Iddo Eliazar

Lévy-Driven Langevin Systems: Targeted Stochasticity

Langevin dynamics driven by random Wiener noise (“white noise”), and the resulting Fokker–Planck equation and Boltzmann equilibria are fundamental to the understanding of transport and relaxation. However, there is experimental and theoretical evidence that the use of the Gaussian Wiener noise as an underlying source of randomness in continuous time systems may not always be appropriate or justified. Rather, models incorporating general Lévy noises, should be adopted. In this work we study Langevin systems driven by general Lévy, rather than Wiener, noises. Various issues are addressed, including: (i) the evolution of the probability density function of the systems state; (ii) the systems steady state behavior; and, (iii) the attainability of equilibria of the Boltzmann type. Moreover, the issue of reverse engineering is introduced and investigated. Namely: how to design a Langevin system, subject to a given Lévy noise, that would yield a pre-specified “target” steady state behavior. Results are complemented with a multitude of examples of Lévy driven Langevin systems.

Searching circular DNA strands

We introduce and explore a model of an ensemble of enzymes searching, in parallel, a circular DNA strand for a target site. The enzymes performing the search combine local scanning—conducted by a 1D motion along the strand—and random relocations on the strand—conducted via a confined motion in the medium containing the strand. Both the local scan mechanism and the relocation mechanism are considered general. The search durations are analysed, and their limiting probability distributions—for long DNA strands—are obtained in closed form. The results obtained (i) encompass the cases of single, parallel and massively parallel searches, taking place in the presence of either finite-mean or heavy-tailed relocation times, (ii) are applicable to a wide spectrum of local scan mechanisms including linear, Brownian, selfsimilar, and sub-diffusive motions, (iii) provide a quantitative theoretical justification for the necessity of the relocation mechanism, and (iv) facilitate the derivation of optimal relocation strategies.

A unified and universal explanation for Levy laws and 1/f noises.

Lévy laws and 1/f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals. The signals are considered to follow, statistically, a common—yet arbitrary—generic signal pattern which may be either stationary or dissipative. Each signal is considered to have its own random transmission amplitude and frequency. We characterize the amplitude-frequency randomizations which render the system outputs stationary law and power-spectrum universal—i.e., independent of the underlying generic signal pattern. The classes of universal stationary laws and power spectra are shown to coincide, respectively, with the classes of Lévy laws and 1/f noises—thus providing a unified and universal explanation for the ubiquity of these classes of “anomalous statistics” in various fields of science and engineering.

Polling under the randomly timed gated regime

Polling systems under the Randomly Timed Gated (RTG) regime are studied and analyzed, and various performance measures are derived. The RTG protocol operates as follows: whenever the server enters a station, a Timer is activated. If the server empties the queue before the Timer‘s expiration, it moves on to the next node. Otherwise (i.e., if there is still work in the station when the Timer expires), the server obeys one of the following rules, each leading to a different model: (1) The server completes all the work accumulated up to the Timer‘s expiration and then moves on to the next node. (2) The server completes only the service of the job currently being served, and moves on. (3) The server stops working immediately and moves on. The RTG protocol defines a wide class of time-based control mechanisms: under rule (1) it spans an entire spectrum of regimes lying between the Gated and the Exhaustive, while under rule (2) it spans an entire spectrum between the 1-Limited and the Exhaustive protocols

The M/G/∞ system revisited: finiteness, summability, long range dependence, and reverse engineering

We explore M/G/∞ systems ‘fed’ by Poissonian inflows with infinite arrival rates. Three processes – corresponding to the systems state, workload, and queue-size – are studied and analyzed. Closed form formulae characterizing the systems stationary structure and correlation structure are derived. And, the issues of queue finiteness, workload summability, and Long Range Dependence are investigated.We then turn to devise a ‘reverse engineering’ scheme for the design of the systems correlation structure. Namely: how to construct an M/G/∞ system with a pre-desired ‘target’ workload/queue auto-covariance function. The ‘reverse engineering’ scheme is applied to various examples, including ones with infinite queues and non-summable workloads.

Facilitated diffusion in a crowded environment: from kinetics to stochastics

Facilitated diffusion is a fundamental search process used to describe the problem of a searcher protein finding a specific target site over a very large DNA strand. In recent years macromolecular crowding has been recognized to affect this search process. In this paper, we bridge between two different modelling methodologies of facilitated diffusion: the physics-oriented kinetic approach, which yields the reaction rate of the search process, and the probability-oriented stochastic approach, which yields the probability distribution of the search duration. We translate the former approach to the latter, ascertaining that the two approaches yield coinciding results, both with and without macromolecular crowding. We further show that the stochastic approach markedly generalizes the kinetic approach by accommodating a vast array of search mechanisms, including mechanisms having no reaction rates, and thus being beyond the realm of the kinetic approach.

The Snowblower Problem

A snowblower is circling a closed-loop racetrack, driving clockwise and clearing off snow in a constant snowblowing rate. Both the snowfall and the snowblowers driving speed vary randomly (in both space and time coordinates). The snowblowers motion and the snowload profile on the racetrack are co-dependent and co-evolve, resulting in a coupled stochastic dynamical system of ‘random motion (snowblower) in a random environment (snowload profile)’. Snowblowing systems are closely related to continuous polling systems – or, so-called, polling systems on the circle – which are the continuum limits of ‘standard’ polling systems. Our aim in this manuscript is to introduce a stochastic model that would apply to a wide class of stochastic snowblower-type systems and, simultaneously, generalize the existing models of continuous polling systems. We present a general snowblowing-system model, with arbitrary Lévy snowfall and arbitrary snowblower delays, and study it by analyzing an underlying stochastic Poincaré map governing the systems evolution. The log-Laplace transform and mean of the Poincaré map are computed, convergence to steady state (equilibrium) is proved, and the systems equilibrium behavior is explored.

Gated Polling Systems with Lévy Inflow and Inter-Dependent Switchover Times: A Dynamical-Systems Approach

Abstract We study asymmetric polling systems where: (i) the incoming workflow processes follow general Lévy-subordinator statistics; and, (ii) the server attends the channels according to the gated service regime, and incurs random inter-dependentswitchover times when moving from one channel to the other. The analysis follows a dynamical-systems approach: a stochastic Poincaré map, governing the one-cycle dynamics of the polling system is introduced, and its statistical characteristics are studied. Explicit formulae regarding the evolution of the mean, covariance, and Laplace transform of the Poincaré map are derived. The forward orbit of the map’s transform – a nonlinear deterministic dynamical system in Laplace space – fully characterizes the stochastic dynamics of the polling system. This enables us to explore the long-term behavior of the system: we prove convergence to a (unique) steady-state equilibrium, prove the equilibrium is stationary, and compute its statistical characteristics.

On the discrete-time g/gi/∞ queue*

The discrete-time G/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following an arbitrary input process X; job sizes are general independent and identically distributed random variables. The systems output process Y (of job departures) and queue process N (tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic maps X↦ Y and X↦ N are explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in the L2 sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsic L2 structure of general stationary G/GI/∞ systems.

From Polling to Snowplowing

We study the limiting behavior of gated polling systems, as their dimension (the number of queues) tends to infinity, while the systems total incoming workflow and total switchover time (per cycle) remain unchanged. The polling systems are assumed asymmetric, with incoming workflow obeying general Lévy statistics, and with general inter-dependent switchover times. We prove convergence, in law, to a limiting polling system on the circle. The derivation is based on an asymptotic analysis of the stochastic Poincaré maps of the polling systems. The obtained polling limit is identified as a snowplowing system on the circle—whose evolution, steady-state equilibrium, and statistics have been recently investigated and are known.