Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in

Computer Networks

Dah-Ming Chiu and Raj Jain

Presented by Yao Zhao

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

Motivation (2)

• Fig. 1. Network performance as a function of the load. Broken curves indicate performance with deterministic service and interarri

val times

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

Control System

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

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(

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(

Criteria for Selecting Controls

• Efficiency– Closeness of the total load on the resource to the kne

e 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 goa

l state from any starting state

)(

)(2

2

i

i

xn

xFairness

Responsiveness and Smoothness of Binary Feedback System

• Equlibrium with oscillates around the optimal state

Vector Representation of the Dynamics

)(2

)(2

22

1

221

xx

xxFairness

Example of Multiplicative Increase/ Multiplicative Decrease Function

Example of Additive Increase/ Multiplicative Decrease Function

Convergence to Efficiency

• Negative feedback–

– So

– 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

Convergence to Fairness (1)

where c=a/b (6)

c>0

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

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

Truncated Case

•

•

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)

Vectorial Representation of Feasible conditions

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

Optimal convergence to Efficiency

•

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

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.

Practical Considerations

• Non-linear controls

• Delay feedback

• Utility of increased bits of feedback

• Guess the current number of users n

• Impact of asynchronous operation

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 e

very 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