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Analysis of the Increase and Decrease Algorithms for Congestion

Avoidance in Computer Networks

• Paper by: Dah-Ming Chiu (Digital Equipment Corporation)

Raj Jain (Digital Equipment Corporation)

• Published in: Computer Networks and ISDN Systems (1989)

• Presented by: Max Podlesny

•Discussion Leader: Michela Becchi

CS7701: Research Seminar on Networkinghttp://arl.wustl.edu/~jst/cse/770/

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Outline

• Problem• Linear Controls• Optimizing the Control Schemes• Nonlinear Controls• Conclusion

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Problem• Mismatch of arrival and service rates

– Increased queuing in a buffer of a router– Packet drops

• Does TCP congestion control scheme have a theoretical basis?

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What is congestion?

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Criteria for selecting controls

• Distributedness• Efficiency• Fairness• Convergence

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Control system model

• n users share the resource• Time is divided into small slots• The i-th user’s load is xi(t) at time slot t

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Control system model

• The total load at the bottleneck resource is xi(t)

• The state of the system is characterized by

x(t)={x1(t), x2(t), …, xn(t)}

• Xgoal is the desired load level

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Control system model

• Users receive the same feedback• Users receive feedback at the same time

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Control system model

• Binary feedback y(t),

• xi(t+1) = xi(t) + f(xi(t), y(t))

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Control system model

User 1

User 2

User n

xi > Xgoal ?

x1

x2

xn

y

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Control functions

• MIMD (Multiple Increase/Multiple Decrease)

– bI > 1, 0 < bD,< 1

• AIAD (Additive Increase/Additive Decrease)

– aI > 0, aD,< 0

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Control functions

• AIMD (Additive Increase/Multiple Decrease)

– aI > 0, 0 < bD,< 1

• MIAD (Multiple Increase/Additive Decrease)

– bI > 0, aD,< 0

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Distributedness

• System does the minimum amount of feedback

• The following information is assumed to be unknown:– The desired load level– The number of users sharing the resource

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Efficiency

• X(t) > Xgoal or X(t) < Xgoal are considered to be inefficient

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Fairness

• 1/n<=F(x)<=1• Independent of scale(unit measurement)• A continuous function• Equal sharing by k users of n users, k-n users do

not receive any resource, F(x)=k/n

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Convergence

• Time taken till the system approaches the goal state from any starting state– Characterized by:

• Responsiveness• Smoothness

– Tradeoff between responsiveness and smoothness

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Vector representation

• 2 users, so n=2• x0 ={x10, x20}

• The fairness at any point (x1, x2) is equal

to

User 1’s allocation

User 2’sallocation

Fairness Line

x0

Efficiency Line

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AIAD

User 1’s allocation x1

User 2’sallocation

x2

Fairness Line

x0

Efficiency Line

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AIMD

User 1’s allocation x1

User 2’sallocation

x2

Fairness Line

x2

x0

x1

Efficiency Line

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How to find control function?

• Convergence to efficiency– The principle of negative feedback:

• if y(t) = 0 then xi(t+1) > xi(t)

• if y(t) = 1 then xi(t+1) < xi(t)

• Convergence to fairness– F(x(t)) 1 as t , so: F(x(t+1)) = F(x(t)) + (1-F(x(t))) (1- xi

2(t) / (c+xi(t))2),

where c = a/b

• Distributedness – if y(t) = 0 then xi(t+1) > xi(t) i

– if y(t) = 1 then xi(t+1) < xi(t) i

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Requirements for control function

• The linear increase policy should have an additive component, and optionally, a multiplicative component– aI > 0

– bI 1

• The linear decrease policy should be multiplicative– aD = 0

– 0 bD < 1

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Optimal Convergence to Efficiency

• Given– n states for n users, i.e. xi(t+1) = a + bxi(t),

i = 1, 2, …, n.– State of the system X(t) = xi(t)

– X(0) - initial state

– Xgoal – optimal state

• One can calculate– te - responsiveness

– se - smoothness

• te, se are decreasing functions of a and b

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Optimal Convergence to Fairness

• F(x(t+1)) – F(x(t)) is a monotonically increasing function of c = a/b

• aD = 0, so decrease steps have no affect on

Fairness

• The optimal value of bI is its minimum value, i.e.

bI =1

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Linear control conditions

• Increase policy should be additive• Decrease policy should be multiplicative

x1

x2

Fairness Line

x0

Efficiency Line

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Nonlinear Controls

• Control function

• A lot of complexity

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Conclusion

• Best control condition is AIMD within the described model

• AIMD is used in TCP congestion control• It is not the case in real networks• MIMD can also be a good decision

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Questions?