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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-25,

NO. 1 ,

JANUARY 1977

The Throughput of Packet Broadcasting Channels

NORMAN ABRAMSON,

FELLOW, IEEE

117

Abstract-Packetbroadcasting is a form of data communications

architecture which can combine the features

of

packet switching with

those of broadcast channels

for

data communication networks. Much

of

the basic theoryof packet broadcastinghasbeenpresented

as

a

byproduct in a sequence

of

papers with a distinctly practical empha-

sis. In this paper we provide a unified presentation of packet broad-

casting theory.

In Section I1 we introduce the theory

of

packet broadcasting data

networks. In Section I11 we provide some theoretical esults dealing

with the performance

of

a packet broadcasting network when the users

of

the network have a variety

of

data rates. In Section

IV

we deal with

packet broadcasting networks dist ributed in space, and in Section

V

we derive some properties of power-limited packet broadcasting chan-

nels,showing that the throughput

of

such channels can approach that

of

equivalent point-to-point channels.

I

INTRODUCTION

A .

Packet Switching and Packet Broadcasting

T

E transition of packet-switched computer networks from

experimental [ l ] o operational 2] ta tus during 1975

provides convincing evidence of the value of this form of com-

munications architecture.Packet sw itching, or statistical multi-

plexing [3 ], can provideapowerfulmeans of sharing com-

munications resources among large number of data communi-

cation s users when th ose users can be characterized by a high

ratio of peak to average data rates. Under such circumstances,

data from each user are buffered, address and control informa-

tion is added in a “head er,” and the resulting bit sequence, or

“packet,” is routed through a shared communications resource

by a sequence of node switches [ 4] , [5 ].

Packet-switched n etwo rks, however, still emp loy poin t-to-

point communication channels and large multiplexing switches

for routing andflow contro l inafashion similar to conven-

tional circuit switched netw orks. In some situatio ns [6] - [101

it is desirable tocom bin e he efficiencies achievable by a

packet communications architecture with other advantages ob-

tained by use of broadcast com mun ication channels. Among

these advantages are limination of routin g nd etwork

switches, system mod ulari ty, and overall system simplicity. In

addition, certain kinds of channels available to the communi-

cations systems designer, not ably satellite channels, are basic-

ally broadcast n thei rstructure. In suc h cases use of these

Manuscript received January 19, 1976; revised Jun e 11 1976. This

work was supported by The ALOHA System, a research project at the

University of Hawaii which is supported by the Advanced Research Pro-

jects Agency of the Department of Defense and monitored by NASA

Ames Research Center under Contract NAS2-8590. The views and con-

clusions contained in this paper are those of the author and should not

be interpreted as necessarily representing the official policies, either

expressed or implied, of the Advanced Research Projects Agency of the

United States Government.

Theauthor is with The ALOHA System, University of Hawaii,

Honolulu, HI 96822.

channels in theirnaturalbroadcastmod e can lkad to sig-

nificant system performan ce advantages [

1

13

,

[121

.

B.

Outline

of

Results

Packet roadcasting is a form of data omm unicatio ns

architecture which canombineheeaturesf packet

switching with those of broadcast channels for data communi-

cation netw orks. Much of the basic theory of packet broad -

castinghasbeen presented as a byprod uct in asequence of

papers with a distinctly practical emphasis. In this paper we

provide a unified present ation of packet broadcasting theo ry.

In Section I1 we introduce he heory of packetbroad-

casting as implem ented in the ALOHA System a t the Univer-

sity of Hawaii; also in Section I1 we explain a modifica tion of

the basic ALOHA method, called slotting. In Section

Ill

we

provide some th eoretical results dealing with the performance

of a packet broadcasting hannelwhen the users of the

channel have avariety ofdata rates.

In

Section IV we deal

with packet broadcasting networks distributed inspace, and

present some incom plete results on the theore tical prope rties

of such networks . Finally, in Section V we derive some prop-

erties of power limited packet broadcasting channels showing

that he hroughput of such channelscan approach hat of

equivalent point-to-point channels.

This

result is

of

importance

in satellite systems using small earth stations ince it’implies h at

the multiple access capabilityand the comp lete connectivity

(in the topological sense) of packet b roadcasting channels can

be obtained at

no

price in average throughput.

11. PACKET BROADCASTING CHANNELS

A .

Operation

of

a Packet Broad casting Channel

Consider a num ber of widely separated users, each wanting

to tran smit short packets over a comm on high-speed chan nel.

Assume tha t the rate at which users generate packets is such

tha t he average timebetween packetsfrom a single user is

muc h greater th an the time needed to tran smi t a single packet .

In Fig. 1 we indicate a sequence of packe ts transm itted by a

typical user.

Conventionalime orrequency multiplexing metho ds

(TDMA or FDMA) o r some kind of polling scheme could be

emplo yed to share the channel among the users. Some of the

disadvantages of these method s for users with high peak-to -

average data rates are discussed by Carleial and Hellman

[

131

.

In additio n, under certain cond itions polling may require un-

acceptable system complexity and extra delay.

In a packet broadcasting system the simplest possible solu-

tion to this multiplexingproblem is emp loyed . Each user

transmits its packets over the common broadcast channel in a

completelynsyn chron ized (from one user to another)

manner. If each individual user of a packet broadcasting chan-


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118 IEEE TRANSACTIONS ON COMMUNICATIONS, JANUAR

t ime

-

Fig. 1. Packets from a typical user.

ne1 is required to have a low duty cycle, the prob abilit y o f a

packe t from one user interfering w ith a packet from anoth er

user is small as long as the otalnum ber of users on he

com mon chann el is no t to o large. As the num ber of users in-

creases, however, the num ber of pa cket overlaps increases and

theproba bility hat a packe t will be lost due to anoverlap

also increases. The question of how ma ny users can share such

a chann el and the analysis of various me thod s of dealing with

packets ost due to overlap are the primary concern s of this

paper. In Fig.

2

we show a packet broadcasting chann el with

two overlapping packets. Since the first packet broadcasting

channel was put into operation in the ALOHA System radio-

l inkedcomputernetworkat he University

of

Hawaii [6] ,

they have been referred to as ALOHA channels.

B.

ALO HA Capacizy

A transmittedpacke t can be received incorrec tly

or

lost

completely becauseof two different ypes of errors:

1)

ran-

dom noise errors and 2 ) errors caused by packe t overlap. In

this paper we assume that he irst ypeof rror can be

ignore d, and we shall be concerne d only with errors caused by

pack et overlap. In Section 11-D we describe several met hod s of

dealing with the problem of packets lost due to overlap, but

first we derive the basic resultswhichell ushowmany

packets can be transmitted with no verlap.

Assume that the start t imes of packets in the channel com-

prise a Poisson point process with parameter h packets/second.

If each packet lasts

r

second s, we can define the normalized

channel traffic

G

where

G

=

AT. (1)

If we assume that only hose packets which do no t overlap

with any other packet are received correctly, we may define

A

<

h as the rate of occur rence of those packe ts which are

received correctly.Then we define he normalized channel

thruput S by

s

=

h'r.

( 2 )

The probability that a packet will not overlap a given packet is

just the probability that no packet starts T seconds befo re or

T second s after the start time o f the given packe t. Then , since

the point process formed from the start times of all packets in

the chann el was assumed Poisson, the pro bability that a packe t

will not overlap anytheracket is

e-2h*,

or

e - 2 G .

Therefore

and we may plot the channel throughp ut versus channel traffic

for an ALOHA channel (Fig. 3).

From Fig. 3 we see that as the channe l traffic increases, the

through put also increases u ntil i t reaches its maximum at S =

1/2e

=

0.184. This value of throughput is known as the capac-

overlap

I

t ime-

Fig.

2 .

Packlsts from several users on an ALOHA channel.

;2e

Fig. 3. Channel throughput versus channel traffic for an AL

channel.

S,

channel thruput

i ty of an ALOHA channel, and it occurs for value of c

traffic equal t o 0.5. If we increase the chann el raffic

0.5, the throughput of the channel will decrease.

C. Appl ica tion o fan ALO HAChannel

In order to indicate the capabilit ies of such a chann

use in an interactive network of alphanumeric computer te

minals, consider the 960 0 bits/s packet broadcasting ch

used in the ALOHA System . From the results of Secti

we see that the maxim um average throughpu t of this c

is 960 0 bits/s t imes 1/2e , or about 1600 bits/s. If we a

the conservative

[141

figure of

5

bits/s as the average da

(includingoverhead) from each active1 termin al in th

work, this channel can handle the traffic of over 300

terminals and each terminal will operate at a peak data

960 0 bits/s. Of course, the total number of terminals in

a network can b e much arger than 300 since only a frac

all terminals will be active and a terminal consu mes no

resources when it is not active.

D. Recovely of Lost ackets

Since the packe t broad casting techniqu e we have des

will result in some packets being lost du e to pac ket ov

it is necessary to int roduce some technique to com pens

this loss. We may list four differen t packe t recovery tech

for dealing with the problem of lost packets. The -firs

make use of a feedback channel to the packet transmit

the epetitionof ostpackets, while the fourth is bas

coding.

1 ) PositiveAcknowledgments POSACKS): Perha

most irect way tohandle ostpackets is to requi

receiver of the pack et to acknowledge co rrect receipt

packet. Each packet is transmitted and then tored

transmitter'sbufferuntil a POSACK is received fro

receiver. If a POSACK is not received in a given a mo

time, the transmitter can repeat th e transmission and co

to repeat until

a

POSACK s received oruntil omeoth

criterion is met. The POSACK can be transmitted on a

l A terminal is defined as active from the time a user trans

attempt to og on until he transmits log

off

message.


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ABRAMSO N: THROUGHPUT

O F

PACKETROADCASTING CHANNELS

119

rate channel (as in the ALOHANET

[6]

) or transmitted on the

same channel as the original packets (as in the ARPA packet

radio ystem [

151).

An

errordetectioncodeand a packet

numbering system can be used to increase the reliability of this

technique.

2 )

Transpon der Packet Broadcasting: Certain communica-

tion channels-notably com mun ication satellite channels-

transmit packets on one frequency to a transponder which re-

transmits th e packets o n a second frequency. In such cases all

units in a packet broadcasting network can receive their own

packet retransmissions, determinewhether apacket overlap

has occ urred, and epeat the packet if necessary. This tech-

nique has been employed in ATS-1 satellite experiments in the

Pacific Educational ComputerNetwork (PACNET)

[16]

and

in the ARPAAtlantic NTEL SAT IV packetbroadcasting

experiments

[171 .

3 Carrier SensePacket Broadcasting: For ground-based

packetbroadcasting netw orks where the signal propagation

time over the furthest transmission path is muc h less than the

packet dur atio n, it is feasible to provide each transmission unit

with a device to inhibit packet transmission while another unit

is detecte d t ransm itting. A carrier sense capability can increase

the channel throughput, even if these conditions are not

met, when used in conjunctionwithother packet recovery

meth ods. Carrier sense systems have been analyzed by Tobagi

[

181

and by Kleinrock.andTobagi

[

191

.A comprehensive yet

com pact analysis of such systems is provided in

[42] .

4 ) Packet Recovery Codes: When a user em ploys a packet

broadcasting channel to transm it long files by breaking them

into large number s of pack ets, it is possible to encode the files

so tha t packets ost du e o broadcasting overlap can be re-

covered. It is clear that some of the existing classes of multiple

burst rror-correcting odes

[20]

and cyclic product codes

[21]

can be used for pack et recovery in transmissions of long

files. It is also clear that thesecodesare not as efficient as

possible for packet recovery and that considerable work re-

mains to be done in this area.

E.

Slotted Channels

It is possible to mo dify th e com pletely unsynchronized use

of

the ALO HA channel described above in order to increase

the maximum throughput of the channel. In the pure ALOHA

channel each user simply transmits a packe t when ready w ith-

out any attemp t to coo rdinate his transmission with those of

oth er users. While this strategy has a certain elegance, it does

lead to som ew hat inefficient channel utiliza tion. If we estab-

lish a time base and require each user to st art his packet only

at certain fixed insta nts, it is possible t o increase the max imum

value of the chann el t hru pu t. In this kind of chann el, called a

slotted ALOHA chann el, a central clock establishes a time base

for asequence of “slots” of the same durati on as a pack et

transmission

[41].

Then when a user has a packet to transmit,

he synchronizes th e start of his transmission to the start of a

slot. In this fashion, if tw o messages conflic t they will overlap

com pletely , rather than partially.

To analyze th e slotted ALOHA channel, define

G ,

as the

probab ility hat he th user will transmit a packet insome

slot. Assume that each user operates independently of all other

users, and that wheth er or not a user transmits a packet in a

given slot does not d epend up on the s tate o f any previous slot.

Ifwe have n users, we candefine the normalizedchannel

traffic for the slottedchannel

G

where

n

G = C

Gi.

i=

(4)

Note that

G

may be greater than

1

As before, we can also consider the rate at which a user

sends packets which do not experience an overlap with other

user packets. Define

S i <

Gi as theprobability hat a user

sends a packe t and hat this packet is the only packet in its

slot.

If

we have

n

users, then we define the no rmalized channel

throughput for the slotted channel S where

n

s = s i

i = l

Note that S is less than or equal to 1 and S


4/12

120

IEEE

TRANSACTIONS ON COMMUNICATIONS, JANUARY

6 1 \ \

184=1/2e 368=l/e

S, ih rupu i

Fig.

4.

Traffic versus throughput for an ALOHA channel and a slotted

ALOHA channel.

111. PACKET BROADCASTING WITH MIXED DATA RATES

A .

Unslotted Case: Variable Packet Lengths

In Section I we were concernedwithhe analysis f

ALOHA chann els carrying a hom ogene ous mix of packets. If

some chann el users have a higher average data rate than others,

however, he high rate usersmusteither ransmitpackets

more fre quently or transmit longer .packets. In this se ction we

shall analyze theunslotted ALOHA channel whencarrying

packets of different lengths, and we shall analyze the slotted

ALOHA channelwhen heprobab ility of transmitting ina

given slot varies from user t o user.

Let us assume an unslo tted ALOHA channe l with two dif-

ferent possible pa cket dura tions , 7 and

7 , .

Assume

7 2 r l ,

and therefore we refer to the two different length packets as

long packets and short packets, respectively. Assume also the

start times of the ong packetsandshortpackets orm wo

Poisson point processes withparameters

h2

and

h,

packets/

second, and that the two Poisson point processes are mutually

independent.Then we candefine, he normalized channe l

traffic for those packetso f duration

ri:

Again assume that only hose pack ets which do no t overlap

with any other packet are received correctly and define hi <

Xi

as the rate

of

occurrence of those packets of duration

ri

which are received correctly. Qefine the normalized through -

put of packets ofdurat ion

i

as

S i

=

Xi'ri,

=

1 ,

2. (12)

Since we assumed two independent, oisson point processes,

the prob ability that a short packet will be received correctly is

1

3)

becomes

exp [-2G1

G2,

G,]

.

Therefore

S 1

= G1 exp [--2G, G21 G,] (

and, by a similar argument, the throughp ut of long pack

S i = G2xp [-.G,, G , G2 ]. (

Fo r any given values of hl and h, we maycalculate

G,, G ,

,, and Gzl ; substitution of these values into (16a)

(16b) will allow calculation of the hrou ghp uts' S 1 and

Therefore (16a) and (16b) may be used

to

define an allow

set of throughpu t p?&s

S1,S2)

n the (S,,S,) plane.

To determine the boundary of this egion we define

a L .

2

71

Note that a 2

1.

We may rewrite (16a) and (16b) in term

a, the ratio of long packet duration to short packet dura

S , =

G1 exp[-2Gl

(

+:) G,]

S 2 = G 2xp [-(1 +

a)G,

G 2 ] . (

The bou nda ry of th e set of allowable

Sl,S2)

airs in

(S1,Sz) plane is defined by setting the Jacobian

equal to zero. A simple calculation shows that the Jacob i

zero when

Note that this checks forG1 = 0 and for a = 1 .

We need only :substitute this expression for G2 into (

and (18b) to obtain wo equations for S, the short pac

throughput , and

S 2 ,

the long packet throughput, in term

the single parameter G , ; an d as G, varies from 0 (all

packets) to1 /2 (all short packets), we will trac eout h

bou ndar y of the achievable values of throu ghp ut in the

S,)

plane. These achievable throu ghp ut regions are indic

for several values of

a

in Fig. 5.

The basic conclusion f this analysis is t hat he ota l

channel hrough put can undergo asignificantdecrease i

packets are not

of

the same length. Thus if the two differ


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ABRAMSON T H R O U G H P U T O F P A C K E T BROADCASTING C H A N N E L S

,104

S I

ong

packel c ha m/ thruput

Fig. 5. Achievable throughput regions in an unslotted

ALOHA

channel.

packe t lengths differ by a large factor, it is ofte n preferable to

break up long packets into many shorter packets s long as the

overhead necessary t o trans mit the text in each packe t is small.

Ferguson [23] has generalized theseesults to show that

channel throughput is maximized over all possible packet

length distributions w ith ixed length packets.

In view of this discouraging resu lt, we might conclude tha t

an inhomogeneous mix of users inevitably leads to a decrease

in the maximum value of channel throughput. Surprisingly,

this conclusion is not warran ted, and we shall show in Section

111-B

hat a mix of users of varied data rates can lead to an in-

crease in the maximum values of channel throughput.

B.

Slotted Case: Variable Packet Rates

In the section we shall consider a slotted

ALOHA

channel

used by n users, possibly with different values of channel traffic

Gi.

From (6) we have a set of n nonlinear equations relating

the channel traffics and the channel throughputs for these n

users:

Define

n

CY=n

(1 -Gj ) ;

j =

1

then (21) can be written

Forany set

of n

acceptable traffic rates

Gl ,

G2, ...,

G,,

these

n

equati ons define a set of channel throughp uts

S1, 2 ,

-, S, or a region in an n-dimensional space whose coord inates

are the Si. In o rder to find the boun dary of this region, we cal-

culate the Jacobian:

Since

~n

, i = l

1 f J . k

after some algebra we may write th e Jacob ian as

Thus the condition for maximum channel throughputs is

p i = l

i

121

(25)

This condition can then be used to define a bound ary to the

n-dimensional region of allowable throu ghpu ts

S1, 2 ,

-., S,.

Consider the special case of two classes of users with nl

users in class

1

and

n 2

users in class 2:

L e t SI a n d GI e t h e t h r o u g h p u t s and traff i c rates for users in

class 1 , and le t S 2 and G2 be the throughputs and traffic rates

for users in class 2. Then the n equations (21) can be written

as the two equations

s

=

Gl ( l

G l) n l - l ( l

G2y2

( 2 9 4

s, = G,(1 G2) 2-l ( l G 1 p .

(29b)

For any pair

ob

acceptable traffic rates

G,

and C,, these two

equations define a pair of channel throughputs

S

and

S2

or

a

region in the S ,

, S 2

plane.

From (27) we kno w hat hebou nda ry of this region is

defined by the con dition

nlGl

+ n,G2 =

1 .

30)

We can use

30)

to substitute for

Gl

in (29a) and (29b) and

obtain two equations for S1 and

S 2

in terms of a single param-

eter

G,.

Then as

G 2

varies from 0 to 1, the resulting

(S1,S,)

pairs define th e bound ary of the region we seek. These achiev-

able regions are indicated for various values of

nl

and

n2

in

Figs. 6 and 7.

The important point to notice f rom Figs. 6 and

7

is that in

a lightly loaded slotted ALOHA cha nne l, a single large user can

transm it data at a significant percentage of the tota l channel

data rate, thus allowing use of the channel at rates well above

the limit of

l / e

or 37 percent obtained when ali users have the

same message rate.

A

through put data rate above the l / e imit


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122 IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY

n l users at rate S,

n2 users at rate S

(nl'n,)

I

e

/

3

Fig. 6. Allowable channel throughputs.

I

o

nl users at rate SI

n, users at rate

S,

in , ,n ,

I

e

I3

Fig. 7.

Allowable

channel throughputs.

has been referred to as excess capacity [24 ]

.

Excess capacity

is impo rtant for a l ightly loaded packet broadcasting network

consisting f many nteractiveermin al users and a small

number of users who send large but infre que nt files over the

channel. Operation of the chann el in a lightly loaded cond i-

tion, of course, may not be desirable in a bandwidth-limited

channel. For a omm unication s satellitewhere the average

power in the satellite transponder limits the channel, however,

operatio n in a lightly loaded acket-sw itchedmod e is an

attractivealternative. Since the satellite will transmit power

only when it is rela9ing a pack et, the duty cycle in the trans-

pond er will be smail an d the average power used will be low

(See Section VI.

Finally, we no te , ha t t is possible t o deal withcertain

limiting cases in mo redetail , toobtain quat ions or he

bound ary of the allowable (S1,S2) region.

I For n1 = n 2 =

I :

Upon using 30) in is), e obtain

bl

S I =

e

Additional deta.ils dealing with excess cap acity nd

delay expe rienced with this kind of use of a slotted ALO

channel may be foun d in [ l I ] and [25]

.

A different vie

the use of a slotted packet broadcasting for different so

may be found in [4.3].

IV. SPATIAL PROPERTIES

OF

PACKET BROADCAST

NETWORKS

A . Packet Repeaters

In thissection we deal withcertain spatial prope rti

packet broadcasting netw orks. Not long after the initial

of t he ALOHA System went nto oper ation , t was rea

that the range of the network could be extended beyon d

range of a single radio link in the network (about 2 00 km

the use of packet repeaters. A packet epeateroperates

much the same manner as a conventional radio repeater w

onemajor excep tion. Since radio transmission n a pa

broadcasting netw ork is interm ittent, a pack et repeater ca

ceive apacketand etransmit hatpacke t in the same

quen cy band by urning off ts receiver during a retran

sion burst. Thus a packet repeater can sidestep many

o

frequency allocation and spatial cell problems [26] of

ventional land-based repeater networks.

The use of packe t repeaters eads to he con sider ation

packet broadcasti:ng networks withmore than ne entra

station istributed over very large areas. Users transm

pack et, and if the packe t canno t be received directlyby

destination, i t is forwarded to its destination by one

or

packet repeatersaccording to some routingalgorithm 27

The s tudy

of

such networks has led to he analysis of

com mun ication heory issues related

to

theperformance

thenetworks: 1) captureeffectand 2) thedistribution

packet traf fic and p acket thro ughp ut in space.

B.

Capture Ef fec t

Up to hispo int we have analyzed packet broadca

channels under the pessimistic assu mp tion that if two pa

overlap at he receiver, bothpackets are lost.

In

fact,

assumption provides a lower bou nd o

the

performanc


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ABRAMSON: THROUGHPUT OF PACKET BROADCASTING CHANNEL S

123

real packet broadca sting channels, since in many receivers the

stronger of tw o overlapping packets may capture the receiver

and may be received withouterror. Metzner [ 4 0 ] has used

this fact o derive an interestingresult, showing tha t by di-

viding users into tw o groups-one transm itting at high power

and the other at low power-the max imum throu ghput can be

increased by about 50 perc ent. This esult is of impo rtanc e

for packet broadcasting ne tworks with a mixture of data and

packetized speech traffic.

In orde r to include the effect of capture in a packet broad-

casting netw ork , we consider a distribu tion of packet genera-

tors over a two-dimensional plane and a single packet bro ad-

casting receiver which receives packets from these generators

[ 4 1 ] The receiver then may be viewed as a “packet sink” and

the packet generators as a distribu tion of “pac ket sources” in

the plane. We assume that the rate of generation of packets in

a given area depends only on r , the distance from the packet

sink, and s independent of direction 0 .

Then we may define a trafficdensity anda hroughput

density analogous to the normalized traffic G and normalized

thoughput S defined in Section 11-B.

C ( r )= normalized p acke t traffic per unit area at a distance

S(r)

=

normalized packe t thruput per un it area at a distance

The traffic due to all pack et generators in a differential ring

r .

r .

of width

dr

at a radius

r

is

G( r) 27rr dr. (3 4 )

We assume that pac ket s from differ ent users are generated

so

that the packet starting times of all packets generated in the

differential ring constitute a Poisson point process. Then since

the sum of tw o ndep ende nt Poisson processes is a Poisson

point process, if users in different rings are indepen dent, the

star t times of all packets generated in a circle of radius

r

also

con stitu te Poisson point process, andheotal traffic

generated by all users within a distance r of the cente r is

G( x) 2n x (3 5 )

If we assume that a p ack et f rom a ser at a distance r from the

cente r will be received co rrectly unless it is overlapped by a

packet sent from a user at a distance ar or less a > l) , then

using th e results of Sect ion 11-B the proba bility h at such a

packet will be received cor rectly is

exp

[

4 a l r C ( x ) xx ] . ( 3 6 )

Any packet generated from a packet source in the circle of

radius ar shown in Fig. 8 will interfere with packets generated

from a source in the circle of radius r . A packet generated out-

side the circle of radius ar will not nterfer ewith packets

generated fr om a source in the circle of radius

r .

We can elate the normalized pac ket hrou ghpu t o he

normalized pa cket traffic in the usual way:

Fig. 8. Regions

of

interferingpackets.

2nrS(r) r

=

27rrC(r)exp -4n C(x)x x r

- [ l r 1

or

S(r) = G(r) exp [ 4 7 r l r G (x )x d x ]

.

( 3 7 )

If we take a derivative of (37) with respect to r and use ( 3 7 )

to subs titute for the ex ponentia l, e get

S’(r)G(r)

=

G‘(r)S(r) nra2S(r)G(r)G(ar).3

8)

We have not fou nd a general solution of ( 3 8 ) for relating

S(r)

t o

G( r )

in the presence of capture. We have been able to

analyze tw o special cases, however.

C.

Tw o

Solutions

In the first of these special cases we assume a constant traf-

fic density G r).We can then show that the through put density

S(r)

has a Gaussian form, due to the fact that those packets

generated fur ther from the receiver will be received correctly

less frequ ently han thosepacketsgenerated close to the re-

ceiver.

In the second special case analyzed we assume a consta nt

packe t hrough put density S(r) and perfect capture

(a =

1 .

Under hese assumptions, hepacket trafficdensity will in-

crease as the distanc e from he receiver increases. We show

that there exists a radius

ro

such th at the packe t traffic ensity

is finite within a circle of radius ro around the receiver, while

the packe t traffic density becomes unbo unde d on the circle of

radius y o .

For he mpo rtant case of a packet broadcastingchannel

distributed over some geographical area and using apacket

retransmission policy Section 11-D), this result has an in-

teresting inte rpretatio n. In such a situation any packet trans-

mi tted from a terminal lo cated within the circle of radius ro

will be received correctly with prob ability one (after a finite

number of retransmissions), while theexpectednumber of

retransmissions required for a packet transm itted from a ter-

minal further rom hecenter than

ro

willbe unbounded.

Thus hereexists a circle of radius ro such that terminals

transm itting from within this circle can get their packets int o

the central receiver, while terminals transmitting from outside

this circle spend all their time re transm itting their packets in

vain. We call ro the Sisyphus distance of the ALOHA channel.

I

Constant Packet Traffic Density:

Assume the density of

normalized packet traffic is constant over the plane


8/12

124 IEEE TRANSACTIONS ONOMMUNICATIONS, JANUARY

C ( r )

=

Go (39)

and define the distance r1 as the radius of a circle within which

the total packet traffic is unity:

rrl

2Go

n 1.

Then

38)

reduces to

4ra

r1

Y ( r )

= S(r)

with the boundary condition

so =

Go

(41b)

so

that the packe t through put density is

and the totalnormalized packet thruput from a circle

of

radius

r

is

S

=[ (r‘)2rr‘ dr‘

=La2

(1

- e x p [-2(

:)‘I)

and

1

lim

S

=

.+- 2a2

( 4 3 )

( 4 4 )

Note hata otal hroughputwhichcan be supported by a

single packet sink with “perfect capture”

(a

=

1)

is equal to

one half.

2 ) Constant Packet Throughput Density: Another case of

interest where we have found a solution for38) is that of con-

stant packet throughput density in the plane. Assume

S(r)

= S o

( 4 5 )

over the region in the plane where S ( r ) and C ( r )are bounded.

Then ( 3 8 ) becomes

G’(r)

=

4nra2G(r)G(ar).

(46)

For the case of a

=

1 (perfect capture), ( 4 6 )becomes

G ’(r )

=

4 n rG2 (r )47)

with the boundary condition

G 0)

=

s o

so tha t

( 4 9 )

Fig.

9.

Region

of

constant packet throughput So for a single pa

sink.

for

where

and ro is the Sisyphusdistance mentioned nSection

Note tha t the Sisyphus distance also has the property tha

1

nro2So=

2

As in the previous case, the total packet throughput w

can be supportedby single packet sink operatingw

perfect capture is one half.

V. PACKET BR.OADCASTING WITH AVER AGE POW

LIMITATIONS

A . Satellite Packet Broadcasting

In

previous sections we have analyzed the perform anc

packet broadcasting channels and compared the performa

of these channels to th at

of

conventional point-to-point

nels op erating at the same peak data rate. Such a compa

is of interest in the case of channels limited by multiple a

interfe rence rather han noise, since an increase n the

mi tted power of such channels will not lead t o improved

formance. But juut as the average data rate of a packet b

casting channel can be w ell below its peak data rate when

operated at a low duty cycle, the average transmitted pow

a pack et broadcasting channel can be well below its peak t

mitted power.

In this ection we analyze the hrough put of apac

broadcasting channel when com pared

to

that of a conven

point-to-point channel of the same average power. This an

sis s of interest in the case of satellite informa tion sys

employing thousands of small earthstations.Fora sat

system the undamen tal imitation in thedownlink i

average power available in the satellite tran spond er rather

the peakpower. Our results show hat in the imit of

numbers of small earthtations,hepackethroughput

approaches 100 percent of he point-to-point capacity.

the multiple access capability and he comp lete connec tiv

(in the topological sense) of an ALOHA channel can b

taine d at no pric.e in average through put. Furthe rmore ,


9/12

ABRAMSON: THROUGHPUT OF PACKET BROADCASTING CHANN ELS

125

transpo nder (while the average power is kept con stan t), he

small earth tations may use smaller antenn asand simpler

receivers and mod ems than would be necessary in a conven-

tional system.

In existing satellite systems the TWT out pu t power in each

transponder cannot be varied dynamically.

In

such systems the

advantages impliedby our analysis may be realized by re-

quency-division haring a single transpo nder mon g several

voice users and a single chann el, operating in an

ALOHA

mode

or some other burst mode, and occupying a f requency band

equivalent t o one or more voice users. The typ e of operatio n

implied by our analysis

also

suggests investigation of high peak

power satellite burst transponders (perhaps employing power

devices similar t o those used in radar systems) fo r use in infor-

mation systems omposed of large num bersof ultra-small

earth stations.

B. Burs t Power and Average Power

The capacity of a satellite channel can be calculated by the

classical Shannon equation

c =

Wlog 1 + -

(

3

where

C

is the capacity in bits (if th e log is a base two loga-

rithm), W is the channel bandwidth, P is the average received

signal power at the earth statio n, and

N

is the average noise

power athe ar thtation.Equation 53) expresses the

capacity of the satellite channel under the assumption that the

t r a n s p o n d e r t r a n s m i t s c o n t i n u o u s l y .

If the channel is used in burst mod e the transpo nder will

emit power only when a data burst occurs, and he average

power out

of

the ranspo nder will e less than heburst

power.

Let

D

be the ratio of he average power transmitted

to he power transmitted during adataburst.Fora linear

transponder D will equal the channel traffic G , and for a hard-

limiting transponder D will equal the duty cycle of the chan-

nel. For both the unslotted and slotted

ALOHA

channel the

dut y cycle is

1

C G . Thus for a linear transponder2

D =

G,

( 5 4 4

while for a hard-limiting transp onde r

D = 1 - e -G. (54b)

Note tha t in the case of a hard-limiting transpo nder with small

values of channel traffic , the duty cycle approaches th at of a

linear transponder.

If

we retain

P

as the not atio n for the average signal power

received at the earth sta tion , the power received during a data

burst will be P/D. Thus

53)

should be modified in t wo ways.

*Our analysis is o f significance only

for G

< 1 . The analysis is for-

mally correct, however , for all

G ,

even though the designation of the

power ransmitted during bursts as “peak powe r”becomes inappro-

priate for the linear transponder case when

G

> 1 . (In such a situation

the “peak po wer” is less than th e average powe r.)

‘ 9

“ h s i a n a l - t o - n o i s e ratio db)

.2 A 6 8 1.0 1.2

1.4

1.6.8

2.0

channel trafflc

Fig. 10. Linear transpond er;unslottedchannel.

1)

We replace

W

by

SW

to account for he fact that he

channel is only used intermittently.

2 ) We replace

P

in

53)

by

P/D

to k eep the average power

of the channel fixed at

P.

We should note that wh en we make these changes, we are

assuming that the packet len gth of the system is long enough

so

that the asym ptotic assumptions which are used to derive

(53)

still apply . In practice, this is not a problem.

With these two changes then , we have fou r different cases.

I )

Uns lotted channel, linear transponder:

C = Ge-2G

Wlog

(I

+ ) .

2)

Uns lotted channel, limiting transponder:

3 Slotted channel, linear transponder:

C, = Ge-GWlog (1

2 )

4) Slotted channel, limiting transponder:

C,

= Ge-GWlog

We have calculated the normaliz ed capacities Ci/C for i =

1, 2 , 3,

4 for different values

of

P/N, the signal-to-noise ratio

of he earth station when the ransponder operates continu-

ously . The normalizedcapacities are plotted in Figs.

10, 1 1 ,

12, and 13 for PIN equal to -20,

-10,

0,

10,

and

20

dB. Of

particular nterest n these curves

is

the fact th at the highest

values of

Ci/C

occur just where we wouldwant them o

occur-for small values

of

channel traffic

C)

and or small

earthstations (low

P I N ) .

In the limit we have (fora fixed

value of

G)


10/12

126 IEEE TRANSACTIONS ON COMMUNICATIONS, JANUARY

c iwnne l t ra f f i c

Fig. 11. Limiting ransponder;unslottedchannel.

:

r stgnpl-to-noiseatio db)

I

Z A .6

.E

I b 112

14

16

118

2 OG

channel traff ic

Fig. 12. Linear ransponder;slotted channel.

ci

s

C.

D '

lim -

i

1 , 2 , 3 , 4

EO

so that

1) unslotted channels, inear transponder

Cl

ZO

C

l im e--2G

2) unslotted channels, limiting transponder

3)

slotted c hanne l, linear transpo nder

4) slotted channel, imiting transponder

C

Ge cG

lirn

0

C

(1 -ecG)

1

signal-to-noise ratio

db)

O L

I ~ I ~ I ' I ~ I ' I ' l

.2

A .6

8 1.0

1.2

1.4 1.6

1.8

2.0

G

channel traffic

Fig. 13. Limiting ransponder;slottedchannel.

and in all cases

Ci

lim lim

1.

C

G - 0

N + O

Thus his multiplexing techn ique allows a netw ork of s

inexpensive ear th stat ion s o achieve the max imu m valu

channel capacity,at he same time providing comp letec

nectivity and multiple access capability.

VI. BACKGROUND AND ACKNOWLEDGMENT

The term packet broadcasting was first coined by Ro

Metcalfe in his Ph.D. dissertation [ 28 ]. As is often he

with simple ideas, the concep t of combining burst transmis

and Poisson user statistics to provide rand om access to a c

nel has occurred independ ently to a numb er of investiga

(56) The first attempt at an analysis of such a system of which

aware is contained n an internal Bell Laborato ries mem o

dumbySchroeder [2 9] , suggested by an arlierpaperby

Pierce and Hop per [30]. Two other early related papers

written by Costas [31] and Fu lton [32]. Of course, a the

ical analysis is not necessary in o rde r to build suc h a syst

and anyone who has sat in a tax i listening to th e stac catov

(57a) bursts of a radio dispatche r and a set of taxi drivers shar

single voice channel will recognize theoperationofa v

packet broadcasting channel using a carrier sense proto

And even after an analysis is available, the con cep t of pa

broadcasting may be suggested witho ut reference to

The first papers analyzing packet broadcasting in the f

implemented in the ALOHA System [6] assumed fixed pa

throu ghp ut and a retransmission ,prot ocol as described in

tion 11-D-1). This approach leads toanumb er of ques

involving optimum retransmission policy [2 8] , t he behavio

the channel with a finite number

of

users [39.]

,

stability o

channel

[

131

,

and transmission of long files by means of

ous

reservationschemes [34]

,

[44]

.

A comprehensive t

ment

of

these as well as other interesting packet broadcas

question s may be fou nd in Kleinrock 1421

.

In this pape

(57d)

have takenadifferentapproachby assuming a given pa

traffic rather than throughput. With such a starting point

(57b)heory331.


11/12

ABRAMSON: THROUGHPUT OF PACKET BROADCASTING CHANNELS

127

questions mentioned above donot assume key importanc e

in theheor y, lthoug h their ractical importanc e is not

diminished.

Much of the theory of packet broadcasting was developed

in two working groups sponsored by the Advanced Research

Projects Agency of the Departm ent of Defense. These groups

circulated a private series of working papers-the ARPAN ET

Satellite System otes (ASS notes) and the Packet Radio

Tem porary notes (PRT notes)-where man y of the theoretical

results described o r referenced n his paper appeared for the

first time. Unfortunately, the several references to ASS notes

in papers subseque ntly published in the open iterature may

have produced some confusion in the minds of those trying to

trace th e references. Am ong the mo st significant of the ASS

not e rind PR T ote results was the first erivation of the

capacity of a slotted ALOHA channel and the first analysis of

the use of the capture effect in packet broadca sting, both by

Larry Roberts.Thatnote has since been republished n the

open l i terature [41].

The results ofSec tion 111-A dealing wit h wodifferent

packet lengths were suggested by an ASS note w ritten by Tom

Gaarder, and the results of Section 111-B dealing with the

excess capacity of

a

slotted channel w ere suggested by an

ASS

note written by Randy Rettberg. Other problems which were

first analyzed in ASS note s or PRT notes bu t not emphasized

in thispaper nclude various packetbroadcasting reservation

systems [22] [35],

[ 3 6 ] ,

carrier sense pac ket roadcasting

[181

,

[

191

,

and questions dealing with packet routing and

protocol issues in anetworkof repeaters [ 37] .The eader

interested in theoreticalnetworkprotocolquestions hould

also see Gallagher [38] , although this work did not originate

in an ASS note’or PRT note.

The first system to em ploy packet broadcasting techniques

was the ALOHA System co mpu ter netw ork at the University

of Hawaii in 1970. Subseq uently, packet repeaters were added

to

the etwork and acket roadcasting by satellite was

demo nstrated in the system. Som e of he people involved in

themplem entation and evelopment f the system were

RichardBinder, Chris Harrison, Alan Ok inak a, nd David

Wax.

The historical relevance of [ 29 ] and [ 3 2 ] was pointed out

to me by Joe Aein, to whom

I

am indebted, in spite of my

embaras sment at having forg otten I was thesis supervisor on

the second of these papers.

REFERENCES

L. G. Roberts and

B.

D.Wessler, “Computer network develop-

ment t o achieve resource sharing,” in 1970 Spring Joint Comp ut.

C o n f , A F IP S

Conf

Proc., vol. 36. Montvale, NJ: AFIPS Press,

1970, pp. 543-549.

L.

G .

Roberts, “Data by the packet,” IEEE Spectrum, vol. 11,

pp. 46-51, Feb. 1974.

W.

W.

Chu, “A study of asynchronous ime division multiplex-

ing for time-sharing computer systems,” in 1 9 6 9 Spring Joint

Comput.Conf: ,AFIP S Conf Proc., vol. 35. Montvale, NJ:

AFIPS Press, 1969, pp. 669 47 8.

F. Heart, R. Kahn, S. Ornstein, W. Crowther, and D. Walden,

“The nterface message processor for the ARPA computer net-

work,” in 1970 Spring JointCompu t .ConJ, AFIPS Con f

Proc., vol. 36. Montvale, NJ: AFIPS Press, 1970, p. 51-

567.

H. F h k , I. Frisch, and

W.

Chou, “Topological considerations in

the design of the ARPA computer network,” in 1970 Spring

Joint Cornput. Con , AFIP S Con Proc., vol. 36. Montvale,

NJ: AFIPS Press, 1970, pp. 581-587.

N. Abramson, “The ALOHA system-Another lternative for

computer communications,” in

I 9 7 0

Fall Jo in t Compu t ; C on f ,

AFIP S Con Proc., vol. 37. Montvale, NJ:AFIPS Press, 1970 ,

R.E. Kahn,“Theorganization of computer resources into

a

packet adio network,” in Nat.Comput.Confi , AFIPS Con&

Proc.,

vol.

44,May 1975, pp. 177-186.

N. Abramson and E. R. Cacciamani, Jr., “Satellites: Not just a

big cable in the sky,”

IEEE Spectrum,

vol. 12, pp. 36-40, SePt.

1975.

I. T. Frisch, “Technical problems in nationwide networking and

interconnection,”

IEEE Trans. Commun., vol.

COM-23, pp. 78-

R. M. Metcalfe and D. R. Boggs, “ETHERNET:Distributed

88, Jan. 1975.

packet switching for local computer networks,” Commun . Ass.

Com put. Mach., to be published.

N. Abramson, “Pakket switching with satellites,” in Nat. Com-

p ut . C o n f , AFIP S Confi Proc., vol. 42, 1973, pp. 695-702.

R. D. Rosner, “Optimization of the number of ground stations

in a domestic satellite system,” in EASCON’75 Rec., Sept. 29-

A. B. Carleial and M. E. Hellman, “Bistable behavior of ALOHA-

typesystems,” IEEE Trans. Commun. ,

vol.

COM-23, pp. 401-

410, Apr. 1975.

P. E. Jackson and C.D. Stubbs, “A study of multiaccess com-

AFIPS ConJ Proc.,

vol. 34. Montvale, NJ: AFIPS Press, 1969,

puter

communications,”

in 1 9 6 9 Spring Joint Conzptrt.

Con&,

pp. 491-504.

T. J. Klein, “A tacticalpacket adio ystem,” in Proc. Nat.

Telecommun. Con&,New Orleans, LA, Dec. 1975.

K. Ah Mai, “Organizational alternatives for a Pacific educational

computer-communicationnetwork,” ALOHA Syst. Tech. Rep.

CN74-27, Univ. Hawaii, Honolulu, May 1974.

R. Binder, R. Rettberg,and D. Walden, TheAtlantic Satellite

Packet Broadcast and Gateway Experiments. Cambridge,

MA: Bolt Beranek and Newman, 1975.

L. Kleinrock and F. A. Tobagi, “Packet switching in radio chan-

nels: Part I-Carrier sense multiple-access modes .and their

throughput-delaycharacteristics,” IEEE Trans. Com mun ., vol.

COM-23, pp. 1400-1416, Dec. 1975.

F. A. Tobagi and L. Kleinrock, “Packet switching in radio chan-

nels: Part 11-The hiddenerminal problem in carrier sense

multiple-access and the busy-tone solution,” IEEE Trans. Com-

mun. , vol. COM-23, pp. 1417-1433, Dec. 1975.

R. T. Chien,

L.

R. Bahl, and D. T. Tang, “Correction of two era-

sure bursts,” IEEE Trans. Inform. Theory, vol. IT-15, pp. 186-

187, Jan. 1969.

N.

Abramson, “Cyclic code groups,” Problems of ln fo rm . Trans-

mission, Acad. Sci. USSR, Moscow, vol. 6, no. 2, 1970.

L. G. Roberts, “Dynamic allocation of satellite capacity through

packet reservation,”

in

Nat. Comput.

Con5

AFIP S Con Proc.,

vol. 42, June 1973, pp. 711-716.

M. J. Ferguson, “A study of hnslotted ALOHA witharbitrary

message lengths,” in Proc. 4th Data Commun. Symp. . Quebec,

Canada, Oct. 7-9, 1975, pp. 5-20-5-25.

R. Rettberg, Random ALOHA with slots-excess capacity,”

ARPANET Satellite Syst. Note 18, NIC Document 11865,

Stanford Res. Inst., Menlo Park, CA, Oct. 11, 1972.

N. Abramson, “Excess capacity of a slotted ALOHA channel

(continued),” ARPANET Satellite Syst. Note30, NIC Docu-

ment 13044, StanfordRes. Inst., Menlo Park, CA, Dec. 6 , 1972.

L. Schiff, Random-access digital communication or mobile

radio in a cellular environment,” IEEE Trans. Commun., vol.

OM-22, pp. 688-692, May 1974.

H. Frank, R.

M.

Van Slyke, and

I

Gitman, “Packet radio net-

work design-System considerations,” in Nat . Compu t . Con f ,

AFIPS Con& Proc., vol. 44, May 1975 , pp. 217-232.

R.

M.

Metcalfe, “Packet communication,”Rep. MAC TR-114,

Project MAC, Massachussetts Inst.Technol., Cambridge, July

1973.

M.

R. Schroeder,“Nonsynchronous imemultiplexsystem or

speech transmission,” Bell Lab. Memo., Jan. 19, 1959.

J. R. Pierce and A.L. Hopper, “Nonsynchronous time division

with holding and random sampling,” Proc. IRE, vol. 40,pp.

1079-1088, Sept. 1952.

pp. 281-285.

Oct. 1, 1975, pp. 64A-645.


12/12

IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. COM-25, NO. 1 , JANUARY

J. P. Costas, “Poisson, Shannon, and the radio amateur,” Proc.

I R E , vol. 47, p. 2058, Dec. 1959. ‘

F.

F:

Fulton, Jr., “Channelutilizationby intermit tent trans-

mitters,’’ Tech. Rep. 004-2, Stanford Electron. Lab., Stan-

ford Univ., Stanf ord, CA, May 12, 1961.

K. D. Levin, “The overlapping problem and performance degra-

dation of mobile digital communication systems,” IEEE Trans.

Commun. . vol. COM-23, pp. 1342-1347, Nov. 1975.

R. Binder, . “A dynamic packet-switching system for satellite

broadcast channels,” in Con Rec., Int. Con Commun. , vol. 111,

June 1975, pp. 41-1-41-5.

S.

S .

Lam and L. Kleinrock, “Packet switching in a multiaccess

broadcast channel: Dynamic control procedures,” IEEE Trans.

Commun. , vol. COM-23;pp. 891-904, Sept. 1975.

L. Kleinrock and S.S. Lam, “Packet switching in a multiaccess

broadcast channel: Performance. evaluation,” IEEE Trans. Com-

mun. ,

vol. COM-23, pp. 410-423, Apr. 1975.

I.

Gitman, “On the capacity of slotted ALOHA networksand

some design problems,” IEEE Trans. Commun., vol. COM-23,

pp. 305-317, Mar. 1975.

R. G. Gallager, “Basic limits onprotocol nformation in data

communicationnetworks,” IEEE Trans. Inform.Theory, vol.

M.

J.

Ferguson, “On the contr ol, stability, and waiting time in

a slotted ALOHA random-access system,” IEEE Trans. Com-

mun. , vol. COM-23,pp. 1306-1311,Nov. 1975.

J J . Metzner, “On improving utilization in ALOHA networks,”

IEEE Trans. Comm un. , vol. COM-24, pp. 447-448, Apr. 1976 .

L.G . Roberts, “ALOHA packet system with and without slots

and capture,”

Compu t . Commun .

R e v . ,

vol.

5 ,

pp. 28-42, Apr.

L. Kleinrock, Queueing Systems, Volume

2:

Computer Applica-

tions. New York: Wiley, 1976 ,pp.,360-407.

I.

Gitman, R.

M.

Van Slyke, and H. Frank, “On splitting random

accessed broadcast communication channels,” in Proc. 7th

Hawaii Int. Con& Syst. Sei.-Suppl.

on

Comput.Nets , Western

Periodicals, Jan . 1974, pp. 81-85.

IT-22, pp. 385-398, July 1976.

1975.

[44] W. Crowther, R. Rettberg,

D.

Walden,

S.

Ornstein, and

F.

“A system fo:r broadcast communication: Reservation ALO

in Proc. 6th Hawaii Int. Con Syst. Sci., Western Perio

Jan. 1973, pp.371-374.

*

A

Multiaccess Model for Packet Switching with a Satellite

Having Some Processing Capability

Abstract-A

multiaccess model for packet switching with a satellite

having the capability

of

interrogating the uplink header and creating

the downlink header

is

proposed. The satellite broadcasts slot assign-

ments,

based

on

the users’ reported queue status, o th e users for trans-

mission in the next frame. With the protocols being done at both the

earth tationsand at he satellite, the proposedmultiaccessmodel

avoids collisions that areprevalent in schemes

of

the

ALOHA

type.

Manuscript received February 27, 197 6; revised July 13, 1976. This

paper has .beenpresented at heThird nternational Conference on

Computer Communication, Toronto, Ont., Canada, August 3-6,1976.

This work was supported in part by the National Research Council of

Canada under Grant A7779 and in part by a Graduate Scholarship.

F. W. N g is with the Department of Electrical Engineering and the

Computer Communica tions Networks Group, University

of

Waterloo,

Waterloo, Ont., Canada.

J

W. Mark is with the Department of Electrical Engineering and the

Computer Communications Networks Group , University of Waterloo,

Waterloo, Ont., Canada. He is currently on leave at the IBM Thomas

J

Watson Research Center, Yorktown Heights, NY

10598.

The actual model is too complex to handle analytically. We d

analytical equa tions for a two-gioup model. Calculated and sim

buffer overflowprobabilitiesasa function of traffic ntensitya

buffer size are compared. We alsoevaluate theperformance o

actual model in terms of average system delay as a func tion of.

intensi ty by means

of

computer simulation.

I.

INTRODUCTION

A

data traffic grows, the demand on computer commu

cation using satellites,which ffer wide transmi

bandw idths over long distances, will continue to increase

trend has been to findmore efficient chemes forchan

sharing. Synchronous time-divisionultiplexing (ST

represents an attractive scheme which permits many use

share the same chann el. One of the serious drawbac ks a

ated

with a snychronous transmission system is that it as

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