Network flow-control using asynchronous stochastic approximation

We propose several stochastic approximation implementations for related algorithms in flow-control of communication networks. First, a discrete-time implementation of Kelly's primal flow-control algorithm is proposed. Convergence with probability 1 is shown, even in the presence of communication delays and stochastic effects seen in link congestion indications. This ensues from an analysis of the flow-control algorithm using the asynchronous stochastic approximation (ASA) framework. Two relevant enhancements are then pursued: a) an implementation of the primal algorithm using second-order information, and b) an implementation where edge-routers rectify misbehaving flows. Next, discretetime implementations of Kelly's dual algorithm and primaldual algorithm are proposed. Simulation results a) verifying the proposed algorithms and, b) comparing the stability properties are presented.

A quantum stochastic Lie-Trotter product formula

A Trotter product formula is established for unitary quantum stochastic processes governed by quantum stochastic differential equations with constant bounded coefficients.

Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

Stochastic simulations of the origins and implications of long-tailed distributions in gene expression

Gene expression noise results in protein number distributions ranging from long-tailed to Gaussian. We show how long-tailed distributions arise from a stochastic model of the constituent chemical reactions and suggest that, in conjunction with cooperative switches, they lead to more sensitive selection of a subpopulation of cells with high protein number than is possible with Gaussian distributions. Single-cell-tracking experiments are presented to validate some of the assumptions of the stochastic simulations. We also examine the effect of DNA looping on the shape of protein distributions. We further show that when switches are incorporated in the regulation of a gene via a feedback loop, the distributions can become bimodal. This might explain the bimodal distribution of certain morphogens during early embryogenesis.

Optimal threshold policies for admission control in communication networks via discrete parameter stochastic approximation

The problem of admission control of packets in communication networks is studied in the continuous time queueing framework under different classes of service and delayed information feedback. We develop and use a variant of a simulation based two timescale simultaneous perturbation stochastic approximation (SPSA) algorithm for finding an optimal feedback policy within the class of threshold type policies. Even though SPSA has originally been designed for continuous parameter optimization, its variant for the discrete parameter case is seen to work well. We give a proof of the hypothesis needed to show convergence of the algorithm on our setting along with a sketch of the convergence analysis. Extensive numerical experiments with the algorithm are illustrated for different parameter specifications. In particular, we study the effect of feedback delays on the system performance.

Hybrid moving block bootstrap for stochastic simulation of multi-site multi-season streamflows

The Hybrid approach introduced by the authors for at-site modeling of annual and periodic streamflows in earlier works is extended to simulate multi-site multi-season streamflows. It bears significance in integrated river basin planning studies. This hybrid model involves: (i) partial pre-whitening of standardized multi-season streamflows at each site using a parsimonious linear periodic model; (ii) contemporaneous resampling of the resulting residuals with an appropriate block size, using moving block bootstrap (non-parametric, NP) technique; and (iii) post-blackening the bootstrapped innovation series at each site, by adding the corresponding parametric model component for the site, to obtain generated streamflows at each of the sites. It gains significantly by effectively utilizing the merits of both parametric and NP models. It is able to reproduce various statistics, including the dependence relationships at both spatial and temporal levels without using any normalizing transformations and/or adjustment procedures. The potential of the hybrid model in reproducing a wide variety of statistics including the run characteristics, is demonstrated through an application for multi-site streamflow generation in the Upper Cauvery river basin, Southern India. (C) 2004 Elsevier B.V. All rights reserved.

Use of Abel integral equations in water wave scattering by two surface-piercing barriers

In this paper the classical problem of water wave scattering by two partially immersed plane vertical barriers submerged in deep water up to the same depth is investigated. This problem has an exact but complicated solution and an approximate solution in the literature of linearised theory of water waves. Using the Havelock expansion for the water wave potential, the problem is reduced here to solving Abel integral equations having exact solutions. Utilising these solutions,two sets of expressions for the reflection and transmission coefficients are obtained in closed forms in terms of computable integrals in contrast to the results given in the literature which,involved six complicated integrals in terms of elliptic functions. The two different expressions for each coefficient produce almost the same numerical results although it has not been possible to prove their equivalence analytically. The reflection coefficient is depicted against the wave number in a number of figures which almost coincide with the figures available in the literature wherein the problem was solved approximately by employing complementary approximations. (C) 2009 Elsevier B.V. All rights reserved.

On some integral properties of hypergeometric functions

In this note certain integrals involving hypergeometric functions have been evaluated in convenient and elegant forms. © 1971 Indian Academy of Sciences.

A fourier-integral solution for the elastic quarter-plane

The problem of an elastic quarter-plane with arbitrary loadings on the boundaries has been solved using a Fourier-integral approach.

Optimal non-linear reinforcement schemes for stochastic automata

Two optimal non-linear reinforcement schemes—the Reward-Inaction and the Penalty-Inaction—for the two-state automaton functioning in a stationary random environment are considered. Very simple conditions of symmetry of the non-linear function figuring in the reinforcement scheme are shown to be necessary and sufficient for optimality. General expressions for the variance and rate of learning are derived. These schemes are compared with the already existing optimal linear schemes in the light of average variance and average rate of learning.

Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space X. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of global solution. The existence and convergence of the Faedo–Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator A. Finally we give an example to illustrate the applications of the abstract results.

Anti-windup Schemes for Proportional Integral and Proportional Resonant Controller

Problems like windup or rollover arise in a PI controller working under saturation. Hence anti-windup schemes are necessary to minimize performance degradation.Similar situation may occur in a Proportional Resonant(PR)controller in the presence of a sustained error input.Several methods can be employed based on existing knowledge on PI controller to counter this problem.In this paper few such schemes are proposed and implemented in FPGA and MATLAB and from the obtained results their possible use and limitations have been studied.

Performance analysis of a suprathreshold, stochastic resonance based nonlinear detector

In this paper we introduce a nonlinear detector based on the phenomenon of suprathreshold stochastic resonance (SSR). We first present a model (an array of 1-bit quantizers) that demonstrates the SSR phenomenon. We then use this as a pre-processor to the conventional matched filter. We employ the Neyman-Pearson(NP) detection strategy and compare the performances of the matched filter, the SSR-based detector and the optimal detector. Although the proposed detector is non-optimal, for non-Gaussian noises with heavy tails (leptokurtic) it shows better performance than the matched filter. In situations where the noise is known to be leptokurtic without the availability of the exact knowledge of its distribution, the proposed detector turns out to be a better choice than the matched filter.

A four-timescale algorithm for constrained stochastic optimization of RED

The overall performance of random early detection (RED) routers in the Internet is determined by the settings of their associated parameters. The non-availability of a functional relationship between the RED performance and its parameters makes it difficult to implement optimization techniques directly in order to optimize the RED parameters. In this paper, we formulate a generic optimization framework using a stochastically bounded delay metric to dynamically adapt the RED parameters. The constrained optimization problem thus formulated is solved using traditional nonlinear programming techniques. Here, we implement the barrier and penalty function approaches, respectively. We adopt a second-order nonlinear optimization framework and propose a novel four-timescale stochastic approximation algorithm to estimate the gradient and Hessian of the barrier and penalty objectives and update the RED parameters. A convergence analysis of the proposed algorithm is briefly sketched. We perform simulations to evaluate the performance of our algorithm with both barrier and penalty objectives and compare these with RED and a variant of it in the literature. We observe an improvement in performance using our proposed algorithm over RED, and the above variant of it.