“Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks”

“Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks”

by Dah-Ming Chiu and Raj Jain, DEC

Computer Networks and ISDN Systems

17 (1989), pp. 1-14

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Motivation (1)

Internet is heterogeneous Different bandwidth of links Different load from users

Congestion control Help improve performance after

congestion has occurred

Congestion avoidance Keep the network operating off the

congestion

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Motivation (2)

Fig. 1. Network performance as a function of the load.

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Power of a Network The power of the network describes this

relationship of throughput and delay: Power = Goodput/Delay This is based on M/M/1 queues ( 1 server and

a Markov distribution of packet arrival and service).

This assumes infinite queues, but real networks the have finite buffers and occasionally drop packets.

The objective is to maximize this ration, which is a function of the load on the network.

Ideally the resource mechanism operates at the peak of this curve.

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Power Curve

Optimalload

Load

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Motivation (2)

Fig. 1. Network performance as a function of the load.

Power = {Goodput}/{Response Time}

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Relate Works

Centralized algorithm Information flows to the resource managers and

the decision of how to allocate the resource is made at the resource [Sanders86]

Decentralized algorithms Decisions are made by users while the

resources feed information regarding current resource usage [Jaffe81, Gafni82, Mosely84] Binary feedback signal and linear control Synchronized model What are all the possible solutions that converge

to efficient and fair states

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Control System

))(),(()()1( tytxftxtx iiiii

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Linear Control (1)

4 examples of linear control functions Multiplicative Increase/Multiplicative

Decrease Additive Increase/Additive Decrease Additive Increase/Multiplicative Decrease Additive Increase/ Additive Decrease

Decreasetyiftxba

Increasetyiftxbatx

iDD

iIIi 1)( )(

,0)( )()1(

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Linear Control (2) Multiplicative Increase/Multiplicative Decrease

Additive Increase/Additive Decrease

Additive Increase/Multiplicative Decrease

Multiplicative Increase/ Additive Decrease

Decreasetyiftxb

Increasetyiftxbtx

iD

iIi 1)( )(

,0)( )()1(

Decreasetyiftxa

Increasetyiftxatx

iD

iIi 1)( )(

,0)( )()1(

Decreasetyiftxb

Increasetyiftxatx

iD

iIi 1)( )(

,0)( )()1(

Decreasetyiftxa

Increasetyiftxbtx

iD

iIi 1)( )(

,0)( )()1(

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Criteria for Selecting Controls Efficiency

Closeness of the total load on the resource to the knee point

Fairness Users have the equal share of bandwidth

Distributedness Knowledge of the state of the system

Convergence The speed with which the system approaches

the goal state from any starting state

)(

)(2

2

i

i

xn

xFairness

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Responsiveness and Smoothness of Binary Feedback System

Equlibrium with oscillates around the optimal state

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Vector Representation of the Dynamics

)(2

)(2

22

1

221

xx

xxFairness

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Example of Additive Increase/ Additive Decrease Function

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Example of Additive Increase/ Multiplicative Decrease Function

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Convergence to Efficiency

Negative feedback

So If y=0: If y=1: Or

).()1(1)(

),()1(0)(

txtxty

txtxty

ii

ii

).( 0)()1(

),( 0)()1(

txandntxbna

txandntxbna

iiDD

iiII

)(1

,)(

1

tx

nab

tx

nab

i

DD

i

II

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Convergence to Fairness (1)

where c=a/b (6)

c>0

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Convergence to Fairness (2)

c>0 implies:

Furthermore, combined with (3) we have:

(9) 0 0

(8) 0 0

D

D

I

I

D

D

I

I

b

aand

b

a

or

b

aand

b

a

)10( 10 ,0

,0 ,0

DD

II

ba

ba

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Distributedness Having no knowledge other than the feedback y(t) Each user tries to satisfy the negative feedback

condition by itself

Implies (10) to be

)11( . )()1(1)(

, )()1(0)(

itxtxty

itxtxty

ii

ii

)12( 10 ,0

,1 ,0

DD

II

ba

ba

.0)( 0)()1(

,0)( 0)()1(

txtxba

txtxba

iiDD

iiII

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Truncated Case

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Important Results

Proposition 1: In order to satisfy the requirements of distributed convergence to efficiency and fairness without truncation, the linear increase policy should always have an additive component, and optionally it may have a multiplicative component with the coefficient no less than one.

Proposition 2: For the linear controls with truncation, the increase and decrease policies can each have both additive and multiplicative components, satisfying the constrains in Equations (16)

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Vectorial Representation of Feasible conditions

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Optimizing the Control Schemes

Optimal convergence to Efficiency Tradeoff of time to convergent to

efficiency te, with the oscillation size, se.

Optimal convergence to Fairness

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Optimal convergence to Efficiency

Given initial state X(0), the time to reach Xgoal is:

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Optimal convergence to Fairness

Equation (7) shows faireness function is monotonically increasing function of c=a/b.

So larger values of a and smaller values b give quicker convergence to fairness.

In strict linear control, aD=0 => fairness remains the same at every decrease step

For increase, smaller bI results in quicker convergence to fairness => bI =1 to get the quickest convergence to fairness

Proposition 3: For both feasibility and optimal convergence to fairness, the increase policy should be additive and the decrease policy should be multiplicative.

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Practical Considerations

Non-linear controls Delay feedback Utility of increased bits of feedback Guess the current number of users n Impact of asynchronous operation

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Conclusion

We examined the user increase/decrease policies under the constrain of binary signal feedback

We formulated a set of conditions that any increase/decrease policy should satisfy to ensure convergence to efficiency and fair state in a distributed manner We show the decrease must be multiplicative to

ensure that at every step the fairness either increases or stays the same

We explain the conditions using a vector representation

We show that additive increase with multiplicative decrease is the optimal policy for convergence to fairness