Poisson Hole Process: Theory and Applications to Wireless Networks

1.1 MB·

by  Zeinab Yazdanshenasan

1
Poisson Hole Process: Theory and Applications to
Wireless Networks
Zeinab Yazdanshenasan, Harpreet S. Dhillon, Mehrnaz Afshang, and Peter Han Joo Chong
Abstract—Interferencefieldinwirelessnetworksisoftenmod- modelingofwirelessadhocandsensornetworks,e.g.see[6],
eled by a homogeneous Poisson Point Process (PPP). While it is [7],ithasrecentlybeenadoptedfortheanalysisofcellularand
realisticinmodelingtheinherentnodeirregularityandprovides
heterogeneous cellular networks as well [8]–[10]. Irrespective
meaningfulfirst-orderresults,itfallsshortinmodelingtheeffect
of the nature of the wireless network, the interference field is
ofinterferencemanagementtechniques,whichtypicallyintroduce
6 some form of spatial interaction among active transmitters. In almost always modeled by a homogeneous PPP to maintain
1 some applications, such as cognitive radio and device-to-device tractability. While this leads to remarkably simple results
0 networks, this interaction may result in the formation of holes for key performance metrics, such as coverage and rate, it
2 in an otherwise homogeneous interference field. The resulting is not quite suitable for modeling the effect of interference
interference field can be accurately modeled as a Poisson Hole
n management techniques, which often introduce some form
Process (PHP). Despite the importance of PHP in many applica-
a tions, the exact characterization of interference experienced by of spatial interaction among transmitters. In this paper, we
J
a typical node in a PHP is not known. In this paper, we derive focusonspatialseparation,whereholes(alsocalledexclusion

7 several tight upper and lower bounds on the Laplace transform zones) are created around nodes/networks that need to be
of this interference. Numerical comparisons reveal that the new
protected from excessive interference [11]. In particular, we
] bounds outperform all known bounds and approximations, and
T assumethatthebaselineinterferencefieldisaPPPfromwhich
are remarkably tight in all operational regimes of interest. The
I key in deriving these tight and yet simple bounds is to capture holes of a given radius are carved out. When the locations of
.
s thelocalneighborhoodaroundthetypicalpointaccuratelywhile the holes also form an independent PPP, the resulting point
c simplifyingthefarfieldtoattaintractability.Ideasfortightening process is usually termed as a PHP, which is the main focus
[
these bounds further by incorporating the effect of overlaps in of this paper.
2 the holes are also discussed. These results immediately lead to
v an accurate characterization of the coverage probability of the
0 typical node in a PHP under Rayleigh fading. A. Related Work and Applications
9
Index Terms—Interference modeling, stochastic geometry,
0 PoissonPointProcess,PoissonHoleProcess,coverageprobability. We first discuss a few of possibly numerous instances in
1 wireless networks where PHP is a more appropriate model
0 for node locations. In particular, we discuss how PHP has
.
1 been used to model cognitive radio networks, heterogeneous
0 I. INTRODUCTION cellular networks, and device-to-device (D2D) networks. The
6 main objective of cognitive radio networks is to improve
The received signal-to-interference-plus-noise ratio (SINR)
1
spectrumutilizationbyallowingunlicensedsecondaryusersto
: is known to be a strong indicator of the performance and
v use licensed spectrum as long as they do not cause excessive
reliability of a wireless link. Several key performance metrics
Xi of interest, such as outage probability, ergodic capacity, and interference to the licensed primary users. One of the ways
to ensure this is by creating exclusion zones (holes) around
r outage capacity, are strongly dictated by the received SINR.
a primaryusers,wheresecondarytransmissionsarenotallowed.
By definition, the SINR distribution depends upon the joint
This spatial separation was elegantly modeled by using a
distribution of the received powers from the serving node
PHP in [12]. In particular, the locations of both primary and
and the interfering nodes, which ultimately depend on the
secondary users were first modeled by independent PPPs.
network topology. Therefore, accurate modeling of the net-
Assuming secondary transmissions were not allowed within a
worktopologybecomesakeysteptowardsmeaningfulperfor-
given distance from the primary users, the locations of active
mance analysis of wireless networks. Owing to its tractability
secondary users were then modeled by a PHP.
and realism in modeling irregular node locations, stochastic
geometry has emerged as an important tool for the realistic The PHP has also been used recently to model inter-tier
analysis of wireless networks [2]–[5]. Initially popular for the dependence in the base station locations in a heterogeneous
cellular network in [13]–[15]. Modeling the macrocell loca-
H.S.DhillonandM.AfshangarewithWireless@VT,DepartmentofECE, tionsbyaPPP,itwasassumedthatthesmallcellsaredeployed
Virgina Tech, Blacksburg, VA, USA. Email: {hdhillon,mehrnaz}@vt.edu. farther than a minimum distance from the macrocells, i.e.,
Z.YazdanshenasaniswithWireless@VT,DepartmentofECE,VirginaTech,
outsideexclusionzonesofagivenradius.Insuchacase,small
Blacksburg, VA, USA and with School of EEE, Nanyang Technological
University, Singapore. Email: [email protected]. P. H. J. Chong is cellsformaPHP.Thismodelintroducesrepulsionbetweenthe
with School of EEE, Nanyang Technological University, Singapore. Email: locations of macro and small cells, which is desirable due to
[email protected]. This work has been submited in part to the IEEE
several reasons, such as interference mitigation at macrocells
InternationalConferenceonCommunications2016[1]. Manuscriptlast
revised:January8,2016. due to small cell transmissions, and the higher advantage of
2
deploying small cells towards the cell edges of macrocells, TABLEI
especially in the coverage-centric deployments. NOTATIONANDNETWORKPARAMETERS
Similarly, for underlay D2D communication in cellular
Symbol Description
networks, inhibition zones may be created around cellular Φ1;λ1 IndependentPPPmodelingthelocationsofholecenters;densityofΦ1
links where no D2D transmissions are allowed, thus saving Φ2;λ2 IndependentPPPfromwhichtheholesarecarvedout;densityofΦ2
Ψ PHPformedbycarvingoutholeswithcentersΦ1fromΦ2
cellular links from excessive D2D interference. The active D RadiusofeachholecarvedoutfromΦ2
D2D transmitters outside the holes form a PHP [16], [17]. In PLcI,(γs) CLaopvlearcaegetrapnrosfboarbmiliotyf(Ii,ndteefirmnesdoafsSEIR(cid:2))e;sSII(cid:3)Rthreshold
thisregard,cognitiveD2Dcommunicationincellularnetwork C=b(y,D) BallofradiusDcenteredaty
hx Fadinggain;hx∼exp(1)forRayleighfading
when transmitters are powered by harvesting energy from the α Path-lossexponentforallthewirelesslinks
ambient interference is studied in [18]. In [19], a Poisson P;r0 Transmitpower;servingdistanceforthelinkofinterest
Cluster Process (PCP) and a PHP are merged to develop a
new spatial model for integrated D2D and cellular networks.
In particular, a modified Thomas cluster process is used to Approaches to incorporate the effect of overlaps in the holes.
model device locations where instead of modeling the cluster In the first set of bounds discussed above, we carefully
centers as a homogeneous PPP, they are modeled as a PHP to circumvented the need for incorporating the effect of overlaps
account for the inhibition zones around cellular links. between holes. While these simple and easy-to-use bounds
Despite the importance of a PHP in modeling wireless net- are tight, we also provide ideas for incorporating the overlaps
works, the exact characterization of interference experienced betweenholes,whichtightentheseboundsevenfurther.Inthe
byatypicalreceiverinaPHPisachallengingproblem.There first result, we generalize the lower bound discussed above by
are two main directions taken in the literature for the analysis considering two closest holes in the interference field while
of wireless networks modeled by PHPs. The first approach, incorporating the exact effect of overlap between them. In
termed first-order statistic approximation, approximates PHP the second result, instead of trying to incorporate the exact
byahomogeneousPPPwiththesamedensity[3].Thesecond effect of overlaps, we propose a new procedure for bounding
approach ignores the holes altogether to approximate the PHP the overlap area, which allows us to derive a provable lower
by its baseline PPP. This overestimates the interference and bound on the Laplace transform while considering multiple
the accuracy of the approximation is a function of system holes in the interference field. In the third and final result, we
parameters [12], [16]–[18]. Besides, the PHP is sometimes propose a new approach that allows to incorporate the mean
approximated with a PCP by matching the first and second effect of overlaps in the holes.
order statistics [12]–[14]. The resulting expressions for per- New insights. Our results concretely demonstrate that for
formance metrics are usually more complicated in this case accurateanalysisofinterferenceinaPHP,itisveryimportant
compared to the above two. While all these approaches are to preserve the local neighborhood around the typical point.
reasonable, they are typically not accurate beyond a specific For instance, we show that considering even a single hole in
range of system parameters. In this paper, we take a fresh the interference field results in a tighter characterization of
look at this problem and derive tight upper and lower bounds interference power at the typical point of a PHP compared to
for the Laplace transform of interference experienced by the seemingly more refined prior approach of first-order statistic
typicalnodeinaPHP.Themaincontributionsaresummarized approximation in which the PHP is approximated by a PPP
next. with the matching density. This is because by considering a
single hole, the local neighborhood around the typical point
B. Contributions
is accurately captured, whereas it is distorted in the other
New approach to the analysis of a PHP. Unlike existing approach due to independent thinning involved in the density
approaches that approximate a PHP with either a PPP or a matching of a PPP.
PCP,wedevelopanewapproachthatisamenabletoshot-noise
analysis and leads to tight provable bounds on the Laplace
transform of interference experienced by a typical node of II. NETWORKMODEL
a PHP. A lower bound is first derived by overestimating
A. System Model
interference by ignoring all the holes except the closest one.
Weprovideanequivalentinterpretationofthisresultinwhich We consider a wireless network that is modeled by a
the closest hole is dissolved in such a way that it results in a PHP in R2. A PHP can be formally defined in terms of
tractablenon-homogeneousPPP.Theresultingboundisshown two independent homogeneous PPPs Φ1 and Φ2, where Φ2
to be remarkably tight. Extending this approach to multiple represents the baseline PPP from which the holes are carved
holes, we derive an upper bound on the Laplace transform out and Φ1 represents the locations of the holes. Let the
of interference by carving out each hole separately without densities of Φ1 and Φ2 be λ1 and λ2, respectively, with
accounting for the overlaps between them. This leads to the λ2 > λ1. Denoting the radius of each hole by D, the region
removalof somepointsfrom thebaselinePPP multipletimes, covered by the holes can be expressed as
thus underestimating the interference power experienced by
(cid:91)
the typical point. This bound is also shown to be remarkably ΞD (cid:44) b(y,D), b(y,D) z R2 : z y <D .
≡{ ∈ (cid:107) − (cid:107) }
tightacrossavarietyofscenarios,includingtheonesinwhich y∈Φ1
(1)
the holes exhibit significant overlaps.
3
The points of Φ lying outside Ξ , form a PHP, which can where (a) follows from the fact that h exp(1), and (b)
2 D
∼
be formally expressed as fromthedefinitionofLaplacetransformofinterferencepower
(s)=E[exp( sI)]. Note that for this setup, it is sufficient
Ψ={x∈Φ2 :x∈/ ΞD}=Φ2\ΞD. (2) LtoIfocus on the −Laplace transform of interference in order to
It should be noted that the PHP Ψ has also been known as a study coverage probability. In general, accurate characteriza-
Hole-1 process in the literature [20]. tion of I(s) is the first step in the analysis of more general
L
We characterize the interference experienced by a typical classes of wireless networks, including cellular networks [9].
node in Ψ due to the transmission of the other nodes of Ψ. Therefore, we will focus on I(s) in the technical sections of
L
Due to the stationarity of the process, the typical node can be thepaperwiththeunderstandingthatthecoverageprobability
assumed to lie at the origin o, and due to Slivnyak’s theorem, can be easily derived for our setup using (5). We begin our
we can condition on o Ψ without changing the distribution technicaldiscussionbysummarizingtwokeypriorapproaches
∈
of the rest of the process [3]. This equivalently means that used in the literature for characterizing I(s) in a PHP.
L
the interferers are modeled by the PHP Ψ with the typical For the ease of reference, the notation used in the paper is
receiver being an additional point placed at the origin. Note summarized in Table I.
that Slivnyak’s theorem for a PPP is applicable here because
conditioned on the locations of the holes (i.e., conditioned III. KEYPRIORAPPROACHES
on Φ ), PHP Ψ is simply a PPP of density λ defined on
1 2
R2 Ξ . Since the typical point is located outside the holes In this section, we summarize two popular approaches that
D
\ havebeenusedintheliteraturetoderivetheLaplacetransform
by construction, there are no points of Φ within a disk of
1
of interference in a PHP. At the end of the Section, we also
radiusDaroundthetypicalpoint.Forthisreceiver,weassume
provide insights into the strengths and weaknesses of each
that the serving transmitter is located at the fixed distance
approach.
r . It should be noted that we could have considered more
0
sophisticated models for the serving link of interest but we
chose to consider this simple setup because our emphasis is A. Lower Bound by Ignoring Holes: Approximating Ψ by Φ
2
on characterizing interference in a PHP.
The first approach is to ignore the effect of holes and
For the wireless channel between points x and y, we
approximate the interference field Ψ by the baseline PPP Φ
considerastandardpowerlawpath-lossl(x y)= x y −α 2
with path-loss exponent α. All the wireless−links ar(cid:107)e a−ssum(cid:107)ed of density λ2. By construction, this approach overestimates
the interference power and hence leads to the lower bound on
to experience independent Rayleigh fading. All the transmit-
the Laplace transform of interference [12]. This well-known
ters are assumed to transmit at a fixed power P. The received
result is stated below for completeness.
power at the typical node from its transmitter of interest is
therefore P = Phr−α, where h exp(1) models Rayleigh Lemma 1 (Lower bound). Ignoring the impact of holes
r 0 ∼
fading. Similarly, the interference power experienced by the (approximating Ψ by Φ ), the Laplace transform of aggregate
2
typical receiver located at the origin is interference I =(cid:80) Ph x −α is lower bounded by:
x∈Ψ x(cid:107) (cid:107)
I = (cid:88)Ph x −α, (3) (cid:20) (sP)2/α (cid:21)
x
(cid:107) (cid:107) (s) exp πλ . (6)
x∈Ψ LI ≥ − 2sinc(2/α)
where hx exp(1) models Rayleigh fading gain for the link Proof. See Appendix A. (cid:4)
∼
from interferer x Ψ to the typical receiver. For this setup,
∈
we define coverage probability next. The tightness of the above bound will be demonstrated in
the Numerical Results section.
B. SIR and Coverage Probability
Using the received power over the link of interest and the B. Approximating PHP by a PPP with the Same Density
interference power defined in the previous subsection, the
The second approach to the derivation of the Laplace
signal to interference ratio (SIR) can be expressed as
transformofPHPisthefirst-orderstatisticapproximation[3].
SIR(r0)= (cid:80) PhPrh0−αx −α. (4) Isnucthhisthaaptptrhoeacrehs,uthlteinbgasdeelninseityPPoPf tΦh2eiPsPinPdiespethnedesnatmlyethasintnheadt
x∈Ψ x(cid:107) (cid:107) of the PHP Ψ, which we denote by λ . The first step in
PHP
DenotetheminimumSIRrequiredforsuccessfuldecodingand
thisapproachisthereforetoderiveλ intermsofthegiven
PHP
demodulation at the typical receiver by γ. A useful metric of
system parameters, which was done in [3]. For completeness,
interest in wireless networks is the SIR coverage probability
we discuss its proof briefly below. To derive λ , we first
PHP
P ,whichistheprobabilitythattheSIRatthereceiverexceeds
c need to derive an expression for the average number of points
the threshold γ. Mathematically, ofthePHPΨlyinginagivenset R2,whichbydefinition
(cid:26) γrα (cid:27) is B ⊂
P = P SIR(r )>γ =P h> 0 I
c { 0 } P  
(=a)E(cid:20)exp(cid:18) γr0αI(cid:19)(cid:21)(=b) I(cid:18)γr0α(cid:19), (5) E (cid:88) (cid:89) (1−1b(x,D)(y))
− P L P x∈Φ2∩By∈Φ1
4
  
=(a)EΦ2 (cid:88) EΦ1 (cid:89) (1−1b(x,D)(y))
x∈Φ2∩B y∈Φ1
(cid:34) (cid:18) (cid:90) (cid:19)(cid:35) y
(=b)E (cid:88) exp λ 1 (y)dy D
Φ2 x∈Φ2∩B − 1 R2 b(x,D) r ✓(✓r()r)
=(c)|B|λ2exp(−λ1πD2), r1
where (a) is due to the independence of point processes Φ
1
andΦ ,(b)followsfromtheprobabilitygeneratingfunctional
2
(PGFL) of a PPP, and (c) follows from the Campbell the-
r
2
orem [2]. From the above expression, we can readily infer
that λ = λ exp( λ πD2). Now to derive the Laplace
PHP 2 1
−
transform of interference in this case, we just need to replace
Fig.1. Illustrationoftheinterferencefieldwithasinglehole.
λ in the result of Lemma 1 with λ . The result is stated
2 PHP
below for completeness.
this case is non-isotropic due to the fixed location of the
Lemma 2 (Approximation). The Laplace transform of in-
terference power I = (cid:80) Ph x −α when PHP Ψ is hole. The Laplace transform of the interference power at the
x∈Ψ x(cid:107) (cid:107) origin from the nodes of Φ outside is characterized next.
approximated by a PPP with density λPHP is 2 C
This intermediate result will be used later in this section to
(cid:20) (sP)2/α (cid:21) derive upper and lower bounds on the Laplace transform of
(s) exp πλ . (7)
LI (cid:39) − PHPsinc(2/α) interference experienced by a typical node in a PHP.
Remark1. Boththeapproachesdiscussedaboveapproximate Lemma 3. Let I =(cid:80) Ph x −α, the Laplace
ΨwithahomogeneousPPP:theonefirstwithdensityλ2 (the transform of interferencxe∈Φco2∩nbdcit(iyo,nDe)d onx(cid:107)y(cid:107) is
baseline PPP), and the second one with density λ < λ (cid:107) (cid:107)
PHP 2 (cid:32) (cid:33)
(the density of the PHP Ψ). While the second approach is a (sP)2/α
(s)=exp πλ
seemingly more refined approach, a careful thought reveals LI|(cid:107)y(cid:107) − 2sinc(2/α) ×
that in order to match the density of the PPP with that of a
(cid:32) (cid:33)
PHP,thebaselinePPPhastobeindependentlythinned,which (cid:90) (cid:107)y(cid:107)+D 2πλ(r)
exp rdr (8)
disturbs the local neighborhood of points around the typical 1+ rα
(cid:107)y(cid:107)−D Ps
point, thus resulting in a loose bound. On the other hand,
(cid:16) (cid:17)
approximating Ψ simply by the baseline PPP Φ2 preserves whereλ(r)= λ2arccos r2+(cid:107)y(cid:107)2−D2 , =b(y,D)denotes
the local neighborhood resulting in a relatively tighter ap- π 2(cid:107)y(cid:107)r C
the hole centered at y with radius D.
proximation. More insights will be provided in the numerical
results section. Proof. See Appendix B. (cid:4)
Before concluding this section, it is important to note that Remark 2 (Dissolving the hole). The above result has an
there is one more fitting-based approach in which the PHP interesting interpretation that will be useful in visualizing the
is approximated by a PCP with matching first and second proposed results. Note that since received power is a radially
order statistics. The Laplace transform of interference and symmetric function, it solely depends upon the distance of
other performance metrics are then studied using the fitted the transmitter to the origin. Therefore, we can in principle,
PCP. Since the formal description of this technique requires dissolve the hole as long as the number of points lying in a
significantly more details compared to the techniques dis- thin strip of radius y D r y +D and vanishingly
cussed above, we refer the reader to [12]–[14], where this small width dr is n(cid:107)ot(cid:107)ch−ange≤d. P≤lea(cid:107)se(cid:107)refer to Fig. 1 for an
approach has been used for the analysis of cognitive radio illustrationofthisstrip.Takingacloserlookattheinterference
and heterogeneous cellular networks modeled as PHPs. We originating from this strip, we note that the only thing that
now discuss the proposed approaches next. matters is the number of points distributed in the part of the
strip which is outside the hole. The area of this region is
(cid:16) (cid:17)
IV. PROPOSEDAPPROACHESTOLAPLACETRANSFORMOF 2rdr(π θ(r)),wheretheangleθ(r)=arccos r2+(cid:107)y(cid:107)2−D2
INTERFERENCEINAPHP is defin−ed in Fig. 1. Therefore, the number of i2n(cid:107)tye(cid:107)rrfering
We now introduce our proposed approach to characterize points lying within this strip is Poisson distributed with mean
the Laplace transform of interference in a PHP. In the first λ 2rdr(π θ(r)). Since the exact locations of these points
2
−
intermediate step, we model the locations of interferers by within the strip doesn’t matter, we can dissolve the hole
a homogeneous PPP Φ of density λ from which only one and redistribute the points uniformly inside the whole strip
2 2
hole of radius D is carved out at a deterministic location. of area 2πrdr. This means, the PPP with a hole can be
Let thCe location of the center of this hole be y R and equivalentlymodeledasanon-homogeneousPPPwithdensity
∈
hence its distance from the origin be y . The resulting setup λ (1 θ(r)/π), where the λ θ(r)/π term (defined as λ(r) in
2 2
(cid:107) (cid:107) −
is illustrated in Fig. 1. Note that the interference field in Lemma 3) captures the effect of hole.
5
Deconditioning the result of Lemma 3 with respect to the
distribution of the distance to the closest hole derived above,
the proposed lower bound is derived below.
D Theorem 1 (New Lower Bound 1). Let I =
v1 (cid:80)x∈ψPhx(cid:107)x(cid:107)−α, the Laplace transform of interference is
lower bounded by
(cid:32) (cid:33)
D (sP)2/α
(s) exp πλ
LI ≥ − 2sinc(2/α) ×
(cid:90) ∞
exp(g(v ))2πλ v exp( πλ (v2 D2))dv , (10)
1 1 1 − 1 1 − 1
D
FDiga.w2a.yIlflruosmtratthioentysphiocwalinpgoitnhtatotfhteheclPosHePstΨpo.intofΦ1 isatleastadistance where g(v1)=(cid:82)vv11−+DDarccos(cid:16)r2+2vv121−rD2(cid:17)1+2λrP2αsrdr.
Proof. See Appendix C. (cid:4)
Using this intermediate result and the above insights, we Remark3. SincethePHPisapproximatedbyΦ bc(y ,D)
2 1
∩
now derive tight bounds on the Laplace transform of interfer- in the above result, the resulting lower bound presented in
ence experienced by a typical node in a PHP. Theorem 1 is by construction tighter than the known lower
bound presented in Lemma 1 where the interference field was
approximatedbysimplyΦ .Onthesamelinesasdiscussedfor
A. Lower Bound on the Laplace Transform of Interference in 2
Lemma 1 in Remark 1, the above approach captures the local
a PHP
neighborhood of the typical node accurately, thus leading to
Beforegoingintothetechnicaldetails,notethatduetopath-
a remarkably tight lower bound in Theorem 1. This will be
loss, the effect of holes that are close to the typical point will
demonstratedlaterinthissectionandinthenumericalresults
bemuchmoresignificantcomparedtotheholesthatarefarther
section.
away. Therefore, to derive an easy-to-use lower bound on the
Laplace transform of interference, we consider only one hole;
the one that is closest to the typical point; and ignore the B. Upper Bound for the Laplace Transform of Interference in
other holes. Denoting the location of the closest hole by y , a PHP
1
the interference field in this case is (Φ2 bc(y1,D)) Ψ, To derive an upper bound on the Laplace transform, we
∩ ⊃
whichclearlyoverestimatestheinterferenceofPHPandhence extend the above approach to all the holes. To maintain
leads to a lower bound on the Laplace transform. Note that tractability, each hole is carved out individually/separately
in Lemma 3, we have already derived the conditional Laplace from the baseline PPP Φ using the above approach. Note
2
transform for the case when there is one hole and its distance that since the centers of the holes follow a PPP Φ , there
1
to the origin is known. To derive a lower bound, we simply will obviously be overlaps among holes. Therefore, when we
need to assume this hole to be the closest point of Φ1 to the remove points of Φ2 corresponding to each hole individually
origin and decondition the result of Lemma 3 with respect to (without accounting for the overlaps), we may remove certain
the distribution of V1 = y1 . To this end, we first derive the points multiple times thus underestimating the interference
(cid:107) (cid:107)
probability density function (PDF) of V1 next. field, which results in an upper bound on the Laplace trans-
form of interference. In the next section, we show that any
Lemma 4. The PDF of the distance V = y between the
1 1
(cid:107) (cid:107) reasonable attempt towards incorporating the exact effect of
typical node at the origin and the closest point of Φ is given
1
overlaps leads to a significant loss in tractability. Fortunately,
by
the bounds derived in this section without incorporating the
f (v )=2πλ v exp( πλ (v2 D2)), v D. (9) effect of overlaps are remarkably tight and can be considered
V1 1 1 1 − 1 1 − 1 ≥
proxies for the exact Laplace transform.
Proof. As discussed in Section II, the typical point of a PHP
liesoutsidetheholesbyconstruction.Therefore,asillustrated Theorem 2 (New Upper Bound). The Laplace transform of
inFig.2,theminimumdistancebetweenthetypicalpointand interference experienced by a typical node in a PHP is upper
the closest hole (closest point of Φ ) is D. Using this fact bounded by
1
along with the properties of a PPP, the distribution of V can (cid:32) (cid:33)
1 (sP)2/α
be derived as follows: (s) exp πλ
LI ≤ − 2sinc(2/α) ×
P(V1 >v1)=P(Number of points of Φ1 in the set (cid:18) (cid:18)(cid:90) ∞ (cid:19)(cid:19)
{b(0,v1)\b(0,D)}=0) exp −2πλ1 D (1−exp(f(v)))vdv (11)
=exp( πλ (v2 D2)), v D.
− 1 1 − 1 ≥ where f(v)=(cid:82)v+Darccos(cid:16)r2+v2−D2(cid:17) 2λ2 rdr.
Theresultnowfollowsbydifferentiatingtheaboveexpression. v−D 2vr 1+rPαs
(cid:4) Proof. See Appendix D. (cid:4)
6
address this question by deriving three results for the Laplace
2
C
transform of interference that incorporate the effect of holes.
In the first result, we generalize the lower bound derived in
Theorem 1 by considering two nearest holes instead of a
y
2 single hole. The overlap among the two holes is explicitly
incorporated in the analysis. The setup is presented in Fig. 3,
v
2 where =b(y ,D) and =b(y ,D) denote the first and
w 1 1 2 2
C C
the second closest holes to the typical receiver, respectively.
�
Theanglebetweeny andy isdenotedbyφ.Theinterference
1 2
v1 y1 D fiwehlidchinovtehriesstcimasaeteissthmeoidnetelerfderbeynce(Φp2ow∩er{Can1d∪heCn2c}ec)lea⊃dsΨto,
alowerboundontheLaplacetransformofinterference.Before
going into the main result, we first need to evaluate the joint
C1 PDFofthedistancesbetweenthefirstandsecondclosestholes
to the origin, which are denoted by random variables V and
Fig. 3. Illustration of the setup used in Theorem 3 where only two holes 1
closesttothetypicalnodeareconsidered. V2, respectively. Using the same arguments as in Lemma 4,
the joint PDF can be derived as [21]
For the same reason as the tightness of lower bound of fV1V2(v1,v2)=fV2(v2|v1)fV1(v1)
Theorem 1 discussed in Remark 3, the above upper bound is =(2πλ )2v v exp( πλ (v2 D2)). (13)
also remarkably tight for a wide variety of scenarios. More 1 1 2 − 1 2 −
Using this distribution, the Laplace transform of interference
details on the tightness are provided in the next subsection
for this case is derived next.
and in the numerical results section.
Theorem 3 (New Lower Bound 2). The Laplace transform
C. Ratio of the Proposed Upper and Lower Bounds of interference experienced by a typical node of a PHP Ψ is
lower bounded by
To study the tightness of the proposed upper and lower
(cid:32) (cid:33)
bounds, we derive a tight approximation on the ratio of upper (sP)2/α
(s) exp πλ
and lower bounds, and show that it is close to one. LI ≥ − 2sinc(2/α) ×
Proposition1. Theratiooftheupperandlowerboundsonthe (cid:32) 1 (cid:90) ∞(cid:90) ∞(cid:90) π (cid:32)(cid:90) v1+D2πλ (r) (cid:33)
Laplace transforms derived in Theorems 2 and 1 is Lu(s) exp c1 rdr
Ll(s) ≈ 2π D v1 −π v1−D 1+ srPα
(cid:90) ∞ (cid:20) (cid:90) ∞ (cid:21) (cid:32) (cid:33)
D exp −2πλ1 v1 (1−exp(f(v)))vdv fV1(v1)dv(11,2) exp (cid:90)v2v−2+DD21πλ+c2sr(Pαr)rdr exp(−λ2B(v1,v2,φ))×
(cid:33)
where Lu(s) and Ll(s) denote the proposed upper and lower fV1V2(v1,v2)dφdv2dv1 (14)
bounds, given by Theorem 2 and Theorem 1, respectively.
(cid:16) (cid:17)
FPurorothf.erS,efe(vA)p=pe(cid:82)nvdv−+ixDDEa.rccos r2+2vv2r−D2 1+2λrs2Pαrdr. (cid:4) wh(vere,vλ,cφi()r)is=giveλπn2abrycc(o2s1)(cid:16).r2+2vvi2ir−D2(cid:17), for i = 1,2 and
1 2
B
This approximation can be interpreted as the Laplace trans- Proof. The key idea behind this proof is to consider (Φ
2
∩
form of interference power removed by all the holes except b(y ,D) b(y ,D) c) Ψ as the interference field, which
1 2
{ ∪ } ⊃
the closest hole from the homogeneous PPP Φ after ignoring clearly overestimates the interference and hence leads to the
2
theeffectofoverlaps.Thisratiowillbeshowntobetightand lower bound. Complete proof is provided in Appendix F. (cid:4)
close to one across wide range of parameters in the numerical
As evident from the above result, incorporating the
resultssection.Next,weexploreafewwaystoincorporatethe
effect of overlap, even among two holes, results in a
effect of overlaps in the holes that was ignored in the results
significantly more complex expression compared to the
derived in this section.
bounds presented in Theorems 1 and 2. This shows that
incorporating the exact effect of overlaps does indeed lead
V. INCORPORATINGTHEIMPACTOFOVERLAPSINTHE to a significant loss in tractability. Therefore, instead of
PROPOSEDAPPROACHES trying to incorporate the exact effect of overlaps, we now
The key lower and upper bounds reported in Theorems 1 propose a new procedure for bounding the overlap area,
and 2 in the previous section carefully circumvented the need which allows us to derive a lower bound on the Laplace
for considering the overlaps in the holes explicitly. This was transform while considering multiple holes in the interference
to maintain tractability. However, it is quite natural to wonder field. This tightens the result of Theorem 1, where only one
howmuchtractabilityisreallylostifwetrytoincorporatethe hole was considered. In particular, we consider k closest
effectoftheoverlapsintheholesaccurately.Inthissection,we holes from the typical receiver, as shown in Fig. 4. In order
7
In the numerical results section, we will show that con-
sidering k = 2, i.e., only the two closest holes from the
typical point, results in a remarkably tight bound. Note that
y for k = 2, the expression is also fairly tractable. For brevity,
3
that expression is not stated separately.
Now,inTheorems3,4,wehavetriedtohandletheoverlaps
insuchawaythattheresultingexpressions:(i)canbeclaimed
y as lower bounds to the Laplace transform, and (ii) tighten the
2
lowerboundprovidedbyTheorem1.Onelastquestionthatwe
addressbeforeconcludingthissectioniswhetheritispossible
y
1
to handle the effect of overlaps in the average sense. For this,
we revisit the upper bound of Theorem 2, which was derived
bycarvingoutholesfromthebaselineprocessΦ individually
2
without caring about the overlaps between them. This led to
the possible removal of some points multiple times, thereby
Fig.4. IllustrationofthesetupusedinTheorem4wherek holesclosestto leading to an underestimation of interference. We compensate
thetypicalnodeareconsidered.
thisover-removal,byrescalingthesecondtermofTheorem2,
which is the one that captures the effect of removing points
from Φ . The rescaling term is derived by estimating the
to claim the result as a bound, we bound the union of 2
k-closest holes, i.e., k = k b(y ,D), with Ω = averagepairwiseoverlapareabetweencircles.Moredetailsof
k b(y ,D)(cid:84)b∪ci(=01,Cmiax( ∪yi=1 +iD, y D)o, the approach are provided in the proof in Appendix H. Note
wC1h∪erie=2Ω{ i k . The se(cid:107)t Ωi−1(cid:107)careful(cid:107)lyi(cid:107)av−oids}the that the resulting expression in this case is the same as that of
o ⊂ ∪i=1Ci o Theorem 2, except a scaling factor of 1 min(λ1πD2,1) that
overlapping part and hence does not result in over-removal − 2 2
appearsinthesecondterm.Thekeydownsidetothisapproach
of the points from the baseline point process Φ . The
2
contribution of ith hole in Ω is d(y ,D)= compared to Theorem 2 is that it results in an approximation
o i
unlike Theorem 2, where the result was shown to be a bound.
(cid:110) (cid:92) (cid:111)
b(yi,D) bc(0,max( yi−1 +D, yi D) . (15) Proposition 2 (New Approximation). The Laplace transform
(cid:107) (cid:107) (cid:107) (cid:107)−
of interference at a typical point in a PHP can be approxi-
InFig.4,thesetd(yi,D)correspondstotheunshadedpartof mated as I(s)
L (cid:39)
aaphporloeacthhartedsuoletssninotthoevreermlaopvwalitohftlheessopthoeinrtshofrleosm. Cthleeabralys,eltihnies (cid:32) (sP)2/α (cid:33) (cid:20) (cid:90) ∞(cid:18) (cid:18)
exp πλ exp 2πλ 1 exp
process Φ compared to a PHP, which leads to a lower bound − 2sinc(2/α) − 1 −
2 D
on the Laplace transform of interference in a PHP. This lower (cid:18) (cid:18)λ πD2 1(cid:19)(cid:19)(cid:19)(cid:19) (cid:21)
bound is presented in the next theorem. f(v) 1 min 1 , vdv (17)
× − 2 2
Theorem 4 (NewLowerBound3). TheLaplacetransformof where f(v)=(cid:82)v+Darccos(cid:16)r2+v2−D2(cid:17) 2λ2 rdr.
interference is bounded by v−D 2vr 1+rα
Ps
(cid:32) (cid:33) Proof. See Appendix H. (cid:4)
(sP)2/α
LI(s)≥exp −πλ2sinc(2/α) × Comparison of this result with the bounds derived in this
paper shows that handling overlaps in the average sense may
(cid:32) (cid:32) (cid:33)
(cid:90) (cid:90) (cid:90) (cid:90) v1+D2πλ (r) not work particularly well, especially when the overlaps are
... exp c1 rdr
1+ rα significant.Thisisbecausesuchresultsdonotcapturethelocal
D<v1<v2<...<vk<∞ v1−D sP
(cid:32) (cid:33) neighborhood of the typical point as carefully as the bounds
(cid:90) v2+D 2πλ (r)
exp c2 rdr ... derived in this paper. We now move on the numerical results
max(v2−D,v1+D) 1+ srPα section, where more such insights about the relative accuracy
(cid:32)(cid:90) vk+D 2πλ (r) (cid:33) of bounds and approximations are presented.
exp ck rdr
1+ rα ×
max(vk−D,vk−1+D) sP VI. NUMERICALRESULTSANDDISCUSSION
(cid:33)
There are three main system parameters that determine the
f (v ,v ,..,v )dv dv ...dv (16)
V1V2..Vk 1 2 k 1 2 k interferenceexperiencedbythetypicalpointinaPHP:density
λ oftheholes,densityλ ofthebaselinePPP,andtheradiiD
1 2
(cid:16) (cid:17)
where λci(r) = λπ2arccos r2+2vvi2ir−D2 for i = 1,2,...,k, owfethideenhtoilfeys.foBuarsmedaionnntehtewroerlkatcivoenfivgaulureastioonfst,hwesheicpharaarmeeiltleurss-,
and joint density function of distances of V ,V ,...,V is
1 2 k trated in Fig. 5. We define the possible configurations as LD-
f (v ,v ,..,v ) = (2πλ )kv v ...v exp( πλ (v2
V1V2..Vk 1 2 k 1 1 2 k − 1 k − SH: configuration with low density of holes and small holes;
D2)).
HD-SH: configuration with high density of holes and small
Proof. See Appendix G. (cid:4) holes; LD-LH: configuration with low density of holes and
8
(a) (b) (c) (d)
Fig.5. ThePHPnetworkmodel(a)Firstconfiguration:LD-SH,(b)Secondconfiguration:HD-SH,(c)Thirdconfiguration:LD-LH,(d)Fourthconfiguration:
HD-LH.
1.07 1
LD-SH: Actual ratio
LD-SH: Proposition 1
1.06
HD-SH: Actual ratio 0.95
d
n HD-SH: Proposition 1
ou1.05 LD-LH: Actual ratio Pc
ower b1.04 LHHDDD---LLLHHH::: PAPrcrootpupoaosls iritatiiootinon 11 bability, 0.9
L o 0.85
d/ pr
un1.03 ge
per bo1.02 Covera 0.8 PPNrreiioowrr llaoopwwpeerror bxbioomuuanntddio :1n L:: eTLmheemmoamr ea1m 2 1
p New upper bound: Theorem 2
U1.01 0.75 New lower bound 3: Theorem 4
New approximation: Proposition 2
Simulation
1 0.7
6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4
SIR threshold, γ (dB) Hole radius, D
Fig.6. RatiooftheproposedupperandlowerboundsderivedinTheorems2 Fig. 8. Analytical and simulation results for the coverage probability as a
and1,respectively. functionofholeradiusD (λ1=0.1).
1 0.95
0.9
0.95
c
obability, P 00.8.95 probability00.8.85
pr e
Coverage 0.8 PPNrreiioowrr llaoopwwpeerror bxbioomuuanntddio :1n L:: eTLmheemmoamr ea1m 2 1 Coverag00.7.75 PPNrreiioowrr llaoopwwpeerror bxbioomuuanntddio :1n L:: eTLmheemmoamr ea1m 2 1
New upper bound: Theorem 2 New upper bound: Theorem 2
0.75 New lower bound 3: Theorem 4 0.65 New lower bound 3: Theorem 4
New approximation: Proposition 2 New approximation: Proposition 2
Simulation Simulation
0.7 0.6
0 0.2 0.4 0.6 0.8 1 6 8 10 12 14 16 18 20
Density, λ1 SIR threshold, γ (dB)
Fig.7. Analyticalandsimulationresultsforthethecoverageprobabilityas Fig.9. AnalyticalandsimulationresultsforthecoverageprobabilityinLD-
afunctionofthedensityλ1 (D=1). SHcase(λ1=0.05andD=0.6).
large holes; HD-LH: configuration with high density of holes (LD-LHcase)ismorebenignthantheconfigurationwherethe
and large holes. Clearly, the configuration where the holes holes are both large and dense (HD-LH case). Therefore, the
are small and sparse (LD-SH case) is more benign than the result that works well in the HD-SH configuration is expected
configurationinwhichtheholesaresmallanddense(HD-SH). to work in the LD-SH configuration as well. The same is true
Similarly, the configuration where holes are large and sparse for LD-LH and HD-LH cases.
9
0.95 otherwisespecified,wesetthenetworkparametersasfollows:
λ = 1, α = 4, P = 1, r = 0.1, γ = 10 dB. We compare
2 0
0.9 ourproposedboundsandthenewapproximationwiththefirst-
orderstatisticapproximationgivenbyLemma2,andthePPP-
bility0.85 based bound given by Lemma 1 where Ψ is approximated by
a Φ . Before going into more details comparisons, we demon-
b 0.8 2
pro strate the tightness of the lower and upper bounds derived in
e Theorems1and2byplottingtheirratioanditsapproximation
g0.75
a
er Prior lower bound: Lemma 1 (derived in Proposition 1) in Fig. 6. In addition to validating
v
Co 0.7 Prior approximation: Lemma 2 thetightnessoftheapproximationgivenbyProposition1,this
New lower bound 1: Theorem 1
New upper bound: Theorem 2 result shows that the ratio in all cases of interest is close to
0.65 New lower bound 3: Theorem 4 one,whichdemonstratesthetightnessofboththebounds.Note
New approximation: Proposition 2
Simulation that, as expected, the ratio is comparatively higher when the
0.6 holes are large and dense (HD-LH case).
6 8 10 12 14 16 18 20
SIR threshold, γ (dB) We now compare the proposed bounds and approximations
with the numerical results and the known approaches in terms
Fig. 10. Analytical and simulation results for the coverage probability in
HD-SHcase(λ1=0.2andD=0.6). of coverage probability. As demonstrated in (5), the coverage
probability for our setup is simply the Laplace transform
0.95 of interference evaluated at s = γr0α. Note that when we
P
substitutes= γr0α inanyoftheLaplacetransformexpressions
P
0.9 derived in this paper, we notice that the resulting expression
is independent of P. This is expected, because the SIR(r )
0
y0.85
bilit expression given by (4) is indeed independent of P.
a
b 0.8 In Fig. 7, we plot the coverage probability of a typical
o
pr receiver as a function of hole density λ assuming all other
e 1
g0.75 parameters are fixed (we assume D = 1). Small values of
a
ver Prior lower bound: Lemma 1 λ1 result in LD-SH configuration, whereas high values result
Co 0.7 PNreiowr laopwperro bxiomuantdio 1n:: TLheemomream 2 1 in the HD-SH configuration. In Fig. 8, we conduct the same
New upper bound: Theorem 2 study but instead of varying λ , we now vary the hole radius
0.65 New lower bound 3: Theorem 4 1
New approximation: Proposition 2 D while fixing λ1 = 0.1. The low values of D result in
Simulation HD-SH configuration, whereas the high values result in HD-
0.6
6 8 10 12 14 16 18 20 LH configuration. Comparison of the proposed results with
SIR threshold, γ (dB) the simulations reveal that all the bounds and approximations
proposed in this paper are surprisingly tight, even for the
Fig. 11. Analytical and simulation results for the coverage probability in
LD-LHcase(λ1=0.05andD=1.5). extreme configuration like HD-LH, where the overlaps are
significant.Thelowerboundsworkevenbetter.Wealsonotice
1 that the prior results (given by Lemmas 1 and 2) deviate
significantly when the overlaps in the holes are significant. In
0.95
particular,theapproximationgivenbyLemma2becomesvery
loose.AsdiscussedinRemark1,thisisbecauseitsderivation
0.9
ability0.85 idnivstoolrvtesdthinedleopceanldneenitghthbionrnhionogdooffththeeinttyeprifcearelnrceecefiiveeldr., which
b
o
pr 0.8 We now plot the analytical and simulation results for the
ge coverageprobabilityofthetypicalreceiverasafunctionofthe
a
er0.75 Prior lower bound: Lemma 1 SIRthresholdinthefourpossibleconfigurationsinFigs.9–12.
v
Co Prior approximation: Lemma 2 Fig. 9 shows the results for LD-SH case with small holes of
0.7 New lower bound 1: Theorem 1
New upper bound: Theorem 2 radius D = 0.6 and low density of λ = 0.05, while results
1
New lower bound 3: Theorem 4
0.65 for the HD-SH case are shown in Fig. 10 with parameters
New approximation: Proposition 2
Simulation D = 0.6 and λ = 0.2. Further, Fig. 11 shows results for
1
0.6
6 8 10 12 14 16 18 20 the LD-LH case with large holes of radius D = 1.5 and
SIR threshold, γ (dB) low density of λ = 0.05, while the results for the HD-
1
LH case with parameters D =1.5 and λ =0.2 are shown in
Fig. 12. Analytical and simulation results for the coverage probability in 1
Fig. 12. The plots again confirm the accuracy of our results
HD-LHcase(λ1=0.2andD=1.5).
and show that the first-order statistic approximation given by
Lemma 2 is rather loose while our proposed bounds lead
Simulations are performed over circular region with radius to tight upper and lower bounds in all cases. Interestingly,
40m and results are averaged over 5 104 iterations. Unless all the proposed results work so well that they are nearly
×
10
1 1
0.98 0.98
0.96 0.96
c c
P P
y, 0.94 y, 0.94
bilit 0.92 bilit 0.92
a a
b b
o 0.9 o 0.9
pr pr
ge 0.88 ge 0.88
a Prior lower bound: Lemma 1 a Prior lower bound: Lemma 1
ver 0.86 Prior approximation: Lemma 2 ver 0.86 Prior approximation: Lemma 2
o New lower bound 1: Theorem 1 o New lower bound 1: Theorem 1
C C
0.84 New upper bound: Theorem 2 0.84 New upper bound: Theorem 2
New lower bound 3: Theorem 4 New lower bound 3: Theorem 4
0.82 New approximation: Proposition 2 0.82 New approximation: Proposition 2
Simulation Simulation
0.8 0.8
5 10 15 20 25 5 10 15 20 25
λ /λ λ /λ
2 1 2 1
Fig.13. Analyticalandsimulationresultsforthecoverageprobabilitiesofthe Fig.15. Analyticalandsimulationresultsforthecoverageprobabilitiesofthe
PHP users as a function of λ2/λ1 under configuration LD-SH (λ1 =0.05 PHP users as a function of λ2/λ1 under configuration LD-LH (λ1 =0.05
andD=0.6). andD=1.5).
1
1
0.9
0.9
c
c P
bility, P 0.8 ability, 0.8
a b
b o 0.7
e pro 0.7 ge pr
g a Prior lower bound: Lemma 1
era 0.6 PPrriioorr laopwperor xbiomuantdio:n L: eLmemmam a1 2 ver 0.6 Prior approximation: Lemma 2
v o New lower bound 1: Theorem 1
o New lower bound 1: Theorem 1 C
C New upper bound: Theorem 2
0.5 NNeeww ulopwpeerr bboouunndd :3 T: hTehoeroermem 2 4 0.5 New lower bound 3: Theorem 4
New approximation: Proposition 2
New approximation: Proposition 2
Simulation
Simulation
0.4
0.4 5 10 15 20 25
5 10 λ /1λ5 20 25 λ2/λ1
2 1
Fig. 16. Analytical and simulation results for the coverage probabilities of
Fig. 14. Analytical and simulation results for the coverage probabilities of
thePHPusersasafunctionofλ2/λ1underconfigurationHD-SH(λ1=0.2 tahnedPDHP=u1se.5rs).asafunctionofλ2/λ1underconfigurationHD-LH(λ1=0.2
andD=0.6).
Figs. 15 and 16 depict results for the LD-LH and HD-
indistinguishableinallconfigurationsexceptthemostextreme
LH cases. Here we consider D = 1.5 and λ = 0.05,0.2 .
1
one of HD-LH (Fig. 12). In this configuration, we first notice { }
As was the case in the above results, our proposed lower and
that both the lower bounds given by Theorems 1 and 4 work
upper bounds provide a remarkably accurate characterization
equally well, which means considering even a single hole
ofcoverageprobability.Thisisbecausethelocalneighborhood
in the interference field accurately is good enough for the
of the typical node is carefully preserved while deriving these
accuratecharacterizationofinterference.Asexpected,wealso
bounds. On the other hand, the prior results, in particular the
noticethattheapproximationderivedbyhandlingtheoverlaps
first-orderstatisticapproximation,leadstoafairlylooseresult.
intheaveragesenseinProposition2doesnotworkbetterthan
the proposed bounds in the extreme configuration of HD-LH
(Fig. 12). This is because any average-based arguments do VII. CONCLUSIONS
not necessarily capture the local neighborhood of the typical In this paper, we have focused on the accurate performance
point as well as the bounds do. characterization of a typical user in a wireless network that is
Forcompleteness,wealsoplottheanalyticalandsimulation modeled as a PHP. This model is of particular interest in sce-
results for the coverage probability of the typical receiver in narios where interference management techniques introduce
a PHP as a function of λ /λ in the four configurations in spatial separation among active transmitters in the form of
2 1
Figs. 13–Fig. 16. In Figs. 13 and 14, design parameters are holes or exclusion zones. In terms of the technical results,
setinordertosimulateLD-SHandHD-SHcases,respectively. we have provided new easy-to-use provable lower and upper
In particular, we assume D = 0.6 and λ = 0.05,0.2 . bounds on the Laplace transform of interference experienced
1
{ }