chapter 4 multiple users access the same communication channel

Chapter 4

Multiple users access the same communication channel

Multi-access Communication

Multiple users access the same communication channel

Multiple access protocols can be classified according to the amount of coordination between users.

Multi-access Communication

ALOHAS-ALOHA…CSMACSMA/CD(Ethernet)Reservation-basedprotocols

Most coordination

Overhead

Least coordination

Section 4.2

Slotted ALOHA

4.2 Slotted ALOHA1. Slotted System : fixed length packets

packet transmission time = 1 slot, all transmit are synchronized.

2. Poisson arrivals : arrival to each of the m stations is an independent Poisson process with rate /m.(because close to real world situation, and easy to analysis)infinite # users, each user generates a small fraction of total traffic, the aggregate traffic → Poisson

In fact,Slot time較 transmittime 大

4.2 Slotted ALOHA

3. Collision or perfect reception : If 2 or more nodes send a packets in a slot, then collision. If exacting one node transmits, then it is received correctly.

4. 0, 1, e: immediate feedbackmore than one

5. Retransmissions : Collided packets will be retransmitted.

4.2.2 Slotted ALOHA Simple throughput analysis

Let G = expected # of transmit in a slot = + rate of retransmitsAssume(incorrect) : the # of transmit in

a slot, K, is Poisson.

...2,1,0,

!)(

kk

eGkKP

Gk PoissonRate of re-txSo never Poisson for G

4.2.2 Slotted ALOHA

)(

)1(

successPGe

kPThroughputSe

# of successPer second

G : attempt rate : # transmit/slotS : # success transmit/slot

0.368 1/e

S

G

4.2.2 Slotted ALOHA Delay Analysis : Markov model(6a:assume no-bu

ffering p.276) Backlogged node is node with packet to be retransmi

tted. Unbacklogged node is node that may generate new tr

affic. After a collision, a backlogged node waits L slots befo

re re-transmit, where P(L=i)=(1-qr)i-1qr, i=1,2,… An unbacklogged node, will transmit a new packet wi

th probability qa=P(at least one new packet in a slot)

me

1

is total arrived rate# users=m

4.2.2 Slotted ALOHA

State of Markov process is the # of backlogged nodes.

Let Nt=# of backlogged nodes at the beginning of slot t.

0 1 2 n m不可能 …… ……

4.2.2 Slotted ALOHALet Qa(i, n)=probably of i unbacklogged nodes transmit i

n a given slot, given N=n

nmigqi

nm inma

ia

,...,1,0,1

Let Qr(i, n)=probably of i backlogged nodes transmit in a given slot, given N=

n nigq

i

n inr

ir ,...,1,0,1

4.2.2 Slotted ALOHA

1),,1(),0(

0,),1(1),0(),0(),1(

1,),0(1),1(

2),,(

|

,

1,

inQnQ

inQnQnQnQ

inQnQ

nminiQ

P

nNinNPPLet

ra

rara

ra

a

inn

ttinn

4.2.2 Slotted ALOHA {Nt, t=0,1,2,…} is an irreducible a perio

dic Markov Chain. Thus, limit probability {n, n=0,1,…,m} exist

m

ii

iijij P

0

1

Can find delay?(using Little’s theorem)N=T

直覺上， leave qr 大 ? But for large m? heavy backlog very long

4.2.2 Slotted ALOHA Let Dn = drift

= expected change in backlog over one slot time give

n state n. = (m-n)qa-[Qa(1,n)Qr(0,n)

+Qa(0,n)Qr(1,n)] increase in backlog

decrease in backlog

=Psucc. (4.5)

4.2.2 Slotted ALOHA Let G(n)=expected # of attempted tx

(new+backlogged) in a slot, given state n = (m-n)qa+nqr

Let A(n)=expected # of new tx in a slot, given state n = (m-n)qa

Homework #2 Due 11/15 3.9, 3.16, 3.37, 4.3 , 4.5 共 5 題

4.2.2 Slotted ALOHA When qa and qr are small, PsuccG(n)e-G(n)

)6.4()5.4()1(

...)!2

1()(2

Fromx

xxee

y

yyxxy

4.2.2 Slotted ALOHA

4.2.2 Slotted ALOHA

0 mn

mqa

mqr

G(n)

(qr>qa)

4.2.2 Slotted ALOHA

Adjust qr such that attempt rate G=1

n

mqa

mqr

G(n)

adjustqr=>

4.2.2 Slotted ALOHA

1. Psucc : at most e-1 for large m.2. qr ↑ , delay in re-tx ↓, but G(n)↑with n

同樣的 n ， qr ↑ 相對 G(n) 較大 Psucc 圖形會被壓縮。上圖 U 點向左移較少 n 即到達 U

3. 反之 qr↓, retx delay↑, but only one state point.

4.2.3 Stabilized Slotted Aloha

Change qr dynamically to maintain G(n)=1, n = estimate n.

[Pseudo-Bayesian Algorithm] (Rivest) Assumptions:

Slotted, Poisson Arrivals, collision or perfect reception, immediately feedbacks.

Infinite # of nodes – each newly arriving packets arrives at a new nodes.

All nodes with a packet(new or old) transmits in a slot with probably qr


Let nk = # of backlogged nodes at the beginning of slot k ( 全部看成 backlogged, if 有 packet(new or old))

Ik = the event slot k is idle Sk = the event slot k is success Ck = the event slot k is collision Ak = # of new arrivals in slot k


nk+Ak if Ik

nk+1= nk+Ak-1 if Sk

nk+Ak if Ck

Let nk be the estimate of nk computed by each node.

Each node assumes nk is Poisson dist.

nk nk+1

Slot k


Algorithm: Each backlogged node transmits in

slot k with probably.

Each node updates its estimate by

k

r nq

ˆ1

,1min Try to getG=1

2

1ˆ

1ˆ,maxˆ 1

en

nn

k

k

k

If Ik or Sk 減少 qr↑

If Ck 增加 qr↓


Properties of the Algorithm: Assume : nk is Poisson distributed with m

ean nk 1

(1)

,...2,1,0,

!

ˆ)(

ˆ

ll

enlnP

knlk

k

,...2,1,0,

!

1ˆ

)|()1ˆ(

1

jj

en

IjnP

knjk

kk


1ˆ1ˆ

ˆ

0

0

!

ˆ

ˆ1

1

|)(

eee

i

en

n

inPinIPIP

kk

k

nn

nik

i

i

ikkkk


!

)1ˆ()

ˆ1

1(!

ˆ

)(

)()|(

)1ˆ(

1

ˆ

l

en

e

nlen

IP

IandlnPIlnP

k

k

nlk

l

k

nlk

k

kkkk

Poisson


)(

)|()|(

)|()|(

)1)(1ˆ(1

)1()1)(1ˆ(1

1

PoissonstilleZ

eeZ

nconvolutioIAPInP

IjAnPIjnP

zn

zzn

kkkk

kkkkk

k

k


(2)

Poissonj

enSjnP

Poissonj

enSjnP

e

inPinSPSP

jnjk

kk

njk

kk

ikkkk

k

k

!

)1ˆ(|

)!1(

)1ˆ(|

1

)()|()(

)1ˆ(1

1

)1ˆ(1

1


(3)

以上 (1)~(3) Assume Poisson is OK, and alg. to estimate nk+1 is reasonable.

Poissone

nuwhere

j

eCjnP

eSPIPCP

k

uj

kk

kkk

2

1ˆ

!|

211)(

1

#


(4)

1ˆ1

ˆ)(

,ˆ

kkr

kk

nnnqnG

thennnIf

# of userswith packets

Probably of trans.

What we wantMaximum throughput.


(5)

)()|(

)()|(

)()()(

thenlarge,ˆ If

1

1

11

kkkk

kkkkkk

kkkk

kk

SPSnnE

CIPCInnE

nnEnEnE

nn


01

)(

)()|1(

)()|()()|(

)()|1(

)()|(

eSP

SPSE

SPSAECIPCIAE

SPSAE

CIPCIAE

k

kk

kkkkkkkk

kkk

kkkkk

Mean nk+1 gettingsmaller e states


01

)2

1)(2

1(

2)1()ˆˆ( 1

e

ee

ennE kk

estimate. accurately large, ,n̂n if

backlog thereduce to tendssystem The

kk

P(Ik+Sk)

P(Ck)


njnnj

n

nG

nnnG

jn

k

j

k

kk

...2,1,0 ,)ˆ1

1()ˆ1

(j)P(R

1)() E(R tx,of #Let R

1ˆ1

ˆ)(

)( 1n̂n if

k

kk

kk

估計不正確，會如何？


1)P(C2)P(R

0)P(S1)P(R

0)P(I0)P(R

1) E(Rsince

kk

kk

kk

k

kk

k

nnnn

nn

ˆ)ˆE(-)E(

thenˆ IfShow

k11k

k


39.12

1

)ˆ()ˆ()(

C 2

1ˆ)ˆ(

)()(

11

k1

1

e

nnnEnEe

nnE

AEnnE

kkkk

kk

kkk

幾乎為

accurate. more

is estimate that means this


[Approximate delay for Pseudo-Bayesian] ( 之前 prove, estimate is accurate G(n)→1,

max throughput stable) How about delay?

n̂n and

large is n backlogged then

,e

1λ large Assume


1,2,3...ifor

n̂1

1

i!

)n̂(

1)n|iP(n

Poisson0,1,...ii!

)n̂( i) P(nk,slot For

ˆk

ˆ

ˆik

kk

ˆik

k

kk

k

k

nn

n

n

ee

e

e

P(nk=0) P(nk=1)


kk

k

nn

nik

i

i kk

eei

en

nni

ˆk

ˆ

ˆ

1

2kk

n̂1

1

!

)ˆ(

)ˆ1

1(ˆ1

1)n|P(S

P(1 tx)


11)n|P(S

1

n̂1

11

n̂1

1

1)!-(i

1)-n̂(1

kk

ˆk

ˆ

)1ˆ(

2ˆ

kˆ

)1ˆ(1-ik

eee

e

e

ee

e

e

kk

k

kk

k

nn

n

inn

n


Wi = delay from the i-th arrival until the beginning of the slot of the i-th departure

Qi = # of backlogged pkts(excluding possible successful trans) at the instant before the i-th arrival.

假設 FCFS, 因為 average is the same.Qi

i-th arrivalWi

rit1 t2 i-th departure

S

tQi yiL


Let yi = # of slots from the Qi-th successful tx (end of slot) until the beginning of the i-th successful tx

i

Q

jjii ytrW

i

1

Where tj = # of slots needed for the j-th successful tx after the i-th arrival


e

yEW

yEWeW

yEtEQEREWE iiii

1

)(21

)(2

1

)()()()()(

W2

1

succP

1


Let L be the slot #, immediately following Qi-th success tx. Suppose nL=1, then E(y| nL)=0 Suppose nL>1, then E(y| nL>1)=e-1

yi+S S success


P1 = fraction of slots in which n=1 and pkt is successfully tx.

= fraction of slots in which there is a successful tx

= fraction of packets successfully trans. From

state 1

)1()1(0)(

)ˆ(Let P

1

k

L

L

nPePyE

knP

1P


= fraction of packets successful transmitted from higher state #

11P

S S S S S

)1)(1()( 1

P

eyE

statesother ,1

1 state ,0

ymean

) y(

i

i

e

really

長度每個定義


slot previous inarrival

100

101

1

)(

1)1(

)1)(1()(

no

stateprevious

ePPP

ePPP

PeyE


fig4.6 ,1)-1)(e-(e-1

1)-1)(e-(e-

e-121

-eW

1)-1)(e-(e-1

1)-e)(e-(1get P tosolve 1

see

4.3.1 Tree Algorithm

Tree Algorithm Using labels Collision resolution interval : split

left/right side When traffic highTDMA ,best for

high traffic Throughput =

0.387(1/e=0.368,ALOHA) Although,throughput is improved a

little But “STABLE”

Tree Algorithm

Tree Algorithm At each slot k,each node computes T(k)

and (k) The interval [T(k), T(k)+ (k) ) is called

the allocation interval for slot k The interval from [T(k)+ (k), k ) is

called the waiting interval for slot k All nodes with packets that arrived in

the allocation interval transmit in slot k; All nodes with packets that arrived in the waiting interval wait.

Tree Algorithm At time k+1,each node does the

following:i. If feedback is collision ,then

T(k+1)= T(k)(k+1)= ½[(k)] (k+1)=L

(k)=L statusor R left or Right

Tree Algorithm

ii. If feedback is successful and (k)=LT(k+1)= T(k)+ (k)(k+1)= (k) (k+1)=R

iii. If feedback is idle, and (k)=LT(k+1)= T(k)+ (k)(k+1)= ½[(k)] (k+1)=L Masseyo` Improvement

Tree Algorithm

iv. If feedback is idle or successful, and (k)=R

T(k+1)= T(k)+ (k)(k+1)= min[0, k+1- T(k+1)]

(k+1)=ROptimal 0 =2.6

0.4871=throughput

Tree Algorithm

Tree Algorithm Markov Chain

(L,i) = system has status L and have been i splits (L:status , i:how many counts )

(R,i) = system has status R and have been i splits

Tree Algorithm

Tree Algorithm Whenever success at right side

finish CRI(Collision Resolution Interval) initial interval = 0

In state (R,0): Let Gi=E(number of packets in an interval

that has been split i times) =*(length of the interval) =* 0 *2-I

G0= 0 , G1=2-10 …………

PR,0=e- G0 + G0 e- G0 =(1+ G0)e- G0

Tree Algorithm In state (L,1) (XL:#of arrivals in left

side)

0

11

G-0

G-G-1

1,

e)1(1

)e-)(1e(G

2

11

2

2,1

2|1

G

XXP

XPXP

XXP

XXXP

XXXPP

RL

RL

RL

RLL

RLLL

Tree Algorithm

1

1

G-

G-1

1,

e1

eG

1

1

1

1,1

1|1

2,1|1

R

R

R

RR

LR

RLLRR

XP

XP

XP

XXP

XXP

XXXXPP

Tree Algorithm In general

)24.4(1

)23.4()1(1

)1(

,

1, 1

i

i

i

ii

G

Gi

iR

Gi

GGi

iL

e

eGP

eG

eeGP

Tree Algorithm

otherwise 0

visitedis i)(L, if 1i) x(L, where

x(R,2) x(R,1)

x(L,2)x(L,1)1

E(k)) find( CRP ain slots ofnumber k

(R,0) to visitsobetween tw interval the

CRP

Tree Algorithm

iRiL

iL

R

i

i

PiRPPiLPiLP

PiLPiRP

PLP

iRPiRP

iLPiLPwhere

iRxPiLxP

iRxEiLxEkE

,,

,

0,

1

1

1),(1),()1,(

),(),(

1)1,(

(R,0) toreturning before visitedis ),(),(

(R,0) toreturning before visitedis ),(),(

)),(()),((1

)),(()),((1)(

Tree Algorithm We next evaluate the change in T(k) from

one CRP to the next(how fast it move?)

Assume that the initial allocation interval is of size 0.I.e.[T(k), T(k)+0]

k

k

0

T(k)

T(k)

Return to W. int

Tree Algorithm The change is at most 0

If the left hand interval has collisions, the corresponding right hand interval is returned to the waiting interval

Let f be the fraction of 0 returned to the waiting interval over a CRP, then 0 (1-f) is the change in T(k)

Tree Algorithm

))(1()(

))(in (change-CRP) of(length

))( backlog in time change(D

(4.30) ),(2),(|

),()),(|()(

)intervalsright left()eG2(1-1

)eG(1-1

2 containsright &left |2contains intervalleft ),(|

0

1

1

2G-i

G-i

i

i

fEkE

kTEE

kTkE

iLPiLeP

iLPiLfEfE

PiLeP

i

i

i

Tree Algorithm

0

0

0

0

offunction is r.h.sin everything

offunction is while offunction are

(4.32) )(

))(1(

))(1( )(

stable" 0D if

ii GGE(f),E(k)

kE

fE

fEkE

Tree Algorithm

Right -hand-side is maximized at 0 =1.266 choosing 0=2.6 , =0.4871

Condition for negative D is satisfied for all <0.4871

I.e. max. throughput is 0.4871

0.4871

1.266 0

4.4.1 CSMA slotted ALOHA Focus on throughput & stability

analysis Slotted system, length of slot = Packet Tx time = 1

Tx time = L/C Propagation and detection delay <<1

Otherwise CSMA doesn't make sense 0,1,e feedback with max.delay

CSMA slotted ALOHA If a packet arrives during an idle

slot, it is transmitted at the beginning of the next slot

A packet is considered backlogged if It arrives during a transmission , or It was involved in a collision

CSMA slotted ALOHA Each backlogged packet is re-

transmission with probability qr<<1 after each subsequent idle slot

1 1

I II

Earliest time at which another Tx can occur

Trans. Or collision

CSMA slotted ALOHA Throughput analysis

State transition time = ends of idle slot

Time between transition = +1 if success or collision if idle

Let Tn=time until the next transition given that n backlogged packets

I

xn backlogged packets Next idle slot

Tn

CSMA slotted ALOHA

nr

nr

qeP

qe

PP

)1(1busyslot next

)1(

itnot transm do packets backlogged

slot, in this arrival noidleslot next

CSMA slotted ALOHA

1

arrival new 0arrival new 1exactly

)1()1(

)in ion transmisssuccessful(1)in departures ofnumber (

)1(1)in arrivals ofnumber (

)1(1

))1(1)(1()1()(

nrr

nr

nn

nrn

nr

nr

nrn

qnqeqe

TPTE

qeTE

qe

qeqeTE

CSMA slotted ALOHA

nr

ngngn

nqnr

nrr

nrr

nr

nr

nn

n

Tnqng

engeD

eqqq

qnqeqeqe

TETE

D

r

in Tx ofnumber expected)( where

)()1(

)1()1( , 1 since

)1()1()1(1

)in departure ofnumber ()in arrivals ofnumber (

drift expected

)()(

1

1

CSMA slotted ALOHA

)n transitiostate ofduration (1

)n transitiostateper departure ofnumber ( )(

0Dstability,For

)(

)(

n

Ee

Eengng

ng

CSMA slotted ALOHA

CSMA slotted ALOHA

g(n)

Slope= qr

n

4.4.3 CSMA Unslotted ALOHA

Unslotted system After a packet arrival if the channel is sense

d idle, transmission begins immediately Each backlogged packet

Re-transmission are repeatedly attempted at random times separated by independent exp. Delay with pdf f (0)=xe-x 0 E()=1/x

CSMA Unslotted ALOHA Detection delay =

Let In= idle period, given n backlogged packets. G(n)=attempt rate during In =+nx

n backlogged packet

Next transmissionsbusy

I

CSMA Unslotted ALOHA

New packet started

P(collision)=1-e-

G(n)

Attempt rate =+nx


Previously backlogged packet started

P(collision)=1-e-G(n-

1)

Attempt rate =+ (n-1)x

CSMA Unslotted ALOHA Assume x very small, then

e- G(n) e- G(n-1), and in both cases , P(succ) = e- G(n)

idle idle

I B

Expected duration =[1/G(n)]+1+


2

11)( when

21

1rate departure max.

1)(

1rate(n) Departure

)(

nG

nG

e nG

CSMA Unslotted ALOHA Unslotted CSMA is also unstable

How does on stabilized it?

4.4.2 Pseudo-Bayesian for CSMA slotted ALOHA

All packets are considered backlogged

At the end of each idle slot, each backlogged packet is independently transmitted with probability qr

backlog estimated theis where

2,2

min)(

n

nnqr

Pseudo-Bayesian for CSMA slotted ALOHA

At the end of each slot, each node updates its estimate by

collision if )1(2

success if )1()(1

idle if )(1

1

k

krk

krk

k

n

nqn

nqn

n


!

)2(|.2

!

)(1

||.1

mean on with distributiPoisson is Suppose

)2(

)(1

j

enCjnP

j

enqn

SjnPIjnP

nn

k

krk

njk

kk

nqnj

krk

kkkk

kk


Delay analysis of this protocol Let Wi be the delay from the i-th arriva

l until the beginning of the i-th success transmission

Let ni be the number of backlogged packets the instant prior to i-th arrival, excluding the possible success transmission


in

jijii ytRW

1

Ri

i-th departureWi

i-th arrivalNi backlogged

1st succ

2nd succ

Ni-th

succ

yi


Where Ri =residual time until the end of the next idle slot tj =time for (j-1)th success transmission until the

j-th success transmission yi =time for nith success transmission until the be

ginning of the i-th success transmission Assume tj are iid.(in fact, not, nj,more collisi

oninfluence tj ) E(t)=E(y)+(1+)


Averaging over all I and by Little`s Formula

To simplify analysis, we assume number of attempted transmission in an interval is Poisson with mean g

(4.45) )(1

)()(

tE

yEREW


2)1(2

)1(2

1

)(2

1)(

2

1)(

2

)()|()()|()()|()(

)1(1

)1()(

(4.44) 1

)(

)(1)1()1())(()(

CPSPIP

CPCRESPSREIPIRERE

ge

etEE(y)

ge

etE

tEgeegetEetE

g

g

g

g

gggg


(4.48) ) 21(12

22

minimize 2

minimizingby minimized is

(4.47) )(12

121

0

)1()1(1)(

min

2

W

E(t)g

E(t)W

tE

β)(E(t)β)λ(W

eeCP

4.5.2 Slotted CSMA/CD (Ethernet Protocol)

Minislots of duration If minislot is idle, all nodes wait until

beginning of next minislot If exactly 1 node transmits in minislot, all

other nodes detect this transmission within and keep quiet until they detect end of transmission

If 2 or more nodes transmit, then these nodes detect each others transmission within and cease transmitting. Then the next idle slot ends after 2

Slotted CSMA/CD (Ethernet Protocol)

B

B

B

2

In

In

1

Idle

collision

success

In:interval between ends of idle slots


0)()(

cycle onein departure ofnumber average -

cycle onein arrivals ofnumber average

backloggedn Drift when

)(12

)()1()(

)(

)()(

)()(

ngn

ngng

ngngn

engIE

enge

engeIE


77.0)( when 31.31

1

, )( where

(4.65) ))(1(1)(

)(

or

max

)()(

)(

ng

nqng

engeng

eng

r

ngng

ng

Unslotted CSMA/CD When unbacklogged node has a packet

If it senses the channel idle, it transmits immediately

If it senses the channel busy, it becomes backlogged

Backlogged nodes choose a backlogged delay which is exponentially distributed then sense the channel again

Unslotted CSMA/CD

B

2 collision

Everyone will sense channel idle

y

Next packet ready to be transmitted

Success transmission

Unslotted CSMA/CD

g

g

ecollisionP

esuccessP

collisionzE

successzE

gyE

xE

zEyExEIE

1)(

)(

2)|(

1)|(

ones backlogged & arrival new including 1

)(

)(

)()()()(

Unslotted CSMA/CD

6

113 when

2.61

1

)1(21

max

gS

eeg

eS

throughput

gg

g

chapter 4 multiple users access the same communication channel

Documents

slotted aloha4

slotted alohacollision

slotted aloha0mnmqamqrgnqrqa4

slotted alohapsucc

given slot

slotted alohaadjust

state n

slotted alohalet gn