January 1985
LIDS-P-1428
A PERSPECTIVE ON MULTIACCESS CHANNELS*
by
R. G. Gallager**
ABSTRACT
The Information Theoretic Approach and the Collision Resolution
Approach to Multiaccess Channels are reviewed in
terms of the Underlying
Communication Problems that both are modelling.
We give some perspective
on the strengths and weakness of these approaches and argue for the need
of a more combined approach focused on coding and decoding techniques.
*This research was conducted at the M.I.T. Laboratory for Information and
Decision Systems with partial support provided by NSF under Grant NSFECS-8310698 and by DARPA under Contract ONR/N00014-84-K-0357.
This paper
is an expanded version of the Shannon Lecture at the International Symposium on Information Theory at St. Jovite, Quebec, in September, 1983.
**Room No. 35-206, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139.
F'RSETIVE
A
1 a.. i.i"1
Fo!r
br -di
r-l
fsome
t he I.ast
rtua i. solat:i.
.s th er e h a. e
multiaccess
t.e
o,. from
and to
vr -
-en
oni
f<iodjelti.
del -.
4'r'!.::
C)-
RO..
T JCT
. it JN
o-.f reseearch
f
di terent
addres
j.yive somi.e persnpective
rk i.: d
7"
of
:i.s i
,..i orruF:':i
L?-
tnie
a: sEfi r !et
i
both
.
' Eac
:::
r(eIcei' e
,ve
.,-
.
and 'i-ther¢
'ani'sm:
r.:
ra.:.:n
.
n etwri LCr
I
te. ephone
line.
ft
netio4.:rks
~~-"
re'',
-r.r
ng-nai
r
=
is
l
t
hese
re
i.
to
r
i
r
-dies
bj
t:o prov'ide
needed
tie
in, orm.1
i
tionrr
~
tharr.
t- r
ee
theora
lI tip
rec:ei v-ds
he
three
hese
'
r'-e
There
rr'
i
i,, r,
d
si
.5
-e:-idbac-::
].
~
also
a common
"
-
in
channel
-
-
f a.i vr
~
most
i::, ,: I:o-u..ur
t.?
.
t
of
-'
c rtra .
is
node
-
tcie r
ir-.Or
forfpo the
h
u. l i nk
-'
networ
rere Woneik where
adectuIate
.r'.i
,nr
.
the
ino rm-a ti'o.
on
d=
e s
. -a
h
1.
tr-a'ffic to th,:e ce ntra
is
et.eHen
fedbrack
m,:-s,!
i
'L:v"
i
bfy an
a
arreas.
.e
b
"t.-,
n
ri-iisro i at
rad io
is
am.oun:t
th ev.rii?
l]L. it'
7 :ar
a
whi-nere
~
t i err-s
s.a:l
i.
t.e'r-.,
i
or
and f or- tih e
...
1:
mdel
repe a.i-er,
st.:udvir'n
tci
ct.ali
shal
tuation
t r ansmi
iom
o.
y.'--.
7This 'ieo.f
ite
tsi.n
o c.ntras
.
ruitiaces
nt'.i fr' . a t.i. oin
'l-hereir is -some
Lu.ilv
s atel
i
We
f.i
h
time.
.e
Lr-noceedin
i
n.oi-e arnd b'ii mu't=ual
.iterterencei
y
t,r ,-T an., d e.a i
r'l(:]i
01..
each
on vihat
areas
co mmuns cat:I.
i
r
1tra-:
e .in
d
!,"a"fR1
r- f i t 'ter. s'.
lt ,
elast three
s.pectr-'"rm.
i ':r.' er."
snd
t-r'-ans-r'!e.
each
L,:. i,7,ere
areas as coll.ision- resolut ion
The
i-.
at.
been
channiels
ti-hers and
T?-e
unLifi-:,.'iftion. bethwteean
and spread
CH:ANN
H
ELS
i-MULTfIA=CCESS
Oi
on a
multi dro
for'
rep.ects.
aljlows all nodes to hear
~
-
-
-
a.L other
nodes
nodes,
a
andri
C.ommon ex..amoies are a cab e
'L.v.iy connecrted
The beoinninl
communi.z-i iot
mutltaess
network:
Camle
n 1
rn
trais..
niri te
tr
transmi;m. rstter
other
ai.l
was
in
tr-ansmitting
encre w ould pre venit
er
inI:terfF
recei'ved.
tihe
cyclic
acknCowt.
ecig ement
a.gain
latLer;
avi:f-id
th
ie
wai'te-d
an
over-laLp.ing
ar'e
( RC ) would
.izeci,,
-
a
sion
if
r-eso:l u'Lio.n
use,
Se.:rt:i.c:,"
r
c'Lt'-r
t..h
i si o
i tter
Ia:na si5
'
pr-ocess:
rece:i.ved
chosein to
ti:ransi.r-itters
4r'''ovid
s
andi
'iai_
io-
var
in'i-;erest-?.
e
t-he
pI'"Co.]
as1 a wa
a message
su
aR
.e
mf s.
s -F:" use:i
iri't"er'erenc.._
ono
i
Erled
co'_lllision
piroblems
!.%o-.n
l.-.cLi:
iathi
'mios:-
ignores the
of
0 -
ir dt- ro uct:. on to: ti"eszde'
'ic ih
aoene ra I I. v
he abse-nce
an
. nein
a-st
i o'n
e r-e,
r
o.nlv that
s numer
A
;-ai
e.ir i ii
r e.. f.: the
r'
t. his ap.proach
as..sunming
ay
't'.pic
o-j
etene-rai
r. ol!i si on
chall
-!'-
n± messa..e.
but:)
A h a.
many
iany
. -:
r'e. t:
ar'r":i.,vals a
criener rally
try
has beeni-mpriove:i
the
E
,.'-.F ':i' ....i::]''i-l
plo."'fi l,:L em.s ,:an.di:nm'..
i::)'..-rs.tv
in
arnd
has pro-vided
m
::-r'i: resea.-rch
CoI
bnth
-cno
ch
heck: ,
w ould
be pseudorandomly
crr'l.i.
another
a nal yze
3r'F, wi. ticiespr"ead
ti
noa t
s.ame tinme.
te
: evf::t
e.::j
in!..ervati.,
and the transmit ter
time woJuld
certair'.tv nf'
networki.
I¥f anotiher;
e +rom beina co rrr ectly 1
r edundarcy ch rck:
the later
_or
s1mpny be tran.-mit-tecd
the r
O,'ver t:he-e ars. thi.s basic svtraec.'iey
genera
a messare
in
t-h e me s cr
w ouid be sent
approach to
.ni.' Alioha.
th ibra.mso
-in.9',:]:,i
it. wOU.li
arrived at a transmitter,
1;-rgn-
reol:ution
'The idea here was that whenever
1] ,
oacki-e'i-
radio network.::
the collisioin
o
connectino many
ani-i
en
betf.':'ene
th
r, e noi s.e.e-
oar e
.unc1er:lyino communicat.ionr
ltr.ansmi-ssiori
and
ilncorrectyi
is
corr-ectly
receiv.,.
ed
-1.3(]': i" 5,
e.r w.i s e.
Thi-e
mu
it
i ac;
i normation
mu!
.. rbecan
ia"cess
in
manyil
an d
ways
fE
r-,bl. ems.i
give n
are
ii nterf'rei-r-..!ce
io.'
t:he
random
f.'-:ire proreeding.r i
i
Io
t-fh
ihe
e
ri
e.
.i..
.::-i
s sse
alrtay
I] (] J. !"1"t
poin·t
h rq n
t, i
channel.si,
~_i' i'1
t,'.- or - t
rie,-::'..:.';
rl:-:il.::j'L'j-
"i:-he
r
.rec i ra
.
l n e 1.,
i i -l7
n
e r-e, er-vi-r
i i
.
l;_ir
thiat i't i i
is
rwl-e5r-'l
Fi
rr
'
j...-£-
ns-r i
'1::r'
i:=.~
iic: {:
wr !li.~r
c(:i)':ier
th~e
1-ci.i t
un r
t
i.a' i t
t rar
r
c.cl.
it'
:i.,
cr.a
)o
i
, .- u ir"
:: 1.:m
1.:
,n
ore+: c
t.
.(sd
ur-s:tsy messace arr--i val s
·rfor
mul
a ti
sources
'-ess
cf-hannels,
aind can and
shou
:c:ld
have
Z:
r
nf
v-i L7
wtL
-'
nave
.
f or point
i
c ana n l
-fi.or
-.'-ce
-
F.orl
detail
to
:
it.-.
fif adti
h- ns.
i"-forml
i'
there
-iommun ti c a
-s
" e
nioth iingc
The arri
be deal t
ic' m Tt -
fi
s i then IIo
out h3er'
to do wi th c-ordini
with
c:h
: theZf
shar
n
th
i ,
iv'! 'i-A
c)oi n-it
vals:
,I.
The
-tE!
so p".'o cti si].i i~aci:
i,.--.k
ciat-:,a minitr
a s
Fo:)r mi. ti a;ce' cha Cnnels
::.n
'''.r
to
1
nr
furt.hermore
I~ ;n a
The p robi.:,M
ari. iw o u .d
-
pr-actl-icai
ter'ss.a; h!-
mu i. t:i. as,
i
'
a mi.i
i::i.,r
.a.: r'i,:::d
o'~n tv a ±few are bu~svy
:,.
,is
.
tro be t rr-ai:F.-itted.,
ai.
,.
r-
o
r
:i.i-'liired*
ner
ofr:- :l.t'a
.
chnnle'
r!.:
the~'ofm'j~de' i
desi
ar rivals
ag
es
Forfntsend
i
rc .-
a pr opri.a"ei¥
usa].ly
IunrFeasorn-.-1
.
sa
rF
are
-ter
c
and
to undersitand
i .. i 1
u
re r-
me.ssa..es ai--e
-ne
m
dr
noise
channel
mportan
is
resear-ch
c.
of thi_
the
4t-.ii
i
aitsnd com mulnicat i on
L1v
.9.rti
infiri
i
-a:
5ies
if
ancih wv
-t.hi-,
a -j::.
r c,f;
i_
gfsni'
5.
i- accsI
mutl
aIrrvals
it
.
a'.cri rc'-ar'i-
ienr.eral j. i5
been
uisp a se iarate a*rea
in !:hi .s
a..erstr."t
u t, thei-
fmo
:)deli
.,e:u,:
i,-lhas alo,
siummari.s and descr ipions
[4,576,
to
theorern developi-ed by
This w-or-:
op
lened
Ih as
ilent
i
approach
197:3 with a c:dingl
in
:[2] arid Liao [:3J
Aihi 1.rswede
theoretic
have
to dco
throui
t
C
.l heor-ems
ri -th the
soutr-ce codinga.
e
-i.4-
without.
i--sven;
s'o:...r-ce coGd.
r
,r
thi erl. o-v er ti-he ar-bi trar
ih
-ieorems
th e
i.-:lt3at.
i...!"_er
no.Iot
sa].I.v
jc
:r-a
iim:i.
ch an;nre
ci eat1er-
CIa n._m.-ine tii
:::'i
i
to-, be smoo:',c'thed out)
tha -ofismenoh,:'_~w
Sourlc""c2e
prec.i
J.
-:onf_~ .
Lu.'ai:,
ij'.i:-'
a.t-rvii r
m-i:
·u:cs-i1u
]
l
i:hen sme
'
,.
t'.he a.:':'E:i.v'
eri
ini
ceret
i.s
C.
aDpear
smotihed
ca.i
i
(iCe.
v s
the po'
all er
hibilit'
t e
t- -e iim::it iCs
ini-ervai
o.t
are
oirv
Ler
ective
calse-
achievabie.
i time
on
nt;
i i
b
b ot I
i
`r'articaiiC
1 he
ef
p
uudes
ad dinI
ccss
decocdirlgc
simult
ianeously
i
thi' . ,-.
0
a neiw
.y
r~ar'n"smi
s
'e
thre
't-ime
foor
t'han trh e
t
iCmoosing
nOl
qu red
ca-iy
fpi
a
For"
r
h-se'
e
of
- ar
I:tim
i-nter'vai
no=
I.i
the
lera
bie
ongo
delays
.
on
]
i
n'
tL ih i s
t
arse a., it
the presence
red
seq.e-nc
i
s
af:
n ew
several
since otinerwise
orQ.-wave-:tlr'.,
:)eir.'i:'
wo£u._
ti.-a.na1:
d
i titeJ
i op
t3 to
t
de
- v ei
i: i ri s-.
to coding theoernem)
This understanidinii
o-f
each
or
that' toL devel op
seem"s:.
-od3i ng. (as ouposed
envirionment .
inr
[i:
on
si._u-c:e- ,
o.
i created
-frJ:s
rnoi'se ani-d whati.ever' was
~J
L:ers.
ar':'
LK
i
..number
iiio
i.Crve",d
when' the m'essa ej i :r
The
i'teo. r-ecei
of
I ii"te
transmit er
ei
ien
i. ; the coi
used
a r in
-sum.e
eri
t
tr.fu,'ct:i.ion
wnich- is
a-i
e.to
ano
.iessa
:-e
unrici-erstand i. noC, of:
mu! iltiai
is
is
that
V
i.e'i
ulidl
be
sin03].e
i.
thie
(nes.sareS.s.
L rine atppr oach
r esl,.
.. ew
c
Ce
er-abie delav.5,
'ci- to
'i
i~t-r
e cd:in-i
a chi- evablee a nd b*
ecaus-
Co-
arr-ivat-
in
.J:orLc
no i_ mat-er:
ar e
-t howe er-
s.__- '....c
d7ore
-
or
er"
i.me -inter-vals usLed
po1.
i' i
i
what
oin
-:c:r t-ne
recqi"-
t-ca
or-oces
ies
'aiL
arr--J
wll
ar .f rom winat
t;-:
.
c ~ss
t
1_
c.- ti e
t
the
arrival-s
c,'.:-F-,;n-?O
tnil
r
t-ieo'e1-s
m "l..'
iy
i
bi_:rst¥
mo re
Fromaf
if
messaues
the pr-ob
shouid
being
in
s:
a
infvolve
trarnsmitted
em simply reduces
to
e
counlict
tnie
Li
resolution
absene cf
mo"re ode'ta ii
and
co-da3ing,
svstems
we
tacklin
in
section
.,
multiacess
coding
in
.kn-'own
the discussion is restricted to
the si1mpl est-.
interest
with
for this
is
context
to
before
many sourci-ls and
m t ters.
spectrum approrachi
not bee discuss'ed
.Ji.il
d1
scussed
ir
munliti.access
c:oi i.
si
'-,
or der
Ymore,
t
i :Imes
in
m:d ..i l ti,itd
than
military
il ook
se seen
iv
tC.. t2
ac:
:' o
can
seiquen'Ices:
others-.
multi
rIie
broad
band
naoise
-the
tCO
i:'he rece:iver' af ter de,::duation
The sJigrnai
f r e , Le
nc y
broader
jamim.er
t o
!:t
.the
s signal
s-i
nal
wi 1.i
to
b.andi
jammer
does
wi l
and
the
be r-educed
by
i:' -.
o:'.rl multi
so::uies,
]. ike
the
invent-i anai
S-pread
;-Q
environment.
AssuL.minortha
br.ieflyi
Posf-ibilities
ginal I y d c-Y'
vel ped
over a much
necessarv,
is
out-side -the c o
ori
no't.ij know tnie fr-i.duiatiln:seIuencei
essentialy
the t-ypes oCf
lie,
catio n
a
bt
theorny appr!acnes.
,.:cdiin
amming
i
in- thi:s papere
that
i78]j
tultf-access channels
to illustri:ate
a.nci
i. t ted ia.
to
any detail
comimunic-at;ion
aoai nst
be tl-irntm
P.
in
is a mode o:.fcomnuni
prote'ct
sav
in
here
resc..onr
u
pe
'.ctru
a
ion
thieory
i -ttleis
what
The rationale
in
.
tihe
mr'obiem ot real
i nf'-ord matio
i
we discuss
In both sections,
The spread
i;lf:i
for reli .able transmis
mul.1 ti ac ,e-.7
discu.ss
with only two sources,
understan-d
tr a n
added
coding.
con-fiictrs.
recrt ion 2.
in
abLout
with
access
'i-rar'-
mit
c ommunnication
using
at
diifferent
once
and each will
If
;irin,
we compare
usinig
this
usinq
. ook
like
type
p frequency
snspread spect rum,
broad
m,::dul atino.
band noise to the
o-f system
bands.
it
several
to frequency
aoupears at f irst
thiat
spread siectrum is not' a very
transmit- tooaetfer
goo(d idiea.
using spread
spectrum:
cronsicierable.. a.nd the resultingg
in
terms: ot capaci ty.
typically ma;ny
femwemr
3undamental
the same
reiso-iution
appr,'
to eachi
in
mesc;ag
other
problem handled
Since many
ach).
seq,ences can be chosen
noi.
se
tirmes
tihat are aimo-st
spread
-Thi.s soiLt ioni
st:il. n-eeds;
at
oncei:
t:.-wev
.'er.
r:, m.:def;
approah
th
ian
i
._ (i
co
.L
:i.
s nfot
the decoding is
miajr
aI
ra
to
.
send,
of
if.t
a,-
es
on r,:'
i ,-ut
-
needed
are
Oi
o-'.
ri-
there
or-1
-
or thoonal
r
(t-lhi
modulatiion
andI.
liike
to the busy
since one
it
s
transmit-es
i
1us
namely tihilat
n et
ii'ook
an automatic
c:for
mfiulti access
. la
is a
transmitters
more t-ihan
too many
-
it
by the collision
compllex
'iti"
is
paper
tiat
but
enltirely satis-factory
very
to FDM
is
ystems
tihe channel
cis:L.'n i-onresol uti.'~on wien
anc
the
spect rum provides
solution to the problem of allocating
user,
clearly interior
the frequencies to thie busy
probliemi all.iocating
is
p with
thian
noise becormes
with FDM:i however,
transmitters
p
p transmitters
tihe seilf
system is
The problem
there are matny more than
When
endi
rates
a bet-ier- set
cLR,mmun icat.on
r y a l on e.
The
2.
in-If orimati on Tin'eoretic
Tihe codi nu tiheor ems
of
holw
set
much
assumed
send.
thiat
in -ormation
reliably
nave a
the sources
ouestion
,'"if
'
T'hre ci ass
Fig
Each
Ii
.
svmbooi
x
channeis
unnit
of
.i
from an
tr:asmi'itter
prob.3iii
and
s i
'i wzWi.
' 4J 4'
'X a=
i
ilit':.' as
gnment.
s3i
FP
The ,h-annel
n
i7i-
,
;i.
i
each
(o
nris
i
is
is
an
output
sen ds
i the e
c -o:i
e
e"n'ds a
if.j
niat
4
w'.
wi.
same way
nX
x
ii
11
I
We assume
j. t
a
nin the
j.,,'twt.o
t'iransmi'tters
and
resoecti
veiy vvi ngver'
units,':her'
thei :i's one
ie probabiliLy
of recei
y N successivye
-(y'i
.
gio'evn Xw,
F:'y xw:.
;:y
Ti F.v,.::
w
bL
.'i
Y and
oi ilfnputs
:-.'.
sense
n.i
a
alpabet
:.determ:.n
-i.
v;Y for e ac:h
r
in
i lust:rated
transmi't(
l
'y
me
moryless
e
vl
s
i in
I'
e
syvtem*s
transmi
'ata
tted
ransmitt er
second
Th er e
eeivinn
.ii-i ri
to dbe
O-i
alpohabet- W.
pro-i.bi
'
X and tnie
toacitly
data ir
io-
m ulkaccess
rst
is
it
area do not-- addr-ess
to be con-sidered
let e
one pointr Itor
reservoir
this
in
the questi on
nts.
empty
in
ti-imes
ti
i
From an alohabet
'.!'::bol
.w'
I'
o
never
arises
tne
'ii
random arrival
f-ro3m
orpoiset o¥
results
the delay that
theory treat -
crmmuricatedC
oi nt,
Thu-s the theor-etical
be cause
. 'fi
can be
to another
poii nts.
f-
the
data
of
Ao.oroacri
or" the
.j
So.l
time
be
oovi(3
as -for- singl.e
indicated
nAs
sour ces which
bei no that
in
.--.
input
the
are encoded
tha
t he al
t
this
p--abets
ca n
be
ar'e all
r]enrn.=..r
time the.
'for
.
di scri.-e.
zii-a
17
cl-ianrnel s
there
.figure,
ar-e two
independently
independent
in-to the two
channel
T'-i
.
i.niuts.
W.Con.ider- bl ockP
ais
inu'nd I{!-_t . }
code
_
{w i
w
+or
L....
}
channel
codino
ib4th
a i
ven
block
for t ransmitte r
s
transmitter
2:
each
paramet+ers as an
(ii'l.HL)
code .4ord is
The rates
code.
N
r.ing
word s
i and L code
a seouence
o. N
(_-ode wi i'h tihese
For convenience wi areieto
jnouts.
lenth
of -tie
two sources are
de i ned as
R
(i n
)l /N,
Each N4 uni.its o.f
R
time.
1ser
i
unfi-forinv
i o'Li-
I-
in =-
-,-_r
o
!ini.mized
.'i.
ieah,_-i y
as i<
iite
:, ; cas
(n t
fworof
s
ouif
' ou.'sce
L'';n.{
,
*i-P
assi
-~~~~~,n~~~~~~~~~~.P1----·--~---~---
'ee
n:men'ts
1
-----
h
nro . eF-:'
ech
'~ou: t p ut-'L
to
result
e
ma,,it a r"
[23
ffhlswede
asiognmentiS on the
o-f rate
ac,
~
,..'
.
satisfy
· --
pairs
~·----··---·---
·
IF-.
3._.
i:.
at
t.i e
i
e
4: c r:de
decoder -ieui---:-w
"o.,.
tn ese can]..r],:i '
i".3
E .i
Lei:t
X and W input
rate
3.
i
(x)
i-ihe
and
alp-habtet'c
r-egion R as the co nve'x
(R 1 ,,.i '.,which .for
each otI the
. ....
e;:-rr.or
- or
a
choosiC-no
o- h-
and L au
a
f
:ti'erw.w.-r-e
m1:-in
F'i
outputC
r ini
ia-
ur n.ko
abou'i
7
messa.ae
dlcoder-
e
or'
o
channel
decodino
L t hat
rt<n--m.
De-firne thle achievable
the
.
i;
ni..lu
i
damelnt-al
be probabili'ty
of
o"?
'
h
ima:n
rfun.
due
m
io
and
ma. 'imum ii
tkel ii'jhood
;'
oener-ates anr-
decoded
Wii_-o are knownIti
oure
n -ce
..
tn,:ei
respectively.
hull
by a
minunliormlv
Thie transmit ters
is
decodfino
fcorrect-
lq
.
t i he,
1orem
c:odi in
i(w!
.
,
into a
mapomed
The pr_.bab'.i.i
c hoe n wi
be
rios't:
"he
t
the
i.
inteoer
1 to L.
_~r-C0Hilbl~lit%!,
"s,
r'
,
W
i:(y x
rom
ie
OCCU-S.
occ'-:-ur
an
and the corres o on
and
d'Jec:':odin-,-o e,:.'-r'-or
"
distriiiute
ithe decroder and'l is
y enters
2
source 2 independently
ri-iVy
rr
xwmand
L)N /N..
1 oenerates
source
dist-ributed .from 1 t-o HI and
intecer
(In
=
some choice
inequalities:
of
i-'i. "r W
*~.
lF- h.>..
I-
k iy
F
W Y
W
*
..
'-W,
iras
yFP
io i
_<
' | rsf
;1
~
iZsr3 1'"iI~
i'}r ioi:· {
i~
'-' '. -,
J::'': ':¢' il
'
jr
u
bound
'
co"::
n ci:i i:i i
.i.
exp
i.,i
l
anr d
In
possible
an
. ·
.
'i, .
...
byL'
2.:)
suzcn
q R 2"
codes
e ffe ect
Ri
P(
( i
w
,, (y
·
'
i
i
'
-2
ftor
I: 1-'
gc )
ex'p
ac t
l
(2.F4
i" '
i -
{
'C iiv
i
(t
Jh
*'
w are r!d:F
alsoe
...
C,:
i
i"
r'd"el'n'
i"iO.L'n
o..
' L( -i
v.
>
each
alli
>':- ,
E
N
(N.,I',L)
>' 'No,
code
L
-the interior
-
exp
)
6
i~t
witih
suc h
NiRi+6),
in
.4
:--w
..
Y
K
f
I£
ar-nd
x anrdW
t he thi-eor-em s ays th
hat
for- source rates
i'n.
S
' n s-: . i:]h
.
f'" r
a'
iven;
e bre'.:k point.
o-f
there
fexists
witlh iM
i
i.
F'or-
t:hai~.i:
Is
ws.
.
(y
d
i_ it' Xjare
Nlisince r --a
Get"ieFa'
equaiiiv
i¥tt
iier'e exists an
iiR)
(Fi i
P t y
w
w)F~~~y~xw~~ln
.
w
.
,Aih5sw'ede,
Li aoi:
iH,
t.
'
nJ es! ,.'::end en t i or ve i
u a
i" K.it,...
;
(Nfi,M,L)
L
5.i
ex:Lsi.
6> ,;;:
' w;·'
O
dd'
is:i iihee.'
th
b
- :LX:'
v
..Y'JIn
i'
iheL:iT.,'::)m
n·ler",e
s't
fL
i' L-i: ini"( q i.3si'.--"
e--a
Gr
y '
:iX;[..
7
i'
g)
'i"he
;. iX'"e3i
.
,.i'
in '-lto
fr
_ei
~
w
r fn(w
xV4
"
i
"'x_-
L
wAW;Y
(':~
e;. p
FP
tha-t
'
(Ri2
"R
N(Rz-6),
For%
FP'-
i
eacn
N(R,,+&).n
thte achievable
'
L
E for al
r-el iable communrication
of
s
is
<-
-2.4is
reoion and
Sleoian
and Wolf
a thi.rd
sou::,rce
tr ans:i.
t -eThiey
r- 5,
-lshowed
tc1iat
outside of
impossible
[9J latier
that
used
in
.-his ran-dom coding
cdCinig
insi.ui.i-
intoi
throuih
thie arr1iument
Let Q i (x)
and
Codinci
is
a very
both
arument
which
with
N and
behavior.
si mpie extensior. ofgives a gr-eat
access channels,
we now
deal
oft
oo
assianments
and consider
respectively
codes wnhere
each coCe
ia
Theorem
be probability
Q., w.
W alphabets
selected accordi-
for
or tihr
e cwo sour-ce case.
Mul tiacc ess
2,i.
by considering
eXrponen+rialiy
input channelis and
muitiple
codinrLo
1for
result
a r-andom cod inog
argument
single
*tor-
this
tChat most codes nave this
a sense-
random
reoion,
achievable
jointly
Pe can be made r-o decrease
s7howed also.
i nrce
eneralized
be encoded
could
.als
the
x ,
to te
i
m
cro aob
itv
orn the X and
an ensemble o-i(N:MiL)
i1i, isiwordind eoendrn'tI
as
. si( nment-
N
and
eiach
c1ice
a c [:oir di r]
n]
D..
wor-cd wq?
i
_ ,
ID._ (.w
eh t1 d
code in
likelihood
vaslue
F' e
i
L is inde
,pen
w
=.-w
.....
W1,
i
e
the
entl
y
seiected
t c(
(w_
1
For each
I S
o+
in
ense
the ensemble.
decoding.
F'E
) ,
tfor this
deod
the decoder
and we want
ensemble.
i,,_
uses maximum
to upper- bound the expected
LDeine an error
event
to
bIe
of
type
the de co ded pair
-
1 if
(mn -
sat.is-fy
.
and
i
12
1),
and
fi!
=
m.
mi
is
of
type
be the probability.
.*-
Consider
encoder,
(Hi-) i
pairs,
3 if
hm
that
we can directly
Xi,,
0 ,
p,
<:[ (H-i) (L-i) ]F' T'
and
the
· 1-he statement of
words are chosen
(xfit
.W
)
fiii
with
(x rW) as
input alphabet
theorem 5.6.1
of
X".'',W
[_l1-],
1l
(:
x.
and P1, E's.
rates
in
theorem 5.6.
(2.2) ,
of
independently, but
m,
i2, or
independent of
rmecarding
this
2 F• 1 .
(2.-'1
2.
2.7
si.mpli.fies, t o
[i-0] assumes that
all
code
the proof only uses pairwise
independence between the transmitted word
word
Z i
For each such pair
Thuss
'
for
('.y)
ffr r m OT f.
dl,.' fi -- tni tion oDf
1
error
choices
is statistically
w
input channel
that for all
t he product
'ei,
a type i
(L-1i
a type 3 error.
yield
y
Let
ri = m
(mk)) enters the
Note that when
-_single
2 if
tyvp
+
apply the coding theorem,
which asserts,
Uis n
r
of
source pair
e 'i
wq o ver the en semble of condes
input to
is
'12
f 1.
m and
choices for ,T and
the code word pair
combined
event
An error
.
PeiPe+
,
first.
there are M-i
L-1)
(niTi),
Xmq
P,;
and the original
over the ensemble
obviousl.Y Pee =
event:
(im,i )
(Xm,w) and each other
e
i+F' N
...I
C..+
L-(v
Y
er
dconi
Ne;-'t
n
F'
this
en-teringi-
w,) transmitted
viev;- the cihannel as a
transition
ma1k
:e
F .y |x m.w
m--. im
inc
this
'e
evern t
u p p etr r..t
bo
li;
meust
by the
at
thi
seconrd input.
e
input
decoder
t;:
occure
IW t
1 error
T:akingl
the expected
=
We
1
code with a
.iven w.,
with input:
si nge
that
Wie
can
x Iand
w:i
one
m.-
1 error
on
~~~---------·------------~
~
~
~
~ ~- ~u-.n
~
any
j
y !
i
Q(xi
C(-
PF
[.' 1. :
o-f
fer
xw
over- w
the--·-·---·~--·---·-~
-
p.~
saent
Z
vf on
a mi o i
th
4hieor:-'.n
m
(2. _11)
-
or
carm
W, beinq,
.,e ihave,
,
Y tL
o+
c hannel
P m.
e r-r r
tiv.f
ising
n,
(M-I) P
input
uous)- if
rrondit:ional
pr foD
a iij.
value
for
wh,.ene,-er- a ty'e
i±nput channel
P'Type
of
channelr
leas.t
at
-
inpuFlei't chanre'.e
o1t ....'i
s si.n1gie
-
mess-age
a particular
a choice
a itypep :.-Ci,
f
nd e d
-'.,tf:v n-sl.:
W )
.F
¢ m ani
d
and
(,o-rbe am.big
P '--'y yX
!]
P(.yx nw .)i .
i hood
an-, err-o-r
:prob-.Dbi j. ii.t
a.
singi
probabil ities,
A ma--i mum3. lii kel
,1i
on
probability
the sec-nd encoder,
partiicullar
.
the probabilit y that
ol
'
-
first condition
X (14A
U
0
]
is-
'-?h e
aga.irn
!::_
1
2
and thien using t h
r
-
11)
-
h
it
:ol.-n
ri'-of-duct
Q
again.
1
and
.t-,
L.1
R
F'
Applying
same arigument-i-to type
the
. 1+
w :,
.
2
J
--
'
r!
.-
±p)
for all
errlors,
p,
0
p
S
f
_.,
PF
p RIp...F.IP
L- x e p C
e.
'
2.. 1,)
[.'...'
2
(w)
Puttting
'"'-wF
(2..
2.)
(i
on N.
e.por'nenti al
d en
T-hieorem
(SiepiaJn-Wc.!f):
2.2
..n
:nd....;
d
"']onseln
h .i h
accor-dino
.--
F'
to
. f.p
a
in
F'
- i-Fi
ei
nb 1 e
'sense;
(22.
for-
sa:.tis-f
the
i.
-
('):i.
the
(N MLA
indep.,n
ib.t
iver-i te J'r-ba
A-
e.'-pected
i t vy
er'r('3r
i-4l
;'
.
f
*,enenti
ie
:'
-t
eto
emphasize
an en'semble of
*la-re
Ti(:..
rhen
1.-.
form
we have:
Confsider
and
i
zen.
_'.w)
:'
3)
1
and
i tv over- thine
pr'.,ibabi 1
i'
dence
; ' (Y
-
all.I
I'
1
F' L
.,
1!1
_
R
f
ln
R= R._
t
L
R-=
R.,2
R,
zf·-1
01
-i~~i~
.
2
iEj t.P ( -,+
=
-in
in'.e
-
of
~(]~~~~~~~
*sfm.
Lv
Pc.~ii·- · 7- n
p
it
and
ore
Lve
a nd
Q6
5.c
I
4
et,
.,
i.
i
mna-
decr eaia -ii
oi-U
1:
ince''"-'.::iegieb'
n
N.
iFi
~he
.
e
t
r
i
t linctfi
th-e
s
.
f!..
iol-ervi o-
10
w)
. .
--
Ex.1
ipp
-
i
th e,
L s
"F:~P~·_1~~~~~~
t
rei
-1i
i mum
c).
i-Q.
Of
in
h n P
*
[ .(v .
O
expres;i
for- p r-ocs)
I
s+ w
',*v
C'.f
E
i+.R
-
-
n,;.ric Inn
I( pG.)
1P
for
tpo~:
-p,.
ve
c .
ovRr
Ps
Thehr-em
-'Li
I0
2en2
r
1
ipL
Ip
;
ee
theorems
theni asserts
decr-eses epoeritia.
to
1..y
- ith
5. Ob
thi.:
it
There are two questions we want to explore in
this
section.
First.
priobabil ityt
anid second,
practicality
of- coding
as upper bounded
s-econd
source
messaig
aliso clearly
a
F-:- is
to P',
errors,
W
ur
is
ieo
et rem 52
problem.
a.
the b est-
c rs
d
':
"idle'
is
we shall
X
t. e pair
:?
1(l-)
,Q2(w,
or
i
is
errors a--re
aided error
the bound f-or
inputs with
for a
probability
the bound on PF
ii that
tie given ensemble o
s ee t hrougCiCh
tihe
a set of
n
always repr ese! .tat iv
in
ci
odes.
'the
t ia-
e. amples,
ri t he'l
e'lsem
i :-
s,
or, Channel
K}
and
if
0 is
and W
"collision
i
()j
.
K}
,:c.
input
for- a
-
the
. i mil
m(
ar l
.
F-irst consider
the two
soon
if
' C:' and wJ
n .1i
thn,
'ight
are not
{:,i,.
=
input,
Finaily
rIr
it
quite
"':"IThe
CIrIioii:i
Let
concl usi on
about
input random coding bound
sinqle
single code of rate R R.i+
2 using combnei
i.(x)
ii
type
Similarly,
Pe .
in
probability
so tnhat when
the genie
t.he
probability
the decoder
error
the
To e' plorer
the error
i
in-formed
approximates
the conventional
t gi ve of
the terms PF
This genie aided
P.
cause of
closely
(2. 12)
"genie3'
lower bound
thie predominant
probability
e
inter-pret
in
o-f
bound on error
multiacceSS channels.
we first
wic'uld resuilt if a
that
this
what ind ication does
i
for
question 0o. tightness.=
Pi-
is
how tight
the rest
i¥
ie n
w input':
t"::
hhe._ ou. tputt i
w
the ou tpt
This is
W
We
regard C) as an
is
a special
SSioWn
i
in
Siig.
it is easy to see that,
conditional
inputs are statistically independent;
y
(U
y
the achievable rate region.
then
symbo.:l
fr.];
fr
c
K-
For any given
on the
output
thus I(X;YW)
=
y,
iX:Y)
rec-ta n l-Ie
(2.5
is
tJe nexnt
e,2
.n-Hd
}LV,(wsr
of
union
i r
he
ri,=
or]i abi
union;'
rates
exn!: onent
all i
n
. !murli.
t:l
e.
2
it
nroC,
C rone:
·9.3..r)ii
s
-
is
-reoion.
p
s
d
I (XW Y)
max;imization
-CQ
was fir-st
is,
a
bu
i.i
1 t.!
ai. r- i
the
each-
relij iabe
T h ui's-i. :
eore
i!
n
uc
in
unc.
: ti
c.ornve
j
D n1. -
...
'n
hifhi
r
:,w.
Q't 'f
..'e recion.
either
over
ext.i-erma can
i
i-)71
oa
a.s-- a. .
n-trcocate
non
rtregeir. n i
tunion
i r
maxnimization
Either way, multiple isolated
int.erior-
o-tr rates
pa i r-
and Q-, can be viewed
or a concave
that
d: i. visi,-
rcanrve
a
of
e- m 2i,
e t ::vr-
G-. is
nTi
u.
inter
C. C.
co nav. ve e as a
a"
r e s ul i
i.n f thie
the
thatthe
s
tihe
shown by
the set
"ratepa.ir ,
fa
f or sor me Q1)
over
t ii me
absh t hor-
'x
pr-o
mbabit i
orf
in
i ven r aci
a.
t
t
.
s. t h e
'_.w
pai.rr's
i
nf
chioc-s
non-convex..' (the non-
pa.ir
tihe
ia
set
st-e
s,
rr
id
s,,
Threiorem 2.2 assures
I (X W Y )i
t
2C.4
4-
region
the .
i vei fg
tha
all
channels
eeilioni
.- o r
0 r and
ifn
over
li.ne be'i'_-e
unin
. effi .
CA$)(,ir'' (h-2
n.ax i nmiz i nr
concave
1
be c...ar
x
need
nr
rat eas a r-e a chi evable
o7n a straioitt
\:, : -,-'-'.'.
! r.iFc'. i: L'5...
n
b'i,
is
1-1nv01 eIn rate
i...
!". e
6 1
rates
all
for
.(2:-
aE'i.t
sh.ul
It
fth'
F
t -his-un4ion
ror
r-t ote
n:.
t,,(::c:.:
r
) ,
U(A.,
o-
ratiher
'sur!orhisino
rir-st
f-'u c ...ion
: (:)t oi.
ii:.i.'.,r.j
oF
-theoremi2
i f
:.ns ac,:
_e'.::
,,x
*:..R '
h ul.
-i
)
codes. fo
no bet-een
se
e:ir.
co: ':un1
.!'
i ,::
cai:-'i
2
and i
multi-access
or of'.e tih
eri
i:C
i
(2:3)-2
g
-')
'the set
rec:. on
lL!. i s
i
::!
n
n
v deer-
t:tm
c:;onx
fi;r
0
rates
F. F
should be constant
satisfsviJ
. t J
O
of
choice
.i.
- orms
so
-
ant
for certain
achitevable
ao-¥
nst
m'The
con vex,
[1I.] ),
some
C!t
rK
r'
p- a.iiit
y
convex
c
,,)-(25)
(
f ind
h
trhe setftesr-ao
nt
¥of
-f.ri
satifiE.d.
t mmecrvV t t ha.
from
.
the
o.- rates satisfvino
and the set
Tin.us-
as a nona non-convex
e;xist and
there3
s
o-- the
no anairo
u t
Arim oto-lah
also rmight be surprising
colli sion
the
..
!C;i
u..using"
th-e o'ther
r''
eison
is
ra.-ctcal
./
For
aI:I
hfa nne
i.n
we
t.Crh-i
:
we"can
C
i. ieva
.'
n
p r. L nc
C
in
it...
,ind
this
-
"C I...
in
R-.'-,': i:;."
iin
tihe
~ ......
t':)
')
1-
..
E.
n
-.::can
' :L
=
a
2.r
'The
f .@-p
I
b
be
-ax imi :c
.d
th
rinouhC
..
in.rtormati3.on.
isorthe ti
di cttr:.
thre
in
-
Djor
user
bu'tI.
i3'.
(tx. w)
1orm:
2
.ver
pr.-.cm.
.ict
'b
creates an exponent
principle
u. nion
har.e
i
oi i n
(a
e'--'tlra
aiin . np
- ut
ir
one
to be i.dl.e also
error- probabii t-i
i
between
idle).
is
s olut
Es
1,'_~scs thJ.
.J.;
e.r-,
.
pen
1]o..?
leS. r
8:
rnot
le.ittipn
r..:l
1 arge KI)
,ti'--
nr-
by
i.e. .
=0
t.he other
wihether
r-
at
fr--
reqgionr
by imiultipie.i-ing
Qi._-
i,
while
chiannel
the
co.ns
-i
der-
::.l..i. os.'-J.i
7
n
(
=
rl:.i
n- -rmatio.'tn . and t h e
emi nent].'-.p
-
aid
i
.
t. hat the choice
connve:s
i
:N;ext
=
i
that the achievable
not achieved
is
channei
alg orithm th
13]
rate reoion,
can be used to -find the achiev-abl.e
It
12]
reqion
m.{
!]~-
.
c+
.'.-.
::[i.-f-9i
a.4,
( ':......
of
'I'The
error decay
same
kinds
of
+or
each
oroblem
e--
acln:i;eve
s;.-:.
*-l
- *-. Af
the
:
by time
and for
R
a
f .... ibie
are
cfo' i. n,::exp-o'. net
e r' sem J1 -I i
diownin
+' i --r n:'1H..
'
.
'u;".
.:U
:
a "-ine
r-eR-.t'
bi'
')
Rx defined
ori
tR ..-
o
dr.
lfthe
".
;
p robabi
-as
i .
b I oc: V
k
This
wou
,
r i-
and
r-2,
d
ftound
to. achiieve ex.ponentia.l
~1-.1-___ 1---__~___
-------
/I
+ (-X)E'
approachi
in
as.
tfrm i-the
.i
F'e for
all
radm
th e
codi.
i g rin.q t
y
eadead':
r
replaced
a
by
g,
a s-,
(2.25)
enilarges the
sufficient:
rate
.s
tL-,he
o i
approach
is
r-Fando'ml
a
c h odi c. e's o.f
where this
decays
.::s
nin(
ci3 fO
be.rig
:
Si f-ef
d:
i: t,-ar
rn
this
i-!
dep
er
:i.'
i,
above,;
to
F it
=
pair-s
in
R-.
R
-st
R. ,R,
+ or which.
Ru' shr
!
a
the tim-e and
r{-ate
is_
i,,
. --
- ap' r.ah
r2. .5)
No examples have been
."~" ^~~-~-~~-
time.
a
in
for
codes
lenqgth
±rac-tion Ankof
t
-iesrom
-
'i on
us
in
,::
iS
andri ER
F
block'
(
.l-the
other-
of
on;
.
:w.e.-i oh';:j
i tde d:-tvea.
,
w,.i
et
E.
equal
,1
.
to
severa.l
-,ereti
].ett er
are two rate
~c or mo-re can be achiev-ed,
)
,, . i.i..'xc
s e .,( culd i
1oni .:ti
there
''
to
we wan.i
2. 2
code a
,a
wt
a-s a
if
:
,a-i a p isi r C
re-:i.:. on R
i."re
t
.s
between
sharing
'ir_
E:,
-istin
:e
pair '.
an exponent of
ne- nivex
'
iulli
c-.,and if
sRu -=ch th.at
'.*-"
..
....
...1'R
th-at
t-hat
Note
decay
qq
.racrtion
!
Ie (i-
~.t.rns
man
·"i:it
regions
R
us-ina the first
,R'-
seco nd
t h i
;;-
for any rate'i.
*i1
sens
rei3
rniow
.5nd
then,
t.h
na this m ax lza.
ti. on a
a givein exponnential
JJpa
l F _.q
wi
i
thie feasibie
b
ind.'iin'4
.' ,
inrertormi
ist
howeverl
pairs in
th7e
i
in terior
oi'
R.
Another
whici'h
approach
successiv.e
Co .1i.1jsioi'
¥:::,r
ui--bl1ocks ,- f -'ur
?.:
--e
e:ui
l.a
:J ni
proibability.
probabl y
each
) t1o
t(: have either'
.
:p r
Ri.
Finally,
chosienf from
sui.b-blck
of
length
x
t he blo
i:
and
f)orm
the
ii b
wi,,:
x
(-)i
.
i s),:
or
to a noc
sciheme -for
:!.sy'.l'.croni
f
ale
rather
or
i.-:
:::'
s
w.it I
"the
ab."ove
mu
.. .i t
spher'e
somewil-at
c hia rie
n s
t.lhere .Si s
acc,"e ss ch anni
codingi
pac'-kin1
-the tw(o enrc:ders
bo.::)und.
b1(tnd
loose since
for
s
it
for
[1' 4 ])
The
-for the
m arginal
random
in
t
mul..i iaccess
does not accountl
[15]
h a-ln
s ame e
but
s.rle
low.er
r r ,r
t
butt
e...-
it
al s
b .o
A
un.
t:)
i
i ne n i
has developed
channels,
for
tohe
mo7
re c ,rmpl
i ' _.:.
ica
s pher e pack-ing
that .1 v el ds t i e
f- i n'
channell,
or s iln
AruLt-yunyan
ve
was notiP to
collision
t h i.s
s
pro babilities.
are f{ar
the type 3
:rrors.
vchannei
resulting
e'. pone
i
. t'
iimp le
a nd
thinan
hthne t-P·hatI
thcrw-,ss wher-e
disus-_sio:]n
t -.i:
a-n.
r.:o
codin
t he same
ti.tlZ.
'i:r-S
_e why errlr
.ta'i. .st-.str why
as th-ie randim
o.:
n
i.~r,'"{'r
e:.'.:poneern:ts:. achievable
to...,. :
.r.
achannels
cei.arl..iy
o-f
n ulti ancL.i:::es s
:i.. iu
is
i.pen
nde
.:
Tfhie- p.-'"p3ipsfe
1arcges'.
Massey's
coeii. sion
i
ze.i
,.'::.::.t:in~
no:-po:neln'it
Tcet.ters
b
with
iseless
e. at.mpie
Ieste,
the
.arr.angement
this
with 21K inp ut
su_
int o
zei
'-a ry
w are independent ly
val ent
tihe
we
(i iw,i .w
or
in
di.vided
is
Wj.-ikth 2`'K input"s:, and
n
a noiseiless wJ channel
was
r
F-
:.C-
hS.~
With
eui
'.L,-
iFn
i.r;.
{ 1 .K).
.2
4 is
ensembles
Within
.i si each sub--bl~ck:s
each,
) has
..
-; it her thne form
t''W2.. . ..
equal
r3s
coding
derpendent.n
ly
sfupp
.
o.=.
ex.a mpI e
le-tt-.-
"o?'2.',:.'
.- ,n,I :.::
* '( ,;:,
to consider random
letft].er-s are stati.stic-al
ch anr'eJ.?
choose
is
a type
is
the separation
o-f
-2142.1
Addit:ive
We
now turn
imp-fortancen
n L ct
t
whlere
Su
ippoe
I.
rea:L
White
anothei
r
of
exaFmple
the random
te1
N)oise Channel
W,
codino
art.1
and the output
num!nbers--
y
Gausrsian
somewhat
exrtonents
Y apih
y is
(AWGN;'
a zero mean Gaus:sian random variable
+
x and
The .
indepiendent
otf
Constr-Lrained
to have mean
w,
channel
at
values
consider. the
I
inplDut
input and w
square
Fj-+
we see that'
wit
iih the
channel
S1
most
is
inot
ar
variance
and S,
as a cascade
cia.nnei
I(XW:Y)
of
-26
inpu't_are each
.xiand w +ollo;wed
nor:Lse.. ess channel. adding
the s:i. n-le
si1_
tihe
(2.
is
Gaussian
each
by
given
'..' +
.we
respec:veley.
practica.-i
ftr-out mfore
4:-
are
abt:j-s
=
.l-where z
reatrf-
.ingl1 e
by a
of
a
input
at most the capacitv of
ned to ener.y
cnt -strai
~~:1i~~~~.
~
I· is
-i
a so_
o-mat
ion
eaI v
t c'
ee
bt~e
·
that
and
I--X
v
:LL
I W
i.n.. t..e
+
·
;i
the
is
ab.sene
a
o.f
w
remtIuS
als
.. Thus
.
''.
~22
-;
iT +
I(sWaY:X)
-.
ihese Finec:ual ties
l
are
s
satisfied
elo
for
tu229)
2
al.l
independen ths
di.,stribiu
tions
on .vxand w and are all satis-fied
and Lw arte independent' zero mean Gau.ls.ian
ihus
respec :i vely.
the
i r't
S
-
R
ri:i
S
w d.s.tribution
is
regi on
,
v
conve
. alread.y
is
it iis
the achievable rate
o!n R.
Th'ii.
s:i
r
2t
LT
2
_
and R
2
12
*_i rnce thliis
Si
i-f-
1
-
+
5.7c
and
+
variances
-_Fr .1)-(2.
which
relion
for some independent- )
satisfied
with
with equLality
l:.L
reiorin R
='"
--S 2
tr:i anqul ar
a
code
are limited
i:.the axis is
"iDMl
o;ar s'.cal.l
Aq
-TD)Mi s
is
moist clear-l.y
and
our model)
limitation
is
almrost
1 arge A ,
- oar
r ce..i"p,
' ts:=
"
,:
A,
(Rt;.o)
region
a code
(C,R)i
th-e
while
it..e boupndair.ry
al most
-for
then
irst
quite
as od
:.3:
-The
inferi)r.
t he case R 1
wastes hal-f
transmitter
transmitting.
2
if
f
i. g2t
reason -Cor
hal.f
t
t
its
i6}Tus
whiel -reas
this
power,
stays within
rate.s
betweenl-
can
Alternating
the available
Losing
ht line
-i
TD
le
codi ng.
bes-ti
-= R.
R
u-e s
one
the ach
ievab
rstr-ai
Si.e
as- th e
arla--Fsr small
rectangu
.wt. then
by the
o,
case where
Note thiat
or
Earious value s
o:r
S + i-the
S =
to the recii. f-in bounded
:i
fig,, 25
in
A =
-For x and
·fo:r::. 1,arge
seen
S/
the
tethat
ote
anid a:l fmst
biet,- weer--
ratios
r
ni- s
to
is s:etched
be
between
since
(by
power
the available
power
Fraction
oinlv a small
loses
whereas.
.for smnal.l
goino
to a
continuous
model
here
ancd cn-.ing
achie vli rg
the
a
A.
larae
model
frequency
power.
available;=
fraction
time
the same simpli
available
of the
lost.
r-ather than
division
of- t reqCluencyi
between
the discrete
the resulting
2.6 shows
rthe
tr-ansmitterS:
s
large A
suggests
time
thiLus
mUi.itiplex-infi
assuming tihe same power- i¥or- each transmitt-:er
splrit
for
This
as TDM, but beingi able
city
Figure
is
capacity
-io:
use all
raLte region
and the optiimal
as a
function of tne
r-ates.
Niext
Usino
consider
the above Gaussian
cal culate
Ea. :
I-a I
ini e:
the random coding
from
P)
".Ther-esult
C:
S...
Q)-p
P: i
we can easily
sums with
S
+
i+PJi
It
channels.
i
Si/
we can m.,
,
ic i
',at
t.t
imize
io. r
l
In.I
2
_g;
A.
these
is
over' p to oCet the parametr
i
2
replacing
i2.19
(2. 17)-
Letti-i-ilnq
+
for
for- x and w.
distribution
F'+
whert-,
exponent
=
li
2
i
~
+
1
i+Ai
1
+h
2.s
t e h34)
For rates
lower than those where
Pi
=
1,
-2.17-
.
r=i El .1 }
. in
1 n
-
-
-}
for
iAsi
i (2.
2)
and
the minimum of
thjree
do minan
t.
Ei
ons
subrgi
A-is
for
source
1 cr
i2
For
a
small
single
thing
resi.ul ts
t-i
having an
in
c- nsi
.
of
rates
fig.
ria r
R di vi des
(Ri)
Er
for
type
dominated
s
(R1R
ri
for
each
p1roboabili
tyb£
iy
i
into
i .
i
of+ type :-
1 and 2 errors..
by errors
in
bothn.
addi
jtiv e
Ga_ ssian
noise
con-t'1raint;
very clo
se
is
to NS
i
en.-sembile
code w.or-ds.
di st
for the first
is
set
s
u-si:
-ii
to aaii.n
r-ai
t
ini. erprei.-tat..ion
n
of
on
type
by tlhe pr-obabi
ity
with the additive
the second set
a N -ri'·
exp[-NE lri
i
phere
c apac-ri-t V.
--;h-ell
h 11
upperbounded
.ee
as the
reasonable
a
nci worr.d
ribuion
to F'
toi
of code words
noise but without
we
we,2,
have PF'
i--1,~,
~~~~~~~ei
t
From the- rq:eni.Fet
we see that-
t-he
are
-chosen
lon the re' iti
clo.en
i't seit
chain nel
not qui'e
code word
ihis
ly
is
choC.i
i
ng
The bes.t di-tr-ibuti On
that---
su.-.ff iieC '
oding
channel .
-a.~sian
i d5i.stribiul, lnn
yieldls ti he same ex'ponenr
dom
E
region
2.7 where
that
2i
i
e:,-ponenti
Te
n di s.tr-i.but i on conditiona
a.
for- i for
1 ,2,..
than
w'i-h the f
der
Gaussian
i5
the bound is
input
a-mu il tl r,-i'
error
in
=
more rapidly
bol iunid f -or rates
and 2 errors,
i
do f-or error exponenLts.
energy
of
i
ver
as shown
F'or
each set
+
-.
the random coding
(Ri )
-ec:,:..~':':i.
on 7.. 4,:i])
pa-.cr:
.A
[
n;
(2. 2"),
from a sihe 1
wit. h a iau.ssia
<1
bu. nort
i--odina ens-emblie
beis-Lt
R.
~ R.
'
>3
I1-----Z
the rate-s decrease. the error
error's. decreases
so that
t ++
ri~r
Lroi
| Ei:.
R
)
of
code words.
Thus.
],
where from s-ection
i
t'4
-if
a
[i1i:}]
Et
t
(.
i
is
,
constant
a
=
+,
and
Ei
1
1
'
L
i
+
'+
'i
~
')J
i
V
r
~~~-1
'ri
.--
R+
t 17, a
t h e
"
Ks
raties
-
-t
satis:-'yincg
sinFgle inpu.t c ha.nrel1
R(.i
:
t-`2
1+-.,-- .
1
F'o--r
n
'
T.
.i
by:
(
W rai
E. '.sr
given
,
1
l;
,.r~r - i i/.
is
(R
i )
Iin
I
1
.3.
i
-'-
I
i.
--.
,
*
1Ai
i
....
:+
,
(2-.54)
4
the sphere pack:ing bfound. fior
gives.
f
. lower
bot-undi
the
-2.
:pE
L
Pei
combined code words x
wei thei-n
,cons
which
(on
corresponds
e
multipi
in
thqe cii;ode words x,I
(iX)
=
'
(2.'i.
t.i'
(tii
ttppel;
-
for
..
:
-
'
n
act
if
-tohave enerv-y TN')
, we would
Q
then
be
1ito be orth3ogonal
ai shel
Choose each x
and each w usinci
i-lhe
to
itime the
freqit..ecncry
Oiusly.
-.usin.r
on Fi:.,
a bound
ano" w(.
previ
N.
simple si'nc
c hannel
continuous
discussed
i.'
We n('::!W cieveloip
ciersit
a
N(Si+S.
the two codes
the code words iof
division
less
+ w are not constrained.
x+w to have energy
cinstrained
raining
is
the situaition
increasing
x to have enerQy N.S1 and w
con'-traininl
after
i=1 2
]
approahes 0 with
' errorsq
-.
type
F-or
+ o (N
ri(Ri i
1or ai
iaN" cod es, where
19-
constreaint-. on
independently
the density
t-. (w)
using
the
w1here
a::p
1
NS
6<i
1<
,.i
otherwi se.
wh.-ere iEPis an:tarrbi trarv; poisi
number,
constan.t
to 1.
t
(,Ax)and
to make CiX(x)
into
kYw)
upper bou'nding
'-=~
+
~
xl
intearate
(2.19)
by
b
replacing
and
Substituting
is
a normalizino
(2.442)
sums with integrals.
far
and
( xx
w,-e
o.f
inJ.
r
expi
that
(2. )
Af{ter
[10].
P3
'1
6 +
1
i i
U
inn
some tedious - intepratior',
_
,.
+rr
.-
Fo r
t he
ca-ri ad
.
· ·
Fs
.
.1
vi.
ntly
v
i mp,3rtan t
1ui
eX
i
r
i ver'
car-n
wih er'-e
'e
L
, r.
i i
for all
andAi
A.
·-- ·--
e
-
A1
b y sy
.for
is
inrntrt y,
4 .:rV
any
by aN"
'i
ation
'-
i ,:al
n
valid,
1 and
and
for
F':, r:-.
optimiz
also
X-
it
v.er
o't"i-miz:ed
'I
a solu ti on
sUsinq
simply bou.nd
b_
1,
ca.s
+F
proportio na.l. to
6,:so we
The e;.:oonent
anld suc h
--
a-y
'1n.1
S
4: (2 is
r,-, and
are equal,
~ap~------~-~--~~----
ar
7..
--
. +p
irsE: term in
sEuitable a.
elLqu
section
-i
.-
ci yVen i,-hoi ce o,- r,
(t3'-
in
11
[he
:.fL-,
(as
+
+
44)
'
:
E
*
ri
we g et,
!J. in+A..-'-~+-
i+F')
:.*.,.:p,
1
1
1
a product -form
into
bre-aks
iex p I 165(r-
r (F:,r~
........
r, .
,
.-
can
be
i
an d
but not optim al,
aA f+or A
A
?+
-_7
21-
E
the e xponnt-
Optimi zin
1
E
n.
( + F.
..
-
[
we Find
1
Li
+
that
R
In
| -4
J:
E
t2-1|
I
(-'(1
I
+
_+
p
) +
i
2
+:t
4
I
i
1--+
..
:1
For R-:.less tha.n
2
1
the
n
I
.- I
A-
(2 5
A+
:
ptrI, P-f
..
[.
2
I -A)
++
-:Z
lower
4i
or
71.
I
2.t
+
/21
'-'
bound
in
(2.49),
22
E(Ri-)
2_1n
=
+
_i
n(
i
-
derived
exponent without
a
constraint
shell
c hannel
,i
with
a
shell
way between
and the exponent with
constraint
would resul-t
to noise ratio
the previously
for a single
A
input Gaussian
that
(i. e.
given
by
s(2 .,
)-
41)
(
=
w:e
1,2L 3
samr-e
.as
i-he
in
d.ifferent
.R
FR..')
r.
f:
(2.6
,oCdes.
ne
were
l-ne
We hw
i.
n
For
I.:no
o ..
vanisnhes.~
:;)ptimum
w.hiter
ie
is
i'.2
:i)
the symmetric
wiheneTver
ti-e
ex?)nont
cet..
is,
ip
es
the co;nmin'ei
an
tisi..omias
.s
givevn b'
Ri r-at-e
.-
pair t
rae
i
nr
rate-. r
u'
l
ate
aal
R:o
.here R..
R.?ta the:::tn
e,?n
ai
t
r
-For optimr
the poJi nnt
w:ir
critical
the recgin
R.2
into
5ons look
i
ar e -s.om
he
f or
l? br-eak s
t hv,
ab-ve
:
.po..i
i2n:]:7-2,
thh e cri tical
nal
and
l
(Ri
Eri
1-re
t
ndeed the ex-'ponent
R2
Rif
crbti:cal
.
si
hnv
that
(or f
Ri
n
l
case where ='
R1
enough
abnh.
e
R:,- v-anis
hwei.eever
,..R;
and
-,.. ishe.:: fonr smal
bound.
a-l thouQh n umeriaii
(or
R.
t'henS
eacih
2.77
-i
exponents
tind tihat the achievable regioin
again
e ir;ns
I
subr
o-f the three
we takle the minimum
When
i
5
A+42
+
that
-sig nal
-_R.F'i4
-t
+
in(i
1]
+
lies roughly half
exponent
Th-iss
3 -
-
t.he
thle
i?:S phenomen?.lron
n.t
occturs
to noise ratiio A-i is bel ow about
;
Coding
3.
Techniques
channels
for multiaccess
is
that
technolcw,
gy
which a
of
of
coding
a
transmitter-s
to the simpler
of-
problem of
always have
where both sources
fig. 2 .1
rding.
simultaneously use the channel.
ourselves
we restrict
the two input channel
subset
is
needed
is
large set
a
for
applicable
has been
little
for multiaccess=.
what
introduction?
but variable
small
however,
Here,
ithie
in
As pointed out
techniques
coding theorems
very
is quite advanced.
to general
done with respect
of
development
While the theoretical
sormethinCg- to send.
.we observe
First
evaluated
linear
6.2
in
The arqument
codes..
[1C:.
each
-1
any desired
achievincg
in13Lt
p
bounds for
for- t(!-he s
ng
fr'-om.n sinqle
Il withi
C1
.
ie
:Lnt
np
r.nenei
t
rn
di.gitf-
or
be
thu,o-r
modelling
to the hearts- of
en .1'ar-edex
i s
e
Thus t herTin
input channels,
dij.Cussing decoding,
is
to m
codes have
the
in
order.
a
is
brief
The discrete
als-.:
t:iacci'.e'ss
poanent
as ,occur
i em
n;"~ pr-b
or convor lutionaa.
:
.ci
4i.her bli ock
as with sing le
Bee:or-e
channel
section
iilzation
convor.iutional
input channrleis
:he s-rlne type ofi
gen er at ing cr:tood codes
].em.t
prob
as in
e tter-.
been generalized
;
uti
of
of
to ensembles
input alphabet.?
trequency
relative
bounds
codes can be generat ed
tfhes-e binary
thie channel
ling
-id.arldom cod
clhanrnelis
the same
is
linear
sub-bc
iflJ.rnr
into
to one
many
:i.
t- :s
and
S
well
equally
for this
binary
n general,
transmi tter
mapped
apply
last section
thie
probability
that the error
T'he
with decoding.
discussion of
time channels dear
information theorists implicitly assume that
the channe-il
iL
element.
Furthermore?
u sualii
'
assume
-v
is
channels
it
ch
rInnel
a
mul' tiacces
increiase
o
the
icord-s
F.
ra-
can be viewed a
der
.r-ei3s .-. -F.er4back:
v
i 1mposls:i.
I
e
ie t
thi
one
C:-.'i"
At h
r.
tters
transmiti
and anqroxxi
:cr
1 y ImTar-i
n.:r
pepr
i .ai:
c..:p:
ph.-ase
chal nel
t ie
f'fc
t -"
t
and baud l evel
however,
,
the
continuousi ?
time narr-:owiwband
shelel
cr'onstrai-.7nt)r
ie
of
be
notfl
.
thainr
inln
t.
.a it
carr- ier
o
phase
..
pase
t'it
fron
syn:,'aronami.s
reas o,-ab
-- Ic,=i wvi
em ine
ri-l'
't-hI
.,
Ienera
i ttl.e of. a
For- ti-he spec< :if
Gaussian
on a
receiver
m b etJe en
-fie-e
is
Cituat
i on is
arnen a ctino
(al.ong with
i ndeoFefdeln,
nt
a t.ynchoni..mn
to qenera..te
rain
re'-e'ived
reason.abl.e
:n
mcI ni*.
i cai-i.
i.:
: fc
ap i:':.ear-s
-
cofn
nave 'nti
r
E.
'7ti
exceeds
the additional
-y-'mb
oi svnchrin i-
,.v-chr-cnIssm
r :.a!
n
r 3'.a! . e..?e da
!'
be fsai.. id ,ab..u
e s
a
c an
to S.I and S-.,
While thi s
rOo abhl
be
sligShtly.vmore
ate h
'"h e r e
*.ie
t
n
a n!T nn
f rom antenna
ex-ect
shioul.ic-
C:
iarasmi-tter
iv
is
and that
not
e dback!ia.
the receiver
tin
cC.rl'Jination.
Typical liv
frrom
energy at
r ming
3i
certait
i.-.
is
thiat
n t s+
Firgy
two transmit
the
- i-I]
*are limited
as a phased array
essentiial l
energy
means- trhat
the AWGii.
-'Fnor
that achieva'ble by a s.rine
ond
transmitters
the signal
buT
T.his
Sl+SZ.
n eir,-
Et ri
individual
respectively,
knowwn [t]
'
t"is
f or mu lti acces-
F or e'xample
Ri+R-b
2 beyo nd
tih'-e
achievable
sx. urcee
1well,
is
bu t
of
ithe-se
o.f
i npuzt channel s
r-eas,
e na
quesi.Tionable.
it
perormance
ideal
iior s ngie
.1
usua.ly
persectiy
i.s o-ten
physical channels are p.rt
in
model
srp aration
ti--e
time
phase and sampiin.j
carrier
t
a4ii urre thai
.'a
ees
...een
h."i
i(:: case
mnu ch simplerD.
en--semble
code wordsC,
~of¥
r::r c=
can
!
at
a. t' AWGN
Usi nrq a
(wiith cor
tlhe discrete
without a
time rcode
I-
*
wrords o-f
lat1.:.ection can
the
the b i,:ck ol- 13
i'-'i
o
with i. ternate
_:
re.
r
['uLks
rom for
wl.l,h-hane
st-ati
bio ck
i
i..1'r_
bi
r.
.....
fn ,rl
r3
sF-' y
'- een
'i;-he
"! i
.i_.
.
n]v.y
I.-Ify
.
;
.,-
a tni
-,,
d
..
-F .
c:me
i
(1
mu L:.
,i-'i acess
b
7
rtt']i o(':
L
r-. ,r-f.a:.7.
i.u
.:5
'y..(.:
..[.. ....
:
chilangea t:,'hel
i.:Frlm.,i4
musrt
f
at
t. hr
but
b ut
tJ,
hi
et.-e r Tn
m.d
onlyj
oFt,:mh.
n I vy
is
i
-,f,
-
ireooa
!o
h r -
le
of:
bi oc:: :s
.:1-F
4t-L,o! tr annsmiter-s
t-he
4: --
m
ef-nds
the reliat ive-
to b e.syn:c.o
f-it,.m..(=;.t_-._] ti
a'et
.--
ca
!n
-i
i
a;Jeqte'ie
-luatd t qFied
d-
;ast.
'i.e
tn .. ti ;lt
l
e frr r
transmi
r-
w
r.t'l
'i.
.
£eedback
.e
b;-C.c:
p
are
r-
"tCrtt
.: in
- ai
b ei
t n os
urnceritainty
e
sim.
roan:
r
u'_~.'
e
hull..
is
,m
wee
.'-"Ln '
as w. z h
i'9 thrant
t
i.
pe-
-
ct
t
-fi:
d
r a; d -i
r:i r .-r:,
a1
hnr
sy r:hr-'o
i n
For co.MlpLete
iti: has.:
beenn shiown
The essenti::iaLl
of fig,
idea here is . that
ri-at
f
t.:--
Im
me.
(i
l-:o utC
e ute.a :s.'..
i b ';-
i
of
C
zat i on
e,
the tunion region
tha'i
i .t ..... 3 t, ^i
b....
c,'_ain]: const'ira.ini: so i ar-ge
hand,
i o n f o.r
r- s.oo
h....-
b.:ecomes i n gl iigi
+easible re:jion
HI S'
si
a
on 'lthe other
n chr,oni zJat
t-k_
e r_,r :.
J/ 'i '''ir iim . inz a '_.I o.n
b
'.
ar"::
is
..
...................................
t
Ess.F'enl]t:i..al I1y
c:(]lovex
t
.
].em of
'-L
r e.:i:
i oin: P
R i:.'
is
timingr,
for each
i-s
r- a r-e c :-a .s e -
t-e
s- ..
"is
*
..
cianl .i...-..
c:sL'e
nd
e' m
tim.e
not
ti::ne
's-':.n c-hr '":.r i ,
sym bc:,
sh. a-. ,
Tihe decrider
.%] C.?
v .=. .octn,-hrr.:ni..-..-ii.
-i r cin i
and saa.am,'1,
end eff''ec
f3i
or
non--
e
Tne
h
pro
rof
i
phace
i me
n c.de b,.ut will
fi,-r
t
t
;_.r
f .:i.q-t.
r], s.'.
-ti.u-e.si
i
ample
ph ee.
c ihoseL
n wJav -F irm
i
r and
o:f
pro ress
Gau s.ian
p hasr-- and o ui- o
vt-r'er
carrier
ig n : r-e).
and
'i j--} rr***t 1.s j
sU.r
rmma4r\'
we
',:t,n oj ned
se
s;
set
ly
i. :aside Esome
from
ic
whici7
ri.-r
phaser
car;
o¥ ree·
thie di s--crt:r.
en semne.:.
in
a gfien
-
co de~wor3Yd~:!a cha
rrtien
ti i
a aro
-r1 w band s at.ionary
l.eter
.nnitn
compc)nents.
cons.idered as time samples over
be
th-tat
.theii. ti m:i no
unc.ertai nty
[55]
[20]
2 4 rather
i-n
that
thani
he
its
time shlar-ing rrann ot
the total
in
be used
absence o-
betw.!.een
timing
relaitive
the
transmi titers.
Having car-.i oned tihe reader
inherent
in
channer-l.,
a
discrete
tLime miemorviess
return
w,,
.e now.
t
to
modael
his
prblem-s
a.bout the model in
mui- tiacce- ss
modei
asaid
see wi-iiat can be
4tio
about codingI
First.,
cani
n
codes using
.i
I3i ties
ensrnembl1e o.F codes
-eet o
deco3der
tr aranslition
for a
Fniy
w i"
sin le
and
F (v lw
ror i th e
pr
vary with
tus
s !'quential
deco'dirng
chiannel
n.
a
bu..i.t
n.umner ol
Viiterbi
int er:i. or- o-f thle achievable
pairs,
one of
Wihicih,
(R'1
reogion to
i:' iv3en
can decodie
us .in
satisfies
,
fepend
ie-
decodi0[
;i
(23
any
')5.-
s'uf.:'"
with this
in
(RiRi)2
'
or
of
a .i
-'for
on
tecChniques
as a convex, combination
R..)
!'nood':
w reliably
decodi ng catn deal
g
_lhe
somewhat uncolnventio.na
itr i :n p ro babii
'fi
As can be seen f:rom
,Q..-,
can be represented
input channel
dec; odinq is
Xnthe trans
andci
cain-f decorde
with
'
precisely
a
..
by
Over- the
Tinhv
s
=. ir om w to
-'
rL
channei
input
is
th is
W
en=.-mnhle
w)
encoder
a singf.e
sc3i-ind
i n that:
rs
t
i'-lputi
sin-gle
Ti-iis
inou.' ts
j.
Over ithe
.s
'P(y
1i
rabi.
decoder- for
anotier
w,
Q!,(w)
i (ix)
and assumino- a single
code word
proabi
iti
R|
.that
< i x
a decoder can deco de the w code word
-.
tthe x
igionoringoi
transi
assignment
-For some
iY)
source decod:tn.g
sa Jsi
(R 1R.)
Firs-i supose
prioblems
th at
a pproach
tihe decodiing problFem to -several single
redue
* (L.
si.mFple general
fair-iy
a
thiere is
nhe
v en
two rate
"r]ias
F.i I
and the other
tor
each of
F-'i nal :.v
z-rate
Ion'-e'.in is
ich.i
sat
: is. ies
two
thi-e .interi:oLr o.f
a't mst
4 c des. n
resp
vvo ]....
Q
and
:i
i ti:.
:for joist:i;
error
that
inrll-)-
decodinig
of
decoding
cecoding..
tri
is
iNote.
by
,) can be decodedi
of
the
(._
however,
o-f
which
s
i
i
b:et"weer'
atih
error
1js
-f:or two
e-poneilreln.'
ts
tocgether
decoding
(aL
;:ac'f'-cr'
muc:1 smaller
simpler
that
p4oint
satisf
'
the two code wor-ds
potentially
crsle U.! -1
t-yfr
an arbirry
rando'csding
m
of
e
one of
by' t im e
two
are often
pairs,
leq
- uali3
i
ent:irel-y
at
measure
the achievabi
twou rate
Thu
. s
ot her-
not
-'
-:his appr oacih
ex.porne"lts as a crude
joint
on. o
*'
-'
thPe
is
a.th.at
.i:irst
in
is
(iR..
1'
the in terior
whicih-lyn
u se rats :-.2
T'ini.s approachn
p::lr'.:babt:
in
tiher
h
two of
:X.21
such codes.
R can be reL. iably sl'arina
decood
.
e[.'s.'te
above and
comnonat;';i
some
,
X
I (i Y
swith
flr
:
t-hese rate pairs can be decc.ded
point
a convex
the o her
ecL
a>ny
i[ tiA;Y)
satisfies
Procedure described
between
:
F:!.
iR
I (X
by time sinarina
an
which
R ... :
(odes
the two step
of
..
ti-iW
f'or- ei-rror-(]r{
tihan
If
we u'lse
simpl icitv,
than the a'bove
exponfents
those
can
we see
single
be m.sie.ading
cofmet'imes
as a
the random coding
example.
channrt- el i is rnt-
larae,
to dercoding
guide
for
exponent
to pcrovide much i nsight
sources.
a small
but. urknown subset
decode
the
channel
r-egl.ican
is
(/2
I!n
Ai / (1+A
wlnc.;ine
A..:is
sma'l.
c ase
-r
C''f
a
as a
P
se
a f::ilde.
m
'du
uen.,:e or
)
.'1.
i
the advantaCe
numbeEr a
,
perhaps
simillarl.y
individual
Aal.yti
case o f
arbi
. i22
,
transmitters
is
seen
ratio-
to
Ior
tihat
'
the ac:hievabie
j
I (X
i (X:YjW)
=
(1/2) lnr[i+
1]
foro
ot
iecodin
.i
joint
general izing
imm. di atel
w.i'Ln an unknownsu
..c56
is
ov
oer
a giien-
d(j i n g
can
where
period
fl u it i a ccess
-bIe
discr-e
r
trary
approacn
.
to
o-
bset
t
t':- e
viL Led
time
i-
ways
in
ito chioose code
such a way that
the use
is
sifmpi
same
erIyless
rr
a
o-f
v an
p ru
s: i t ii
f .A s.
e.
with more than two transmitters
investigate
oFr
SpreEad spec'trum with pn se-eruences ca
t5Cscr:ompiement
p msn
I 2
it 4is
calily
a
to
so:)urces
s
ers.
raptroach
the exponent
err`or
o
ai
to the use oS
tra,:nsmi4
tt
2-.5;
to noise
close
thtis case.
special
i
is
it
i.nridependently
is almost the same as
.ae
For" an
channel,
transmitter
nal
that
joi nt decodini
-:, of
Fr-om fic.
si
are trvial..
is
leiss gieneral,
each
which
co
: I-. t r-ai i i-i: o n t iie e ,nl
I.at
1large
i
but
small
the scurces transmi ting
ise viewed
a
as noise.
In
approach ha
Ls
lhfe
of
Forl
binary
not, gen:
er-a ize
almost r-ectan;i:ul arr.
i:di vid ua l decodin
hhi
-ruom
with
rate
i
does
simpier
reqar dirng the other
the
fAiWGi
certainly
coode wora-ds
approach
into the qtestiornn of
several
A second.
a noiseless
whereas coding and decoding
The oti-her o bji-ection to this
It
complexity.
n
wa V
m
.nltiacc
'ss
one can
word sets
.for t'_he
they are mutual ll
non
1
!-r er i
rrmorre precise
i
decod ed wi'.t
smali
1
code worrld
o!e
po' -sii. :ii
j.
.
possib iiities
chann
el,
word
mIore
A
sets
property
in
pr-obability).
ty,
We have
t1asv;
and
asvincnron'ousuolii
s
time mieoylesv
it-
the non-intr'i"--ferencen
srt fymboil
that
ma, t-c
i
d
av.ar-
c-f
in
this
ao-..-.ther
i 1
i
[14]
ari-e known
c hian nel
will
.I 1
al:::i:
i:.
a
r ::.ir;i::.:i!;ned:
poss.ib.e
number
oC:f
'tates-,, s
i.
(..,-
s,
o-.fal gebraic
Vi.ter-bi
thi-i s
objecti-,'e;
at
discreite
gener'al
i-not
coins:i. der a
stat:es
The pr-obiem
With morle
is
tie
I
be
am n:)ot
geerral
r:r
convolutional
codes
very- promi-si ng
decoder
is t1ha--.t tlhe
.i- ich
iod
ier
of-: both enc
vhichi-
and
hess
appear
igebraic
must
st
-ructure
decod ino of
joint
true
consider
charact eristic-
approar:
does not
buot
o:"f sta'tes
p roducet
is
as a
shoui.'--:
.ea ds to a
i ndi vi.dual.
oi-I tni-e
than t wo trans-.mi'i tters
the probiLem
-
Finally
4
approac-h
qso
in teThnique.
tnum.iT-:rs
' evenl'
to
-ense
.the channel
iit
possibi0
t:i.
:decod
for' the
f-or
to decoding,
c .o
ode w o rd-s
any e...¢amples
.j::i.'i:
be done -for the
ca;
ch-annels.
e s
mnl.Ai
t access
m betiween
s yncrhron
is
i
'Ii clchni q ues here since analgebrai
oerciii-i
that
co e
to cinoose the
also achieves
no, approaches
As a thc-Iird approach
decofdin,.'' of
problem is
codina scheme
on channel
si
inowev er,
'
seen
ot-er
AWGN
lack or
o¥
.ithin a
as we have seen for the
as to maintain
the presence
AWG3N c'.hanlneil
Time sharing
depending on the channel,
r elated
difficult
tter-.
biu t
be preferabie.
su.ch a ,.ay
:in
thie transmin:i
i:prese:.t
err-r
migh.
they cran be indijviduai.v
so that
sequential
to mui'i access
lack of
block
decoding appears
joi nt decodi n
synchronization
is
and it
to be a general
has been
not a serious
shown
C23Vr
impediment
to
Un-torun
irts operatio.
il
to meake sequentia
expl i
n t+he d ific ity
sequence
oar sie rcihes
nt
f"l1metriic:
backwi.ward
st1ocastically
tkhe actuai
Fiti.L owin:
t.he d ecoder
now,
ini
thr,~ee ways
in
secti
of
the d
ti-,
recent.
Y
srme
i:,or-
erfa"_ure:
E:
: ,,- y =
_
c:ael cul.:ete
de'::-deer
thie
d~~eA
as
is.
decoder can nIn oI-'f
thle
te
anId
madei
ypoes of
tiree
errors
to use
metric
inowltedoqe is
this
the
Arikan
(eJe)3n}
,
cdirrQii n g
c--insiders
ih a
L=
With proba1bilii:y
.
wIiher eas with probabili.t.y
i-x...w)
input,
with
i i iuLt
[24]
Ii
,.Jher-e
I3
--
u.i::i).,')
i.'
? i 3)
=
e qu i p: r : b,a. b e
a
when
problem with sequentia
e rX
ra u;-'e
(.3
cnhanneis
beino
error
discovered by Ari kan
.ce.s
binary
ihaInei
independent ot
the encoded
downward
o
1the approprl i
ilundair-iental
bveen
mu .L__:La- i
to a
fcoder.
Another
has-.
up
the val.ue o.
and drifts
to ti=
r.-resondin
jdepends on the type
uiirnown
on
T
is a
sequence
by e)x'tending
thi
is'that
U! ir.F-oirtu-nately
2.
or
i a
nne
,h.w
tr
o-fi-theiack..
ets
The problemn
track
lear-
upward when the decoder
sequence
encoded
not
cha
iaccess
thle encoded
deoending
drifts
is
sequen tiall decoding
proceeds forward
aind either
pc:.int
thet
ecall
that h-ypothesizes
search pro-edure
gi.:ven
r
-for a m ul
work
dec.ding
imeq i.
ti
i
nahis
at ely
. f:_~eah
-:'
i.n
(ee
ef'fect
tof'f
Cnalascomp
I cu ~er
co:)mputeaiti
1:'d-7
ti-
t wo
sures Ln.iiera
ng
correl.ated
we
: ransmi
we have
w
reriion R
cean
4-o r.'l
for a
I y1
jo:nt
_;W-..
i---',
a
41
rCj
ol
F._
'
i
E .
-'
1~_:5
·
~,
I=fi-iLn
1 )
(
|
4
I
' E '
R-
4 )
-(:3.5)
we notfie that
j
[n
>
rihus
.
foi r- R '
1nr
-
= R. ,.
i
is
35)
;
E
all
the active
0<
(3.6)
r
constraint,
1.
::
and evean
the mcetir-ic problei-ims discussed above.(.
,
3.5)
th1-e achievable
rate
However,
U s Ing
s eq uen-i
withinout
ary
se D-parat
e
(.3
*.
..
ma k:
th e
Ihi e
s it-ati:ie,
co
lmputati
icna
-t io-
cnvoi
ai
J':L n
riesi:
cnahnnel
iut i ona
e
:i.n
't.h
'?.'.'-
i- hIe
potential
which
tt:or
encoded
are as
f'
se.
we
r-ate
,Q- a
likely
digits
and decoders
hI
i gi-e
i h'e r- r.
partijul. L
:ofcodes
ws
n
as two binary
r
ens.:emble o-f cortvol utlri
pair;s
tl e
twor
I aiel
ati on i we can ac
cui
encoiders
a'c :ii ev'
or
decoding.
es
eve
th e
binary
as the
pairs o'f
/- '-] 4
Inpurt
np
t,a'i - -rnary
q
to the
anld using
separa:te
the two di
qits,
m
i s-.
si 1-gi
1 --
bei ng
inputs) , the expected
(or
higQ Iher rat-ies
i n I ( it .
ere
n- t
d.:
Over
inpi:ut
number
sequences)
transmitted sequence
we;e can
used.
cofdes 'for the quaternary
sequences
and
t4h a t -
d .i i:' , i c i
a igor h i
search
,llal
'h
t:or
limits
transmit ters
-1owever,.r by regardingq the inpui-ts
er'asure c hannel.i
~~quate!·c.v
sequential
al decoderis
e corr
nd
.F
"ti:3
with joint
th!'
S'3Lr ei
:
ci
igr
-ri
a .so3
o-f
at
(or- pair)
the
(o3
of
length N
is
in
exponentially increasing
-in[
is
t(li-:')
;4]J
Thle conclusion
not really
a
same exampile,
discussed
fundamenft.al
by iMassey
much
F:rit3oi.wiedge
e:· i i~%
about
paramet-er
coding
is
cess
rate
reach
in
excess o-it
is. that Romp
oft communicatlion.
channel , has been
26].
ary before any chesive
and decoding
s
'Ti
approaches to decoding,
previousL
ithe
must
photonr
the
and Humblert
25]
more researc-i
one
that
the contex.t o-i
in
-ummrarizinng
that
N for- ainy combirned
for
multiacces.s
we see
body
channels_
of
wi
l3.l
4, ,
COLLIS-3ION RE7SLHUTiUrNi
iThie coilision
co;i'municajt·- i_4ior,
the
channrel
i'(t
vsites
-E.
approach
resolution
as menti'oned
among a
ias
in
large set-
the weakness
ormmutniCtcatji
assump iions. t(hat
of
users
on
ClOCL:-e
alloicatin
at different
oT es-sertiaiv
ly
the class
limit
mt].tiaresfs
section
i,
of the problem.
aspeits
tc
of
transmitirin
t-he
gnoring
We start
svsternms
by a
we will
set o-Tibe
cronsi er ing
liotte
a.)
starts
transmission
small
ti me
of
clotEks,
for
slot)
lonlig pac-ket- s
is
ends before
Not'-e t'-hat
packets.
th:is assumption
d
o short.
car-rie-r
ng ,
sensir
sytems can be undersr'lZ.ood
r "i
which
the nexti:
secon
arind
,-vat
dsci
af ter tihis
sitrnlmpl'y
eacih
icult
,d,a
thir
d,
precludes
both t'
e
t
r-ster
make
pack1et.:
of
too dif
usuall y not
betweenr
spac
to be
transmission.
f-eedback from the receiver,
timing
aEnd of
time and
zation
QLuard
of in send
F.possi
.f:i'liy
Suchi
synchironih
a smail
aimi.,.:(nt
s~t!i:able
at an integer
Sci
n
first,
gCivi en,
(a
(pacl:ket)
hhers are synchronized so tnhat the reception
tratnimit
i
i l..teerr
each message
that
f¥,its into one time unit
transimitted
l1
We assume
S ystem:
or
ons
Ltss
la_ter
as-ic
madei
is
undd e r- s t' oo d:.
b)
Collision
i
oCn':i.:. '-l'*"ansm:. tn*i iter
co 1 i. si on
or oriCiin?
sends a
or
. p Ferf ect _ R eception
a
s
send s
a
r a:ke
c
an irthe receive.r
of
packet
the assumption
i- n a
cets
eino
..
llme
a slot,
that
·from t ihe problem;
it
is
i n. orfrat
i
If
received
fhW a':
i-hen i:her'e
sl ot.,
the transmitted packets.
in
i
on about
itfh
: mo-r'e
j.s a
t'he contents
jlus.t one tra nsmit'ter
with no error-s.
removes the noise and communication
it_
allows
collision
than
resolution
This is
aspects
to be studied in
th e simplest context
strateoies and
c:)
Inf inite
but
also severely limits the class oi-
tradeoffs
of
Set
that can be considered.
Tr-ansmi tters:
Assume that each
arrivino
parket arrives at a transmitter that has never previously
This precludes
received a. packet.
transmitters
queueing
iLthat,
a finite set of
regarding each
co-ncept-iual
packet
transmitter.
on a
.inite
set
is
sionir'1ficant
ronl
u
sef,l 1 or
at
one transmii tter;
Coll i si on
sfet of tran.smittters
i n this
The maaximum
al. g or
inm
an
a vod d
dt
i mpi"t-ovemenl-its
achievable
d)
iAr-rivals:
IAs'sunme
PoF3i.3sson
Poisson
at
independent
e)
(,
provioes
an overall
arrival
i
trans mitti-er
wl-hen queueino
1, c Immediate Feedback:
the
p1ri marn.R
pac:t-c. rarely
iy
quteue
with
up
a finite
o-f an algorith-im under- the
of ithi-fe g'oodnei
measure
un d amental
less
proceses at
.waiti
nq at
approx i iated by that . wi t h an
s a q u al]iitativie
r-ate X.
the difference
t ihe pert
e rforman c:e
thrcugthput
that
that
al gori thims ar e
reg tcin
i e
case bound
a worst
paickets
are
multiple
sh ould be
l wel
:i.ri-if:
i. i-te se't assuimptiio
(:',-f tie
rc
resolution
'1ere
a separate
colliding multiple packets.
two or more
ratesr.
to
this case, a physical
assumption
when
low inut
set.
in
of trans*mitters and, second4
· ane tr-ansmit ter.
inin:t
nt
the transmitters send
as corresponding
simultaneous,
that
first,
shoiws,
T-his
an
transmitters can use the same algorithm,
arrival
wo(uld sometimes send
is
point of view. but note
gqiven any alagorithm determining when
pa ,:kets?
individual
This
and precludes the use of TDi'i.
unreasonable assumption from a practical
at
tinhrt
occurs at each
hrip ut
transmitter.
nekw. pac.ket ar-rivals
This is
reasonable
the individual
are
given
nodes.
Assume that by the end of each
s
.-
each
slot,
learns
transmitter
(c
more than one packet
slot.
This is
immediate feedbi-ack
algorithms
c
of
is
benefit
e:
dj .scrusCseCi
in
wnhat
It
f+or
n
Some alternative
feedback;
compliicat-es
of
i,
i'l
the transmitters
idle
also'-
of
resolution
assumption
f-The
between an
is;teni
col.ision
however
inigh:i
(or
gets
with delayed
feedback:
e
in
transmitter
that
and a collision.
slot
i-iat
implf-
thi
feedbacki
s
forms s
of
idle
, w. fich
feedback
is
will
not
be
foiJ.lows.
SLOT.ED ALOHA
The
as'.mrit.ionsr
of
form
stimples-1t
aab]v'
is
desct~-ribedi
arrivyes
at
ne:.t slot.:,
pac-::et
involved
remains
nbacklogged
sLuch
backlogged
ofd:
the
In
the
untij.
packet
with some fi:.ed probability
of other packets.
A.l oha
Arloha
is
it
iS
is.
saiRi
0.
C
to
t
is
in
tran'E--t.yi't-ted
a slot
be
backloogged
independent
be
a packet
in
each
't ted
wil.
iy
I
and
Eacih
subsequent
of past
in
each
I[y transmitted.
suc-essful
transmitted
p
whenever
occurs
Sli
, [i2which
t-hat pack-e
a coiliion
1g
i ' in
(Rober-ts L2''1].o
(AIbramson
slot'ted
collisiosi"on
strat
resolution
':tra.nsmitt-ers.
Whenever
in
isi on
pur-e Ar]. oha
subsequentlVy.
one
col
Sl otted
Alioha is a var iati on 0
th
e
but
the receiver
are alwavs
alway-s desirabl
th-e
th
not alwavs reasonable
tranmisMtte rs
4:, 1
in
delay
in
The assumption
to deal
be modified
can distinguish
themselves)
each
often unrealistic.
implies that
feedback
whlichi
is
little
with
that
or
1 packet.
were transmitted
collision)
of packets elsewhere.
can usua.ll
the intr(oduction
analysis
for
the only information
the e<ist-ence
about
C) packets,
whether
slot
slots and
Note that if p were 1, backlogged packets
would continue colliding and no more packets would ever be
-4. 4successu-n tril
ansmitted.
-effect. iveiv
resolved
irlfinite
of
a known set
staie at
number
slotted
Aloha
pa.cl-:ket
is
n'"e,,.
i
p.roackt
ne,/ pack::et
and
k+i
ttate, trans
1
paci k:et
artrives
be the state
=
anyv
in
at
t+l.
is
l
rly
=i
if:
same way,
s
probabilities fk.
k,+
are given
by
i
L-p )
[
1-kp( -p)
El .4e
A
e
(i2
e
k
=.'+
ot
i1
eitler
o
rs
ino
or :Cone
transmitted.
the
cases
is the
back1 oogcgged
orne
> i in
the
i
:ftt
ttfthatLLs
ti'ransmission
successfu.lly
but does
the number
and
Sim
nila
pack:ts
to t
Thus
1 es
-f{t-'i
as a
Let k be the
r,{t+!? r
in
{t Ct+l
in
and no successful
:: p
P
like
time t
from t-l
to t.
in [It- ,t)
arriives
ar'rives and
ition
d;etermined
of backlogged
the slot
from t-l
(if
transmi.ii tted
Analy7'zing
be
can be used
'The state at
in
new packet arrivals
no nev
slots
can be analyzed
using the number
o.F new parck-et arrivals
if
of
cannot
Such strateiles
as the state.
su,-,'-~='=~uf
l
tran.smissions
= -1
the
looading.
time t
time t
of
and can be made to behave
includes packets that collided
n-0ot include
some number
transmitters
M'arkov chain,
at each integer
because
the collision
the transmitter.
can be seen that
homogeneous
that
transmitters.
waiting
of
TDMA under heavy
it
of
bv transmitters
by tile identity
with
set
also
Note
(l-p)
_
that. t;e he
ee
i
e
-,
3
i
1A
(e
Ž2 (4' .
-4.i5-
at the
In understandin(g how this chain behaves. we look first
drift,
D
(i-e.
at
as the expected
defined
k.
the e.pected
-[(i--p
.
..
at
+
on k
conditional
the state
between
on the state
t conditional
-
difference
i
value of
at
t+1
and that
t).
kp(i-p)
]
e
The first
term
departure
rate or
throutghOpt.
p
will
positive for all sufficiently larce i-::. This
.:. CD.
k
is
.x
be
the arrival
rate
term is
and the second
>: (
f: rn any X
N ote that
the system becomes sufficierntly bac.kloqedd
drif-ts in the
direction of
a.i
Js..,") gives a
noi
th e
when
but
i
simple
unsti-able
legarnt
,ge .t
ly
suffic ient
this
otf
slotted
oha
.l
approach
especial
:lly
[28
,hain is
type of
thi,-at
on almost
i-.ap lan
. are
proo-f
instability
resolution
collision
rl
modii-i:i ed
since collisions occur
i.e.
'i
non--ergod
ic)
IDesfpite the
use'uftt
backlo
it
more and more backlogged;
becoming
be surpri si ng
the
and, any
means that if
thi-s shoul.d
,
(
tc:avoid or
5Using a.-. small
c-aLas'trcph.ic
value
r'cover
oi:
b3ehavior
l;iC
.from the
heavil'y'
the syste -m .is
i-f
ibac-loqqged
p helps pos'tponre tihe on.set of
iabove,
and
for' simall
p,
be a
can stili
it
ii(4 2.
state.
the
ca-i
be
weil
an p 1r ,! i mza'lt-i d: b y
h: 3
'.X+p
*I..
-4..-Xpk3)
-~.
Fig. 4.1
-
::
..
illustrates this equation.
> 0 'for all
k
For .X .:: e-,
For
X > e
-
there is a range of
we see that rD.
k:
for which
D
and
a '0- the size
as p dcreases.
riather
would
oi
retrials
ot this
range
Unfortunatlely
not:
X is
the aririval
p is
stable
recgion.
chiange
p as the backloqg
solution
The engrineerinjg
k::changes.
this
solution
is5
-tunc:
it
snoi.w.ed
ims in
f: the
Is
i
,
..
nl' y a
c.o':!:d wn^hen p
p is
functions
twou.ICiij:e to
ei
For small
we see
Dk
that
a prox
pn- i nmat
ion
:is sm all
.D.
su fic:Lenti
.-.
Aloha is
unstable in
Thus.
y
lari
k must
each,,islit
be
p must
a cl.ass
si m1
for
be
o
a1
in ormation.
feedbaci
anyrt
e
and
f.or
ao
l /e,
k no matter
this
(4. 2
P mu st be s
.::>
smaller
Dk
is
i:
T 'r
e
si
a'
X
..
1 /e
if
ut the app r-o
all.
I
when
k
positive
s
t -o
is
i mat ionis
1lar..e
k:
is
ti .
-for
all
chiosen: so tha't
how p is
case even
delays,
w...henever
i - posi tive
the next subsection we s-mhow that
and presumably
,: on tlhe
The problem with
'f29]
have analyzed
ca, n be chosen
t
For
= :.,
s yis-.5tem sta5ble
mi(r i:niz'
I
In
we
or an appropriate value of
,utda-ted at
t he
almos-t obvious--
and either
s p and the
p '' vir'-
:- !he-.
r e-ultino3
-'riom
-r:- i
'hiis
unknow.
feedbiack:
which-,
t;h-,a--.t su..ichi
ima:ei t
t'hati k is
I--iajek::
and VanLoon
'ion
packets i n
Dk, which occurs at pk+.;X(-l-p)
,from the
et-.tim a t ed
air.?
-
p makes it
p is iarij..e and thus delay is small,
hiand,
estim;aa;ted
is
Ideally.
larqe i.:,this miamintains a t-hrouSqhputf
otheri-
between
delay
ied reo ion,
large delay for coli ded
small.
1 p ca u ses
adj:st p to-iminim:ize
which v.:e
rate
lar-e
very und-eirable;
very easv to ernter the hunstabl.e heavily backlogi
w er ea
and
pac1---:e.
a collided
in
as X decreases
L p means i. ar-ge
and smari
decrease,
This tr:adeoff
increases
slot-ted
known.
much
higher
-hroughputs,
are possible when newly
arriving
-4,7pac.lets.1
a-.re sometimes
'iioh
Slotted Aloh
!no-- requ.iring
all
many phi-v-ical
muitia accs-s chiannesi-
a collision
uish
slotti-ed
that- this
:~,how.n that-
slo-..
tuted
thanrl 1/e(
.
he:never
mo
rdel
oe .ed
tihe
ou.!r ass1,mpti in
t'
Iha t
[1:31
ot
eac h
h
' 'li.
:m-.di.t
n
:tateit
a
dispersi.ve
from idle
chil-ack,:
is
,to
or collision,
suf+icient
and
for
tihe more sopiisticaLted
much
miore
dif
ic_.ult tro erstimate
Cruz
[4.'i].
hnowever,
-for throu. hpu-s
ican be modelled
rmat
ioughion
slot
passed
tihr
capaci.ty:
and
as the
a
inas:
.1=-ess
idle,p
discrete
the ca-se
ahbovie '-an
wayv.
was the precur--rsor
slt'o-ted
pac ket:
svstem.
requires
EacLh newly
up on arrival
!v
an expolen -ial
y
coll:ision
it i s
Fior
usually
feerdback
for
eedbark
g:.)ver- i. api. c i. pac ::e ts
afi:terr
transmission
i
4we h a ve a-- sumed
a cvr-lic redcundanrv
of positive
i n thi--j
F-Pure Alr.oha
a)
use of
is
Aloha ccan be stabilized
c ,annel
,
and
It
i. n more
inantage of
nauJSh.from
a n idle
dii
type of-ieedback.
c i t'F ::.nsi
or
memorivis-s--:2
I,
-.
that .
tf
kind of
rith this
the bac*kloog
as.ul.me
-fic. ut
Unfortunatr''iely
has the ad v
parJticiularly
Al.oha but not su.fficient
't.. .ate-..3tem-.
ob
throuih
a success.ful
it can be seen
· i . - ' -.. f
h h-owever
with high reliabil.ity.
strai;-ht-orward
d:istingo:§.
d
are resolved
infnormat-ion
fee-iback
c hannel-., i..
t is
i.no
t
the
colli -ions
up and
ra,
wavs
istiica-ted
iheld
c. :ili
de,
star-ting
anywiere
at
nre thie
than
i
ed del.ay.
in
time t
interval
a slotted
will collide
(t-
lt+l).
we continue
nf
as
' sumi ,
and backl oaaaed
.
i.s hignher here
transmission
that
arr I ved p ac ket
av-oiids
unit. for tran'mI si:,:i...
o n.
i m
and
Alo ha andf:
slotte
ait'horgh
onr e
di stribut
startinO
0
z
c ,
ions
i
t rar ans
i
mi't t
ac-et-::ts
ar-e
tiran:-mi tte'd
The prol:-a:i.li
syst-em;
a
ty
of
packet
with other
packets
The upper bound on
throu:h,.-
h:oui: becoomes
arise
as t,-r
pure Aloha.
die-,erent
In
-
is
however,
ALGO6RITHMS
slotted Aioha,
is
upper bounded
by lie
were a:lwa/-s transmitted
and
arrival.
thi-e
tr-ansmi
iton f
cofnrsi::-:er
a slot"
arri-vals ,
were netl
ru unt..il
c,'ot-h~er
:ri.
t. er!'at,-
p
i en-5
.
'L
si.:: t'-
iy
*5t?
t
then,
1
prbi.
S
i- n:iio
W':i.
w-..'.
f.
e, t ran . f
i:]r-oa
io'
ti.i. i-../ 1/2,
t e d
aind
in
sn
th e
until
'The
resolution
yields an effective
period.,
wh
t":hen a
coll ii ding pa.cket:.
in
i
this
a.i . ot
to-
tv1./2yi
the
arn
slot.
ca'e,
enn
:tre
W:i.th
ur-s
c
another colis
io.n or
an i, Jle
each pa.k
ti
n
n.
v
et
tnth
w.
the two pack:ets ar,
e..pec(teed
y transmit tihe two pac.-ke:ts
.to s ucctessfull1
which
no
idea
.ole.,
the -f;ol1c.w.p.n
in
fo1.] oni
wi
f.nr-t:h
'
. Y t ransmitted
.
were
H;..A
ain,
lot.
ive
If - t(he newfi.
rans ision
1/2.
tbili'iy
]1](:i
s
intit.
nsl-wiwnith
oprob-..iit
su
a
j
their
af.ter
a singile
cn1 J. iide
ts
each
be for
w.would
w :itH.
wL.:.. l.b Ci
sC:-
the ne>xt slot
rcollision
fi.,oultd be tran*smi tted
.ac'e-t
vei
the
ofallog
ti:he
; :!.1!.es'
/2,
new
should sometiimes be postponed,
tlw oia,.
,
packe
whif'h
in
nmiL:.
tism:n'(:
pr - b ab:-. iiy
of collided
To get an
new ar-i.val
reaso~..na.-b.i.z.estlrategy
re'.
].n
retransmissionn.
.
the strategy
iupondepended
back loggled packets
that
p for
parafn-eter
of
regardl-ss
the
imposed by the restrictrion..n that
This bf-und was
arrivals
we saw that
n probability
used to adjust the retransmissio
packlets.
to handle packets of
.
our discussiion o+
throuigiput
issues
advanta.]e of
practical
Ai major
ability
its
iencths [-2;0,-1]
st-abi i. ity
annd the same k:ind .s o.f
systemy
the slotted
FSPLITTING
4..2
(2e)
throutghput of
number
is
of, Slit.S
required
to be
easily seen to'.,
2/:3 during the collision.r
This concept
of
involved in
pac-,kets
non-transmittirg
idrea
!
of
achieve
thiro.-ih'r
we call i
these
:or
a:il owing
ti r'ans.i-t_
of
la rer
.involve
MIi
t et khai I 'v
i
[I]4]
i
et weenri-.w.
tw-:
cc::llis.ior'i
C:::),t
3,. iLi,
f·ir,
on
.i:
m.
We
r::,::r
o.i -. i n:
i1o
t".:"'.,am
i.
'.'.
t
all
lmode
s
v
g.'
:
.+ r-ir h .ri_: i.- :o i
i '
e
t
is
is
1slccessrful
or' idie,
in
t1hie folilo wing slot.
tihe collision
thietree al
of
t
on-].r'i-
I'ticrn
general:
ii.-!
s1lot,
of
t'hin :
u bb s e ' :sw i t -,j ef-::
oif
.
t ie
. r .n
?..
i-c
i
S
h
.
l : c. f u ri_!i-le r
T'he firs't
this
the subsets is
whenever
.1
:a.-i
i ti e
4.2).
pi
and it
_.
a b-ra
*)
'i sseu,:e
b:.e
:e
t he
branch.
next
the netax
nvl]
( ee
ee f.
sub s e t s
moo.de=
-gro irto
!
in'--i
o
f. i tr s
and
.ltl.-ernates
, .i..
c.or-r-es.-pondi n
tr-
iithe second
In
thee
sY-stem
..
.......
c:c. i. is:li s onj occr
i1"l the
oiri itih 'ms
and Tsybak::ov
r
pac
:et's
all
thlh·e
j--
the rul.es
in
a corllision'tLo
in
.a.
i_
as
tr,o
These
.wait unt-:Ji.
'-Liews..
tL o s .::sa
i.l .er
.I.
tr-anrlmittetd
ithms.
new arr:i.valis
b:i.i ar'y
hEtt
r
?. I
ode
o ne
th.at
assumotions a)
tral'si-..
' i ttrs
haches
. rQow:i.G
g fr.'rm:;inal
the or i
subi~:e'{:s
al.gorithms
ali 'i
land
s
an
.n
the
ved
ac
mis
mode,
i
ie
i.-
ed
Hayes [33],
ns im
nrmi
iiC:3r-inmai
u.
re .iso
1]2]
ti 'ir he r-'oo't:- a,': a. reoft:ed
s':.1::,:
c t
t:)r
--- nr
i rIt
fitd e . ].E d e
.i.
i nvol
.-
and a
is
wait
f-or splittino
used
thiese.
allionritms
:tm
|
* .::i d
reso.1tution
io nI7 .'i.i
i.t
'tr-ai.niisl-.i
agor
al.gorit inms were
a..:'C:]. i si on
:3. oc-rs
1 :
When
proi
-i:)a
modne
ngi set
resolu.tion
sp
littin
co-l. i si on is
In
i.......
transmitti
mor'e tihan two packets)
:and
by-, Capetanakis
(Af-
the seht- of
1/e while u.-inc
pack::ets not
t.e
splitting
other packeits
colljision
the rule-s
in
The f:i.rst sp:itt i
developed
into a
tfhan
algorithms
waitiing
a t. er
sion
while makin:ii
s
h mi_riht
(i'thi,
i
a. variety
aig orita-ins dji.ffer
set
a col
seet
-centra
e)
probabilistically
o'f
these
tr-arnsmis.:.oion
transmitted
thie tranrsmissior';
o-f
-4.
a subset results
branches
1of
a
the tree
transmission
is
in
of
a
transmit
subset
onel
mode is
again
resolving
eacerly be awaiting
will
re-sounding
collision
packets
oT
the preceding
number i oe
pac.::ets-.
br-anqcihes
en'sue.
rrisino
t. irou',:,u'i
set
-
(.
incre.ased
even
is
to
period,
is
in
is
that
number oF
mode;
randomiily
with thie lenqgth
period s o that
of
onen
the expectei
'The
rom the root and two
node.c
algiori thm
or- all
attainable
input
has a maximum
ra tes
:less
t
simple improve
The first
split
slot
an
with tree a.griorithms
ment
into
two subsets of
following
was
first
Note what the algorithm does when
is
the
at
resolution
packet
inicreasess
this
stable
that many
Thus the
a new collision
on the order
non-leaf
a collision
empty.
1.
and each waiti.ng
(0.46due to a
[35].
worse is
by eliminating the normal
sih,_owed
: an d
by lMavi-xssey
involved
the first
is
kp branches ri si ng
from each
mode and a
ollow until the expected
7.4~ maxi:...
T4The um througq-hput
suggested
What
resol.tioni
has
a .petaaIak:is [1.:2]3
later
many new arrivals
the return to normal
per suseti
correspond:i. n... triee
exhausted,
algorithm spends many
this
h-ie n umber-r
collision
subset
used to
subsets have been
if
entered
..oins one of
,k:: k s:-ubset=a.
is
t-ihe
entered.
res olution
is- imme
diatel. y
t
so1
a subset becomes on the order of
in
a collisioi-n
period
the next
Whenever
(i.e. the
successu-i
When all
will
1
algorithm can be improved
if
packet) ,
will
e collisions
sucessive
enld
idle or
collision, then typically
a
sp.it and two new
flrom the old branch.
should be apparent that
Itr
is
the subset
is
the nevxt subset.
the normal
slots
collision,
are grown
empty or coan tains
10-
the collision
the
which
is
-4. 11then
is a
idle and the ne-xt
in the first collision.,
collision, involving
all
the packets
Massey's improvement was to avoid this
predrictable collison by resplitting the second subset of a
set whienever the first
cllision
The ne;xt improvement
and somewhat
and
later, with7
Miklhailov
entiirely.
since
is
the tree
First.
al
nuimber
orithm.
ofi
packets.
'ubt- hen,
sets,
pac,' kets
i
t.-us,
in
as far
tati.-,iisti,-al
aid
t hz
'Lt
1i,
t le
l.y
we view it
of
collision
nnc
get
the
two subsets
is
inroved
i-,
e ntit
The easiest
waiting
If
all
packets
it
the packets
by meanrs of
that
arrived
of
e
li
th e
ti
ng
of
distributed.
new arrivals,
than
waCting a
wai ti
ng
ne w
how to do this
sec ond
r
subset
is
and
subset is
issue of
divided,
as a
then we
to consider.
way to do the bookhkeeping
is
this
t ime the f i rs-t s.u
Lbset
more than two subset.s
subset
the number
PoiF-. -on
ra.ther
t.; 'the b-ookkepirna;h
Sepa rate
have
in
to ;0_some tinme int erval
v
be
bu,-:d
o 'ite two r-esul
concerned,
but note 'that i-f we eli mi na
te
never
FPoi sson di stri
on these colli.3si ons,
.hortLy,i
every
modrif ication ot
for such a
the first
we simpl y tr eated
s er
We will
as another
occurs
in
as the algiorithim
i identi-ali
later,
than the tree
contains a
subsets
cond-.it:i onal
algorithm would
ar ri''ls.
to analyze
occurs
-second
structure
the ewid of a collision resolution period,
if' a
col lision
the tree
algorithm precisely
easier
the kI-niewly found
tihen anti'lther
i
At
analysis? by Tsvybak:ov
elimirnating
describe this
however,
empty..
was du.e to Gallager [36],
in throughput
involved
considerably
al-g or i thfin.
is
a more complete
this
We shall
it
eac.ih of
[37];
suo.bset
the arrival
in
a
given
concerning
times of
su5bsets and
the paczkets.
time interval
are
i.s
- p:i:i.t
two
into
suhinterval
iir-t
a
in
alw..ays sent
time t
packets
that
vi,eC4
sizer
t"he pack
ac:
e't:s a:-one
tr3X~qS d..!ari
'i ndt
.
....
..
atFr: a
.i1r1i
me
l ' ime
I'i--A:t
,
a
ortI-
i].f:.1,a'.
. r.iti.
an
tr.7-nsM'i '
the wai
ni-
is
te
hi n o: t o do sin',
best
.s. o t
'-ino
p'ro"
o .ab
th
al.l o,:.s
a
hi
i;
r
it
a-
i(t,-E:...ii2,
anrd
i;c:
t
wee
'-
cn c
-ha
or
ir
si si5....
e
is
.in
a
orc)the
tributaed
ept-...
s1lot
i
t
i
qLuee
5.a
.s
tf
t ihat
ssoi :.sh
t e.
:.-c.i on
.t
ne'::
i.n th e
i n:har'th.
no:e
int
ri.nO
s*oii
slo'i ,
f:ut
terms
:
ts
attemp
one
ibut t-,fhe
the
queu
..
.
the
to
t he qu ee ued
I
a cn'ci
s!-uc ce
in
based
dL
the
succ:eedi
the
and
help-ffu
:... ,
..'F.
t he
given below determines
it(r1::.
.fteedba!ck
arrived
,e
jt
als possibl
v
se-e-n
in
throughput
he
for the sl.ot Ett+l)
the
!.
i
nac V,.:l.=-.
i.ti
o
eni.
e
e r ron tt
at t .
i
IMit
n]hus th-e a ].go r.thm
'SS
le wi th an i dle slo i
i bib
o
Th'xe algorithm
1)
....
l p ac(:ketI
hat....-
:t i.:erva-l
t]h.e
i..iii
i:art+nron]'i.:
r an-,.
W.'-i
r
.........-..
einT
ts
at;r"t
t.
li k e -ito a 1. 1ccat e
We v,outiI
4,3)
tjr.-
;r ar' sm:i. t- ter
,ma.'..'
i m i .z
[ s
'Tt
at
transmitter
by= eiac-h
time
.-(to
ee,]bari.::
the
arri-,a.ai
-ffig
queCe c-.ee
that
L. .e cali
nhe time T(t)
are transmitted.
(t) are determined
-_lf
ert
Sup.pose
the packe-ts that
all
ttt+i-)
the slot
ilistor y
on tihe
s
(not necessari:iy
some time T(t)
before
bet.4.;een r.(t) and T(t)+F(t)
the interval
ordrF
are
transmitted all
has successIuilv
the algorithm
arrived
In
inte er),
(iLFC S
precisely.
express tihe algorithm
integer
piacik:et s
approach,
serve
t
the
in
algorithm.
a FCFS splitting
We noni
fir
come
ihis
With
t.ir Si-.
the
tubset- and those
ir-'-t
as the
are regar-ded
int'r--ival
in
and the pack::ets
suibintervals
'eq,,a.l
then the
a coili sion resuL tsr
and
the second subsI.et.
second as
this
a siot
in
tir-a-nslmitted
p(t)t,
of
r
T(t)
'
p(t-1)
[t-i.,t).
annd
T(t
Th.e
-
)
-
state
Q(t)
cr-n fideration.
,
_
'
Lt-,t
I
has been
'Ti'
r
otherwi.se
resolutio.
.:imum s tablie
intterv-a L
thata
-iaor
is
w.a
i
foar
itin,
itf
i -fetadback
1R
i'f3:
t:
1n
--.
as, o
a i 'i,': ';:'
'i . : c,
i-
sb ib
i. :_-.r, i- r vl.a i. .,
e:5.:
i sten·ce
·
-i
1
2
..
aof
=ubi
nt e.rvi a
th-en
-
(t
:2
these
tw]
t
s
":orrgeft
"
iabout
part of
the waiting
z.
'I6.
4*
.
t
;
-
a
4t IL:
- t4
t
. +- -. -'Tr n[t -a1
--
,
thei
it at this
queue.
f
Eq.
nto
n Lt
t o
'is,ia
( t-1 ) +2
.(tl-
there
f:
(".44;,
two e
J
4 -h..
-e
il.-
i "r
e r mmeth er
h
t
w.as a previous
the algorithm
regar.rding that
pointed
!'-'5
..
-vi ( t- I·ii
If:x
point,
As
1,
s.
subinterval.
t-1(
,.5....
f-!t)
lILi1, nin
=r''v'.
t ( t-i
..
-
rn
tim=-s
a rrival
_i.
f
,and
ni.(t-=
F;
..-
-
on
- iJin
~~ ~ ~ ~ ~
rIti-'
o-
oc ati
i:-.-t3n
thV-
;
· 4..~~~
.-
the ai
i n ter-,ai
L ad.t-,
r-
-!.:i-pt(':--'5=
min....!i... , --,t
T(t)
iT( 3-3= T~t-)+p~t-);
-I
....
thhat to achieve
at
N teth
2t
on
ien
-
f.D:eedbac5:
=
.
to
ra1.nsissmioLr n
( i,- = 5' .t,
.'+-) =r
t
s
It
t - him:
t
eedb f= ,.:'-:
"
and
a-ter a cllisi
out
It turns
=
t !
_e
t - i..
to be used
completed
iy
the intervals ,-or s.
o,
iunder-
a p.arametera".t tlh--ht determines
has
interval
t t i Ini- ,:mi a
i.,
.lanl
f
l:i msited by t-T
-(t)
rt-e still
F'F
-1 S
i thin
crtrrentliv
.
one
i n--to 2 - or
tir 1-ui. oput
.i-.,
is
to 2 i
set
d5ivideid
period
n
i s
alLca ti on
c-
-ithe size
Ta
1ofh
sub-' .
i .-e.--e -In _-.
thi-e nLumbrl,
Qr;(.-
out before,
subinterv..l
the
number
.as
oSt
-4' 14packet'=i.
PF'oiss:n
is
sub'sequent
8a.
o-f the queue
head
t i-
sn nd
rr
or! a4.
i.s
r-vI:
.. i.i..-(4_~ .-/!<.:7i
nr- ithn
;a!o4
w hen -
i d lee
.:
J.s
col
i l owed
The FiCFS
homo
I a. e.n , o.:i-s
'mt
t.,"r
tie
'
w,,: re a
: :i. :-
t
o,,r
!e
-
!..
c-i,
::10.:1
]. otn
::i
-.
of
t_1
ent t.he f ecorndo
l
. i1. s i o
r
"i 4
a
-3
ef..ct
....-.
i
ts-
I
s
,
i i ti
p ic- i
, ,t
i -:! r
J.s3i.
C:..
-i:
ori··
_is li:
fo
cha:i n
ar't time 't
s tate
at
~i
51l
time
re
ti is ' r
i
':
'i..r-"s_.. p: :F
t,-I T'i-t
I I ii
.* 4
d-'.
so
2
Ivte for- some i
t
byj
desc(ribed
i
where
o'. . -c: ii o n r
'i.
r
i.
=
r
:, C
'
satet
ea
i
r:
: 3.L..:Y.
to:IL
s so
iS
i
; ie-1i -A'
. nn.
n,1
rier
I,.ti
i
ios
r.
is =,; i i" .t n :i.. . ..,....
the
e
T'he st.te
!(t) and
Fi i ure 4 4 shows the posQsib1e
a-_ `!
...
.
t:i:
'5'-t,'lh r
_ i.
and
by Qf(t)
j
evea
nij,
de '. s-.Co i- a - j
.3
-
s
s pe'"r
ee 'i 1
eaSchl
is
tree
"IC
nor'r e.pondino
casee ar:.e
'P
i.,
r r-o
r'!
f3
r' 'l41t
~4lt- , nr-t;"i
i
...
i-.f..
ta] :
s'u interv
....r' ,-'
i] o 4, -i',-.J
.'j-.:
e- o e'n t:therwi':e
!,. o-,.
;_d
3 I t o ,, , 1 e¢:rno
's
i o:i.
n,
r eso
n.::i].f.,::or' '. it -. .-..
the
'..t
r'-.ee.,.= w...._,here
.ibu'
der t-he~m, period
(.:or~nei ote
;-0-f
t-Tbeinin,
t.i.-)
r',end en-'t::.
· id e-n.r.-ir
:=
nr-,
per hlps
'i-t1
,
hn:-t t.
.,...
'ni: s!e O
i
i
u
.
a'-s a
i t ? i
-'t|.,- _4r
o_j.
_ - r±l -
r
(4.6)
second seubin'terval.
r t ,
i
i
I-.. -t
Triov
-.
onf thEli
.iran....
n
(o
i.die
I
t...
_
,n .i.
ino
r-
can be anal
-id
'Ci T_
,...
n
firs:
i
imprtovemen:
rl
the
no
t-'-i
---.
:=-.
l
!....s o.
the
of
i
.ind move
ch a iv r,I,,
'i'I',
I:
v'
ence,
li i '-:
ion
1 kon!'
l
i mp
l.
Eq.
inter-viX.
on ta iiowed by an
e:
....
a n'-,
.mis
by spi.ittring
ir Ma
kCi
..
r
.or.
i..to:.r
i-= !'
all1oca.tes
nJ. 1anor:_~i
'.-ii
tt in
.=.:~..,. .s.*t':-'1.!
~ r) ._ i
te
....
a.nd
il
hisoryt
r-els .-t t
no coi. ision. alnd c
d with
per-i
tO
Ma.ee.
.rrs'-·es
nnd·~.
.t.'i n--5p 1.i ~-0~.r
r ; 'e
,^
c- iJi Tion
i
the ends o!
±
coil ii i oa
i_Ci
L
.Fi.
-teedbrc
the
X.U (t - 1 ).
.. l_ce.-i:
fra.n;
s
on
c onditional
'ho
('4
.- .l1"' -orr'ee~nrd'
pericd or
r-
,=--
paraa
w-ith
E:-:
the
ubint rval 1
tha.t
in
the.
we d:eno't'e
such
transition
of
,ne
arme
-the
pi(t)
that
eas
-4. t15di':imned ' by'
4 4)
4:
t. i
3o.ii'?ii
l1 eads
1eads
-a
1r-r
to
to complete
Ini
prbabiiis
,at.
t*
te,
;eacn
j-'-ize
h
i ai.
r
Pii
sie
the chain
c-
The
is
;X..i
rconditional
on the sum o0f the number
a
tIi
being
o
_i.
isrt
ihe
ths
gC
two or
nmoreL.
number
is
pait - i
of
; j.n
with para9.
mfi-i r
iin
packei-s
The transitiion
>-actiy_one
acces
u
issuin
s .nterva
the
hv e t4 wo
o-f
._btinterva
iiont'iiains
ir-st
transmission
p r ob a.n
C-
From
to calculate
P'oisson random variable,1
tie two sulbinterv;-als
i-f the
is
we
e
sLbinterval
the
or
iasres
i
eai.
c_
oc: cusl
an
Fo
remains
subintier.'ilvas
ii
i
that
transitir
From SaIwt1hereas
- rEds
S..i
S.-.
...+t*
i :i
...-
H:i
4,7).
.idie
t'Si
(i
pa*ic:
,_e
'
Te
thien
L 1. e
e
. f'.-
'ini
sta.te
i.i-C
't':~..(.SrDirtb.ntei"V..;i].!
,
-sub :.
Lt,.1val
is
tPhe sum be i n(.
.:''ic
1.t.- y
se
fi'd
one
ex;actly
)'
i: . i0 son: i
,:
tin
"is
is
of
sire
pc-I
meas
t-
ter
so
-ra.
nsmi tt
pacne-s
nterval
nu mbeer
the
to S. .
o:
'.eton
. :e
i
cond i tona.
.arae
.i:
:First.
tihat
t
riP~iji'
umber
is
then
achI,
bc'z h
oni-
-oLntaini:nr
ot
on be i n:oI
!oi
F.'oi 'sson cD ndil i ional
t
a transition
orne packe't-
abou.:t
i
TheF
m,'z-re and the
p aC:e
sui ID
:i. nt'ieiva
Th-e probi..bility:
of
:eachi-
two
4e are
1.
paci::ets
i e
ofrf
in
mol-re
the probab:i..i'i.iy
tie
-4.
1,--
I -e
of
the probbiii',tyI
Fin.aly
from S!-:1 to Si
t ra nsition
dir ect
I
,
is
P.
+
"'he numbaer
Te
oT:- slots.
'Thhe i1 queue
,:i]
'.:i.s:i.
on
t--T (it)
peri
i
resolution'l
1'
c.,eriAOd
i
ess 'ie·iT();
change
i
redidue-d b-y
Ibut. is
the5
;=~
(.V.)
in?
=C5remen'.
CaIl.
i...
in
ev
it
l-rI
t:'..']r" e.:-h
.-
ke .ee
s
now ..
o .
. ai
in
tis i - re
perifjd
oJf
thile alqori thm
period
.:::* ,,4S71,
a
=
.:
ihat
i fhe dr .ft
i
i t
is
la
t:
h. .
periods st'artingr
with a queue
the
o¥f
Thi
i
J.
alg 5ror
:i.
riod
P rl '
s
4
ae
mum
i.s
thif
is
a. b
i5
d ef
inle
colt 1 i.si on
:
-y and
sion reso:luti on
queue
lengths
i.i
max i mu.m
noaI.
(
length t:-T (ti
i
V be
i
length
it
string o f
the nex3t coll
'TThe sequence
..
A --. i.n
0.
!
the
aF: mrst
M,:-i
X( a n d
mo e3p-e.:se. i
Inr-
a consec,:uti vee
with t-r(t)
r
that
i-le
ah
up 'to the beC1ginni i nof
resol
V
in
ettinc9
.:o i:
iie
s"io-s
is
i.n the que-?ue
b
.asi
p
ion
vwhich
c
and
o-.
SLi
fLni r
a
X for
..
in
t
over
a
in- "F( t
colis
.i.
a s
max-imum
i i on'L'
!
n
remenit
inc
to the number
over
peri ord is.
-:oore
the
ir-st r-etu-n
co ll.siIibon occurs
-
..........
"
runnlingi
a
i
the cJhane
i cal
a
c:
is-
hth-ere
*od equal
ien 'ih
ne:-ive
resolution
a
'i- i-f
beO
has an
n
':_eue
i
-
Si ncE-e
* 5es
en-er'
ngit'h
4
e1
l.'J.ision reso-lu-tio.ion
a
in I
t Isn'ii
of
uinb
'Lt
.impLi
i
at the
-4,
beriinning
witih
of each colli-sion
ai-n abs-rbing barrier
queue lercgtit
and
(the
pt(]
F'.-r r
I.ef3ngth
a moment-l
gcfenerating
cier-ayving
f:-r
period
i-
i n1.
as: an
487
X
*
in
a busy period
si
s
in
is
is
also
di.it.
st'r -i bUt ioT
ft
(;p..
._e 'i.t pier'iaps
-th l.h
v
.' o
se.
·
n
"
"i. *i' 4 U.:.:
p p"'~
er.. r. an-csp
i 1i i
delav
i.Un t.f
r
The FCFS splittino
ot- the possibility
and
of
on iprio
i
i , di tsir'
..
n::rl
.f.-3
i bb u- t
stable,
/2'
s:lo,ts
an optimal
more than
sl.t..
f
the n umntuber ot
y dec:av'i n
[here-tore
'. v'v. i e
-
a
f unct ion
-
o7-f
t' I1 ' i,t
dT
.W
i.
periods-tin
in S'i..-'='-"i
; -i..v
b
ivals
l arr
nusv -·s! per
then
o.
l
-i
f
thus has
,-
it
t.hat. the
,
F.:i
variai::ie=
r
' iior7n
there-Fore
pC
tne
r
n6
int
i
'.r
T:ijf'
ii
somewhat
way a-fter collisions.
two packets
~::ct
- suT
s.
1/f..s a ind abiout
be improved
ai*e
oac
oo '1n- I:ui l ..r
. t. J.i
i mu 1. at
--- J on
a
algorithim can
in
reso
!. .t
distr-i:i
as'iin.'berva1.
t..-
-. j i as
'-i.
:is aboiu::ti:- 5
inr'tervals are split
inuLmbeir
iF' i nal'i.
.'
4 ' .3".'; .
i
:4 - . .:nmcs
iJ- '. faa
If3 i-y
deli
v-H
,48'i. v':s
1and].]. ;-then icie,-3.
at,:,coi ¢'er~ '"
ihm i.s
feJe
::edt
decayv ing!7
easy
tne.T:
that
r
randomf
the
oz_
iii si,:n
a]. so hias an
1- exnn
i-,
ction
initial
(4.
10)
icollision resolution
a't most hNp.).
a busy
i
perio
wn.iere the
From Wald's identity,
e.f' prane.'ia.i.y
It
Tihe
distribution lutnctiion and
function,
walk
are identical ].y
and
(4,9) and
ver a
ol]
l.ows t-ia th f-he numheir
'fi of
bi-us'y
thie busy period.
_**:,:,
i
This
.
means. ithat
i ncrement
exf''ponernt-iali. v
forms a random
period
increment
Observe f-rom
i
as
i
queue
has anl
n
at the end of
.iortihe first
ex'epi?3'Ut
is less Lhan
: i/
resolutio
increfments are indeiendent
dis'tribuhted
p(t.)
17-
a collision:
s:l.,:ii
if
ti-he
Beecau'se
ei-qual.
-4
subi
nterva.tS]..
't.
T-syba k
are
v aniid
su
bi. 'i-r'"'.v'a
Recei'zl
s
V ie-.-denska
I37]7'i
i ncrea.e-
t_ hE
in.tere-.a
C ons
- i.der" E:.'ie
t "1-ie
mai.-imulm
k:fnown
iS
i
to
(j.-58'7
Pi.Fpencer' '
au(nd
res-ult
sho·nwst that if
to'
i
f'l;v
ne
n-h
eg-. iat:i.
!bi.-:: ..o
*R-.
r
si: :,'iat-ie-
i
a':E l
o:.ln 1. / si:i)i.p i:' .i'
r
·f eedb:as,:'-
tiherei
ar'sd
of
44.4
.itn :i s
smai.
also of
i_
each
Tie
-thne
tightest-
bound
[F46],
since- he
interest
incre.sed
co.1 iisi
on.
of
u ppe r' bounds
ievd using
particu-_lar
is
:ew-nterva].--l
i
any
4S61.
o- feedback
l
it
.
from the prin ciple
acl:
45.
b-v
tade
rm
olnfinding
cain be
t he
n,:::i. s'
spil
l'
J. 'i7.
i
as
If
a.z co
C
.
i nuer
i
t
to
then
results
paper by Tsybak:ov
[,481
li'-
anvy
the
e
-_hrogh
ihput.
of
.lgnroithms
are
+or
'.
1
it imes
in
i lot'
.
t_ h en
.is ]
t-hie
ii e
snar
be s
v
va
it
'.ter
.a
:'
r ke
Iistj,
alg ri
ir
Tihe
known
[35
4'7,
48].
reviews. manv variations
a variety of
other
th m
andsa
l .er
:
'frs',
re1ei
a row. on'd rc"-:
in
al so-
ieir
th
.
Si
s oni i
rand
thi-
b
C
pt
erneral
sti 1 not weel 1 understood., butv
is
also
.grit.
f r.s
an idle .
- l
3ii
'r-vyinn. the entire
feedh..-ck:
in
sc.F
ii Ljt t
l. ri:_b
p
.
em
e ni-/ number
lit
a.g -iti.r -ns
Tree
feedbacr-k,
number- of partial
reso1 ,uion
is
tr .ns.'in i 'i-ters
ln
t
secon jd su~,i
i:: i
4.3.
.f11,:-'F:evF
r:"J
' c C)
w.:i.']
departs
ve .aspect of I-Cf-
nois;y
it
ttioh,e
htihat
been
.,-hi(:.e(d"
si.. -:ept
.y b
tfede
-aha.s
tihis a
been
f
C0
pe
i nvol ved in
be
it
a4*nd
a
;4z:~7'
C
due tko Hikhailov' and Tsvbaio::o)v
i mproveS.m
it
ent
,ni Q' o-i
s
subl::jec:t
is
the amount
cire
as seyv
ugh(-iput
[42,1
- lC, 7
.i. .
iaion o'o-ore trving
ha s
[42]
r'!.tff! b(.'r o'F psaac ket.:
up
coL i i
ei
hput to
A il tihougch
s.inc:e
e ffort
t-hr
.u (i i) -asj ors
t
onie
ti Cir
440
chnoosing--the ooti.m
inui
mum thr.]ua
ma..i
aid i Humbl.et
rMcseiy
that
E414i
aind Fdi
,in:ern
ai ay.-'- re-- resol. v.vi!c
ti mal.
s-.howi
er i fpr._'vemert
of tiheo.reti:-ica
I
-
o
quite
Mikhai.1ov
a.r ii.j
v
no
i8..
Th
onr
a
r eview
re
collision
s, i1ssumptions..
-4
alno.-:ithms
T'-he so3litting
transmitters
to =:+n.'-
interesting
to i n1fvesi-: l
a
e
Flajoret
i [-49] - have develpe
atr--act-i:
ve bothr
feec-i7ac.:
FCF'S spl ittin
Very
which
but
j
ma:. ni t ai ns t he
also
.-_":
muti.
i.i
-1
f
a...i v
::r
or ace- ulI
simp
li ity
can be modified
also has t-hi.s
modi-fy
t:
nta;.C:vr
ile
ofn¥
sysi-te.ms
tnu.mber
e
firi; .
tran'sitio"('n
tbcti-.
s
:
5_.ause
,"!'"F1_t3
u.%
Lr
,[5
l'
of.
i-ErA
5R ].
i nos
r
a
nf
bete
]i- :.'-4
ileis;ad.'e..
.
oe
ven
i r7
'i-i'-r':
in
thnttsisc
cabp
i.
of
.iliiy
users,
it
is
orithims so as to
t-r'ansmitt'P+'ers
sr)
rp.i. JtijT
the
l
las-t
come fir'st
number
[i..5:-I
that
- 487.
resolution
1:2a ln-A1I el
;'1'
o fh
is
a ins
sensino
t:
aprach-
-OaC::1h i n
as!7.
':....
int 1.
1r 'Tihei ap
test
0'san d
limited
te
and
and
has sfhown
intoa a
s'plit'rii rl
Sp.c:i..ic
'i._?.'d: .'ii(- S. .
.i.
no
thes-e
only
capabi.lity
n
[=55]
a finite
from collision
arl'-i.-,; J. r'-ate Inle.es
ds i
wii.h
is
r
i=.
with a ma-.liMrunM stable
sensi
imb.
et
samie mai murm thirouo
i
iacc-ess
terest
Hui
so it
Mtathys
send
to
all
trimes.,
,al
and robuistness
y,
recentl
algcorithin
serve al*orithm
pac:i
an aioritrithm
td
at
.whichsen-_in
in
this limite d
for its
Ci: ierrors.
For
has a
4 that
o+
Iha
feedback
ai,: oriithms
transmitter
after
)
discused so far- require
the channel
required
throUoihu,+1
19-
to
and
TDM.i
this
s
.; o
f
kt-.
t!e
as
are
i s teraJ i.so o
jn
tr
-ir
e-
: ii
thm
r i .t i i-,A4 Cnill:7'5. l :L
e y
S
i !
antd
a;
;C
'.- .
t.,. ,',
, ,SENF:,
s. i HC':,,:
rinot the other
i
7....
a-
c .spei!
rCrSJ.,,r.Fi...
i n ....
"-i1'5tic
f4, i ir~~ t
t.
.tr
caEn
i ,I
jn
Not
· r'. so '.on,it I
en d
does
not
the..
i:
b... .i-
nece:r
. t
.1
i
?
icldn]
i.......
.:
i.mo. vt
he
n
effectivelyan
dn.htiiir
..
t
~t.
,
.n
...
H..
arc
t v
i
r'aci:iJ 'e-d
'..n.u et
........
cu.rrent>.
ermino
use o
-A
the
infinite:? numriber
-
of
en i
-~
b tsense
oy.,
,
carr
.er-
sense_,
C a-ir
e,
but
. i. Ci i .I
n
!.
o
O±'
e
send:.
-I-ter
Carri
i
c
n-
t
the mi dd' -
i
i
s=o u-r
ur-c:s t to..
s
"t i.. b.i.
.. : dJ:
:
h ri t-sd ' th
.
th,
n
:
...
,
mo
,:i wen
be1t :i.
..e-' ets
rre i i -'.ii r
t .... J
I :i.
in ns:cictenat
J.
s,":._ch
ts
to sIibe
onst.
ofc, cot].'J.
].-s.ri
r. 'v
't
tmri a.- i u ,en-d,
at. .i.i
.CJ
(::i.. tt .aC
e ·
idle
t.
number
Tihe
?i
me
in
a) ndJ.
rI
[5.
]
the.r' t
L'
i
b
,h.,
r
hej,-enstriC.t Cti
Tob
b a i
tm7f ro Ui
ir;.
_
si n
to :end
,-k--.I-t.
a i
_
transmi'~
tt.r
ie_.s
-this
Ci'the
io
r-educe
KI. c=i n ro ci
ti:.t.1?..
t-aGrIt
other
s,
t=o
,,
'
t
; ,rs t
f
mi..Crr.>.
h i.::t.,on
-d
;,,
"nI m:!.
a'!-.
cot
w.......ni .
·.....
t.r-rs.mittcr-
pr-f
j a*i-i-
Pesi
no.._.
We
e, t.-erm:i ne
1i.n.t,. et.: thb'i
n"d
b . .r-.-nr-Y.s tci..
ct i
Jsm
.. :, it
y..- .
..
c hi
ot,i
souirces..
We
btet:in
first
loEtted Al-ha for 'this
new situation and1 the.n mod-'
I
..... T
. .aJ.l
r. t h m
i
.
modifyv
the
1he
ma.i
or-dJ. nRr
n:
Jdu4-t:in
ri r ve=
r,.. './ a
C
;_r;'",
ine
r.e
1.
t
is
pacl. et
wi t-th
lot
Fiet
rth
e.
hs.].'I-h~
:is that.
in
...
. , ~: ...
,,
i
ec
t
t,
d
ID 1-1
Ii.. oh
w:i. t. i-
.n
s-e
id
thef.3
.o
of
1 e
ci. nt
T
ti. h-
s5ci
clt
si
,r i
p- obb.
ib.t
tth
ult u -sed
q:i. . en
ab f ore
i.a.
'ty
:-
'.
-:. a
ex
r -ec -.eRd
r
ji d.i.
:. ,-l1- f a ,
.
i vl.,
ti
tt:
hh.3e
i:t
C
i
js~h
tr
led.
::r-omi
,
two
1_
.
t
rn;.
wiB.
'I
d simn
r h
y
i. S.I i,
c,:.i s.r--.t
:
be
F:rIne
a
'i
i:srm
;
-i.
s:i
...
.i- .t
ni-
-
ens i.t
r'..s
tr-n
trh
:37cirn
i. .
i
tnc
st.c
t_
i
on
te t.r:-n;--:-i;.
'F. ,ei .,
t new 1:;i -he mode.
ittwl-; - i, t. fort
EFe:--' 0
hti·
,Cit
t 4- i[lie:.. ih
: n .:- t '-e i
: i.: ,.i. j,
'41i..rt i .. i t,,..r1
j:i.:
h
n ....t
9r-i-citCi
*t
r 'ncn:i- tte-
:i. .li. i cme
;- j
ben.7::'L t;i',i. ,f.
s'jbseu':enti i.i-td
, - :-
"p.n
ot
tne
jir
was cl
r-i-
t
i"
i.i.;,
C
...ercj.stent
.
b..:'.
'
4hi-
. Ib
:n
i a -- i;
i.r -r
7
rn:e
..
tD·C
i f..' e -nt
di
these i·r.. ;:;;r :- 'i nj et.;
:.1 no.re
e
tIn h
-.robi i.*i. t
ri
.
.
r o;.arr
i ;' p
:ot
'ter
qi.
ID-
i
i
Daciket
?
r
and bes
..
t~chni
"l"i-.
In one,,
et J.. on
var.
.,ons.
.ar:I.
i
*'d C j..
l
r.tr r-,- i...- i r . t
att.empt!snc dur.. sc,
s:
t. ra n'=-m.F i1ion
mt
:i.die
ann
inF 'sti
-['.:'_l
nttn-persi ct.rnit- CSt-ri
J..:_
s_.].o;i
ae+ter ;ach
d'3.n,-i
uu.
.i a
n
.
oged paci.::et
Iroabi ti . tv
arriv-:i.n(i.
c,acl-:etr
slc,:t;
,,h'
- hve
nrihw
lots
tr.....mi
~.
bac :l
as a
re.a.rded
tranin7 mission
:in
ference
clthtr
di
'T' he
cs u a
sit.-tted Al oh.a C,'.IM'l4 and
idle
in th t
Aii.oih
'.tted
,een
Tf
erence bet
'.r di.
i o
1
I or,,
t._,it
nun ei.n
n
l i
n,; ..
i.s
iiri,,
-i
- nCc ote
.h
c oS3
c h
- ere
f. I
iiir- F
thieni'
.
1.5!
h! htlt.
rr -
. of e ,,,-,
r'
n
ji,
the
vet~
c
In
*,,r.
h . .
j1.....p
.,''
s'L
.si.
:,
'.i.,'t.
:.
'e .,,.'3
ht.....,, yi F,i5.-.
...
nd wit-ith
,
i o.: L'::';Cd
el
ends
i7eri
-
b
c
;.-
1
'is ,,
v:2
t.o t
t
It
a
we
l.
..uct
Lhe
correspondina
arrivals
transi't-lonThe
model~~:
wias~3-e':l
id].;-~ si'~
.............
~
_ b,:-
s-. i mrq
-f-
p,
4--
fthe
4-
~-o!1owing
i4
i t
t.his
·
~Ie d._
n
nL~~mb~.'.::.~O±
departures
__ ....
p,?r
0~~~~~~~~~~~~~~~~~~~~~~
(]i
ol
r
rn..ton
ui-I
--
"i~mi].-,n,
small
s
l
P
p~~~~~~~~~~~~~
J., =,:,
-
k
F·~·:0!'" ';"tl........
h d nt a
4 ,7;
art'
cou·.l~rd
be cha-noe u ,_o e'limin!~at
e F..o b.- .'..DiI i .i.
.)'v i:
_
as
.l~rene isE
'""'"'
:_ed
i
p
01-
-.-.
ir.:
4~~~~~(
'hhi t~~~~~~~~~~~~~~~~~~~~~~~~~~~
ise mlnim~~iTC_;
...... -- C----fValiSO
P
it
~ as:- v~'i t,-h
dr
(1
~ rLFv sEort t ed1
.t hi
'L
e
ptima
.t..t
.,~~~~~~ vat
~ f~ ~p ~
._then sati..s.es
sn con
n:
.r
2......
I.
.....
hv~~~·...
rr~~or~~
I n C:;
i.I.: he
t:
Peedbs~~~~c:
.:
o
o~ne
:i.
Ien
m __:_"__.._-'...
1. C)v
' C.:~ssi: eC
,_F
m ~ or
F'
....
s3 sminai].i:rif',/a].'_
'~-',=ifL:~~::;
e
- ,4
th
ba. time
a
i-ckJ.n a:J,
·
;R I.--:C
3I~ a power
~
~
~
~
~
~
~
~~~~(~.
tI~
v
ex:pand
·
in
-J.
(4.
I
s
e+r::
in
I
J.
e
id
l-t
.- ! vi-F
is,/ari
..-..
.;.n.m.,el
-.,.h
(:-ci~.4nich
. -r i.s ....
p( +,'')3.
~ c~i.n
I p .:ar
Di::....;.
,_,:k
.....,, r....1.u~p
hpeF.
t h at
t h . b~
rars
.......c me'
t.c'o
:..~ l.=~r
ll~~
.t w (.:--,L"=.7 the t:pe t i~s
m a ] "~
r. ,
(:) i' i.i:~ ~re:.-(imt)
i.,erq4e
r --. (*.-r,:-:,
4-{l~
...omi
cll::-- ec- too
I
'.".
it
~!ueteop.I
h
o..,.iai
a=-.
:,·I.u01
..'; i;
L ±oreh
p
·.
...i
.tI
hhi
(;
:
n
h, s e
'
-i
C'T·he
d
thti-.fo..,r
si
'
' ' the
i =-. syste
:.m
n e{:~~~~~~~~~~~~~~.,,
vei: a't:.i.j~
c),
B\;'·
.e>'panrdirng
(4 1)
-
is
.-.
stabL--e
or at.
n a po~.~er" series in
trfhai- +orP small1 ,:,; the .syst. em i~s
. .I
h.a.n
!.i::..0''~'.:n.I.
stablite
+ori
al!
1 ... ,l'o:. "/he optima! valJ...tm o+~ m then sati.s-,~.1. e
t....Awer
,
A· :les-s
p
''-'
-{'lind
thanl
!c:.
lit
4 ,.
23
:i.
n.1
tJ
initerest-i
ti.e T
the
-
whertire
at.. s ...
tir-
:d:L --
cci4
_.
ps
!t..L:thoh..l
i.
:i.r
Ti'·..
.
-ti
shoIi
i..-.
;-e
:sp.
]. i -_ h
3
.'
.i
]
n
eR i
hDtavde
nit
Ser
an
imn
.:
3.- -in
-
he
'
f
.:,
the
G
-ic03
tl . o
a h"
ti
sbirterval.s.
d_.J
. p so,
n4t- .'.-r-..
rr.;
ej.c
-fl
ivi
" ..
nii
I s
orh
..y:_ n......,..,
.r.,s.
nrte
,:.; 4t-7i
tri,,,,
h
th-
Si'ce
,s.i
d.: I
ti
.
-
n.d..w
!
not
i"ons
3 i..i1t t:
c,t. ',. I
I~,fl.
O sh~o].d
.n b
should
co
'i h
e
rrr
;r '-4 ev
-2
s a..m od :i..!
1 n' i ,14- s
.-
t
Je
_nc
ision
i.t
preasei
m:i.
.s
;-+:, if
' t'i
(4
;
4
-'
t :)
mI._.ch
s:
...
1h~e -..be
h I"jr
.t
.
.c.l...
withf
'
e qL. .!
-....
...-7-hl
..
I.']
. t?,':J e
-:L. Sd
h=
:a.q.:-'!.t:.thuqh
prameer
t
i
..
e 7-fiCfl -
ti.
ide trsLFr= LfioFl,
es2 ii c. a
:h
i.s
a
-,
.spi f oh_
. t t . .n c,_ l.-' s- or it-
jFCF
ps.-etr... t. r,
l inter-v
rttl
t4
4 ... pJr i.
.
L
.
my
l ote sI
.
r-: .deIfi -u
r
thsn
:. t : ri
c-.
i..
si, ;
t:ra.en
'.L
o.i. equ,:+i
itt
timert
n '..: t
h
i
t
,..:r-s
c:-
c-:i.n
' i r.. tv
c~-_,- frrj.:i.j h ,..uh. incrE- . E
l - C,
...
4.
ons.i
I
Pfd or
(J'S-.'.
je-j...
trn-mi
.
pf_C.i
iidc:Le
d
, t!.fi
,,fl't .
t. il . tv .t't
-
sei.,j.
r r..
._
n
_
.m
h-.,dE
er
_
the.4
Ld prohbai::
qI
Co
4r-ft-i .x ..
.t
ar
i-.
j
_.E i- LtJ.l
r
,
i
-. th,.n a ,
pcrai-rrD .
p,
.
.1
* il.
te:yi
n.Fp3.
n ne r'"-.-i
.
r1 J. 7J.
..
i.n..r
_4
pr-
the
hisa;oci
d-nt. t
.., c
..
o]i
'
s:i-.e i. h -. 4-
w
-
rJ
s-
J
:'.
,I
chn(:lct,:and
trc
ste
.r
onAbr
be
{d.-
r
.
I.'"'.i. i,.-
t',
',t.i,'
L ta
s
w
hou.. ld
e
ch s
er-i
is
an a.ys:i.s
s:i
mpler
We first
S.LL-:..
0.,
i isi
i.o,
E-
;a
rnl1
or
i de
co
in
3
ei,.i
-
,
r
co
t t'
-f lot
p
I. oc.l
i
rn.
.-r,e
j.'
re:o.
sion
i :I. -
f
..........here
'letion
- .
tr
1r
. 1' o00 t ti:'i.
im(.=
]. 'i..;":.
4;.~
S:
al
V~ 1j
-i
. ei-i.!i.
s:.,,-4
-=
v..s
e .
fr..
e.o,
.....
I
ec
i:
7
l t.de
c '\/,, ..... ].n
j4
' l_
-t cr
.,.fiii I
~. i
9m-7..
t:\..e
now...
We
now)i
/
ic:
f.
*t
,
+a_.t.
.
..s..'.::. s .......
e't t.,
'
...
.-
(:.....·.
.
si
.
hnad
the
:":ih for
' ro:+
..
.
i
....
riqht
.
"
+......(.~,2
t
d.iia
ini
o
ti +
m aximize
e
?..d
I.~e
i. t
.
.,
"[Ond -': ,: i '
-h .aa'n
*T'-
.1!, m4 ce...
o.
an
e
.'.
r*.
i.-
o
e....
J
";'~
:i. |
I::,:i J.
J
y:i.ei.;diian
j
atr~a" o-f
t-he.... end
-,..
-].,'::
r oa b
r c ,rb a , . IJ. t.
.i, i th
iri
em pt-y
I
i
t :i.
a.
idct,'...{,din.
f
.
_j
roJ-
,
4-
h
a.
num: er
.
e-.dpected
;- -es
aerete
iJr t. . .1.t.
Lfis
the
4-.T
4-
e
thi
iT
t..i on
ot
thus
d-
t
de
:i.
t'
(..i
as
on'
i i ntervl
r
re
}-.,
iii
a nd
pen.:
4r
14~r.:..~d
peri
od
Ll,
e-,
probe:3b:i
end
c
i.' ..
.e]C.
a.
:....
neoJ.:iia3le
be
coi=.!-i
t....on
that
in tur-n
thari before;
,.
o.;
a~e
fmore
:Lfc,.ui
i
re:-...
i
n
This mieans
ti.-
-Findr
=.'S
=.i.
ll .
two n pl::et=
t.ha
4W:i.th
sm
side
)
.
I +
,o -F
( 4,T1'
o v4 1
(, .,*,
o.ve
:,....y mp
,. p"m
orj
t
5.s
i
U
a.ymptc,tic
*
a ter
ril
1is.on
En
i.n terva.
cm
iut-.es.
. si
.eme
:.=..
ri.
cG'..e:
u..=.ed
> i-
.itj i;..
5 te
i
..
.
.* -
n.
h
mu
thr-.hp
j t
-
3'
i.
:-1 t- ,n, --i,. .o.i.
;or
.f:irst
1.
.T
I·
Iff
.
m
. |n....
same maF'.
. t he
c
n
inor
C.-;M
::
1"L
t'h
+-i--':
t
, .. o t;
as
'¢ F" I::^:C;;si
h
.1.loted ,
e.
F-
o' :
ion
n i n terv!i
j.
wi th
v'
,u
;
1-he
e
co
:.
!d
t.h
t
... o
..,h ee
bset
:,: -. n
,
t ..
e
LrLetd
e
:
i-
i.
sn,,3
et
'
s.
4
'
.r " t
e
OCCUrled.
the
diL=
,.
.,_
ha-
in
e-san
rt1eF-f,'
hfsi
...'
.J
i t
onTue. i.
ri
a>
*'
3t
5, "'f ) ..
! 1rt::i...
e
i
FCF-4r:t3
.-- we q
5
e--pre ssiOnr
': :.=:'
!roLa
':
--
... '.n
the i.imi
.
- ii
....
.
-,:_h
]
:.*, ...
o '-i
.
* t-i -
T.i.C
-h
t
.:R
iont
.
r"
in
ioth
o
.:i. t :i.ih7 n m
o
Aloh-a.
ti-
i ./.
:.
j :i.q o r
This
is
t
h
not-
ttit
and
wih t
surpri
-in
equal:
col I isi ons
,f,,
at
n
o.
sut.tant:i.al
sam5fe
van
m
:.-: 4..
The
idle
or
(.
ttroh
,;..:.t
e>·ieted
e:-
time
i..:
wn:i
t
l i ci-. n
Ft'e ra·
a.n
o'
..t
dtra·t.ion
cI,
on
du..;e to a
req.
whereas
,:
5 t '.
forr
iThus
a-..:
I
T.
i. nj .^
io.
ve
in
a
tranl
on
i d.e
i
Li
3
--
orni.
an
t.hn
deIs: .
i dle
ei.isaeand (a ,
is
ationr,
.,
is-. the.
the same
same as
as 4t.he
the aove
above rserv
reservation
e
c- t .m
win.-.h:in
sutccesses have dra.:tioun
and
.L c,
;o'-s
en.
-.
:"
r
_
a
sF.ots
.
wou.ltd have
a; orth-inm
h t
w,._f h
success
ii r
.... res..rvation.
i.
C.SI! ,
in
.t
he re-,,rvat:i on-.,
c.
and
d Lura.tri n
.nd d l z sl
ol'3sion
i:i
rn
a
. d have the d urat
aredic
c,
or
,
ouri
caI 1. that
Ri
te n,,
o'm).
[ 53]
Qeneraiie
to
..
'r o.
r l'
to a
- var-ie:/--o.-- otvhater
It.
7 xtt
t-r
th'i-e E:.th.net
tThedn systre
limited
d ue t:o M iasse
and S=,.
i . ue,::
- os:
sti
and
In
:
a reservation
I9+,
=lot.
ta
r...
:i
.i.
t
alq orithm
nwif.4ith
to el.:to allow arbitrary
col
at
usetd here can be u sed
s5
ssems
is
somew
cnl isi
nson
o-f
b vHumbtl eL: f54.] is
eerva
es't
ion p.:acket,
Ctsa
durat 4 ,i.
ion It+o:,,.
I' t a;
t trar;
·
n the nLumTer
.idea, orio-iin aly
sl - ts had dur.tin
i.. on
t he FCFES'
< then we are
analysi
o
.idle.. success,
had d.:iuriatic,n
coil.
:
.
asum.mtions a)
ir.f-'cRr...........................
tL
.
i
T'his d e radation
type
-:turther,- d eveIoped
ft{or
=12:
mlticcess
ui
ati
c on d i ti.ons.
ori i.
j..:.l
ie,,
increase
tie
iu_.
: e
that
sub.iinterval s
throu. hp....ptt
reser;
tirj.;-at
voccur-'
rarely
and
.:stem..
c
t
X5
unita
i:..,ih .d
:i :;
,: th ,,
as
in
have
the throu:utto....!tn
psc,tem, a
REFERENC ES
1)
*Abramsont,
t'-.."iThe Alo-hna
Syvsttem--'inother Alternative
for
Computer Cionmmun nications", Falt.
1 Joint
Comp....ter Conf+.erenc
e
37, 197nnAFIFPS Conf. , Vol.
."2
Aihnlswede?
.t·
')
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n
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,
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"H"ACodina
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al-s:o
EE.,
"-Pful'J
Unii v
i-heorem
fo'r
Symp
Ln-r
ii ns:. lTnt
piIe Rc -eass Channelsi
i
19
.
I-772
H.ai
tul.piple Access
mrtion
Thn
A-sil omar,
PiF. D.
Tinesi s
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1k'97'
o.f
4)
El Gamal,
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1 98- .
r5i
va n der il ulen,
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·;Jan,
19'77.
C1.,
"
196i-9/6":
E
i_--iisz ar, I . and ..
K
hieorems f r [.:-'iscret e
.
S:)eci;.l
/7
Itssue
on
'. mmu n i. cl t . o n s
6)
F',urslet
ic- c k eiReport
91
:3.e12.pn.
Ac.::-es s
Te:.hi
C
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and
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f'
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I.
Ci an nels in
.
'o o . IT'-
Inform atii o n Theor-:y:
essin- Systems, Ci iap.
.prrctrum,
IEEl
Coding
:i::
; Acade
Transac'rtionsc;
on
2.
Sat-ie ii *
, in .
"F'Frequency-lHop Transm-isi.'i1on fr
c k e'.
tad i
Net o r.w.t
ks
S i t c i. i n. t
d
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Tera
r isiei t r a i'a
i"--i 44, Coordir.nated Sciencee'L:Laboratory.
Univ _
i 11
1)
..- 7C
1rner
1
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IEEE -Trans.
Meorl
Spread
ay
Sr
T-2'5,
:::
wf. .th
l Vtl
VJo
tr
f.
Wol.f,
"A' Codi n,
ei ateod
F,':rr't"
1
p'7-0
nformatio
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tn'.
John
i
Wi e y
i 9 68 .
. Wa. imeler
MuiullC
tcces.:
Chaln
.iEEE
nl.e"
H-.
t
p.
Theorem for
tu It i p1I e
S vystem
-.e I
ources
Sept 1.........
anrF Reliabie
n::, e
on the Ca ::)
T"lr-ans?,i o'n
.a
yit
i rul
4 34.
t2)r i moti os
S:
n &I
A"i0onrithm for
C omlpu't ti ng thIe Caparc itity of
t'lemoryless Channels": IEEE Trans. on
D.iscrete
Arbitr-ary
,i Jan 19'72, pp. 14-2(0
.IT-18i' No.
Infotrmati:ion 1"ibhery, Vol3.
B:lahut:,
Distortion
Vol.
R.
.T'-l
"Computation orf Channel Capacity
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pp. 46(0-4 C70
No, 4, Ju.y 1972,
E.,
and
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14)
Massey,
J.
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,.5)
tArutyunyan,
E.
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the
Error
ilultiple-Ai
ccesss
Channels", Problemy
Peridachi
Vol. i, ,io.
2.
' 2:p.-.-.6,
April 1'75.
L. and iP. Mathys,
"The Collison Channel
IEEE 'Trans. IT,
this
issue.
without
of
Probability
In formatsii,
1-.
Peterson, R. L. and I. J . Costello, "Error Probability
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Free Distrane
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IT
T,
ol. n
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6.
5pp
8- 67o
Nov. i98.
i7)
Gaarder,
N,.
T.
and JT..
K.
Wolf,
"The Capacity
Region
ra-
a
lllui-iple Access Discre-;ete Memroryless Channel. Can Increase
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IEEE Trans. IT
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1975.
Jan.
,1)
.
zarow, Li H.
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VolIT
.
-3
flyv
62-`;-u29,
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IT,
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July 1981.r
.ui
..NHi P
Pnr
&.
Humblet,
-.
"Th e
Totarl 1y A -ynchro
noucs tMultiple Acrs:
"i" ,
J1
Capac ity Region ot tihe
Chann' ,
I:EEE T rans.
i-ss'ue.
A. R., ,. A.
Heller,
and A. J.
fecininique fort.A-chron.ous
Muii
iip,
e
IEEE
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i"om.
Te.
fch ,
Vol. CM0f-1-9,
Cohen,
19 7:.
i
1this
IEEE Trans.
Viterb-i.
"
pp
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NewJ
Con.di nQ
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849-855f
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Apri
COi-l-6, pp 222-2:2.2
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Rate_
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Detect:i.
pp.
26)
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: Priva
P
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Communication.
.
"Capaci.
Cuttof Ra te, and Codin
n Opt.ical
Cihannel",
IEEE 'Trans. Comm..
16i
5-1622,
LNov.
,
fmr a i:i-rect
Vol
COri'-29!:
1983I.
Humblet, P.,
"Error Ex-tponents -for a Direct Detection Optical
Channel", Report
LIDS-P-1: 57, Laboratory
for Information and
Decision Systems,
M. I.. T.
Oct.
198,.;
,27
.
t:'pl,
28i
..- ChiaiW
.forCoflnd1ition
"A S uFi'rient
IEEE Trans. iTT. kV .
ani
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rac.ket S*ystem Witn and Wi thou.t Si1
an2d f.-ct rf- H-ob-c Note 8, ARFPA Network in-fo. Ctr ,
FR i
Stanord. Cal i I
June 1972.
Try
N.
B.
ul tiple
,
1hm"
P
Aorit.
pp.
65-79i
"Airi
P'eridach:i.
198Fi.
Upper
BFound
Frobtlemy
Channel
3c t-.
N4, pp. '76-84,
ACCs.-ss
1i3
Vo1li.
F
Fren
roblemv
Oct.
Liikhanov,
..
t
ov-
__-
-
on
1 9803,
.:-'
*3
Huain
i
a
J -CiiT. and
rt e er,
'Di
e - Ai - visi s ofi the Modifi iJ d
0in2
487 Contention Riesoi.r ution Alzzorithm". Cornell Univ.
iReport, submitted to IEEE Tr ans. on Co.mm.
Moseiy:
*.J. and P. H-lumbiet,
"A Class of Ef ici ent C ont ntion
R-esi ..iti
"L
F,-i
Alg:
or MfGultiple Accerss ChraTnels
i'
t
appe ar in iE
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rI'rta-., ,in Coimin.
also Repo 1r-4 lIS-R--3194.
Labo
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ir- iln'
r
trmai.L n and Dei1
on
stI e s
i, T.
r
-
March
LiDS-TH-T
].982--Reporrt
5 LR
I-o-ai
Informatir"
aarv:or
41.
V'edena, s k
N.
D.
anD M. .
F'insi-::er
'- on
- o oi
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of
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!Papers?
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pP
141-14'F, l1983
42.
Pippengerr
N
"'Bontis
'
n
on the Pertformance of F'rrtircol
iult-i plIe Acci -ss Bradc-ast Channel",
iEEE Trans
TT- .
-2'7,
MIlarfci
ir' 8
tiie
-for a
Junte
19i8,
4.4) -.
L,
"On the Capacity
i
of In-fini e Pr
pulat
o
tcce'r
roto~ol:rs'
IEEE Trans.
VITa,
o
I'T-25.,
:,-:--4()'1 ',M;
vay l"?F2
(s'...bjmi tted
l.ay 'lib8)
i-'-,._
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ltipe A
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Con -f -erenc- e,
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Many
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198i: ai1ro Laboratorvy -or
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Sy-tems
R eort
LIDS-F-i1 .'31
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CISMA Modes a niD
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COM-23,
pp.
V5ol
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st Ciimm. , IEEE TraIns
1i,'s
i-i ar ac'ri
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. 75 _
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List of Figure Captions
Figure 1.1:
Multiaccess Channel
Figure 2.2:
Multiaccess Channel With Two Transmitters
Figure 2.2:
Rate Region of Eqs. (2.3)-(2.5)
Figure 2.3:
Output y = (y ,y ) as a Function of x, w for Collision
Channel with K = 2
Figure 2.4:
Achievable Rate Region for Collision Channel
Figure 2.5:
Achievable Rate Region As Function of Signal/Noise A
Figure 2.6:
Comparison of TDM (as modelled) and FDM
Figure 2.7:
Regions Ri Where Er i
Figure 3.1:
Timesharing Construction to Obtain Any Achievable
Rate Pair Without Joint Decoding
Figure 4.1:
Dk as a Function of X,p,k
Figure 4.2:
Tree Algorithm for Collision Resolution
Figure 4.3:
Record of a Collision Resolution Period for FCFS
Splitting Algorithm
Figure 4.4:
Markov Chain for Collision Resolution Period
Dominates Error Bound; for A=1
T4
-(T3
rn
N
SOU1RCE rEN
E
SOURCE
CHANNEL
E
2 ENCODER
p(y/xw)
N
(Y 47 *ytq DECODER
I(W;YIX)
I(W;YIX)
I(W;Y)
RR
R
R2
I(W;Y) -
I(X;Y)
I(W;Y)
I(X;YIW)
I(X;Y)
R.
R1
NORMAL CASE
Rl
SPECIAL CASE
I(X;Y) =I(X;YIW)
0
0
(0,)
1 (1,0)
2
1
I(X;YiW)
2
(1,0) (2,0)
(c,c) (c,c)
(2,0) (c,c) (c,c)
SPECIAL CASE
I(X;Y) = 0
R
\
UNION REGION
I
II-
I
R1
\
R2
=10
(BITS)
(1.29, .44)
(.5,.29)
A=.4
(.3,. 1)R
A=2
1(BITS)
R 1 (BITS)
ACHIEVABLE
BOUNDARY
R2
REGION
FOR FDM
BOUNDARY
FOR TDM
R1
_0_.51
0.5
-~
(.3,. 5)
(R2
(BITS)
T)
(.5,.3)
(.238,.238)
E=O
0.238)
R1
(BITS)
0.5
I(W,Y/
'X)|I
®'.%
I(W:Y)
-
I(X,Y)
QI
I
]
I
R
I(X, Y/W)
REGION 1: DECODE X,THEN
W GIVEN X
TIMESHARING OF
POINT R IN CONVEX
REGION 2: DECODE X,W
HULL BETWEEN R'
INDEPENDENTLY
REGION 3: DECODE W THEN
X GIVEN W
REGION 4: USE TIMESHARING
BETWEEN 1,3
AND R'"
~
.
-
x =X+pk
SUCCESS
SUCCESS
NSUBSETZ
121-/SUBSET
SUCCESS \/
COLLISION
COLLISION
SUBSET
1 /SUBSET
IDLE
\ ~UBSE/UBSET
SUBT/ S2
COLLISION
COLLISION
ORDER OF TRANSMISSION
AFTER INITIAL COLLISION
1) SUBSET I
2) SUBSET II
(SUBSET OF SUBSET 1)
3) SUBSET 12
(OTHER SUBSET OF SUBSET 1)
4)
5)
SUBSET 121
SUBSET 122
6)
SUBSET 2
FRONT OF
TRANSMISSION
TRANSMISSION QUEUE
BY TIME t
ARRIVAL TIMES
OF WAITING PACKETS
t
ILOTt
\
t[t
INTERVAL
SLOT It,t+1)
(tl)_
"X
X'
SLOT[t t+
TIME
ALLOCATION
T(t)
)
time -
,SECOND,
t+,
t+2
-
XX'
L r;SUBSETI
CURRENT
T(t+l)
TIME
SLOT [t+l,t+2)
SUCCESSFUL BY
TIME t+2
_
_
4(t+2)
tt+2
CURRENT
T(t+2)
TIME
SLOT[t+2, t+3)
I
P2,
P2,2
Pi, 0
P1,2
P2,3