Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless Communications

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Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless

Communications

Bartlomiej Blaszczyszyn, D. Yogeshwaran

To cite this version:

Bartlomiej Blaszczyszyn, D. Yogeshwaran. Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless Communications. Advances in Applied Probability, Applied Probability Trust, 2009, 41, pp.623-646. �10.1239/aap/1253281057�. �inria-00288866v2�

SURES, SHOT NOISE FIELDS AND SOME APPLICATIONS TO

WIRELESS COMMUNICATIONS

BART LOMIEJB LASZCZYSZYN,

INRIA/ENSandMath. Inst. UniversityofWro law

D.YOGESHWARAN,

INRIA/ENS

Abstrat

Diretionally onvex (dx) ordering is a tool for omparison of dependene

strutureofrandomvetorsthatalsotakesintoaountthevariabilityofthe

marginaldistributions. Whenextendedtorandomeldsitonernsomparison

of all nite dimensional distributions. Viewing loally nite measures as

non-negative elds of measure-values indexed by the bounded Borel subsets

of the spae, in this paper we formulate and study the dx ordering of

random measures on loally ompat spaes. We show that the dx order

is preserved under some of the natural operations onsidered on random

measures and point proesses, suh as deterministi displaement of points,

independent superposition and thinning as well as independent, identially

distributed marking. Furtheroperationssuhas positiondependent marking

and displaement of points are shown to preserve the order on Cox point

proesses. We also examinethe impat of dx orderon the seondmoment

properties,inpartiularonlusteringandonPalmdistributions. Comparisons

of Ripley's funtions, pairorrelation funtions as wellas examples seem to

indiatethatpointproesseshigherindxorderlustermore.

As the main result, we show that non-negative integral shot-noise elds

with respetto dx ordered random measures inheritthis ordering from the

measures. Numerous appliations of this result are shown, in partiular to

omparisonofvariousCoxproessesandsomeperformanemeasuresofwireless

networks,inbothofwhihshot-noiseeldsappearaskeyingredients. Wealso

mentionafewpertinentopenquestions.

Keywords: stohasti ordering, diretional onvexity, random measures, ran-

domelds,pointproesses,wirelessnetworks

2000MathematisSubjetClassiation: Primary60E15

Seondary60G60,60G57,60G55

Postaladdress: ENSDITREC,45rued'ulm,75230Paris,FRANCE.

Postaladdress:ENSDITREC,45rued'ulm,75230Paris,FRANCE.

Point proesses (p.p.) have been at the entre of various studies in stohasti

geometry, both theoretial and applied. Most of the work involving quantitative

analysis of p.p. have dealt with Poisson p.p.. One of the main reasons being that

harateristis of Poisson p.p. are amenable to omputations and yield nie losed

formexpressionsinmanyases. ComputationshavebeendiÆultingreatmanyases,

evenforCox(doublystohastiPoisson)p.p..

Comparisonofpointproesses Toimproveuponthissituation,qualitative,om-

parativestudiesofp.p. haveemergedasusefultools. Therstmethodofomparison

ofp.p. hasbeenouplingorstohastidomination(see[18,20,32℄). Inourterminology,

theseareknownasstrongorderingof p.p.. Whentwop.p. anbeoupled,oneturns

outtobeasubsetoftheother. Thisorderingisveryusefulforobtainingvariousbounds

andprovinglimittheorems. However,usingitoneannotomparetwodierentp.p.

withsamemeanmeasures. An obviousexampleis anhomogeneousPoisson p.p. and

astationary Coxp.p. with thesameintensity. The questionarises ofwhat ordering

is suitable for suh p.p.? This is an important questionsine it is expeted that by

omparing p.p. of the same intensity one should ahieve a tighter bound than by

oupling. Forsomemoredetailsonstrongorderingofp.p. andneedforotherorders,

seeremarksin [29,Setion5.4andSetion 7.4.2℄.

From onvex to dx order Tworandomvariables X andY with thesamemean

E(X) = E(Y) an be ompared by how "spread out" their distributions are. This

statistial variability (in statistial ensemble) is aptured to a limited extent by the

variane, but more fully by onvex ordering, under whih X is less than Y if and

only if for all onvex f, E(f(A)) E(f(B)). In multi-dimensions, besides dierent

statistial variability of marginal distributions, two random vetorsan exhibit dif-

ferentdependene properties of theiroordinates. Themost evidentexample hereis

omparisonofthevetoromposedofseveralopiesofonerandomvariabletoavetor

omposed of independent opies sampled from the same distribution. A useful tool

foromparisonofthedependene strutureofrandomvetorswithxedmarginalsis

thesupermodular order. Thedxorderisanotherintegralorder(generatedbyalass

of dxfuntions in thesame manneras onvexfuntions generatethe onvexorder)

thatanbeseenasageneralizationofthesupermodularone,whih inadditiontakes

intoaountthevariabilityofthemarginals(f[29,Setion3.12℄). Itanbenaturally

extendedtorandomeldsbyomparisonofallnite dimensionaldistributions.

extension that onsists in dx ordering of loally nite measures (to whih belong

p.p.) viewed as non-negativeelds of measure-valueson all bounded subsets of the

spae. Weshowthat thedxorderispreservedundersomeofthenaturaloperations

onsideredonrandommeasuresandpointproesses,suhasindependentsuperposition

and thinning. Also, we examine the impat of dx order on the seond moment

properties, inpartiularonlustering,andPalmdistributions.

Integral shot-noise elds Many interesting harateristis of random measures,

bothinthetheoryandinappliationshavetheformofintegralsofsomenon-negative

kernels. We all them integral shot-noise elds. For example, many lasses of Cox

p.p., with the most general being Levy based Cox p.p. (f. [14℄), have stohasti

intensityelds,whihareshot-noiseelds. Theyarealsokeyingredientsofthereently

proposed,so-alled\physial"modelsforwirelessnetworks,aswewillexplaininwhat

follows(see also [1,8,11℄). It is thus partiularly appealing to study the shot-noise

eldsgeneratedbydxorderedrandommeasures.

Sine integralsare linearoperators on thespae of measures,and knowing that a

linearfuntion of avetoris trivially dx, it is naturallyto expet that the integral

shot-noiseeldswithrespettodxorderedrandommeasureswillinheritthisordering

fromthemeasures. However,thispropertyannotbeonludedimmediatelyfromthe

nite dimensional dx ordering of measures. The formal proof of this fat that is

themain resultofthis paperinvolvessomeargumentsfrom thetheoryofintegration

ombined with the losure property of dx order under joint weak onvergene and

onvergeneinmean.

Ordering in queueing theory and wireless ommuniations The theory of

stohastiorderingprovideselegantandeÆienttoolsforomparisonofrandomobjets

andisnowbeingusedinmanyelds. Inpartiularinqueueingtheoryontext,in[33℄,

Rossmadeaonjeturethatreplaing astationaryPoissonarrivalproess inasingle

server queue by a stationary Cox p.p. with the same intensity should inrease the

averageustomer delay. Therehavebeenmanyvariationsof theseonjetures whih

are now known as Ross-type onjetures. They triggered the interest in omparison

ofqueueswithsimilar inputs([6,25,31℄). Thenotionofadxfuntion waspartially

developedandusedinonjuntionwiththeprovingofRoss-typeonjetures([21,22,

34℄). Muhearliertotheseworks,aomparativestudyofqueuesmotivatedbyneuron-

ringmodelsanbefoundin[16℄. Alsoomparisonofvarianesofpointproessesand

breproesseswasstudiedin[36℄andheneitanbeonsideredasaforerunnertoour

artile. TheappliabilityoftheseresultshasgeneratedsuÆientinterestinthetheory

andStoyan([29℄). Asmostworksonorderingofp.p. weremotivatedbyappliations

toqueueingtheory,resultswereprimarilyfousedonone-dimensionalpointproesses.

An attempt to retifythe lakofworkin higher dimensionswasmadein [24℄,where

omparisonresultsforshot-noiseeldsofspatialstationaryCoxp.p. weregiven. The

resultsof[24℄ arethestartingpointofourinvestigation.

Our interest in ordering of point proesses, and in partiular in the shot-noise

eldstheygenerate,hasrootsin theanalysisofwirelessommuniations,wherethese

objetsareprimarily usedto modelthesoalled interferenethat isthetotal power

reeivedfrom manyemitterssatteredin theplaneorspaeandsharingtheommon

Hertzianmedium. Aordingto anewemergingmethodology,theinterferene-aware

stohasti geometry modeling of wireless ommuniationsprovidesa way of dening

and omputing marosopi properties of large wireless networks by someaveraging

overall potential random patterns for node loations in an innite plane and radio

hannelharateristis,inthesamewayasqueuingtheoryprovidesaveragedresponse

timesorongestionoverallpotentialarrivalpatternswithinagivenparametrilass.

These marosopi properties will allow one to haraterize the key dependeniesof

the network performane harateristis in funtion of a relatively small number of

parameters.

In the above ontext, Poisson distribution of emitters/reeiver/users is often too

simplisti. Statistis show that the real patterns of users exhibits more lustering

eets (\hotsspots")thanobserved in anhomogeneousPoisson pointproesses. On

theotherhand,goodpaket-ollision-avoidanemehanismsshemeshouldreatesome

\repulsion" in the pattern of nodes allowedto aess simultaneouslyto the hannel.

Thisrisesquestionsabouttheanalysisofnon-Poissonmodels,whihouldbetosome

extent takled on the ground of the theory of stohasti ordering. Interestingly, we

shallshowthat thereareertainperformaneharateristisinwirelessnetworksthat

improvewithmorevariabilityintheinputproess.

Theremainingpartoftheartileisorganizedasfollows. Inthenextsetion,

wewillpresentthemaindenitionsandstatethemainresultsonerningdxordering

of the integral shot-noise elds. Setion 3 will explore the various onsequenes of

ordering ofrandom measures. Theproofs of themain resultsare given in Setion 4.

Examples illustrating the use and appliation of the theorems shall be presented in

Setion 5. Setion 6 will sketh some of the possible appliations of results in the

ontext of wireless ommuniations. Finally, we onlude with some remarks and

questions in Setion 7. Thereis anAppendix (Setion 8) ontainingsomeproperties

ofstohastiordersandtheirextensionsthatareusedin thepaper.

TheorderonR n

shalldenotetheomponent-wisepartialorder,i.e.,(x

1

;:::;x

n )

(y

1

;:::;y

n )ifx

i y

i

foreveryi.

Denition2.1. We say that a funtion f : R d

! R is diretionally on-

vex(dx)ifforeveryx;y;p;q2R d

suhthat px;yqandx+y=p+q,

f(x)+f(y)f(p)+f(q):

Funtionf issaidtobediretionallyonave(dv)iftheinequalityintheabove

equationisreversed.

Funtionf issaiddiretionally linear(dl)ifitisdxanddv.

Funtionf =(f

1

;:::;f

n ):R

d

!R n

issaidtobedx(dv)ifeahofitsomponent

f

i

is dx(dv). Also, weshall abbreviateinreasing and dx by idx and dereasing

anddxbyddx. Similarabbreviationsshallbeusedfordv funtions. Moreover,we

abbreviatenon-negativeandidxbyidx +

.

Inthefollowing,letFdenotesomelassoffuntionsfrom R d

toR. Thedimension

disassumedtobelearfrom theontext. Unlessmentioned,whenwestateE(f(X))

forf 2FandX arandomvetor,weassumethattheexpetationexists,i.e.,foreah

randomvetorX weonsider thesub-lassofFforwhih theexpetationsexistwith

respetto(w.r.t) X.

Denition2.2. SupposeX andY arereal-valuedrandomvetorsofthesame

dimension. ThenXissaidtobelessthanY inForderifE(f(X))E(f(Y))for

allf 2F(forwhihbothexpetationsarenite). WeshalldenoteitasX

F Y.

Suppose fX(s)g

s2S

and fY(s)g

s2S

are real-valued random elds, where S is

an arbitraryindex set. We say that fX(s)g

F

fY(s)g iffor everyn 1and

s

1

;:::;s

n

2S,(X(s

1

);:::;X(s

n ))

F (Y(s

1

);:::;Y(s

n )):

Inthe remainingpartofthe paper,wewill mainly onsider Fto bethelass ofdx,

idxandidvfuntions;thenegationofthesefuntionsgiverisetodv;ddvandddx

orders respetively. If F is thelass of inreasing funtions,weshall replaeF by st

(strong)in theabovedenitions. Thesearestandardnotationsusedin literature.

As onerns random measures, we shall work in the set-up of [17℄. Let E be a

loally ompat, seond ountable Hausdor (LCSC) spae. Suh spaes are polish,

i.e., ompleteandseparablemetrispae. LetB(E) betheBorel-algebraandB

b (E)

bethe -ring of bounded, Borel subsets (bBs). Let M =M( E) be thespae of non-

negativeRadonmeasuresonE. TheBorel-algebraMisgeneratedbythemappings

7!(B)forallB bBs. Arandommeasure isamappingfromaprobabilityspae

(;F;P)to(M;M). Weshallallarandommeasureap.p. if2N,thesubsetof

ountingmeasuresinM. Further,weshallsayap.p. issimpleifa.s. (fxg)1for

allx 2E. Throughout, weshalluse for anarbitraryrandom measureand fora

p.p.. Arandommeasure anbeviewedasarandomeldf(B)g

B2B

b (E)

:Withthis

viewpointandthepreviouslyintroduednotionoforderingofrandomelds,wedene

orderingofrandommeasures.

Denition2.3. Suppose

1

() and

2

() are random measures on E. We say that

1 ()

dx

2

()ifforanyI

1

;:::;I

n

bBsin E,

(

1 (I

1 );:::;

1 (I

n ))

dx (

2 (I

1 );:::;

2 (I

n

)): (1)

Thedenitionissimilarforotherorders,i.e.,whenFisthelassofidx=idv=ddx=ddv=st

funtions.

Denition2.4. LetSbeanysetandE aLCSCspae. Givenarandommeasureon

E andameasurable(intherstvariablealone)responsefuntionh(x;y):ES !

R +

where

R +

denotestheompletionofpositivereal-linewithinnity,the(integral) shot-

noise eldisdenedas

V

(y)=

Z

E

h(x;y)(dx): (2)

Withthisbriefintrodution,wearereadytostateourkeyresultthatwillbeproved

inSetion 4.1.

Theorem2.1. 1. If

1

idx(resp.idv)

2

, then fV

1 (y)g

y2S

idx(resp.idv)

fV

2 (y)g

y2S .

2. LetE(V

i

(y))<1,forally2S,i=1;2:If

1

dx

2

,thenfV

1 (y)g

y2S

dx

fV

2 (y)g

y2S .

Therstpartof theabovetheoremfor theone-dimensionalmarginalsofbounded

shot-noiseeldsgeneratedbylowersemi-ontinuousresponsefuntionsisprovedin[24℄

for the speial ase of spatial stationary Cox p.p.. It is onspiuous that we have

generalizedtheearlierresulttoagreatextent. Thismoregeneralresultwillbeusedin

manyplaesinthispaper,inpartiulartoproveorderingofindependently,identially

marked p.p. (Proposition 3.2), Ripley's funtions (Proposition 3.4), Palm measures

(Proposition 3.5), independently marked Cox proesses (Proposition 3.7), extremal

shot-noise elds (Proposition 4.1). Apart form these results, Setions 5 and 6shall

amplydemonstrateexamplesandappliationsthat shallneedTheorem2.1.

WeshallnowgiveasuÆientonditionforrandommeasurestobeordered,namely

thattheondition(1)inDenition 2.3needsto beveriedonlyfordisjointbBs. The

neessityistrivial. Thisisamuheasieronditionandwillbeusedmanytimesinthe

remainingpartofthepaper.

Proposition 3.1. Suppose

1

() and

2

() are two random measures on E. Then

1 ()

dx

2

()if and only if ondition (1)holds for all mutually disjoint bBs. The

sameresultsholdstrue for idxandidvorder.

Proof. We need to prove the 'if' part alone. We shall prove for dx order and

the same argument is valid for f being idx oridv. Let ondition (1) be satised

for all mutually disjoint bBs. Let f : R n

+

! R be dx funtion and B

1

;:::;B

n be

bBs. We an hoose mutually disjoint bBs A

1

;:::;A

m

suh that B

i

= [

j2Ji A

j for

all i. Hene (B

i ) =

P

j2Ji (A

j

): Now dene g : R m

+

! R n

+

as g(x

1

;:::;x

m ) =

( P

j2J1 x

j

;:::; P

j2Jn x

j

): Theng isidl andsof Æg is dx. Moreover,f((B

1 );:::;

(B

n

))=fÆg((A

1

);:::;(A

m

))andthustheresultfordxfollows.

3.1. Simple OperationsPreserving Order

Point proesses are speial asesof random measures and as suh will besubjet

to theonsidered ordering. It is known thateahp.p. onaLCSC spae E anbe

representedasaountablesum= P

i

"

X

i

ofDirameasures("

x

(A)=1ifx2Aand

0otherwise)insuhawaythatX

i

arerandomelementsinE. Weshallnowshowthat

allthethreeordersdx;idx;idvpreservesomesimpleoperationsonrandommeasures

andp.p.,asdeterministimapping,independentidentiallydistributed(i.i.d.) thinning

andindependentsuperposition.

Let:E !E 0

beameasurable mappingtosomeLCSCspaeE 0

. Bytheimageof

a(random)measurebyweunderstand 0

()=( 1

()). Notethat theimageof

ap.p. byonsistsin deterministidisplaementofallitspointsby.

Let = P

i

"

x

i

. By i.i.d. marking of , with marks in some LCSC spae E 0

,

we understand a p.p. on the produt spae E E 0

, with the usual produt Borel

-algebra, dened by

~

=

P

i

"

(xi;Zi)

, where fZ

i

gare i.i.d. random variables (r.v.),

soalled marks,on E 0

. By i.i.d. thinningof ,weunderstand = P

i Z

i

"

xi , where

Z

i

arei.i.d. 0-1Bernoulli randomvariables r.v.. The probability PfZ =1gis alled

theretentionprobability. Superpositionofp.p. isunderstoodasadditionof(ounting)

measures. MeasuresonCartesianprodutsofLCSCspaesarealwaysonsideredwith

theirorrespondingprodutBorel-algebras.

Proposition 3.2. Suppose

i

;i=1;2are random measuresand

i

;i=1;2arep.p..