1

Interference Model Similarity Index and
Its Applications to mmWave Networks:

arXiv:1710.02659v1 [] 7 Oct 2017

Extended version
Hossein Shokri-Ghadikolaei, Student Member, IEEE,
Carlo Fischione, Member, IEEE, and Eytan Modiano, Fellow, IEEE

Abstract
In wireless communication networks, interference models are routinely used for tasks such as
performance analysis, optimization, and protocol design. These tasks are heavily affected by the accuracy
and tractability of the interference models. Yet, quantifying the accuracy of these models remains a major
challenge. In this paper, we propose a new index for assessing the accuracy of any interference model
under any network scenario. Specifically, it is based on a new index that quantifies the ability of any
interference model in correctly predicting harmful interference events, that is, link outages. We consider
specific wireless scenario of both conventional sub-6 GHz and millimeter-wave (mmWave) networks
and demonstrate how our index yields insights into the possibility of simplifying the set of dominant
interferers, replacing a Nakagami or Rayleigh random fading by an equivalent deterministic channel,
and ignoring antenna sidelobes. Our analysis reveals that in highly directional antenna settings with
obstructions, even simple interference models (such as the classical protocol model) are accurate, while
with omnidirectional antennas, more sophisticated and complex interference models (such as the classical
physical model) are necessary. We further use the proposed index to develop a simple interference model
for mmWave networks that can significantly simplify design principles of the important procedures for
wireless communication, such as beamforming, interference management, scheduling, and topology
H. Shokri-Ghadikolaei and C. Fischione are with KTH Royal Institute of Technology, Stockholm, Sweden (E-mails: {hshokri,
carlofi}@kth.se).
E. Modiano is with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge,
USA (E-mail: modiano@mit.edu).
A preliminary version of this paper [1] has been accepted for presentation at the IEEE International Conference on
Communications (ICC), 2016.

2

control. Our new approach makes it possible to adopt the simplest interference model of adequate
accuracy for every wireless network.

Index Terms
Wireless communications, interference model, performance analysis, millimeter wave networks.

I. I NTRODUCTION
Due to the shared nature of a wireless media, interference plays a critical role in the design
and performance analysis of wireless networks, where the intended signal is combined with
other undesired wireless signals transmitted at the same (time, frequency, spatial) channel. The
receiver typically decodes the received signal by canceling parts of the interference and treating
the rest as noise. Successful decoding at the receiver depends on the desired signal strength, the
ambient noise level accumulated over the operating bandwidth, and the interference level. Signalto-interference-plus-noise ratio (SINR) is a common metric to evaluate the outage probability
(or the probability of successful decoding) of a transmission. However, performance analysis
using the SINR expression is complex as it depends on the transmission strategies (transmission
power, antenna pattern, and medium access control (MAC) protocol), often unknown or hard to
estimate random channel attenuation, receiver design, and the (often partially unknown) network
topology. Due to this overwhelming complexity, the design and analysis of wireless networks
based on the actual SINR expression, while being accurate, is very challenging. This difficulty is
further exacerbated in millimeter-wave (mmWave) networks, where penetration loss, first-order
reflection, and antenna pattern introduce further elements of randomness [2]–[4]. This motivates
developing different techniques to mathematically model (abstract) various components of the
SINR, e.g., the transmission strategy, wireless channel, and network topology.
A. Related Works and Motivations
Define an interference model as a set of deterministic or stochastic functions that model various
components of the SINR expression. There have been many attempts in the literature to design
interference models (equivalently, to approximate the SINR expression) that accurately capture
the effect of interference, while being tractable for the mathematical analysis. These interference
models largely try to answer the following questions under various network settings:

3

Q1. How can we model the set of interferer whose contributions in the aggregated interference
term are dominant?
Q2. How can we simplify the transmission/reception and propagation models to enhance tractability of the interference model with marginal loss in its accuracy?
Answering Q1 demands a careful balance between the accuracy and the simplicity of the interference model. Considering the effects of more interferers in the SINR model generally increases the
accuracy but also the complexity. In this regard, the simplest model is the primary interference
model [5], wherein an outage event occurs only if two communication links share a common
endpoint. In other words, the only interference component in this model is self-interference that
leads to a half-duplex operating mode. Interference range model (IRM) is an attempt to improve
the accuracy of the primary interference model [6], where an outage event occurs if the closest
interferer is located no farther than a certain distance of the receiver, called the interference
range. By setting this distance to 0, the IRM can be reduced to the primary interference model.
A modified version of IRM is the protocol model (PRM), formalized by the seminal work of
Gupta and Kumar [7]. The only modification is that the interference range, instead of being a
constant value as in the IRM, depends on the received power from the intended transmitter and a
minimum SINR threshold for successful decoding. Although the IRM and PRM are very simple,
they fail to capture the effect of interference aggregation (i.e., the sum of the interference power
from multiple interferers). It might be that, while there is no interferer inside the interference
range, the aggregated interference from several transmitters outside the interference range downs
the perceived SINR below the threshold. Thus, these models are generally considered to be overly
simplistic. Nonetheless, due to their mathematical tractability, the IRM (including the primary
interference model) and PRM are extensively adopted for the performance analysis and for the
system design; e.g., transport capacity [7]–[9], delay [10], [11], fairness [12], throughput [13]–
[15], topology control [16], [17], routing [18], and backoff design [19].
To alleviate the aforementioned problem of IRM and PRM, the interference ball model (IBM)
considers the aggregated impacts of near-field interferers, located no farther than a certain
distance. The price is higher complexity of the IBM compared to the IRM and PRM. Nonetheless,
the IBM has been extensively adopted in the performance evaluation of wireless networks [4],
[20]–[22]. The topological interference model (TIM) [23] is a natural extension of the IBM that

4

considers the aggregated impact of all the transmitters whose individual interference level at
the receiver side is not below a certain threshold. In other words, this model neglects weak
links based on the “topological” knowledge. The TIM is adopted for capacity and degreeof-freedom analysis [23], [24]. The most accurate and complex answer to Q1 is the physical
model (PhyM) [7], which considers the aggregated interference of all transmitters in the entire
network.1 The PhyM, also known as the SINR model, is adopted mostly at the physical layer;
e.g., beamforming design [26]–[28], capacity evaluation [7], [29], [30], power control [31], [32],
coverage analysis [4], energy efficiency characterization [33], and spectrum sharing [34].
The answer to Q2 depends heavily on the transmission and reception strategies and propagation
environment. For instance, approximating the random wireless channel gain with its first moment
(average) is a common technique to simplify the SINR expression and to design MAC and
networking layers [14], [17], [21], [35]–[38]. Reference [39] replaced a Nakagami fading channel
by a Rayleigh one for mathematical tractability and numerically concluded from its Fig. 5 that
such approximation preserves the main properties of the rate coverage performance. Yet, the
impact of these mathematical approximations on the accuracy of the performance analysis is not
well understood. Recently, [40] considers the impact of such approximation on the scheduling.
In particular, the authors show that, if we design scheduling for n transmitters based on a proper
non-fading channel model (deterministic approximation of the random channel gain), the network
throughput will be within O(log n) of that of the optimal scheduler, designed based on the actual
random channel gains. This result, however, is limited to the Rayleigh fading model. As another
example, for mmWave communications with many antenna elements, [14] and [41] assume no
emissions from the antenna sidelobe, which affects the SINR distribution. This assumption is
relaxed in [4], where the antenna sidelobe is modeled by a small constant value, adding further
complexity into the interference model. As a result, the final derivations, while being more
accurate, are less tractable and provide less insights. However, without having a mathematical
framework that allows assessing the impact of neglecting antenna sidelobes, it is not clear which
approach better balances the simplicity-accuracy tradeoff of mathematical analysis.
1

Under very special network settings (e.g., homogenous Poisson field of interferers exhibiting Rayleigh fading channel), the

PhyM may be mathematically more tractable than both PRM and IBM [25]; however, the PRM and IBM are yet more desirable
models for protocol design and for network optimization [22].

5

The proper choice of interference model depends on many parameters such as the receiver design,
antenna directionality, network topology, channel model, and the choice of medium access
protocol [6], [35], [36]. To the best of our knowledge, there has been no systematic method
to analyze the accuracy of various interference models, choose the proper interference model,
and quantify the amount of error due to adopting other interference models for a given network
scenario. The accuracy of different interference models has been mostly evaluated qualitatively,
without fully understanding the mutual impacts of different parameters of the physical, medium
access, and network layers. This qualitative analysis, however, is often overly simplistic, and may
result in the use of interference models that are only marginally more accurate, yet significantly
more complex than needed. As we will show throughout this paper, in certain settings of relevant
practical interest, even the simplest interference models are sufficiently accurate and can be used
to provide significant insights into the network performance and to enable efficient protocol
design.

B. Contributions
In this paper, we substantially extend the preliminary version of this study [1] and propose a new
framework to assess the distance of two arbitrary SINR distributions. We use this framework to
develop an interference model similarity index that takes on real values between 0 and 1, where
higher values correspond to higher similarity. This index builds a universal method to assess
the accuracy of any interference model under any network scenario. In other words, instead of
introducing a new interference model or a new approach to analyze SINR distribution, we propose
a novel framework to investigate the accuracy of the existing interference models. Therefore,
our study is complementary to the rich literature of interference analysis.
To exemplify the abilities of the proposed index, we mathematically evaluate it for the PRM and
IBM under three scenarios: (i) Rayleigh fading channel and omnidirectional communications (a
typical sub-6 GHz system); and (ii) Rayleigh fading channel and directional communications; and
(iii) deterministic wireless channel, directional communications, and existence of impenetrable
obstacles in the environment (a typical mmWave system). Although the applications of the
proposed index is general and goes beyond the examples provided in this paper, we use these
examples to illustrate fundamental properties of this index and also to provide insights on the

6

mutual effects of various network parameters on the accuracy of the interference model, thus
commenting on the proper model for a given network scenario.
In the first example scenario, served as a baseline, we derive a closed-form expression for
the accuracy index. We show that the accuracy of the IBM monotonically increases with the
interference range, at the expense of an increased complexity. In contrast, we show that there
is no such monotonic improvement in the accuracy of PRM. Thereby, we find the optimal
interference range that maximizes the accuracy of the PRM.
In the second example scenario, we show that both the PRM and IBM are significantly more
accurate with directional antennas. Further, in the third example scenario, we show significant
accuracy improvement of both PRM and IBM due to deterministic channel, directionality, and
also blockage. As these conditions hold in mmWave networks, we show that the PRM can be
used in the analysis of mmWave networks to significantly improve the mathematical tractability
of the problem, with a negligible loss in the analysis accuracy. We further use this index to
observe marginal impacts of the first-order reflection and sidelobe transmissions on the accuracy
of the interference model, which inspire us to propose a tractable and accurate interference model
for mmWave networks.
Furthermore, we use the proposed framework to investigate the feasibility of modeling a random
fading channel with a deterministic channel. We show that if the spatial distribution of the
transmitters follow a Poisson point process on the plane and if the path-loss exponent is 2, then
the average of the fading random variable2 is among the best constant approximations of the
random fading channel to analyze any ergodic function of the SINR (e.g., transport capacity,
throughput, and delay).
Throughout the paper, we show how the proposed index can increase our understanding of the
mutual interactions among the accuracy of the performance evaluation and various network
parameters and modeling techniques. We also signify how we can rigorously develop simple
interference models of adequate accuracy to simplify design principles of the main functions
of wireless communications such as beamforming, interference management, scheduling, and
topology control.
2

Rigorously speaking, the fading should be an absolutely continuous random variable, which holds for almost all wireless

channels.

7

The rest of this paper is organized as follows. In Section II, we introduce our interference model
similarity index, and investigate it under various network scenarios in Sections III–VI. Future
works are presented in Section VII, and the paper is concluded in Section VIII.
II. I NTERFERENCE M ODEL S IMILARITY I NDEX
A. Interference Model
We define a link as the pair of a transmitter and its intended receiver, where transmitter (receiver) i
refers to the transmitter (receiver) of link i. Without loss of generality and for brevity, we assume
that there is no interference cancellation, so all unintended transmitters act as potential interferers
to any receiver. Consider a reference receiver and label its intended transmitter by subscript 0.
Denote by I the set of its interferers (all active transmitters excluding the intended transmitter),
by pi the transmission power of transmitter i, by σ the power of white Gaussian noise, by di the
distance between transmitter i and the reference receiver, and by giCh the channel gain between
transmitter i and the reference receiver. We denote by giTx the antenna gain at transmitter i toward
the reference receiver, and by giRx the antenna gain at the reference receiver toward transmitter
i. Then, the SINR at the reference receiver is
p0 g0Txg0Ch g0Rx
.
γ= P
pk gkTxgkCh gkRx + σ
k∈I

The SINR depends on the transmission powers, antenna patterns, set of active transmitters,
channel model, and network topology. Let β > 0 denote the SINR threshold corresponding to
a certain target bit error rate. An outage on the reference link occurs when γ < β. Different
interference models attempt to approximate the outage probability by ignoring certain components of the interference (see questions Q1 and Q2 in Section I-A). In particular, the IRM, PRM,
IBM, TIM, and PhyM characterize the set of interferers I. Neglecting various components of
the channel model translates into different distributions for giCh . Power allocation affects pi , and
various scheduling protocols further affect I.
B. Formal Definition of the Similarity Index

Consider reference interference model y under a given set of parameters/functions describing the
wireless network. Define γ y as the SINR of a reference receiver under this model. We define a

8

binary hypothesis test, where hypotheses H0 and H1 denote the absence and presence of outage
under reference model y, respectively. That is,


H0 , if γ y ≥ β ,

H ,
1

(1)

y

if γ < β .

We consider a test interference model x under any set of parameters/functions describing our
wireless network, which are not necessarily equal to those of the reference model y. These
differences result in possible deviation of the SINR of the reference receiver under x, denoted
by γ x , from γ y . From the outage point of view, irrespective of the differences between individual
parameters/functions of x and y, we say model x is similar to model y if it gives exactly the same
outage result as y. Assume interference model x is a detector of outage events under y. To evaluate
the performance of this detector compared to reference model y, we can use the notions of false
alarm and miss-detection. A false alarm corresponds to the event that x predicts outage under
hypothesis H0 (i.e., y declares no harmful interference); whereas a miss-detection corresponds
to the event that x fails to predict outage under hypothesis H1 . Now, the performance of any
interference model x can be evaluated using the false alarm and miss-detection probabilities,
x|y

x|y

namely pfa and pmd . Formally,
x|y

pfa = Pr [γ x < β | γ y ≥ β] ,

x|y

pmd = Pr [γ x ≥ β | γ y < β] .

(2)

The false alarm and miss-detection probabilities quantify the similarity of any interference model
x in detecting outage events compared to any reference model y. Next, we define our index to
be a convex combination of these probabilities.
Definition 1 (Interference Model Similarity Index). For any constant 0 ≤ ξ ≤ 1, any SINR
threshold β, any test interference model x, and any reference interference model y, we define
similarity of x to y at β as




x|y
x|y
x|y
x|y
Sβ,ξ (xky) = ξ 1 − pfa + (1 − ξ) 1 − pmd = 1 − ξ pfa − (1 − ξ) pmd ,
x|y

(3)

x|y

where pfa and pmd are given in (2). Notice that random variables γ x and γ y must have a
common support.
Sβ,ξ (xky) is a unit-less quantity ranging within [0, 1], where higher values represents higher similarity between x and y in capturing outage events at SINR threshold β. Setting ξ = Pr [γ y ≥ β],
x|y

x|y

ξpfa +(1 − ξ) pmd is the average error in detecting the outage events; therefore, Sβ,Pr[γ y ≥β] (xky)

9

shows the probability that interference model x has similar decision as reference interference
model y in detecting the outage events.
Remark 1 (Accuracy of an Interference Model). Let reference model y perfectly capture the
outage events in reality, namely the model y does not make any approximation/simplification.
The accuracy of any interference model x is then Sβ,ξ (xky), and we call it the accuracy index
throughout the paper.
The proposed index is a universal metric that can be used to quantify the accuracy of any
interference models, proposed in the literature, as we exemplify in the following sections.

C. Comparison to the Existing Statistical Distance Measures
Interference model similarity index, formulated in (3), is measuring the distance3 of the PDF
of γ x compared to that of γ y . Let fX denote PDF of random variable X. In the following, we
highlight three main advantages of using our index with respect to the existing standard distance
measures, such as the Bhattacharyya distance and the Kullback-Leibler (KL) divergence [42].
First, the existing standard distance measures mostly map the distance between fγ x and fγ y in
their entire support to only one real value. It might be that two distribution are very similar in the
meaningful ranges of the SINR values (0–10 dB), but very different outside this range. Still, the
classical statistical distance measures may result in a high distance between two distributions,
as they compare fγ x to fγ y in the entire SINR range. This is indeed a misleading result that
may mistakenly avoid the use of the simplified interference model x in practice. However, our
similarity index allows us to investigate whether or not x is accurate at any given SINR threshold.
Second, both the Bhattacharyya distance and the KL divergence may fail in a comparative
analysis. In particular, fγ y might be more similar to fγ x than fγ z with point-wise comparison,
but the Bhattacharyya distance and the KL divergence of fγ x from fγ y become higher than that
of fγ z from fγ y , as shown in the following toy example.
Example 1. Consider discrete random variables X, Y, and Z with common support of [1, 2, 3]
with probability mass functions
3

Rigorously speaking, our similarity index is not a distance measure, as it does not satisfy the subadditivity property. Moreover,

we are measuring the similarity, which could be in general a decreasing function of the distance.

10

t

fX (t)

fY (t)

fZ (t)

1

0.05

0.1

0.25

2

0.25

0.45

0.2

3

0.7

0.45

0.55

Then, we have the following metrics (fX is the reference in the KL divergence):
Distributions

fX ,fY

fX ,fZ

Euclidean distance

0.324

0.255

Bhattacharyya distance

0.033

0.045

KL divergence

0.059

0.098

In this example, neither the Bhattacharyya distance nor the KL divergence can identify higher
point-wise similarity of Z to X than Y to X.
Last, but not least, unlike the existing statistical distance metrics that are not necessarily intended
for communication systems, our similarity index is developed for these systems so that it has a
physical meaning and can provide practical insights. Specifically, setting ξ = Pr [γ y ≥ β], our
index Sβ,ξ (xky) evaluates the probability of correct decision of outage events under interference
model x.
Note that other distance metrics may still be useful to evaluate the accuracy of an interference
model, and they may also have some relationship to our proposed index; see the following
remark as an example.
Remark 2 (Relationship to the Bhattacharyya Coefficient). Let ξ = Pr [γ y ≥ β]. By noting that
Sβ,Pr[γ y ≥β] (xky) is the probability of having no hypothesis detection error and following [42,
Equation (48)], we get
r
p
3
1
− ξ − ρ ξ (1 − ξ) ≤ Sβ,ξ (xky) ≤ 1 − ξ +
− ξ (1 − ξ) ρ2 ,
2
4
R
where ρ = fγ x (t)fγ y (t)dt is the Bhattacharyya coefficient.

(4)

D. Applications of the Interference Model Similarity Index
In the following, we provide two class of illustrative examples where our index can be used
either to simplify the mathematical analysis or to justify the existing interference models. Use
cases of our index, however, goes beyond these examples.

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1) Simplifying the Set of Interferers: This is one of the first steps in choosing an interference
model for performance analysis, protocol design, and network optimization. With omnidirectional
transmission/reception and without interference cancelation, an outage occurs under
•

PRM: if there is an active transmitter no farther than an interference range rPRM = (1+∆)d0 ,
where ∆ is a constant real positive value [7];

•

IBM: if its SINR due to all active transmitters located no farther than an interference range
rIBM is less than β [22];

•

TIM: if its SINR due to all active transmitters with strong links (with individual channel
gains higher than ε) toward receiver i is less than β [23]; and

•

PhyM: if its SINR due to all active transmitters is less than β [7].

To present a unified view, we associate three random variables aPRM
, aIBM
, and aTIM
to the link
k
k
k
between each transmitter k ∈ I and the typical receiver. aPRM
is set +∞ if dk ≤ (1 + ∆)d0 ,
k
and otherwise 0. aIBM
is set 1 if dk ≤ rIBM , and otherwise 0. Finally, aTIM
is set 1 if gkCh > ε,
k
k
and otherwise 0. We define a virtual channel gain for those interference models as
gkx = axk gkCh ,

for interference model x ,

(5)

where x is a label denoting PRM, IBM, TIM, or PhyM, and aPhyM
, 1. Despite the virtual
i
channel gain, all other parameters of interference models x and y are identical. The SINR at the
typical receiver under interference model x is given by
p0 g0Tx g0Ch g0Rx
γx = P
.
pk gkTxgkx gkRx + σ

(6)

k∈I

The design of many key functions of a wireless network such as scheduling [43] or power
allocation [31] need an estimate of (6). To this end, a receiver may need to coordinate with a
set of interferers to estimate their individual instantaneous contributions to the SINR expression,
namely pk gkTxgkx gkRx for all k ∈ I. The PhyM may imply that every receiver should coordinate
with all the interferers in the entire network (global information) whose cost, complexity, and
delay may be unaffordable in many networking scenarios. Using IBM implies that each nodes
should coordinate with all transmitters within a certain radius (local information), and the PRM
necessitates coordination only with the closest unintended transmitter, which are appealing from
energy and protocol overhead perspectives. Our proposed index gives a quantitative insight on
the accuracy of various interference models, used for protocol development and for network

12

optimization, and allows the use of the right interference model for a given channel model and
network scenario.
2) Simplifying the Channel Model: Our accuracy index can be used to adopt tractable channel
models (gkCh for every transmitter k) of adequate accuracy. This is specially important for
mmWave networks, where LoS and non-LoS conditions have different channel models, nonLoS (blockage) probability follows a rather complicated function, the LoS channel may follow
a Nakagami fading in general, and realistic antenna patterns might be a complicated non-linear
function. Various researches tried to simplify those complications without rigorous analysis
on the validity of such simplifications. For instance, [14] assumed impenetrable obstacles (so
communication only in the LoS conditions) and neglected antenna sidelobe, [4] approximated
the non-LoS stochastic function by a deterministic LoS ball in which there is no obstacle within
a certain range of the receiver and there is no LoS links outside the circle, and [39] replaced
the Nakagami fading channel by a Rayleigh fading that facilitates mathematical analysis. Due to
lack of a systematic approach to simplify the channel model, the understanding of the cross-layer
dynamics between MAC and physical layers of most of the existing standards is a largely open
problem, and the existing frameworks such as the one in [44] are not usually mathematically
tractable.
In the following, we illustrate the utility of our index for four example scenarios. Although
our index poses no limitation to these example scenarios, we may simplify some parameters of
the system model to avoid unnecessary complications. In the first three examples, we focus on
simplifying the set of interferers for various network settings and derive closed-form expressions
for the accuracy index to highlight its fundamental properties. In the last example scenario, we
use our index to numerically assess the accuracy of various approaches in simplifying the channel
model.
For the rest of this paper, without loss of generality, we assume ξ = Pr [γ y ≥ β], so Sβ,ξ (xky)
evaluates the probability of correct decision under interference model x.

13

III. E XAMPLE S CENARIO 1: R AYLEIGH FADING C HANNEL

WITH

O MNIDIRECTIONAL

C OMMUNICATIONS
Consider a wireless network with Rayleigh fading channel and omnidirectional transmission/reception.
Assume that the PhyM can perfectly capture the outage events. In this section, we evaluate the
accuracy of IBM, PRM, and TIM (see Section II-D where we recalled the definition of these
prominent models) for such scenario.
We consider a reference receiver (called the typical receiver) at the origin of the Polar coordinate,
and its intended transmitter having geometrical/spatial length d0 . We consider a homogeneous
Poisson network of interferers (unintended transmitters) on the plane with intensity λt . We assume
that all the transmitters are active with transmission power p (no power control), and that there is
no interference cancellation, which are natural assumptions in personal and local area networks.
With omnidirectional transmission and reception, there is no antenna gains, so gkTx = gkRx = 1,
k ∈ I ∪ {0}. Note that, under these assumptions, the PhyM is more tractable for coverage
and rate analyses than other models (PRM, IBM, and TIM) [25]; however, we still use this
example to derive closed-form expression for the new accuracy index and thereby illustrate its
fundamental properties that hold in general. Nonetheless, even in this network setting, the PRM
and IBM are more appealing than PhyM for protocol design and for network optimization [22].
We define by B(θ, rin , rout ) a geometrical annulus sector with angle θ, inner radius rin , and outer
radius rout , centered at the location of the typical receiver (origin of the Polar coordinate). To
model a wireless channel, we consider a constant attenuation c at reference distance 1 m, a
distance-dependent attenuation with exponent α, and a Rayleigh fading component h. To avoid
the physically unreasonable singularity that arises at the origin under power law attenuation, we
change the path loss index to α1B(2π,0,a) , where 1· is the indicator function assuming value 1 over
set · and zero otherwise. This modified power law model implies that the signal of all transmitters
located outside a disk with radius a will be attenuated by traditional power law method; however,
the transmitters inside this disk will observe no channel attenuation. Therefore, the channel gain
−α1B(2π,0,a)

between transmitter i at radial distance di and the typical receiver is giCh = chi di

.

To avoid unnecessary complications while illustrating the utility of our index, we eliminate the
shadow fading from our channel model.
We are now ready to illustrate the utility of our proposed index using the SINR expression (6).

14

A. Accuracy of the Interference Ball Model
For mathematical tractability, we assume that rIBM ≥ a and d0 ≥ a, and the extension to the
general case is straightforward. The false alarm probability can be reformulated as

 

 IBM
 Pr γ IBM < β Pr γ PhyM ≥ β | γ IBM < β
IBM|PhyM
PhyM
. (7)
pfa
= Pr γ
<β|γ
≥β =
1 − Pr [γ PhyM < β]
Although the PhyM considers the impacts of all the interferers in the entire network, the
IBM considers only the effects of the near-field ones. Consequently, γ PhyM ≤ γ IBM , and thus


IBM|PhyM
Pr γ PhyM ≥ β | γ IBM < β = 0 in the nominator of (7). This results in pfa
= 0.

For the miss-detection probability, we have




IBM|PhyM
pmd
= Pr γ IBM ≥ β | γ PhyM < β = 1 − Pr γ IBM < β | γ PhyM < β

 



Pr γ IBM < β Pr γ PhyM < β | γ IBM < β
Pr γ IBM < β
=1−
= 1−
,
(8)
Pr [γ PhyM < β]
Pr [γ PhyM < β]
where the last equality is 
from γ PhyM ≤ γ IBM . In Appendix A, we have derived






α


−α
−σβd0
α 
α
2
− πλt Eh a2 1 − e−βd0 h + rIBM
1 − e−βd0 hrIBM −
Pr γ IBM < β = 1 − exp

pc












2
2
−α
2/α
2/α
α
α
−α
α
−α
α
−βdα
ha
2

0
,
+ (βd0 h) Γ 1 − , βd0 hrIBM − (βd0 h) Γ 1 − , βd0 ha
a 1−e

α
α


(9)

and


Pr γ

PhyM






σβdα0






−βdα
ha−α
2
−βdα
h
2
0

0
< β = 1 − exp −
− πλt Eh a 1 − e
−a 1−e

pc







 

2
2
α 2/α
α
−α 
α 2/α
− (βd0 h) Γ 1 − , βd0 ha
, (10)
+ (βd0 h) Γ 1 −

α
α




where Γ (·, ·) is the incomplete Gamma function, Γ (·) is the Gamma function, Eh denotes
expectation over random variable h, and the probability density function of h is fh (x) = e−x .

Substituting (9) and (10) into (8), the miss-detection probability can be found. Also, from (3),
the accuracy of the interference ball model Sβ,ξ (IBMkPhyM) is derived. A simple extension of
our analysis gives the accuracy index when d0 is a random variable. Recall that the purpose of
this section is to illustrate only the utility of our index, and investigating more practical system
models is a subject of our future work; see for instance [45].

15

Result 1 (Perfect Interference Ball Model). For any constant 0 ≤ ξ ≤ 1 and any β,
Sβ,ξ (IBMkPhyM) → 1 as rIBM → ∞.
IBM|PhyM

Proof: We know that pfa
= 0 for any constant 0 ≤ ξ ≤ 1 and any β. Moreover, as rIBM
 IBM



IBM|PhyM
increases, Pr γ
< β tends to Pr γ PhyM < β . Considering (8), Pmd
asymptotically

goes to zero as rIBM → ∞. With zero false alarm and asymptotically zero miss-detection
probabilities, the proof is concluded from (3).

Result 1 indicates that the IBM becomes more accurate with higher rIBM , and it can be arbitrary
accurate for sufficiently large rIBM . The price, however, is more complicated IBM as its approximations at a receiver demands coordination with more interferers.4 Also, negotiation with other
transmitters (e.g., for MAC layer design) within this larger rIBM becomes more challenging in
terms of power consumption, signaling overhead, delay, and processing overhead.

B. Accuracy of the Protocol Model
We now consider the PRM and first note that




1 − Pr γ PRM < β 1 − Pr γ PhyM < β | γ PRM ≥ β
PRM|PhyM
,
(11)
pfa
=1−
1 − Pr [γ PhyM < β]
and that

 

1 − Pr γ PRM < β Pr γ PhyM < β | γ PRM ≥ β
PRM|PhyM
pmd
=
.
(12)
Pr [γ PhyM < β]
In the last two equations, note that Pr[γ PhyM < β] is derived in (10). In the following, we


derive Pr[γ PRM < β] and Pr γ PhyM < β | γ PRM ≥ β .

Event γ PRM < β occurs if there is at least one interferer inside B(2π, 0, rPRM ). As I is a
homogenous Poisson point process with intensity λt , we have



2
Pr γ PRM < β = 1 − exp −λt πrPRM
4

.

(13)

Note that for special settings of this section, considering the impact of all interferers (PhyM) simplifies the analysis. However,

this does not hold in general, e.g., if we change the spatial distribution of the interferers to a determinantal point process.

16

In Appendix A, we have also derived 



 −σβdα


−α
α
0
2
− πλt Eh  − rPRM
1 − e−βd0 hrPRM
Pr[γ PhyM < β | γ PRM ≥ β] = 1 − exp

pc






 



2
2
α 2/α
α 2/α
α
−α
+ (βd0 h) Γ 1 −
− (βd0 h) Γ 1 − , βd0 hrPRM  . (14)

α
α



Substituting (11)–(14) into (3), we can find Sβ,Pr[γ PhyM ≥β ] (PRMkPhyM) for Rayleigh fading
channel with omnidirectional transmission/reception.

Result 2 (Miss-detection–False Alarm Tradeoff). Consider the protocol model of interference
with Rayleigh fading channel. Increasing the interference range rPRM reduces the false alarm
probability and increases the miss-detection probability. Decreasing the interference range increases the false alarm probability and reduces the miss-detection probability.


Proof: Pr γ PRM < β is a strictly increasing function of rPRM , see (13). Considering the equations

of the false alarm and miss-detection probabilities given in (11) and (12), the proof concludes.

Result 3 (Asymptotic Accuracy of the Protocol Model). Consider Equations (3) and (11)–(13).
For any 0 ≤ ξ ≤ 1 and any β > 0, we have the following asymptotic results:
PRM|PhyM

→ 0 , pmd

PRM|PhyM

→ 1 , pmd

rPRM → a, a → 0 ⇒ pfa
rPRM → ∞ ⇒ pfa

PRM|PhyM

→ 1 , Sβ,ξ (PRMkPhyM) → ξ .

PRM|PhyM

→ 0 , Sβ,ξ (PRMkPhyM) → 1 − ξ .

Result 3 further confirms the tradeoff between the miss-detection and false alarm probabilities.

C. Numerical Illustrations
To illustrate the accuracy index in Scenario 1 with Monte Carlo simulation, we consider a spatial
Poisson network of interferers and obstacles with density λt and λo per unit area. Length of
the typical link is d0 = 20 m. We simulate a traditional outdoor microwave network [4] with
average attenuation c = 22.7 dB at the reference distance a = 1 m, path-loss index α = 3.6, and
noise power σ = −111 dBm (around 2 MHz bandwidth). We consider p = 20 dBm transmission
power and β = 5 dB minimum SINR threshold. For the ease of illustration, we define the notion

17
1
0.9

PRM: Pfa , dt = 30

0.8

PRM: Pfa , dt = 80
0.6

Accuracy index

Detection performance

1

PRM: Pmd , dt = 30
PRM: Pmd , dt = 80

0.4

IBM: Pmd , dt = 80

0.2
0
20

PRM: dt = 30

0.8

IBM: dt = 80

0.7

PRM: dt = 80

0.6
0.5

40

60
80
100
Interference range [m]

120

140

0.4
20

40

60
80
100
Interference range [m]

(a) Error probabilities

120

140

(b) Accuracy index

Fig. 1: Impact of the interference range on the accuracy of interference models under Rayleigh fading channel and omnidirectional
communications.

√
of the average inter-transmitter distance as dt = 1/ λt . This distance directly relates to the
inter-site distance in cellular networks, and also shows the transmitter density in a network.
Fig. 1 illustrates the impact of the interference range on the accuracy of both IBM and PRM
PRM|PhyM

under Scenario 1. From Fig. 1(a), increasing rPRM increases pfa

PRM|PhyM

and reduces pmd

,

highlighted as the tradeoff between the miss-detection and false alarm probabilities in Result 2.
This tradeoff may lead to increment (see dt = 30) or decrement (see dt = 80) of the accuracy
index of the PRM with the interference range. The IBM has zero false alarm probability, not
IBM|PhyM

depicted in Fig. 1(a) for sake of clarity of the figure. Moreover, as stated in Result 1, pmd

decreases with rPRM , leading to a more accurate IBM, as can be confirmed in Fig. 1(b). Note
that with the same transmitter density and interference range, the PRM has lower miss-detection
probability than the IBM; however, better false alarm performance of the IBM leads to less
errors in detecting outage events and therefore higher accuracy index, see Fig. 1(b). The TIM,
not depicted in the figure, has a very high accuracy in all simulations. In particular, with
ε = −130 dB, its accuracy is about 0.99. However, the corresponding TIM considers many
interferers inside an irregular geometrical shape, which substantially decreases the tractability of
the resulting interference model.
Fig. 2 shows the accuracy of the IBM and PRM under Scenario 1 against the average intertransmitter distance. Again, we can observe an enhancement in the accuracy of the IBM with
rIBM , whereas the accuracy index of the PRM shows a complicated behavior as a function of
rPRM . By adopting the optimal rPRM that maximizes the accuracy index, as shown in Fig. 2(b),

18
1

1

Accuracy index

Accuracy index

0.9
0.8
0.7
rIBM = 20
rIBM = 60

0.6

0.8
0.6
rPRM = 20
rPRM = 60
rPRM = 120
optimal rPRM

0.4

rIBM = 120
0.5

0

50
100
150
Average inter-transmitter distance [m]

(a) Interference ball model

200

0.2

0

50
100
150
Average inter-transmitter distance [m]

200

(b) Protocol model of interference

Fig. 2: Impact of transmitter density on the accuracy of the interference models under Rayleigh fading channel and omnidirectional
communications. The accuracy of the TIM with ε = −130 dB is higher than 0.98.

we can maintain a good performance for the PRM. Both interference models are very accurate
at extremely dense transmitter deployments. The main reason is the very high interference level


(ξ = Pr γ PhyM ≥ β is almost 0 in this case), implying that the accuracy index is determined only

by the miss-detection probability. Increasing the transmitter density through reducing dt decreases

the miss-detection probability for both IBM and PRM, see Fig. 1(a), improving their accuracy.
For ultra sparse transmitter deployments, again, both interference models work accurately, as ξ
goes to 1 in this case and therefore only the false alarm probability determines the accuracy
index. This probability is zero for the IBM, and it gets smaller values (asymptotically zero) for
the PRM with higher dt , see Fig. 1(a). Finally, the TIM with ε = −130 dB, not shown in Fig.2,
has a very high accuracy in modeling the interference. Its accuracy for the same ranges of dt is
higher than 0.98.
Fig. 3 shows the KL divergence of fγ IBM (x) from fγ PhyM (x) and also their Bhattacharyya distance
for the same setting of Fig. 2(a), where lower values translates into higher accuracy of the IBM.
From this figure, both the KL divergence and the Bhattacharyya distance can identify higher
accuracy of the IBM with rIBM = 60 m. However, they both fail to show that the performance
of IBM with rIBM = 20 m converges to that with rIBM = 60 m once the network gets sparser.
Moreover, calculating these measures entails almost the same mathematical/numerical complexity
as our similarity index. Due to these reasons, we investigate only our accuracy index for the
rest of the paper, though one may incorporate those metrics in our proposed interference model
similarity analysis framework.

19

0.08
KL: rIBM
KL: rIBM
BD: rIBM
BD: rIBM

Distance

0.06

= 20
= 60
= 20
= 60

0.04
0.02
0

0

50
100
150
Average inter-transmitter distance [m]

200

Fig. 3: The KL divergence (labeled by “KL”) of the distribution of γ IBM from that of γ PhyM and their Bhattacharyya distance
(labeled by “BD”) corresponding to the accuracy index values of Fig. 2(a). Lower values translate into higher similarity between
two distributions.

1

Accuracy index

0.8
0.6
0.4
0.2
0
-10

IBM:
IBM:
PRM:
PRM:
-5

dt
dt
dt
dt

= 50
= 100
= 50
= 100
0

5
10
SINR threshold [dB]

15

20

Fig. 4: Impact of the SINR threshold on the accuracy of interference models under Rayleigh fading channel and omnidirectional
communications.

Fig. 4 illustrates the accuracy index against the SINR threshold. Increasing the SINR threshold
generally increases the sensitivity of the interference model to any approximation error in x.
IV. E XAMPLE S CENARIO 2: R AYLEIGH FADING C HANNEL , D IRECTIONALITY,

AND

O BSTACLES
In this section, we analyze the accuracy of IBM and PRM in modeling a wireless network with
Rayleigh fading channels, where all transmitters and receivers use directional communications
to boost the link budget and to reduce multiuser interference. We also consider impenetrable

20

obstacles. The application areas of this scenario include modeling and performance evaluation
of mmWave networks, where directional communication is inevitable and extreme penetration
loss due to most of the solid materials (e.g., 20–35 dB due to the human body [46]) justifies the
impenetrable obstacle assumption. In Section VI-B, we will comment on the impact of assuming
impenetrable obstacles on the accuracy of the interference model.
Note that the interference is not the primary limitation of mmWave networks specially if we
take an average over all possible realizations of a random topology [4], [41]. However, even
if mmWave networks are noise-limited in a statistical sense (that is, taking an average of the
interference over some time or some topologies), there are significant realizations of network
topologies at given times where some transmitters can cause strong interference. We cannot use
noise-limited arguments, which are valid over some time horizons, when we have to optimize
in real-time resource allocations or routing. In the following two sections, we show that special characteristics of mmWave networks, such as blockage and deafness, can be exploited to
substantially simplify the interference model, so as to develop efficient scheduling and routing
algorithms, which may otherwise be impossible. In fact, our results provide, for the first time,
mathematical justifications for the use of simpler interference models in mmWave networks, as
extensively done in the literature [14], [15], [17], [47]–[51].
We assume a homogenous Poisson network of interferers as in Section III. If there is no obstacle
on the link between transmitter i and the typical receiver located at the origin, we say that
transmitter i has line-of-sight (LoS) condition with respect to the typical receiver, otherwise it
is in non-LoS condition. We assume that transmitter of every link is spatially aligned with its
intended receiver, so there is no beam-searching phase [52]. We model the antenna pattern by
an ideal sector model [4], where the antenna gain is a constant in the main lobe and another
smaller constant in the side lobe. We assume the same operating beamwidth θ for all devices
in both transmission and reception modes. Then, the antenna gain for each transmitter/receiver
is [52, Equation (3)]



 2π − (2π − θ) z ,
θ

z,

inside the main lobe
(15)
inside the side lobe ,

where 0 ≤ z ≪ 1 is the side lobe gain. For mathematical tractability, we assume negligible
side lobe gain (i.e., z = 0) throughout this section, and numerically assess the impact of this

21

simplification in Section VI-B.
Consider the link between transmitter i and receiver j with distance dij . It is shown that
with a random number of obstacles, each having random location and size, this link is in the
LoS condition with probability e−ǫλo dij , where λo is the intensity of the obstacles and ǫ is a
constant value that depends on the average size of obstacles in the environment [53]. Due to
the exponential decrease of the LoS probability with the link length (also see [54, Fig. 4]), very
far interferers are most likely blocked. For mathematical simplicity, we assume independent
LoS conditions among the typical receiver and all other transmitters, and also impenetrable
obstacles. Nonetheless, the following analysis can be extended for more realistic blockage
models, introduced in [41]. Notice that we are using this simplified model to investigate the effects
of directionality and blockage on the accuracy of the interference models and to characterize
fundamental properties of the proposed accuracy index. The exact value of the accuracy index
with a more realistic mmWave channel can be readily numerically calculated under any system
model, as we highlight in the next sections.
To evaluate the accuracy of IBM and PRM, we first notice that an intended transmitter can cause
a significant interference contribution to the typical receiver if: (a) the typical receiver is inside
its main lobe, (b) it has LoS condition with respect to the typical receiver, and (c) it is inside the
main lobe of the typical receiver. Due to random deployment of the transmitters and receivers, the
probability that the typical receiver locates inside the main lobe of a transmitter is θ/2π. Moreover, we have independent LoS events among the typical receiver and individual transmitters.
Therefore, the interferers for which conditions (a)–(b) hold follow an inhomogeneous Poisson
point process I with intensity of λI (r) = λt θe−ǫλo r /2π at radial distance r. Condition (c) reduces
the angular region that a potential interferer should be located to contribute in the interference
observed by the typical receiver. We note that I ∩ B(θ, 0, rPRM ) is the set of potential interferers
inside the vulnerable region of the PRM, shown by red triangles in Fig. 5, and I ∩B(θ, rPRM , ∞)
shows the set of potential interferers outside that region, shown by green circles in Fig. 5. Also,
I ∩ B(θ, 0, rIBM ) is the set of potential interferers for IBM (near-field interferers).

22

U350

Fig. 5: Illustration of the vulnerable area.

A. Impact of Directionality and Blockage
Before deriving the accuracy of IBM and PRM, we first evaluate the impact of directionality
and blockage on the number of the interferers. We define by ΛB(θ,0,R) the measure of the region
B(θ, 0, R), i.e., the average number of interferers inside the region. We have
Z R

θ 2 λt 
−ǫλo R
ΛB(θ,0,R) = θ λI (r)r dr =
1
−
(1
+
ǫλ
R)
e
.
(16)
o
2πǫ2 λ2o
0
Then, for any real R > 0, the number of potential interferer inside the region B(θ, 0, R), denoted
by NB(θ,0,R) , is a Poisson random variable with probability mass function
n
Λ
B(θ,0,R)
Pr[NB(θ,0,R) = n] = e−ΛB(θ,0,R)
.
(17)
n!
Result 4 (Impact of Directionality). Consider (16), and let ǫλo → 0. The average number of
potential interferers converges to



θ 2 λt 2
θ
θ 2
.
(18)
R =
λt
R
4π
2π
2
To interpret Result 4, with no obstacle in the environment (ǫλo → 0), we will have a homogenous
Poisson network of interferers with density λt θ/2π. Therefore, the average number of interferers
over B(θ, 0, R) is the product of the density per unit area and the area of B, which is θR2 /2. It can
be concluded that adopting narrower beams reduces the average number of potential interferers
within a certain distance R; however, it still tends to infinity almost surely as R → ∞.
Result 5 (Impact of Blockage). Consider (16), and let R → ∞. The average number of potential
interferers converges to
θ 2 λt
,
2πǫ2 λ2o

(19)

which is less than infinity almost surely if ǫλo > 0.
Result 5 implies that any receiver observes a finite number of potential interferers almost surely if
there is a non-negligible blockage. This unique feature holds for the mmWave bands, as most of

23

the obstacles can severely attenuate the signals.5 Therefore, not only the farther transmitters
will contribute less on the aggregated interference (due to higher path-loss), but they will
be also thinned by directionality and blockage such that only a finite number of spatially
close transmitters can cause non-negligible interference to any receiver. Note that, these fewer
interferers may still cause strong interference, if they are located very close to the receiver. The
point is that the thinning process due to directionality and blockage makes the SINR distribution
under PhyM closer to that of the IBM, which considers only the near-field interferers. To elaborate
more, we characterize the average number of far-field interferers in the following.
Proposition 1 (Measure of Far-Field Interferers). Let θ be the operating beamwidth, λt be the
density of the transmitters, λo be the density of the obstacles, and ǫ > 0 be a constant. Then,
the average number of interferers located inside B(θ, R, ∞) is
θ 2 λt
(1 + ǫλo R) e−ǫλo R ,
ΛB(θ,R,∞) =
2
2
2πǫ λo
and the probability of having no far-field interferer is

(20)

Pr[NB(θ,R,∞) = 0] = e−ΛB(θ,R,∞) .
(21)
R∞
Proof: To prove, we only need to compute R θλI (r) r dr, and (20) follows. Moreover, by
substituting ΛB(θ,R,∞) into (17) with n = 0, we conclude (21).

From Proposition 1, the average number of far-field interferers will be decreased exponentially
with distance. Consequently, from (21), the probability of having no far-field interferers increases
exponentially with the distance. Fig. 6 shows the probability of having at least one far-field
interferer as a function of the distance. By defining any arbitrary minimum threshold κ on this
probability, we can find a distance Rκ after which the probability of having far-field interferer(s)
is arbitrarily close to 0 (less than κ). This suggests that by setting rIBM = Rκ , IBM can capture,
at least, (1 − κ) fraction of the total number of interferers for any arbitrary small κ. Recall
that the neglected interferers, if any, are far-field and their contributions to the total interference
term are suppressed by the significant distance-dependent path-loss. All these facts result in the
following conclusion:
5

In the conventional microwave systems where the transmission is less sensitive to blockage, the number of potential interferers

is almost surely infinite, as highlighted in Result 4.

24

Probability of having far−field interferers

0

10

−1

10

−2

10

c = 0.5, λo=0.25, λt=1
c = 0.5, λo=0.25, λt=0.44

−3

10

c = 0.5, λo=0.44, λt=0.44
c = 1, λo=0.44, λt=0.44

−4

10

0

20

40
60
Distance [meters]

80

100

Fig. 6: Probability of having at least one far-field interferer as a function of distance. Simulation parameters are similar to those
of Fig. 1.

Result 6. Directionality and blockage can substantially increase the accuracy of the interference
ball model.
We can argue similar accuracy improvement in the PRM, as we numerically illustrate in the next
subsections.

B. Accuracy of the Interference Ball Model
Assume rIBM ≥ a and d0 ≥ a. Using similar claims as in Section III-A, it is straightforward to
IBM|PhyM

show pfa


Pr γ IBM

= 0. To findthe miss-detection probability, in Appendix B, we derive






2
α
2

 1 − (ǫλo a + 1) e−ǫλo a
σθ βd0
θ
−βdα
h

0
< β = 1 − exp −
−
λt E h 1 − e

4pcπ 2
2π
ǫ2 λ2o





Z rIBM 


−α
−ǫλ
r
−βdα
hr
o
+
e
r dr  ,
1−e 0

a



(22)

25

and


Pr γ PhyM < β = 1 − exp









2

−

σθ βdα0
4pcπ 2

2

−




θ λt
1 − e−βdα0 h 1 − (ǫλo a + 1) e−ǫλo a
E
h
2πǫ2 λ2o



Z ∞

α
−α
2 2
−βd0 hr −ǫλo r

− ǫ λo
e
r dr
, (23)

a



and substitute them into (8). Then, Sβ,ξ (IBMkPhyM) can be found using (22), (23), (8), and
then (3). Similar to Remark 1, for any 0 ≤ ξ ≤ 1, Sβ,ξ (IBMkPhyM) → 1 as rIBM → ∞.
C. Accuracy of the Protocol Model
To derive the accuracy of the PRM, we need to derive Pr[γ PhyM < β], Pr[γ PRM ≤ β] and

Pr[γ PhyM < β | γ PRM ≥ β], and substitute them into (11) and (12). Pr[γ PhyM < β] is derived
in (23).
Event γ PRM < β implies that |I ∩ B(θ, 0, rPRM )| ≥ 1, namely there is at least one potential interferer inside B(θ, 0, rPRM ). Considering (17), the probability of this event is Pr[NB(θ,0,rPRM ) ≥ 1],
thus
n
o


Pr γ PRM < β = 1 − exp − ΛB(θ,0,rPRM) ,

(24)

Event γ PRM ≥ β implies that there is no interferer inside B(θ, 0, rPRM ). Assuming rPRM ≥ a, it
is easy to find Pr[γ PhyM < β | γ PRM ≥
β]:






Z ∞
 σθ2 βdα θ2 λ


α hr −α
t
0
PhyM
PRM
−βd
−ǫλ
r
o


0
.
−
E
Pr[γ
<β|γ
≥ β] = 1−exp −
1
−
e
e
r
dr
h


4pcπ 2
2π
rPRM




(25)

Substituting (23)–(25) into (11) and (12) gives the accuracy index of the PRM. Note that Results 2
and 3 hold here.

D. Numerical Illustrations
To numerically illustrate the accuracy index in Scenario 2, we use the same simulation environment of Section III-C. We independently randomly mark some wireless link to be blocked by
obstacles, with the exponential blockage probability with ǫλo = 0.008 [53]. We then assume
infinite penetration loss for the blocked links, and use the large scale LoS path loss model at

26

Accuracy index

1

0.98

0.96
IBM:
IBM:
PRM:
PRM:

0.94

0.92
20

θ = 20o
θ = 40o
θ = 20o
θ = 40o

40
60
80
Average inter-transmitter distance [m]

100

Fig. 7: Accuracy of IBM and PRM under Rayleigh fading channel and directional communications with obstruction.

28 GHz [2, Table I]. System bandwidth is 1 GHz (noise power σ = −84 dBm). Without loss
of generality, we assume rPRM = 40 m and rIBM = 80 m.
Fig. 7 illustrates the impact of the operating bandwidth and average inter-transmitter distance on
the accuracy index of both IBM and PRM under Scenario 2. As expected, the IBM outperforms
PRM. More importantly, directionality and blockage improve the accuracy of both interference
models. We show in the following section that changing the underlying channel model from a
Rayleigh fading model to a deterministic model further enhances their accuracies. Moreover, the
accuracy of the TIM with ε = −130 dB, not depicted for the sake of clarity in Fig. 7, is nearly
1 in our simulations. Notice that a simplified interference model (e.g., PRM, IBM, or TIM) may
not be of sufficient accuracy for all range of parameters, still it is substantially improved by
directionality and blockage, as highlighted by Results 4–6.
V. E XAMPLE S CENARIO 3: D ETERMINISTIC C HANNEL , D IRECTIONALITY,

AND

O BSTACLES

In this section, we investigate how accurately the IBM and PRM can model a wireless network
with directional communications, blockage, and deterministic wireless channel. The last assumption holds generally in mmWave networks, where sparse scattering characteristic of mmWave
frequencies along with narrow beam operation makes the mmWave channel more deterministic
compared to that of microwave systems with rich scattering environment and omnidirectional
operation [54].

27

A. Accuracy of the Interference Ball Model
IBM|PhyM

Again, it is straightforward to show pfa

= 0. However, unlike previous cases, we cannot

derive closed-form expression for the miss-detection probability, and consequently for the accuracy index. In Appendix C, we have derived upper bounds on the miss-detection probability
using the Chernoff bound.
B. Accuracy of the Protocol Model
Again, deterministic wireless channel prohibits deriving closed-form expressions for the false
alarm and miss-detection probabilities. Nevertheless, we can show that both Remarks 2 and 3
holds here. Moreover, we have the following result:
Result 7 (Zero False Alarm Probability). Under the deterministic channel model, the false alarm
probability is zero for any rPRM ≤ ζ −1/α , where
 2
θ
σ
d−α
0
−
.
(26)
ζ=
β
pc 2π
Proof: The SINR that the typical receiver experiences due to transmission of the intended transmitter and an unintended receiver located at distance rPRM is
2 −α
d0
pc 2π
θ
.

2
−α
pc 2π
r
+
σ
PRM
θ
Comparing the SINR expression to β, we get that any interferer located at distance rPRM less
than
σ
d−α
0
−
β
pc



θ
2π

2 !−1/α

= ζ −1/α

can cause packet loss at the typical receiver, namely γ PhyM < β. Now, if we consider the general
equation of the false alarm probability, we have

 

 PRM
 Pr γ PRM < β Pr γ PhyM ≥ β | γ PRM < β
PRM|PhyM
PhyM
pfa
= Pr γ
<β|γ
≥β =
.
1 − Pr [γ PhyM < β]


For rPRM ≤ ζ −1/α , Pr γ PhyM ≥ β | γ PRM < β = 0; since event γ PRM < β implies that there is
PRM|PhyM

at least one interferer inside B(θ, 0, rPRM ). This interferer ensures γ PhyM < β, so pfa

=0

for rPRM ≤ ζ −1/α .
The following proposition characterizes bounds for the accuracy index for the Example Scenario 3
(mmWave networks):


Proposition 2. For ξ = Pr γ PhyM ≥ β and any 0 < rPRM ≤ ζ −1/α , we have


Pr γ PRM < β ≤ Sβ,ξ (PRMkPhyM) ≤ 1 ,

28



where Pr γ PRM < β is given in (24).

We have provided a proof for this proposition along with other bounds in Appendix C. We have
the following scaling law results:
Result 8 (Scaling laws for the PRM). The following scaling laws are implied by Proposition 2
and inequality ex ≥ 1 + x for any x ≥ 0:
•

Scaling with θ: For any constant rPRM no larger than ζ −1/α , limθ→0 Sβ,ξ (PRMkPhyM) ≥
2

1 − e−θ C , for some constant C ≥ 0.
•

Scaling with λt : For any constant rPRM no larger than ζ −1/α , limλt →∞ Sβ,ξ (PRMkPhyM) ≥
1 − e−λt C for some constant C ≥ 0.

•

Scaling with λo : For any constant rPRM no larger than ζ −1/α , limλo →0 Sβ,ξ (PRMkPhyM) ≥
−2

1 − exp{−C} for some constant C ≥ 0, and limλo →∞ Sβ,ξ (PRMkPhyM) ≥ 1 − e−λo

D

for some constant D ≥ 0.
Due to lack of space and complexity of the analysis, we leave scaling laws of the IBM for a


future publication. In Appendix C, we have used the Chernoff bound to bound Pr γ PRM < β ,
which is the first step to derive scaling laws for the IBM.

C. Numerical Illustrations
Using similar setting as in Section IV-D, Fig. 8 shows the accuracy index of both IBM and PRM
under Scenario 3 against dt . Comparing this figure to Fig. 7, we observe that directionality and
blockage can further boost the accuracy index when we have a deterministic wireless channel.
Surprisingly, the PRM is accurate enough to motivate adopting this model to analyze and design
of mmWave networks instead of the PhyM, TIM, and even IBM. For relatively pencil-beams (e.g.,
θ = 10 ∼ 20◦ ), which may be used in wireless backhauling applications, the accuracy of the
PRM in detecting outage events is almost 1 in all our simulations. Compared to the PRM,
the PhyM and IBM respectively have less than 5% and 2% higher accuracy in modeling the
interference and detecting the outage events, but with substantially higher complexities. These
complexities often result in limited (mostly intractable) mathematical analysis and little insight.
More interestingly, the relative difference between the average rate of the typical link computed
by the PRM and that of computed by the PhyM, namely E[log2 (1 + γ x )] and E[log2 (1 + γ y )] is

29

1

Accuracy index

0.9995
0.999
0.9985
IBM:
IBM:
PRM:
PRM:

0.998
0.9975
0.997
20

θ = 20o
θ = 40o
θ = 20o
θ = 40o

25
30
35
40
45
Average inter-transmitter distance [m]

50

Fig. 8: Accuracy of IBM and PRM under deterministic channel and directional communications. rPRM = ζ −1/α where ζ is
given in (26), and rIBM = 2rPRM . The relative difference between the average per-user rate computed by the PRM and that of
computed by the PhyM is less than 0.002%.

less than 0.002%, implying the accuracy of the simple PRM to analyze long-term performance
metrics (such as throughput and delay).
Fig. 8 together with Results 4-7 support the validity of the previously proposed pseudo-wired
model [14], at least for sparse networks like mmWave mesh networks [55]. This highlights
the importance of having quantitative (not only qualitative) insight of the accuracy of different
interference models we may face in different wireless networks. Thereby, we can adopt a simple
yet accurate enough model for link-level and system-level performance analysis.
So far, we have observed how we can simplify the set of dominant interferers and how much
accuracy loss they entail under three network scenarios. Besides the set of interferers I, computing the SINR expression requires modeling the wireless channel and the antenna patterns.
More accurate models generally reduce tractability of the SINR expression and therefore the
interference model. In the next section, we analyze the possibility of adopting simple models
for the wireless channel and for the antenna pattern.
VI. E XAMPLE S CENARIO 4: I MPACT

OF

OTHER C OMPONENTS

OF THE

SINR E XPRESSION

In this section, we analyze the accuracy loss due to simplifying wireless channel model and
antenna pattern of the SINR expression. In particular, we use the proposed accuracy index to
investigate the feasibility of modeling a random fading channel with a constant value without
affecting the long-term performance of the real system (with random fading). The importance

30

Accuracy index

1

0.9

0.8
Nakagami (m = 9)
0.7

Nakagami (m = 3)
Rayleigh

0.6

2

2.5

3

3.5
Path-loss index

4

4.5

5

Fig. 9: Impact of modeling a fading channel by a deterministic one on the accuracy of the resulting interference model (dt =
80 m).

of this scenario is due to that numerous studies develop protocols and optimize the network
based on deterministic wireless channels, yet no study focuses on the accuracy and validity of
this underlying model. In the following, we comment on what this deterministic channel gain
should be to maximize its similarity to the actual random wireless channel. We then use the
proposed accuracy index to assess the impact of neglecting the reflections, assuming impenetrable
obstacles, and neglecting sidelobe gain of the directional antenna on the accuracy of the resulting
interference model. We consider the PhyM for both x and y throughout this section.
A. Approximating a Fading Channel with a Deterministic One
To design many protocols for wireless networks (such as power control, scheduling, and routing),
it is often preferable to use deterministic channel gains that depend only on the distance among
the transmitters and receivers [14], [17], [36]–[38]. In this subsection, we investigate the accuracy
of approximating the fading gain between transmitter i and the reference receiver (hi ) in y
by a deterministic value c0 in x. After this approximation, the channel gain in x becomes
giCh = ac0 d−α
i , and all other parameters of x are identical to those of y. For sake of simplicity,
we consider omnidirectional communications without blockage, as in Section III.
Using the same simulation setup as of Section III, we numerically find c0 in x that gives the
highest similarity between x and y, averaged over all β ∈ [0, 10] dB. Fig. 9 shows the accuracy
index, obtained by the optimal c0 , for Rayleigh and Nakagami fading. Moreover, we report in

31

TABLE I: Accuracy of the mathematical analysis when we replace fading channels with a deterministic one (dt = 80 m). “AI”
refers to our accuracy index, shown also in Fig. 9. “BC” refers to the Bhattacharyya coefficient of the SINR distributions of x
and y, and “TD” refers to the deviation of the throughput obtained by interference model x from that of y.
Fading type
Rayleigh

Nakagami (m = 3)

Nakagami (m = 9)

α=2

α=3

α=4

α=5

AI

0.68

0.881

0.939

0.956

BC

0.275

0.048

0.014

0.005

TD

13%

9.3%

6.7%

4.5%

AI

0.951

0.985

0.995

0.998

BC

0.01

0.004

0.003

0.002

TD

5.8%

4.1%

3.2%

2%

AI

0.997

0.9991

0.9996

0.9999

BC

0.001

0.0008

0.0006

0.0003

TD

1.4%

1%

0.7%

0.3%

Table I the Bhattacharyya coefficient between SINR distribution of x and that of y, and also the
relative difference in corresponding average throughput. From Fig. 9 and Table I, interference
model x (with deterministic channel) becomes more similar to y (with fading channel) as
the path-loss index grows. This higher similarity manifests itself in higher accuracy indices,
in lower Bhattacharyya coefficients, and also in lower errors in the rate analysis. Moreover,
approximating a random wireless channel gain with Rayleigh fading and a small path-loss index
(outdoor environment) by a constant value6 may lead to a non-negligible inaccuracy in the final
throughput analysis (up to 13% error in our example). However, a Nakagami-m fading channel
with high m can be well approximated by a deterministic channel gain, substantially simplifying
the mathematical analysis and protocol development. The error due to this approximation will
be reduced with m. To highlight the importance of this observation, we note that the directional
communications will be largely applied in future wireless networks [56]. Therefore, wireless
networks with Nakagami-m fading channels will play a major role in future of wireless networks.
For mmWave communications, for instance, we are already using narrow beams [3], [55], which
result in high m in the corresponding Nakagami-m fading channel. The following conjecture
states how we can approximate a Nakagami-m fading channel by a deterministic channel gain.
Conjecture 1. Consider a 2D network. Assume that the wireless channel attenuation consists of
a constant attenuation at a reference distance, a distance-dependent attenuation with path-loss
6

We observed in our simulations that c0 = Eh [h2/α ] = Γ(1 + 2/α) is roughly the optimal constant that provides the highest

similarity index in Rayleigh fading channel. Notice that it is 2/α-th moment of random variable h.

32

index α, and a random fading h. If h has a Nakagami-m distribution with m ≥ 3, the wireless
channel can be well approximated by a deterministic LoS channel without significant drop in
the accuracy of the resulting interference model or in the analysis of the ergodic performance
metrics such as spectral efficiency, energy efficiency, throughput, and delay. If h has Rayleigh
fading distribution, replacing h by its 2/α-th moment, namely Eh [h2/α ], results in sufficiently
accurate analysis of the ergodic performance metrics.
In the following, we further exemplify the proposed index to investigate the accuracy drop due
to simplifying other parameters of the SINR expression.

B. Other Components of the SINR
In this subsection, we focus on mmWave networks and propose a very simple yet accurate
interference model. In particular, we consider a PRM wherein we assume that i) obstacles are
impenetrable, ii) there is no reflection, and iii) there is no sidelobe transmissions/receptions. Although these assumptions do not generally hold in practice, we show that this simple interference
model can be very accurate abstraction of real mmWave networks. Previously, references [14],
[38] used this interference model for performance evaluation and protocol development for
mmWave networks. Therefore, the discussions of this subsection are a complementary study
of those works.
We consider a random number of obstacles in the environment each with penetration loss lo .
The obstacles are assumed to have rectangular shape whose centers follow a spatial Poisson
distribution with density λo on the plane, independent of the Poisson process of the interferers.
To each rectangle, we associate a random width that is independently uniformly taken from
[0, 4] meters, a random length that is independently uniformly taken from [0, 3] meters, and
a random orientation that is independently uniformly taken from [0, 2π]. The obstacles can
represent small buildings, human bodies, and cars. We independently randomly mark some
obstacles to be reflectors with coefficient r ≤ 1. Without loss of generality, we mark the obstacles
as reflectors with probability 0.1. We also assume that the links can be established either by
the direct path or by a first-order reflected path. We consider a large scale path loss model at
28 GHz [2], which consists of a constant attenuation, a distance dependent attenuation, and a
large scale log-normal fading. Besides these attenuation sources, we consider the penetration

33

TABLE II: Effects of assuming infinite penetration loss, no reflection, and no sidelobe gain on the accuracy of the resulting
√
√
interference model. The shown parameters are for reference model y. SINR threshold is β = 5 dB. dt = 1/ λt and do = 1/ λo .
Experiment

lo

r

z

θ

dt

do

Accuracy

◦

1
2
3

10
10
20

0.63
0.74
0.9

-10
-10
-10

20
40◦
40◦

50
30
50

20
20
50

0.9998
0.9992
0.9993

4
5
6

10
20
20

0.74
0.74
0.74

-10
-10
-10

20◦
20◦
20◦

50
30
30

50
50
20

0.9614
0.9856
0.9588

7
8
9

15
15
15

0.74
0.74
0.74

-5
-5
-10

20◦
20◦
40◦

50
20
30

20
50
20

0.9235
0.7090
0.9311

10
11
12

25
15
15

0.9
0.63
0.74

-10
-15
-10

10◦
30◦
20◦

30
50
100

30
50
50

0.8810
0.9473
0.9718

and reflection losses. Consider path k between transmitter i and the reference receiver. Let dik
be the distance of this path (path length), nk be the number of obstacles in this path, lo be the
penetration loss due to any obstacle in dB, and lr = −10 log(r) be the reflection loss in dB. Let
1k denote an indicator function that takes 1 if path k contains a reflector, otherwise 0. Then,
the channel gain in k-th path between transmitter i and the reference receiver is modelled as
Ch
gik
[dB] = −61.4 − 20 log(dik ) + 1k 10 log(r) − nj lo − X ,

(27)

where X is a zero mean i.i.d. Gaussian random variable with standard deviation 5.8 [2]. Note
that the atmospheric absorbtion is almost negligible (0.15 dB/Km) at the 28 GHz [46]. Moreover,
changing the carrier frequency will change the parameters of channel model (27), without
affecting the generality of the results of this subsection. Again, we consider ideal sector antenna
pattern, formulated in (15), at all transmitters and receivers.
We consider a realistic reference physical model y with finite penetration loss (lo < ∞), first-order
reflections (r > 0), and non-zero antenna side lobe (z > 0). We execute several experiments in
which we change the type of the reflectors, type of obstacles, side lobe gain, operating beamwidth,
and the average number of interferers and obstacles. We execute four sets of experiments. For
each experiment, we compute the average accuracy index over 105 random topologies and report
the result in Table II.
Effects of no reflection assumption: In the first set of experiments (1–3), we consider three
materials for the reflectors: drywall with reflection coefficient 0.63, clear glass with reflection

34

coefficient 0.74, and the tinted glass with reflection coefficient 0.9 [57]. All parameters of
interference models x and y are similar (reported in the table), except that r = 0 in x. From
Table II, the accuracy index is near 1 for all scenarios. The accuracy marginally decreases with
the density of the transmitters, yet it is high enough for typical transmitter densities (dt > 30 m in
downlink cellular networks). Increasing the operating beamwidth has similar effect as increasing
the transmitter densities.
Effects of infinite penetration loss assumption: In the second set of experiments (4–7), we
consider different penetration losses for the obstacles. All parameters of x and y are similar
(reported in the table), except that lo = ∞ in x. From the table, the accuracy index reduces
with the density of the obstacles, as more obstacles correspond to higher source of errors in x.
Moreover, the assumption of impenetrable obstacles is more accurate for higher penetration loss
values. Moreover, denser mmWave networks (dt = 30) are less sensitive to assuming infinite
penetration loss. The main reason is that densifying the network increases the probability of
having interferers with LoS condition to the reference receiver. The contribution of those nonblocked interferers in the aggregated interference term dominate that of blocked interferers.
Effects of no side lobe gains assumption: In the third set of experiments (7–9), we investigate
the impact of neglecting antenna side lobes z in interference model x. All parameters of x and y
are similar (reported in the table), except that z = 0 in x. Expectedly, neglecting higher z lowers
the accuracy of x, and this error increases also with the number of interferers in the network.
Unlike the previous parameters, neglecting side lobe gain may lead to a large deviation of x from
y. From the numerical results, not shown in this paper due to the space limitations, if we have
either a typical dense network

7

with dt = 30 or enough side lobe suppression (at least 10 dB),

we are safe to ignore side lobe gains from the interference model. Increasing the operating
beamwidth increases the chance of observing an aligned interferer (which contributes in the
link budget with its main lobe gain). As such interferers have dominant role in the aggregated
interference term, increasing the operating beamwidth can improve the accuracy of x.
Joint effects of all parameters: In the last set of experiments (10–12), we analyze joint effects of
7

Note that using scheduling, we can reduce mutual interference by controlling the number of simultaneous active transmitters.

Therefore, the number of transmitters in the environment is not necessarily equal to the number of interferers; rather, it is usually
much higher than that.

35

all those parameters by considering infinite penetration loss, zero reflection coefficient, and zero
side lobe gain in x. Other parameters of x are similar to those of y, reported in Table II. From
the results, our simple interference model x is sufficiently accurate for typical mmWave network
scenarios. On the negative side, larger number of interferers magnifies the small error due to
neglecting antenna side lobes. This magnified error together with other approximations leads
to 12% error in detecting outage event by the simplified interference model in Experiment 10.
On the positive side, this higher transmitter density reduces the error due to both neglecting
reflection and assuming impenetrable obstacles.
VII. F UTURE D IRECTIONS
Throughout this paper, we highlighted the tradeoff between the accuracy and mathematical
tractability of the interference models, and exemplified the use of our accuracy index to optimize such tradeoff for different wireless network scenarios, with specific reference to mmWave
networks. Although we have simplified system models of the examples to avoid unnecessary
complications, our index poses no limitation to these example scenarios. We have recently
used this index to assess the accuracy of a simple interference model for a mmWave cellular
network [45]. Two future directions can be envisioned from this paper.
First, one may use our accuracy index to simplify the existing and develop new interference
models for various network settings. In particular, illustrative examples of this paper were
more suitable for ad hoc networks, and evaluating the generality of the resulting insights is
an interesting future research line. Moreover, our proposed index can be used to assess the
accuracy of different blockage models like one-ball [53], two-ball [4], cone [41], and queuebased models [58] and even develop novel accurate yet tractable models.
Second, we can extend the index itself. In this paper, we have defined the similarity index for any
interference model x based on its ability to correctly predict the outage events; see Definition 1.
To generalize our approach, one may aim at measuring the similarity based on any other functions
of SNIR. For example, given some alternatives for one function inside SINR (e.g., different set
of interferers or different antenna models), one may use an extension of our approach to identify
which of them better balances the accuracy-complexity tradeoff for a throughput/delay analysis.

36

VIII. C ONCLUSION
We developed a new mathematical framework to address very fundamental questions in analysis
and design of wireless networks: how accurate different interference models are and how to select
the right one. We proposed a new accuracy index that quantifies the ability of any interference
model in correctly predicting outage events, under any network setting. We analytically and
numerically illustrated the use of our index via many example scenarios. In particular, we
evaluated the accuracy of the prominent techniques that model the set of dominant interferers.
We then showed that directional antenna and obstructions (basic characteristics of mmWave
networks) substantially enhance the accuracy of any interference model, making the simple
classical protocol model accurate enough for analysis and optimization of such networks. Furthermore, we measured the accuracy of approximating a random fading wireless channel with
a deterministic channel. We conjectured that a Nakagami-m fading channels with m ≥ 3 can
be well approximated by a deterministic value without introducing a significant gap in the
ergodic performance metrics (e.g., throughput and delay); whereas, such gap is generally nonnegligible under Rayleigh fading channels. Finally, we showed surprisingly high accuracy of a
simple interference model that assumes (i) infinite penetration loss, (ii) no reflection, and (iii)
no antenna side lobes in modeling a typical mmWave network where none of those assumptions
hold.
A PPENDIX A: D ERIVING C OMPONENTS

OF

E XAMPLE S CENARIO 1

A. The Interference Ball Model
Let Ex denote expectation
over random variable x. From (5) and

 (6),



Pr γ IBM < β = Pr 


pch0 d−α
0

P

k∈I∩B(2π,0,rIBM )



−α1
pchk dk B(2π,0,a)



= 1 − Pr h0 ≥ 

X

X

k∈I∩B(2π,0,rIBM )





σ  α
βd0
pc



−α1
σ
hk dk B(2π,0,a) +  βdα0 

pc

−α1B(2π,0,a)

hk dk

k∈I∩B(2π,0,rIBM )

 

= 1 − EI,h exp − 



+σ


< β


+

37

 




X
−α1
EI,h exp − 
hk dk B(2π,0,a)  βdα0 
= 1 − exp


pc
k∈I∩B(2π,0,rIBM )






o
n
α
Y
−α1
−σβd0
EI 
Eh exp −βdα0 hk dk B(2π,0,a)  .
= 1 − exp
pc
k∈I∩B(2π,0,rIBM )
{z
}
|


−σβdα0





A

(28)

From probability generating functionals, we have

A = exp














− 2πλt

Z

∞


h
i
−α1
B(2π,0,a)
−βdα
hr
1B(2π,0,rIBM ) 1 − Eh e 0
r dr













Z ∞



−α1
B(2π,0,a)
−βdα
hr


0
r dr
= exp − 2πλt Eh
1B(2π,0,rIBM ) 1 − e


0











Z
Z



rIBM 
a−
αh
α hr −α
−βd
−βd


0
0
1−e
r dr +
1−e
r dr
= exp − 2πλt Eh


0
a













−α
α
−α
α 
α
(⋆)
2
1 − e−βd0 hrIBM − a2 1 − e−βd0 ha
= exp − πλt Eh a2 1 − e−βd0 h + rIBM



0




 



2
2
α 2/α
α
−α
α
−α 
α 2/α
+ (βd0 h) Γ 1 − , βd0 hrIBM − (βd0 h) Γ 1 − , βd0 ha
,

α
α



(29)

where Γ (·, ·) is the incomplete Gamma function, (⋆) is derived using integration by parts, and


probability density function of h is fh (x) = e−x . To find Pr γ PhyM < β , we only need to


evaluate Pr γ IBM < β at rIBM → ∞, that is,




Pr γ PhyM < β = lim Pr γ IBM < β
rIBM →∞

38

= 1 − exp









−

σβdα0
pc





α
−α
α 
− πλt Eh a2 1 − e−βd0 h − a2 1 − e−βd0 ha





 


2
2
α 2/α
α
−α 
α 2/α
− (βd0 h) Γ 1 − , βd0 ha
,
+ (βd0 h) Γ 1 −

α
α


(30)

where Γ (·) is the Gamma function.
B. The Protocol Model
Event γ PRM ≥ β implies that there is no interferer inside B(2π, 0, rPRM). Assuming rPRM ≥ a
and d0 ≥ a, and following similar steps
as in (28) and (29), we have




Z ∞



−α1
α
B(2π,0,a)
Pr[γ PhyM < β | γ PRM ≥ β] = 1 − exp − 2πλt Eh 
1B(2π,rPRM ,∞) 1 − e−βd0 hr
r dr 

0





σβdα0 
−
pc 









Z
 −σβdα


∞ 
α hr −α
0
−βd
= 1 − exp
− 2πλt Eh 
1−e 0
r dr 


pc
rPRM










 −σβdα


−α
2
0
α 2/α
−βdα
hrPRM
2
0

+ (βd0 h) Γ 1 −
− πλt Eh − rPRM 1 − e
= 1 − exp

pc
α





 

2
−α
α 2/α
α

− (βd0 h) Γ 1 − , βd0 hrPRM
.

α


(31)

39

A PPENDIX B: D ERIVING C OMPONENTS

OF

E XAMPLE S CENARIO 2

A. The Interference Ball Model
We have






Pr γ IBM < β = Pr 


p
P

p


2π 2
θ

 


= 1 − EI,h exp − 



= 1 − exp



2

σθ βdα0
−
4pcπ 2


2π 2
θ

k∈I∩B(θ,0,rIBM )

ch0 d−α
0
−α1B(2π,0,a)

chk dk
X

+σ



EI 

Y

k∈I∩B(θ,0,rIBM )

|


< β


−α1B(2π,0,a)

hk dk

k∈I∩B(θ,0,rIBM )





Eh




σθ  α 
+
βd0

4pcπ 2
2





n
o
−α1
exp −βdα0 hk dk B(2π,0,a)  ,
{z

}

B

(32)

where 






 Z ∞

h
i
−α1
α
B(2π,0,a)
−βd0 hr
B = exp −
1B(θ,0,rIBM ) 1 − Eh e
θλI (r)r dr


0













Z ∞

 θ2
−α1
α
B(2π,0,a)
−βd0 hr
−ǫλo r


λt E h
1B(θ,0,rIBM ) 1 − e
e
r dr
= exp −


2π
0











Z
Z
 θ2


rIBM 
a−
α h  −ǫλ r
α hr −α
−βd
−βd
−ǫλ
r
o
o
= exp −
1−e 0 e
r dr +
1−e 0
e
r dr 
λt E h 


2π
0
a











 Z rIBM 

 θ2


−ǫλo a
1
−
(ǫλ
a
+
1)
e
α hr −α
α h
o
−ǫλ
r
−βd
−βd
o
0
 .
= exp −
+
e
r
dr
λt E h  1 − e 0
1
−
e


2π
ǫ2 λ2o
a




(33)

Also, we have


Pr γ PhyM < β =



Pr γ IBM < β
rIBM →∞




 σθ2 βdα
2

θ λt
0
1 − e−βdα0 h 1 − (ǫλo a + 1) e−ǫλo a
E
−
= 1 − exp −
h

4pcπ 2
2πǫ2 λ2o


lim

40




Z ∞

2 2
−βdα
hr −α −ǫλo r
0

− ǫ λo
.
e
r dr

a



(34)

B. The Protocol Model
We have
Pr[γ PhyM < β | γ PRM ≥ β] = 1 − exp











− Eh θ

Z

∞



−α1
B(2π,0,a)
−βdα
0 hr

1B(θ,rPRM ,∞) 1 − e

0

−








σθ2 βdα 
0
2
4pcπ





λI (r)r dr 








Z


σθ2 βdα0
θ 2 λt  ∞ 
−α
−βdα
hr
−ǫλ
r
o

0
= 1 − exp −
−
E
.
1
−
e
e
r
dr
h


4pcπ 2
2π
rPRM






(35)

A PPENDIX C: D ERIVING B OUNDS

FOR

E XAMPLE S CENARIO 3

In this appendix, we derive bounds on the miss-detection probability in Example Scenario 3.
A. The Interference Ball Model
To derive an upper bound on the miss-detection probability, we substitute a lower bound of




Pr γ IBM < β and upper bound of Pr γ PhyM < β into (8) of the main manuscript. Recall the

definition of ζ in (26).
" For any real positive τ ,
#
"
#
X −α1B(2π,0,a)
X −α1B(2π,0,a)
 IBM
 (⋆)
dk
dk
Pr γ
< β = Pr
1B(θ,0,rIBM ) > ζ = 1 − Pr
1B(θ,0,rIBM ) ≤ ζ
k∈I

k∈I



(⋆⋆)

(

≥ 1 − inf eτ ζ EI  exp −τ
τ >0

|

X

−α1B(2π,0,a)

dk

k∈I

{z
C

1B(θ,0,rIBM )

)

.
}

(36)

where (⋆) is due to (6) in the main manuscript, and (⋆⋆) follows from the Chernoff bound and
the probability

 generating functionals.
Z rIBM 



−α1
B(2π,0,a)
−τ r
λI (r)r dr
C = exp − θ
1−e


0

41









Z rIBM 


 −ǫλo r
−τ
−ǫλo r
−τ r −α

re
dr +
1−e
re
dr
1−e

a





Z −
θ2  a
= exp −
λt

2π
0






 θ2 λ 1 − e−τ + (1 + ǫλ a) e−ǫλo a−τ − (1 + ǫλ r ) e−ǫλo rIBM
o
o IBM
t
= exp −
2
2

2π
ǫ λo






Z rIBM

−ǫλo r−τ r −α
−
re
dr  .

a



(37)

Using similar technique, we use the Chernoff bound to find exponentially decreasing bound on
the tail distribution"of γ PhyM as
#
X −α1B(2π,0,a)
 PhyM

Pr γ
< β = Pr
dk
1B(θ,0,∞) > ζ
k∈I



(

≤ inf e−τ ζ EI  exp τ
τ >0

= inf exp
τ >0









− τζ −

X
k∈I



)
−α1
dk B(2π,0,a) 1B(θ,0,∞) 

θ2 λt  1 − eτ + (1 + ǫλo a) e−ǫλo a+τ
2π
ǫ2 λ2o




Z ∞

−ǫλo r+τ r −α

−
.
re
dr

a



(38)
Note that bounds (36) and (38) are derived using the Chernoff bound. However, easier but looser
bounds can be found using Markov and Chebyshev inequalities. These bounds can be readily
derived by direct application of the Campbell’s Theorem [25].

B. The Protocol Model
We can also find bounds and scaling laws for the accuracy index of the PRM in Example Scenario 3. To this end, we use the following proposition:


Proposition 3. For ξ = Pr γ PhyM ≥ β , we have




max Pr γ IBM < β , Pr γ PhyM ≥ β ≤ Sβ,ξ (IBMkPhyM) ≤ 1 .

Also, for any 0 < rPRM ≤ ζ −1/α where ζ is defined in (26) of the manuscript we have




max Pr γ PRM < β , Pr γ PhyM ≥ β ≤ Sβ,ξ (PRMkPhyM) ≤ 1 ,

(39)

(40)

42

where


Pr γ

PRM


θ 2 λt 
−ǫλo rPRM
1 − (1 + ǫλo rPRM ) e
.
< β = 1 − exp −
2πǫ2 λ2o




Proof: For the IBM, the upper bound is trivial. To derive the lower bound, from (3) of the
manuscript we have
IBM|PhyM

Sβ,ξ (IBMkPhyM) = 1 − ξpfa
(⋆)

IBM|PhyM

where (⋆) is because pfa

IBM|PhyM

− (1 − ξ) pmd

IBM|PhyM

= 1 − (1 − ξ) pmd




(⋆⋆)
= Pr γ PhyM ≥ β + Pr γ IBM < β ,

(41)

= 0 for any rIBM ≥ 0, and (⋆⋆) is due to (8) of the manuscript.

Then, (39) follows.




To derive the lower bound of the PRM, we first note that Pr γ PhyM < β | γ PRM ≥ β ≤ Pr γ PhyM < β
for any rPRM > 0. Therefore, from (12) of the manuscript,


 PRM

θ 2 λt 
PRM|PhyM
−ǫλo rPRM
pmd
≤ 1 − Pr γ
< β = exp −
1 − (1 + ǫλo rPRM ) e
.
2πǫ2 λ2o
Then, from (3) we have
PRM|PhyM

Sβ,ξ (PRMkPhyM) = 1 − ξpfa
(⋆)

(42)

PRM|PhyM

− (1 − ξ) pmd

PRM|PhyM

= 1 − (1 − ξ) pmd
PRM|PhyM

≥ 1 − pmd



 PRM

θ 2 λt 
−ǫλo rPRM
,
1 − (1 + ǫλo rPRM ) e
≥ Pr γ
< β = 1 − exp −
2πǫ2 λ2o

(⋆⋆)

(43)

PRM|PhyM

where (⋆) is because pfa

= 0 for any rPRM ≤ ζ −1/α where ζ is defined in Eqn. (27) of

the manuscript (see Result 7 of the manuscript), and (⋆⋆) is due to Eqn (42). Moreover,
PRM|PhyM

Sβ,ξ (PRMkPhyM) = 1 − (1 − ξ) pmd

 

(⋆)
= 1 − Pr γ PRM ≥ β Pr γ PhyM < β | γ PRM ≥ β


≥ 1 − Pr γ PhyM < β | γ PRM ≥ β


≥ 1 − Pr γ PhyM < β ,

where (⋆) is from (12). Combining (43) and (44) results in (40).

(44)

43

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[58] R. Congiu, H. Shokri-Ghadikolaei, C. Fischione, and F. Santucci, “On the relay-fallback tradeoff in millimeter wave
wireless system,” in Proc. IEEE International Conference on Computer Communications (INFOCOM) Workshop, 2016,
pp. 622–627.

1

Accuracy index

0.8
0.6
0.4
0.2

θ = 10o (exact)
θ = 30o (exact)
θ = 10o (bound)
θ = 30o (bound)

0
-10

-5

0

5

10

SINR Threshold [dB]

15

20

0

10

θ = 30o
θ = 15o

−1

Accuracy index

10

θ = 5o

−2

10

−3

10

−4

10

−5

10

−4

10

−3

10

−2

10
Obstacle density

−1

10

0

10

0

10

−1

Performance of the detector

10

−2

10

−3

10

−4

Pmd

10

Pfa (upper bound)
Pmd (upper bound)

−5

10

Accuracy index
Accuracy index (bound)

−6

10

−6

10

−4

10

−2

10
Transmitter density

0

10

2

10

1

Accuracy index

0.9
0.8
0.7
0.6

θ = 10o (exact)

0.5

θ = 20o (exact)
θ = 10o (bound)
θ = 20o (bound)

10 -3

10 -2

10 -1

Transmitter density

10 0

10 1

0

10

Performance of the detector

Pmd (UB1), λt=0.1
Pmd (UB2), λt=0.1

−1

10

Pmd, λt=0.1
Pmd (UB1), λt=1
Pmd, λt=1

−2

10

Pmd (UB1), λt=10
Pmd, λt=10
−3

10

−4

10

0

10

20

30
40
50
60
70
Operating beamwidth [degrees]

80

90

100

Accuracy index

1
0.8
0.6
rPRM = 20
rPRM = 60
rPRM = 120
optimal rPRM

0.4
0.2

0

50
100
150
Average inter-transmitter distance [m]

200

Accuracy index

1
0.98
0.96

λt = 0.1, λo = 0.02
λt = 0.1, λo = 0.04

0.94

λt = 0.1, λo = 0.07
λt = 1 , λo = 0.04

0.92

5

10

15

20
25
30
35
Penetration loss [dB]

40

45

Accuracy index

1
0.999
0.998
λt = 0.1, θ = 20o

0.997

λt = 1 , θ = 20o

0.996

λt = 1 , θ = 40o

0

0.2

0.4
0.6
Reflection coefficient

0.8

1

Accuracy index

1
0.9
0.8
0.7
0.6
-30

λt = 0.1, θ = 20o , β = 5
λt = 1 , θ = 40o , β = 5
λt = 1, θ = 20o , β = 5
λt = 1, θ = 20o , β = 10

-25

-20
-15
Sidelobe gain [dB]

-10

-5

1

Accuracy index

0.99
0.98
0.97
0.96

IBM: θ = 10o
IBM: θ = 30o
PRM: θ = 10o
PRM: θ = 30o

0.95
10

15
20
25
Average inter-transmitter distance [m]

30

1

Accuracy index

0.95
IBM: rIBM =30, λ t = 0.11
IRM: rIRM=30, λ t = 0.11

0.9

IBM: rIBM =10, λ t = 0.0001
IRM: rIRM=10, λ t = 0.0001
IBM: rIBM =10, λ t = 0.11

0.85

IRM: rIRM=10, λ t = 0.11

0.8
-10

-5

0

5
10
SINR threshold [dB]

15

20

1
IRM: β = 1, λ t = 0.11
IBM: β = 1, λ t = 0.11

Accuracy index

0.95

IRM: β = 10, λ t = 0.0001
IRM: β = 10, λ t = 0.11
IBM: β = 10, λ t = 0.0001

0.9

IBM: β = 10, λ t = 0.11

0.85

0.8
0

20

40
60
Interference range [m]

80

100

1

Accuracy index

0.8

0.6

IBM: rIBM = 30, λ t = 0.0001
IRM: rIRM = 30, λ t = 0.0001
IBM: rIBM = 10, λ t = 0.0001

0.4

IRM: rIRM = 10, λ t = 0.0001
IBM: rIBM = 10, λ t = 0.11
IRM: rIRM = 10, λ t = 0.11

0.2
-10

-5

0

5
10
SINR threshold [dB]

15

20

1
IRM: β = 1, λt = 0.0001

Accuracy index

0.8

IBM: β = 1, λt = 0.0001
IRM: β = 10, λt = 0.0001

0.6

IRM: β = 10, λt = 0.11
IBM: β = 10, λt = 0.0001

0.4

IBM: β = 10, λt = 0.11

0.2
0
0

30

60
90
Interference range [m]

120

150

