in LEO Satellite Networks: The Single Orbit Case
A. Halim Zaim George N. Rouskas Harry G. Perros
TR-2000-06
June 19,2000
Abstrat
WestudytheproblemofarryingvoieallsoveraLEOsatellitenetwork,andwepresentananalytial
modelfor omputingall blokingprobabilities for asingle orbitof asatellite onstellation. We have
devisedamethodtosolvetheorrespondingMarkovproesseÆientlyforupto 5-satelliteorbits. For
orbits onsisting of a larger number of satellites, we have developed an approximate deomposition
algorithmtoomputetheallblokingprobabilitiesbydeomposingthesystemintosmallersub-systems,
anditerativelysolvingeahsub-systeminisolation usingtheexatMarkovproess. Ourapproahan
aptureblokingduetohand-osforbothsatellite-xedandearth-xedonstellations. Numerialresults
demonstratethatourmethodisaurateforawiderangeoftraÆpatternsandfororbitswithanumber
ofsatellitesthatisrepresentativeofommerialsatellitesystems.
Keywords:Lowearthorbitsatellitenetworks,allblokingprobability,hand-os,deompositionalgorithms
DepartmentofComputerSiene
NorthCarolinaStateUniversity
Currently,wearewitnessinganinreaseinthedemandforabroadrangeofwirelesstelephoneandinternet
servies. Satellitebasedommuniationisposedtoprovidemobiletelephonyanddatatransmissionservies
onaworldwidebasisinaseamlesswaywithterrestrialnetworks. Satellitesystemsareloation-insensitive,
andtheyanbeusedtoextendthereahofnetworksandappliationstoanywhereontheearth.
Satellitesanbelaunhedindierentorbits,ofwhih,thelowearthorbit(LEO),themediumearthorbit
(MEO), and thegeo-stationaryorbit(GEO), arethemostwell-known. LEO satellitesareplaedin orbits
at analtitudeoflessthan2000kmabovetheearth. Theirorbitperiodisabout90minutes,andtheradius
of thefootprintareaofaLEOsatelliteisbetween3000km to4000km. Thedurationof asatelliteinLEO
orbitovertheloalhorizonofanobserveronearthisapproximately20minutes,andthepropagationdelayis
about25ms. Afewtensofsatellitesonseveralorbitsareneededtoprovideglobaloverage. MEOsatellites
are plaedinirular orbitsat analtitudeofaround 10000km. Their orbitperiodis aboutsixhours,and
thedurationofasatelliteinMEOorbitovertheloalhorizonofanobserveronearthisafewhours. Fewer
satellitesontwoorthreeorbitsisenoughtoprovideaglobaloveragein aMEOsystem. Propagationdelay
in aMEOsystem isabout125ms. GEOsatellitesare also in irular orbitsin the equatorialplane at an
altitudeof35786km,withanorbitalperiodequaltothatoftheearth. AsatelliteinGEOorbitappearsto
bexed abovethe earth's surfae. Thefootprintof aGEOsatellite oversnearlyonethird of theearth's
surfae (between 75 o
south to 75 o
north). Therefore, a near global overage an be obtained with three
satellites,but thepropagationdelayis250ms.
A LEO orMEO satellite systemis a set of idential satellites, launhed in several orbital planeswith
theorbitshavingthesamealtitude. Thesatellitesmoveinasynhronizedwayintrajetoriesrelativetothe
earth. Suh asetof satellitesisreferredtoasaonstellationofsatellites. Theposition of allthesatellites
inrelationtotheearthatsomeinstaneoftime,repeatsitselfafterapredeterminedperiodwhihisusually
severaldays,while asatellitewithin anorbitalsoomesto thesamepointontheskyrelativetotheearth
after aertaintime.
InaLEOorMEOsatellitesystem,satellitesanommuniate diretlywith eah otherbyline ofsight
using intraplaneinter-satellitelinks (ISL)whihonnetsatellitesin thesameorbital planeandinterplane
ISLs whihonnet satellitesin adjaentplanes. ISLsintrodueexibility in routing,theyanbeused to
build in redundany into the network, and they permit twousers in dierent footprints to ommuniate
without theneed of aterrestrialsystem. To improvethe bandwidthand frequenyeÆieny, the satellite
footprintareaisdividedintosmallerells. Foreahellwithinafootprint,aspeibeamofthesatelliteis
used. Aonstellation ofsatellitesmayprovideeithersatellite-xedelloverageorearth-xedelloverage.
In therstase,thesatelliteantennasendingthebeamisxed, andasthesatellitemovesalong itsorbit,
its footprintandtheellmoveaswell. Intheaseofearth-xed elloverage,theearth surfaeis divided
intoells,asin aterrestrialellularsystem,andaellisserviedontinuouslybythesamebeamduringthe
entiretimethat theelliswithin thefootprintareaofthesatellite.
objetsonearth. Therefore,thenumberofhand-osduring atelephone alldepends ontheallduration,
the beamsize,the satellitefootprintsize and thesatellitespeed,while theloationand mobilityof auser
only eets thetime ahand-o takes plae. Forexample, foraall durationof 3min., theustomer of a
LEO satelliteonstellation with anelevation angle of10 o
, willexperienetwohand-os. Inanearth-xed
system,eahbeamisassignedto axed ellontheearthwithinthesatellite'sfootprint. Duringasatellite
hand-o,allbeamsarereassignedtotheirrespetiveellsintheadjaentfootprintarea. Therefore,inthese
systems, bothbeamandsatellite hand-osourat thesametime. Inasatellite-xedsystem,ausermay
behandedotothenextbeaminthesamesatelliteorthesatellitebehind,astheelldenedbythebeam
moves away from the user. The newly entered beam, may or may not have enough bandwidth to arry
the handed-o traÆ. Inthe ase of satellite-basedtelephony, the new beam maynot beable to arry a
handed-otelephone all,inwhihase,the allwill bedropped. Ingeneral, hand-osin satellitesystems
imposeabigproblemfrom thepointofqualityofservie.
ThereareseveralLEOsystemsurrentlyinoperation,suhasArgos,VITAsat,ORBCOMM,and
Glob-alstar. Severalothers, suh asLEOone,SkyBridge, andTeledesi,aresheduledtostartafter 2000. These
systems dierin many aspets, inluding thenumberof orbitsand the numberof satellitesperorbit, the
numberofbeamspersatellite,theirapaity,thebandtheyoperate(K-Band,Ka-Band,L-Band,et.),and
theaessmethodemployed(FDMA,TDMA,orCDMA).Also,thesesystemsprovidedierentserviesand
theymayormaynothaveon-boardswithingapabilities. Forinstane,Iridiumprovidestelephoneservie
only, LEOone will provide data ommuniations only, while most other satellite systems are designed to
provide multimedia servies. Iridium and Teledesi haveon-board digital proessing and swithing, while
other systems,suhastheGlobalstar,atasabentpipe. Despitethese dierenes,from thepointofview
of providing telephony-basedservies,the priniplesof operationare verysimilar, andthus, theanalytial
tehniques tobedevelopedin theproposed work willbeappliableto anyLEO satellitesystemthat oers
suhservies.
1.1 Related Researh
Despitetheimportaneofsatellitesystems,theirperformanehasnotbeenadequatelyevaluated. Atypial
wayof modeling asatellitesystemin theliterature isto representeahellasan M/M/K/Kqueue. This
approahpermitsthealulationofvarioususefulperformanemeasures,suhastheallblokingprobability.
However,thistypeofmodeldoesnottakeintoaountthefatthattheamountoftraÆinoneelldepends
ontheamountoftraÆin oneormoreotherells. ThistypeoftraÆdependeniesaretakenintoaount
in ourmodelsdesribedinSetion 2.
In [6℄, Ganz et al., investigated the distribution of the numberof hand-os and the average all drop
probabilityforLEOsatellitesystems. Bothbeam-to-beamandsatellite-to-satellitehand-osweretakeninto
aount. EahellwasmodeledasanM/M/K/Kqueue,where Kdenotes thenumberofhannelsperell,
assuming that thenumberof hand-oalls entering aellis equalto thenumberof hand-oalls leaving
throughput and average delay was alulated. See also [9, 7℄. In [12℄, Pennoni and Ferronidesribed an
algorithm toimprovetheperformaneofLEOsystems. Theydenedtwoqueuesforeahell,onefornew
allsandoneforhand-oalls. Theallsareheldinthesetwoqueuesforamaximumallowedwaitingtime.
That is,theyaredroppediftheyarenotservedwithinthis time. Thequeuefornewallshasamaximum
waitingtime equalto20 se. Thequeueforhand-oallshasamaximumwaitingtime equalto the
ross-over time of the overlappingzone of two adjaent ells. The hand-o queue has higherpriority than the
newalls queue. Simulationresultsshowedthatthisalgorithmdereasedthealldroppingratedrastially.
In [14℄,Ruiz etal., usedteletraÆtehniquestoalulate thebloking andhand-oprobabilities. Various
hannel assignmentstrategies were investigated. In [5℄, Dosiereet al., dened amodel foralulating the
hand-o traÆ rate. Theauthors divided astreet of overage into small piees where eah piee is equal
to the footprint area of a satellite. Given the total arrival distribution for the whole street of overage,
thearrivalrateofeahsatellitewasobtainedbyintegratingthatdistribution forthesatelliteinterval. The
hand-oratewasthen alulatedthroughaseondintegration. One thehand-oratehasbeenobtained,
theblokingprobabilityanbealulatedusingtheErlanglossformula.
Anumberof authorshavealso dealt withtheveryinterestingproblemof routingin asatellite system.
In [18, 19℄, Werner et al., proposed adynami routing algorithmfor ATM-based LEO and MEO satellite
systems. Duetothefatthatsatellitesmoveinorbitsandorbitsslowlyrotatearoundtheearth,thenetwork
topologyanbeseenasonsistingofaseriesoftopologieswhih ontinuouslyrepeatthemselves. Foreah
topology,end-to-endroutes arealulated. Subsequently, anoptimization proedureis arriedoutoverall
thenetworktopologieswithaviewtominimizingtheourreneofhand-osbetweensuessivetopologies.
In [11℄, Mauger and Rosenberg proposed the virtual node routing algorithm for ATM traÆ. Users are
mapped onto virtualnodes, and eah virtual node is served bya satellite. When the satellitepasses, the
nextsatellitetakesitsplaeandservesthevirtualnode. Routingisperformedaordingtothetopologyof
thevirtualnodes.
Changetal.,proposedthenitestateautomaton(FSA)modelin[3,1℄tosolvetheISLlinkassignment
problem inLEOsatellitesystems. Thetotaltimeittakestheposition ofallthesatellitesovertheearthto
repeatitself,is dividedintoequallengthintervalsduring whih thevisibilitybetweensatellites,thatis the
networktopologyofthesatellites,doesnothange. GivenatraÆmatrixforeahinterval,alinkassignment
algorithm is runwith a viewto maximizing theresidual apaity of the bottlenek links. Theresult is a
tablethatshowsonnetivitybetweensatellitesforeahinterval. Thesetablesanbestoredineahsatellite,
and duringthe real-timeoperationof thesystemtheinter-satellitelinks areestablishedaordingtothese
tables. Furtherrelatedresearhanbefoundin[4,2℄.
Uzunaliogluetal.,suggestedin[16,17℄aonnetionhand-oprotoolforLEOsatellitesystems. First,a
minimumostrouteforaonnetionbetweentwopointsontheearthisobtained. Thisrouteisusedforas
longaspossible. Whenahand-ooursateitherend oftheonnetion,theprotoolsimplyaddsthenew
link to thepath. This ontinuesfor apredetermined amountof time, when the protool omputesa new
1.2 Contributions and Organization
In this paper we study the problem of arrying voie alls over a LEO satellite network and we present
an analytialmodel for omputingall blokingprobabilities for asingle orbitof asatellite onstellation.
We rstderiveanexatMarkovproess,and orrespondingqueueingnetwork,forasingleorbitunder the
assumption that satellites are xed in the sky (i.e., there are no hand-os of voie alls). We show that
thequeueingnetworkhasaprodut-formsolution,andwedevelopamethodforomputingthenormalizing
onstant. Intermsoftimeomplexity,ourmethodrepresentsasigniantimprovement(whihwequantify)
overabrute-fore alulation,however,it anbeapplieddiretly to orbitswithat mostvesatellites. For
a system with a larger number of satellites, we then presentan approximate deomposition algorithm to
ompute allblokingprobabilities bydeomposing thesystem intosmallersub-systems, and solvingeah
sub-systeminisolationusingtheexatsolutiondesribedabove. Thisapproahleadstoaniterativesheme,
where theindividual sub-systemsaresolvedsuessivelyuntilaonvergeneriterionissatised.
Next,weintroduehand-osbyonsideringthesystemofsatellitesastheyorbittheearth. Foranorbit
with earth-xed overage,wethenshowthat there isno bloking due to hand-os,and thus, the solution
(exat orapproximate) obtained under the assumption that satellites are xed in thesky an be used to
omputeallblokingprobabilitiesinthisase. Foranorbitwithsatellite-xedoverage,ontheotherhand,
bloking due to hand-os does our. In this ase, we show how the queueing network desribed above
anbeextendedto modelall hand-osbyallowingustomers to movefromone node toanother, andwe
derivetherateofsuh node-to-nodetransitions intermsof thespeedofthesatellitesandtheshapeof the
footprints. We also show that the new queueing network has aprodut-form solution similar to the one
under theno-hand-os assumption, and thus, the exatand approximate algorithms developed abovean
beapplieddiretlytoomputeall blokingprobabilitiesunderthepreseneofhand-os.
Thepaperis organized asfollows. InSetion 2 we develop anexat Markovproess model under the
assumptionthatsatellitesarexedinthesky(i.e.,nohand-ostakeplae),andin Setion3wepresentan
approximatedeompositionalgorithmforalargenumberofsatellites. InSetion4weextendourapproahto
modelhand-osforbothearth-xedandsatellite-xedoverage.Wepresentnumerialresultsin Setion5,
andinSetion6weonludethepaperbydisussingpossiblediretionstowhihthisworkmaybeextended
in thefuture.
2 An Exat Model for the No Hand-Os Case
Letus rstonsidertheasewherethepositionofthesatellitesinthesingleorbitisxedin thesky,asin
the aseof geo-stationarysatellites. Theanalysisof suh asystemissimpler,sinenoalls arelost dueto
hand-osfrom onesatellitetoanother, aswhenthesatellitesmovewith respetto theusersontheearth.
Satellite 1
Satellite 3
Satellite 2
ISL 1-2
ISL 3-1
ISL 2-3
Figure1: Threesatellitesinasingleorbit
earth-xed andsatellite-xedoverage.
Eahup-and-downlinkofasatellitehasapaitytosupportupto C
UDL
alls,whileeahinter-satellite
link hasapaityequalto C
ISL
alls. Letusassumethat allrequestsarriveat eah satelliteaordingto
aPoissonproess,andthatall holdingtimesareexponentiallydistributed. Wenowshowhowtoompute
blokingprobabilitiesforthe3satellitesinthesingleorbitofFigure 1. Theanalysisanbegeneralizedto
analyze k> 3satellitesin asingleorbit. Forsimpliity, we onsider onlyshortest-path routing,although
theanalysisanbeappliedtoanyxedroutingshemewherebythepathtakenbyaallisxedandknown
in advaneofthearrivaloftheallrequest.
Let n
ij
be a random variable representing the number of ative alls between satellite i and satellite
j; 1 i;j 3, regardless of whether the alls originated at satellite i orj. Let
ij
(respetively, 1=
ij )
denotethearrivalrate(resp.,meanholding time)ofalls betweensatellitesiandj. Then,theevolutionof
thethree-satellitesysteminFigure1anbedesribedbythesix-dimensionalMarkovproess:
n = (n
11 ;n
12 ;n
13 ;n
22 ;n
23 ;n
33
) (1)
Alsolet1
ij
denoteavetorwithzerosforallrandomvariablesexeptrandomvariable n
ij
whihis1. The
statetransitionratesforthisMarkovproessaregivenby:
r(n ;n+1
ij
) =
ij
8 i;j (2)
r(n ;n 1
ij
) = n
ij
ij
8i;j; n
ij
>0 (3)
The transitionin (2) isdue to thearrivalofaallbetweensatellitesi and j,while thetransition in (3)is
due totheterminationofaallbetweensatellitesiandj.
Duetothefat thatsomeoftheallsshareommonup-and-downandinter-satellitelinks,thefollowing
onstraintsareimposedonthestatespae:
2n
11 +n
12 +n
13
C
UDL
(4)
n
12 +2n
22 +n
23
C
UDL
13 23 33 UDL n 12 C ISL (7) n 13 C ISL (8) n 23 C ISL (9)
Constraint(4)ensuresthatthenumberofallsoriginating(equivalently,terminating)atsatellite1isat
most equalto the apaity ofthe up-and-down link of that satellite. Note that a allthat originatesand
terminates within the footprintof satellite 1aptures two hannels, thus the term2n
11
in onstraint(4).
Constraints (5)and (6) are similar to (4), but orrespond to satellites2 and3, respetively. Finally,
on-straints(7)-(9)ensurethatthenumberofallsusingthelinkbetweentwosatellitesisatmostequaltothe
a-paityofthatlink. Notethat,beauseof(4)-(6),onstraints(7)-(9)beomeredundantwhenC
ISL C
UDL .
Inotherwords,thereisnoblokingattheinter-satellitelinkswhentheapaityofthelinksisatleastequal
to theapaityoftheup-and-downlinksat eah satellite 1
.
It isstraightforwardto verify that theMarkovproess forthe three-satellitesystemshownin Figure 1
hasalosed-formsolutionwhihisgivenby:
P(n) = P(n
11 ;n 12 ;n 13 ;n 22 ;n 23 ;n 33 ) = 1 G n11 11 n 11 ! n12 12 n 12 ! n13 13 n 13 ! n22 22 n 22 ! n23 23 n 23 ! n33 33 n 33 ! (10)
whereGisthenormalizingonstantand
ij =
ij =
ij
; i;j=1;2;3;istheoeredloadofallsfromsatellitei
tosatellitej. Asweansee,thesolutionistheprodutofsixtermsoftheform n
ij
ij =n
ij
!; i;j=1;2;3;eah
orrespondingto oneof thesix dierent typesof alls. Therefore, itis easily generalizableto a k-satellite
system,k>3.
AnalternativewayistoregardthisMarkovproessasdesribinganetworkofsixM/M/K/Kqueues,one
for eah typeofalls betweenthethree satellites. Sine thesatellitesdonotmove,thereare nohand-os,
and asa onsequeneustomers do not move from one queue to another (we will see in Setion 4.2 that
hand-osmaybemodeled byallowingustomersto movebetweenthequeues). Now,theprobabilitythat
there are n ustomers in an M/M/K/K queue isgivenby the familiar expression ( n =n!)= P K k =0 k =k! ,
andtherefore,theprobabilitythatthereare(n
11 ;n 12 ;n 13 ;n 22 ;n 23 ;n 33
)ustomersin thesixqueuesisgiven
by(10). Unlikepreviousstudiesreportedintheliterature,ourmodeltakesintoaountthefatthatthesix
M/M/K/K queuesare notindependent, sinethenumberofustomersaeptedin eahM/M/K/K queue
dependsonthenumberofustomersin otherqueues,asdesribedbytheonstraints(4)-(9).
Ofourse,themainonerninanyprodut-formsolutionistheomputationofthenormalizingonstant:
G = X n n11 11 n 11 ! n12 12 n 12 ! n13 13 n 13 ! n22 22 n 22 ! n23 23 n 23 ! n33 33 n 33 ! (11) 1
Whentherearemorethan threesatellitesinanorbit, allsbetweena numberofsatellitepairsmaysharea given
inter-satellitelink. Consequently,theonstraintsofak-satelliteorbit,k>3,orrespondingto(7)-(9)willbesimilartoonstraints
(4)-(6),inthattheleft-handsidewillinvolveasummationoveranumberofalls.Inthisase,blokingoninter-satellitelinks
mayourevenifC
ISL C
omputethenormalizingonstantGin aneÆientmanner.
WeanwriteP(n)as:
P(n 11 ;n 12 ;n 13 ;n 22 ;n 23 ;n 33
) = P(n
11 ;n 22 ;n 33 jn 12 ;n 13 ;n 23 )P(n 12 ;n 13 ;n 23 )
=P(n
11 jn 12 ;n 13 ;n 23 )P(n
22 jn 12 ;n 13 ;n 23 )P(n 33 jn 12 ;n 13 ;n 23 )P(n
12 ;n 13 ;n 23 )
=P(n
11 jn 12 ;n 13 )P(n 22 jn 12 ;n 23 )P(n
33 jn
13 ;n
23 )P(n
12 ;n 13 ;n 23 ) (12)
Theseondstepinexpression(12)isduetothefatthat,onethevaluesofrandomvariablesn
12 ;n 13 ;n 23 ,
representingthenumberofallsineahoftheinter-satellitelinks,isxed,thentherandomvariablesn
11 ;n 22 ; andn 33
areindependentofeahother(refer alsotoFigure1). Thethirdstepin(12)isduetothefatthat
randomvariablen
11
dependsonn
12 andn
13
,anditisindependentoftherandomvariablen
23
;similarlyfor
randomvariablesn
22 andn
33 .
Whenwexthevaluesoftherandomvariablesn
12 andn
13
,thenumberofup-and-downallsinsatellite
1isdesribedbyanM/M/K/Klosssystem,andthus:
P(n 11 jn 12 ;n 13 )= X
02n11CUD L n12 n13 n11 11 n 11 ! (13)
SimilarexpressionsanbeobtainedforP(n
22 jn
12 ;n
23
)andP(n
33 jn
13 ;n
23
),orrespondingtosatellites2
and3,respetively. Weannowrewriteexpression(11)forthenormalizingonstantasfollows:
G = X 0n 12 ;n 13 ;n 23 minfC UD L ;C ISL g n12 12 n13 13 n23 23 n 12 !n 13 !n 23 ! 2 4 0 X
02n11CUD L n12 n13 n11 11 n 11 ! 1 A 0 X
02n22CUD L n12 n23 n22 22 n 22 ! 1 A 0 X
02n33CUD L n13 n23 n33 33 n 33 ! 1 A 3 5 (14)
Let C = maxfC
ISL ;C
UDL
g. Using expression (14) we an see that the normalizingonstant an be
omputed inO(C 3
)timeratherthantheO(C 6
)timerequiredbyabruteforeenumerationofallstates,a
signiantimprovementin eÆieny.
Onethevalueofthenormalizingonstantisobtained,weanomputeblokingprobabilitiesbysumming
upalltheappropriateblokingstates. Considerthe3-satelliteorbitofFigure1. Theprobabilitythataall
whiheither originatesorterminatesat satellite1willbeblokedontheup-and-downlinkofthat satellite
isgivenby:
P UDL 1 = X 2n11+n12+n13=CUD L
P(n) (15)
whiletheprobabilitythataalloriginatingatsatellitei(orsatellitej)andterminatingat satellitej (ori)
P
ISLij =
0; C
ISL >C
UDL
P
n
ij =C
ISL
P(n); otherwise
(16)
One the bloking probabilities on all up-and-down and inter-satellite links have been obtained using
expressionssimilarto(15)and(16),theblokingprobabilityofallsbetweenanytwosatellitesanbeeasily
obtained. Wenotethatexpressions(15)and(16)expliitlyenumerateallrelevantblokingstates,andthus,
they involvesummations overappropriate parts of the state spae of the Markov proess for the satellite
orbit. Consequently, diret omputation of the link bloking probabilities using these expressions an be
omputationallyexpensive. Wehavebeenabletoexpresstheup-and-downandinter-satellitelinkbloking
probabilities in awaythat allows us to omputethese probabilitiesasa byprodut of the omputationof
thenormalizingG. Asaresult,allblokingprobabilitiesin asatelliteorbitanbeomputedinanamount
oftimethatisequaltothetimeneededtoobtainthenormalizingonstant,plusaonstant. Thederivation
of the expressions for the link bloking probabilities is a straightforward generalization of the tehnique
employedin (12)andis omitted.
3 A Deomposition Algorithm for the No Hand-Os Case
Let k be thenumberof satellitesin asingle orbit,and N bethenumber ofrandom variablesin the state
desriptionof theorrespondingMarkovproess, N =k(k+1)=2. Usingthemethod desribedabove,we
an omputethe normalizingonstant Gin time O(C N k
)asopposed to time O(C N
)needed bya brute
fore enumerationofallstates. Althoughtheimprovementintherunningtimeprovidedbyourmethod for
omputing Ginreases withk, thevalueof N will dominate forlargevalues ofk. Numerial experiments
withtheabovealgorithmindiatethatthismethodislimitedtok=5satellites. Thatis,ittakesanamount
oftimeintheorderofafewminutestoomputethenormalizingonstantGfor5satellites. Thus,adierent
method isneededforanalyzingrealistionstellationsofLEOsatellites.
Inthis setion we present amethod to analyzea singleorbitwith k satellites, k >5,by deomposing
the orbitinto sub-systemsof 3orfewersatellites. Eah sub-systemis analyzedseparately, and theresults
obtainedbythesub-systemsareombinedusinganiterativesheme.
In order to explain how the deomposition algorithm works, letus onsider the ase of a six-satellite
orbit,asshowninFigure2(a). Thisorbitisdividedintotwosub-systems. Sub-system1onsistsofsatellites
1,2,and3,andsub-system2onsistsofsatellites4,5,and6. Inordertoanalyzesub-system1inisolation,
weneedto havesomeinformationfrom sub-system2. Speially, weneed toknow theprobabilitythat a
alloriginatingatasatellitein sub-system1andterminating atasatellitein sub-system2will bebloked
duetolakofapaityinalinkinsub-system2. Also,weneedtoknowthenumberofallsoriginatingfrom
sub-system2andterminatinginsub-system1. Similarinformationisneededfromsub-system1,inorderto
analyzesub-system2.
In view of this, eah sub-system is augmented to inlude two titious satellites whih represent the
1
2
3
4
5
6
augmented sub-system 1
augmented sub-system 2
(b)
(a)
1
2
3
6
5
4
N1
S1
N2
S2
sub-system 2
sub-system 1
Figure2: (a)Original6-satellite orbit,(b)augmentedsub-systems
and S1,asshownin Figure2(b). Aall originatingat asatellitei;i=1;2;3,andterminating atasatellite
j;j =4;5;6,will be representedby aall from i to oneof the titioussatellites (N1 orS1). Depending
uponiandj,thisall mayberouteddierently. Forinstane,letusassumethati=2andj=4. Then,in
our augmentedsub-system 1,thisall will berouted tosatellite S1throughsatellite3. However,ifj =6,
the allwill be routed to satelliteN1 throughsatellite 1 2
. In other words, satelliteN1 (respetively, S1)
in theaugmentedsub-system 1is thedestinationfor allsof theoriginalorbitthatoriginatefrom satellite
i;i=1;2;3andareroutedtosatellitej;j=4;5;6inthelokwise(respetively,ounter-lokwise)diretion
inFigure2(a). Similarly,allsoriginatingfromsatellitej;j=4;5;6,tosatellitei;i=1;2;3,andareroutedin
theounter-lokwise(respetively,lokwise)diretion,arerepresentedinsub-system1asallsoriginating
fromN1(respetively,S1)toi. Again,theoriginatingsatellite(N1orS1)forthealldependsonthevalues
of iandj andthepaththeallfollowsintheoriginal6-satelliteorbit.
Sub-system2islikewiseaugmentedtoinludetwotitioussatellites,N2andS2(seeFigure2(b)),whih
representtheaggregatebehaviorofsub-system1. SatellitesN2andS2beometheoriginanddestinationof
allstravelingfromsub-system2tosub-system1,andvieversa,inamannersimilartoN1andS1desribed
above.
A summary of our iterative algorithm is provided in Figure 3. Below we desribe the deomposition
2
A 6-satellite orbit is deomposed into two 3-satellite sub-systems as in Figure 2. Sub-system 1 onsists
of satellites 1to 3in the original orbit plustitious satellitesN1 and S1, while sub-system 2onsists of
satellites4to 6oftheoriginalorbitplustitioussatellitesN2andS2.
1. begin
2. h 0 //Initializationstep
//p
ij
(h)istheprobabilitythataninter-sub-systemallwill beblokedinsub-system 1
//q
ij
(h) istheprobabilitythataninter-sub-systemallwillbeblokedin sub-system2
q
ij
(h) 0; 1i3<j6
3. h h+1 //h-thiteration
4.
ij
(h)
ij
; 1ij3 //Sub-system1
1;N1
=(1 q
16 )
16
+(1 q
15 )
15
1;S1
=(1 q
14 )
14
2;N1
=(1 q
26 )
26
+(1 q
25 )
25
2;S1
=(1 q
24 )
24
3;N1
=(1 q
36 )
36
3;S1
=(1 q
34 )
34
+(1 q
35 )
35
Solvesub-system1toobtainnewvaluesforp
ij (h) 5. ij (h) ij
; 4ij6 //Sub-system2
N2;4 =0
S2;4
=(1 p
14 )
14
+(1 p
24 )
24
+(1 p
34 )
34
N2;5
=(1 p
15 )
15
+(1 p
25 )
25
S2;5
=(1 p
35 )
35
N2;6
=(1 p
16 )
16
+(1 p
26 )
26
+(1 p
36 ) 36 S2;6 =0
Solvesub-system2toobtainnewvaluesforp
ij (h)
6. RepeatfromStep3untiltheblokingprobabilitiesonverge
7. end ofthealgorithm
ij
alls betweensatellitesiandj. Foranalyzingtheaugmentedsub-systemsin Figure2(b), wewillintrodue
the newarrivalrates
i;N1 , i;S1 , N2;j , and S2;j
, i =1;2;3,j =4;5;6. Speially,
i;N1
(respetively,
i;S1
)aounts for all alls between satellite i;i = 1;2;3, and a satellite in sub-system 2that are routed
in the lokwise (respetively, ounter-lokwise) diretion. Similarly,
N2;j
(respetively,
S2;j
) aounts
for all alls betweensub-system 1and satellite j;j =4;5;6that arerouted in thelokwise (respetively,
ounter-lokwise)diretion.
Initially,wesolvesub-system1in isolationusing:
1;N1
=(1 q
16 )
16
+(1 q
15 ) 15 (17) 1;S1
=(1 q
14 ) 14 (18) 2;N1
=(1 q
26 )
26
+(1 q
25 ) 25 (19) 2;S1
=(1 q
24 ) 24 (20) 3;N1
=(1 q
36 ) 36 (21) 3;S1
=(1 q
34 )
34
+(1 q
35 ) 35 (22) Quantityq ij
;1i 3<j 6, representstheurrentestimateof theprobability thata allbetween
a satellite i in sub-system 1 and and satellite j in sub-system 2 will be bloked due to lak of apaity
in a link of sub-system 2. For the rst iteration, we use q
ij
= 0 for all i and j; how these values are
updated in subsequentiterationswill bedesribed shortly. Thus, the term(1 q
16 )
16
in (17)represents
theeetive arrivalrateofalls betweensatellites1and 6,asseenbysub-system1;similarlyfortheother
termsin(17){(22).
The solutionto the rst sub-system yields an initial value for the probability p
ij
;1 i 3<j 6,
that aallbetweenasatelliteiin sub-system1andasatellitej insub-system2willbeblokedduetolak
of apaityin alink of sub-system 1. Therefore,theeetive arrivalratesof alls between,say, satellite1
and satellite4,that isoeredto sub-system2anbeinitially estimatedas(1 p
16 )
16
. Weannowsolve
sub-system2inisolationusing 3
:
N2;4
=0 (23)
S2;4
=(1 p
14 )
14
+(1 p
24 )
24
+(1 p
34 ) 34 (24) N2;5
=(1 p
15 )
15
+(1 p
25 ) 25 (25) S2;5
=(1 p
35 ) 35 (26) N2;6
=(1 p
16 )
16
+(1 p
26 )
26
+(1 p
36 ) 36 (27) S2;6
=0 (28)
3
In(23)wehavethat
N2;4
=0beauseweassumethatallsbetweensatellitesinsub-system1andsatellite4arerouted
S2;4
satelliteinsub-system1andsatellite4,asseenbysub-system2. Expressions(23){(28)anbeexplainedin
asimilarmanner. Thesolutiontotheseondsub-systemprovidesanestimateoftheblokingprobabilities
q
ij
;1i 3<j 6, that alls between satellitesin the twosub-systems will be bloked due to lakof
apaityin alink ofsub-system2.
The newestimates for q
ij
are then used in expressions (17) to (22) to update thearrival ratesto the
twotitioussatellitesofaugmentedsub-system1. Sub-system1isthensolvedagain,andtheestimatesp
ij
are updated and used in expressions (23)to (28) to obtainnew arrival ratesfor thetitious satellitesof
sub-system 2. This leadsto aniterativesheme, where thetwosub-systems aresolvedsuessivelyuntil a
onvergeneriterion(e.g.,in termsofthevaluesoftheallblokingprobabilities)issatised.
Orbitsonsisting of any numberk >5 of satellitesan be deomposed into anumberof sub-systems,
eahonsistingof3satellitesoftheoriginalorbit(thelastsub-systemmayonsistoffewerthan3satellites).
The deomposition method is similar to the oneabove, in that for sub-system l, the remaining satellites
are aggregated to two titious satellites. Eah sub-system is analyzed in suession as desribed above.
Thedeompositionalgorithmdesribedaboveissimilarin spirittothedeompositionalgorithmsdeveloped
for tandem queueing networks with nite apaity queues (see [13℄). We note that when employing the
deompositionalgorithm,theseletionofthesub-systemsizewilldependonthenumberofsatellitesin the
originalorbitandhoweÆientlyweanalulate theexatsolutionoftheMarkovproessassoiatedwith
eah sub-system. It is well known in deomposition algorithms that the largerthe individualsub-systems
that have to be analyzed in isolation, the better the auray of the deomposition algorithm. Thus, as
wementionedabove, wehavedeided to deompose anorbitinto sub-systems ofthe largestsize (threeof
theoriginalsatellitesplustwotitiousones)forwhihweaneÆientlyanalyzetheMarkovproess,plus,
possibly,asub-systemofsmallersize,ifthenumberofsatellitesisnotamultipleofthree.
4 Modeling Hand-Os
4.1 Earth-Fixed Coverage
Let us now turn to the problem of determining bloking probabilities in a single orbit of satellites with
earth-xed overage. Letk denote the number of satellites in the orbit. In this ase we assumethat the
earth is divided into k xed ells(footprints) and that time is dividedin intervals of lengthT suh that,
during agiveninterval,eahsatelliteservesaertainellbyontinuouslyrediretingitsbeams. Attheend
of eahinterval,i.e.,everyT timeunits,allsatellitessimultaneouslyredirettheirbeamstoservethenext
footprintalongtheirorbit,andtheyalsohand-ourrentlyservedallstothenextsatelliteintheorbit.
Wemakethefollowingobservationsaboutthissystem. Hand-oeventsareperiodiwithaperiodof T
timeunits,andhand-ostakeplaeinbulkattheendofeahperiod. Also,thereisnoallblokingdueto
hand-os,sine,ateahhand-oeventasatellitepassesitsallstotheonefollowingitand simplyinherits
period(approximately90minutes)dividedbythenumberofsatellites(e.g.,11fortheIridiumonstellation)
weanassumethat thesystemreahessteadystatewithin theperiod,andthus,theinitialonditions(i.e.,
thenumberofallsinheritedbyeahsatelliteatthebeginningoftheperiod)donotaetitsbehavior.
Now, sineeveryT units oftime, eah satelliteassumesthetraÆ arried bythe satelliteahead, from
thepointofviewofanobserverontheearth,this systemappearsto beasifthesatellitesarepermanently
xed overtheirfootprints. Hene,weanusethedeompositionalgorithm presentedaboveto analyzethis
system.
4.2 Satellite-Fixed Coverage
Consider now satellite-xed elloverage. As asatellite moves, its footprinton the earth (the ell served
by thesatellite) also moveswithit. As ustomers move outof thefootprint areaof asatellite, theiralls
are handed o tothe satellitefollowingitfrom behind. Inorder to model hand-osin thisase, wemake
theassumption thatpotentialustomers areuniformlydistributed overthepartof theearthservedbythe
satellitesintheorbit. Thisassumptionhasthefollowingtwoonsequenes.
Thearrivalrate to eah satelliteremainsonstantasit movesaroundtheearth. Then, thearrival
rateofallsbetweensatellitei andsatellitej isgivenby
ij =r
ij
,where r
ij
istheprobabilitythat
aalloriginatingbyaustomerservedbysatelliteiisforaustomerservedbysatellitej.
Theativeustomersservedbyasatelliteanbeassumedtobeuniformlydistributedoverthesatellite's
footprint. As aresult,therateofhand-osfromsatelliteitosatellitej that isfollowingfrombehind
isproportionaltothenumberofalls atsatellitei.
Clearly,theassumptionthatustomersareuniformlydistributed(evenwithinanorbit)isanapproximation.
InSetion6wewilldisusshowweareurrentlyextendingtheresultspresentedinthissetiontoaurately
modelthesituationwhenustomersarenotuniformly distributed.
LetAdenotetheareaofasatellite'sfootprintandvdenoteasatellite'sspeed. Asasatellitemovesaround
theearth,withinatimeintervaloflengtht,itsfootprintwillmoveadistaneofL,asshowninFigure4.
Callsinvolvingustomersloated in thepartof theoriginal footprintof areaA(the hand-oarea) that
is no longer served by the satellite are handed o to the satellite following it. Let A = AL, where
depends on the shape of thefootprint. Beause of the assumption that ativeustomers are uniformly
distributed overthe satellite'sfootprint, theprobability q that a ustomer will behanded o to the next
satellitealongtheskywithin atimeintervaloflengthtis
q = A
A
= L = vt (29)
Dene =v. Then,whentherearenustomersservedbyasatellite,therate ofhand-ostothesatellite
Satellite i
Time
∆
t
∆
L
Hand-off area,
∆Α
Figure 4: Calulationofthehand-oprobability
Let us now return to the 3-satellite orbit (see Figure 1) and introdue hand-os. This systeman be
desribedbyaontinuous-timeMarkovproesswiththesamenumberof randomvariablesasthe
no-hand-os model of Setion 2 (i.e., n
11 ;;n
33
), the same transition rates (2) and (3), but with a number of
additionaltransitionratestoaountforhand-os. Wewillnowderivethetransitionratesduetohand-os.
Consider alls between a ustomer served by satellite 1 and a ustomer served by satellite 2. There
are n
12
suh alls serving 2n
12
ustomers: n
12
ustomers on the footprint of satellite 1 and n
12
on the
footprint of satellite 2. Considera all betweenustomer A and ustomer B, served by satellite1 and 2,
respetively. Theprobabilitythat ustomer A will bein the hand-oareaof satellite1but Bwill notbe
in the hand-oareaof satellite 2 is q(1 q) =q q 2
. But, from (29), we havethat lim
t!0 q
2
t
=0, so
therateat whih theseallsexperieneahand-ofromsatellite1tosatellite3that followsitisn
12 . Let
n=(n
11 ;n
12 ;n
13 ;n
22 ;n
23 ;n
33
),anddene1
ij
asavetorofzeroesforallvariablesexeptvariablen
ij whih
is1. Basedontheabovedisussion, wethushave:
r(n;n 1
12 +1
23
) = n
12 ; n
12
>0 (30)
Similarly,theprobabilitythatustomerBwillbein thehand-oareaofsatellite2butAwillnotbein the
hand-o areaof satellite1is q(1 q)=q q 2
. Thus, the rateat whih these alls experiene ahand-o
from satellite2tosatellite1that followsitisagainn
12 :
r(n;n 1
12 +1
11
) = n
12 ; n
12
>0 (31)
Ontheotherhand,theprobabilitythatbothustomersAandBareinthehand-oareaoftheirrespetive
satellitesisq 2
are n
11
suhalls serving2n
1
1ustomers. Theprobabilitythatexatlyoneoftheustomersofaallisin
thehand-oareaofsatellite1is2q(1 q),sotherateatwhihtheseallsexperienehand-os(involvinga
singleustomer)tosatellite3is2n
11 :
r(n ;n 1
11 +1
13
) = 2n
11 ; n
11
>0 (32)
As before, the probability that both ustomersof the allarein thehand-oareaof satellite 1isq 2
, and
again,nosimultaneous hand-osareallowed.
Thetransitionratesinvolvingtheotherfourrandomvariablesin thestatedesription(1)anbederived
usingsimilar arguments. Forompleteness,thesetransitionratesareprovided in(33)-(38).
r(n;n 1
13 +1
12
) = n
13 ; n
13
>0 (33)
r(n;n 1
13 +1
11
) = n
13 ; n
13
>0 (34)
r(n ;n 1
22 +1
12
) = 2n
22 ; n
22
>0 (35)
r(n;n 1
23 +1
13
) = n
23 ; n
23
>0 (36)
r(n;n 1
23 +1
22
) = n
23 ; n
23
>0 (37)
r(n ;n 1
33 +1
23
) = 2n
33 ; n
33
>0 (38)
Fromthequeueingpointofview,thissystemisthequeueingnetworkofM/M/K/Kqueuesdesribedin
Setion 2,whereustomersareallowedtomovebetweenqueuesaordingto(30)-(38). (Reallthat in the
queueingmodelofSetion2,ustomersarenotallowedtomovefromnodetonode.) Thisqueueingnetwork
hasaprodut-form solutionsimilarto(10). Let
ij
denotethetotalarrivalrateofallsbetweensatellitesi
and j,inludingatarateof
ij
)andhand-oalls(arrivingat anappropriaterate). Thevaluesof
ij an
beobtainedbysolvingthetraÆequationsforthequeueingnetwork. Letalso
ij n
ij
bethedeparture rate
whentherearen
ij
ofthesealls,inludingalltermination(atarateof
ij n
ij
)andallhand-o(atarate
of 2n
ij
). Also,dene 0 ij = ij = ij
. Then, thesolutionforthisqueueingnetworkisgivenby:
P(n) = P(n
11 ;n 12 ;n 13 ;n 22 ;n 23 ;n 33 ) = 1 G ( 0 11 ) n11 n 11 ! ( 0 12 ) n12 n 12 ! ( 0 13 ) n13 n 13 ! ( 0 22 ) n22 n 22 ! ( 0 23 ) n23 n 23 ! ( 0 33 ) n33 n 33 ! (39)
whihisidentialto (10)exeptthat
ij
hasbeenreplaedby 0
ij .
Theprodut-formsolution(39)anbegeneralizedinastraightforwardmannerforanyk-satelliteorbit,
k>3.WeanthususethetehniquesdevelopedinSetion2tosolvethesysteminvolvinghand-osexatly,
or we an usethe deomposition algorithm presentedin Setion 3 to solve orbitswith alarge numberof
Inthissetionwevalidateboththeexatmodelandthedeompositionalgorithmbyomparingtosimulation
results. Inthegurespresented,simulationresultsareplottedalongwith95%ondeneintervalsestimated
bythemethodofrepliations. Thenumberofrepliationsis30,witheahsimulationrunlastinguntileah
typeofallhasatleast15,000arrivals. Fortheapproximateresults,theiterativedeompositionalgorithm
terminateswhenallallblokingprobabilityvalueshaveonvergedwithin10 6
.
For the results presented here we onsider three dierent traÆ patterns; similar results have been
obtained for several other patterns. Let r
ij
denote the probability that a alloriginating by austomer
servedbysatelliteiisforaustomerservedbysatellitej. TherstpatternisauniformtraÆpatternsuh
that:
r
ij =
1
k
8 i;j (uniformpattern) (40)
where k is the number of satellites. The seond is a pattern based on the assumption of traÆ loality.
Speially,itassumesthat mostallsoriginatingatasatelliteiaretousersinsatellitesi 1,i,andi+1,
where additionandsubtrationismodulo-kforak-satelliteorbit:
r
ij =
(
0:3; j =i 1;i;i+1
0:1
k 3
; j 6=i 1;i;i+1
(loalitypattern) (41)
Thethird patternissuhthatthere aretwoommunitiesofusers,andmosttraÆisbetweenuserswithin
agivenommunity(e.g., satellitesoverdierenthemispheresoftheearth):
r
ij =
(
0:8
k =2
; i;j =1;;k=2; ori;j=k=2+1;;k (2 ommunity pattern)
0:2
k =3
; i=1;;k=2;j=k=2+1;;korj=1;;k=2;i=k=2+1;;k
(42)
5.1 Validation of the Exat Model
InthissetionwevalidatetheexatMarkovproessmodelforthenohand-osasedevelopedinSetion2.
ReallthatweandiretlyomputethenormalizingonstantGusingexpression(14)fororbitsofuptove
satellites. Thus,weomparethebloking probabilityvaluesobtainedbysolvingtheexatMarkovproess
to simulationresultsfora5-satellite orbitandthethreetraÆpatternsdisussedabove.
Figure5plotstheblokingprobabilityagainsttheapaityC
UDL
ofup-and-downlinks,whenthearrival
rate =10 andthe apaityof inter-satellitelinks C
ISL
=10, fortheuniform traÆ pattern. Three sets
of plots are shown: one for alls originating and terminating at the same satellite (referred to as \loal
alls" in thegure),oneforalls travelingoverasingleinter-satellitelink,and oneforalls travelingover
twointer-satellite links 4
. Eahset onsists of two plots,oneorresponding to bloking probability values
obtainedbysolvingtheMarkovproess,andoneorrespondingtosimulationresults.
From the gure, we observe that, as the apaity C
UDL
of up-and-down links inreases, the bloking
probability of all alls dereases. However, for alls traveling over at least one inter-satellite link, the
4
Thesearetheonlypossibletypesofallsina5-satelliteorbitandshortestpathrouting. Furthermore,beauseofsymmetry,
10
15
20
25
30
35
40
45
50
10
−4
10
−3
10
−2
10
−1
10
0
UDL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analysis
1 ISL hop, Simulation
1 ISL hop, Analysis
2 ISL hops, Simulation
2 ISL hops, Analysis
Figure5: Callblokingprobabilitiesfora5-satellite orbit,=10,C
ISL
=10, uniformpattern
10
15
20
25
30
35
40
45
50
10
−2
10
−1
10
0
ISL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analysis
1 ISL hop, Simulation
1 ISL hop, Analysis
2 ISL hops, Simulation
2 ISL hops, Analysis
Figure 6: Callblokingprobabilitiesfora5-satelliteorbit,=10,C
UDL
=20,uniformpattern
10
15
20
25
30
35
40
45
50
10
−4
10
−3
10
−2
10
−1
10
0
UDL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analysis
1 ISL hop, Simulation
1 ISL hop, Analysis
2 ISL hops, Simulation
2 ISL hops, Analysis
Figure7: Callblokingprobabilitiesfora5-satelliteorbit,=5,C
ISL
10
15
20
25
30
35
40
45
50
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
UDL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analysis
1 ISL hop, Simulation
1 ISL hop, Analysis
2 ISL hops, Simulation
2 ISL hops, Analysis
Figure8: Callblokingprobabilitiesfora5-satelliteorbit,=10, C
ISL
=10,loalitypattern
10
15
20
25
30
35
40
45
50
10
−4
10
−3
10
−2
10
−1
10
0
UDL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analysis
1 ISL hop, Simulation
1 ISL hop, Analysis
2 ISL hops, Simulation
2 ISL hops, Analysis
Figure9: Callblokingprobabilitiesfora5-satelliteorbit,=10,C
ISL
=10, 2-ommunitypattern
bloking probability urve attens out after an initial drop. This behavior is due to the fat that, for
smallvaluesofC
UDL
,theup-and-downlinks representabottlenek,thus,inreasingC
UDL
reduestheall
blokingprobabilitysigniantly. However,one C
UDL
inreasesbeyondaertainvalue,theinter-satellite
links beomethebottlenek,andtheblokingprobabilityofallsthathavetotravelovertheselinks isnot
aetedfurther. Ontheotherhand,theblokingprobabilityofallsnotusinginter-satellitelinks(i.e.,those
originatingandterminatingatthesamesatellite)dereasesrapidlyasC
UDL
inreases,droppingtozerofor
valuesC
UDL
>30(beauseofthelogarithmisale,valuesofzeroannotbeshowninFigure5,sothereare
novaluesplottedwhenC
UDL
>30fortheurvesofloal alls).
Figure 6plots thebloking probability for thesamealls asin Figure 5, againstthe apaity C
ISL of
inter-satellite links;fortheresultspresentedweassumethat =10andC
UDL
=20. Inthisgurewean
see thatasthevalueofC
ISL
inreases,theblokingprobabilityofallsusinginter-satellitelinksdereases,
ISL
explainedbynotingthat,asC
ISL
inreases,alargernumberofnon-loalalls(i.e.,allsusinginter-satellite
links) isaepted(sinetheirblokingprobabilitydereases). Sinebothloal andnon-loal allsompete
for up-and-down links,an inreasein thenumberof non-loal alls aeptedwill result in higherbloking
probabilityforloal alls. Butwhenthevalueof C
ISL
exeedsthevalueofC
UDL
(whih isequalto 20in
this ase), the up-and-down links beome the bottlenek, and further inreasesin C
ISL
haveno eet on
blokingprobabilities.
Figure 7issimilar to Figure5exept that thearrivalrateis =5insteadof 10(allother parameters
areasin Figure5). Thebehaviorofthevariousurvesissimilarto thatinFigure5. Themain diereneis
that theblokingprobabilitiesinFigure7aresigniantlylower,aresultthat isexpeteddueto thelower
arrivalrate.
Finally, Figures 8 and 9 show results for the same parameters as in Figure 5, but orrespond to the
loality and 2-ommunitytraÆ patterns, respetively. Again, thebehaviorof theurvesissimilar for all
three gures,although theatualblokingprobabilityvaluesdependonthetraÆpatternused.
TheresultsinFigures5{9illustratethefatthattheblokingprobabilityvaluesobtainedbysolvingthe
Markovproessmaththesimulationresults;thisisexpetedsinetheMarkovproessmodelwedeveloped
isexat. Thus,thismodelanbeusedtostudytheinterplaybetweenvarioussystemparameters(e.g.,C
ISL ,
C
UDL
, traÆpattern, et.) and theireet onthe allbloking probabilities, in aneÆientmanner. We
notethatsolvingtheMarkovproesstakesonlyafewminutes,whilerunningthesimulationtakesanywhere
between30minutesandseveralhours,dependingonthevalueofthearrivalrates.
5.2 Validation of the Deomposition Algorithm
WenowvalidatethedeompositionalgorithmdevelopedinSetion3byomparingtheblokingprobabilities
obtainedbyrunningthealgorithmtosimulationresults. Weonsiderasingleorbitofasatelliteonstellation
onsistingof12satellites,anumberrepresentativeoftypialommerialsatellitesystems. Inallasesstudied,
wehavefoundthat thealgorithms onvergesin onlyafew(less thanten) iterations, takingafewminutes
toterminate. Ontheotherhand,simulationof12-satelliteorbitsisquiteexpensiveintermsofomputation
time, takingseveralhourstoomplete.
Figure 10 plots the bloking probability against the apaity C
UDL
of up-and-down links, when the
arrivalrate =5 and the apaityof inter-satellite links C
ISL
=20, for theuniform traÆ pattern. Six
sets of alls are shown, oneforloal alls,and vefor non-loal alls. Eahset onsistsof twoplots, one
orrespondingtoblokingprobabilityvaluesobtainedbyrunningthedeompositionalgorithmofSetion3,
and one orresponding to simulationresults. Eah non-loal all for whih resultsare shown travelsover
a dierent number of inter-satellite links, from one to ve. Thus, the results in Figure 10 represent alls
betweenall the dierent sub-systems in whih the 12-satellite orbitis deomposed by the deomposition
algorithm.
UDL
lessthan 20, these links representabottlenek. Thus, inreasingthe up-and-downlink apaityresultsin
a signiant drop in the bloking probability for allalls. When C
UDL
>20, however, theinter-satellite
links beome thebottlenek, and non-loal alls donotbenet from further inreasesin theup-and-down
link apaity. Wealsoobservethat,thelargerthenumberofinter-satellitelinksoverwhihanon-loalall
must travel,thehigheritsblokingprobability,asexpeted. Theblokingprobabilityof loalalls,onthe
otherhand,dropstozeroforC
UDL
>20sinetheydonothaveto ompeteforinter-satellitelinks.
Figures 11 and 12 are similar to Figure 10 but show results for the loality and 2-ommunity traÆ
patterns, respetively. For theresultspresentedweused =5andC
ISL
=10, andwevariedthevalueof
C
UDL
. We observethat the valuesof the allblokingprobabilities depend on theatual traÆ pattern,
but thebehaviorofthevariousurvesissimilartothatinFigure10. Finally,inFigure13,wexthevalue
of C
UDL
to 20, andweplotthe all bloking probabilitiesfor the2-ommunity traÆ pattern againstthe
apaityC
ISL
oftheinter-satellitelinks.
Overall,theresultsinFigures10{13indiatethatanalytialresultsareingoodagreementwithsimulation
overawiderangeoftraÆpatternsandsystemparameters. Thus,ourdeompositionalgorithmanbeused
to estimateallblokingprobabilitiesinLEOsatellitesystemsinaneÆientmanner.
6 Conluding Remarks
We have presented an analytial model for omputing bloking probabilities for a single orbit of a LEO
satelliteonstellation. Wehavedevised amethod forsolvingtheexatMarkovproesseÆientlyforupto
5-satellite orbits. Fororbitsonsistingof alargernumberof satellites,wehavedeveloped anapproximate
deompositionalgorithmtoomputetheall blokingprobabilitiesbydeomposingthesystemintosmaller
sub-systems, anditerativelysolvingeahsub-systemin isolationusing theexatMarkovproess. Wehave
also shownhowourapproahanaptureblokingdueto hand-osforbothsatellite-xedandearth-xed
orbits.
Weareurrentlyextendingthedeomposition algorithmto handleanentire satelliteonstellation.
As-sumingthattheonstellationonsistsofRorbits,anaturalapproahistodeomposeitintoRsub-systems,
eah representing a single orbit. We are also planning to analyzethe ase of heterogeneous traÆ. One
approah to aount fordierentgeographi arrivalrates, isto segmentthe band of earth overedby the
satellites into xed regions, eah with a dierent arrival rate of new alls. This approah givesrise to a
10
15
20
25
30
35
40
45
50
10
−3
10
−2
10
−1
10
0
UDL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analytical
1 ISL hop, Simulation
1 ISL hop, Analytical
2 ISL hops, Simulation
2 ISL hops, Analytical
3 ISL hops, Simulation
3 ISL hops, Analytical
4 ISL hops, Simulation
4 ISL hops, Analytical
5 ISL hops, Simulation
5 ISL hops, Analytical
Figure10: Callblokingprobabilitiesfora12-satelliteorbit,=5,C
ISL
=20,uniformpattern
10
15
20
25
30
35
40
45
50
10
−3
10
−2
10
−1
10
0
UDL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analytical
1 ISL hop, Simulation
1 ISL hop, Analytical
2 ISL hops, Simulation
2 ISL hops, Analytical
3 ISL hops, Simulation
3 ISL hops, Analytical
4 ISL hops, Simualtion
4 ISL hops, Analytical
5 ISL hops, Simulation
5 ISL hops, Analytical
Figure11: Callblokingprobabilitiesfora12-satelliteorbit,=5,C
ISL
=10,loalitypattern
10
15
20
25
30
35
40
45
50
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
UDL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analytical
1 ISL hop, Simulation
1 ISL hop, Analytical
2 ISL hops, Simulation
2 ISL hops, Analytical
3 ISL hops, Simulation
3 ISL hops, Analytical
4 ISL hops, Simulation
4 ISL hops, Analytical
5 ISL hops, Simulation
5 ISL hops, Analytical
Figure12: Callblokingprobabilitiesfora12-satelliteorbit,=5,C
ISL
10
15
20
25
30
35
40
45
50
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
ISL Capacity
Blocking Probability
Local Calls, Simulation
Local Calls, Analytical
1 ISL hop, Simulation
1 ISL hop, Analytical
2 ISL hops, Simulation
2 ISL hops, Analytical
3 ISL hops, Simulation
3 ISL hops, Analytical
4 ISL hops, Simulation
4 ISL hops, Analytical
5 ISL hops, Simulation
5 ISL hops, Analytical
Figure13: Callblokingprobabilitiesfora12-satelliteorbit,=5,C
UDL
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