Approximate delay analysis Stabilized slotted Aloha with - - PowerPoint PPT Presentation

Approximate delay analysis

Stabilized slotted Aloha with pseudo-Bayesian algorithm Assuming arrival rate λ is known Probability of successful transmission Ps = 1/e if backlog n ≥ 2 and Ps = 1 if n = 1 Let Wi be the delay from arrival of ith packet until beginning of ith successful transmission We can assume that the average of Wi over all i is the expected queueing delay W Let ni be number of backlogged packets at the instant before i’s arrival (not including any packet currently being successfully transmitted)

Information Networks – p.1/22

Approximate delay analysis

Let Ri be the residual time to beginning of next slot and

t1 the subsequent interval until next successful

transmission is completed. Similarly tj the interval from the end of the (j − 1) subsequent success to the end of the jth subsequent success. After ni successful transmissions yi is the remaining interval until the beginning of next successful transmission, then

Wi = Ri +

ni

• j=1

tj + yi

Information Networks – p.2/22

Approximate delay analysis

Wi = Ri +

ni

• j=1

tj + yi

For each interval tj the backlog is at least 2, thus each slot is successful with probability 1/e and the expected value 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]

Information Networks – p.3/22

Approximate delay analysis

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 a slot boundary If state is 1 at beginning of a slot we always get a successful transmission, thus p1 is the fraction of slots in which state is 1 and a packet is successfully transmitted

Information Networks – p.4/22

Approximate delay analysis

Total fraction of slots with successful transmission is λ, thus p1/λ is the fraction of packets transmitted from state 1 and 1 − p1/λ is the fraction transmitted from higher 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) and successful transmission with probability 1/e give rate of packets transmitted from higher states as (1 − p0 − p1)/e Thus we get λ = p1 + (1 − p0 − p1)/e

Information Networks – p.5/22

Approximate delay analysis

State 0 entered only if no new arrivals occurred in the previous slot and previous state was 0 or 1, thus

p0 = (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)

Information Networks – p.6/22

Approximate delay analysis

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))

Information Networks – p.7/22

Time division multiplex

For comparison, consider the delay in a time division multiplex system with m traffic streams of equal length packets arriving according to a Poisson process with rate λ/m each Time axis divided into m-slot frames with one time slot dedicated to each traffic stream This corresponds to m M/D/1 queueing systems, each with 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/m

1/m = λ

Information Networks – p.8/22

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/2 waiting for the traffic slot for the traffic stream in question Our total average delay from a packet arrival until it begins transmission is

WTDM = m 2(1 − λ)

Information Networks – p.9/22

TDM vs Stabilized Slotted Aloha

0.1 0.2 0.3 0.4 0.5 0.6 0.7 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.

Information Networks – p.10/22

Binary exponential backoff

For packet radio networks and some other multiaccess situations the assumption of immediate (0,1,e) feedback is unrealistic In some systems a node only receives feedback about its own packets, no feedback in about slots in which it does not transmit This limited feedback is insufficient for the backlog estimation of pseudo-Bayesian strategy An alternative stabilization strategy is binary exponential backoff used in Ethernet If a packet has been transmitted unsuccessfully i times the transmission in successive slots is set to qr = 2−i

Information Networks – p.11/22

Binary exponential backoff

When a packet initially arrives it is transmitted immediately in next slot, since the node knows nothing

• f the backlog this is reasonable

With successive collisions any reasonable estimate of backlog would increase which motivates the decrease in retransmission probability qr When qr is reduced the node gets less feedback per slot about the backlog, to play safe it’s reasonable to increase the backlog estimate by larger and larger amounts on each successive collision In the limit as number of nodes approach infinity this strategy is unstable for every arrival rate λ > 0

Information Networks – p.12/22

Unslotted Aloha

The original Aloha protocol was unslotted, in this strategy each node, upon receiving a new packet, transmits it immediately rather than waiting for a slot boundary We omit the slotted system assumption If a packet is involved in a collision, it is retransmitted after a random delay We assume that if the transmission times for two packets overlap at all those packets fail and retransmission will be required We assume that each node, after a given propagation delay, can determine whether or not its packets were correctly received

Information Networks – p.13/22

Unslotted Aloha

If one packet starts transmission at time t, and all packets have unit length, any other transmission starting between t − 1 and t + 1 will cause a collision Assume infinite number of nodes A node is considered backlogged from the time it has determined that its previously transmitted packet was involved in a collision until the time that it attempts retransmission Number of backlogged nodes is n Assume that period until attempted retransmission τ is exponentially distributed with probability density xe−xτ, where x is interpreted as retransmission attempt rate

Information Networks – p.14/22

Unslotted Aloha

With an overall Poisson arrival rate λ, the times of attempted retransmissions is a time-varying Poisson process with rate G(n) = λ + nx where n is the backlog at a given time Let τi be the interval between the ith and (i + 1)th transmission attempt, the ith attempt will be successful if both τi and τi−1 exceed 1 (assuming all packets have length 1) The probability distribution for the interval τi is

G(n)e−G(n)τi thus the probability that τi > 1 is e−G(n)

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

Information Networks – p.15/22

Unslotted Aloha

Attempted transmissions occur at rate G(n), the expected number of successful transmissions per unit time, the throughput as a function of n is G(n)e−2G(n) The situation is very similar to slotted Aloha, except the maximum throughput is 1/(2e) achieved when

G(n) = 1/2

We have assumed that backlog is same in intervals surrounding a given transmission attempt, but whenever a backlogged packet initiates a transmission the backlog decreases by 1 and whenever a collided packet is detected it increases by 1, for small x this error is relatively small

Information Networks – p.16/22

Unslotted Aloha

We have the same stability problems as in slotted Aloha With limited feedback stability is quite difficult to achieve

• r analyze

One advantage with unslotted Aloha is that it can be used with variable length packets, this compensates for some of the inherent throughput loss and gives an advantage in simplicity As for unstabilized slotted Aloha, if we have very small arrival rate λ and very large mean retransmission time the system can be expected to run for a long time without major backlog buildup.

Information Networks – p.17/22

Aloha Summary

Assuming Poisson arrivals, collision or perfect reception, (0,1,e) feedback, retransmission of collisions, either no buffering or infinite set of nodes Already simplistic analysis identifies maximum throughput 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 for number of backlogged nodes, and thus expected number of backlogged nodes and (using Little’s theorem) average delay

Information Networks – p.18/22

Slotted Aloha Stationary Probabilities

5 10 15 20 25 30 −0.2 0.2 0.4 0.6 0.8 1 p0, p1, pm and rejection probability as function of number of nodes m

Information Networks – p.19/22

Slotted Aloha Average Delay

5 10 15 20 25 30 35 40 20 40 60 80 100 120 140 packet delay as function of nr of nodes, qr=0.2,0.3,0.4,0.6

Information Networks – p.20/22

Slotted Aloha Average Delay

5 10 15 20 25 30 2 4 6 8 10 12 14 16 18 20 packet delay as function of nr of nodes qr=0.05,0.1,0.2,0.3

Information Networks – p.21/22

Aloha Summary

Steady-state shows strange behaviour due to instability Analysis of the dynamics via the drift Stabilizing with pseudo-Bayesian algorithm, learning from the analysis of the dynamics Need to estimate number of backlogged nodes n, this is done from the feedback according to Bayesian strategy for idle/successful and approximate Bayesian strategy for collisions Waiting delay until successful transmission

W = e − 1/2 1 − λe − (e − 1)(eλ − 1) λ(1 − (e − 1)(eλ − 1))

Information Networks – p.22/22