MODELING COMMUNICATIONS IN

LOW-EARTH-ORBIT SATELLITE NETWORKS

Peter Gvozdjak

31-Sc., Comenius University: Slovakia, 1993

A THESIS SUBMITTED [il; PARTIAL FLLFILLMENT

OF THE REQUIREMENTS FOR T H E DEGREE OF

DOCTOR O F PHILOSOPHY

in t h e School

of

Cornput ing Science

@ Peter Gvozdjnk '2000

SIMON FR-ASER UNIVERSITY-

-4ugust 2000

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Abstract

This thesis studies cornniunications in low-earth-orbit satellite networks. It develops

simple analyt ical models of networks formed by interconnect ing the satellites. T hen

it uses the models to stud'; efficient communication schemes in these networks.

The use of wireless communications has increased rapicfly over the p s t years. Since

t lie cellular systems are often cost-prohibit ive in sparselj. populated areas, several

consortia are involved in building alternative systems that use low or medium orbit

satellites. Compared to geostationary orbi*, the low orbits achieve smaller delay but

result i n smaller coverage areas recluiring a large n ~ ~ m b e r of sateIlites. The satellites

are in constant motion. which results in high variability in the networks.

This t hesis studies the communication networks t hat result from interconnecting

Iotv-earth-orbit satellites in inclined orbits. The first part of the ttiesis focuses on

mocleling two aspects of the networks-the network topology and the transmission

clelay. It provicles an estensive numerical stucly of the impact of \-arious parameters

on the delay between both directly and indirectly connected satellites. It shows that,

within a certain class of topologies. a new network topology callecl a skewed torus

minimizes the clelay within the network. Due to the complex formula for the exact

value of intersatellite delay, it also introdiices two approximations callecl the linenr

npproxirnnt ion ancl the constant npprosimnf ion. It discusses the clist inct ion between

the links connecting satellites on the same orbit and on different orbits, and proposes

the use of the constant approximation for interorbital links resulting in a simple two-

ungbrnz mode/.

The second part of the t hesis develops efficient communication algorit hms for

toroidal networks under the two-uniform model. The fociis is on all-to-al1 exchange

(gossiping) algorit hms. The t hesis develops algori t hms under tmo transmission-cost

models: the constan t-cost rnodel t hat considers propagation del- onl- and the linear-

cost ntodel that considers both propagation delay and data rate. .An algorithm that

ut ilizes the overlap between propagation delay and transmission t ime on different

links is developed for the linear-cost rnodel. The algorithm improves the time of the

best-known algorithm in the special case of a one-uniform regular torus (a11 links have

the same paramet ers).

Acknowledgment s

First 1 want t o thank J o e Peters, my senior supervisor. I a m deeply thanlifnl for his

help, encouragement. ancl advice throughout my ivhole work on t his thesis. He was

always able t o meet me whenever 1 needed to talk. ancl t he cliscussions were always

very fruitf~il. 1 woulcl also like t o thank rny supervisors Steve Hardy and Tiko Iiarnecla

for their help and suggestions on the improvements of this thesis. Further, 1 ivould

like to espress my gratitucle t o Afonso Ferreira, the eaternal examiner of this thesis.

for the valuable comments and suggestions he made.

1 am also grateful t o the Simon Fraser University for financial support in the form

of scholarships and fello\vships. Joe Peters for siipport in t he form of research assis-

tantships. and the School of Compiiting Science for support in the form of teaching

assistantships. .AdclitionalIy, 1 would like to thank the School of Computing Science.

and al1 i ts faciilty and staff. for providing an excellent stucly ancl research environment.

Last but not ieast 1 woulcl like to thank the Geometry Center. University of Min-

nesota for t he visualization software they clevelopeci ancl macle availablc for public

use. Figures 2.1 and 2.5 iwre generatecl mith SaVi, software written a t the Geometry

Center. University of Minnesota (http://ivr~~iv.geom.umn.ec~u/~~~~orfolk/SaVi/).

Contents

Approval

Abstract

Acknowledgments

List of Tables

List of Figures

. . . Vl l l

List of Symbols xv

1 Introduction 1

1.1 SatelIite cornm~inications . . . . . . . . . . . . . - . . . . . . . . . . . :3

1.1.1 Cr EO satellites . . . . . . . . . . . . . . . . . . . . . . . . .- . . .5

1.1.2 L E 0 satellites . . . . . . . . . . . . . . . . . . . . - . . . . . . 6 -

1.2 Building a LE0 satellite network . . . . . . . . . . - - . . . . . . . . 1

1 .:3 L E 0 constellation esamples . . . . . . . . . . . . . - . . . . - . . . . 9

1.4 Ootline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Related work 12

2.1 L E 0 satellite networks . . . . . . . . . . . . . . - . - . . . . . . . . . 12

2.1.1 c!onstellations and topology . . . . . . . . . . . . . . . . . . . 13

2.1.2 Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Communications in toroidal ineshes . . . . . . . . . . . . . . . . . . . 21

. . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Constant-costmodel 23

2 - 2 2 Linear-cost mode1 . . . . . . . . . . . . . . . . . . . . . . . . . 25

I Models of physical network properties

3 Network topoloa~ 29

. . . . . . . . . . . . . . . . . . . . . 3.1 Geometry of intersatellite links 130

. . . . . . . . . . . . . . . . . . . . . . . 3.2 Length of intersatellite links :34

. . . . . . . . . . . . . . . . . . . . . . . . . . 13-13 S kewed toms topology 41

. . . . . . . . . . . . . . . . . . . . . . . . 3.4 Lengtli of end-to-end paths 45

4 Modeling intersatellite links 56 " . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Modeling link lengths -31

4.2 T~vo-uniform mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

. . . . . . . . . . . . . . . . . . . . . 4.3 Modeling communication modes 6.5

II Coinniiiiiicat ion algorit hms

5 Constant-cost two-uniform toroidal meshes 67

. . . . . . . . . . . . . . . . . . . . . . . 1 Ciossiping in a regular torils 68

. . . . . . . . . . . . . . . . . . . . . . . 5.2 Chxsiping in a skewecl torus 71

6 Linear-cost two-uniform toroidal meshes 82

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Lower bouncls S3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 -411-port moclel 86

. . . . . . . . . . . . . . . . . . . . 6.2.1 Basic gossiping slgorit hm 86

. . . . . . . . . . . . . . . . . . . 6.2.2 Refined gossiping algorithm 108

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 One-port moclel 124

7 Conclusion and further research 129

Bibliography 132

List of Tables

1.1 The footprint size: orbit periocl P , and propagation delays as a f~mc-

tion of the altitude H . The footprint size measured by its haif-sided

center angle d. depends also on the minimum elevation angle <,in. The

horizontal lines separate the approximate ranges of the low-enrth-orbit

(LEO), medium-enrth-orbit (MEO), and high-earth-orbit satellites. The

last two columns show the round-trip earth-satellite-eart h delay and the

clelay between two opposite satellites: respectively. . . . . . . . . . . 5

i3.1 The value of skew per orbit minimizing link lengt hs for given number

of orbits nh and satellites per orbit n, for constellations with inclination

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5". 34

6.1 The r~inning times achievecl using the methocl of the proposed hl l -

duplex gossiping algorithm compared to the bouncl (6.19) for a sample

set of network parameters. . . . . . . . . . . . . . . . . . . . . . . . 1 Z 3

3.19 T h e relative difference between average path length for the skew opti-

mizifig the intersatellite link length and its optimal value. T h e inclina-

tion is a0 = 6.5". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.20 The percentage of cases when minimizing link length does not minimize

the maximum a n d average pat h lengths for nh a n d n, multiples of 5 in

the range 5 t o :30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3-21 The highes t relative difference of the maximum a n d average pat h lengt hs

between t h e values obtained for the skew minimizing the intersatellite

link length and their optimal values. The values of nh and n , are

multiples of 5 in the range .5 to 130. . . . . . . . . . . . . . . . . . . . .Y3

The linear and constant approximations of interorbit al link lengt hs.

Inclination is a0 = 65", n, = S, A 9 = -19'. t h e al t i tude is 1400 km. 5 S

The relative error of the average path lengt h for the linear approxima-

tion. T h e incIination is a0 = 65". t he orbits have eclual phasing, and

the skew per orbi t ko minimizes the intersatellite link lengths. . . . . 59

The relative error of the average pat h length for the constant approx-

imation. T h e inclination is no = 65". the orbits have ecliial phasing.

and the skew per orbit Iro minimizes the intersatellite link lengths. . BO

The relative error of t h e average path length for t h e linear ancl constant

approximatioris as a function of the inclination. T h e values were taken

over a range of values nh and n , . T h e skew per orbit ko minimizes the

intersatellite link lengt hs. . . . . . . . . . . . . . . . . . . . . . . . . 61

The relative error of the masimum path lengt h for the linear ancl con-

stant approximations as a f~inctiori of the inclination. T h e values were

taken over a range of values nh ancl n . . The skew per orbit ko rninimizes

the intersatellite link lengths. . . . . . . . . . . . . . . . . . . . . . . 61

The relative error of the average path lengt h for t h e linear and constant

approximations as a fiinction of the t ime instant. T h e values were taken

over a range of values n h and n,. T h e inclination is a0 = 6s0, and the

skew per orbit minimizes the intersatellite link lengths. . . . . . . 62

The relative error of the maximum path Iength for the Iinear and con-

stant approximations as a function of the time instant. The values were

taken over a range of values nh and n,. The inclination is a0 = 6 j 0 7

and the skew per orbit ko rninimizes the intersatellite link tengths. . -4 t wo-uniform rect angular mes h. . . . . . . . . . . . . . . . . . . . .

Gossiping in a n x 2 torus. (a) initial exchange, (b) forivarcling mithin

cycles. ( c ) final exchange; the messages received by one pair of nodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . are highlighted.

Wrap-around connections of a skewecl toruç modeled by multiple copies

of its nodes placed on a plane. . . schematic shortest path routing from

a node s for the case nhh > kv is also shown. . . . . . . . . . . . . . The general scheine of the message eschange algorithm. The links are

labeled by the times a transmission starts on t hem. . . . . . . . . . . The potential start times trr(2r. 0) for a ti.ansmission from nocle (Lr , O)

in the iip clirection. The shadecl time dots cause a conflict \vith a

transmission in the (a) right and ( b ) left clirection. The parameters are

h = 1. L : = 3 - , a = S . . . . . . . . . . . . . . . . . . . . . . - . . . . . -

The worst-case for the path from n: to y. . . . . . . . . . . . . . . . . The seams formed by the wrap-around of a skewed toms when nh. n .

ancl the skew k are even. . . . . . . . . . . . . . . . . . . . . . . . . . The vertical seam when the skew k is odd, nh? n,, are even. . . . . . .

The time instants that cause a conflict in the aspchronoiis mode ivith

a horizontal transmission in the (a) right and (b) left clirection. . . . .

One transmission step transrnitting a packet of size L over a link with

parameters ,9, T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -An esample of a diagram showing the origiris (ciïcles O) of the messages

known to the highlighted nocle (solid circle) at a given tirne instant.

h.Iessageç known to two horizontal neighbors .zr (crosses x ) and y f circles

O). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The s successive transmission steps performed by one link, . . . . . .

The order of the messages that are received by the highlighted node

a t the beginning of stage one through its four adjacent links-Left.

Right. Top. Bottom. The parameters are gh = 3, = TA = r, = 1.

Each differently shaclecl area represents the messages received during

the t ime taken by one horizontal step. Note that the horizontal links

transmit more t han one message simultaneously. . . . . . . . . . . .

The timing of the horizontal and vertical transmissions in stage one of

. . . . . . . . . . . . . . . . . the algorit hm. ,dh = L?, = ~h = T ~ , = 1.

The shaded area represents the messages received by the hiphlighted

node by the end of the first stage of the algorithm. Note that it shows

the set of the messages received, not the data paths traversed by the

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . messages.

The origins of the messages the highlightecl node is aware of a t the end

of stage two. The lightly shaded area represents the messages learnecl

during stage one: and the darker shacled area the messages learnecl

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . during stage two.

The origins of the messages the highlightecl nocle is aware of a t the encl

of stage three. The darkest areas represent the messages learnecl during

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the third stage.

-4 schematic clepiction of a possible transrriission arrangement for stage

four. The horizontal lines represent the origins of niessages sent to

the highlighted nocle t hrough its horizontal neighbors, ancl the vertical

lines the origins of the messages sent Ihrough its vertical neighbors.

The shacled areas represent the origins of the messages the highlightecl

node learnecl during the first three stages. . . . . . . . . . . . . . . . The reason the vertical links may be iclle at the beginning of stage two.

The circles O clenote the origins of the messages that r sends to y, the

crosses x the origins of the messages that .E cari sencl to y. . . . - . .

The modification of the lwt step of stage one to recluce the idle tirne of

vertical links. x denotes the origins of the messages the node r sends

to y through the connecting vertical link at the beginning of stage t ~ o .

... S l l l

6.13 The messages learned by the highlighted node (circles O) and its bottom

neighbor (crosses x ) through the gossips of stage one and tmo of the

all-port half-duplex algorit hm. . . . . . . . . . . . . . . . . . . . . . 1 19

6.11 The tree that one message is broadcast along during the gossips of the

first two stages of the ail-port half-duplex algorithm. . . . . . . . . . 1.20

6.1.5 The origins of the messages known to the highlighted node after the

first three stages of the all-port half-duplex algorit hm. Each differently

sliaded area represents the origins of the messages learned diiring a

different stage of the algorithm. . . . . . . . . . . . . . . . . . . . . . 121

6.16 The broadcast tree for one message in the gossiping algorithm for a

. . . . . . . . . . . . . . . . . . . . . . . one-port full-duplex torus. 125

xiv

List of Symbols

inclination

instant aneous latitude

propagation delay of a link

longitude of a descending node; angle betmeen twvo descending nodes

inst antaneous longitude

propagation clelay of a horizontal link

propagation del- of a vertical link

diameter of a network

minimum length of an interorbital link

maximum length of a n interorbital link

length of a linli at time t

linear approximation of a link length

constant approximation of a link lengtli

elevat ion angle

grai-itat ional constant

angle bet~veen twvo satellites

greatest common divisor

propagation clelay of a horizontal link for the constant-cost mode1

altitude

skew of a torus

skew per orbit

L

lcm

hl

!Ir

message size

least cornmon multiple

earth's mass

nurnber of nodes in a network

nurnber of orbitsr horizontal dimension of a torus

number of sat,ellites per orbit: vert.ica1 dimension of a torus

orbit period

phase of a satellite/ link

phase ofTset of an intersatellite link

phase offset of interorbital links

phase offset of intraorbital links

optimum phase offset of an interorbital link

half-sided center angle of a footprint

radius of the earth

number of horizontal transmission steps

n~irnber of vertical transmission steps

total running tinie of an algorithm

time a horizontal link is idle cluring an algorithm

time a vertical link is idle during a n algortihm

time to transmit one unit of data

time to transmit one unit of data on a horizontal link

time to transmit one unit of data on a vertical link

propagation del- of a vertical link for the constant-cost niocle1

svi

Chapter 1

Introduction

In recent years? a new phenornenon in the area of persona1 communications-wireless

comm~inications-has been introduced. Wireless telephone systems give their users

the opportiinity to use their phones oiitside of their homes and/or offices. e.g.. while

visiting another city or traveling on a highway. Esamples of the benefits of the possi-

bility to reach or be reachable by other people almost any t ime are increased business

procluctivity due to recluced tirne overhead and contacting ernergency services in the

case of an acciclerit. The traditional phone service is being augmented by da ta ser-

vices ancl lnternet connectivity making it possible t o read and write electronic mail

messages. to connect persona1 cligital assistants to large ch t a servers, or to bsowse the

information on the WorId Wide Web without the need for a fised. wirecl connection.

Most of the briilt systems are based on cellular phone technology that ernploxs rel-

atively regularly-spacecl base stations to wliich t he mobile phones can connect. The

major feature of wireless communications? the mobility of users, introdtices variability

into the network in both variable delay and variable topology as mobile phones recon-

nect to a new base station cluring a hancl-over. The cellular systems are successfiil in

highly populated areas or along frequently travelecl highways but tliey do have their

limitations. C:~irrently there is no globally adoptecl cellular phone standard meaning

t hat the use of cellular phones is limitecl to a certain country or a region. Furt hermore,

more remote areas, like Canada's northern parts, a re often cost-prohibitive to cover

dile t o the low density of population and potential subscribers. In order to provicle a

truly worldwide connectivity to the subscribers. an alternative to earth-based cellular

systems is the employment of satellites wit h much larger coverage areas.

Geostationary (GEO) satellites a t high altitudes have been used for telecommu-

nication services for several decacles. The high altitude orbits (approximately 36.000

km above the ecliiator) are chosen so that the satellites remain stationary relative to

the surface of the earth. The high orbits permit large footprints which cover large ar-

eas, but t h e associated long delays of more than 230 msec. for an earth-satellite-earth

round trip. and high power recplirements. restrict their uses.

In recent years, there have been several proposals to use networks of satellites

in low-eart h-orbi ts (LEO) for communications. -4 big advantage of LE0 satellite

networks over G E 0 satellites is miich smaller delay. LE0 satellite orbits are less than

1.500 km ahove the earth and typical earth-satellite-earth round trip delays are less

than 10 msec. Hoivever, the lower altitudes also introdiice disadvantages incliiding

much shorter orbit periods, typically a few heurs: and smaller footprints. This resdts

in constant motion of the satellites with respect to the earth's surface. and the neecl

for more satellites to provicle fiill coverage of the earth. LE0 satellites can be iisecl to

supplement terrestrial networks by providing links between points on the surface that

ivoulcl be clifficuit or too espensive to connect with terrestrial links. -4 more ambit ious

approach is to replace large parts of terrestrial networks by iising intersatellite links

to interconnect a number of satellites into a L E 0 satellite network. The clesign and

use of such a network involves some interest ing challenges. some of rvhich are explorecl

in this thesis.

The remaining parts of this chapter discuss general issues in satellites. and low-

eart h-orbit satellite communications. Section 1.1 disc~lsses sateIlite cornmunications

in general. Some issites relatecl to building LE0 satellite networks are coverecl in

Section 1.2. while Section 1.3 provicles some examples of L E 0 satellite networks t hat

were proposed ancl/or built. Section 1.3 provides an outline of the thesis with a

summary of its main contributions.

1.1 Satellite communications

This section provides a general introduction to the use of satellites for communication

purposes. It covers the basic geometry of satellite coverage, namely. the size of the

coverage area. orbit period, and propagation clelay. Shen it compares geostat ionary

satellites to medium and low-earth-orbit satellites.

\.\-e start wit h general geomet ry of satellite coverage following the paper by Werner

et al. [ilï]- The geornetry is shown in Figure 1.1. The two main parameters deter-

Earth station

Satellite

Earth's center

Figure 1.1: The geornetry of a satellite's footprint.

mining the size of the area covered by one satellite, the so-called footyrinf, are the

C'HA P TER 1. IXTR OD uC2"OiV

altitude H of the satellite and the minimum elevation angle Emin- The angle E,, is

the minimum angle between a tangent to the earth's surface at a covered point and the

satellite. The smaller the angle, the larger the attenuation of the signal between the

satellite and the covered point due to the eartli's atmosphere. Therefore, to rninimize

the power requirement, one would wish to use as Iarge an elevation angle as possible.

However. t hat limits the size of the covered area. To see the impact. we will measure

its size bjr the half-sided center angle ~ of the footprint. i.e.. the angle between the

nmst distant covered point and the satellite as seen from the center of the eart h (see

Figure 1.1). The relationship between the minimum elevation angle and $ is given by

where R is the radius of the earth. Table 1.1 s h o ~ s the values of ~ for sorne sample

values of <,in and H.

The orbi t periocl ancl propagation delay clepend on the satellite's altitude as well.

The orbit period P is given by the following formula [-21:

ivhere 11 = 398, 600..5 krn3/s2 is a constant equal to the product of the gravitational

constant G and earth's mass d l . The values of the period P , meastirecl in minutes,

are incluclecl in Table i.1.

=\part from the size of the coverage area ancl the orbit period, another important

issue for telecomm~mication purposes is the propagation delay between a satellite

and an earth-station, ancl-in the case of intersatellite links-between two satellites.

These depend on the altitude N and are shown in the last t ~ v o columns of Table 1.1.

In part.icular, the table shows two delay-related figures. The secorid last colomn shows

the round-trip propagation delay betweeri a satellite at a given altitude and an earth

station clirectly itncler the satellite assuming the signais propagate at the speecl of

light. The Iast colurnn shows the propagation clelay fiom one satellite to the most

distant satellite at the same altitude-the satellite above the opposite point of the

earth. The table assumes that the signal is relayecl along a circular path at a constant

Table 1.1: T h e footprint size. orbit period P , and propagation delays as a function of the al t i tude H . T h e footprint size measured by its half-sided center angle iL, de- pends also o n t he minimum elevation angle E,;,. T h e horizontal lines separate t he approximate ranges of the low-earth-orbit (LEO). medium-enrth-orbit (MEO). and h igh-earth-orbit satellites. The last two coiumns show t h e round-trip eart h-satellire- eart h clelay and the delay between two opposi t e satellites, respect ively.

degrees) (degrees)

alti t ucle wi t hou t any aclcli t ional delays. The iength of t he circular path is an upper

borincl on t h e actual path if relayecl throiigh satellites at the same altitiide. In the

following siibsections we discuss these parameters for both G E 0 and L E 0 satellites.

1.1.1 G E 0 satellites

The gmstntionn ry (GEO) satellites orbit the eart h once per 24 hours. This is achieved

by their high al t i tude of approsimately 36,000 km above the eart.h surface. If their

orbit iç above t he ecluator: they remain stationary over a fised point of the surface:

which is a major advantage. Another advantage is the large footprint due to the high

altitude. From equation (1.1) we see that the half-sided center angle. of the footprint

increases with increasing H . This is demonstrated in Table 1.1. If we consider the

minimum elevation angle of 10 degrees, which is a typical value: the value of ~ is just

over il degrees. That means the angle hetween two most distant points covered is

over 1-40 degrees. Theçe features make them well suited for broadcast services to large

areas. e-g.. TV broadcasts. They also make it easy to connect two distant points of

the earth by simply boiincing the signai off the satellite. However. if interactivity is

an issue. we encounter a major drawback. which is the relatively long delay due to

the hi& alt i tude-the round-trip d e l q is more tlian 230 msec. This is not a problem

for TL' broaclcasting biit it is non-negligi ble for interactive applications.

1.1.2 L E 0 satellites

To eliminate the long round-trip clelay, one has to position the satellite in a much

lower orbit. The low-eart h-orbit satellites operate a t altitiides of 1,500 km or less ( the

range from 1.500 to approsimately 7,000 km is not suitable for telecommunications

due to the racliat ion in the so-called inner Van :illen belt ). From Table 1.1 we see t liat

the round-t rip clel- is less t han 10 nisec and the propagation clelay to the opposite

point of the earth is uncler 85 msec, which is still less than the round-trip delay for

a G E 0 satellite making L E 0 satellites more suitable for connecting clistant points

on the surface if intersatellite links are employecl (see below). bledium-earth-orbit

(3IEO) satellites operat ing in t h e approsimate range of 7.000-14.000 km are anot her

option. Hoivever, the issues involveci are sirnilar to the L E 0 satellites, ancl therefore

ive clo not cover them esplicitly. The range 147000-20.000 k m is again unavailable due

to the outer Van ,Allen radiation belt.

The L E 0 satellites reduce the delay biit there is a price to pay. First. clue to

Iower altitude. the footprints of the satellites are much srnaller. For altitude H of

1,.300 km and minimum elevation angle 10 degrees, the d u e of t !~ is 27 degrees (see

Table 1.1). This forces the use of a larger nurnber of satellites. If we want to avoid

terrestrial networks for communication between two distaat eart h-stations, we have

to e m p l o inf ersntellite links. creat ing an intersatelli t e network. Second: the lower al-

titude decreases the time to orbit meaning that the satellites constantly move against

the surface of t he earth. and usually also against each other. The results are freciuent

hand-overs even if t he earth-station is fised, and dÿnamic changes in the distances

between satellites. These things are among the main new networking challenges in

building a telecommunication network connecting L E 0 satellites. The following sec-

tion elaborates further on the main issues involved.

1.2 Building a L E 0 satellite network

In this section we briefly mention a list of issues related to building L E 0 satellite

networks. particiilsrly the ones that cliffer from G E 0 satellite systems. The issues tha t

are of particular relevance to the topic of this thesis are further coïerecl in Cliapter 2.

Here we mention the following issues:

0 constellation.

0 frecluency reuse,

a inillt ipIe access met hods,

a intersatellite network topology.

a intersatellite network routing.

-4 criicial issue in the clesign of a LE0 satellite network is the constellation of t he

satellites-their total nurnber ancl their positions. This includes the nurnber of orbits

and t heir type. T h e two main types are polor and indined orbils. The former introcluce

less relative motion between satellites but create a seam between two orbits rotating

in opposite directions. The latter create a homogeneous network but \rith higlier

relative satellite motion. A combination of polar and incIined orbits or orbits with

CH-4 PTER 1. IiVTROD liCTIO?ï

varying inclination is also possible. T h e choice of a constellation is further discussed

in Section 2.1.

Among other things, the choice of a constellation most take into accoiint the cle-

sired coverage-the way the surface of t h e earth, o r i ts parts, are covered by satellite

footprints. An extensivestudy of the topic is Rider [37]. It cliscusses the best constella-

tions. in terms of t h e total number of satellites. with respect t o t h e recluired coverage.

which may b e the whole earth surface o r just certain regions. Relatecl t o coverage

is frequency re use. Satellites whose footprints overlap must use non-overlappirig fse-

quency bands t o avoid interference. - i n important design goal is t o masimize the

available bandwidth by maximizing the reuse of frequency bands in non-overlapping

footprints. T h e w q s the frequency spect rum is sharecl among users within one

foot print are referred to as multiple access methods. Esamples include frequency cli-

vision mult iplexing? tirne division rnultiplexing. code division multiple access. or t heir

combinations. AS we mentioned above? t h e L E 0 satellites move against the earth

surface resulting in hnnd-ocers even for a stationary user: -4Lyilcliz e t al. [l] survey

the hancl-over methocls proposecl in the contest of L E 0 satellite ~ietworks. Another

source of l-rancl-overs is wit hin the intersatellite network clue to the relative mot ion

of the satellites. To fincl a proper replacing satellite. the effects on t h e clelay ancl

its ~ariaticrn ( j i t ter) have to be considered. T h e issues of coverage. frecjuency reuse.

multiple access ancl hancl-overs are beyond the scope of this thesis, a n d Ive rvill not

ctiscuss them furt her.

F inal le we mention the issues of choosing t h e neiu*ork topologg ancl rozrting for

it. ivhich are ciiscussed in more detail in Section 2.1. T h e network topologj- is cleter-

minecl by t h e choice of intersatellite links, ~vh ich is mainly depericlent o n the clistances

between pairs of satellites and their variations. It is significantly affected by the type

of orbits-polar o r inclinecl. Most of the proposed systems assume a fised topology

but dynarnic changes are possible as well. T h e roiiting methoci determines the best

i v q s to cleliver the d a t a through the network. T h e optimal methocl m a - clepend not

on1~- on the networl.: topology but on the distribution of the consiclerecl traffic 10x1 as

well.

1.3 L E 0 constellation examples

In this section we feature a few esarnples of L E 0 satellite systems for telecornmuni-

cations. hIore cletails can be found in the surve-s by Comparetto and Ramirez [IO],

Miller [35] and Wood [dg]. We start with systems that do not use links between satel-

Iites. One of the most basic ones is the OrbComrn systern which uses 13.5 satellites a t

altitude 825 km t o do primarily transmission of short messages? emergency alerts ancl

position determination with typical throughput between 2.3 and 9.6 kbps per user.

A more ambitious system is the Globalstar system which consists of 4s satellites in S

inciined orbits with 52 degrees inclination. The altitude of the satellites is 1.400 km.

The service offered is voice or da ta transmission a t a rate of up to 9.6 kbps. A system

targeted a t high da ta rate neecls is SkyBridge that plans to use SO satellites in 20

orbits with inclination 513' at an altitucle of 1,469 km. It assumes fixed-site terminals

with da ta rates between 2 and 100 Xlbps. -An interesting constellation is proposed for

the Ell@so system. It consists of ï satellites in the eqiiatorial plane a t the altitucle

of 8,050 km and 10 satellites in two highly elliptical orbits. Tlieir highest altitucle is

7.60.5 km, the lowest 6:X3 km. the former heing reachecl oves highly poptilatecl areas.

The ptanned da ta rates are betrveen 9.6 ancl 64 kbps.

There are two main systems that eniploy intersateIlite links. The first one is the

Iridium system' (see Hutcheson and Laurin [28 ] ) . The system uses 66 satellites in 6

polar orbits at aItitucle f S O km. Each satellite is connected to its four neighbors-two

in the same orbit, and two in the neigl-rboring orbits. The system has to deal wi th the

seam problenl mentioned above. ancl the satellites have to be turned off in the polar

areas due to their high concentration. The whole system relies on terrestrial/cell~ilar

netwosks: and uses the satellite links prin-iarily when terrestrial ones are rinavailable.

The intendeci ch ta rate is in the 2-3 kbps range.

.A competing system is Tdcdes ic (see S t~ i r za (411). It also uses a polar constellation

but with a higher nomber of satellites-810 satellites in 21 orbits planned originally.

'.Ifter the launch of its commercial service, the systein suffered financial losses, and is currently in the state of bankruptcy seeking potential buyers.

later reduced to 288. The higher number of satellites orbiting at an altitude of ap-

prosirnately '700 km resdts in smaller footprints thus increasing the elevation angle,

which increases the potential data rates. Each satellite is supposed to connect to two

neighbors in each of the four orthogonai directions, i.e.? S links per satellite. The

typical clata rates are 16 kbpç with a potential of up to 2 Mbps for mobile terminais.

and up to 1.2 Gbps for fised-site terminais.

1.4 Outline of the thesis

This thesis explores some of the issues related to LE0 networks. It uses an extensive

numerical study of the L E 0 network parameters to develop models for studying com-

munications in t hem. Then it uses the developed models to design efficient algorithms

for certain typical communication patterns.

The numerical studies explore the impact of various clesign parameters on the

network performance. particularly, on the communication delay. The models acidress

two characteristics of LE0 networks. We demonstrate that a modified toroidal mesh

interconnection topology callecl a skewed torus is a natural consecluence of choosing

inclinecl oibits (ivhich lie on planes that are a t an angle to the plane that inclucles the

earth's asis of rotation. ancl enable more homogeneous networks). In a LE0 netirork.

there are two types of intcrsatellite links: intrnorbitnt links between satellites in the

sarne orbit. and interorbitni' 1irzX.s between satellites in acljacent orbits. Li'hen inclinecl

orbits are usecl. the directions and lengths of interorbital links are constantly changirig.

F\ye propose two moclels of the variable-length interorbital links. l\ye clemonstrate that

a siniplified tzco-un$orm mode/ in which al1 intraorbitai links have the sanie length.

ancl ail interorbital links have the same length: provicles a goocl approximation of the

average pat h lengt hs in inclined orbi ts.

The second part of the thesis clevelops efficient communication algorithms for

toroiclal network topologies uncler the two-uniform model. The focus is on an all-

to-al1 communication pattern callecl gossiping in which each transmitter broaclcasts

information to al1 other stations. This type of cornmiinication pattern can rnodel the

eschange of control information witliin the network and is an example of a pattern

that exercises al1 parts of the network. The thesis first develops algorithms for the

consfnnt-cosf mode1 which takes into account only propagation delay and disregarcls

the data rate of the links (i.e., the transmission t ime is constant regardless of the

message lengt h). The main contributions are a n algorit hm for the one-port half-duplex

mode1 (each satellite can be involved in a t most one transmission a t a tirne) rvhose

one-iiniforrn special case (al1 links have the same delay) improves the best kno~vn

upper bound for regular toroidal meshes? and an algorithm that takes advantage of

the smaller diameter of the skewed torus. Then it develops all-port and one-port

gossiping algorithrns for toroidal rneshes under the linenr-cost mode1 which consiclers

both the propagation delay and data rate. T h e all-port mode1 allows one satellite t o

use al1 its links simiiltaneous1~-, and our algorithms utilize the overlap between the

transmissions in the two orthogonal directions of a toroidal inesh to reduce the time

ivasted due to propagation clelay. Its full-duplex (bidirectional links) one-uniform

special case improves t he time of the best-known algorithm For a regular torus from

( N - 1 ) ( ~ / 4 ) + D,B to (A- - l ) ( r / 4 ) + (D/2)/3 + O(1og D). where 4 is the propagation

del- l / r the clata rate, M the number of nocles (satellites). ancl D the dianieter

of the torus. The half-cluplex \-ersion (unidirect ional links) recluces the upper bound

from (1' - i)(r/i) + 2D,O to ( N - i ) ( r / 2 ) + ( D l ? ) @ + C?(log D). Tlien the thesis

develops one-port gossiping algorithms for trw-uniforin tori. in rvhich one satellite can

use only one Iink a t a time. At the end it compares the all-port and one-port rnoclels.

The rest of the t hesis is organizecl as follorvs. Chapter 2 covers relatecl results from

the literature. The results of the thesis are diviclecl into two parts. Part 1 acldresses

modeling of the physical properties of L E 0 satellite networks and comniunicat ion

links. In Chapter 3 it presents a numerical s tudy of the impact of various parameters

on the netrvork performance. and introcluces t h e skewecl torus topology. Chapter 3

aclclresses approximations of the communication delay and proposes t lie two-uni form

rnoclel. Part TI uses t he models cleveloped in Part I to stucly the gossiping problem

in the networks. Chapter 5 consiclers the propagation tlelay only, while Chapter 6

incorporates both the propagation cklay and the da ta rate of the communication

links. T h e thesis conclucles in Chapter 7 ivith final rernarks and a list of problems for

furt her research.

Chapter 2

Related work

This chapter reviews publishecl literature t h a t is rnost relevant to the topic of this

thesis. It s tar ts in Section 2.1 with papers on LE0 satellite networks covering major

design issues like t h e interplay between satellite constellations ancl network t o p o i o g ~

ancl issues relatecl to d a t a transfer and routing. -4s it turns out, and as Chapter 13 of

this thesis d e t n ~ n s t r a t e s ~ typical LE0 satellite topologies are relatecl t o rectangular

meçhes and their wrapped around toroida! variants. Section 2.2 covers resrilts on

cornniunications in these types of iietworks. kIost of t h e resiilts originate in the

context of rnultiprocessor networks but , when abstractecl, t h e interconnection patterns

are sirnilar to those of LE0 satellite networks.

2.1 L E 0 satellite networks

This section covers results from the literature t h a t discuss t h e major issues in builclirig

LE0 satellite net~vorks. Subsection 2.1.1 cliscusses the major types of satellite con-

stellations, and t h e interplay between the choice of a constellation and t h e network

topology. Subsection 2.1.2 presents published results on t h e transfer of d a t a in L E 0

satellite networks and on t h e network routing techniclues.

CHAPTER 2. RELATED WORK

2.1.1 Constellations and topology

This subsection cliscusses t,he network topology of LE0 satellite networksl i.e.' the

interconnection pattern among LE0 satellites forming a communication network. The

choice of a topology is strongly affected by what is perhaps the most important design

issue for a L E 0 satellite network, namely, the choice of a constellation-the number

of satellites. the number of orbits: and their inclination, i.e.' the angle between the

orbital and equatorial planes. The choice of orbital planes and the number of satellites

directly affects their distance, its variations. and, hence, the possible connections. All

of the constellat,ions that Ive will consider (and most of the proposed networks) are

regzllnr constellations t hat place the same niimber of uniformly spacecl satellites in each

orbit: position the orbits iiniformly around the earth, and use the same inclination

for al1 orbits.

The inclination of orbits is either polar o r inclinecl. Polar orbits lie on a plane that

includes the eartli's asis of rotation, so satellites in polar orbits cross over hoth of the

eartli's poles during each orbit as illustrated in Figure 2.1. In a polar constellation.

Figure 2.1: -4 polar constellation (Iridium).

the satellites in one "hemisphere" move north, while those in the ot her hemispfiere

move south. This creates a seam where t h e satellites of two neighboring orbi ts a re

moving in opposite directions. This is best seen in a polar vie~v shown in Figure 2.2.

Fised links across the seam a re impossible, which is n*hy virtiially al1 proposals omit

Figure 2.2: -4 polar view of a polar constellation showing the seam.

t k m . As a result. t he intersatellite topology of a polar constellation is not uniform.

tvhich cornplicates the intersatellite routing.

To select interconnections between satellites. a natiiral choice seems to be t o con-

nect a satellite to its two neighbors in the same orbit using an ir2traorbitnl l ink T h e

satellites keep a fixed relative position ancl distance rneaning the transniitters and re-

ceivers d o not need any steering. To reach the full connect ivi ty over the tvhole globe,

it is necessary to employ interorbital links as well. Due t o the relative motion of satel-

lites in clifferent orhits t h e choice is somewhat niore comples. T h e Iridiiim system uses

links t o t h e closest satellites in each of the two neighboring orbits resulting in four

intersatelli t e links in to ta l (see Figure 2.:3). The interconnectecl pairs remain fised

al1 t h e time. T h e relative motion of the satellites reyuires the steering of transmit-

ters/receivers but for t h e CO-rotat ing orbi ts, t he impact is relat ively small. Hoivever,

the satellites in t h e counter-rotating orbits move in opposite directions making it irn-

possible to keep t h e connectioii between a fixecl pair of satellites. T h a t is why the

Iridium system omits intersatellite links between thern. This, of course, introdiices

irregularities into the network.

Figure 2-13: The interconnection pat terri of t he Iridium system.

Cavish and Iialvenes [21] compare this t o three other topologies that use four

links per satellite. They s tudy the shortest-path routing and the worst-case encl-to-

end clelay. The results show that the interconnection pattern shown in F ig~i re 2.4

outperforms the pattern of the Iridium system. Note, however: that it uses links

Figure 2.1: T h e interconnection pattern of Gavish and I<alvenes [21].

between more distant satellites. Other interconnection patterns can be considered as

i\.ell. Wood [XI studies topologies using 2: 4, 6, and 8 links per satellite. The Teledesic

systern uses eight links per satellite: two in each clirection. Four of the links connect

to t he four closest satellites in the sarne orbit, while t h e o t her four links connect t o

satellites in the four closest orbits. One can consider three links as well. e.g.. two

intraorbital and one interorbital.

T h e situation is sornewhat different for inclined co~istellations which have orbits

t hat lie on planes t hat include the center of the earth but not the poles. An esarnple

is shown in Figure 2.5. One advantage of inclined constellations is the absence of a

Figiire 2.5: An inclined satellite constellation (Globalstar).

seam. so more regtilar topologies are possible. as mentioned in Werner et al. [46]. In

some cases, inclinecl constellatioiis can acliieve the same coverage with fewer satellites

than polar constellations. particularly when the polar areas need not be cov-ered [37].

*-llso, they freqiientiy result in double coverage of certain areas-having more than

one sat.ellite covering the area a t a time. The impact of this needs more stucly. It

can lead to complicat.ions in frequency reuse but it can also provide more routing

Rexibili ty. possibly reclucing the pat h lengt h between two satellites.

-\ nminor disadvantage of inclined constellations is poor coverage a t the earth's

poles. A more serious tlraivback is higher variability in t h e relative positions of satel-

lites in clifferent orbits. This gives rise to the need for beam steering and effects

result ing from Doppler shift t hat reduce the available bandwidt h [20]. Nevert heless.

as reported in [46]. a proposecl system called 34-Star assumes such interconnections.

Since there is not enough information on the system, the paper [4G] clisciisses possible

interconnection pat.terns. The intraorbital links seem to be straightforward-the two

neighboring satellites. As for the interorbital links: the- consicler the closest sateHi tes

in the two closest orbits as shown in Figure 2.6(a). For further studies. they choose

Figure 2.6: The intersatellite links (a) considered, and ( b ) proposecl in 6i7erner et al. [4G].

one satellite from each of the closest neighboring orbits (see Figure 2.6(b)). As one

can clerive? the resiilting nettvork is a toroidal mesh skewecl by half the number of

orbits (see Section 2.2 for a forma1 definition of toroiclal meshes). Hotwiler. since the

constellation stiidied in the paper consists of 12 orbits with 6 satellites per orbit. the

skew is equal to the number of satellites per orbit resdting in an ordinary torus.

M'hen a net,work topology is chosen, an important question is how to route the chta

in the network in the most efficient way. Obviously, the answers can have an impact

CHAPTER 2. REL-ATED W O R K 18

on the choice of the topolo0~- A simple study of routing suitability is presented in

iVood [SOI. The work considers networks with 2. 4, 6 and 8 links per satellite. I t

studies minimum hop count paths in the network, i.e. it disregards the actual link

length and the resulting propagation delay I t calculates their number between each

pair of nodes, and uses the resiilts to estimate the load on the network Links- T h e

work is clone for seamless/infinite grids since the calculations become more cornplex

ot herwise.

.L\ shortest-path routing rnethod for polar consteIlations taking the propagation

clel- into account is proposed in Werner [G]. It assumes a fixecl network topology.

e-g.. the one of the Iridium system. The method is based on the fact that the posi-

tion of the satellites periodicallÿ changes with the period ecfual to the orbit time. It

suggests to cliscretize the t ime n-ithi~i one period into fisecl-length intervals, and to

precornpute the shortest paths between each pair of satellites for ewry time interval

iising Dij ksi, ra's shortest pat h algorit hm [13] . For robost ness piirposes. i t act iially

suggests to compute several link-disjoint paths for each pair. Dile to the constant

motion of the satellites. the shortest path routes change between two time inter\-aIs-

Anot lier source of reqriirecl changes are the off-periocls of sateIlites over the poles mod-

eled after the Iridium systern. In addition to t.he minimization of the path length.

the paper also investigates the minimization of the delay jitter-the clifference of t he

path lengths clue to the rerouting. The paths are precornputecl off-line and stored

in tables iised during the actual connection set-iip. While the work of W r n e r [45]

was done for connection-oriented path set-up. Ekici e t al. [l?] iise a similar approach

for connectionless transn-iission. They characterize t h e shortest patlis in a polar con-

stellation rvith interorbital links turned off in the polar regions and across the searns.

Eacli sateIlite stores the direction of the nest roiiting hop accorcling to the location

of the destination satellite. Each packet is routed independently according to the

stored tables. In case of a link congestion or failure, they are reroiited thro~igh an

orthogonal link. Note that t he algorithm does not a t tempt t o balance the load to

rediice or prevent congestion, which is particularly likely to occur on the interorbital

links closest to the polar areas that are turnecl off.

The n-orli of Werner [45] was estendecl for inclinecl constellations in a paper by

iVerner e t al. [46]. The paper presents simulation results comparing the trvo types

of constellations. The' suggest t hat inclined constellations provide smaller del- jit-

ter during hand-overs even if the routing for them is optimized for shortest paths,

while for polar constellations it is optimized for minimum jitter. Another extension

is stuclied by Lverner and Maral [G]. T h e - consider polar constellations7 and com-

pare the shortest-path routing t o a roiiting met hod aimed at balancing the load over

network links. (Note that the maximum link load dictates the minimum da ta ra te

of a link.) The routing algorithm used is the distributed Bellman-Ford (DBF) Algo-

rithm [6] which adaptively recomputes the routes based on the current network load.

They perform simulations based on the voice traffic pattern studies of Violet [ad]

which give statistical distribution of voice calls around the globe. -4s expected. t he

DBF algorithm achieves lower maximum link 1 0 x 1 thiis enabling the use of less bancl-

width ancl/or power. Moreover. the results show that the delay does not increase

significantly.

:\ similar approach based on precomputing routes for fixed t ime intervals is pro-

posed in Chang et al. [ô]. T h e main ditference is that this paper suggests to precom-

pute. and c1';narnically change, the intersatellite links as well resulting in a dynaniic

network topology. The objective is to rninimize the maximum link load ivith respect

to the offered traffic. The algorithm is an iterative application of topology changes and

routing optimization stated as linear programming problems ( the former mixeci inte-

ger) and solved by the simulatecl annealing method [:IO]. T h e work of Chang et al. [9]

stuclies an aclaptive version of the roirting algorithm n-hich changes the routes rvithin

one tinie interval. It compares it to the static. precomputed version from the point of

view oof cal1 blocking probabilities. The resiilts shoiv that the static roiiting performs

better achieving lower blocking probabilities. This may sound surprising compared

to the results of Werner and hlaral (181. Hoivever, the algorithms of Chang et al. [9]

use link loacl minimization in static routing and sliortest paths in adaptive routing?

while in [G] it is the other may around, so a direct comparison is not possible.

Algori t hms for rerout ing connections during hand-overs in polar constellations

are proposed in Uzunalioglu et al. [43] and Uzunalioglu [42]. Uzunalioglu e t al. [43]

focus on hancl-oïers between a ground station and the satellite covering the area in

which the station is located as a result of the satellite motion. The paper proposes a

Footprint Handover Rerouting Protocol (FHRP) based on the fact that the position

of the satellites periodically repeats once a successor satellite takes the position of

its predecessor. The proposed routing algorithm uses a shortest path algorithm to

establish the initial route p for a new call connection. When the ground-satellite link

of one of the endpoints iç hancled over t o the next satellite. the new satellite is added

to the original route p: note that this usually causes non-optimality. This augmented

route is replaced by a new route p' after the gsound-satellite link of the ot her endpoint

is hancled over as rvell. The new route p' consists of the successors of al1 satellites

in the original route p. Note that , when ignoring link length variations within one

hand-over period, the periodic repetition of satellite positions implies optimality of

the new route p' given the optimality of p.

The hancl-overs among the interorbital links are addressed in üz~~nalioglu [12].

Similarly to the above-mentionecl paper. this paper also assumes a polar constellatiori

whose interorbital links are turned off in the polar regions and across the seanis.

Due to the motion of the satellites dong their orbits. a link contained in a previously

establishecl end-to-end path m a - be turned off as it moves into a polar area resiilting in

a link hnnd-ocer and the need for a rerouting. The paper presents an algori t hm callecl

Probabilistic Roiiting Protocol (PRP) aimecl at rninimizing the niimber of link hand-

O\-ers. The algorithm modifies the initial route selection diiring the establishnient

of a new call connection in order to minimize the probability of a link hand-over

occurring on a selectecl link prior to the termination of the call and to the hand-over

of one of the ground-satellite links. Since the former is unknoivn. ancl the latter

clepends on the actiial location of the gound stations within their footprints. the

algorithm assumes a probabilistic distribution for each of them. Based on the assumed

probabili ty distribution function. it eliminates from consiclerat ion t hose links t hat

woulcl result in an early link hand-over. Disregarding these links, it uses a standard

shortest path algorithm to set up the shortest path for a new call. The simiilation

results show that the methocl clecreases the rerouting due to ISL hand-O\-ers at the

expense of increased blocking probability of new calls.

-A hierarchical architecture consisting of satellites a t different altitudes, and a

corresponding routing protocol a r e proposed by Lee and I iang [33]. T h e proposed

architecture, cailed Satellite over Satellite (SOS) Network, corisists of several layers

of satellite constellations~ each a t a clifferent ait itude. T h e satellites are connected

with ISLs both within one layer and t o the next higher ancl lower layer(s). The

satellites in t h e lomest layer are responsible for connecting t o t h e ground stations and

for routing d a t a mithin smaller distances. T h e satellites in the higher layer(s) are used

to route t raffic to more distant areas of t h e ear t h to reduce the switching d e l a - at the

erpense of a n iricreased propagation delay. T h e paper siggests a system containing

tu-O layers-a L E 0 layer consisting of several polar orbits and a ME0 l q e r consisting

of one ecluatorial orbit. T h e a c t d split between the amount of traffic routed throiigh

each layer depends on the relative values of t h e swi tching and propagation clelays.

2.2 Communications in t oroidal meshes

Section 2.1 clescri bes proposed topologies for L E 0 satelii t e networks which often re-

sult in mesh-li ke topologies. typically with wrap-aroiincl connections. T h e rectangular

mesh has been a popular interconnection topology for multiprocessor networks. -4

fon i s is obtainecl from a rectangiilar mesh by wrapping around connections in both

horizontal and vertical cliniensions. Foïmally, a n orclinary nh x n , torus has nocles la-

beled by integer pairs (i. j ) , O i i < n h - l ? O < j < nt,-1. The four neighbors of nocle

( i , j ) are ( ( i & 1 ) mocl nl,, j ) and (il ( j + 1) mocl n,). T h e connections are illustrat~ecl

in Figiire 2.7. Abstracting from t h e physical realization. the interconnection graphs

share a great cleal of sirnilarity with t h e proposecl LE0 satellite networks. There

has been a lot of ii teratare devoted to t h e abstract s tudy of communications in this

type of networks. particiilarly to the clevelopment of algorit hms for freqiient ly usecl

communication patterns. This section cIiscusses two common patterns-broadcasting

ancl gossiping (also called an all-to-all eschange). In the broadcnsting problern, there

is one node in t h e network that holds a piece of information t h a t neec1s to be clis-

tributecl among al1 nocles of the network. In the gossiping problem? each nocle starts

with one piece of information tha t has t o be distribiited t o al1 o ther nocles. A typical

application of these comrn~inication pat terns in the context of L E 0 satellite networks

Rf, X n,,

Figure 2.7: -4 4 x 6 torus.

might be the distribution of control informationl e.g.. link loads for routing purposes.

Given the problem. one can make various assiirnptioris about the properties of

the links ancl the transrnissiori times on them. Several models of the transmission

time have been proposect and stuclied in the literature. -4 very simple approach is

to assume the constant-cost rnodel. in which the transmission of one message takes a

constant amount of time regarclless of its size. The terni unit-cost is commonly iised

in the literature. LV-e prefer the term constant-cost because our moclels sometimes

Lise more than one constant. The constant-cost assumption ma. be reasonable for

the hroaclcastirig problem but is less realistic for gossiping. where the message size

usuallj- grows as several of the original messages are combinecl and sent together.

- ln approacli that takes into accoirnt both the propagation delay of each link and

the size of the messages is to assume the linenr-cosi moclel [29, 38: 401. The time

to transmit a message of size L over one link is ,8 + L r . where 13 is the propagation

del- of the link (incloding the processing time to prepare the message to sencl); and

Ï is the time to transmit one unit of data: i.e.. l/r is the chta rate (see Figure 2.8).

In the contest of rnultiprocessor networks, where the propagation clelay is negligible

compared to the processing time to prepare the message to sencl, ,8 is typically callecl

the start-up time. while T is calleci propagation time (assurning that there can be

source destination

Figure 2.5: T h e linear-cost model.

onlj. one bit on the link a t a t ime). T h e work on multiprocessor networks iisually

assunies the sarne parameters ,3 and r for al1 links in the network. Other models

of the transmission del- tliat have been proposed in the contest of m~~ltiprocessor

networks include the Postal model [ 3 ] and LogP model [12]. Horvever. these moclels

mode1 the total encl-to-end delay instead of modeling individual links. In essence, tliey

view the interconnection network as a **black box" in which the total end-to-end clela-

is independent of the number of hops traversecl. Their justification is basecl on the

assumption tliat the propagation clelays on t he links are negligible comparecl to the

processing and transmission time. which is typically valid for multiprocessor networks

but not for wicle-area networks lilie the L E 0 satellite networks. Another aspect of

modeling related to transmitting messages alorig commiinication links is the W ~ J - the

messages are relayeci ( routecl) over several consecot ive links. Throiighou t t his work we

will assume the store and forwxzrd model in which each message must be 1 ~ ~ 1 1 ~ receivecl

by the link's endpoint before the endpoint can start formarcling it over the nest link.

, in alternative is the cil-cuit-stvitched moclel in tvhich the initiator first creates a circuit

between the source ancl destination (similar t o a telephone circuit) over ivliich the bits

of a message are pipelined. If t he linear-cost mode1 is used. the t ime t o transfer a

message of size L over d links using circuit-switching is cr + d6 + L r , ivhere a is

an initial start-up cost. 6 is the switching delay per link during the circuit set-up

(incliiding the link's propagation delay): and l / r is the data rate of the links. There

are ot her variations of the routing methods, e-g. virtual cut-t hrough or wormhole

rout ing. which lie somewhere between t hose two es t remes.

We mention a few more parameters that deal with the capacity of nodes and

links. The dl-port mode1 assumes that one node can comm~inicate through al1 its

links simultaneously. The one-port model allows only one link at a time. In the f d I -

duplex model, one link can comrnunicate in both directions simultaneously. wfiile in

the hav-duplex mode1 in one direction only.

Given the problem and a particular network moclel~ one can ask several questions

regarcling t h e algorithms solving the problern. The ones that have been most com-

monly studied in the literat lire can be roughly categorized into t lie following major

categories:

O finding the lower bouncls for the time of the algorithms solving the pi-oblem \vit h

no restrictions on the network topology,

O finding the topologies that enable one to solve the problem in optimal (mini-

mum) time.

restricting oneself to certain midespread topologies (or their classes):

- finding the lower bounds for the problem on the given topology.

- fincling (optimal) algorithms for the problem on the given topology.

In the following siibsections we mention some of them, particularly the ones that

are most relevant for the work of this thesis. hrlore details can be found in the sur-

veys of Hedetniemi et al. [%]' Fraigniaiid ancl Lazard [ l i ] , Hromkovif et al. [ZT] and

Grammat ikakis et al. [23] .

Assuming that transmission of each message takes one unit of time, a trivial lower

bouncl on both broadcasting ancl gossiping in the constant-cost mode1 is the diameter

CH-4PTER 2. RELATED WORK

D of a network. i.e. the number of links (liops) separating the tivo most distant nodes.

which is ecpal to L$ j + 121 for a nh x n , torus (note that we assume a store and

forward routing). This bound is easily achieved in the all-port full-duplex nioclel by

copying each incoming message t o each oiitgoing link b - every node of the network

( a methocl also known as flooding [39]). It applies not only to broadcasting but ais0

to gossiping since. under the constant-cost model. the transmission time is constant

regarclless of the message size. For broadcasting. the half-duplex mode1 has the same

t ime cornplexi ty as the full-duples one.

Entringer and Slater [15] show that the restriction to the one-port full-duplex

model results in at most one additional step in a ?-dimensional torus. For the gossiping

problem. an algorithm for the all-port half-duplex niodel can be based on the work of

Iionig et al. [ 31 ] on the so-called orientecl diameter of a torus giving a running tirne

of a t most D + 2 [l'Tl. For the one-port moclel. there is only one link available to an-

node at an\; given time. Farle- and Proskurowski (161 show an upper bouncl of at

most D + 1 for a one-port full-duplex torils. Botli the upper and lower bounds the-

derive ckpend on the parities of the dimensions of the torils, a n d their algorithms are

within at most t ~ v o steps of the lower bounds. .\ larger gap remains for the one-port

half-duplex model. Iirumme et al. [32] present an algori t hm for a d-dirnensional torus

that gossips in time D+ lSd+N. This thesis improves this upper bound to D+3d+5.

i-e. to D + 11 for a 2-dimensional torus.

IVhile the constant-cost algorithms mentionecl in the previoiis subsection solve the

broaclcasting and gossiping problems ivithin a srnall additive constant. the situation

is different for the linear-cost model. Fraigniaud and Peters [IS] give a lower bound

on one-port full-cluples linear-time gossiping in a complete graph tliat is tight for an

even number of nocles but no tight bound is known for toroiclal meshes.

First. ive review results on broadcasting. If the original message cannot be split

and has to be sent in its entirety over every link, ive end iip with a case ecfuivalent to

t h e constant-cost model (with the constant cost P+LT per transmission). O n the other

C'HA PTER P. R EL4TED WORK

hand, splitting the original message into smaller pieces and sending them separately

can reduce the transmission t ime subst antially Under t his assumpt ion. a lower bound

cornbining both /3 and r can be obtained as follows. Consider a pair of nodes separated

by D links. It takes D(,d+r) t ime units until the first bit of the message is received by

the destination. hloreover. the destination can receive a t most four bits by this t ime L since t here are only four incoming links. After t his. it takes a t least yr = ( - l)r

t ime units to receive the rect of the message ( the rate is a t most four bits per t ime r ) .

Therefore. the total time is a t least D(/3 +r ) + (4 - l)r = 0/3+ (D - 1 + $)T . htost of

the efficient algorit hms for broadcasting in the linear-cost store and forwarcl model are

based on the combination of two techniques: (i) arc-disjoin t apnnning treea-split the

message into several pieces and broadcast them dong spanning trees that are miitiially

arc-disjoint (in the full-dupIex model, the trees can share the same link in opposite

directions: in the half-cliiples moclel. t hey cannot, nnd are callecl edge-disjoint ), ancl

(Li) pipelining-split each piece into several packets and pipeline the packets along the

branches of the spanning trees. T h e optimal size of t he packets depends on the relative

vaiiies of 0. r_ L. ancl the depth of the trees. klichallon et al. [:34] find four arc-disjoint

spanning trees of a torils of clepth D + 1. Splitting the message into four pieces, and

using the optimal packet size (ivhich recluires L 2 9 to guarantee the packet size is

a t least one) results in total broadcast time (m+ dm)' = D,B + ,/'D1dL7 + 9- This niethod can be modifiecl for the one-port model t o obtain the runriing t ime of

a t rnost (,/m.+ &G)2 [17]. A pair of different edge-disjoint spanning trees

vas used in Bermonci et al. [Ij] t o broadcast in the all-port half-duplex moclel in time

a t most ( + JLT / . ) ) ' . Again, the algorit hm can I,e modifiecl for the one-port

mode1 obtaining an upper bouncl of (Jm + JF)'. In general. rnatching

upper and lower bouncls for linear-time broadcasting in tori are not known.

The situation is s h i l a r for the gossiping problem. If AÏ = nh x n , denotes the total

nunlber of nodes in an n h x n, torus, one can clerive the follou-ing two lower bouncls

for the gossiping problem: (i) D ( P + Ï ) since each bit must traverse at least D links.

ancl each transmission takes a t least ,fi + r time units, (ii) ( N - l)? since each nocle

m u t receive N - 1 messages of size L in total: and it can receive at most ?/r data

units per time unit. -4 resiilting lower bound is max(D/?+ Dr- (Ar - l)%). Fraigniaucl

and Lazard [17] describe a n algorithm for an n x n square toms that gossips in t ime

+ (!V - 1 ) y = 013 + ( 1 i T - 1)$, Le., its running t ime is equal to the sum of

the lower bounds for ,O and r. An initial algorithm described in [17] consists of two

stages-a vertical one and a horizontal one. The verticai stage uses the vertical links t o

perform gossips within each vertical cycle forwarding one message a t a tirne, ivhile t he

horizontal links are idle. T h e horizontal stage distributes t he accumulated messages

within each horizontal cycle ( t he vertical links are idle). To avoid the idle links, they

split t he original messages into two halves. and perform two sirnuitaneous orthogonal

gossips. In this schenie, the vertical gossips along vertical links are complemented by

horizonta1 ones dong horizontal links. Both of the gossips have the same niimber of

transmission steps and, hence? propagation delays. This is followecl by a second stage

forwarding the accumulated messages with vertical and horizontal gossips swapped,

again with an eclual number of transmission steps. For n odd. each step in the first

stage takes ,fi + $T - t ime units. and each step in the second stage takes ,3 + $T tirne

iinits. Since there are 151 = steps in each stage, the total running time is

For n even the schenie needs a slight modification but the running tinie is the same.

-4 corollary of a resolt of Section 6.2.2 of this thesis gives an algorithm running in t ime

:,LI + (:\- - l)? + f (Iog D) for arbitra- rectangular sizes of the torus. Fraigniaud

anci Lazarcl [l'TI clescribe Iiow to apply the idea of tlieir all-port full-cluples algorit hm

to the one-port niodel resulting in t ime n,û + (Ai - 1) L r = D,L? + ( N - 1) L Ï for n

even; note t hat the lower boond for the one-port mode1 is rnas (DP + Dr. (1\ - 1) L r ) .

T h e - also siiggest t o simulate one step of a full-duplex algorithm by two steps of a

half-cliiples algorit hm.

Part 1

Models of physical network

propert ies

Chapter 3

Network topology

This is the first chapter of Part 1 of this thesis that proposes simple rnodels for

in\-est igating communications in inclined low-eart h-orbit ( LEO) satellite networks.

-As mentionecl in section 2.1.1. one advantage of inclinecl constellations is the absence

of a seam. so more regiilar topologies are possible. LVe assume that al1 of the inclinecl

orbits use the same angle of inclination relative t o the earth's asis of rotation. Our

moclels address two parameters of LE0 networks-t he seIection of intersatelli t e links.

ancl the rnocleling of link lengths.

This chapter focuses on the selection of intersatellite links and the resolting net-

work topology. It stiidies the impact of various parameters like inclination. the num-

ber of orbits ancl satellites. and the interconnection pattern between satellites on the

communication clelay. Since the inclination ancl the nurnber of orbits and sateIlites

are most iikely t o be determinecl b ~ - the clesirecl coverage. the main focus is on the

interconnection. It shows how the cielay varies when interconnecting different pairs

of satellites. T h e chapter finds an interconnect ion pattern t hat minimizes the clel-

between both directly connected satellites and arbitra- pairs of satellites. and clemon-

strates t hat inclinecl orbi t s nat urally lead to a modified toroiclal mesh interconnection

topology called a ske a-ed torirs.

Our st iidy of the impact of ISL connections on the communication clelay (given by

the length of the links) starts with general satellite geometry in Section 3.1. Then the

chapter considers two criteria for selecting interorbital links. The first one, coverecl

CHA PTER 3. NETWORK TOPOLOGY

in Section 3.2. minimizes the length of the links themselves. Section 3.:3 discusses

the network topology t hat results from the optimal select ion. Section :3.3 considers

minimization of the total length of end-to-encl paths. Our numerical results show that.

within a certain class of topologies. it usually Ieads to the same kind of connections

and network topology.

3.1 Geometry of intersatellite links

To clerive formulae for the length of intersatellite links: we will first introduce some

terminology relatecl to satellites and their orbits. The inclination cro of a satellite orbit

is the angle between the orbital plane and the ecl~~atorial plane: note that al1 orbital

planes intersect the center of t h e earth. The descending node of a satellite is the point

a t ~vhich the satellite crosses the ecliiator in the direction from north to south. The

angle betu-een the clescencling node, the center of the earth, and the satellitees position

will be called phase 9. Note that the satellite reaches the maximum latitude a0 at

y = -90". ive will consider a coordinate system centerecl at the center of the earth

~vhose sy-plane coincides with the ecluatorial plane. To simplify the formulae. terms

like latitude and longitude d l refer to the fisecl sphere associatecl with the coordinate

si-stem. not t o a rotating earth. with 0" longitude in the direction of the .r-mis.

The first lemrna can be derived ~ising elementary geometry (see Figure 3 . 1 ) .

Figure i3.1: The lengtl-i of an intersatellite link.

Lemma 3.1 Corzsider two sate l l i fes SI and S2 nf equnl alt i tude H oaer the enr th k

s u ~ j n c e . Let R be the radius o j the earth, a n d let -/, be the angle bettceen SI' S2 a n d

the center of the earth at a time instant t . Then the length of n strnight link between

.Y1 and S2 nt ti.me t is *,

21 = ~ ~ ( R + H I J=. clt = 2 ( R + H ) Isin -, - In the rest of this section ive develop a formula for cos-y,. Since we are interested

in the angle between two satellites. we can simplify the calculations assurning that

the radius of the orbi ts is eqiial to one. The following lemma gives the instantaneous

coordinates of a satellite given its phase and orbital inclination (see Figure 3 .2 ) .

highest 1

latitude = I

descendi ne node

Figure :3.2: General position of a satellite.

Lernma 3.2 Let S be n sutellite irz an orbit uith unit radius and inclil

;; be the instantaneous phase of S at a giceiz tinte instant.

1. If the longitude of the descending node o j S is 0': then i t s coordinates in the

J b e d coordinnte s p t e m are

(cos 9, cos cro sin 9: - sin û o sin p) .

-3. If the longitude of the descending node o f S is ,Bo? then the coordinates are

(cos cos i; - sin 90 cos cro sin y. sin ;30 cos y + cos ,Bo cos a. sin 9. - sin no sin p) .

C'HAPTER 3. .NET'IVORK TOPOLOGY

Proof: First assume that the longitude of the descending node is O O . Let the coordi-

nates of the satellite be (x: y, z ) , and let d, denote the distance of the satellite from

the x-asis. Then, referring to Figure 13.2. one can see that

and

This implies that

sin (-9) = -- 1

- - - sin a. sin p- H -

Further. one can see that

which implies -. -

y = -- = COS ctO sln p. tan a0

The s-coorclinate cari be obtained from the eqiiality .zZ + + z2 = 1:

2 = 1 - cos2 n o sin2 p - sin' oo sin

Since. accorcling to our orientation, s is positive i f and only if y € ( -go0. !JO0) . Ire get

.C = cosy .

The second part of the theorem can be obtained by rotating the position by angle

in the horizontal plane. i

To c!isciiss the lengt h of intersatellite links. we clefine the phase of a iink connecting

tmo satellites Si ancl S2 with phases pl and p s as y = (9, + q 2 ) / . Xote that t lie two

satellites clo not liave to be in the same orbit. The o f se t of the connected satellites

will be definecl as A,- = 9 2 - pi. The following lemma gives a formula for the cosine

of the angle bet~veen two satellites Si and S2 in two orbits with arbitrary angle ,L&

between t heir descencling nodes ancl wit h arbitrary clifference of their phases Ai;.

Lemma 3.3 Let S1 and S2 be tu70 satellites with phase oflset A? that are in two orbits

rcith unit rad ius and eqval inclination ao. Let be the angle between the descending

nodes of Si a n d S2- ./Lssurne thnt the phase of n link connecting satellites Si and S2

nt tirne t is q,. Then the cosine of the angle -it bettaeen the satellites SI and .Y2 at

f i n e t i.s

1 +, sin2 no ( 1 - cos ,Oo) cos Ay + cos ,go cos A y - - cos û o sin Bo sin 39.

Proof: The value cos +-it is equal to the inner product of the vectors cl and c1 from the

center of the eart h to the locations of SI and S2 at time t . The phases of the satellites

Si and S2 are pi = yt - Ap/2 and p2 = pl + i lp /2 . Without loss of geiierality. we

can assume that t he longitude of the descending node o î Si is Ool and the longitude

of the clescencling node of S2 is ,Oo. Then: applying Lemma 3.2.

~ 1 2 = (COS jgCos y2 - sin .& COS n o sin 9 2 .

sin,do cos p2 + COS^^^ cos ct.0 siny2.

- s inao s inp2) .

The inner procluct of t h e two vectors cl ancl u2 is

cos y, = cos ,Bo cos pl cos 172 - COS CIO sin #Oo cos sin 9 2

+ cos cro sin Po sin pl cos y2 + cos2 a0 COS sin ql sin 9 2

+ sinZ n o sin 151 sin 9 2 .

2 Using c o s h o = 1 - sin 0 0 . this is eqiia1 to

cos yt = cos ,Oo (cos pl cos p2 + sin pi sin 9 2 )

2 - sin cro cos ,Bo sin y1 sin y2 + sin2 û.0 sin 91 sin y,

+ cos a0 sin ,fi0 (sin (îl COS p2 - COS pl sin y2)

= c o s , & c ~ s ( ~ ~ ~ )

2 + sin û o ( 1 - cos .do) sin pl sin

+ cos a0 sin Po sin (vI - 9,) .

1 sin 9 1 sin i 72 = r; (COS (pl - p2) - COS + p)) :

and

ive get the desired equality. i

3.2 Lengt h of intersatellite links

In this section, we study the impact of various parameters on the length of inter-

satellite links. Consicler an interorbital link that connects two satellites with phase

offset A+ We start tvith the length variation with time t . Figure 13.3 plots the length

variation of tmo sample interorbital links as a function of instantaneous phase q,.

Roth links connect trvo satellites in orbits witli inclination no = 65' that are ,do = 3 6 O

apart, and with altitucle 1400 km. The phase offsets Ap of t,he links are -18' ancl

+ 18'. respectively. The plots show a significant variation of the iink lengtli oïer

t ime. According to Lemmas 3.1 and 3 .3 tlie link achieves its minimum when cos Zp,

is miriimized. i.e. wlien 2q t = f 1SOo7 or 9 1 = f 90'. This occurs when the linli

peaks arouncl its minimum and maximum latitude. Similarly, the length achieves the

maximum wlien the link crosses the equator. These positions are shown in Figure 3.4.

The nest parameter ive fociis on is the phase offset between the interconnected

satellites Ay. Figure 3.5 plots the length of an intersatellite link a t its minimum and

maximum positions as a function of Aily. These positions correspond to vt equal

. .

Delta phi = +18

0 1 1 1 , I 1 1 -180 -135 -90 4 5 O 45 90 135 180

phi-t

Figure 13.3: T h e length variation of interorbital links as a hinction of their instanta- neous phase vt. T h e parameters are: a0 = G 5 0 7 go = :36", ancl a l t i tude is 1300 km. The phase offsets of the links are -18" and +lSO; respectivel.

I min

-180 -135 -90 -45 O 45 90 135 100

Figure 3.4: The minimum and maximum length of an intersatellite link.

CiH_APTER 3. XET'IVORK TOPOLOGY

Delta phi

Fig~ire :3..5: T h e impact of Ap on the lengt ti of intersatellite links at their minimum and masiniurn positions. T h e inclination is cro = 6.j0, the separation between the orbits is = :3G0, and t h e altitude is 1400 km.

to 90" and 0". respectively. T h e parameters a re a. = 6.5'. ,JO = 3 6 O ancl al t i tude is - -0 l 4 O O km. lVe note that phase offsets A;7 < - ,b and Ay > 37' a r e unrealistic since

for t liese values the intersatellite angle is too large. and one can show t hat the beam

~voiilcl interçect the ear th surface when t h e link crosses t h e eqiiator. T h e figure shows

a fairly strong impact of the phase offset on t h e link length. -4 natural objective

in mininiizing intersatellite distances is minimizing the length of t h e links. Nest ive

present a t heorem t hat gives the phase offset tliat achieves tliis n i in im~im. To make the

presentation clearer. we first int rodiice a lemma t hat gives the instantaneous latit ucle

and longitude of a satellite.

Lemma 3.4 Let S be n satellite in art orbit with inclinntiorz ao7 and let the lorzgi-

tiide o j its descending node be 0'. Let th.e instnntaneous phase o j S be y. Then its

in-dnntnrzeous latitude a , and longitude /3', sntisfy

0, = - arcsin (sin a0 sin 9)

Proofs W i t h o ~ ~ t loss of generality, Ive can assume that the radius of the orbit is one.

Let f r . y . 2 ) be the instantaneous coordinates of the satellite. The instantaneous

latitude ct; can be clerived from the fact that sin ct; = 7 = - sin a0 sin p accortling t o

Lemma 3.2.

Let c f p be the distance between the origin of the coordinate system and t h e pro-

jection of the sateliite's location ont0 the xy-plane. Then the longitude ,8, satisfies

sin,& = *. Using the equalities d:, + z2 = 1: sina, = 2: and Lemma 3.2. Ive obtain df

cos a0 sin 9 sin,d, =

J=

Theorem 3.5 The iength of an intersatellite link betîüeen t tro st~tellifes in orbits u:ith

inciinntiorz no and with the angle betroeen their descending nodes q u a i to & i-i a t an9

tirne irzsfnrzf rninimized for the phase oflset

cos 9 Apopt = -2 arccos

2 & JI - sin2 a. sin .,

Proof: By Lemma 13.1. minimizing link Iength is ec1iiivalent to masimizing cosyt. By

Lernnia :3.:3? for any time instant t : cos yt is masimizecl for the phase offset -19 tha t

masirnizes the espression

1 - sin2 oo (1 - cos ,Oo) cos Ap + COS Po cos Ap - COS a. sin ,Bo sin AP. 3

Since this expression does not clepencl on time t , it is sufficient to fincl the value

Aisopt that minimizes the link length at any given time instant. Consicler the time

instant when pt = -90' and the link length achieves its minimum. The link length

a t this position is minimal when it is eclual to zero. This occurs when the connected

Figure 3.6: T h e geometry for optimal phase offset A;s.

satellites cross the intersection of their orbits at the same t ime (see Figure 3.6).

T h e instantaneous phase of the satellite moving south a t this moment is eyiial to

p = -90' - l y O p , / 2 . -4ssuming t h a t the Longitude of its tlescencling nocle is O". al1

ive need t o do is to gitarantee tha t the longitude ,Ov a t phase 9 is -90' + ,B0/2.

To get the longitude at phase 9. we plug in s i n 9 = sin(-90" - AqOpl/2) =

- cos(AyOPt/2) into the formula for instantaneous longitude ,JG from Lemma 3.4:

w here

The latter irnplies

COS a0 ,$i = - arcsin (i A90Pt

COS - cos Q, 1 - 1 ) -

+opt sin a, = sin a0 cos - 3 3

and

CH-M'TER 3. NETWORK TOPOLOGY

Plugging this and Pp = -90' + - 2 into (3.11, we obtain

PO - COS Qo - arcsin +opt -90" + T) - cos -) . - 3 -

This implies

sin (900 - $) = COS 00 Af O#

COS - Aaopt JI - sin2 0 0 cos2 3

.d

Bo COS 0 0 A p o p t COS - = COS -

-2 Avopc 3 . JI - sin2 a0 cos2 -

By squaring and rearranging t h e terms: this can b e turned into

Apopt cos2 - ( 2 D o

3 cos2 00 + sin2 a0 cos2 5) = cos - - - -? -

2 h y o ~ t 1 - sin' a. cos - ( -2 -

Since both and 2 a re in [-90°, 90°]_ we get

+op: 30 COS 2

- COS - - .

Finally, since the condition tha t the satellites cross the intersecting point of tlieir

orbits sirn~dtaneously reqtiires t h e phase offset t o be negative? ive get

00 - COS 2

Apopt - -2 arccos 2 & JI - sin2 a. siri ,

a

To discuss t h e selection of interorbital links for a particulaï satellite constella-

tion, let n h denote the number of orbits and n,. t he number of satellites per orbit.

If the satellites are regularly spaced in their orbits, we iisuallÿ cannot achieve the

optimum phase ciifference Ayopt for satellites on a11 orbits simultaneously (note tha t

,fio = :360°/nh). For the sake of simplicity, we will assume that al1 orbits have the

same phasing. If we den0t.e by = 36O0/nw the phase difference betmeen two

successive satellites in one orbit, then the phase difference of interorbital links Av;,~,,

m u t be a multiple of Apint,,. i-e.. ilvinter = x 360°/n. for some integer ko we

cal1 skei~? per orbit. To achieve i lp in te , = Apopt , one neecls Aisopt = ko x 360°/n, or

k o / n , = i190pi/3600. This means that unless ~+,~/360O is a rational nuinber. it is

impossible to achieve the exact value of Ap,,,. If both n , and ko can be arbitrarily

chosen. one can approxirnate the value AyOp1/36O0 with the ratio l&/n,. T h e accu-

racy of the approximation increases with increasing nu. Hoivever. increasing nu means

putting more satellites into orbits increasing the cost of the system. In practice. the

value of r 2 , . or a t least its range, is likely to be determineci b - the desired coverage

and other technological and commercial issues. Then. to minimize the interorbital

link lengtli. the skew per orbit can use the approximate valoe by rounding off the

fraction:

The values of ko obtained by rounding off thiç value for constellations with inclination

6.7" ancl n h and n, in the range of 5 to 130 are shown in Table 13.1 in t h e appenclis

to this chapter. T h e values of nh or n, smaller than 5 result in beams tha t intersect

the earth surface even for altitude 1500 km. which is approsimately the maximum

altitude of L E 0 satellitesS regarclless of the choice of interorbital connections. .As

a result: they are not physically possible. One can see fsom Theorem 3.5 that , in

genera.1, the minimum link recluires a non-zero phase oEset. -As shoivn in 'Table :3.1,

unless n , is small relative t o T Z ~ ? this results in a negative skew per orbit b. The

impact of the negative sketv per orbit on the resulting topology of t he network is

discussecl in the nest section.

In the case when the choice of constellation is a t least partially flesible, one can

also stucly the impact of nh ancl n , on the length of the shortest achievable link. i-e.,

when optimal ko given by (3.2) is itsecl. The plot in Figure 3.7 shows the length of

the link a t its minimum position as a function of nh ancl n, in the range fronl 5 t a

30. The two "valleys7' in t he bottom-right corner correspond to the cases when the

value of the fraction in formula (3 .2 ) is close to the integers -1 and -2' respectivel~r.

Figure :3.7: T h e length of interorbital links a t their minimum position when optimal skew is used as a function of nh and n,. T h e inclination is a0 = 6.5" ancl the al t i tude is 1400 km.

If a n objective is to minimize t h e number of satellites, one can t ry to approximate

t h e optimal phase offset mit h t h e smallest possible skew per orbit ( in terms of

absolute valiie) = -1. From (:3.2), we get

These values are plotteci as a function of nh for constellations with 6-5' inclination in

Figure 3.8.

Finally. i re mention the impact of t h e inclination o n the interorbital link lengths.

Figure 3.9 shows the link length a t its mininitim position as a function of t lie phase

offset A y for several values of ao. LVe see that t h e actual values Vary slightly bu t ,

~~ual i ta t ive ly . the shape of the curves is similar.

3.3 Skewed torus topology

The prcvious section discussed t h e distance between two satellites in two different

orhits. It gave the formula in Theorem 3.5 that says mhat the phase offset between

Figure :3.S: T h e optimal value of n, for skew per orbit ko = -1 as a function of nt, for constellations with inclination 6.5".

phase otlset

Figure 3.9: The impact of inclination no on the iengtli of intersatellite links a t their minimum position. The considerecl values of inclination cro a r e 6.5' and 55': and of the angle Po between the clescending nocles are 90° and 36'. T h e altit ilde is 1400 lim.

the satellites should be if ive want to minirnize their distance. This section discusses

the impact of this choice on the topology of the interconnection network. We focus

on fixed topologies in mhich each satellite always maintains links to the same set of

satellites. 'vloreover, we assume that the topologies are symmetric. i.e.. that the set

of the connected satellites is defined in the same way for every satellite. The number

of links per satellite can vary. ancl topologies ranging from 3 to 8 links per sateHite

have been proposecl in the studies rnentioned in Section '2.1.

In what follows. ive restrict ourselves to four links per satellite. two intraorbital

links connecting to satellites in the sarne orbit. and two interorbital links, one con-

necting to a satellite in each of the neighboring orbits. The satellites connected by in-

traorbital links are in fixed relative positions and clistances, so the links clo not require

steering of the transmitters and receivers. -4 natural choice seems to be connecting a

satelLite to its preclecessor and successor in the sarne orbit. These connections appear

in virtually al1 stuclies. and will also be assumed throughout this work. The focus of

this section is on the choice of interorbital ISLs. The choice. particularly in inclined

constellations, is more comples than that of intraorbital links. There are severaI cri-

teria that can be ~ised including minimizing ISL length, average or maximum path

lengt h. clel-- ji tter. ease of hancling off connections. and steering of the heams. One

of the fiinclamental parameters of communication networks in general is the distance

between nocles (the satellites in t his case). Lve concentrate on t his parameter. and its

impact on the netivork topology.

Let n h denote the number of orbits. and r 2 , the niimber of satellites per orbit.

The intraorbital links create n h cycles. one per each orbit, each containing nt, nodes

(satellites). The choice of interorbital Iinks determines the way these cycles are iriter-

connectecl. If each satellite is connectecl to the satellites with the same phase in the

two neigliboring orhits: traversing interorbital ISLs forms cycles of length n,,. The

resulting topology is an orclinary torus of size n h x n , (see Section 2.2). However, the

previous section shows that the clistances between satellites in two different orbits is

minimized when their phases cliffer. If formula ( 3 . 2 ) is used and I& # O then traversing

interorbital ISLs "shifts" the position of the satellite by ko positions with each orbit.

CK4PTER 3. NET'Ci'ORK TOPOLOGY

-4s a result: after nh links, the position is shifted by k = nh x /Q positions. This re-

sults in a modified toroidal topology we cal1 a skewed torus. -4 k-slzezued torus of size

nh x n, is formed in a similar w - to a regular nh x n, toms with the esception of the

horizontal wrap-around connections that are "skerved': the edges [(nh - 1, j ) , (O- j )] .

O 5 j 5 n,-1 are replacecl by theedges [(rih-1: j ) : (O, ( j + k ) mod n , ) ] , O 5 j 5 nu-1.

Conceptually, we cut the torus vertically hetween columns nh - 1 and O and reconnect

each node in colurnn nh - 1 to the node in colurnn O that is k rows up. See Figure 3-10

for a n esample. The 1-skewed toms kvas introduced and studied in Armitage [2].

nu n h x nu

Figure 3.10: -4 3-skewed torus of size 4 x 6.

The k-skewed torus is also related to Miclimew netmorks [dl. a class of clouble-loup

circulant graphs [jl]. ..A rectangular Midimew netivork is a special case of a skewecl

torus with skew equal to nh - 1.

In the contest of L E 0 satellite networks? the skeiv of a skeived torus is cletermined

by the choice of interorbital links. If the phasing of al1 orbits is eclual then skew per

orbit ko is an integer, and the skew k = nh x ko is a multiple of the number of orbits.

If n h x Aso > r z , then the topology is ecluivalent to a skewed torus with skew eclual to

(nh x ko) - n,. If the phasing of orbits is clifferent, fract ional multiples of nh are also

possible. One proposa1 for an inclinecl constellation witli 12 orbits. 6 satellites per

orbit? and different phasing in the orbits investigated in Werner et al. [16] is described

in section 2.1.1. The topology in [?6] is a 6-skewed toms but. since the total skew

is the same as the number of satellites per orbit, the resiilt is a n ordinary torus.

Figure :3.11 shows the interconnections for a n inclined satellite network which has 4

orbits rvith the same phasing, 10 satellites per orbit? and skew 3. FVë note that four

Figure 3.1 1 : An ititerconnect ion for an inclined satellite constellation.

orbits are too few to achieve a universal coverage; ive use it for clarity of the figure

only.

3.4 Length of end-to-end paths

This section studies the distances between arbitraïy pairs of sateHites defined as the

sum of the lengths of the links on the shortest path connecting the two satellites: i.e..

the length is measurecl in kilometers, not nuniber of links (hops). Because of the in-

terorbital link lengths varying accorcling t o the cornples forrnulae given in Lemma :3.:3,

ive were not able to obtain analytic solutions for end-to-encl distances o r path lengths.

Insteacl. we obtainect nunierical results implementing the Floyd-LVarshall afgorit hm

for all-source shortest path lengths [II]. We will first demonstrate t h e impact of ko.

nh and n , for constellations rvith inclination a0 = 6.5" and eclual phasing of orbits.

The path lengths were calculated for t he t ime instant when one of t h e links achieves

its minimum length, i.e. when it peaks at its maximum latitude. Then we discuss the

CHAPTER 3. L\~ET\~'ORII;: TOPOLOGY

impact of relasing these conditions. We calculated the maximum pat h Iengt h (diam-

eter) and the average path length between satellites for constelIations mith niimber

of orbits nh aricl number of satellites per orbit n . ranging from 5 to 30. .A major

result of this section shows that the skewed toms topology typicaily minimizes not

only the distance between directly connected satellites but also the total end-to-end

pat h lengt hs.

Similarly to the previous section. ive start with the impact of interorbital connec-

tions. Figure 3.12 shorvs the maximum and average path length as a function of ko

for a constellation with nh = 10 orbits and n , = 130 satellites per orbit. The altitude

of the orbits is 1100 km: note that the path lengths are defined as the surn of lengths

of their links. not the number of links. We note that only the values of between

x l0* alpha-O = 65. phi-t = 0. rel.phaçe = O. n h = 10. n-v = 30 1% I 1 I 1 p. t

Figure 3.12: The impact of the skew per orbit ko on the maximum ancl average path length. The inclination is a. = 65". nh = 10. n , = 30, and the altitucle is 1400 kni.

-6 ancl + 3 avoid the beams intersecting the eartli surface. The figure shorvs that a

proper choice of the interorbitai connections can have a significant impact on hoth

the maximum and average path lengths. At the end of this section ive compare the

values of ko that minimize the link length ancl path lengths, respectively.

The impact of the constellation parameters nh and n, is shown in Figures 3.1:3

and :3.14. Figure 3 . 1 3 shows t h e maximum path lengt h? while Figure 3-14 shows

Figure 3.13: T h e masimiirn path length as a function of nh ancl n, when t h e optimal skew per orbit ko is iised.

the average path length. In both cases: the skew per orbit ko was chosen t o minimize

the particular parameter. The relatively flat surfaces in the left portions of tlie fig~ires

correspond t o cases when the ko = O is optimal due to the low riumber of satellites.

-As the riumber of satellites increases: t h e skew per orbit ko can approsirnate more

closely the optima1 value of interorbital phase offset. T h e figures also show t hat the

average path lengt h is equal to about one half of tlie masiniiim path lengt h.

Xoiv Ive tiirn to the other parameters tha t have impacts on the path lengths. In

our calculations~ we first ftxed the parameters rzh = 10. n , = 30. ancl ko t o t h e value

minimizing a particiilar parameter. T h e n ive varieci the remaining parameters. i.e.,

the time instant of the path length computat ion, inclination: ancl relative phasing of

0rbit.s. T h e results are plotted in Figures 3.15 t o 3.17. Figure :3.15 shows t h e impact

of the t ime instant when the path lengths are calculated. The inclination is a0 = 6.j0,

and al1 orbits are equaIly phased. I t shows tha t it does not influence the results

in a significant way. T h e impact of inclination n o shoivn in Figure 3.16 is somewhat

more significant; liere, the time instant corresponds to tlie moment when one of the

Figure 13-14: The average path length as a fiinction of nh and n , when the optimal skew per orbit ko is used.

,,* alphag = 65. re1,phase = O. n-h = 10. n-v = 30, k g = minimimg

Figure 3-15: T h e impact of time instant on the average and maximum pa th length. The inclination is cto = 65", nh = 10, n, = :30: and t h e altitude is 1400 km.

CHAPTER 3. NETki'ORK TOPOLOGY 49

links achieves its minimum length. The figure shows that the path lengths slightly

phit = 0. ?el-phase = O. n-h = 10. n-v = 30. k-O = minimizing

Figure 3.16: The impact of inclination a0 on the average and maximum pat h lengt h. The nurnber of orbits is nh = 10. the nrrmber of satellites per orbit is nu = :30- and the altitude is 1400 km.

increase with increasing %. the variation from 2.5' to S.3' being about 12.5%. Hoivever:

one shoitlcl note that the inclination is more likely to be determinecl by the desirecl

coverage rat her than minimizing pat h lengths. FinalIy, Figure 13-17 shows the impact

of relative phasing of the orbits: in this case a0 = 65" and the time instant again

corresponds to the moment when one of the links achieves its minimum. The s-asis

shotvs the relative shift of the phase bet~veen t r ro neighboring orbits as a portion of the

angle between two corisecrrtive satel!ites in one orbit A;s;,,,,; tlie value O corresponcls

to equal phasing of orbits. The figure suggests that tlie path leiigths clecrease with

increasing phase shift reaching a minimum for shift equal to -lpint,,/2. The issue

deserves more attention in f ~ w t her stuclies.

We end this section ivith a cornparison of minimizing the length of single interor-

bital links and minimizing the total path lengths. The choice of the skew of a slewed

torus satellite topology enables one to affect trvo parameters of the network: (i) the

length of interorbital links (measured in kilometers), and (id the number of links

Figure 3-17: The impact of relative phasing between orbits on the average and masi- mum path Iength. The inclination is a0 = 6.5". the number of orbits is nh = 10. the number of satellites per orbit is n, = :30, ancl the altitude is 1400 km.

(hops) necessary to reach t he most distant nocle ( the former being a subject of this

chapter, while the latter heing disc~issecl in Chapter 5 ) . The actiial path lengths re-

sul t from the interplay of these two factors. We start the comparison with inclination

cto = 6.5" and eclually pliasecl orbits. The optimum skew minimizing the link lengths

is given hy (i3.2). Figure 13-18 compares the maximum path length (cliameter of the

network) for ko from (13.2) t o t he minimum possible value obtainecl from oiir niimerical

stuclies. T h e fig~ire shows tha t in most of the cases t h e minimum Iink length also

minimizes the maximum path length. The clifference in the remaining cases is less

than 7% of its optimal value. The results are similar for the average path length. the

difference being ind der 1.5% of the optimal value as shown in Figiire 3.19. T h e very

close match of the resiilts is cliie to the fact that niinirnizing the above-mentionecl

criteria (i) and (ii) leacls t o approxirnately the sanie skew per orbit for inclination

cro = 65"- This is not necessarily the case for other inclinations. Figures 3.20 and i3.21

siirnmarize the results for inclinations between 2.5" and 8.5". T h e values of nh anct r ~ ,

usecl are mtiltiples of 5 in t he range from .? to 130. Figure :3.IO plots the percentage

Figure 3-18: The relative difference between maximum path length for the skew opti- mizing the intersatellite link length and its minimal value. The inclination is a0 = 6.5".

Figure 3.19: The relative difference between average path length for the skew opti- mizing the intersatellite link length and its optimal value. The inclination is a0 = 6J0.

of cases when t h e minimal link lengt h does not minirnize t h e maximum and average

path length, while Figiire :3.21 plots the highest relative difference for both the maxi-

mum and average path length. Our inspection of the particular cases reveals tha t

X . max .

Figure 13.20: T h e percentage of cases when minimizing link length does not minirnize the maximum ancl average path lengths for n h and n , m~ilt iples of 5 in the range 5 to 30.

the higher discrepancy for the lower inclinations is typically due to taking aclvantage

of a skew tha t significantly recluces t h e number of links necessary t o reach the most

distant satellite. T h e impact of this reduction is stronger on t h e diameter (maximum

path length) of the net~vork than o n t h e average path length. It is quantifiecl cising

an approximation to the interorbital link lengths in Theorem 5.4 of Chapter 3. Nev-

ertheless, the plots suggest that minimizing interorbital link lengtlis is usualIy a goocl

heuristic for achievint; low end-to-end path lengths.

Figure 13.21: The highest relative difference of the maximiirn and average pat h lengths between the values obtained for t h e skew rninimizing t.he intersatellite link length ancl their optimal values. The values of nh and n., are multiples of 5 in the range 5 to 30.

CHAPTER 3. I W T W O R K TOPOLOGY

Appendix

Table 3.1: The value of skew per orbit ko minimizing link lengths for given number 6.5'. of orhits nh and

nu

10 11 12 13 14 1.5 16 17 18 19 20 21 S'L 23 24 2.3 26 27- 28 29 :]O

satellites per orbit n , for constellations with inclination nh

.5 6 7 8 9 10 11 12 13 14 1.5 16

- 5 0 0 0 0 0 0 0 0 0 0 0 0 6 - 1 0 0 0 0 0 0 0 0 0 0 0 7 - 1 - 1 0 0 0 O O O O O O O 8 - 1 - 1 - 1 0 0 O O O O 0 O O 9 - 1 - 1 - 1 O O O O O O O O O

-1 -1 -1 -1 O O O O O O O O -1 -1 -1 -1 -1 O O O O O O O -1 -1 -1 -1 -1 -1 O O O O O O -1 -1 -1 -1 -1 -1 -1 O O O O O -1 -1 -1 -1 -1 -1 -1 -1 O O O O -1 -1 -1 -1 -1 -1 -1 -1 O O O 0 -2 -1 -1 -1 -1 -1 -1 -1 -1 O O O -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 O O -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 O -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 - 1 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -3 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 - 3 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 - 3 -2 -2 -2 -1 - 1 -1 -1 -1 -1 -1 -1 - 3 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1

Chapter 4

Modeling intersatellite links

In m~~lt iprocessor networkç and many terrestrial networks, the clifferences among the

propagation delays associated with the links are small enough to be ignored. so it

is reasonable to use models that assume al1 links have the same lengt h. We call

such a model a one-.irn$orm model. In a LE0 network. it is reasonable t o assunie

that al1 intraorbital links have the same length. but what about the interorbitai links

which have lengt lis constantly changing accorcling t o the complex formulae clerivecl

in the previous ctiapter:' This chapter focuses on fincling tractable rnodels of the

interorbital link lengths. a n d introcluces simple rnocleIs that approsirnate the actual

\ dues . Throughout t h e whole work. it assumes t hat inclinecl co~lstellations are mecl

ancl tlmt al1 of the orbits use t he same angle of inclination.

Section 3.1 clefines two ways of approximating the interorbital link length-linenr

arid constnrzf. The nunierical results show that the typical average error is under 5%

for the linear, and under 20% for t he constant approximation. -4s an initial stiicly. it

siiggests the use of the constant approsimation leacling to a model ive call t u o - u n i j o m .

in which al1 intraorbital links h a ï e t he same length and al1 interorbital links have the

same lerigth. The model. clisci~ssed in Section 4.2, neglects the tirne-variability of the

network but. at least partialiy, it c a p t u e s i ts space-variabili ty. Section 4.:3 disciisses

another aspect of intersatellite linlis tliat is relatecl to the handshaking protocol iised-

namely. the t ime a sender is occii pied transmit ting one message.

4.1 Modeling link lengths

This section discusses mocleling and approsimating the length of intersatellite links.

Since two satellites in the same orbit keep a constant relative position and distance.

the length of intraorbital links does not varjr with time and position of the sateliites.

If al1 orbits contain an equal number of regularly spaced satellites, the length of al1

intraorbital links is the sarne. Therefore. we moclel them by just one value.

The lengths of interorbital links va r - as the satellites orbit the earth. The corre-

sponding forrnulae were developed in Lemmas 3.1 and :3.:3. Because of the complexity

of the expressions, ive propose two ways to approxirnate them. The first method lin-

earlq- approximates the curve between its minimum and maximum, while the second

one uses a constant value equal to the midpoint between the minimum and maximum.

From Lemma :3.:3 we see that an interorbital link acliieves its minimum iength

when cos '2pt =

minimum lengt h

-1, and its maximum length when cos '2vt = +1. Therebre, the

of the link is eclual to

l +- sin2 a. ( 1 - cos Po) cos A p + cos ,Oo cos Ay 3 d

- cos a. sin ,go sin J.F.

Similady, the maximum length is

w here

1 +;, sin2 û o (1 - cos Bo) cos A p + cos ,Oo cos l p &

- cos 01, sin ,ao sin ilp.

-180 -135 -90 4 5 O 45 90 1 35 180 phase: ph,-osphi-t

Figure 4.1: The linear and constant approximations of interorbital link lengths. In- clination is = 6.5'. n h = S: Ai, = -18'. the altitude is 1400 km.

The linenr approximation uses a linear segment between each pair of consecutive

minima and maxima as slionm in Figure 4.1. The corresponcling forrnulae can be

?;ote t hat t here are arternative ~vays of defining a piece~vise linear approsirnat ion. For

esaniple. one could ilse a larger nurnber of linear segments or a fit ting teclinique such

as least square error fit iiisteacl of connecting the local minima and maxima with a

straight line.

The constant rrpp.ro.t.imntion uses a constant value

for al1 time instants t . An example is shown in Fig 1.1. This method disregards the

rime-varying aspect of the network. However, ive tliink tha t it is a natural initial

step for studying this type of networks since it does capture the difference between

the two types of links-interorbital and intraorbital. Mûreover, as we will see below,

in some cases it outperforms the linear approximation in terrns of approximating the

path lengths between nocles of the network.

The rest of this section discusses the error introcluced by the proposecl approxima-

tions. First we show sample plots of the relative error for fixed values of inclination

00. relative phasing, and tirne instant as a filnction of the number of orbits nh and

satellites per orbit n,. Figures 1.2 and 4.3 show the relative error of the average path

Iengt h for 00 = 6.5". eclually phased orbits. and the time instant when one of the links

achieves its minimum Lengt h. The ranges of nh and n, are 5 to :30. Basecl on the

Figure 4.2: The relative error of the average pat ii length for the linear approximation. The inclination is cuo = 6.-jol the orbits have eyual phasing. ancl the skew per orbit ko minirnizes the intersatellite Iink lengt hs.

results of section 3.4, the skeiv per orbit ko was chosen to rninimize the ISL length

accorcling to formula (3.2). From the figures we see t hat the error for bot h t,he linear

and constant approximations is less than 15%.

Nest: we illustrate the impact of the inclination and the time instant on the

accuracy of the approsimations. We calculated the error of the approsimations for

Figure 4.3: The relative error of the average path lengt h for the constant approsima- tion, The inclination is a0 = 63", the orbits have equal phasing, and the skew per orbit l;o minimizes the intersatellite link lengt hs.

the values of nh and n . that are multiples of 5 in the range between 5 and :30 for

varying values of inclination and t ime instant. respectively. Then ive obtainecl bot h

the niasimum ancl mean error taken over al1 valries of nh and n , . Figure 4.4 shotvs

the maximum ancl average of the relative error of the average path lengt h for bot h the

linear and constant approximations. -4 similar plot for the maximum path length is

shoivn in Figure 4.5. The plots support the expectation that , on average. the Iiriear

approximation outperforms the constant one. It also shows that the error of the

constant approsimat ion s harply increases for higher inclinations, part icularly above

6.5". However. high inclinations are impractical since they result in large COL-erage

overlap between two opposite orbits. Figures 4.6 ancl 3.7 show similar plots as a

hnction of the instantaneous time when the pat hs were calculated (the inclination is

fised a t EO). As espected, its impact on the error is much smaller.

avg. path lengths: rel-phase = 0. phi-t = O. k-O = min.

25

8

8 avg 8

. - -.- lin

Figure 4.4: The relative error of the average path length for the linear ancl constant approximations as a function of the inclination. T h e values were taken over a range of values nh and n,. The skew per orbit ko minimizes the intersatellite link lengths.

max. path !engths: rel-phase = O. phit = O. k-O = min.

455

rnax

/ r i

. - se-- -'- - - lin

25 35 45 55 65 75 85 alpha-0

Figure 4.5: The relative error of the maximum path lengtli for the linear ancl constant approximations as a function of the inclination. T h e values were taken over a range of values nh and n,. The skew per orbit ko minimizes the intersateliite link lengths.

avg. path lengths: rel.phase =O. alpha-O = 65. k-O = min-

lin: max

- -.- - - -.-. _____________-l_-___

- - _ _ - - - - - --- - - - - - _ - - - - _ _ - - - - - cmst: avg

phi-t 1 dghijntra

Figure 4.6: T h e relative error of the average path length for t he linear and constant approsimations as a function of the t ime instant. The values were taken over a range of values IL^ and n,. The inclination is cro = 65": ancI the skew per orbit /& minimizes the intersatellite Iink lengths.

The previoiis section proposed t.wo approximations to the length of interorbital ISLs.

This section discusses t heir use for rnocleling the whole interconnection topology. The

topology is moclelecl by a labeled graph whose nocles (vertices) correspond to satel-

lites and edges t o intersatellite links. Each edge is labeled by the lengt h of the corre-

sponcling link. -4s the length of intraorbital ISLs is fised. we assume that the eclges

corresponding t o t hem are labeled by t heir exact lengtli. The labels of the eclges cor-

responding to interorbital ISLs depend on the cliosen approximation. If ei ther exact

values or the linear approximation is iisecl, the labels must be time-clepenclent. We

propose to use t he constant approximation as an initial step toward stuclying this

type of networks. In this case, the interorbital ISLs are labeled by the average of

t lieir minimum and rnasimum lengt hs. The constant approximation does neglect the

time-varying aspect but it captures the fundamental difference between the tmo t.ypes

of links-intraorbital and interorbital. kforeover, as seen in the previous section, in

CWAPTER 4. î . 1 O DELkVC: INTERSATEL LITE LlNKS

Figure 4.7: The relative error of the maximum pat h lengt h for the linear and constant approximations as a function of the time instant. The values were taken over a range of values .nh and n , . The inclination is a0 = 6s0 , and the skeiv per orbit ko minimizes the intersatellite link lengths.

some cases it outperforms the linear approximation in terms of approsimating the

path lengths between the nodes of the networks.

If a four-link mesh-like topology described in Section 3.:3 is used, it results in a

moclel that ive cal1 t w o - u n f i r m . In this moclel al1 intraorbital links have the same

lengtli, and al1 interorbital links have the same length. Hence. the topology is a

(wrappecl-around) rect angular mesh ivi t h tivo different values of link lengt h-one.

denoted 'Dy r : for vertical links. and the other, denoted by h , for horizontal links (see

Figure 4.8). In the test and figures below. the vertical links ivill represent int.raorbital

IS Ls, while the horizontal ones represent interorbi ta1 IS Ls.

The constant approsimation can also be applied to other types of interconnection

patterns. A hexagonal mesh mith sis links per satellite was proposed in Woocl [-XI.

One can again moclel al1 interorbital ISLs with one value resulting in a two-uniform

moclel. Hoivever: since there are two types of interorbital ISLs with two different phase

offsets and two different average lengths, a more natural choice is to mode1 each of

Figure 3.8: A two-uniform rectanguIar mesh.

them with a clifferent value (eclual to the average of their minimum and maximiim

lengths). This resuIts in a three-uniform model: one length for intraorbital ISLs and

two for interorbital, one for each type. .A four-uniform model coulcl be appliecl to the

topology chosen for the Teledesic system [al] that uses eight links per satellite-four

intraorbital and four interorbital. The properties and use of r-uniform moclels with

r- 2 3 are left for fiirther studies.

The above disc~ission focusecl on the variations of propagation clelay due t o changes

in the lengths of intersatellite links. However, the varying link lengt hs have an impact

on the clata rate as well [L9]. A larger distance between the connected satellites means

larger atteniiation of the signal, and hence lower clata rate if a constant power is usecl.

Varying distance between satellites also causes Doppler shifts which limit the sviclth of

the frecluency bands tliat can be allocatecl to comm~inication channels. -4s a residt, it

is also reasonable to clistingiiish between the intra- ancl inter-orbital links wit h respect

to the data rate. If adopting the linear-cost model described in Section 2.2: we can

introcluce a two-uniform model with four parameters: f ih , the propagation clelay

and transmission time of a horizontal link, ancl B,, 5-t he propagation cleIay and

transmission tinie of a vertical link.

4.3 Modeling communication modes

This section adds one additional parameter t o the parameters of communication links

discussed in Section 2.2. It deals with t h e tirne a sencler is occupied transmitting

one message, which is related to the handshaking protocol used. We distinguish two

modes. The synchrone us mocle is a connect ion-oriented type of hanclshaking protocol

in which two satellites establish a connection, remain connected while the information

is transferrecl, ancl then break the connection. Depending on whicl-i type of link is

used. the total tinie for a synchronous communication in the two-tiniform mode1 is

eit her h or u t inx units for the constant-cost model, and ,Oh + Lrh or 13" + LÏ,. for the

linear-cost model. It moclels protocols like t he binary synchronous control protocot [il

that reclriire a n acknowleclgment before a sender can send another message. This type

of protocol reduces the utiIization of networks with hi& propagation delays induding

LE0 satellite networks. Therefore, we relax the condit ion in the asynchrone u s niocle

which we now introduce. In the asynchronous mode, t lie sending satellite transmits

the information without first establishing a connection. ancl can then ssvitch to other

transmissions wit hout wai ting for the information to be receivecl or acknowlecigecl.

-4 satellite is busy only while transmitting or receiving regarciless of the latency of

the link: this is one time ~ in i t for the constant-cost model. and L r t ime rinits for the

linear-cost model. Note t hat a Bow-cont rol-based protocol falls somewhere between

the synchronous ancl asynchronous modes.

Part II

Communication algorit hms

Chapter 5

Constant-cost two-uniform toroidal

meshes

Part I of this thesis introduced the skewed toriis topology and the tivo-uniform ap-

proximation to the transmission time on intersatellite links as simple mociels of L E 0

satellite networks. This and the nest chapter use these models to study efficient

communications in these networks. T h e main focus is on a stanclarcl communication

pattern called gossiping or all-to-al1 eschange. This is a commiinicat ion pattern in

ivhich each node (satellite) of the network needs to distribute its piece of informa-

tion to al1 other nocles. as can be needecl in the clissemination of control information

t hroughout the network. This chapter. whose results appeared in [?dl. considers the

propagation clelay of the links only, i-e., it assumes the constant-cost moclel in wliich

the time t o transmit one message over one link is inclependent of the message size.

ancl is equal to the propagation time of the Iink. Folloaing t h e notation introcluced

in Section 4.2. the clel- on the intraorbital links will be clenoted by u : and the clelap

on interorbital links hp h. Corresponclingly, in figures, the former ivill be drawn \-er-

tically, ~vliile the latter horizontally. The nest chapter enhances the mode1 by taking

into account the da ta rate of the links, and makes the transmission time dependent on

the message size. ive start t h e presentation of gossiping algorithms for the constant-

cost moclel wi t h an algori t hm for regiilar, unskelveci toroidal meshes. Its one-uniforrn

special case (al1 links have the same clela>-) improves t he best known upper bound

for the problem. Then me turn to skewed tori. and present an algorithm that takes

advantage of their smaller diameter. Note that the diameter of a two-iiniform network

clepends on the lengths of the links; not just number of hops. Lve present algorithms

for both the synchronous ancl asynchronous transmission modes. 8 1 1 the algorit hms

are developed for the one-port half-duplex model. Their running times are mithin

additive constants of the diameter of the network, which is a natriral loiver bound. As

a result. the same running time provicles an upper boond on the gossiping problem

for the less restrictive models too. We mention though that utilizing the full-duplex

links or all-port nodes can reduce the constant.

5.1 Gossiping in a regular torus

In this section. ive present a gossip algorithm for ordinary (itnskewed) toroidal meshes.

IVe assume the one-port half-duplex model, and begin with a version for the syn-

chronous one-uniform moclel in ivhich al1 links are of length 1. We present a gen-

eralizecl. il-dimensional, version of our algorithm t hat recl~ices the upper bound of

Iirumme et al. [:32] on the gossiping problem for a (1-dimensional one-uniform toroidal

mesh wit h cliameter D from D + 1Scl+ :39 to D + :kl+ 5 . Then we adapt the algori t hm

for both the synchronous and asynchronous tmo-unilorm models. The algorit hm sim-

tilates gossiping ivithin a full-duplex cycle by two neighboring half-duples cycles. It

works in cl stages distributing the messages along one dimension at a time.

Theorem 5.1 Gossiping in n one-port hctlf-duplex cl-dimensional toi-ils with diclmeter.

D and d l link lengths q u a l to one, takes tirne at niost D +- :3cZ + 5 .

Proo f: Assume an ,n l x 722 x x nd tocoidal mesh. The algorit hm works in cl

stages: eacli stage distrib~iting the messages within one dimension. During the k-th

stage. neighboring cycles running in dimension k are paired up to simulate a gossipirig

scherne for a full-duplex cycle 1161. The dissemination of messages ivithin the two

cycles takes [nk/Z1 + 2 time steps. Summed over al1 dimensions. the total t ime is at

most xf=, ( rni/2] + 2) 5 D + 3d; note that the diameter of a cl-dimensional torus

d is &=, Lnk/2J. The additional five steps account for the initialization and handling

odcl size dimensions.

We start with the simulation of a full-duplex cycle by two interconnected half-

duplex cycles: i.e. \vit h gossiping in a n x 2 torus in time rn/Z1+1; ive note t hat stages

2.3 , . . . : d rvill use only rn/Z1+2 steps instead of rnli?] +1. First consider n even, and

denote the cycles C and Cr, and their nodes Co' C l , . . . .C.',-i and C';,Ci:. . . :CA-,. respectivel- In the first two steps each Ci and Ci exchange their messages (see

Figure a . l (a)) . Then the cycles forivard al1 messages in opposite directions. say C

Figure 5.1: Gossiping in a n x 2 torus. (a ) initial eschange, ( b ) forwarding rvithin cycles, ( c ) final exchange; the messages received by one pair of nodes are highlighted.

right and Cf left, for n/2 time steps. After each step, the nodes alternate tetween

transrnitt ing and receiving (see Figure 5.1 (b)) . iAÏe set the transmissions so that a node

Ci is transmitting whenever Ci is receiving and vice versa. Withoiit loss of generalit-

assume that Ci was receiving in the last step. Then, after n / 3 steps? Ci received the

messages from Ci-+ - , . . . , C,- and C': from C:+2, . . . . C:+nIz-I ( t lie

indeses are taken rnod~do n). Since each C, exchanged its message with C: the

collective knowleclge of C; and Cf contains the messages from the whole n x 2 toriis.

In aclditional two time steps they exchange their part. If n iç odcl. one nocle from

each cycle has to be idle during every step of the forivarcling phase. If ive let it move

from CL,/?, to C,-I in C , and froni Ci,/,,, to Ci in Cf. after [n/" steps, the nodes

C,-i7 Cf1; receive messages from their rnl-1 - 2 neighbors, CL-,; Ci from ri2/211 and

al1 other nodes from either rn/" or rn/Z] - 1 neighbors. Hence? every pair Ci. Cf is

informecl of al1 -Ln messages. The time bound [n /2] + 1 to gossip follows.

The algorithm for the nl x n;! x . - x n d toroiclal mesh starts with stage one when

we pair iip cycles running in dimension one that are neighbors in dimension tmo.

Each pair performs the nl x 2 gossiping in [nl /2] + 4 steps. -4fter that, in each

stage k = Z , 3 . . . . , cl, ure pair up cycles running in dimension k that are neighbors in

dimension one' and they perform the nk x 2 gossiping. Xote that a t the beginning of a

stage k > 2, the neighboring nocles aheady have the same information, and the initial

tivo step eschange is omitted. If bot h nl axid nz are even. this completes the gossiping

in tirne 2 + xi,, ( rnk/21 + 2) 5 D + 3d + 2. If n2 is odd' one cycle remains unpaired in

stage one. and we insert one step before and after the forwarding phase. During them

the nodes of the unpaired cycle send and receive al1 of their messages to and from

t heir neighbors. respectivelp. Finally? if ni is odd, during stages 2,3, . . . , cl, nodes with

the first coordinate equal to nl - 1 remain unpaired. We postpone informing them

until the very end of the algorithm. Their messages were already distributed among

other nodes diiring the first stage. Stages 2: 3 , . . . , d clescribed above disregard these . .

nodes. -4fter stage d is finished: every node (ni - 2: 12 : 1 3 . . . . id) informs its neighbor . *

( n i - 1 : 1 ~ ' 1 3 : . . .:id) completing the gossiping algorithm. Shus the total gossiping

time is a t most D + : 3d+ 5 . i

The same algorithm can he also applied to two-uniform tori. The diameter D

of a two-uniform torils is defined as t he total sum of the lengths of the links on the

shortest path connecting the two most distant nodes? i.e.? D = 121 h + 121 U. The

order of the gossips d o n g t lie tivo dimensions in the following tivo corollaries depencls

on the relative values of h and o.

Corollary 5.2 Gossipiny in ci one-port hay-duplex synchronoes t uv - uniform t x o -

dimensional toriis takes tirne n / most D + I ( v + h ) + 3 min{r., h ) .

Proof: .Assunie c 5 h. Let the horizontal links represent diniension one. and the

vertical ones dimension tivo. i i ïe use the algorit hm from the proof of Theorem 5.1.

Each time st.ep in the proof must be replaced hy c or h time units obtaining total

time of at most rn l /2] h + [nz/21 c + 62,- + :3h 5 D + ï u + -Ih = D + l ( h + u ) + :3c. In

case 2: > h , we swap the two dimensions- i

Corollary 5.3 If h > 1 and v > 1 then gossiping in a one-port hnlf-dirpler nsyn-

chronous two-uniform two-dimensional torus takes time nt most D+~?+h+rnin{u, h)+

:3 .

Proof: .4ssume v < h. apply the algorithm froin Theorem 5.1. In this case, the

exchange of the messages between the paired-up nodes of two cycles can be accorn-

plished in u and h tirne: respectively. since the ttvo nodes can transmit the messages

simultaneously without causing a coilision. If the dimensions are odd, the unpaired

cycles can be handled in just one additional time iinit instead of waiting iintil the

entire message has been received after 27 or h time units. Finall~., t he forwarding

wit hin the cycles needs Lnk/?] steps even for odd-lengt h cycles because al1 the nodes

can transmit simultaneoiisly-there is no need to work in pairs and no resulting idle

nodes. This results in ' 3 ~ ' + h time units for the exchanges, :3 time units for handling

the unpaired nodes. and [ni/2] h + Ln2/2j v = D time units for the forwarding within

cycies. r

5.2 Gossiping in a skewed torus

In the previous section we showecl a gossiping algorithm that runs in time D + c.

wliere D = LY]h + 1% - l u is the cliameter of an ordinary two-dimensional torus. ancl

c is inclepenclent of nh and n , . The cliameter provicles a trivial lower bound. and the

nest theorem shows that the dianleter Dr of a k-skewed torus is smaIler than that

of an unskewecl one of the same climensions. The smaller tliameter results from the

rediiced niimber of vertical links necessary to traverse due to the skew of the torus.

The theorem assumes that k < n u / ? since if I; > nu/-. the network is equivalent to a

skewed torus trith skew k' = n , - A- < 2, /2. In this section ive clevelop an algorithm

t hat utilizes the smaller cliameter. and gossips in time D' + cf mith cf independent of

hoth the climensions and the skem k of the toius.

Theorem 5.4 As.sun2e k < 9. .. The d i a m e t e r 01 the tmo-unqoorrn k-skcued torils of

size nh x n, is D 5 y h + -o.

Proof: Since the network is vertes-transitive? it is siifficient to find the most distant

nocle from an arbitrary fixecl node S. To visualize the ivrap-aroiind connections. we

can represent the skewed torus as a r-,h x n , planar rectangle surrounded with copies

of itself that cover the wliole plane [2] . Figure 5.2 shows the original rectangle sIiac1ed

and surrounded bu the six neighboring copies. Ali possible paths from s t o a node

Figure 5.2: CVrap-aroiind connections of a skewed t oms mocleled by muIt iple copies of its nodes placed on a plane. -4 schematic shortest path routing from a node s for the case nhh > kr: is also shoivn.

t are then eclitiva!ent t o paths from al1 possible copies of s to t h e original copy of

t . Consider the copy s' with planar coordinates ( n n . n , - k ) relative t o the original

copy s = (0:O). Diie t o k 5 n , - k. one can see tha t no node in the shaciecl area

can be more distant from s than the midpoint between s ancl sr. Its coorclinates are

t = (2. e). - and the distance t o s is at rnost 2 h + +r. hence the theorem. The - arguments for the other five copies of s are similar. rn

If nhh > k c , me can show a lower bound of )h + mas{-, $ } u - h - 1, close t o

fi']. [ y ] ) . The the iipper bound of Theorem 5.4 by considering the node t = ( [y - shortest path ro~ i t ing in this case is schematically shown in Figure 5.2. The routing

is clifferent from the shortest path roiiting in a regular. unsken-ed torus. In a rtgiilar

toriis. the shortest path from the source s = ( O t O ) t o a node .r = (i, j ) depends

only on the cluadrant t o which r belongs (ive assume lil 5 Lnh/2j. 1 jl 5 Ln./-J). If

both i > O ancl j > O then a shortest path from s t o .t. consists of i horizontal links

traversecl right and j vertical links traversecl up. T h e other cluaclrants are sirnilar.

The shortest path in a skewed torils is different for some clestinations. -Assuming nt. -k i > O : j > 0' the shortest path t o the nocle x = (i'j) satisfying ih +Jv > 2 h + T U

does not follow i horizontal links right and j vertical links iip. Instead, there is a

shorter pat h in the left-bottom direction that uses the wrap-around. Ii follows ni, - i

horizontal links and n, - k - j vertical links. One can see that ih + jo. > 2 h + e u 2

implies ( n h - i ) h + ( n , - k - j)v < i h + jv. T h e situation is analogoiis for the other

cluadrants. except in the bottorn-right (i > 0: j < 0) and top-left (i < O, j > 0)

quadrants one replaces n, - k with k , i-e., the wrap-arouncl connections are used

whenever lilh + l j l u > 2 h + su. The case nhh < kv implies that it is shorter to

circle horizontally around the torus than to traverse k vertical links. In certain cases

it may be shorter to follow two or more horizontal circles than to follow vertical links.

The actual number of horizontal circles that is worth traversing. and the shape of the

shortest paths, depends on the relative values of the parameters.

We sstart the presentation of t.he gossip algorithms with a message exchange al-

gorithm for a synchronous infinite mesh. The algorithm is based on the algorithm

of Iirumme et al. [:El for t he one-uniform moclei. The main idea of the algorit hm

is forwarcling messages without any delays along straight lines; the only delays are

esperienced when turning corners, which are bounded by a constant. The straight

lines are alternately divided into lines transmitting messages either only left or only

right (for horizontal lines). or either only up or only clown (for vertical lines). The al-

gorit hni describecl in [32] repeats a four-step cycle. ivhich tloes not cause any conflicts

for the one-uniform case. This is in general not true for the two-ilniform case, and

the nest t heorern shows how to arrange t h e transmissions to eliminate the conflicts

by increaçing the length of the cycle but preservilig the constant bo~ind on the del-

experiencecl by messages turning corners. /Ifter the description of the algorit hm for

an infinite mesh. ive show how to moclify it for a skewecl toriis. The algorithms for

t lie asyncIironous motle are analogous. As aiways t hroughou t t his chapter, Ive assume

a constant-cost one-port half-duplex rnodel. In the proofs we use the following no-

tation. The clistance d(z, y) between the nocles .z. and y is the length of the shortest

path connecting s and y; d(z. 9) is defined as the s u m of the link lengths, not the

nuniber of hops. The greatest common clivisor of integers a and b will be cleiioted by

gcd(a, b ) ? and t heir least common multiple by lcm(n, b).

Theorem 5.5 In the one-port hnlf-duplex synchronous t u ~ o - u n i f o n i+nite rne-sh,

there is a message exchange scheme such thnt the data frorn nny node 3. arriz.es at

any node y in time a t most d(x , y ) + c. where c is a constant independent of x and y .

Proof: Instead of giving an exact description of an algorithm satisfying the given time

bound. me will show that there exists such an algorithm mhich perioclically repeats

every p time units. The period p depends on h and r:, and will be defined later. LVe

start with the general striicture of the algorithm, and the conditions its transmission

times nlost satisfy. Then we present one way of choosing the transmission times t hat

sat isfy the conditions.

Our algorithm is based on the scheme of Iirumme et al. [32] for the case h = u = 1.

Each horizontal line forwards messages in either left or right direction orily without any

clelays along the line. Similarly, eacli vertical line transmits either iip or clown o n l -

The algorithm described in [32] repeats every four steps which cannot be achieved

for arbitrari h and v due to conflicts between horizontal ancl vertical transmissions.

Beloa we present one 1n.y of setting the transmission times that avoicls the conflicts

and can be repeatecl every p time units for a suitably definecl constant p.

W-e start with a non-perioclic scheme that avoicls the conflicts. Since oiir algorithm

cliains transmissions within each straight line. it can be fdly described in terms of

the starting times of transmissions at the horizontal and vertical axes. These will be

clenoted by tR(O, j ) . j even, and tL(O: j ) , j odd, for the transmissions from a nocle (O, j )

in the right and left clirection, respectively. Similarly7 for a node (i, O ) , the s ~ m b o l s

tci(i. 0). i even, ancl t D ( i . O) . i ocld, denote the starts of the transniissions in the iip

and down direction. respectively. See also Fig~ire 5.3. Consiclrr a nocle ( i , j ) with both

i ancl j even. It is busy receiving/transmitting in the vertical direction during the 2v

time slots starting at trr(i, 0) + ( j - l ) u . Using the notation x + S = { s + yly E S): the hosy slots can t>e written as tLi(i. O ) + jv + {-o.. -u + 1? . . . 7 t i - l}. The horizontal

transmissions occupy the time slots tR(O, j) + ih + { -h , -h + 1,. . . . Ii - 1). To avoid

conflicts a t (i: j ) , these two sets must be disjoint, which is ecluivalent to

ta ( i .O) t R ( O : j ) - j ~ + i h + { - h - v + l , - h - v + 2 . .... h + v - l }

for i, j even.

Figure Z.3: T h e general scherne of the message exchange algorithm. T h e links are labelect by t h e times a transmission starts on them.

Siniilar conclitions can be derivecl for i ancl/or j oclci:

t . O ) B; t L ( O . j) - jv - i h + { -h - (7 + 1: -h - v + 2, . . . . h + u - 1)

for i even. j odd,

t D ( i . O ) B; t R ( O . j ) + j v + i h + { - h - v + 1 .- h - ~ ' + & ...: h + o - 1 )

for i odcl, j even,

t D ( i . O ) B; I L ( O . j ) + j u - i h + { - h - r + L , - h - o + 2 '.... h + c - l }

for i, j ocld.

Nest we show tha t one can satisfy these conclitionç with values of t R ( O , j ) ancl

t L ( O , j) tha t increase linearly with j , i.e. tR(Ol j ) = c ~ j : tL(07 j ) = « j : wliere CL is

a suitable constant. Denoting B = {-h - z? + 1: -h - v + 2, . . .; h + a - 1) a n d

stibstituting i = 2r for i even. i = Z r + 1 , i ocld: j = Z s , j even and j = 2s + 1: j ocld,

t h e conditions for i, j both even and odd can be wri t ten a s

for al1 integers r: S.

We d a i m that for a = Lh + :3u7 ive can choose a valid value of tci(2r, 0) and

tD(2r + 1.0) for an- integer r. T h e union BLiWR(+) of the right-hand sides of (5.1).

taken over ail values of s, consists of a set of (Lh + 2u - 1)-tuples that are offset

by 2 (n - 21) = 4h +- l u time units with respect t o each other. The union BcrqL(r) of

the right-hand sides of (5.2) consists of a similar set of (Zh + 227 - 1)-tuples that is

offset by Zh + 2 u - ?rh with respect to BLTR(rm). This is illustrated in Figure 5.4.

It means t hat in total any interval of length 4h + l z : contains a t most 4 h + 4u - 2

Figure 5.4: The potential start times tLi(Zr- O ) for a transmission from nocle (2r . O ) in the up direction. The shaded t ime slots cause a conflict irith a transmission in t he (a) right and ( b ) left direction. T h e parameters a re h = 1. v = '2. « = S.

elements of BLLR(r) U B L i T L ( ~ ) ? and we can choose tu(2r . 0) to be one of the reinaining

two tirne slots. The si t~iation for t o (2r + 1,Q) is similar. The unions B D , ~ ( I . ) and

B D S L ( r ) of the riglit-hand sides of (5-:3) and (5 .4) , respectively, are sets of (2h+%- 1)-

tuples offset by 4h + 80 meaning that in an- interval of length 4 h + St;? there are a t

most 412 + - 4 ~ - 2 time slots occupieci. ive can now factor al1 the time slots moclulo

p = lcrn(4h +4v. l h + S u ) = ? (h + c ) ( h + 2 u ) / gcd(h, TI), ancl we cari repeat them every

p time units without causing any conflicts or delays d o n g straight lines.

As the last step, we show how the bounci of d(.-c, y ) + c for a suitable constant c

follows. Consicler an arbitrary pair of nocles x and y. It is easv to verify that there

is always a path p(.ro 9) from .C t o y in our message exchange sclieme with the total

lengt h of the traversecl links a t most d(s , y ) + 3h + 2.v. The worst case is shown in

Figure 5.5 when both links leading from x point away from y: and both links leading

from y point toward 2. If a message is transrnitted along p ( x , y) , its propagation can

Figure 5.5: The worst-case for the path from x to y.

be clelayed ordy at the beginning in the nocle .z: or when turning corners which occurs

at rnost three tirnes. It can be shown that the former is boiinded by p - 1 tirne iinits,

while the la t te r by p - h - .LJ units each. Therefore. the total time to reach y from .r.

is at most d(s. y ) + 2h + 2~: + ( p - 1) + :3(p - h - u ) = d(x, g ) - h - .r? + 4 p - 1. i

The nest theorem shows that the algorithm for the infinite pIane can be aclapted

to a skewecl torus in a straiglitforwarcl way. The wrap-around can introcluce conflicts

but they resiilt in an aclditive constant only.

Theorem 5.6 Gossiging in rc one-port hrtlf-duplex synchr-onous two-uniform k-skerred

toru.s of size nh x n , f a h time nt most D + c. whcre D is the dinrnete 1% of the skerred

torus and c is n con.stc~nt independent of nh , n , nrzd k .

Proof: The proof applies the scheme with period p from the proof of Theorem 5.5- If

al1 of nh: T I , and X: are even numbers that are multiples of p, no conflicts occur clue

to wrap-arouncl, ancl the time bound D - h - v + 4p - 1 follorvs. -4rbitrary even r zh ,

n , ancl k may cause timing but no clirectional conflicts in the srhenle. The rvrap-

around creates two circular seams, one in each dimension, crossed by links orthogonal

to the seam (see Figure 5.6). Consicler the vertical seam and one horizontal link e

CHAPTER 5. CONSTANT-COST TCVO-CjiVlFORM TOROIDAL AfESNES 7s

verücai sesm

i : t

Figure -5.6: The seams formecl by the wrap-around of a skewed torus when n h , n, and the sketv k are even.

crossing it, the horizontal seam being similar. The transmissions on the ttvo adjacent

vertical links on one side of the seam occupy 2: time slots causing a conflict with

2 ~ ' + 2h - 2 slots for a potential transmission start on e. An adjacent horizontal

link occupies /2 slots conflictirtg wi th 3h - 2 start times. Consiclering both sicles of

the searn. a t most 10h + -Iv - S starting times cause a cotiflict. One can shoiv that

10h + 4u - S < 4(h + u ) ( h + 2v)/gcd(h, c ) . the period p of the scheme. meaning

that the transmission on e can avoid any conflicts. -4s apparent from the proof of

Thcorem 5.5. the transmission paths of the scheme cross each seam a t most twice.

The delays at the encl nocles of t.he link crossing a seam are trivially bounclecl by p.

hence a total of a t most 4 p additional idle time slots. Therefore, the time to transmit

a message betn-een two nodes is bounded by D - h - c + S p - 1.

In case n h : nv even and k odd, the horizontal links crossing the vertical seain

cause directional conflicts when forwarcling messages dong a horizonta[ line across

the seam. In Figure S.:, this is shown a t nocle s when the message needs to be

forwarded right. It is easy to see that such a message can be rerouteci ackling only a

constant number of additional links. In the esample shoivnt the message will traverse

one vertical link frorn nocle x, and then continue in the right direction. At the end,

it may need one additional horizontal and vertical link, the worst case being shown

Figure 5.7: The vertical seam when the skew k is odd, nh, nu are even.

in Figure 5.7. ivhen either nh or nu are odd- the links parallel to the seam point

in the same direction on both sicles of the seam. This may pre\?ent a message from

traverçing a straight line parallel to the seam in the clesired direction. However, this

can he hanclled sirnilady as oclcl skew k resiilting in n constant ntimber of additional

tirne units. i

The riest two theorems develop similar algorithms for the asynchronous case. The

synchronous algorithms could be appliecl directly, however, t he asynchrorious mode

enables one to recliice the transmission time. In particular, the wai ting delay experi-

encecl by a message when turning corners in the moclifiecl algorithm presented in the

nest theorem is independent of the link c l e b s h and v. The valries of h and L: appear

in the additive constant only as a result of traversing the extra links at the source

ancl/or destination for the pairs of nodes whose links point opposite to the direction

of a rilessage that woiild follow the shortest path between them.

Theorem 5.7 h the one-port hnv-duplex ctsynchrorzot~s two-uniforrn i n h i t e me&,

thel-e is n message erchange schenze such thaf the dntn from n n y node x arrives nt

a n y node g i r z tirne nt nzost d ( x , y ) + 2h + 2 u + 41.

Proof: The algorithm follows the same scheme as the one for the synchronous case,

just the conditions get relaxecl due to more flesibility in the nodes. In particular,

the right-hand sides of (5.1)-(5.4) are replacecl with a series of triplets insteacl of

(2h -+ 2t. - 1)-tuples regardless of h and 27:

This is illustrated in Figure 5.8. If we choose a = 5.c: the union BW(r) of the right-

Figiire 5.8: The time instants that cause a conflict in the asynchronous mode mith a horizontal transmission in the ( a ) right and (b) left direction.

hancl sicles of ( 5 . 5 ) consists of triplets offset by Su with respect to each other, and the

union BLrL(?.) of the right-hancl sicles of ( 5 . 6 ) consists of triplets that are offset by

4 ~ : - 4rh \vit h respect to BLjVR(r). That means that no period p that is a nlultiple of 4

will cause conflicts. The unions BDVn(r) and BD,L(r) OC the right-hancl sicles of (5.7)

and (5.8) are triplets that are offset by 1Zu. That means that if ive factor al1 time

slot,s moclulo 12, there are at most sis time dots occupied out of possible twelve. -4s

a result, choosirig p = 12 parantees that there are rio conflicts for either upwarcl or

clownward transmissions regarclless of the values h and v.

U'e note that in some cases a smaller period can be chosen. For esample. i f the

parity of h and v iç the same, we can choose n = t. and p = 4. Then the sets BLr,R and

BLrL contain one triplet each, which are offset by -4rh O (mod 4 j with respect to

each other. The sets B D . ~ ( ~ ) and BDqL(r ) are sets of triplets offset by 4.v O (mocl 4).

and the set BDSL(r) is offset by Zv - 4rh - Zh O (mod 4) mith respect to BD.R(T).

-4s a result: out of four consecutive time slots, there is always one slot unoccupied.

Hoivever: it can be shown that 12 is the mallest perioci indepenclent of h and o. i

Corollary 5.8 Gossiying in the one-port haljdupler asynchronous two-unqbrm k-

sketued forus of size n h x n , tnkes time nt most D + c, where D is the diameter of the

.ikewEd torus and c k a constant independent o f n h , n , and k.

Chapter 6

Linear-cost two-uniform toroidal

meshes

This chapter adds the data rate parameter to our transmission mode1 making the

transmission tirne dependent on the message length. The model is a trvo-uniform

extension of the linear-cost model (see Section 2 2 ) , in which each link is associated

with tivo parameters: ,O-the propagation clelay. and T-the time to transmit one unit

of data (i-e., l / r is the data rate). The tirne to transmit a message of size L is equal

to ,a + LT. Since oiir two-irniform mode1 assumes two types of liiiks-horizontal and

vertical. we end up \vit h foiir parameters in total: Ph and for a horizontal link. and

,B. and ru for a vertical 1ink. The following sect.ions stiidy the gossiping problem for

toroiclal meshes under the two-uniform linear-cost model. Section 6.1 adclresses the

loiver bounds on the total time of a gorsiping algorithm for a two-iiniform toroidal

mesh of size nh x nu. Sections 6.1 and 6.:3 develop algorithms for the all-port ancl cjne-

port models. respect ively. Throughout the chapteq nh and n , denote the horizontal

and vertical sizes of the torus? and Ar = nh x n , denotes the total number of nodes.

-411 the results assume the synchronous transmission mode. hloreover. Ive assume that

the original size of a11 messages is eclital to one, and our algorithms do not split them

into any smaller pieces. For full-duplex algorithms, terms like "number of messages

sent by one link" will always niean number of messages sent in one direction (similarly

for number of steps, time taken by one transmission, etc.). Since our algorithms will

C'HA PTER 6. LINEAR-C'OST TWO- I;'\'IFORM TOROIDA L MESHES

be symmetric, this will not cause any ambiguity- T h e asymptotic symbols O, 0, O

will rekr to fiinctions of nh and n,. FinaIly: t h e logarithm log r without specifying

the base will altvays be base 2 .

6.1 Lower bounds

First. ive clerive three lower bounds for the two-uniform gossiping problem based on

the three techniques introducecl in Ho [26] for t he oncuniform broadcasting problem:

root dominance-the t ime required for the source (root) t o send the data. lalency

dominnnce-the propagation delay to reach the furthest destination. and 6andwidth

dominance-the total bandtvidth reqiiirecl divided by the total bandtvidth available.

Lat eiicy dominance

In the two-uniform moclel, due t o propagation clelay, traversing one link takes a t least

,dh or 3, time units, depending on the type of the link. As a result, the minimum time

t o reach the f~irthest destination is the diameter of the nettvork with eclges labeled by

either $h or ,a,. For a regirlar torus, this is

since the most clistant nocles are horizont.al and vertical links apart. T h e

ctianieter of a skewecl two-uniform torils was derived in Section 5.2. We note tha t ive

coulcl get a slightly better bound by laheling links with ,3h + Ï ~ ancl ,b'. insteacl of ,Oh

ancl 3,. since each transmission step rnust carr'; a t least one unit of data. However, Ive

keep it in the simpler form for the sake of cornparison of the lower bouncl cornponents

introclucecl by the propagation clelay ancl data rate.

Root doiiiiiiaiice

For the gossiping problem, we modify the root dominance technique by consiclering

one destination instead of one source. The minimum time to receive al1 messages

is q u a 1 to the total ~ i u m b e r of messages divided by the maximum rate of receiving

CHA PTER 6. LINE-4 R- COST TIW- U N I F 0 Rh1 TO ROIDA L MESHES

data. The former is equal to !V - 1: while the latter depends on the particular model

of communication. The maximum rate at which one iink can transmit data is l/rh

data units per time unit for a horizontal link; and I/Ï, for a vertical link. Since each

node is connected to two horizontal and two vertical links. for t he all-port full-duplex

model the maximum total rate a.t which one node can receive data is 2(* + $) . If we consider one particular node under the all-port half-duplex model. its maximum

rate of receiving data is the same as for the full-duplex model. However, since each

link can be active in only one direction a t a time, there rnust be at least one node in 1 1 the netn+ork whose overall receiving rate is at most one half of thiç value, Le. + G .

For the oncport model, the rnasirniim possible rate of receiving data is mas($. 5) . To sirnplify the formulae belon--, we will assame T ~ , 5 rh obtaining l/~,, . For the haIf-

cluples model, ive use the same reasoning as for the all-port moclel obtaining a nocle

whose receiving rate is a t most l/(L?r,). Putting it al1 together we obtain a lomer

boiincl for each of the four cases:

0 all-port full-duplex lv - 1 i\- - 1 Ï h TV!

0 orle-port full-cluples

( N - 1) ÏL,?

0 one-port half-cluplex

-2 - ( A'- - 1 1 TL,.

The bounds (6.1) and (6 .5) will be supersecled using banclwidth dominance.

Baiidwidt 11 doininance

Using banclwidth clominance, we obtain a lower bound by dividing the total required

banclwiclth by t h e bandu-idth available during one time unit. The minimum band-

wiclt h reqiiired to broaclcast one message to al1 ot her nocles in a network is eclual t o

t h e total weight of the minimum spanning tree of t he network obtained by labeling

edges with r h or r,,: depending on t he type of the Link. For a regular torus, one can

assume ru 5 ~h rvithout any loss of generalite obtaining the weight of the minimum

spanning tree equai t o

(nh - 1) Ï,+ + (nt, - L j n h r ,

If r, 5 r h , the value is the same for a skewed torus. Otherwise. the weight of t h e

minimum spanning tree of a k-skewed torus depends on the relative values of the skew

k and t h e vertical size n,. T h e total bandwidth required for the whole gossip is equal

t o the rveight of the minimum spanning tree multiplied by the number of sources N .

T h e bandwidth available during one tinie unit is equal tü the total number of links

t h a t can be sirnriltaneorisly active. We can write a general forrn of the lower bound:

# (sources) x tu (~vIST~, , ,~ , ) # (links simultaneously active) '

Pliigging in the maximum number of links that can be simultaneo~isly active for each

of the four models, and ass~iming Ï,, 5 ~ h : ive obtain the following lower bounds (for

both regular ancl skewed torus):

dl-port f~ill-cluples

all-port half-cluples

one-port full-duplex

one-port half-duplex

We note that these bo~inds reduce to (6.2)-(6.3) under the one-uniform model, and

that (6.S): (6.9) are tighter bounds than (6.4): ( 6 . 5 ) .

6.2 All-port model

This section disciisses the gossiping problem in toroidal meshes under the all-port

transmission model. The main results are efficient gossiping algorithms for an all-

port torus (regular or skewed) that improve the best-known upper bouncls in the

one-uniform special case. The first algorithrn is for the f~lll-duplex model. It can

be adapted for the half-duples model by simulating each bidirectional cycle by two

neighboring unidirectional cycles. For the sake of cIarity of presentation. ive start

in subsection 6.2.1 with a basic version of the full-duplex algorithm that is easier

to understand even though it cloes not achieve the desired running time clue to iclle

periocls. T h e algorithrn is refined in subsectiori 6.2.2 to eliminate the iclle time. For

the sake of simplicity, rve disregarcl the fractions when calculating the number of steps

of the algorithms but rve point to the ways of hanclling thein.

6.2.1 Basic gossiping algorit lim

This subsection presents the basic version of oui. algorithm for gossiping in an all-port

full-duplex two-iiniform toroidal mesh. In the algorithrn. the nodes often combine

se\-eral unit-size messages into a larger pncket. and transmit them as one block. The

transmission of one packet over one lirik will be callecl a trnnsrnis.l;.ion s t e p (or simply

a step). If the transmitted packet contains L iinit-size messages then the time taken

11y one transmission step over a link rvith parameters ,B and r is /3 + LT: ,8 time units

are occupiecl due to propagation clelay and LT time units due to chta transmission.

This is illustrated in Figure 6.1. We will refer to transmission steps perforrned by a

horizontal link as horizontal steps, and to transmission steps performecl by a vertical

link as certicnl sfeps. Note that the cluration of a transmission step varies clepencling

on the packet size and the link type. The horizontal and vertical transmission steps

of our algorithm are esecutecl in parallel. They will not be synchronized, and one of

CHAPTER 6. LIiVE.4 R-COST T l 4 0 UXIFORM TOROID-4L MESHES

transmission step -

time

Figure 6.1: One transmission step transinitting a packet of size L over a link with parameters ,8: r .

them will typically take substant ially longer t han the ot her.

\lie will describe oiir algorithm as a gnthering scheme. This means that ive d l

clescribe the orcler in which one node receives messages from the rest of the network.

This is different from a broaclcast scheme in which one clescribes the data paths

traversed by one message. To visualize the algorithm we will use cliagrarns that

depict the origins of t h e messages known to one node at a particiilar tirne instant

(informally. Ive will often say that the diagram clepicts the messages insteacl of their

origins). An example is shown in Figure 6.2. ?Ve empliasize that the cliagram depicts

Figure 6.2: An example of a cliagram showing the origins (circles O ) of the messages knotvn to the liighlighted nocle (solid circle) at a given t ime instant.

the s e t of messages only, and cloes not show t 11s clata paths traversed by the messages.

CH-4 PTER 6. LINE-AR-COST TWO- UNIFORLU TOROIDAL MESHES

Superimposing diagrams showing messages knomn to two neighboring aodes tells us

the collective knowleclge of the two nocles, and which messages they can exchange.

This is illustrated in Figure 6.:3. The figure uses crosses x and circles O to depict the

Figure 6.3: b,Iessages known to two horizontal neighbors z (crosses x ) and y (circles

0)-

messages knonn to two horizontal neighbors r and y. respectively. The node r can

send to y the uncirclecl crosses, while the node y can send to rç the uncrossecl circles.

The overlapping crosses and circles are known to both .c and y. so there is no neecl to

exchange them. The full-tliiplex algorithm will be symmetric For a11 nocles implying

that the shape of the messages known to one node is the same for al1 nodes in the

network in the same ivay that the shape covered \vitLi crosses in Figure 6.3 is the same

as the shape coverecl wit h circles.

A consecluence of the symnletr- of otir algorithm is the fact t hat al1 horizontal

links perform an eq~ial niimber of horizontal steps with an ecliial total length: and al1

vertical links perform an equal number of vertical steps with an eclual total length.

The follon-ing tlieorem derives the riinning time of such an algorithm.

Theorem 6.1 Consider an ail-port full-duplex gosslping algo rit h m for a t wo-unijorm

jske tced) t o m s of size nh x n, that sntisjîes the jollowing conditions:

O al! horizontal l inks perfoorm the snme number of steps sh. spend the snme amount

C H A P T E R 6. LIiVE4R-C'OST T WO- UIWFORR/~ TO ROIDA L MESNES

of tirne 'Th,idle idle, and transmit the sume total number of unit-ske messages in

ench. direction?

al1 vertical links perform the same number of steps s,, spend the same nmount

of t ime TWidle idle, and transmit the same total number of uni t -s ix messages in

ench direction,

no message is sent to one node more thnn once.

the running t i m e of the algorithm i-s

where X = n h x n, is the total nwmber of nodes.

Proofr Let Nh be t he total n ~ i m b e r of unit-size messages t ransmi t ted during t h e

whole algorithm by one horizontal h i c , ancl let X , be the total number of messages

t ransmit ted by orle vertical link. T h e n t h e running t ime of the algorithm is eclual t o

Each node receives N - 1. messages in total. Since it receives messages thro~ig1-i two

horizontal and two vertical links. assuming tha t no message is sent to the same node

Subst i tut ing into (6.10) leads to

This rneans t.hat the total gossip t i m e is eclual to

I

Similar reasoning can be applied to t h e all-port half-doplex model. For the half-

duples model, t h e symbols sh and s, denote the number of s teps performed by one

link in both of its directions.

Corollary 6.2 Consider an ail-port halj-duplex gossiping algorith,m for n two-unqorrn

(skewed) torus of size nh x n , that satisfies the follozoing conditions:

0 ail ho~izontal links perform the same total number of steps s h , and spend the

snrne am0,llnt of ThVidle idle.

0 al1 horizontal links transmit right the sante number of unit-size nzessnges, and

d l horizontal links transmit lefl the snme number of unit-size messages,

0 ni1 vertical links perform the sume nunaber of steps s u , ancl spend the snme

amount of tirne Tu,îdre i d e ,

nll ~.'ertical links transmit irp the same nurnber 01 unit-size messages. and ail

uertical links f rclnsmit doion the snme nzrmber of unit-six rn esaages.

0 no message k sent to one node more thnn once.

Then the running tinze of the algorithm is

ruhere !V = nh x n o is the total nwmber of nodes.

Proof: Let -Yhwrighl denote the total number of ~init-size messages one horizontal link

sends in t h e right direction' and let A'h,r,rt denote the total number of messages it

sends in t h e left direction. Sirnilarly' let ArL..,, and denote t h e number of

messages one vertical link sends up ancl clown. respectively. Then the riinning time is

Since t h e values i vh , r igh t : and lVu,d,,n are eq11a.1 for al1 links- ive get t h e

ecluali t y

N - 1 = !vh,rirrht t !Vh,/ell + 1\iv,up +

The rest of the proof is similar to the proof of Theorem 6.1: and we omit i t. i

The calcdation of the running tirne of our algorithm uses the following technical

lemma. It ~rovides a lower bound on the total number of transmitted messages in

consecutive transmissions during a given tirne period.

Lemma 6.3 -4ssurne that a given linb zuith parameter-s 0: r perfornzs s succe.ssiue

trnnmission steps tnking nt lenst T t ime units such that the jïrst s - 1 steps talie less

than T ti,me units. Let the size of the packets trans-mitted in the successive steps be

k l > ka > - - 2 ks 2 1- Then the total nurnber of unif-size messages cornpletely

receioed during the fird T tirne unit.5 is ut least

kS- i f the lasi atep finishes at time T , and O + k T

PI-001: The condition of the letnma assumes ~ ~ = , ( , B + k i r ) = T + AT. where O < AT < ,O + k,r. as shorvn Figure 6.4. Since each k; 2 lis, and is an incrcasing

Figure 6.4: The s successi\:e transmission steps performecl b - one link.

fiinction of k. ive get k; ks

L , ,13+ksr

If A T = 0' the total nu~mber of the transmitted messages is eqiial to

Otherwise. the iast step does n ~ t finish its transmission, and the total number is

m

The efficiency of our gossiping algorithm results from the small number of trans-

mission steps performed by one Iink. The basic version of the full-duplex algorithm

for a torits of size nh x n , borinds the number of horizontal transmissiun steps s h

by 121 + (7(log nh + log nu) and the number of vertical transmission steps su hy

[ y ] + (7(log nh + log n,). This is lower than the number of steps of the algorithms of

Fraigniaucl and Lazard [l'il that use nh horizontal and n , vertical steps (nh = 1 1 , for

their algotithms). The refinecl version of the algorithm bouncls t h e i d e time of an-

i n b ) iVe present the algorithm in terms of a regiilar torus but it is eas - to

see t hat it applies to a skewed torus as weli. LVe note, hocvever. t hat it does not take

aclvantage of the smatler diameter of a skemecl torus, and its running t ime is the sanie

as for a regular one.

Our algorit hm consists of four stages. The first t hree stages perform alternating

partial gossips-vert ical. horizontal ancl vertical. A special feat lire of the part ia1

gossips is tha.t the nodes clo not drst,ribiite in the direction of the gossips al1 the

messages t hey are aware of. Instead, they send only a subset of the messages that

guarantees t hat every message reaches a location t hat is at most o ( log n h ) links from

any node in the network. In the meantirne, the links orthogonal to the direction of the

partial gossip transmit al1 the messages they are aware of at a given time instant. FVe

will show that this guarantees an esponential growth of the size of the transmitted

packets resulting in a logarithmic number of steps. The fourth and last stage of the

algorit hm uses O(1og nh) horizontal and vertical steps to dist ribute every message

to al1 nocles that are not aware of it. iVe emphasize that information is moving

both horizontally and vertically in al1 stages: and tha t the transmissions in the two

directions are simultaneous a n d asynchronoiis.

As derived in Theorem 6.1, t h e total t ime of the gossiping algorithm can be caicu-

iated from the number of transmission steps performed by each link and from the t ime

each link is iclle during each of the four phases. iVe will do the calculations through-

out the description of the algorit hm. T h e algorithm reqiiires certain assumpt ions on

the relative values of the dimensions of t h e torils. They will b e derived diiring the

description of the algorithm. and discussed a t the end of t his subsection.

Basic algorit hm

Stage one: During the first s tage of the algorithm the vertical links perform vertical

gossips distributing each message within the whole vertical cycle (column) that its

originator belongs to. T h e gossips s tar t with every nocle exchanging its message with

botli of its vertical neighbors. Then each message is forwarded in both directions

of the CJ-cle ( u p ancl clown) in unit-size packets. one link a t a time. T h e size of the

packets transmitted by vertical links is always one, so the t ime taken by everh- vertical

s tep is 3 , + 7, . :Ifter 1% - j vertical steps each message is distributed within the whole

cj-cle of length nu. Therefore. t h e number of vertical transmission steps in stage one

is s(l) , = 121. ancl the total t i m e of stage one is [+] ( ,d, + ru). - In t h e meantirne. every node uses its horizontal links to t ransmit to its horizontal

neighbors in the two neighboring columns the messages accumiilatecl in the previous

horizontal and vertical transmission steps. In every step. it combines al1 the niessages

the otl-ier end is not aware of into one packet. ancl sends the packet over one horizontal

link. Below we show t hat due t o the combination of messages accumulated from the

pre\-ious horizontal and vertical steps. the length of the transmittecl packets increases

esponentially. -4s a resiilt. t h e number of horizontal steps in s tage one is logatithniic

in the total t ime of the stage. T h e process is illustrateci in Figure 6.5 which shows

the order of messages as they a r e receivecl by one highlightecl nocle througli its four

adjacent links. T h e nodes of t h e network are labeled by the label of the link that

deliveis their message to the highlighted nocle and the number of the link's step in

Figure 6.5: The order of t h e messages that are receivecl by the highlighted nocle at the beginning of stage one throiigh its four adjacent links-Left. Right, Top. Bottom. The parameters are ,3h = (3, = î-h = 7,. = 1. Each clifferently shaded area represents t h e messages received cloring t he t ime taken by one horizontal step. Note that the horizontal links transmit more t han one message simultaneously.

which they were delivered. While t h e two vertical links always transmit one message

in one step, the size of the packets transrnitted through the two horizontal links grows.

The resulting timing is shown in Figure 6.6. The syrnbol Af;), i 2 1- in the figure

vertical step M

I horizontal step

Figure 6.6: T h e t iming of the horizontal a n d vertical transmissions in stage one of t h e aIgorithm. ,ah = P,, = ~h = rL, = 1.

denotes the size of the packet transmitted i n t he i-th horizontal s tep (as mentioned

above, ive will mean by tliis the niimber of messages sent in one direct ion). and TL-',) the t ime taken b?; t his step. Figure 6.7 scheinat ically shows the origins of the messages

known to one node a t the end of s tage one; we will augment this figure after stages

two. three, and four.

Xest we formally demonstrate t h e esponential growth of the packets transmittecl

horizontal l- The size :\hH of a packet transrnitted in the i-th horizontal step is eqtial

to the number of messages knomn to one node r a t the beginning of t h e i-th step that

are not known t o i ts horizontal neighbor y. Since al1 the messages .t. was aware of a t

the beginning of t h e ( i - I )-st horizontal s t ep were delivered to y by t h e end of the

( i - 1 )-st s tep, !\hi) is equal to the ntirnber of new messages t hat .t. learned cluring the

time T TL:)-^ taken by its ( i - 1)-st horizontal s tep. The node 1 received new messages

not knoivn t o y through one horizontal link (opposite to y ) and through two vertical

links. T h e horizontal link delivered one packet of size during the t ime period ( 1 ) ( 1 )

= k t 4 is a n integer then each of t h e vertical links perforrned esactly ~hf,', If s;T;,

C'HA PTER 6. LI!YE,4R-COST TWO- UNlFORibf TOROIDAL Ad ESHES

Figure 6.7: The shaded area represents the messages receiwd by the higliiighted nocle by the end of the first stage of the algorithm. Note that it shows the set of the messages received, not the chta pat hs traversed by t h e messages.

vertical transmission steps. each clelivering a packet of size one ( the fractiorial case is

cliscussed belon-). Therefore? the niimber of messages learned from the two vertical

neiglibors between the starting times of the (i - 1)-st and i-th horizontal steps is eqital TL"-

to '- - , S ~ + T ~ - - Since the size of the packet transrnitted in the first horizontal step is one. ( 1 ) and since each ThTi = ,.3h + !Lht).rh7 we obtain the following recurrence:

and

Solving the recurrence results in

and

Xote that the forniulae are valicl for a11 i > 1. The horizontal transmissions continiie

until the vertical gossip iç finished. Tha t rneans that the nurnber siL' of horizontal

steps performed in stage one satisfies the equality

Since. according to (6.1 3).

we obtain

( P h + ~h ) ( 9 L f + 5u) 2 rh

+

13, + TV

hence,

and

= (3 (log n . )

for a suitahle constant cl tha t depencls on ,3,: rh a n d T, (and For sufiiciently large n,.). If ( 1

is not an integer then the recurrence (6.11 ) does not liold. and the exponential O,+r,

growth of consecutive values !Vif;) cloes not follow. Hoivever, in tliat case ive can

compare !\:if) to a value more steps. say m. hacl<. For a sofficiently large constant rn:

the vertical steps performed dtiring t h e t ime t aken by m consecut ive horizont al steps ( 1 ) ( 1 )

mTtt.e-rn will cleliver nt least 1 3u+r , > T h . t - r n

J - ,3"+~7 new messages fsom each direction (note t hat

( 1 1 ( 1 1 ( 1 ) ThVj > Th,i-m for each j 2 i - rn since 2 Nh,i-m) giving an ineqiiality:

This demonstrates an exponential growth? though a t a slower rate' meaning that a

suitable constant cl can be founcl.

To guarantee tliat t h e nodes alivays receive a new message from their horizontal

neiglibors, our algoritlim needs the assumption sr) 5 121. A sufficient condition is

cl log n , 5 [y] which ive will assume in the rest of the algorit hm. We will discuss

this condition further a t the end of this subsection.

Stage two: the main task of tlie second stage of tlie algorithm is t o distribute mes-

sages across horizontal cycles (rows). Since ei-ery node accumulated al1 n , messages

from its whole column during stage one. one could employ gossips within each hor-

izontal cycle: and distribute packets of size n,. However. this would not utilize the

vertical links increasing the total running time. Instead of that. in our algorithm. each

nocle distributes within its horizontal cycle only a subset of the messages it is aware

of. The chosen subset consists of those messages that originate a n integer multiple

of clog nh rows away from the node's row: we will cal1 these rows designated roim:

the constant c will be definecl below. StatecI more formaily, one node sends to its

horizontal neighbor in one horizontal step a packet consisting of the messages that

the neighbor is not aware of and tha t are an integer multiple of clog nh r o m an'ay

from the node's location. .-\fter a t most LFJ horizontal steps al1 t he messages from

the clesignatecl rows are clistri buted ivi t hin the wliole horizontal cycle. so the nurnber

of horizontal steps in stage two satisfies sf) 5 [r?p]. Let the nurnber of the clesig-

nated rows be clenoted k ( 2 ) : note tliat k ( 2 ) -$&-. Since a packet transmitted in one

horizontal step contaiiis a t most one message from each clesignated row. its size is at

most k(". Therefore, the total t ime taken by stage tbvo is a t niost ( p h + X - ( ' ) T ~ ) .

The vertical links work in a ivay similar to horizontal links in stage one. Each

node sencls in one 1-ertical step to its \.ertical neighbors a packet that consists of

the known messages that the other endpoint is not aware of. However. to siniplifi-

certain calculations below Ive make one modification comparer1 to stage one. A node

.r sends to its vertical neighbor y only those messages that originate a t a node that

is closer to r than to y. This is best seen in Figure 6.8 which shows the origins of

the messages a node is aivare of a t the end of stage two; the differently shadecl areas

represent the origins of messages the highlighted notle learned during stage one aricl

tivo, respectively. We emphasize that Figure 6.8 sliows only the original locations of

the messages received by the highlighted node. It does not provide any information

about the paths that those messages take to reach the higlilighted nocle. During stage

two the highlighted node receives from its bottom neighbor only the messages that

are "below the designated rows" in the bottom half of the torus, and from its top

neighbor only the messages that are "above the designated rows7' in the top half of

the torus. Ot herwise? t here is no restriction on the origin of the messages transmi ttecl

T c log nll

Figus stage and t

me 6.S: The origins of the messages the highlightetl node is aivare of at the end of trvo. The lightly shadecl area represents the messages learnecl during stage one, he darker shaded area the niessages learned cluririg stage two.

through vertical linlis. They niay originate at an arbitrary roiv or column. not just at

t h e designatecl rows.

As it turns out. there m a - not be a siifficierit number of messages available for

a vertical transmission cluring the time taken by the first horizontal step, ancl the

vertical links m q r be idle during a part of this period. Since treating this woiiltl

complicate the presentation of the algorithm. ive leave the vertical links idle cluring

the time taken hy the whole first horizontal step in this version of the algorithm, and

will discuss this issue in its refined version described in the next siibsection.

Nest we show that the size of the packets transmitted in vertical steps of stage

tivo grows esponentially. similarly to the packets transmitted horizontaliy in stage

one. Let denote the size of the packet transmittecl through one vertical link

during its i-th transmission stepo and let T':) denote the duration of this step. i.e.:

T(-) v-1 = ,O, + ~\Y:)T~,. During the first vertical step, a node 1: transrnits to its vertical

neighbor y messages contained in the packets received in the first horizontal s tep of

stage two. The size of the packets received from each of the two horizontal directions

is equal t o /d2). Since t h e k(*) messages contain the messages originating in the sarne

row as 2: whose origins are closer to x than to y. a t least one half of the /d2) messages

originate a t a node that is closer to :c than to y. Therefore, the size NL!! of the first

packet transmitted vertically is a t least ; (Zd2)) = P('). Similarly to horizontal steps

in stage one. the size i~$) of the packet that node z sends to y in its i-th vertical st,ep

is eq~ial t o the number of nesv messages x Iearned from one vertical and two horizontal

neighbors during the tirne T':', taken by the (i- 1)-st vertical transmission step. The

number of new messages learned from the vertical neighbor is eclual to the size of the

packet received in the ( i - 1)-st vertical step, which is 1\7$)i - The nurnber of new

messages learnecl from one horizontal neighbor is eyual to the surn of the sizes of the

packets that s receivecl from one horizontal neighbor cluring the periocl T',:-!~. If s is

the nunlber of the horizontal transmission steps performeci in this period, and if the

s-th step f i n i s h a t tirne ~ ' 2 ) ~ then: according to Lernrna 6.3, the surn of the s packet (21 ( 2 )

Tu , - r Tu 1-1 sizes is a t least ks- 2 - where k, is the size of the last packet (we use the J h + T h :

( 2 ) Tu.,-1 fact that k- is an increasing f~inction of k ) . Ot herwise, the transmission of the

last packet does not finish: and the number of messages learnetl from one horizontal

neighbor is lo~ver hy at most x . ( ~ ) . This case can be hancllecl in a IV- similar to the

fractional case in stage one. Sincc the node z sends t o y in its i-th vertical step at

least one half of the messages learnecl from

follon-ing reciirrence:

the tsvo horizontal neighbors. we get the

*

fi" for i > 1. P h + Th

CH-A PTER 6. L INE-4 R- COST T m - UiVIFO RM TOR OiDA L MESHES 102

Solving t he recurrence results in

and

This shows that the length of each vertical çtep increases exponentially. Since t he

total t ime of stage two is a t most [Fj (ph + /d2)7h). the number sy) of vertical steps

perfornied in stage two satisfies

and

= log r r ) ([?]+l) (L+ &,,

for a suit able constant cl ( the constant gets modified i l fractions are handlecl).

To griarantee that the nodes always obtain new messages via vertical links, we need

the ineqiialities sy) < 14J and s r ) < c l o g h . FiTe will assume that c? log nh 5 171

holds true, and choose the value of the constant c > c2. T h e assumption cz log nh 5 121 mil1 be discussed filrther a t the end of this subsection.

During stage two. every message was distri buted across al1 rows whose distance

to Its originator is an integer multiple of clog nh rows. Therefore. every message is a t

most c log nh vertical links away from any destination ( in fact; it is a t most ( c log n h ) / 2

but t hat does not affect the asyrnptotic performance). I t may seem that the algori t h m

coiild be completecl in clog 721, vertical steps. However, this may not guarantee tha t

the horizontal links are able to transmit large enough packets t o perforrn a t most

O(log nh + log n , ) steps wit hout any idle periods during t his process. Therefore. we

adcl a third stage to guarantee tha t every message is not just C3(lognh) vertical links

but also O(log n h ) horizontal links away from every destination. The third stage is

sinlilar to the second one. just the horizontal and vertical dimensions are swappecl.

Stage three: T h e nodes forward through their vertical links messages that are a

multiple of c' log nh colu~nns away from t heir positions ( the constant c' to be tlefinecl

belon-). Through their horizontal links they forwarcl newly learned messages in packets

whose sizes grow esponentially. In this stage a node z sencls through botti the

ancl horizontal links only those messages that originate a t a node that is closer t o .r

than t o the other endpoint. The origins of the messages one node is informed of a t t he

encl of stage three are clepicted in Figure 6.9. The nurnber of vertical steps SU) neeclecl

is a t most c log nh since every message is a t most c log n h vertical links away from any

clestination, ancl it gets closer by one link with every step. LVe will again postpone

the horizontal transmissions cluring t,he time taken by the first vertical step to t he

refinecl version of the algorit hm. -4fter t hat, the horizontal links transmit messages

the other end of the link is not aware of (at least one half of t hem in a pattern similar

to stage two). Let i\h?) denote the size of the packet sent through one horizontal link

during the i-th horizontal s tep, ancl let TL:) denote the cluration of this step. Finally,

let k(3) clenote the size of t he packet sent through one vertical link cluring its first

step (note that the packet contains more than one message from each colurnn that

contri butes messages; typically i t contains approsimately k ( 2 ) / 2 messages from each

suc11 coIiimn). Sirnilarly t o stages one and two, we obtain the following recurrence

CHAPTER 6. LIXE-4 R-COST TWO- UNIFORM TOROIDA L MESHES 104

I c log nh

Figure 6.9: The origins of the messages the highlighted node is aware of a t the end of stage three. The darkest areas represent the messages learned during the third stage.

(fractions are also hanctled similarly):

(3) > p l - ~\~h . 1 -

This results in

meaning t liat

Since the number of vertical steps is a t most clog nht the total time of stage three is

a t most ( c log nh ) (!3, + x-(~)T, ) . Therefore! t h e number s r ) of horizontal steps in stage

t hree sat isfies

implying t hat

LVe choose the value cf so that c' log n h > log(, ) ( c log n h + 1). Note t hat t his is u +TL'

always possible for large enough n h .

Stage four: The algorithm is completed in the fourtli stage. Doring this stage t,he

nodes forward t hroogh t heir horizontal and vertical links the messages the other end

is not aware of. since each message is a t most c log n h vertical and cf log nh horizontal

links away from any destination, at most c log n h vertical ancl c'log n,, horizontal steps

are rieected. The total niimber of messages sent through a horizontal link, denoted

s:'? and a vertical link: clenotecl iq4>, is chosen so that the horizontal and vertical

t,ransrnissions end simiiltaneoiisly:

where :\-(') is the number of messages a node is not amare of at the end of stage three.

We do not specify the particiilar arrangement of transmissions but a possible choice is

schematically clepicted in Figure 6.10. T h e horizontal lines represent messages that

are sent to the highlighted node throiigh t he two adjacent horizontal links. while the

vertical lines represent the messages sent through the two vertical links.

CH-4PTER 6. LINEAR-COST T W - U N I F O R M TOROIDAL MESHES

I c log nh

Figure 6.10: A schematic ciepiction of a possible transmission arrangement for stage four. The horizontal lines represent the origins of messages sent to the highlightecl node through its horizontal neighbors, and the vertical lines the origins of the messages sent through its vertical neighbors. The shaded areas represent the origins of the niessages the highlighted node learned during the first t hree stages.

To obtain the total running time of the whole gossiping algorithrn, ive total the

number of transmission steps in each of t he four stages. For a horizontal link, we

obtain

The number of steps performed by a vertical link is

Xote that the constants hidden behind t h e O-notation depend on gh7 ,cjt.: r h l T u -

Let ~'2;;~~ and ~h:i~~ denote the idle time of the vertical and horizontal links at the

beginnings of stages two ancl three? respectivelyi Then, according to Tlieorem 6.1. ive

obtain the total time of the gossiping algarithm:

( 3 ) ( 2 )

The next version of the algorithm shows how to eliminate the term T h . r d l + r v f T ~ s . i d l e r h

Th +ru

Lire end this siibsection with a comment on the conditions that cvere assumed in

stages one and two of the âlgorithm. Namely, we assumed that

and

CHA PTER 6. LINEAR-COST T WO- UNIFORAf TOROLD-4 L MESHES

for suitable constants cl, c2 introciuced a t the end of stages one and two. These

conditions guarantee that the dimensions of the torus are not too disproportional.

and the nodes always receive new messages from their horizontal neighbors in stage

one, and from their vertical neighbors in stage tmo. As a result? t he length of the

respect ive transmission steps is guaranteed to increase exponent ially due t o the term

~ ( 1 ) h.i-l in recurrence (6.11) and the term iVv,i-l (2) in recurrence (6.15). Howerer. one

can see from (6.12) and (6.16) t hat if > /?,, + T~ or T" > ,oh + ~h these conditions

are unnecessary since the exponential growth is g~iaranteed by the terms 3: TL? 1 and - & + T U

(2) Tu. , - r

û h + ~ h from (6.11) and (6.15)-

6.2.2 Refined gossiping algorit hm

The previous subsection showed a basic version of our algorithm that runs in time

( 3 ) (2) The term Th.8dl.7" +T"..dl. " Th + r u

results from the idle links at the beginnings of the seconcl

and third stages due to the potential lack of messages to send. This subsection N-1 T h T u shows how to replace this term with O(I ) obtaining the total gossip tinie -

2 r h f ~ u +

L ~ J . ~ ~ ~ u + ~ ~ J I ~ v T ~ Th f r u

+ O ( ~ O ~ ? Z ~ + log nu) . bve assume that conditions (6.17) and (6.18)

hold true though, as mentioned at the encl of the previous subsection, in certain cases

they can be omitted. The reasori for the possible idle time of vertical links a t the

beginning of stage two is the potential lack of new messages one node can receive

from its vertical neighbors. The algorithm described in the next theorem increases

t his nunlber by omitting the transmission of messages originating a t every other row

from the packet transmitted in the last horizontal transmission step of stage one to

the given node by its horizontal neighbors. If, for esample, the node receives messages

from even-nirmbered rows at the end of stage one then its vertical neighbors receive

messages from odd-numbered rows meaning that the omitted messages are known to

the two vertical neighbors. They are delivered to the given node through the vertical

links a t the beginning of stage two. The elimination of the idle horizontal links a t the

C'HAPTER 6. LILVEAR-COST TWO- UiVlFO RM TO ROIDAL MESHES

beginning of stage th ree follows a similar pat tern.

Theorem 6.4 Giuen conditions (6.17) and (6.18), ggossiping in an all-port full-duplex

tuw-unijonn (skewed) t o r t ~ s of si-e nh x n , tnkes t f m e nt mosl

where N = nh x n , is the total number of nodes in the torus.

Proofr The gossiping algorithm achieving the t i m e bound is basecl on t h e algorithm

of subsection 6.2.1. T h i s proof shows hoiv t o bound the idle time of al1 links b - O(1).

FVe start with t h e elimination of the idle t i m e a t the beginning of stage two,

the case of stage three is similar. The reason t h e vert.iça1 Iinks were left idle in the

basic version of the algorithm is illustrated in more detail in Figure 6.11. T h e

circles O represent t h e messages containecl in t h e packet tha t is sent by notle 2 t o the

highlighted node y through the left horizontal link in t h e first horizontal s tep of stage

two. The crosses x represent t h e messages tha t a r e available for transmission by node

.r through the bot tom vertical link of nocle y a t t h e beginning of stage two. T h e size

of tlie packet sent throrigh one horizontal link is O(%): while the maximum size of

a packet that can be transmitted in the first vertical s tep is O( logn, ) . -4s a result.

for large nh and nu, tlie horizontal transmission takes signifi caxitly longer t ime than

the vertical one. and keepirig the vertical link biisy requires a large niimber of vertical

steps. This resiilts in a large amount of t ime wastecl due to propagation delays t hat

is not bounded by O(1og n h + log n , ) . Since its elimination is somewhat intricate, i v e

left the vertical links idle in the basic version of t h e algorithm, and postponed the

issrie until t his point.

The goal of t he refinecl algorithm presented here is t o increase the size of t h e packet

tha t can be sent in t h e first vertical step of s tage tivo. To achieve this goal, we need

t o increase the number of messages known to o n e node .r that are not known to its

vertical neighbor y. W e increase this number by reducing the size of the packets sent

t o y from its horizontal neighbors in the last horizontal transmission s tep of s tage one

by one half. In particular, a horizontal neighbor i of y omits messages originating

Figure 6.11: The reason the vertical links may be idle at the beginning of stage two- The circles 0 denote the origins of the messages that z sends to g, the crosses x the origins of the messages that r can send to y.

a t every other roiv from the packet it transmits to y in the last horizontal step. The

omitted messages are delivered to either s or to the other vertical neighbor of y by

their horizontal neighbors. The messages the node x includes in the first packet sent

to y in stage two are shown in Figure 6.12: the figure assumes that .r is the bot tom

neighbor of y. Note that x sends to IJ only those known messages whose originator

is closer to s than to y. The messages originating in the other half are sent to y

by its other verticaI neighbor. The reduction of the size of the packet transmitted

in the last horizontal step of stage one may increase the number of horizontal steps

performed during stage one but we show that the number rernains logarithmic in nu.

We demonstrate two facts: (i) the number of horizontal steps in stage one remains

O(log nu). (ii) the nurnber of messages available to a vertical link at the beginning

of stage two is sufficient to bound by 0(1) t h e iclle time of a vertical link cluring the

time takeii by the first horizontal transmission step of stage tiro.

First. focus on the number of horizontal steps in stage one. Comparecl to (6.11)

from the basic version of the algorithm: the reciirrence for the size of the packets sent

in each step is modifiecl in the last step (we disregard the fractions):

Solving it in a m a y sirnilar to the basic version of the algorithrn, and using T;:) = 41)- + :\ihei , h 7 ive obtain

Figure 6.12: T h e modification of the Iast step of stage one to reduce the idle time of vertical links. x denotes t h e origins of the messages the node z sends to IJ through the connecting vertical link at the beginning of stage trvo.

CHA PTER 6. LINEA R- COST TWO- FORM TOROID.4 L &f ESHES

Equating this to 1% - - I(,& + ru); t h e total t ime of stage one, results in

Nest we compare t h e size of t h e packet sent through one horizontal Iink t o t h e

nurnbeï of messages available for t.ransmission through one vertical link at the begin-

ning of s tage two. Since a horizontal link scnds messages from every (clog nh)-th row

only. the packet sent in t h e first. horizontal step coiitains at most [el,] messages.

T h e messages amilable to a vertical link are direct neighbors of the messages sent

in the last horizontal s tep of s tage one. Subtracting t h e messages sent through a

horizontal link. their nurnber is at least IV") h.s(hl ) - [ C i o g n h - 1. T o show that the clilration

of the idle periocl of a vertical link is bounclecl by 0(1): i t is stifficient to show that ,

for stifficiently large n h 7 it is triie tha t

which is equivalent t o

\Ve n-il1 bound n , in terms of LV~')(,, . We will denote by D I ' ) t h e nurnber of messages vSh

;: sent bp a vertical link during t h e i-th horizontal s tep in s tage one: i.e. DI') = - o u +TU '

Then ive can write two formulae:

T( 1)

(1) ( 1 ) h . y - - !\*(l) - 'h ive obtain since ~k-h,l = 1 and Ds, 1, = &+., h,sp + ' h

Therefore,

for nh s~ifficiently large.

T h e idea behind the elimination of the idle horizontal links a t the beginning of

stage three is the sarne-send only every other message in the !ast vertical step of stage

two: and send the left out messages via a horizontal link in t h e first horizontal s tep

of stage three. Similarly t o stage two, we need to show tha t the ntimber of horizontal

steps in s tage two reniains C3(lognh), and tha t the idle t ime of a vertical link a t the

C H A P T E R 6- LINEAR-COST TW-O- UîVIFORM TOROIDA L MESHES I l 5

beginning of stage three is bounded by O(1). The recurrence for Ni:) becornes

l\$) 2 !\-(? U . L - I + ~ , i - 1 for 1 < i < sr), ,8h + Th

and

s(2) ( 2 ) - ( 2 ) After solving the recurrence, sumrning C i l TL,,i - C::,($. + i\i(:>~~)' and cornparing

to the total time of stage two, this results in

This means that

S y ) 5 log ) ( 2 1 $ j + l ) + ~ (1 + fi;&,

Regarding the number of messages sent in the first step of stage two, the size

of the packet sent through a horizontal link \vas easily espressible in terms of n,.

Here, a similar expression for the packet sent through a vertical link in terms of nh

is harder to obtain due to the messages sent through a horizontal Iink in stage one.

Insteacl, we will express the size of t he packet sent throiigh a vertical link directly in

terms of where D ! ~ ) denotes t he total number of unit-size messages sent via one

horizontal link ciuring the t ime taken by the i-th vertical s t ep of stage two. \ive will

find an iipper bound on the size of t h e packet transrnitted t o one node y from its top

neighbor it. in the first vertical s tep of stage three ( the bot tom case is similar). One

can see that the origins of the messages sent frorn x t o y a t t h e beginning of stage

three are nodes tha t are in a row located above y and tha t a r e vertical neighbors in

every (c' log nh)-th column of the origin of a message sent t o y during stage two (eit her

horizontally o r vertically). Consider one designated row r tha t is not located below

9 and t hat contains messages delivered to y ciuring stage two. Let denote the

number of columns located left of y tha t contain the origin of a t least one message

that was delivered t o y from the "vicinity of the designated row r" during stnge tua

(this excliides the messages deliverecl t o y in stage one). T h e n t he niimber of those

neighbors of these nocles whose messages are conta.inec1 in the packet sent from ;L. t o

y in the first vertical step of stage three is a t most Lctlognh "'.'"' ] 5 , Dr.[- ,,&, r The same

bound holcls true for the columns that are right of y becaiise t he sirnilarly defined

niimber of columns Drvright is equal t o due t,o the syrnmetry between left and

right transmissions. Becaose of the up-down symmet re if we surn the valiies

ancl DrSright throiigh al1 designated rows r located above y, a n d express the sum in ( 2 ) x" D!') +ivt?l

terms of DI" ) we obtain a value t hat is bounded by :, log nh - Hoivever, if we add

the valiies for t he "miclclle" row containing y we obtain a sum t h a t is larges than this

bouncl because the miclclle row coiints twice ( i t coiints for bot h upward and clownward

transmission). Thesefore: nve double this valiie: ancl use an iipper bound on the size ( 2 ) ::, D:') +.VA:/

of the packet sent in the first vertical s tep that is equal to 2 =, log nh . Then t h e ( 2 ) ., C", D ! ~ )

nimber of messages available t o one horizontal link is a t least N ( ~ ) - d ,,,y c1 log nh - That means tha t it is sufficient t o show that

CHAPTER 6- LINEAR-COST T m - U N I F O R M TOROIDAL MESHES 11'1

which is e q u i d e n t to

SimiIarly to the formulae for the transmissions at the beginning of stage two, we can

That means that

for sufficiently large nh. i

The next theorem shows that the algorithm can be aclapted to the half-cluple'c

moclel without cloubling the number of steps. The constants in conditions (6.17)

ancl (6.18) a re moclified but their form remains the same. The idea is to sirnulate each

full-duplex cycle by two neigliboring half-duples cycles, similarly t o the algorithm of

Tlieorem 5.1.

Theorem 6.5 Given conditions (6.1 7) and (6-1s). gossiping in an all-port hnlf-duplex

two-urzzforrn (skezued) tor.tcs of size nh x n , talies t ime nt ,most

,tohere AT = n h x n, is t h e total n.um6e.r of nodes in the torus.

Proof: The algorithm used to obtain the desired bound is a half-duplex adaptation

of the full-duplex algorit hm of Theorem 6.1 that borrows the idea from the half-

duplex constant-cost gossiping of Theorem 5.1 : simulate each bidirect ional cycle by

two neighboring unidirectional cycles in a number of steps that is larger by a srnall

constant only.

At the beginning of the algorithm each node exchanges in two steps its message

with its two horizontal neighbors. The vertical neighbors are idle but the two steps

take ûnly O(1) time units. After that, the vertical columns are alternatel- divicled

into two groups sending data only up or onlj- dorvn. Similarly, horizontal roivs send

data either right or left. The algorithm follows the four stages of the algorithm

of TLieorem 6.1. The first stage performs a vertical --half-gossip" in 12 J steps-each

node sends in each vertical step to its vertical neighbor (either top or bottom) a packet

cont aining the t hree messages it obtainecl in the previous vertical step-one from its

own column, and one from each neighboring column. Each node sends through its

horizontal link large packets consisting of a11 the newly learned messages similarly to

the algorit hm of Theorern 6.1. Since two neighboririg coliirnns transmit in opposite

directions. after 121 steps the collective knowleclge of each two horizontal neighbors

contains messages froni al1 nocles in their two columns. eacli linouring one tialf. In

contrast to the constant-cost algorithm of Theorem 5.1. they do not exchange their

portions so that the vertical links can be prevented from sitting idle for an estenclecl

period of tirne. The eschsnge is unnecessary in the tivo-diniensional case since the

niessages ~vill be distributecl during the horizontal gossips in the following stage (in

fact. it could have been onlittecl at the encl of the fiïst stage of Theorem 5.1).

Stage tkvo of the algorithm performs horizontal -half-gossips' by esecuting 121 horizontal steps by eacli node. Each nocle sends packets either right or left only. de-

pending on the row to wliich it belongs. To guarantee that eve- message is distribicted

in both clirections. a node forwarcls through its horizontal link al1 the messages that

originate either rn(c1og nh) - 1- rn(c1og nh) , or rn(c1og n h ) + 1 rows apart. where m

is an integer, instead of jiist m(c1ognh) roivs as was the case in the full-duplex al-

gorithm. We use three neighboring rows instead of two to take care of an oclcl total

number of rows when two neighboring rows transmit in the same direction. The ver-

tical links complement the t ransrnission in a way similar to the f~dl-duplex algori t hm

(of course, in one direction only). Figure 6-13 illustrates the messages known to one

node and its bottom neighbor at the end of stage ttvo; note that the images of the

messages knomn to the ttvo neighbors are horizontally flipped since the nodes belong

to rows transmitting in opposite directions (right and left: respectively). For the sake

Figure 6.113: The messages learned by the highlighted node (circles O) and its bottom neighbor (crosses x ) through t h e gossips of stage one and two of the all-port half- dri ples algori t hm.

of clarity, the figure shows only the messages learned through t h e vertical and hor-

izontal gossips clisregarding the orthogonal transmissions. If we superimpose copies

of the figure shifted down by even numbers of rows smaI1er than or equal to clog nh,

the nodes coverecl by either a circle or a cross represent the origins of al1 messages

t hat are known to eit her the highlighted nocle or to one of the c log nh nodes t hat are

below it in its column. Since al1 the nodes of the network get covered at least once, a t

most c log nIL additional upward transmissions are sufficient for the highlightecl node

to receive messages from the whole network. This is also seen in Figure 6.14 which

illustrates t h e tree one message is broadcast dong during the gossips of the first two

stages.

Figure 6.11: The tree that one message is broadcast along during the gossips of the first two stages of the all-port half-duplex algorithm.

Stage t hree guarantees t hat every message can reach any destination via a log-

arithmic number of horizontal links as well. It follo~vs a pattern that modifies the

full-cluples version in a way similar t o stage two. =\ nocle sends t hrough its vertical

link niessages originating a t nodes to which it is closer than the link's other endpoint

and that are m(cf log ? z h ) - 1: m(cf log n h ) or nz(ct log nh) + 1 columns apart. Throogh

its horizontal link. it sends messages not known to the other endpoint that originate

at nocles t o which it is closer than the other enclpoint. Figure 6.1.5 illustrates the

messages known to one node after t h e first three stages. Note t hat the images of the

messages known to its vertical neighbors are horizontally flipped, and the messages

known to i ts horizontal neighbors are vertically flipped. By successively superimpos-

ing flipped copies of the figure in either direction: one can see tha t any message can

CHA PTER 6. L lNE-4 R-COST TWO- UiVIFO RM TOROIDA L MESHES 121

Figure 6.1-5: The origins of the messages known to the higl-ilighted node after the first three stages of the all-port half-duplex algorithm. Each differently shaded area represents the origins of the messages learned duïing a different stage of the algorit hm.

reach the highlighted node by traversing either a t most clognh vertical or at most

c'log n h horizontal links in the transmission direction of the link's column or row.

Stage Four splits the messages among horizontal and vertical links according to

the formulae

(c' log nh) ph + +\'P)Ïh = ( C log n h ) + X;,')T~,

!J-;) + 1vJ4) = ~ ( 4 )

Xote that one node receives messages through one horizontal and one vertical link

only. To obtain the total running time, compared to the frill-duplex algorit hm, the

half-duplex version adds a vertical eschange of messages at its beginning. Ottierwise,

the structure of the algorithm remains unchangecl. The initial exchange takes time

O( 1 ), and the number of steps in t he main "gossiping" clirections of each stage remain

the same. Moreover, even thorigh the constants change conipared t o the full-duplex

version: the esponential growth of the sizes of the packets transmitted in the steps

orthogonal to the main gosçipç in stages one: two and t hree is preserved. Therefore,

the total nunlbers of horizontal ancl vertical steps satisfy

The symbols sh . S , represent t he total numbers of steps in both directions for the

half-cluplex case. Using CoroIIary 6.2 gives a total running time eclual to

LVhen restricting ocirselves t o the one-uniform mode1 and a regrilar torus? Ive obtain

the folloiving results by setting = ,O,, = ,L?. i h = Ï~ = Ï, ancl D = LFJ + 111. Thej-

improve the upper bounds of Fraigniaud aiid Lazard [l'il which are YT + D,L? for

the full-duplex mode1 and y r + 2D,B for the half-duplex model.

Corollary 6.6 Gicen conditions (6.1 7 ) a n d (6.1 b'), gossiping in an dl-port one-

uniform regulnr torus zuith N nodes and diclnefer D tnkes t i m e

ut most LV - 1 D

4 T + $'+ (3 (log D ) -

u n d e r t h e full-duplex model, and

nt rnost

u n d e r the half-duplex model.

At the end of this section we address the issue of asymptotic bounds presented

in the previous theorems. LVe focus o n the full-duplex version of t h e algorithm. The

asymptotic bounds imply tha t , for large enough values of nh ancl n,, the algorithm

outperforms t h e bound eclual t o

tha t \vas achieved for nh = n , a n d the one-uniform case in [ I F ] . However. since

the sizes of a LE0 satellite network are typically in the range of a few tens. it is

interesting t o see the performance of the algorit hm for t his range of the torus sizes.

Since the bounds clerived in the t heorems are loose in order to cover al1 steps of the

algorithm. we will compare numerical values ive actually achieved for certain d u e s

of t h e network parameters. Table 6.1 compares the act ual running t ime we achievecl

using the rnethocl of the proposecl algorithm comparecl to t h e boirncl (6.19). The

Table 6.1: T h e running times achieved ~is ing the methocl of the proposed Ml-duplex gossiping algorithm compareci to t h e bound (6.19) for a sample set of network param- eters.

Ih

1 1 -2 -

16

Pu

1 1 2 4

T

1 1 1 1

nh

9 2.5

9 1'7

'

1 1 1 1

n,

:3:3 2.5 :3:3 17

proposecl algorithm

SS 1 1.5 1 0:3 208

bourld (6.19)

93 1SO 1 14 2 3 2

results show that for torus dimensions between 9 and 33 the algorithm can achieve

improvements of up to 10%. The improvement nat urally increases wit h increased

sizes of the torus.

6.3 One-port model

The main feature of the algorithms for the all-port model presented in the previous

section is keeping al1 links of the network constantly busy. This is not achievable

in the one-port model since there can be only one link active for one nocle. -4s a

result, the best one can hope for is keeping al1 nodes constantly busy, by which we

mean that one of their links is active. For the one-uniform model, al1 links have

the same parameters s o there is rio preference of one link over the other. For the

trvo-irniform model, a natural objective is to send as much data as possible over the

links with the higher data rate (i-e.. snialler T-parameter), and t o use as fem steps

as possible over the links with the larger propagation delay (P-parameter). This

section presents one-port gossiping algori t hrns t hat , for the full-cluples moclel, use

approximately 191 - horizontal steps, 121 vertical steps, and t hat maximize the use

of the links with higher data rate. Assuming r, < ~ h , the running time for nh, T L ,

even is ecliial to ( n h - l ) r h + (nu - l )nh5; + [?J,3,., + Ly],ûu, irhich is the sum of the

lower bouncls (6.1) and (6.8). If n h or n, or both are odcl. the t ime is larger by at

rnost 2,dh + 29, + 2 ~ , $ + 2nhr,,. Similarly to the all-port algorit hrns, the algorithms

developecl here apply to both regiilar and skewed tori but do not take advantage of

the smaller cliameter of t.he skewecl tcri. At the end of the sect,ion we compare the

one-port and dl-port moclels.

The first algorithm is for the full-duplex model. It works in two stages. First?

it distributes short messages dong one dimension, ancl then the accumiilatecl longer

messages along the other dimension. T h e idea is to use the clirection wit h the larger

r-parameter for the short messages (horizontal in Our case), and the other clirection

(vertical) for the long messages.

Theorem 6.7 Assume r, < r h . Gossiping in a one-port Jull-duplex two-urziSo-rm

(skewed) torus of six nh x n, tabes ti.me at most

chen both nh and nu are euen! and at most

ofherwise.

Proof: First consider the case nh7 nu even. Our algorithm follows the same pattern as

the algori t hms described iri Fraigniaud and Lazard [lï]? fine-t uned for the two-uniform

model. The messages are disseminated along minimum spanning trees of the network

with links laheled with transmissiori parameters TA and T, (see Figure 6.26) . The

Figure 6.16: The broadcast tree for one message in the gossiping algorithm for a one-port f~ill-duplex torus.

algorithm first gossips short messages along horizontal cycles with larger transmission

time Q. This is clone iising the algorithm of Saad and Schultz [3S] tliat takes 2 steps.

Each node alternately comrnunicates with its left ancl right neighbors eschanging the

messages the other end does not know ( the algorithm was proven optimal for a one-

port full-cluples cycle in Peters [ 3 û ] ) . T h e gossiping within horizontal cycles takes

time ( n h - l)rh + ?ph- Then the packets of length nh containing the messages

accumulated during the horizontal gossips are disseminated within each vertical cycle

using the same algorithm. The corresponding time is (nu - l ) n h ~ , + y,&. This gives

a total gossiping time of

For cases when one of the dimensions is odd we use the algorithm of Peters (361 for

gossiping witliin odd-length cycles. This algorithm takes 141 + 1 steps f ~ r a cycle of

odd length n. In t he first and last steps the nodes send packets of size only one, rvhile

in the ot her r:] - 1 steps they send packets of size two. The running t ime for original

messages of length m and links with parameters ,B and r is ( n + l)mr + ( r:] + 1)S. In

the worst case, bot11 n h and n, are odd. and applying this algorit hm to both horizontal

ancl vertical gossips results in total time

The algorithm can be applied to the half-cliiplex case by simulating each step by

t tvo.

Corollary 6.8 Assume Ï ~ , < rh . Gossipirzg in a one-port hnlf-duplex t u:o- unijorm

(skerued) t0ru.s of size nh x nu tulies t irne nt most

We note that for the one-uniform case these bounds rediice to the upper bounds of

Fraigniaud and Lazard [l;]. However, under the iialf-duplex rnodel one can do bet.ter

in terms of D7s by simulating eacli bidirectional cycle by two neighboring unidirectional

CHA PTER 6. LINE-4 R-COST TWO- UNIFO RM TOROIDAL MESHES

cycles. One can show that if tve use the scheme of Theorem 5.1 for the constant-cost

model. the total running time under the linear-cost model for nh7 nu even is

where D is the diameter, and N the total number of nodes. This is lower than the

bound 2(iv - 1 ) r + 2D,d obtained by directly simulating each step of the full-duplex

algorithm. The method increases the Ï-term for the two-uniform case since, at the

end. it requires an eschange of messages between neighboring nodes d o n g slower

horizontal links.

ive conclude with a comparison of the power of the ali-port and one-port trans-

mission modes. Since our upper bounds are not necessarily eqiially tight with respect

to propagation delays. n-e assume ,dh = ,O, = O. Considering the full-cliiples model, by

esamining the algorithm of Theorem 6.4. and assuming conditions (6.17) and (6.18)

hold true. one can see t hat its running tirne is

(one can also find simpler algorithms achieving this bound under the assumption

J h = ijL< = O). This matches the lower bound (6.2). .Assuming nh? nu e ïen for the

sake of simplicity, the algorithm of Theorem 6.7 runs in time

matching the loiver bound (6.8); note that we assume Ï, 5 rh. Under the one-unif'orrn

moclel ~h = r, = Ï we get

and

C'HA P TER 6. L l X E A R- COST TWO- LWlFO RM TOR O U M L MESHES 128

This means that the all-port model is four times faster than the one-port one. Stated

in other words7 the all-port model with a given paver and data rate per link is equally

fast to the one-port model with transmission power equal to the total power of the

all-port model. The situation is different for the two-uniform case. We get

That means that the asymptotic speed-up of the all-port mode1 over the one-port

rnoclel mith the same parameters is 2(1 + ?). Since O < Ï, 5 rh. we get the bounds

for the speed up:

If we assume that the different values of tlie parameters -rh ancl ru reçult from clifferent

clegradations of the signal, noi cliserent link powers (e.g.. due to different link lengths),

ancl that increasing the link power increases t h e clata rate linearly, the one-port model

wi t h link power equal to tlie total power of the all-port model is actiially more efficient

than the all-port model: its total gossiping time is $ to 1 times the time of the all-

port moclel. This can be explained by the fact that the one-port moclel can clevote

more potver to the more efficient (vertical) links while the all-port moclel distributes

it eclttally. Similar results can be clerivecl for the half-duplex model. From the point of

view of LE0 satellite network design this suggests that the ability to switch between

transmit ters based on the load demancl can lead to increasecl efficiency.

Chapter 7

Conclusion and furt her research

In this thesis, we studied communications in inclined low-earth-orbit satellite net-

works that use intersatellite links to connect the satellites. The main contributions

are new moclels of the networks and efficient commiinication algorithms baseci on

t hese moclels. The rnodels address two main issues-t he network topology int roduc-

ing the k-skewecl torus topologyl and propagation clelay introducing the two-uniform

moclel. The Il.-skewecl torils is a nioclified two-dimensional t.oroida1 mesh that. as our

numerical st udies s h o ~ ~ is a natural consecluence of choosinp inclined orbi ts. Since the

exact delay on intersatellite links is governecl by a comples formula, we proposed two

approsimat ions-linear and constant. The constant approximation Ieacls t,o a niodel

we cal1 two-uniforrn? in which one value is used for links between satellites in the sanie

orbit, ancl another value for links bet,cveen satellites in two crifferent orbits.

The t wo-uniform rnoclel was used to st udy the gossiping (dl-to-al1 eschange) prob-

lem in toroidal meshes. LVe cleveloped efficient st.ore-and-forward gossiping algorit hms

uncler two transmission cost rnodels-constant-cost ancl linear-cost . T h e constant-

cost algorit hms take into account only the propagation delay: while the linear-cost

ones consider bot h propagation delay and data rate. In particular, t lie linear-cost

algoritlims use a new communication pattern that minimizes the total t ime by masi-

rnizing the overlap of propagation delay and transmission t ime on different links. The

cleveloped algori t hrns improve the best-known upper bounds for the gossiping problem

in the special case of one-iiniform regular tori.

At the end we mention some unresolved open problerns and extensions of this work

for possible future research. We organize them according to the two main parts of the

t hesis-modeling and algorit hm design.

Our models can be extended in several ways. Two issues we see pa,rticularly

interesting are:

Dynarnic 6ehnuior. The communication clelay in L E 0 satellite networks peri-

odically changes over time. It would be interesting to find suitable models îo

capture t his behavior. We think that the linear approximation of the cornmil-

nication clelay proposed in this thesis is a siiitable step in that direction.

Diffèrent topologies. Our moclels consiclered topologies in wliich each satellite

has four links. One can consider ot her numbers of links, e.g., topologies ivith :3.

6: or S links have been proposed. Topologies resulting from polar consteliations

can be also considered.

Wit h respect to the design of efficient two-~iniforrn communication algorit hrns. we

see t lie following issues as imrnediate1~- interes t ing:

0 Snczller diameter- of a skeiued torus. An interesting question is the utilization

of the smaller diarneter of a skewed torus for linear-time gossiping. This thesis

develops gossiping algorithms that take advantage of the reducecl diameter of a

skewed torus comparecl to the regular one under the constant-cost rnoclel. The

linear-cost algorithms apply to both regular and skeived tori but mith the same

riinning time. It ivould be interesting to develop an algorithm for skewecl tori

t hat recliices t lie gossip time cornparecl to their unskewed counterparts.

a .-lsynchrono ils linen r-cost algorith rns. The linear-cost gossiping algorit hms de-

veloped in t his t hesis assume synchrono u s communication mode in irhich a trans-

rnitter is occiipied ~ v i th one transmission iint il the receiver successfully receives

the transmitted message. A natural extension is to consider the nsynchr-onous

mode in which it is occiipied oniy diiring the actual data transmission.

Different objectives. One can consider ot her rninirnization objectives t han the

communication clelay- An example is balancing the load instead of minimizing

the detay.

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