1. “Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks”
Lecture Note 7

2. “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

3. Motivation (1) Internet is heterogeneous Congestion control
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

4. Fig. 1. Network performance as a function of the load.
Motivation (2)
Fig. 1. Network performance as a function of the load.

5. 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.

6. Power Curve

7. Motivation (2) Power = {Goodput}/{Response Time}
Fig. 1. Network performance as a function of the load.

8. Relate Works Centralized algorithm Decentralized algorithms
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

9. Control System

10. 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

11. Linear Control (2) Additive Increase/Additive Decrease
Multiplicative Increase/Multiplicative Decrease
Additive Increase/Additive Decrease
Additive Increase/Multiplicative Decrease
Multiplicative Increase/ Additive Decrease

12. 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

13. Responsiveness and Smoothness of Binary Feedback System
Equlibrium with oscillates around the optimal state

14. Vector Representation of the Dynamics

15. Example of Additive Increase/ Additive Decrease Function

16. Example of Additive Increase/ Multiplicative Decrease Function

17. Convergence to Efficiency
Negative feedback So
If y=0:
If y=1: Or

18. Convergence to Fairness (1)
where c=a/b (6)
c>0

19. Convergence to Fairness (2)
c>0 implies:
Furthermore, combined with (3) we have:

20. 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

21. Truncated Case

22. 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)

23. Vectorial Representation of Feasible conditions

24. 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

25. Optimal convergence to Efficiency
Given initial state X(0), the time to reach Xgoal is:

26. 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.

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

28. 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