# On Dominant Interference in Random Networks and Communication Reliability

In this paper, we study the characteristics of dominant interference power with directional reception in a random network modelled by a Poisson Point Process. Additionally, the Laplace functional of cumulative interference excluding the n dominant interferers is also derived, which turns out to be a generalization of omni-directional reception and complete accumulative interference. As an application of these results, we study the impact of directional receivers in random networks in terms of outage probability and error probability with queue length constraint.

## Authors

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## I Introduction

The studies of communication technologies, services and applications for massive Machine Type Communication (MTC) have significantly increased in recent years. Emerging MTC requires significant performance improvements for a communication link, which is either an increase of capacity for vehicular technology, or an increase in high reliability while keeping low latency constraints for tactile internet and factory automation purposes. A key point of these challenges is to accommodate the increasing number of devices with reliable services.

Multiple access schemes, especially non-orthogonal multiple access, have been identified as potential solutions to meet future massive access requirements in wireless networks under various applications [1, 2, 3, 4]. However, fading is modeled as path loss, and hence only the Euclidean distance was taken into account to derive the () dominant interference power in these works. A closed form of the Euclidean distance distribution to the nearest neighbor in networks modeled by a Poisson Point Process (PPP) was derived in [5, 6].

Though path loss is one of the major factors causing wireless signal fading, channel fading is also significantly influencing the strength of wireless signals. Due to the dynamic nature of such fading, the signal power received from a nearer transmitter is possibly smaller than that from a farther transmitter. Thus we want to study characteristics of the dominant interference power as well as accumulative interference in terms of power itself rather than Euclidian distance. It is also of interests to see how these affect communication reliability in wireless networks. In addition, directional reception induces sectorization in cellular networks, which is a typical method for interference management[7, 8]. Thus, directional reception angle also affects the characteristics of dominant and accumulative interference power.

In this paper, we model stationary random networks by homogeneous PPP. We give probability distribution of the

dominant interference power in stationary random networks where receivers have directional reception. In addition, the partial accumulative interference excluding some dominant interferers is studied, which is a generalization of the omni-directional case and also complete accumulative interference. The outage probability and transmission error probability with queue length constraint in Nakagami-m fading are studied.

The remainder of this paper is organized as follows. In Section II, we derive the closed-form distribution of the dominant interference power and Laplace functional of partial accumulative interference power. In Section III, we discuss the communication outage probability and transmission error probability with queue length constraint. In Section IV, numerical results are given to verify the derived results. Finally, we conclude this paper in Section V.

## Ii Dominant and accumulative Interference

We model a random network by a PPP , , with intensity , where denotes the coordinates of node111Term “node” and “point” are alternatively used depending on the context of network or point process. . The intensity measure of is denoted by . For a receiver located at in the random network, the interference power from node is

 Ii=hi||x−xi||−α, (1)

where is the channel gain from to , the Euclidean norm, and the path-loss parameter. We arrange into a descending-ordered sequence , such that is the dominant interference power ().

To study the distribution of , point process mapping and displacement [6] are be applied on . We assume directional reception and the reception is within deterministic angle , where . Since is homogeneous, without losing generality, we define the interferers to node by

 Φ′={x′i=xi∣∠−−→xxi≤ϕ,xi∈Φ}, (2)

where

gives the angle of vector

regarding the horizontal axis on . Then we map to as

 Φ1={yi=||x′i−x||α∣x′i∈Φ′,α>2}. (3)

We further define the point process that takes values from as a displacement of as

 Φ2={zi=yihi∣yi∈Φ1}. (4)

We arrange the points of into an ascending-ordered sequence (), along the power axis . Hence, is the closest point of to origin on the power axis leading the dominant interference power to be .

###### Theorem 1 (The nth Dominant Interference Power)

In a PPP on with intensity , , the inverse of the dominant interference power , to a receiver with reception angle

has the Probability Density Function (PDF)

 fϕzn(z)=2exp{−12¯h2/αϕλz2/α}(12¯h2/αϕλz2/α)nαz(n−1)!, (5)

where denotes expectation supported by PDF of channel gain .

###### Proof:

Combining the definition in (2) and (3) gives

 Φ1={yi=||xi−x||α∣∠−−→xxi≤ϕ,α>2,xi∈Φ}. (6)

Denote the intensity measure of by and its intensity function by . Since is obtained by thinning and mapping independently from , is also a PPP [6], [9]. The intensity is

 Λ1(y)=E[Φ1([0,y])] =∫Bx(y)λ1(∠−→xx′≤ϕ)dx′=ϕλy2/α2, (7)

where is a disc centered at with radius , and is the indicator function. Thus, the intensity function of is

 λ1(y)=∂Λ1(y)∂y=ϕλy2/α−1α,y>0. (8)

Since the point process is actually a displacement of , is also PPP according to the Displacement Theorem [9]. The intensity measure of can be obtained according to the Displacement Theorem. We have

 P(yh

where

is the Cumulative Distribution Function (CDF) of

. The probability kernel of the displacement is

 ρ(y,z)=∂∂z[1−Fh(yz)]=yz2fh(yz). (10)

Hence, the intensity function of is obtained as

 λ2(z) =∫∞0λ1(y)ρ(y,z)dy=ϕλαz2/α−1Eh[h2/α], (11)

where . Then

 Λ2(z)=∫z0λ2(z)dz=ϕλ2Eh[h2/α]z2/α,z∈R+. (12)

We arrange points of into an ascending-ordered sequence . By definition, is the largest interference power from points of . Denote the number of points of within on the power axis by . Then

is a Poisson random variable with intensity

and we have

 P(Nz=k)=(Λ2(z))kk!exp{−Λ2(z)},k=0,1,2,⋯. (13)

Let be the CDF of , we have

 Fϕzn(z)=1−n−1∑k=0P(Nz=k)=γ(n,Λ2(z))Γ(n), (14)

where is gamma function and is lower incomplete gamma function. By taking the derivative of the PDF of is

 fϕzn(z)=2exp{−Λ2(z)}(Λ2(z))nαΓ(n)z. (15)
###### Corollary 1.1

By Theorem 1, the dominant interference power is expected to be

 E[In]=(ϕ¯h2/αλ2)α/2Γ(n−α/2)Γ(n),n>α/2 (16)

As expected, the dominant interference power expectation decreases with path-loss parameter . increases with the increasing reception angle and node intensity . As a function of , decreases with , whereas the expectation of the nearest node’s distance increases with [5].

###### Corollary 1.2

The CDF of the dominant interference power is obtained as

 FϕIn(z)=Γ(n,Λ2(z−1))Γ(n), (17)

where is the upper incomplete gamma function.

Let us define the partial accumulative interference power as

 I(n)=∞∑k=n+1Ik=∞∑k=n+1z−1k,n=1,2,⋯. (18)
###### Theorem 2

[Partial Accumulative Interference Power] In a PPP on with intensity , the partial accumulative interference excluding the first dominant interferers to a receiver with reception angle is characterized by its Laplace functional

 LI(n)(s|zn)=exp{ϕλα¯h2/αqzn(s)}, (19)

where .

###### Proof:
 LI(n)(s|zn)=E[exp{−s∞∑k=n+1z−1k}] =E[∞∏k=n+1exp{−sz−1k}] (a)=exp{∫∞zn(exp{−sz−1}−1)Λ2(dz)} (b)=exp{ϕλα¯h2/α(αz2/αn2+s2/αγ(−2α,szn))}, (20)

where uses the Laplace functional for PPP [9, 6] (, here is non-negative function on ) and substitutes Eq. (12).

###### Corollary 2.1 (Average of Accumulate Interference)

The expectation of (partial accumulate interference without first dominant interference) in Poisson random network is

 ¯I(n)=2α−2(¯h2/αϕλ2)α/2(n)1−α/2. (21)

where is the Pochhammer sequence and .

###### Proof:

According to the definition of the Laplace functional, the first moment of

can be directly obtained by the derivative of its Laplace functional evaluated at on condition that is known:

 E[I(n)|zn]=−∂LI(n)(s|zn)∂s|s=0. (22)

Since the derivative of can be directly calculated as

 ∂LI(n)(s|zn)∂s = −LI(n)(s|zn)∫∞zn1ze−s/zΛ2(dz) = −LI(n)(s|zn)ϕλ¯h2/ααs2/α−1γ(1−2/α,s/zn) (23)

we have

 lims→0∂LI(n)(s|zn)∂s=−¯h2/αϕλz2/α−1nα−2. (24)

Thus we can calculated the average of accumulate interference as follows

 ¯I(n)=E[I(n)]=∫∞0E[I(n)|zn]fϕzn(zn)dzn. (25)

Applying Theorem 1 and substituting Eq. (22) gives us the expectation of

 ¯I(n)=2α−2(¯h2/αλθ2)α/2Γ(n+1−α/2)Γ(n). (26)
###### Corollary 2.2 (Scaling from Omni-directional Reception Case)

In a Poisson random network with density , the partial accumulative interference taken in angle directional reception averagely can be equivalent to the accumulative taken in omni-directional reception, if the interference within is avoided, where

 R=(ϕ¯h2/α2)2α(2−α)((n)1−α/2π¯h)12−αλ1α, (27)

here .

###### Proof:

The accumulative interference without the nodes within is formulated as

 IΦ∖B0(R)=∑xi∈Φ∖B0(R)hi||xi||−α. (28)

The Laplace functional of is

 LIΦ∖B0(R)(s|R) =E[exp{−sIΦ∖B0(R)}] =E⎡⎣∏Φ∖B0(R)Eh[exp{−shi||xi||−α}]⎤⎦ =exp{∫Φ∖B0(R)(Eh[e−sh||x||−α]−1)λdx} (29)

Thus the derivative of is

 ∂∂sLIΦ∖B0(R)(s|R) =LIΦ∖B0(R)(s|R){−λ∫Φ∖B0(R)Eh[−h||x||−αe−sh||x||−α]dx} (30)

Then we have

 E[IΦ∖B0(R)]=−lims→0∂∂sLIΦ∖B0(R)(s|R)=2π¯hR2−αα−2. (31)

Then comparing with gives the equivalent condition.

###### Corollary 2.3 (Lower bound of accumulative interference)

A lower bound on the Laplace functional of partial accumulative interference is

 LlI(n)(s)=exp{n+ϕλ¯h2/αs2/ααEzn[γ(−2/α,s/zn)]}, (32)

where is the expectation of with the support of probability density function of , i.e. .

###### Proof:

The lower bound is obtained straightforwardly by applying Jensen’s inequality. Since exponential function is a convex function regarding , thus we have

 ∫∞0LI(n)(s|zn)fϕzn(zn)dzn ≥exp{∫∞0ϕλ¯h2/ααqzn(s)fϕzn(zn)dzn} =exp{n+ϕλ¯h2/αs2/ααEzn[γ(−2/α,s/zn)]}. (33)
###### Corollary 2.4 (Upper bound of accumulative interference)

An upper bound on the Laplace functional of partial accumulative interference is

 LuI(n)(s)=exp{ϕλ¯h2/ααγ(s,¯zn)}, (34)

where

See Appendix A.

## Iii Applications to Communication Reliability

In this section, communication reliability is studied with Nakagami-m fading model, the channel gain has for PDF

 fh(x)=mmxm−1ΩmΓ(m)exp{−mxΩ},m>12, (35)

### Iii-a Outage Probability

Outage of communication occurs when the Signal to Interference Ratio (SIR) drops below a threshold . The outage probability can be calculated by evaluating the CDF of the SIR at the threshold

 FϕSIR(η)=P(η>hu−αI(n))=EI(n)[Fh(uαηI(n))], (36)

where is the Euclidean distance between transmitter and receiver, denotes expectation regarding and

 (37)

where uses PDF (35) and achieves when is positive integer. Then applying Eq. (5) and (19) gives

 FϕSIR(η)=EI(n)[1−m−1∑k=0(muαηI(n))kΩkk!e−muαηI(n)Ω] =1−m−1∑k=0(muαη)kΩkk!EI(n)[Ik(n)e−muαηI(n)Ω] =1−m−1∑k=0(−muαη)kΩkk!Ezn[L(k)I(n)(muαηΩ∣∣∣zn)], (38)

where is the derivative of .

 ¯h2/α=Eh[h2/α]=(mΩ)−2αΓ(m+2α)Γ(m). (39)

### Iii-B Error probability under QoS Constraint

QoS (Quality of Service) is defined by parameter pair and used to measure communication link quality, where is the tolerable queue length for service data at transmitter and is the violation probability of constraint . [10, 11] give the approximation of as

 ϵq∼exp{−θQmax}, (40)

where is QoS exponent222Larger stands for higher QoS requirement, i.e. smaller or violation probability bound .. For any required QoS, corresponding effective bandwidth [10, 11] gives the minimal data rate to meet the QoS requirement , defined as

 a(θ)=limt→∞logE[exp{θA(t)}]tθ, (41)

where is the cumulative source data over time interval . If transmitter can send data out with guaranteed rate , violation error probability can be bounded by . However, data rate over wireless channel is dynamic and unreliable. The selected rate by transmitter could be failed due to poor SIR. With derived result in Section III-A, the error probability that wireless channel can not provide rate is

 ϵr=P[log(1+SIR)

The overall error probability is due to either queue violation either channel fading and can be approximated as

 ϵ≈1−(1−ϵq)(1−ϵr)=ϵq+ϵr−ϵqϵr. (43)

Hence for a given queue length constraint and chosen transmission , the total error can be approximated by Eq. (43).

### Iii-C Relationship between r and ϵ

The following theorem shows the whether a given QoS specification is possible:

###### Theorem 3

Assume a wireless link with error probability for corresponding link achievable rate . Denote the target QoS specification by (). The target QoS is possible to be met by rate adaptation (increasing ), if the condition

 ϵr(r∗)≤1−2√1−ϵ′ (44)

is met. Here is the root to equation

 ϵq(r)=ϵr(r), (45)

where is the queue violation probability with service rate and maximum tolerable queue length at transmitter . Note that the equation has at most one root.

###### Proof:

As stated the total error is actually a function of the selection rate , which can be formed as

 ϵ(r)=ϵq(r)+ϵr(r)−ϵq(r)ϵr(r). (46)

For an error probability , the selected rate must satisfy

 r>a(θ′), (47)

where

 θ′=−logϵ′/Qmax, (48)

otherwise the packet queue at transmitter would not be stable and there would be no bound on the queue violation error.

We rewrite and have

 ϵ(r) =1−(1−ϵr(r))(1−ϵq(r)) (a)≥1−(1−ϵr(r)+ϵq(r)2)2, (49)

where

uses that arithmetic mean of non-negative numbers is greater than or equal to their geometric mean. The equality achieves when the

. By setting , we have

 ∂ϵq(r)∂r=−Qmaxe−θQmax∂θ∂r. (50)

According to [11], effective bandwidth is an increasing function of , thus we have

 ∂r∂θ>0. (51)

Combining Eq. (50) and (51) gives

 ∂ϵq(r)∂r<0. (52)

Thus, we concludes that is monotonically decreasing function of selected rate , i.e. selecting larger rate () leads smaller queue violation error .

On the other hand, we would like to show that communication link failure error is an increasing function of selected rate . Assuming that the probability density function of SINR is ( in its domain) and using Shannon capacity, we have

 ∂ϵr(r)∂r =∂∂r∫er−10f(x)dx =f(er−1)er >0. (53)

Since is monotonically increasing and is monotonically decreasing when we choosing larger , there is one and only one root to if the condition

 ϵr(a(θ′))≤ϵq(a(θ′)) (54)

is met.

Assume that the condition (54) is met and is the unique root to , according to (III-C), we get

 infϵ =1−(1−ϵr(r∗)+ϵq(r∗)2)2 =1−(1−ϵr(r∗)). (55)

We have the QoS requirement that error is no larger than . Thus, this is possibly be met by choosing better communication rate if

 infϵ≤ϵ′. (56)

Otherwise, we can not guarantee that the initial QoS specification would be met by choosing larger .

Substituting Eq. (III-C) into (56) gives the condition

 ϵr(r∗)≤1−2√1−ϵ′, (57)

where .

Theorem 3 gives the sufficient condition to evaluate if a proposed QoS specification can be reasonably fulfilled with a certain communication link condition. However, it is also possible that we can find a root for Eq. (45) that can not fulfill the condition of inequatlity (44). In this case, we claim that it is possible to find a rate that brings lowest total error but without meeting the QoS requirement . Still, this rate selection gives the smallest error, i.e. .

There is worse situation where . In this case, there is no way for QoS to be met. Due to the monotonic property of and regarding , Eq. (45) does not has root. But it is still possible find rate such that . This could be done by solving the first derivative equation

 ∂ϵ(r)∂r=0. (58)

subject to

 r>a(θ′). (59)

## Iv Numerical Results

This section shows some numerical results (“Sim.”) and their comparisons with analytic results (“Ana.”). We set the parameters as node intensity , path-loss exponent , , and , unless stated otherwise.

As shown in Fig. 1, curves of CDF for different dominant interference power are given numerically and analytically. Since is directly sorted by the interference power, we compare it with the nearest333Here nearest node is sorted out by Euclidean distance. Thus the nearest interferer is the closest neighbor by distance. interferer’s power (“Nearest Sim.”) under same fading context. As shown, the difference between and distribution of “Nearest Sim. n” is larger as increases. In small range of , is smaller than CDF sorted by distance. But in large range of , is larger. That means that distance-based approximation can be overestimation or underestimation depending on . For larger , the Euclidean-distance-based approximation has larger bias.

Fig. 2 shows the outage probability against the reception angle , which matches well with in Eq. (III-A). Here . As expected, reception with larger angle is subject to heavier interference and thus the outage probability increases along with the rising . Additionally, decreases obviously for increasing , when excluding more dominant interferers. The changes of outage probability vary with network setting such as . The Rayleigh fading () is simulated for comparison. In small range of , the outage probability of Rayleigh fading is larger than that of , since signal fading of interest due to absence of direct line of sight (LOS) dominates. But, in large value of , outage of Nakagami-m () is larger since LOS advantage makes receiver suffer more interference as large leads to large number of interferers with LOS.

Fig. 3 shows total communication error that matches well with Eq. (43). Poisson arrival with parameter is used and thus effective bandwidth is . For deterministic rate choice, total error decreases with increase of in small range of but flattening eventually in large range of . This means smaller error can be expected by loosening constraint on when initial is not large. Otherwise flattens around . Choosing larger is an effective way to get lower error before flattens but it could bring larger error in larger range of . However, larger could lead smaller total error even in larger range of .

## V conclusion

In this paper, we studied the dominant interference power in random networks modeled by PPP. Both the dominant and partial accumulative interference power were studied. We showed the bias of Euclidean-distance-based approximation by the nearest interferer numerically. This bias could be large for large . Then, the obtained results were used to evaluate communication link reliability by metrics of outage probability and error probability with consideration of queue length violation. The possible way to decrease outage probability and total error was simulated and discussed.

## Appendix A Upper bound of LI(n)(s|zn)

###### Proof:

For the upper bound of , we will show that is concave regarding asymptotically for growing (decreasing dominant power), i.e. the concavity of as a function of is also preserved for non-dense networks (small ). If this condition satisfied, the conditional upper bound is as follows

 ∫∞0LI(n)(s|zn)fϕzn(zn)dzn≤exp{ϕλ¯h2/ααγ(s,¯zn)}, (60)

where

 ¯zn=E[zn]=(¯h2/αλϕ2)−α/2Γ(n+α/2)Γ(n). (61)

According to Theorem 2, the Laplace functional fo partial accumulative interference is

 LI(n)(s|zn)=exp{ϕλ¯h2/ααγ(s,zn)}. (62)

The second derivative of regarding to is

 ∂2LI(n)(s|zn)∂z2n = LI(n)(s|zn)(ϕλ¯h2/αα)2(∂qzn(s)∂zn)2 +LI(n)(s|zn)ϕλ¯h2/αα∂2qzn(s)∂z2n = LI(n)(s|zn)ϕλ¯h2/αα⎛⎝ϕλ¯h2/αα(∂qzn(s)∂zn)2+∂2qzn(s)∂z2n⎞⎠ (63)

It is obvious that is positive. Then it is the formula inside the parenthese that decides the sign of second derivative of in Eq. (A). Thus we need to analyze the two terms inside the parenthesis to see its sign. The is positive. Since

 qzn(s)=s2/αγ(−2/α,szn)+αz2/αn2, (64)

we have the first derivative of regarding as

 ∂qzn(s)∂zn=(1−e−s/zn)z−1+2/αn (65)

and the second derivative as