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

“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

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

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

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.

Power Curve

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

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

Control System

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

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

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

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

Vector Representation of the Dynamics

Example of Additive Increase/ Additive Decrease Function

Example of Additive Increase/ Multiplicative Decrease Function

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

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

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

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

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 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