Randomized Load Balancing on Networks with Stochastic Inputs

Iterative load balancing algorithms for indivisible tokens have been studied intensively in the past. Complementing previous worst-case analyses, we study an average-case scenario where the load inputs are drawn from a fixed probability distribution. For cycles, tori, hypercubes and expanders, we obtain almost matching upper and lower bounds on the discrepancy, the difference between the maximum and the minimum load. Our bounds hold for a variety of probability distributions including the uniform and binomial distribution but also distributions with unbounded range such as the Poisson and geometric distribution. For graphs with slow convergence like cycles and tori, our results demonstrate a substantial difference between the convergence in the worst- and average-case. An important ingredient in our analysis is a new upper bound on the t-step transition probability of a general Markov chain, which is derived by invoking the evolving set process

Block-Structured Supermarket Models

Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging. This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinite-dimensional system of differential vector equations satisfied by the expected fraction vector, where "the invariance of environment factors" is given as an important result. (2) Introducing the phase-type structure to the operator semigroup and to the mean-field limit, and a Lipschitz condition can be obtained by means of a unified matrix-differential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times. Thus the results of this paper give new highlight on understanding influence of non-Poisson inputs and of non-exponential service times on performance measures of more general supermarket models.Comment: 65 pages; 7 figure

Gossip Algorithms for Distributed Signal Processing

Gossip algorithms are attractive for in-network processing in sensor networks because they do not require any specialized routing, there is no bottleneck or single point of failure, and they are robust to unreliable wireless network conditions. Recently, there has been a surge of activity in the computer science, control, signal processing, and information theory communities, developing faster and more robust gossip algorithms and deriving theoretical performance guarantees. This article presents an overview of recent work in the area. We describe convergence rate results, which are related to the number of transmitted messages and thus the amount of energy consumed in the network for gossiping. We discuss issues related to gossiping over wireless links, including the effects of quantization and noise, and we illustrate the use of gossip algorithms for canonical signal processing tasks including distributed estimation, source localization, and compression.Comment: Submitted to Proceedings of the IEEE, 29 page

State Estimation for the Individual and the Population in Mean Field Control with Application to Demand Dispatch

This paper concerns state estimation problems in a mean field control setting. In a finite population model, the goal is to estimate the joint distribution of the population state and the state of a typical individual. The observation equations are a noisy measurement of the population. The general results are applied to demand dispatch for regulation of the power grid, based on randomized local control algorithms. In prior work by the authors it has been shown that local control can be carefully designed so that the aggregate of loads behaves as a controllable resource with accuracy matching or exceeding traditional sources of frequency regulation. The operational cost is nearly zero in many cases. The information exchange between grid and load is minimal, but it is assumed in the overall control architecture that the aggregate power consumption of loads is available to the grid operator. It is shown that the Kalman filter can be constructed to reduce these communication requirements,Comment: To appear, IEEE Trans. Auto. Control. Preliminary version appeared in the 54rd IEEE Conference on Decision and Control, 201

Steady state analysis of balanced-allocation routing

We compare the long-term, steady-state performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks, to the performance of a path-selection algorithm based on the "balanced-allocation" principle; we refer to this new algorithm as the Balanced Dynamic Alternative Routing (BDAR) algorithm. While DAR checks alternative routes sequentially until available bandwidth is found, the BDAR algorithm compares and chooses the best among a small number of alternatives. We show that, at the expense of a minor increase in routing overhead, the BDAR algorithm gives a substantial improvement in network performance, in terms both of network congestion and of bandwidth requirement.Comment: 22 pages, 1 figur