Weiqiang Sun

PACKET MULTIPLE ACCESS: THE ALOHA PROTOCOL

Shanghai Jiao Tong University

Weiqiang Sun

What is multiple access?

• As opposed to Point-to-point communication medium (or single access medium)

• Shared transmission medium (or multi-access medium)– A receiver can hear multiple transmitters– A transmitter can be heard by multiple users/a single user

• The main problem here is to allocate the channel between users, such that– Optimal system throughput can be achieved– System fairness can be achieved

• Difficulties– Distributed system, each user may not have information

about other users

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Examples of multi-access media

• Examples– Local area networks, Ethernet, in the traditional way– Satellite channels– Wireless radio– Multi-drop telephone

• Ways to multi-access– Fixed assignment

• TDMA: GSM (/w modifications)• FDMA: early analog cellular phones• CDMA: cellular phones

– Contention mechanisms• Random access• Polling• Reservation and scheduling

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Slotted multi-access – a few assumptions

• Slotted system– Transmitters are synchronized– Each transmission needs one slot

• Poisson arrivals– Each transmitter generate i.i.d Poisson packets with rate λ/m

• Collision or perfect reception– If only one transmitter transmits in one slot then transmission is regarded as successful– Otherwise all collided packets are lost

• Immediate feedback– Upon reception, the receivers are able to send feedback to transmitter immediately– Types of feedbacks: ‘1’ for successful, ‘0’ for lost, ‘e’ for collision

• Retransmission of collided packets– Wait some random time and re-transmit, until the packet is successfully transmitted

• No buffer– One node can only hold exactly one packet. If a new packet arrives at a node with one still waiting to be transmitted,

the newly arrived packet is dropped and never transmitted again.

• Infinite number of nodes– Each newly arriving packet arrives at a new node, i.e. no packet drop

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The ALOHA system

• The ALOHAnet and the ALOHA protocol– Invented by Norman Abramson, University of

Hawaii in 1970– Initial motivation is to connect far-flung

campuses with amateur-radio like systems

– Two channels: outbound and inbound• Outbound for broadcasting• Inbound for the hub• Receiver resend upon reception of a packet• Sender resend in case of conflict

– First multi-access system, generated a lot of research on shared medium access systems, like Ethernet, and even GSM

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transmitter

receiver

transmittertransmitter

• ALOHA: affection, love, peace, compassion and mercy

Weiqiang Sun

Slotted ALOHA

• Transmission attempts = external arrivals with rate λ + re-transmission of backlogs• Given the above assumptions, the transmission attempts can also be treated as

Poisson, say, with rate G, G > λ

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(0) Gp e

(1) Gp Ge

( ) 1 (0) (1)p collision p p

Success (1)

Idle (0) Collision(e)

Idle (0) Success (1)

Idle (0) Collision(e)

t

Arrivals and re-transmissions

System backlogs

The probability of zero transmission, i.e., the slot is idle

The probability of 1 transmission, i.e., success:

The probability of collision

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Performances of the idealized system

• e-1 is the maximal achievable throughput

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(1) Gp Ge1e

• λ must be smaller than e-1

λ

GdGedG

ALOHAG

λ < e-1

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Performances of the idealized system

• Define D = expected change in backlog over one time slot

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λ

= λ – Ge-G

D is system drift– When D > 0, backlog tends to increase– When D < 0, backlog tends to decrease

What causes the instability?

Negative drift

Positive drift

Stable pointUnstable point

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Instability of Slotted ALOHA

• Let us take a closer look at G– G is the transmission attempts = external arrivals + retransmission from

backlogged nodes– G = λ + n*p, n: number of backlog, p: probability of re-send in next slot

• Instability is caused by the re-transmission policy, given we are offered a reasonable external arrival rate

• The problem now becomes estimating n, so that we can have a G close to 1, and we can achieve maximum throughput.

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Success (1)

Idle (0) Collision(e)

Idle (0) Success (1)

Idle (0) Collision(e)

t

Arrivals and re-transmissions

System backlogs

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Some examples, p = 0.3

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λ = 0.01 λ = 0.05

λ = 0.1 λ = 0.2

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Some examples, p = 0.1

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λ = 0.01 λ = 0.05

λ = 0.1 λ = 0.2

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Under the finite node assumption…

• Slotted system– Transmitters are synchronized– Each transmission needs one slot

• Poisson arrivals– Each transmitter generate i.i.d Poisson packets with rate λ/m

• Collision or perfect reception– If only transmitter transmits in one slot then transmission is regarded as successful– Otherwise all collided packets are lost

• Immediate feedback– Upon reception, the receiver are able to send feedback to transmitter immediately– Types of feedbacks: ‘1’ for successful, ‘0’ for lost, ‘e’ for collision

• Retransmission of collided packets– Wait some random time and re-transmit, until the packet is successfully transmitted

• Finite number of nodes– Packets arrived at an backlogged node will be discarded and never re-transmitted again

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Slotted ALOHA - Under the finite node assumption…

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…

m nodes in totalamong them n are backlogged

External arrival: λλ/m

-If the total external arrival rate is λ, the arrival rate to each node is λ/m, -The arrivals to the backlogged nodes are dropped-For each un-backlogged node, the probability of no arrival is: / me

Hence one or more arrival, i.e. one transmission attempt in next slot is:/1 me ← Let it be qa, i.e. attempt from new arrivals in a node

We also have qr, re-transmission attempt from each backlogged node

-The number of attempts in a slot, when n nodes are backlogged is given by:

G(n) = (m-n) qa + n qr

backlogged un-backlogged

from new arrivals

from backlogged re-trans.

Weiqiang Sun

Slotted ALOHA - Under the finite node assumption…

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The probability that i un-backlogged nodes attempt to transmit:

( , ) 1a

m n i ia a

m nQ i n q q

i

The probability that i backlogged nodes attempt to transmit:

( , ) 1r

n i ir r

nQ i n q q

i

Given the system state, i.e. the number of backlogged node is n

And, the probability of one successful transmission:

succP (1, ) (0, ) (0, ) (1, )r a r aQ n Q n Q n Q n

Weiqiang Sun

Slotted ALOHA - Under the finite node assumption…

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The probability of one successful transmission:

(1, ) (0, ) (0, ) (1, )succ r a r aP Q n Q n Q n Q n 1 1(1 ) (1 ) ( )(1 ) (1 )a a a

n m n n m nr r rn q q q m n q q q

( , ) 1a

m n i ia a

m nQ i n q q

i

( , ) 1r

n i ir r

nQ i n q q

i

( )(1 ) (1 )

1 1a

n m n arr

r a

m n qnqq qq q

(1 ) , when is smally xyx e x ( ) ( )1 1

arq m nq n ar

r a

m n qnqe eq q

( )( )r anq m n q

r ae nq m n q

( )( ) G nsuccP G n e

G(n) = (m-n) qa + n qr

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Slotted ALOHA - Under the finite node assumption…

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Again we define the system drift: the change in number of backlogged nodes

External arrival (m-n) qa

Departure rate Ge-G

Dn = (m-n) qa - Psucc

Stable point

Undesirable stable point

Unstable point

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Stabilizing ALOHA – Rivest’s Pseudo-Bayesian algorithm

• New arrivals are backlogged immediately

• Maintain an estimation of backlogs, namely , and update the estimation at the beginning of each slot

• All packets transmit with probability qr in each slot

• Update qr according to

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n̂

n̂

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qr and

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n̂

ˆ rG n q ˆ1/rq n

• Since all packets are transmitted with qr , the attempt rate now becomes

Hence, a reasonable estimation of qr is: And since can be 0, i.e. no backlog, qr in this case should be 1, so finally:n̂

ˆmin{1,1/ }rq n• We want to decrease if we find an idle slot, or the previous transmission is

successful. And to increase it in case of collision.n̂

1 1

ˆmax{ , 1}, if idle or sucessˆ

ˆ ( 2) , if collisionk

kk

nn

n e

Decrease oneNew backlogs due to arrivals

To maintain the balance between n and n̂

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Slotted ALOHA

Pure ALOHA

Throughput↓

“Pure” ALOHA (unslotted)

• New arrivals are transmitted immediately• Retransmission after random delay, typically exponentially distributed delay

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E[τ] = 1/xτ: exponentially distributed delay with pdf xxe , x is attempt rate

G(n) = λ + nx

New arrivalPrevious arrival Next arrival

No collision wit previous packet: e-G(n)

No collision wit next packet: e-G(n)Psucc = e-2G(n)

Hence the throughput = G(n)*Psucc = G(n)* e-2G(n)

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Contention Resolution Protocol

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What is the probability that can we successfully transmit them in two time slots?The answer is : 1/2

And what is the probability of 3 slots? And for n slots?

What is the expected number of slots needed to transmit two packets?

Consider a very simple collision resolution algorithm:We have two packets, and in each slot, each packet is transmitted with probability 1/2

The answer is : 1/4

The answer is : 3

The throughput is then:The answer is : 2/3

This very simple collision resolution algorithm provides good throughput, but in reality, we may not know how many packets to transmit.

Weiqiang Sun

Splitting Algorithms

• Tree algorithms– After collision, all nodes involved split into two groups, left (L) and right (R)– Members of L transmit in next slot, members in R wait– If collision occurs, go back to (1). When L is done, transmit R.

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slot Transmission Set Waiting Set feedback

1 S - e2 L R e3 LL LR,R 14 LR R e5 LRL LRR,R 06 LRR R e7 LRRL LRRR,R 18 LRRR R 19 R - 0

↓An example: initially the transmission set is S.

Weiqiang Sun

Splitting Algorithms

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S

L R

collision

idlecollision

success collision

collision

LL LR

LRL LRR

successsuccess

idle

LRRR LRRL

collision will happen for sure, better splitting beforehand

collision will only happen with small probability, can be merged with subsequent periods

Any improvements?

Collision resolution period

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Performance comparison

• Stabilized slotted aloha T = 0.368 = 1/e

• Stabilized pure aloha T= 0.184 = 1/(2e)

• Basic tree algorithm T = 0.434

• Best known variation on tree algorithm T = 0.4878

• Upper bound on any collision resolution algorithm with (0,1,e) feedback

T≤0.568

• TDM achieves throughputs up to 1 packet per slot, but the delay increases

linearly with the number of nodes

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