,j≠ihjiXj+Zi (1)

where denotes the link fading channel between transmitter and receiver , denotes the set of active transmitter-receiver pairs in the unit area neighbourhood of the th receiver at and represents background noise at node during a time-slot. Note if 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., if where is a threshold. This 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 denote the set of possible links in the unit area region centered at , where is the Poisson number of transmitters in this region, as described earlier.

The achievable rate bits of link can be thus be written as

 Ri ≤ Bln⎛⎜ ⎜ ⎜ ⎜⎝1+P|hii|2σ2+∑j∈Ai,j≠iP∣∣hji∣∣2⎞⎟ ⎟ ⎟ ⎟⎠ (2)

where 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 throughout the paper.

In the remainder of the paper our goal is to estimate , the cardinality of with the maximum number of simultaneous transmitting links that can exist at a time when we require that all transmitters in must be able to transmit at a rate greater than . Note and hence depends on , i.e., , is random and depends on the channel gain realization. We show that when is large the distribution of sharply concentrates around a given value modulo constants.

Let us denote by :

 γi(n)=Bln⎛⎜ ⎜ ⎜ ⎜⎝1+P|hii|2σ2+∑j∈Ai,j≠iP∣∣hji∣∣2⎞⎟ ⎟ ⎟ ⎟⎠ (3)

## 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 denote the set of active transmitters. Define the stochastic rate of link as

 γi,m \lx@stackrelΔ= ln⎛⎜ ⎜ ⎜ ⎜⎝1+P|hii|21[|hii|2>h0]σ2+∑j∈Am,j≠iP∣∣hji∣∣21[|hjj|2≥h0]⎞⎟ ⎟ ⎟ ⎟⎠ (4)

Then the maximum number of active links supporting the minimum rate is given by the following optimization problem.

{numcases}

M_n=max—A_m—,  m≤N_n
γ_i,m ≥R_min,  i∈A_m , Clearly, with fixed and , the maximum number of active links is a random variable which depends on the Poisson point process, the channel gains ; 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 denoted by 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 to denote . Without loss of generality we assume that has an exponential distribution with mean 1.

Accounting for both fading and path-loss between a transmitter and receiver for is characterized by a channel gain of the form:

 hij=gijD−αij1{Dij≤1},  α≥2 (5)

where is the fading gain which we assume to be an exponential random variable with mean 1 (Chi-squared with two degrees of freedom and mean 1), is the distance from transmitter to receiver and is the path loss exponent which is typically 3. In the expression (5), 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 applied to a Poisson process of intensity , keeps its Poissonness intact and just changes the intensity to . 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 are assumed to be random and denote the distances from transmitter to receiver noting that the transmitters (and receivers) form a stationary Poisson field with intensity . 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 the density of the exponential random variable with mean 1 representing the fading gain between a transmitter and receiver ( ), and by the random variable with the same distribution as each of the i.i.d. distance random variables of ( ) from transmitter to receiver whose distribution is obtained from the spatial distribution of the transmitter-receiver pairs. We assume that is independent of the random variable whose density is .

###### Lemma 3.1

Fix . For any receiver and transmitter and as above, we have

 c1z−β≥P(hij>z|ti∈rj+S1)≥c2z−β for all z≥b, (6)

where and are constants that depend on but are bounded, i.e. is heavy-tailed.

Proof: Without loss of generality, suppose the receiver is located at the origin, so that there are transmitters in the region which have an effect on the receiver , where is a Poisson random variable of mean as discussed in (b) of the description of the model.

Given , we know that the transmiter is uniformly located in the ball , so that for .with the density of being

Therefore,

 P(hij>z|ti∈rj+S1) = P(gijD−αij>z|ti∈rj+S1)=P(gij>zDαij|ti∈rj+S1) = ∫10P(gij>zuα|ti∈rj+S1)pd(u)du = 2∫10ue−zuαdu

where we have used fact that for .

Now,

 ∫10ue−zuαdu = z−2α∫z0v2α−1e−vdv = Γ(2α)P(2α,z)z−2α

where is the Gamma function and is the Incomplete Gamma function for . Note that by definition . 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 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 and where are i.i.d. Then the following SLLN holds:

 n−1p(Sn−an)→0,  a.s. (7)

for some real constant if and only if .

If satisfy (7) and we can choose while if then

In our context we need a slight generalization of the above result. First we note that

###### Lemma 3.2

Let be a non-negative r.v whose tail distribution is given by (6) with . Then for any , the r.v. is integrable, i.e.

 E[|X|p]<∞ (8)

Proof:

The proof readily follows from the fact that for

 P(Xp>z) = P(X≥z1p) ≥ c2z−2αp=c2z−(1+δ), for some δ>0.

Hence has a finite mean.

###### Corollary 3.1

Let be i.i.d. non-negative random variables, each having a probability density function whose tail distribution is given by (6) with and as in Theorem 3.1. Let be a Poisson random variable with mean , independent of the random variables . Then we have

 N−1pnSNn→0,  a.s.,  ∀ 0

Proof: Note that

 SNnN1/pn=Snn1/p−(Snn1/p−SNnN1/pn).

Now writing as we note that
(i) by Theorem 3.1,

 (Snn1/p−SNnN1/pn)1{|Nn−n|≤√nlnn}≤2∑n+√nlnnj=n−√nlnnXin1/p→0 almost surely as n→∞,

(ii) by Chebychev’s inequality

 P{|Nn−n|>√nlnn}≤Var(Nn)n(lnn)2=1(lnn)2→0 as n→∞

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 and with are i.i.d. and whose tail distribution is given by (6) and the direct channel gain is exp(1) distributed arising from a Rayleigh fading model. Let denote the number of simultaneous transmitter-receiver pairs that can transmit at a rate of at least . Then,

 ηn∼n14  a.a.s. (10)

We prove the result through showing several intermediate results.

Let be a threshold and define the Bernoulli random variables:

 ξj={1if  hii>h00if  hii

and let: denote the number of good or potentially active channels. Let and choose for . Then we can show the following result:

###### Lemma 3.3

Let where are i.i.d. , random variables with where for . Then as

 P(Mn=O(n1−γ))→1. (12)

Proof:

 ξi={1, with probability p00, with probability 1−p0 (13)

for . Then, the number of “good” links has the same distribution as , which satisfies the Binomial distribution .

 p0=P(hii>h0) = exp(−h0) (14) = 1nγ

Hence .

Now from the fact that , using Hoeffding’s inequality, see [10, Example 8, p 477] :

 P(|Mn−Nnp0|>εn) = ∞∑k=0P(|Mn−kp0|>εn)e−nnkk! ≤ ∞∑k=0exp(−ε2n/(2k(1−n−γ)))e−nnkk!.

Now let for some and, for a given , let be large such that for all (i) and (ii) . Thus, for ,

 nlogn∑k=0exp(−ε2n/(2k(1−n−γ)))e−nnkk!+∞∑k=nlogne−nnkk!≤η∞∑0e−nnkk!+η=2η

where we used the fact that for . Therefore we have for . Since is arbitrarily small, we have . The result is established by Slutksy’s theorem and on noting that as , by the strong law of large numbers, and .

Next we show that the minimum rate constraint is satisfied for at least transmitter-receiver pairs in for any .

###### Lemma 3.4

Consider a dipole SINR random graph with transmitter-receiver pairs. Suppose the channel gains are direct channel gains are distributed and the cross transmitter-receiver channel gains denoted by are i.i.d with distribution given by (6). Let denote the set of active transmitter-receiver pairs. Then, asymptotically almost surely, every set of cardinality with can support a mimimum rate .

Define the set:

 Uε,m={ω:1m1p∑j∈Am,j≠ihji1[hjj>h0]≤ε} (15)

Clearly for by Theorem 3.1 . However we need the following estimate of probability of the complement of . First note that the r.v.’s are i.i.d. for and moreover

 P(hji1[hjj>h0]>z)=P(hji>z)P(hjj>h0)∼c1z2αnγ (16)

by independence of and for . This shows that the random variables are also heavy tailed with the same exponent .

Now we use the principle of the single large jump for heavy tailed random variables [18, Chapter 3]

###### Theorem 3.2

Let be a collection of n i.i.d. sub-exponential distributions with common distribution . Then:

 P(X1+X2+⋯+Xn>x)∼P(max1≤i≤nXi>x)∼1−F(x)n∼n(1−F(x)) as x→ ∞ (17)

Applying Theorem 3.2 to for every fixed we obtain:

 P(Ucε,m)=P(Ω/Uε,m) ∼ mP(hji1[hjj>h0]>m1pε) (18) ∼ mc1(m1pε)2αnγ ∼ c1m1−2pαε−2αn−γ→0 as n→∞

Note that and hence

Let us show that if when then the minimum rate constraint is met when .

Let be the (random) rate as defined before in (4). Now, choose . Then, since the conditions of Theorem 3.2 and (18) are met. Without loss of generality let us take the transmit power

 Xi,m1[Uε,n] \lx@stackrelΔ= ln⎛⎜ ⎜ ⎜ ⎜⎝1+hii1[hii>h0]σ2+∑j∈An,j≠ihji1[hjj≥h0]⎞⎟ ⎟ ⎟ ⎟⎠1[Uε,n] ≥ ln⎛⎜⎝1+h0σ2+(m−1)1pε⎞⎟⎠1[Uε,n] ∼ ln(1+γlnnγe−Rminlnn) −a.s.  n→ ∞ ∼ ln(1+eRmin)≥Rmin a.s.   n→ ∞

since by the SLLN given in Theorem 3.2.

Let us now show that indeed transmitter -receiver pairs can simultaneously transmit above the rate provided thus completing the proof of the main result.

First of all, in light of the above result, it follows that:

 {ω:Xi,m

where denotes .

Therefore noting:

 P(Xi,m

from the union bound with

 P(⋃i∈Am{Xi,m

where (20) follows from the union bound and fact that the are identically distributed, (21) follows from (18) and (22) follows by our choice of .

Since denotes the cardinality of the set of “good” transmitters and it implies that , and therefore and we can make it as close to as needed.

The proof of the upper-bound can be obtained by noting that when the direct fading gains are Rayleigh, . Therefore if the cardinality of the ”good” set of probable links goes to zero. From Lemma 12, , Then, it can be seen that our estimate of with 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 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 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 gains are i.i.d. and the interfering channel gains, are i.i.d, Pareto with . The maximum transmitted power was taken as watt (i.e. 15 dBm which is typical power for WiFi). The spectrum bandwidth is MHz (typical for WiFi) and the background noise variance is . Numerical results on each figure were generated by Monte-Carlo simulations.

Figure a shows the number of links supporting a minimum rate of 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 simulation results that are centered around the line where is a constant.

Likewise, Figure b shows the number of links supporting a minimum rate versus the total number of users for Kbps .Once again we see that the asymptotic 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 the additive constant is while for the case , the constant is given by . It is not difficult to see that the constant should be inversely proportional to .

## 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.

### Footnotes

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