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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
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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
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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 ,
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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)
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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.
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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.
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*
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