Bounds on Query Convergence

The problem of finding an optimum using noisy evaluations of a smooth cost function arises in many contexts, including economics, business, medicine, experiment design, and foraging theory. We derive an asymptotic bound E[ (x_t - x*)^2 ] >= O(1/sqrt(t)) on the rate of convergence of a sequence (x_0, x_1, >...) generated by an unbiased feedback process observing noisy evaluations of an unknown quadratic function maximised at x*. The bound is tight, as the proof leads to a simple algorithm which meets it. We further establish a bound on the total regret, E[ sum_{i=1..t} (x_i - x*)^2 ] >= O(sqrt(t)) These bounds may impose practical limitations on an agent's performance, as O(eps^-4) queries are made before the queries converge to x* with eps accuracy.Comment: 6 pages, 2 figure

Trading link utilization for queueing delays: an adaptive approach

Understanding the relationship between queueing delays and link utilization for general traffic conditions is an important open problem in networking research. Difficulties in understanding this relationship stem from the fact that it depends on the complex nature of arriving traffic and the problems associated with modelling such traffic. Existing AQM schemes achieve a "low delay" and "high utilization" by responding early to congestion without considering the exact relationship between delay and utilization. However, in the context of exploiting the delay/utilization tradeoff, the optimal choice of a queueing scheme's control parameter depends on the cost associated with the relative importance of queueing delay and utilization. The optimal choice of control parameter is the one that maximizes a benefit that can be defined as the difference between utilization and cost associated with queuing delay. We present two practical algorithms, Optimal Drop-Tail (ODT) and Optimal BLUE (OB), that are designed with a common performance goal: namely, maximizing this benefit. Their novelty lies in fact that they maximize the benefit in an online manner, without requiring knowledge of the traffic conditions, specific delay-utilization models, nor do they require complex parameter estimation. Packet level ns2 simulations are given to demonstrate the efficacy of the proposed algorithms and the framework in which they are designed

Router-based algorithms for improving internet quality of service.

We begin this thesis by generalizing some results related to a recently proposed positive system model of TCP congestion control algorithms. Then, motivated by a mean ¯eld analysis of the positive system model, a novel, stateless, queue management scheme is designed: Multi-Level Comparisons with index l (MLC(l)). In the limit, MLC(l) enforces max-min fairness in a network of TCP flows. We go further, showing that counting past drops at a congested link provides su±cient information to enforce max-min fairness among long-lived flows and to reduce the flow completion times of short-lived flows. Analytical models are presented, and the accuracy of predictions are validated by packet level ns2 simulations. We then move our attention to e±cient measurement and monitoring techniques. A small active counter architecture is presented that addresses the problem of accurate approximation of statistics counter values at very-high speeds that can be both updated and estimated on a per-packet basis. These algorithms are necessary in the design of router-based flow control algorithms since on-chip Static RAM (SRAM) currently is a scarce resource, and being economical with its usage is an important task. A highly scalable method for heavy-hitter identifcation that uses our small active counters architecture is developed based on heuristic argument. Its performance is compared to several state-of-the-art algorithms and shown to out-perform them. In the last part of the thesis we discuss the delay-utilization tradeoff in the congested Internet links. While several groups of authors have recently analyzed this tradeoff, the lack of realistic assumption in their models and the extreme complexity in estimation of model parameters, reduces their applicability at real Internet links. We propose an adaptive scheme that regulates the available queue space to keep utilization at desired, high, level. As a consequence, in large-number-of-users regimes, sacrifcing 1-2% of bandwidth can result in queueing delays that are an order of magnitude smaller than in the standard BDP-bu®ering case. We go further and introduce an optimization framework for describing the problem of interest and propose an online algorithm for solving it

Router-based algorithms for improving internet quality of service.

We begin this thesis by generalizing some results related to a recently proposed positive system model of TCP congestion control algorithms. Then, motivated by a mean ¯eld analysis of the positive system model, a novel, stateless, queue management scheme is designed: Multi-Level Comparisons with index l (MLC(l)). In the limit, MLC(l) enforces max-min fairness in a network of TCP flows. We go further, showing that counting past drops at a congested link provides su±cient information to enforce max-min fairness among long-lived flows and to reduce the flow completion times of short-lived flows. Analytical models are presented, and the accuracy of predictions are validated by packet level ns2 simulations. We then move our attention to e±cient measurement and monitoring techniques. A small active counter architecture is presented that addresses the problem of accurate approximation of statistics counter values at very-high speeds that can be both updated and estimated on a per-packet basis. These algorithms are necessary in the design of router-based flow control algorithms since on-chip Static RAM (SRAM) currently is a scarce resource, and being economical with its usage is an important task. A highly scalable method for heavy-hitter identifcation that uses our small active counters architecture is developed based on heuristic argument. Its performance is compared to several state-of-the-art algorithms and shown to out-perform them. In the last part of the thesis we discuss the delay-utilization tradeoff in the congested Internet links. While several groups of authors have recently analyzed this tradeoff, the lack of realistic assumption in their models and the extreme complexity in estimation of model parameters, reduces their applicability at real Internet links. We propose an adaptive scheme that regulates the available queue space to keep utilization at desired, high, level. As a consequence, in large-number-of-users regimes, sacrifcing 1-2% of bandwidth can result in queueing delays that are an order of magnitude smaller than in the standard BDP-bu®ering case. We go further and introduce an optimization framework for describing the problem of interest and propose an online algorithm for solving it

Bounds on Query Convergence

The problem of finding an optimum using noisy evaluations of a smooth cost function arises in many contexts, including economics, business, medicine, experiment design, and foraging theory. We derive an asymptotic bound E[ (x_t - x*)^2 ] >= O(1/sqrt(t)) on the rate of convergence of a sequence (x_0, x_1, >...) generated by an unbiased feedback process observing noisy evaluations of an unknown quadratic function maximised at x*. The bound is tight, as the proof leads to a simple algorithm which meets it. We further establish a bound on the total regret, E[ sum_{i=1..t} (x_i - x*)^2 ] >= O(sqrt(t)) These bounds may impose practical limitations on an agent's performance, as O(eps^-4) queries are made before the queries converge to x* with eps accuracy