We present a solution of a class of network utility maximization (NUM) problems using minimal communication. The constraints of the problem are inspired less by TCP-like congestion control but by problems in the area of internet of things and related areas in which the need arises to bring the behavior of a large group of agents to a social optimum. The approach uses only intermittent feedback, no inter-agent communication, and no common clock. The proposed algorithm is a combination of the classical AIMD algorithm in conjunction with a simple probabilistic rule for the agents to respond to a capacity signal. This leads to a nonhomogeneous Markov chain and we show almost sure convergence of this chain to the social optimum.
The AIMD literature is huge and it is not straightforward to discuss the available results in any sort of compact manner here. We refer interested readers to some recent works on this topic in the context of TCP and internet congestion control @cite_22 @cite_15 @cite_11 @cite_26 @cite_19 @cite_26 @cite_6 @cite_49 @cite_48 . Much of this work is based on fluid approximations of AIMD dynamics; the notable exceptions are @cite_42 @cite_1 @cite_2 . The latter of these papers make use of tools from iterated function systems to deduce the existence of such a unique probability distribution for standard linear AIMD networks (of which TCP is an example) under an assumption on the underlying probability model (albeit under very restrictive assumptions). To the best of our knowledge, this paper, along with the companion paper @cite_42 , established for the first time, the stochastic convergence of AIMD networks. However, the window of infinite length considered in this paper goes well beyond the set-up in these papers. In particular, the result presented in this paper may be considered as the limiting case of the results presented in these papers.
@cite_26 @cite_22 @cite_48 @cite_42 @cite_1 @cite_6 @cite_19 @cite_49 @cite_2 @cite_15 @cite_11
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We study communication networks that employ drop-tail queueing and Additive-Increase Multiplicative-Decrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties, such as: 1) fairness; 2) rate of convergence; and 3) throughput, can be characterized by certain nonnegative matrices. We demonstrate that these results can be used to develop tools for analyzing the behavior of AIMD communication networks. The accuracy of the models is demonstrated by several NS studies. This papers analyzes a class of nonlinear additive-increase multiplicative-decrease (AIMD) protocols that are widely deployed in communication networks. It is demonstrated that the use of these protocols guarantees that the system has a unique stable outcome to which it converges geometrically under all starting points. The development is based on a contraction argument and the derivation of explicit bounds on the contraction coefficient of corresponding operators in terms of the network parameters. In particular, bounds on the corresponding rate of convergence are obtained, improving upon known bounds for standard (linear) AIMD networks. .We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks. 1. Introduction. This paper is a continuation of Dai and Lin (2005), in which maximum pressure policies are shown to be throughput optimal for a class of stochastic processing networks. Throughput optimality is an important, first-order objective for many networks, but it ignores some key secondary performance measures like queueing delays experienced by jobs in these networks. In this paper we show that maximum pressure policies enjoy additional optimality properties; they are asymptotically optimal in minimizing a certain workload or holding cost of a stochastic processing network. Stochastic processing networks have been introduced in a series of three papers by Harrison (2000, 2002, 2003). In Dai and Lin (2005) and this paper we consider a special class of Harrison’s model. This class of stochastic processing networks is much more general than multiclass queueing networks that have been a subject of intensive study in the last 20 years; see, for example, Harrison (1988), Williams In this paper we investigate the potential benefits of coordinated congestion control for multipath data transfers, and contrast with uncoordinated control. For static random path selections, we show the worst-case throughput performance of uncoordinated control behaves as if each user had but a single path (scaling like log(log(N)) log(N) where N is the system size, measured in number of resources). Whereas coordinated control gives a throughput allocation bounded away from zero, improving on both uncoordinated control and on the greedy-least loaded path selection of e.g. Mitzenmacher. We then allow users to change their set of routes and introduce the notion of a Nash equilibrium. We show that with RTT bias (as in TCP Reno), uncoordinated control can lead to inefficient equilibria. With no RTT bias, both uncoordinated or coordinated Nash equilibria correspond to desirable welfare maximising states. Moreover, simple path reselection polices that shift to paths with higher net benefit can find these states.