approximate delay analysis - linköping university...approximate delay analysis stabilized slotted...

Approximate delay analysis

Stabilized slotted Aloha with pseudo-Bayesianalgorithm

Assuming arrival rate λ is known

Probability of successful transmission Ps = 1/e ifbacklog n ≥ 2 and Ps = 1 if n = 1

Let Wi be the delay from arrival of ith packet untilbeginning of ith successful transmission

We can assume that the average of Wi over all i is theexpected queueing delay W

Let ni be number of backlogged packets at the instantbefore i’s arrival (not including any packet currentlybeing successfully transmitted)

Information Networks – p.1/22


Let Ri be the residual time to beginning of next slot andt1 the subsequent interval until next successfultransmission is completed. Similarly tj the interval fromthe end of the (j − 1) subsequent success to the end ofthe jth subsequent success.

After ni successful transmissions yi is the remaininginterval until the beginning of next successfultransmission, then

Wi = Ri +

ni∑

j=1

tj + yi



Wi = Ri +ni∑

j=1

tj + yi

For each interval tj the backlog is at least 2, thus eachslot is successful with probability 1/e and the expectedvalue of each tj is e

Little’s theorem gives E[ni] = λE[Wi] = λW

E[Ri] = 1/2, and we get

W = 1/2 + λWe + E[y]



Consider the first slot boundary at which both the(i − 1)st departure and the ith arrival have occurred

If backlog is 1 then yi = 0

If backlog n > 1, then E[yi] = e − 1

Let pn be steady state probability that backlog is n at aslot boundary

If state is 1 at beginning of a slot we always get asuccessful transmission, thus p1 is the fraction of slots inwhich state is 1 and a packet is successfully transmitted



Total fraction of slots with successful transmission is λ,thus p1/λ is the fraction of packets transmitted fromstate 1 and 1 − p1/λ is the fraction transmitted fromhigher numbered states, in total we get

E[y] = (e − 1)(1 − p1/λ) =(e − 1)(λ − p1)

λ

The rate of packets transmitted from state 1 is p1

The probability of state higher than 1 is (1− p0 − p1) andsuccessful transmission with probability 1/e give rate ofpackets transmitted from higher states as (1− p0 − p1)/e

Thus we get λ = p1 + (1 − p0 − p1)/e



State 0 entered only if no new arrivals occurred in theprevious slot and previous state was 0 or 1, thusp0 = (p0 + p1)e

−λ

λe = (e − 1)p1 + 1 − p0

p0 = (e − 1)p1 + 1 − λe

(e − 1)p1 + (1 − λe) = ((e − 1)p1 + (1 − λe) + p1)e−λ

(e − 1)eλp1 + (1 − λe)eλ = (e − 1)p1 + (1 − λe) + p1

(1 − λe)(eλ − 1) = p1((e − 1)(1 − eλ) + 1)

p1 =(1 − λe)(eλ − 1)

1 − (e − 1)(eλ − 1)



Now, combining our equations

W = 1/2 + λWe + E[y]

E[y] =(e − 1)(λ − p1)

λ

p1 =(1 − λe)(eλ − 1)

1 − (e − 1)(eλ − 1)

We get

W =e − 1/2

1 − λe−

(e − 1)(eλ − 1)

λ(1 − (e − 1)(eλ − 1))


Time division multiplex

For comparison, consider the delay in a time divisionmultiplex system with m traffic streams of equal lengthpackets arriving according to a Poisson process withrate λ/m each

Time axis divided into m-slot frames with one time slotdedicated to each traffic stream

This corresponds to m M/D/1 queueing systems, eachwith service rate µ = 1/m

According to M/D/1-formula for queueing delay (3.45)p. 187 the average queueing delay is Wq = ρ/(2µ(1− ρ))

where ρ = λ/m1/m = λ


Time division multiplex

Thus we get average queueing delay

Wq =mλ

2(1 − λ)

In addition to this we have an average delay of m/2waiting for the traffic slot for the traffic stream inquestion

Our total average delay from a packet arrival until itbegins transmission is

WTDM =m

2(1 − λ)


TDM vs Stabilized Slotted Aloha

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

2

4

6

8

10

12

14

16

18

20

Stabilized Slotted Aloha in solid line, TDM with m = 8 in

dotted line and with m = 16 in dash-dotted line.


Binary exponential backoff

For packet radio networks and some other multiaccesssituations the assumption of immediate (0,1,e) feedbackis unrealistic

In some systems a node only receives feedback aboutits own packets, no feedback in about slots in which itdoes not transmit

This limited feedback is insufficient for the backlogestimation of pseudo-Bayesian strategy

An alternative stabilization strategy is binaryexponential backoff used in Ethernet

If a packet has been transmitted unsuccessfully i timesthe transmission in successive slots is set to qr = 2−i


Binary exponential backoff

When a packet initially arrives it is transmittedimmediately in next slot, since the node knows nothingof the backlog this is reasonable

With successive collisions any reasonable estimate ofbacklog would increase which motivates the decreasein retransmission probability qr

When qr is reduced the node gets less feedback perslot about the backlog, to play safe it’s reasonable toincrease the backlog estimate by larger and largeramounts on each successive collision

In the limit as number of nodes approach infinity thisstrategy is unstable for every arrival rate λ > 0


Unslotted Aloha

The original Aloha protocol was unslotted, in thisstrategy each node, upon receiving a new packet,transmits it immediately rather than waiting for a slotboundary

We omit the slotted system assumption

If a packet is involved in a collision, it is retransmittedafter a random delay

We assume that if the transmission times for twopackets overlap at all those packets fail andretransmission will be required

We assume that each node, after a given propagationdelay, can determine whether or not its packets werecorrectly received


Unslotted Aloha

If one packet starts transmission at time t, and allpackets have unit length, any other transmissionstarting between t − 1 and t + 1 will cause a collision

Assume infinite number of nodes

A node is considered backlogged from the time it hasdetermined that its previously transmitted packet wasinvolved in a collision until the time that it attemptsretransmission

Number of backlogged nodes is n

Assume that period until attempted retransmission τ isexponentially distributed with probability density xe−xτ ,where x is interpreted as retransmission attempt rate


Unslotted Aloha

With an overall Poisson arrival rate λ, the times ofattempted retransmissions is a time-varying Poissonprocess with rate G(n) = λ + nx where n is the backlogat a given time

Let τi be the interval between the ith and (i + 1)thtransmission attempt, the ith attempt will be successfulif both τi and τi−1 exceed 1 (assuming all packets havelength 1)

The probability distribution for the interval τi isG(n)e−G(n)τi thus the probability that τi > 1 is e−G(n)

Assuming τi and τi−1 independent gives probability ofsuccessful transmission Ps = e−2G(n)


Unslotted Aloha

Attempted transmissions occur at rate G(n), theexpected number of successful transmissions per unittime, the throughput as a function of n is G(n)e−2G(n)

The situation is very similar to slotted Aloha, except themaximum throughput is 1/(2e) achieved whenG(n) = 1/2

We have assumed that backlog is same in intervalssurrounding a given transmission attempt, but whenevera backlogged packet initiates a transmission thebacklog decreases by 1 and whenever a collided packetis detected it increases by 1, for small x this error isrelatively small


Unslotted Aloha

We have the same stability problems as in slotted Aloha

With limited feedback stability is quite difficult to achieveor analyze

One advantage with unslotted Aloha is that it can beused with variable length packets, this compensates forsome of the inherent throughput loss and gives anadvantage in simplicity

As for unstabilized slotted Aloha, if we have very smallarrival rate λ and very large mean retransmission timethe system can be expected to run for a long timewithout major backlog buildup.


Aloha Summary

Assuming Poisson arrivals, collision or perfectreception, (0,1,e) feedback, retransmission of collisions,either no buffering or infinite set of nodes

Already simplistic analysis identifies maximumthroughput 1/e at attempt rate G = 1

More precise model using a Markov chain with state n,the number of backlogged nodes

We can compute steady state probability distribution fornumber of backlogged nodes, and thus expectednumber of backlogged nodes and (using Little’stheorem) average delay


Slotted Aloha Stationary Probabilities

0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

p0, p

1, p

m and rejection probability as function of number of nodes m


Slotted Aloha Average Delay

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

packet delay as function of nr of nodes, qr=0.2,0.3,0.4,0.6


Slotted Aloha Average Delay

5 10 15 20 25 300

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10

12

14

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20

packet delay as function of nr of nodes qr=0.05,0.1,0.2,0.3


Aloha Summary

Steady-state shows strange behaviour due to instability

Analysis of the dynamics via the drift

Stabilizing with pseudo-Bayesian algorithm, learningfrom the analysis of the dynamics

Need to estimate number of backlogged nodes n, this isdone from the feedback according to Bayesian strategyfor idle/successful and approximate Bayesian strategyfor collisions

Waiting delay until successful transmission

W =e − 1/2

1 − λe−

(e − 1)(eλ − 1)

λ(1 − (e − 1)(eλ − 1))


approximate delay analysis - linköping university...approximate delay analysis stabilized slotted...

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