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

D-M Chiu, R. Jain, "Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks," Computer Networks and ISDN Systems, 17 (1989), pp 1-14.

One line summary: In this paper the authors algebraically analyze a subset of algorithms for congestion avoidance with respect to a number of criteria, including efficiency, fairness, distributedness, and convergence; they are able to conclude that the optimal algorithm uses additive increase and multiplicative decrease.

Summary

In this paper, the authors analyze decentralized algorithms for congestion avoidance based on binary feedback signals, focusing on what they call increase/decrease algorithms. They focus on an abstract system in which there is one bottleneck resource used by n users. They assume that the binary feedback signal indicates whether this resource is overloaded or underloaded, and that the signal is sent by setting a bit in the packet header. They also assume that the feedback and control loop is synchronous across all users. The authors focus on linear control functions. In this context then, increase/decrease algorithms are merely those in which one signal indicates that the resource is underloaded and the load should be increased, and the other signal means the opposite. Examples of such algorithms include multiplicative increase/additive decrease and additive increase/multiplicative decrease.

The criteria by which the authors select controls are efficiency, fairness, distributedness, and convergence. Efficiency is measure by how close the total load on a resource is to the point where any further load would cause a decrease in response time and throughput, i.e. the point of maximum utilization and throughput and lowest response time. Fairness is measured by a function that the authors define which attempts to measure how equal the allocation of the resource is across all users. Distributedness means that the users only have limited knowledge of the system, in particular, whether the resource is overloaded or underloaded. Lastly, convergence is a measure of both responsiveness, or how fast the system takes to reach equilibrium, and of smoothness, or how large the oscillations around the equilibrium are. They choose to examine distributed linear control functions that converge to efficient and fair states.

In order to analyze the set of possible control functions, the authors represent system state transitions in an n-dimensional vector space (representing the load for each of n users) across discrete time slots. The ith user’s load at time t is x_i(t) and the load at the next time interval, x_i(t+1), is a function of the control scheme used with the load at time t and the system feedback. For example, it could be that x_i(t+1) = a + b*x_i(t) if the binary signal is indicates the resource is underloaded. In this function, a is additive and b is multiplicative. With this representation, they are able to deduce algebraically a number of propositions. The first is essentially that the decrease policy should be multiplicative and the increase policy should have an additive component and can optionally have a multiplicative component with coefficient no less than one. The second proposition is that both the increase and the decrease can have both additive and multiplicative components provide that a special step is taken in the case when certain convergence restrictions would be violated. The third and most conclusive proposition is that for feasibility and optimal convergence, the increase should be additive and the decrease should be multiplicative.

At the end of the paper, the authors describe what nonlinear control functions in congestion avoidance algorithms might look like, but reason that because such control is too sensitive to various system parameters, the robustness of the control is decreased. On this note, they go on to enumerate a few practical considerations in choosing an algorithm for congestion avoidance. The first is that it should be independent of hardware and software parameters as much as possible in order to reduce the complexity of the configuration task and make configuration errors less possible. The second consideration is that the resources and allocations must be measured in integers, so this should be taken into account when choosing an algorithm. The last practical consideration they mention is ease of implementation. Given all these considerations in addition to the result of their earlier analysis, they conclude that an algorithm that uses a linear control function with additive increase and multiplicative decrease is optimal.

Critique

I didn’t like everything about this paper. I thought that it was a bit too abstract, and I think in general it is no trivial matter to translate abstract results into real working solutions. However, given that there appear to be so many different algorithms for congestion avoidance, of which they only examine a subset, their analysis does provide some solid guidance in choosing among these, especially if you accept that the conclusions they draw from their analysis of their simplified system still completely hold in the much more complex environment of real systems. Since real TCP uses additive increase/multiplicative decrease, it seems that this could be a safe bet.

I’d be curious to see if algorithms with control functions other than additive increase/multiplicative decrease have been tested in practice. The authors mention that this paper is just one in a series of papers in which they examine a range of mechanisms for congestion avoidance, so an interesting exercise might be to take some of the algorithms they look at in their other papers and actually test them against additive increase/multiplicative decrease, and also maybe examine some algorithms that use nonlinear control functions.

One thing I did really like about this paper is their use of graphs. Whereas some of the algebra was at times a bit tedious or difficult to follow, I thought the graphs helped them provide a nice, clear, and more intuitive explanation for how the control functions worked and how they converged to fairness and efficiency. Although these graphs were further simplified on top of their simplifying assumptions, in that they represented the case with only two users, they were still pretty much key in helping me to understand some of their conclusions.