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

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Analysis of the Increase and Decrease Algorithms
for Congestion Avoidance in Computer Networks
CS7701 Research Seminar on Networking http//arl.
wustl.edu/jst/cse/770/

• 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

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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 users 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(t1) xi(t) f(xi(t), y(t))

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Control system model
User 1
x1
x2
?xi gt Xgoal ?
User 2
?

xn
User n
y
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Control functions

• MIMD (Multiple Increase/Multiple Decrease)
• bI gt 1, 0 lt bD,lt 1
• AIAD (Additive Increase/Additive Decrease)
• aI gt 0, aD,lt 0

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Control functions

• AIMD (Additive Increase/Multiple Decrease)
• aI gt 0, 0 lt bD,lt 1
• MIAD (Multiple Increase/Additive Decrease)
• bI gt 0, aD,lt 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) gt Xgoal or X(t) lt Xgoal are considered to be
  inefficient

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Fairness

• 1/nltF(x)lt1
• 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 n2
• x0 x10, x20
• The fairness at any point (x1, x2) is equal to

Fairness Line
x0

User 2s allocation
Efficiency Line
User 1s allocation
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AIAD
Fairness Line
User 2s allocation
x0
x2
Efficiency Line
User 1s allocation
x1
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AIMD
x1
Fairness Line
User 2s allocation
x0
x2
x2
Efficiency Line
User 1s allocation
x1
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How to find control function?

• Convergence to efficiency
• The principle of negative feedback
• if y(t) 0 then ?xi(t1) gt ? xi(t)
• if y(t) 1 then ?xi(t1) lt ? xi(t)
• Convergence to fairness
• F(x(t)) ? 1 as t ? ?, so
• F(x(t1)) F(x(t)) (1-F(x(t))) ? (1- ? xi2(t)
  / ? (cxi(t))2), where c a/b
• Distributedness
• if y(t) 0 then xi(t1) gt xi(t) ? i
• if y(t) 1 then xi(t1) lt 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 gt 0
• bI ? 1
• The linear decrease policy should be
  multiplicative
• aD 0
• 0 ? bD lt 1

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Optimal Convergence to Efficiency

• Given
• n states for n users, i.e. xi(t1) 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(t1)) 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

Fairness Line
x0
x2
Efficiency Line
x1
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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?