Random Time Instants Research Articles

Stochastic resetting with refractory periods: pathway formulation and exact results

We look into the problem of stochastic resetting with refractory periods. The model dynamics comprises diffusive and motionless phases. The diffusive phase ends at random time instants, at which the system is reset to a given position—where the system remains at rest for a random time interval, termed the refractory period. A pathway formulation is introduced to derive exact analytical results for the relevant observables in a broad framework, with the resetting time and the refractory period following arbitrary distributions. For the paradigmatic case of Poissonian distributions of the resetting and refractory times, in general with different characteristic rates, closed-form expressions are obtained that successfully describe the relaxation to the steady state. Finally, we focus on the single-target search problem, in which the survival probability and the mean first passage time to the target can be exactly computed. Therein, we also discuss optimal strategies, which show a non-trivial dependence on the refractory period.

Tight-binding model subject to conditional resets at random times.

We investigate the dynamics of a quantum system subjected to a time-dependent and conditional resetting protocol. Namely, we ask what happens when the unitary evolution of the system is repeatedly interrupted at random time instants with an instantaneous reset to a specified set of reset configurations taking place with a probability that depends on the current configuration of the system at the instant of reset? Analyzing the protocol in the framework of the so-called tight-binding model describing the hopping of a quantum particle to nearest-neighbor sites in a one-dimensional open lattice, we obtain analytical results for the probability of finding the particle on the different sites of the lattice. We explore a variety of dynamical scenarios, including the one in which the resetting time intervals are sampled from an exponential as well as from a power-law distribution, and a setup that includes a Floquet-type Hamiltonian involving an external periodic forcing. Under exponential resetting, and in both the presence and absence of the external forcing, the system relaxes to a stationary state characterized by localization of the particle around the reset sites. The choice of the reset sites plays a defining role in dictating the relative probability of finding the particle at the reset sites as well as in determining the overall spatial profile of the site-occupation probability. Indeed, a simple choice can be engineered that makes the spatial profile highly asymmetric even when the bare dynamics does not involve the effect of any bias. Furthermore, analyzing the case of power-law resetting serves to demonstrate that the attainment of the stationary state in this quantum problem is not always evident and depends crucially on whether the distribution of reset time intervals has a finite or an infinite mean.

CasODE: Modeling Irregular Information Cascade via Neural Ordinary Differential Equations (Student Abstract)

Predicting information cascade popularity is a fundamental problem for understanding the nature of information propagation on social media. However, existing works fail to capture an essential aspect of information propagation: the temporal irregularity of cascade event -- i.e., users' re-tweetings at random and non-periodic time instants. In this work, we present a novel framework CasODE for information cascade prediction with neural ordinary differential equations (ODEs). CasODE generalizes the discrete state transitions in RNNs to continuous-time dynamics for modeling the irregular-sampled events in information cascades. Experimental evaluations on real-world datasets demonstrate the advantages of the CasODE over baseline approaches.

Suboptimal Filtering Over Sensor Networks With Random Communication

The problem of filter design is considered for continuous-time linear stochastic systems using distributed sensors. Each sensor unit, represented by a node in an undirected and connected graph, collects some information about the state and communicates its own estimate with the neighbors. It is stipulated that this communication between sensor nodes connected by an edge is time-sampled randomly and for each edge, the sampling process is an independent Poisson counter. Our proposed filtering algorithm for each sensor node is a stochastic hybrid system: It comprises a continuous-time differential equation, and at random time instants when communication takes place, each sensor node updates its state estimate based on the information received by its neighbors. In this setting, we compute the expectation of the error covariance matrix for each unit which is governed by a matrix differential equation. To study the asymptotic behavior of these covariance matrices, we show that if the gain matrices are appropriately chosen and the mean sampling rate is large enough, then the error covariances practically converge to a constant matrix.

A Time-Inhomogeneous Prendiville Model with Failures and Repairs

We consider a time-inhomogeneous Markov chain with a finite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F, R). Due to a failure, a transition from an operating state to F occurs after which a repair is required, so that a transition leads to the state R. Subsequently, there is a restore phase, after which the system restarts from one of the operating states. In particular, we assume that the intensity functions of failures, repairs and restores are proportional and that the birth-death process that models the system is a time-inhomogeneous Prendiville process.

On Stochastic ISS of Time-Varying Switched Systems With Generic Lévy Switching Signals

Switched systems in which switching among subsystems occurs at random time instants find numerous applications in engineering. Stability analysis of such systems, however, is quite challenging. This letter investigates the stochastic input-to-state stability of this class of switched systems. The random switching instants are modeled by a non-decreasing, positive, and real-valued Lévy process, which, at every time instant, selects the active subsystem from a family of deterministic systems. No assumption on the stability of subsystems is presumed; they can be stable or unstable. Stochastic properties of the switching signal are coupled with a family of Lyapunov-like functions to obtain a sufficient condition for stochastic input-to-state stability.

Multiclass classification of dephasing channels

We address the use of neural networks (NNs) in classifying the environmental parameters of single-qubit dephasing channels. In particular, we investigate the performance of linear perceptrons and of two non-linear NN architectures. At variance with time-series-based approaches, our goal is to learn a discretized probability distribution over the parameters using tomographic data at just two random instants of time. We consider dephasing channels originating either from classical 1/f{\alpha} noise or from the interaction with a bath of quantum oscillators. The parameters to be classified are the color {\alpha} of the classical noise or the Ohmicity parameter s of the quantum environment. In both cases, we found that NNs are able to exactly classify parameters into 16 classes using noiseless data (a linear NN is enough for the color, whereas a single-layer NN is needed for the Ohmicity). In the presence of noisy data (e.g. coming from noisy tomographic measurements), the network is able to classify the color of the 1/f{\alpha} noise into 16 classes with about 70% accuracy, whereas classification of Ohmicity turns out to be challenging. We also consider a more coarse-grained task, and train the network to discriminate between two macro-classes corresponding to {\alpha} \lessgtr 1 and s \lessgtr 1, obtaining up to 96% and 79% accuracy using single-layer NNs.

Motility and energetics of randomly flashing ratchets

We consider randomly flashing ratchets, where the potential acting can be switched to another at random time instants with Poisson statistics. Using coupled Fokker–Planck equations, we formulate explicit expressions of mean velocity, dispersion and quantities measuring thermodynamics. How potential landscapes and transitions affect the motility and energetics is exemplified by numerical calculations on random on-off ratchets. Randomly flashing ratchets with shifted sawtooth potentials are further discussed. We find that the dynamics and output power of such system present symmetry w.r.t. the shift between the two potentials Δmax + Δmin, which is the sum of the shift between the two peaks (Δmax) and the shift between the two bottoms (Δmin). The mean velocity and output power both reach the optimal performance at Δmax + Δmin = 1, provided that the asymmetry α i of potential U i implies a positive flux respectively, i.e., α i > 0.5 for i = 1, 2.

ImdLMS: An Imputation Based LMS Algorithm for Linear System Identification With Missing Input Data

In many practical systems concerning linear system identification with adaptive filters, the performance of the system is intrinsically limited by the problem of missing data, which might arise either because the input or the output data is missing at random time instants or subsets of input or output data are missing. In this paper the problem of linear system identification is studied with only input data missing at random time instants while output data is obtained correctly at all time instants. To model the input data missing process, the available input data sequence is modeled as the product of the actual (unknown) input data sequence with an i.i.d. sequence of Bernoulli random variables with known mean. A new data sequence, called the imputed data sequence is created from the available data sequence where, at any time instant, the imputed data is either the available data at that instant, if it is nonzero, or is set to a constant factor times the imputed data at the previous time instance. This imputed sequence is then used to propose an LMS-type algorithm called Imputation based missing data LMS (ImdLMS). Performance analysis of the algorithm is carried out where the extreme complications and possible intractability vis-a-vis an exact mean square performance analysis is avoided by finding an upper bound of the mean square deviation of the algorithm. Simulation results demonstrate the efficacy of the proposed ImdLMS algorithm, especially when the actual (unknown) input data sequence is correlated.

Transient temperature distribution and thermal stresses in a thick plate subjected to random pulses

This paper deals with the determination of second-order statistics (the mean of each component and correlations between components) of the quasi-static temperature distribution and thermal stresses on a thick plate under thermal load. The heated thick circular plate with random internal heat generation acting at random instants of time, in addition to sectional heat supply in the form of random impulses on the upper face, while the lower face is kept at zero temperature, and the fixed circular edge thermally insulated in considered for investigation. A type of Laplace transform method is employed to obtain the analytical solutions for the temperature and stresses statistics. Numerical results demonstrate the effects of the magnitude with the statistics of the temperature and thermal stresses and represented graphically.

Coincidence functions and Bartlett spectra of point processes

The second order statistical properties of time point processes (PP) are described by the time coincidence function (CF) and the frequency Bartlett spectrum (BS). For PPs recorded by pulses appearing at random time instants, as in photodetection experiments, the CF can be measured by various physical devices showing in particular the famous bunching effect of photons. On the other hand, for PPs recorded by the intervals between successive points (lifetimes), especially for renewal processes, there is no usual procedure for the estimation of the CF, and the aim of this paper is to describe an approach of this problem. The starting point is a mathematical relation between the CF and the set of probabilities density functions of the lifetimes of any order of the PP. As a consequence, the CF can be obtained by processing the results of the multiple normalized histograms of these lifetimes. In the cases, relatively rare, where the mathematical expression of the CF is known in closed form, the correct behavior of the procedure is verified by an experimental analysis of simulated data. The method is extended in order to verify the relationship between the CF and the BS.

Deterministic and randomized motions in single-well potentials

Newtonian, undamped motion in single-well potentials belong to a class of well-studied conservative systems. Here, we investigate and compare long-time properties of fully deterministic motions in single-well potentials with analogous randomized systems. We consider a special type of energy-conserving randomization process: the deterministic motion is interrupted by hard velocity reversals at random time instants ti. In the 1D case, for fixed initial conditions, the differences in probability distributions disappear in the long-time limit making asymptotic densities insensitive to the selection of random time instants when velocity is reversed. Substantially different probability distributions can be obtained, for instance, through the additional randomization of initial conditions. Analogously, in 2D setups, the probability distributions asymptotically are insensitive to a protocol of velocity reversals.

Multi-Stage Algorithms of Localization of a Random Pulsed Source with a Two-Step Probability Density Function in the Search Interval

Time-optimal algorithms of spatial localization of random pulsed-point sources are proposed, and their parameters are calculated. These sources have a two-step probability density function in the search interval and are detected due to generation of instantaneous pulses at random time instants (delta functions). The advantage of the developed optimal localization algorithms over the search procedures that ignore specific features of the spatial distribution of the detected pulsed-point objects is demonstrated.

Randomized Transmissions for Networked Control Under High-Frequency Jamming

We explore the networked control problem of a linear continuous-time plant under jamming attacks. We introduce a state-feedback control framework where the sensor transmits state information packets to the controller at random time instants. We show that our randomized transmission approach can handle jamming attacks with high occurrence frequencies and closed-loop stability can be achieved as long as the average jamming duration is sufficiently small. We illustrate the efficacy of our approach with a numerical example.

Event-Driven Cooperative Receding Horizon Control for Multi-Agent Systems in Uncertain Environments

We propose an event-driven Cooperative Receding Horizon (CRH) controller to solve maximum reward collection problems (MRCP) where multiple agents cooperate to maximize the total reward collected from a set of targets in a given mission space. In previous work, a CRH controller was developed and in this paper, we overcome several limitations of this controller, including potential instabilities in the agent trajectories and poor performance due to inaccurate estimation of a reward-to-go function. Rewards are non-increasing functions of time and the environment is uncertain with new targets detected by agents at random time instants. The controller sequentially solves optimization problems over a planning horizon and executes the control for a shorter action horizon, where both are defined by certain events associated with new information becoming available. In contrast to the earlier CRH controller, we reduce the originally infinite-dimensional feasible control set to a finite set at each control update event. We prove some properties of this new controller and include simulation results showing its improved performance.

Flame Dynamics Intermittency in the Bistable Region Near a Subcritical Hopf Bifurcation

We report experimental evidence of thermoacoustic bistability in a lab-scale turbulent combustor over a well-defined range of fuel–air equivalence ratios. Pressure oscillations are characterized by an intermittent behavior with “bursts,” i.e., sudden jumps between low and high amplitudes occurring at random time instants. The corresponding probability density functions (PDFs) of the acoustic pressure signal show clearly separated maxima when the burner is operated in the bistable region. The gain and phase between acoustic pressure and heat release rate fluctuations are evaluated at the modal frequency from simultaneously recorded flame chemiluminescence and acoustic pressure. The representation of the corresponding statistics is new and particularly informative. It shows that the system is characterized, in average, by a nearly constant gain and by a drift of the phase as function of the oscillation amplitude. This finding may suggest that the bistability does not result from an amplitude-dependent balance between flame gain and acoustic damping, but rather from the nonconstant phase difference between the acoustic pressure and the coherent fluctuations of heat release rate.

Time-Optimal algorithms of searching for pulsed-point sources for systems with several detectors

Time-optimal algorithms of localization of pulsed-point sources randomly distributed over the search interval and detected due to generation of instantaneous pulses (delta functions) at random time instants are developed. Optimal search procedures ensure prescribed accuracy of localization and depend on the number of receivers in the detector.

Asymptotic Relation for the Transition Density of the Three-Dimensional Markov Random Flight on Small Time Intervals

We consider the Markov random flight $\bold X(t), \; t>0,$ in the three-dimensional Euclidean space $\Bbb R^3$ with constant finite speed $c>0$ and the uniform choice of the initial and each new direction at random time instants that form a homogeneous Poisson flow of rate $\lambda>0$. Series representations for the conditional characteristic functions of $\bold X(t)$ corresponding to two and three changes of direction, are obtained. Based on these results, an asymptotic formula, as $t\to 0$, for the unconditional characteristic function of $\bold X(t)$ is derived. By inverting it, we obtain an asymptotic relation for the transition density of the process. We show that the error in this formula has the order $o(t^3)$ and, therefore, it gives a good approximation on small time intervals whose lengths depend on $\lambda$. Estimate of the accuracy of the approximation is analysed.

Optimization of the service strategy in a queueing system with energy harvesting and customers’ impatience

Abstract A single-server queueing system with an infinite buffer is considered. The service of a customer is possible only in the presence of at least one unit of energy, and during the service the number of available units decreases by one. New units of energy arrive in the system at random instants of time if the finite buffer for maintenance of energy is not full. Customers are impatient and leave the system without service after a random amount of waiting time. Such a queueing system describes, e.g., the operation of a sensor node which harvests energy necessary for information transmission from the environment. Aiming to minimize the loss of customers due to their impatience (and maximize the throughput of the system), a new strategy of control by providing service is proposed. This strategy suggests that service temporarily stops if the number of customers or units of energy in the system becomes zero. The server is switched off (is in sleep mode) for some time. This time finishes (the server wakes up) if both the number of customers in the buffer and the number of energy units reach some fixed threshold values or when the number of energy units reaches some threshold value and there are customers in the buffer. Arrival flows of customers and energy units are assumed to be described by an independent Markovian arrival process. The service time has a phase-type distribution. The system behavior is described by a multi-dimensional Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for key performance measures are given. Numerical results illustrating the dependence of a customer’s loss probability on the thresholds defining the discipline of waking up the server are provided. The importance of the account of correlation in arrival processes is numerically illustrated.

Estimation of Fourier Transform Using Alias-Free Hybrid-Stratified Sampling

This paper proposes a novel method of estimating the Fourier transform (FT) of deterministic, continuous-time signals, from a finite number $N$ of their samples taken from a fixed-length observation window. It uses alias-free hybrid-stratified sampling to probe the processed signal at a mixture of deterministic and random time instants. The FT estimator, specifically designed to work with this sampling scheme, is unbiased, consistent and fast converging. It is shown that if the processed signal has continuous third derivative, then the estimator’s rate of uniform convergence in mean square is $N^{-5}$ . Therefore, in terms of frequency-independent upper bounds on the FT estimation error, the proposed approach significantly outperforms existing estimators that utilize alias-free sampling, such as total random, stratified sampling, and antithetical stratified whose rate of uniform convergence is $N^{-1}$ . It is proven here that $N^{-1}$ is a guaranteed minimum rate for all stratified-sampling-based estimators satisfying four weak conditions formulated in this paper. Owing to the alias-free nature of the sampling scheme, no constraints are imposed on the spectral support of the processed signal or the frequency ranges for which the Fourier Transform is estimated.