stability and fairness of service networks jean walrand – u.c. berkeley joint work with a....

Stability and Fairness of Service Networks

Jean Walrand – U.C. Berkeley

Joint work with A. Dimakis, R. Gupta, and J. Musacchio

Outline

Stability of Longest Queue First Fluctuations can stabilize

Fairness through flow control Control of long term rates

Fairness of multiple access Impatience may help in a crowd

Stability of Longest Queue Firstwith Antonis Dimakis (PhD 5/06)

Motivation Easy Case Subtle Effect

LQF - Motivation Wireless:

Goals: Simple protocol, large throughputTransmission priority increases with backlog

2 3 4 5 1 6

LQF: Motivation

Iterated Longest Queue First (iLQF) [McKeown’95]:Queues are considered in decreasing queue size order.

Maximum throughput?

1112

22

21

1 1

2 2

input 1

input 2output 2

output 1

Q11(t)

Q12(t)

Q21(t)

Q22(t)

Switch:

Stability of LQF: Easy CaseExample:

LQF: 12 – 9 – 8

7 – 9 – 8

9 – 9 – 8

9 – 9 – 8

w.p. 1/2

1 2 3

1 2 3

service vectors

i.i.d. arrivals

w.p. 1/2

Stability of LQF: Easy Case Necessary:

1+2<1, 2+3<1.

Sufficient!

Under LQF, longest queues tend to decrease:Say, Q1¼ Q2>>Q3, for some time.

Then, Q1+Q2 decreases, and so do Q1,Q2.

Key: locally in time, service from common resource pool.

1 2 3

1 2 3

Stability of LQF: Easy Case

Local Pooling:

Assume {1, 2} are longest for some time

Note that {1, 2} are served at constant rate (1)

(We say that {1, 2} satisfies Local Pooling.)

{1, 2} must decrease (because 1+2<1)

longest queue must decrease

1 2 3

1 2 3


Local Pooling:

Assume {1, 2, 3} are longest for some time


(We say that {1, 2, 3} satisfies Local Pooling.)



1 2 3

1 2 3


Local Pooling:

Assume {1, 3} are longest for some time


(We say that {1, 3} satisfies Local Pooling.)



1 2 3

1 2 3

Stability of LQF: Easy Case Local Pooling: Set L satisfies LP if

it has a subset K that LQF serves at a constant rate

Theorem:If every set L satisfies LP and if the rates are feasible,then LQF makes system stable

Proof: Longest queue is a Lyapunov function(Consider fluid limit ….)

1 2 3

1 2 3

Stability of LQF: Easy Case Graphs that satisfy Local Pooling:

Trees3, 4, 5 Cycles

Combinations

Stability of LQF: Subtle Effect Graph that does not satisfy Local Pooling:

3

6 5

1 4

2

{1, 2, 3, 4, 5, 6} has no subset served at constant rate {1, 2, 3, 4, 5, 6} does not satisfy LP

Every proper subset satisfies LP

E.g., {1, 2, 3, 5} longest serve {2, 3} at rate 1

Service Vectors: {1, 3, 5}, {2, 4, 6} {1, 4}, {2, 5}, {3, 6}

Stability of LQF: Subtle Effect Note: Deterministic inputs with

rate close to 0.5 unstable

(LQF serves 2/6 a positive fraction of time)

3

6 5

1 4

2

Theorem: LQF stable if i.i.d. arrivals with nonzero variance

Key Idea: {1, 2, 3, 4, 5, 6} cannot be set of longest queues for a positive fraction of time! LP holds most of the time Longest queue decreases

Stability of LQF: Subtle Effect

3

6 5

1 4

2Key Idea: {1, 2, 3, 4, 5, 6} cannot be set of longest queues for a positive fraction of time!

Assume all queues are longest for a while{2, 3} and {5, 6} served at same rate


(n)

(k + 1)(n)k(n)

L

Max – Min large at k(n) A subset L of queues dominates the others during intervalThis subset satisfies LP Longest queue decreases.

Stability of LQF: Subtle EffectTheorem:

Assume that whenever a set L does not satisfy LP, the corresponding service vectors have rank ≤ |L| - 2.

Assume also the arrivals are i.i.d. with positive variance(and satisfy a large deviation bound).

Then LQF is stable for any feasible arrival rates.

Stability of LQF: Subtle EffectExamples:

1 2

3

4

56

7

83

6 5

1 4

2

Outline




Fairness Through Flow Control(with John Musacchio, UCSC)

Motivation Analysis

Motivation Example

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a1

a2

b1

b2

h: discardthreshold

11

12

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22

1

1

Intuitively: h large enough max – min fair

Long-term average rates max – min for h >> 1

h = discard threshold

Motivation

11

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22

0.75

0.25

0.5

0.5

h: discardthreshold

1

1

0.75

0.25

0.2

0.7

h = discard threshold

Motivation

Fairness Through Flow Control(with John Musacchio, UCSC)

Motivation Analysis

Analysis

11

12

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21

12

22

0.75

0.25

0.5

0.5

nh

n

n

n

n

n

n

Qn(nt): Scale thresholds and speed up

Analysis

11

12

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22

11

21

12

22

0.75

0.25

0.5

0.5

h

1

1

Qn(nt)/n: Scale space

Qn(nt)/n fluid limit Q(t) with suitable rates….

AnalysisRoughly, x(n; t) := Qn(nt)/n uoc fluid limit

Q(t)For t ≥ t0, Q(t) = q* with suitable rates.This implies

Key argument: Most of the time t ≥ 0, x(n; t) ≈ q*

However, we want

Motivation

11

12

21

22

11

21

12

22

0.75

0.25

0.5

0.5

h: discardthreshold

1

1

0.75

0.25

0.2

0.7

q*:

Motivationq*:

Analysis Key argument:

Average throughputclose to max min

Outline




Fairness of Multiple Accesswith Rajarshi Gupta (PhD 5/05)

Motivation Protocol Analysis Simulations

Motivation: Exponential Backoff is Unfair Exponential backoff scheme

(e.g. 802.11b) Nodes pick backoff uniformly in a

backoff range If collision, double the backoff range

Multiple interference domains Node in center sees more

contention and collision It backs off more Gets lesser share of bandwidth

Unfair towards middle nodes in network

Active Link

Rcvd on A

Rcvd on B

Rcvd on X

A 6

A,B 6 6

A,X 3 3

A,B,X 4 4 2

A1

inte

rfer

ence

inte

rfer

ence

A2

B1

B2

X1

X2

Cory HallRoom 273

Cory HallRoom 264M

CoryHallway

All rates in Mbps

Protocol: Impatient Backoff Algorithm

Approach: Nodes that face more contention should get higher priority

Key MechanismUpon collision, nodes decrease their backoff

Need to worry about Stability Fairness Throughput

Protocol: Backoff Update If collision or quiet

Decrease the mean backoff delay b := b/m, where m>1

If successful transmission Increase the mean backoff delay b := bm

Note: Distributed reset mechanismWhen a node’s mean delay falls below threshold, node broadcasts “multiply by K” ….

Protocol: Simplified MAC Model

All packet lengths are same Transmissions occur slot by slot Local synchronization is assumed

Similar to any slotted protocol

No RTS/CTS

Protocol: IBA Mechanism

Backoff Contention Phase Each node has mean backoff b Picks backoff delay B using

exponential variable with mean b Sends out Slot Capture Message

after B backoff mini-slots If a node carrier senses another

message sooner – it keeps quiet

Packet Transmission Phase Starts after completion of Backoff

Contention Phase Nodes with successful Slot Capture

Messages transmit Constant packet length Transmission confirmed by ack

1 5432

interference

1

5

4

3

2

1's Packet Transmission

5's Packet Transmission

BackoffContentionPhase

PacketTransmissionPhase

backoff

slot capture

ack

ack

Collision occurs if two neighbors pick same backoff

Neither hears slot capture

Both try to transmit Packet transmission

wasted

Markov Chain Models Two extreme topologies

Star Topology (unfair) Triangle Clique Topology (symmetric)

Model ratio between mean backoffs Prove stability, fairness

Throughput-fairness tradeoff (in Star) Max throughput = 0 + 41 = 4 But fair throughput = 0.5 + 40.5 = 2.5

interference

interference

Star Topology: Birth-Death Chain

Stable: Positive recurrent for m>1 Strong drifts towards stable state S0

Fair: Expected transmission rate for all nodes is 0.5

n n.m4

n.m2

n.m-2

n.m-4

S0

S1 S2S-1S-2

interference

Star Topology: Varying Neighbors

Model sleeping nodes Every 100 slots, some

nodes go to sleep Fairness = 1 Average success

probability Middle Node(sX)= 0.473

Outer Nodes(sZ)= 0.470

Triangle Topology Markov Chain

111

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1416

11664

1116

1464

116

256

14

256

1164

16464

1 1/9 1/33

8/9 32/33 128/1292/3

1/3 1/21

16/21 64/81

4/21 16/818/9

64/69 256/273

1/9 1/69

4/69 16/27332/33

1/33

256/261

interference

m=2

Prove positive recurrence using Lyapunov function Chain drifts towards bottom-left stable states Fairness is due to symmetry

Simulations on Random TopologyExponential Backoff

Min Throughput = 9% of mean Jain’s Fairness Index = 0.58 Mean Throughput = 0.101

Min Throughput = 49% of mean Jain’s Fairness Index = 0.68 Mean Throughput = 0.102 Circle = Node : Center = Location, Area = Throughput

Impatient Backoff

Variations in Simulation Nodes execute random walk Initial bias against selected nodes Nodes switch between active and sleep cycles

Similar comparisons with exponential backoffComparable throughputSignificantly better fairness

Conclusions

Stability of LQF if eitherLocal Pooling, or L not LP, rank(Service) ≤ |L| - 2 ( L “splits”)

Fairness through flow controlKey idea is long term rates through 2 scalings:

Impatient backoff for fair MAC in ad hoc networks

stability and fairness of service networks jean walrand – u.c. berkeley joint work with a....

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