Non-stationary Poisson Process Research Articles

SGR-like behaviour of the repeating FRB 121102

Fast radio bursts (FRBs) are millisecond-duration radio signals occurring at cosmological distances. However the physical model of FRBs is mystery, many models have been proposed. Here we study the frequency distributions of peak flux, fluence, duration and waiting time for the repeating FRB 121102. The cumulative distributions of peak flux, fluence and duration show power-law forms. The waiting time distribution also shows power-law distribution, and is consistent with a non-stationary Poisson process. These distributions are similar as those of soft gamma repeaters (SGRs). We also use the statistical results to test the proposed models for FRBs. These distributions are consistent with the predictions from avalanche models of slowly driven nonlinear dissipative systems.

OUP accepted manuscript

Type I X-ray bursts in a low-mass X-ray binary (LMXB) are caused by unstable nuclear burning of accreted materials. Semi-analytical and numerical studies of unstable nuclear burning have successfully reproduced partial properties of this kind of burst. However, some other properties (e.g. the waiting time) are not well explained. In this paper, we find that the probability distributions of fluence, peak count, rise time, duration and waiting time can be described as power-law-like distributions. This indicates that type I X-ray bursts may be governed by a self-organized criticality (SOC) process. The power-law index of waiting time distribution (WTD) is around $-1$, which is not predicted by any current waiting time model. We propose a physical burst rate model, in which the mean occurrence rate is inversely proportional to time $\lambda\propto t^{-1}$. In this case, the WTD is well explained by a non-stationary Poisson process within the SOC theory. In this theory, the burst size is also predicted to follow a power-law distribution, which requires that the emission area possesses only part of the neutron star surface. Furthermore, we find that the WTDs of some astrophysical phenomena can also be described by similar occurrence rate models.

Assessing hurricane damage costs in the presence of vulnerability model uncertainty

Probability-based assessment of hurricane damage costs for coastal communities is vital for policy-makers and insurers. The uncertainties associated with hurricane damage costs include both the inherent uncertainty due to the random nature of hurricane process and the model uncertainty of the mathematical representation of hurricane damage (vulnerability model). The hurricane vulnerability model has traditionally been modeled as a deterministic function of hurricane wind speed in the literature, without considering the effect of vulnerability model uncertainty on hurricane damage assessment. This paper develops two methods to assess the hurricane damage costs in the presence of vulnerability model uncertainty. To account for the non-stationarity in hurricane actions due to the potential impact of climate change, the hurricane occurrence process is modeled as a non-stationary Poisson process and the hurricane intensity is assumed to vary in time with time-variant statistical parameters of hurricane wind speed. A case study of Miami-Dade County, Florida, is conducted to illustrate the proposed methods and to investigate the impact of vulnerability model uncertainty on hurricane damage costs.

A probabilistic framework for hurricane damage assessment considering non-stationarity and correlation in hurricane actions

The intensity and/or frequency of hurricane storms may change due to the impact of potential climate change. This paper presents a probabilistic framework to assess the hurricane damage to residential constructions considering the non-stationarity and correlation in hurricane actions. The framework includes a non-stationary Poisson process of hurricane occurrence, a failure rate function of hurricane damage, and explicit formulas for evaluating the mean and variance of annual hurricane damage. The framework is illustrated using a case study of Miami-Dade County, Florida, where the current probabilistic models of hurricane intensity and occurrence rate were estimated by examining hurricane history in this area. The impacts of time-variant hurricane intensity and time-variant hurricane frequency on building damage are assessed individually using the developed framework. The paper also investigates the effects of correlation in hurricane wind speeds on hurricane damage.

Trees outside forests in agricultural landscapes: spatial distribution and impact on habitat connectivity for forest organisms

Various species of forest trees are commonly used for ornamental purposes and are therefore frequently found in nonforest ecosystems. They constitute an important component of the so-called trees outside forests (TOF). Not much is known, however, about the drivers of TOF spatial distribution either in urbanized or in agricultural landscapes since they are generally absent from forest inventories. The present study focused on the spatial distribution of TOF across agricultural landscapes and their potential role in the dispersal of a forest pest insect, the pine processionary moth, Thaumetopoea pityocampa (PPM). All the TOF belonging to the genera Pinus, Cedrus and Pseudotsuga were considered as potential hosts and inventoried within a 22 × 22 km study window. We fitted a nonstationary Poisson process to the empirical data and used the distance to the nearest building as a covariate. Both empirical and simulated data indicated that TOF associated to human artifacts/urbanized areas constituted the main source of landscape connectivity for the PPM in the open fields under study. Because they do not account for TOF, forest inventories dramatically underestimate landscape connectivity and provide an erroneous picture of the PPM habitat distribution. We conclude that TOF, especially the ornamental component, must be taken into account when it comes to understanding forest insect landscape dynamics or genetics. The omnipresence of TOF also suggests a potentially huge role in pest dispersal and invasive species expansion.

Nucleation Studies of Active Pharmaceutical Ingredients in an Air-Segmented Microfluidic Drop-Based Crystallizer

In this work, an innovative experimental platform to study the nucleation behavior of an active pharmaceutical ingredient (API) using a gas-segmented flow produced by a microfluidic T-junction device with fine perfluoroalkoxy alkane (PFA) tubing is presented. In the experiments, a continuous airflow was used to segment the supersaturated liquid phase into nearly monodispersed droplets (with a coefficient of variance about 15% of the drop size distribution) at the T-junction. As the API molecules have relatively low gas/solvent partition coefficients, mass transfer between phases is effectively inhibited. Nucleation then occurred within the drops in the temperature-controlled capillary crystallizer. By treating the probability of crystal nucleation in each drop as a nonstationary Poisson process, the effect of process conditions on lactose and aspirin nucleation was studied. In lactose nucleation studies, the Poisson rate process effectively captures the nucleation behavior, and the estimated nucleation parameters show good agreement to literature values. However, the lactose nucleation rate increased more sharply with a supersaturation ratio than predicted by classical nucleation theory. The validation of aspirin nucleation studies from ethanol demonstrates the application of this newly design system for crystallization of relatively nonpolar organic molecules from nonaqueous solvents to give crystals with controlled attributes.

WAITING TIME DISTRIBUTION OF SOLAR ENERGETIC PARTICLE EVENTS MODELED WITH A NON-STATIONARY POISSON PROCESS

We present a study of the waiting time distributions (WTDs) of solar energetic particle (SEP) events observed with the spacecraft $WIND$ and $GOES$. Both the WTDs of solar electron events (SEEs) and solar proton events (SPEs) display a power-law tail $\sim \Delta t^{-\gamma}$. The SEEs display a broken power-law WTD. The power-law index is $\gamma_{1} =$ 0.99 for the short waiting times ($<$70 hours) and $\gamma_{2} =$ 1.92 for large waiting times ($>$100 hours). The break of the WTD of SEEs is probably due to the modulation of the corotating interaction regions (CIRs). The power-law index $\gamma \sim$ 1.82 is derived for the WTD of SPEs that is consistent with the WTD of type II radio bursts, indicating a close relationship between the shock wave and the production of energetic protons. The WTDs of SEP events can be modeled with a non-stationary Poisson process which was proposed to understand the waiting time statistics of solar flares (Wheatland 2000; Aschwanden $\&$ McTiernan 2010). We generalize the method and find that, if the SEP event rate $\lambda = 1/\Delta t$ varies as the time distribution of event rate $f(\lambda) = A \lambda^{-\alpha}exp(-\beta \lambda)$, the time-dependent Poisson distribution can produce a power-law tail WTD $\sim \Delta t^{\alpha - 3}$, where $0 \leq \alpha < 2$.

Dynamics and Morphological Outcomes in Thin-Film Spherical Crystallization of Glycine from Microfluidic Emulsions: Experimental Studies and Modeling

We present detailed experimental studies, supported by a theoretical model, of dynamics and morphological outcomes in thin-film spherical crystallization of glycine from microfluidic emulsions. Specifically, the effects of droplet size, shrinkage rate, and temperature are studied with the aim of understanding and delineating, from a processing standpoint, crystallization conditions that ensure spherulitic crystal growth in individual droplets, ultimately yielding compactly packed spherical crystal agglomerates (SAs). Our experiments reveal the existence, under all processing conditions, of a critical concentration (supersaturation) beyond which droplet shrinkage due to evaporation is dramatically arrested. The morphological outcome of the crystallization then depends on whether a nucleation event in a droplet occurs before (yielding loosely packed or faceted single crystals) or after (compactly packed SAs) this critical concentration is reached. We analyze our observations within the framework of a simple physical model based on classical nucleation theory and the theory of nonstationary Poisson processes, which accurately captures the overall trends. Experiments were conducted in a temperature range of 45–85 °C, droplet diameter range of 50–160 μm, and film thickness range of 0.5–1.5 mm. We found that smaller droplets and faster shrinkage rates (in thinner films) favor the formation of compact SAs at a given temperature, and lower temperatures generally favor compact SA formation at a fixed drop size and shrinkage rate. This work builds on our recent demonstration of spherical crystallization in microfluidic emulsions and provides valuable guidelines for the design of spherical crystallization processes using this method.

Simulations of Some Doubly Stochastic Poisson Point Processes

Computer simulations of point processes are important either to verify the results of certain theoretical calculations that can be very awkward at times or to obtain practical results when these calculations become almost impossible. One of the most common methods for the simulation of nonstationary Poisson processes is random thinning. Its extension when the intensity becomes random (doubly stochastic Poisson processes) depends on the structure of this intensity. If the random density takes only discrete values, which is a common situation in many physical problems where quantum mechanics introduces discrete states, it is shown that the thinning method can be applied without error. We study in particular the case of binary density and present the kind of theoretical calculations that then become possible. The results of various experiments realized with data obtained by simulation show a fairly good agreement with the theoretical calculations.

Heavy-Traffic Limits for a Many-Server Queueing Network with Switchover

We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.

Heavy-Traffic Limits for a Many-Server Queueing Network with Switchover

We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.

Modeling clustered non-stationary Poisson processes for stochastic simulation inputs

A validated simulation model primarily requires performing an appropriate input analysis mainly by determining the behavior of real-world processes using probability distributions. In many practical cases, probability distributions of the random inputs vary over time in such a way that the functional forms of the distributions and/or their parameters depend on time. This paper answers the question whether a sequence of observations from a process follow the same statistical distribution, and if not, where the exact change points are, so that observations within two consecutive change points follow the same distribution. We propose two different methods based on likelihood ratio test and cluster analysis to detect multiple change points when observations follow non-stationary Poisson process with diverse occurrence rates over time. Results from a comprehensive Monte Carlo study indicate satisfactory performance for the proposed methods. A well-known example is also considered to show the application of our findings in real world cases.

Extraction of statistical properties of the point source response of a reverberant plate and application to parameter estimation (L)

The point source response of a reverberant solid plate is modeled through a nonstationary Poisson process based on the image-source method. The theoretical expectation of the envelope is then derived, taking into account the dispersive nature of plate waves, and validated by numerical results. Least-square curve-fitting applied to an ensemble average over N realizations can then be used to identify useful parameters such as wave velocity, plate surface, or source-receiver distance. It is shown that even values of N down to 1 (no averaging) allow a satisfying identification. Application to the estimation of the source-receiver distance using a single sensor is finally highlighted to illustrate the promising potentialities of the measurement principle proposed.

Optimal outpatient appointment scheduling with emergency arrivals and general service times

In this paper we study the problem of deciding at what times to schedule non-emergency patients when there are emergency arrivals following a non-stationary Poisson process. The service times can have any given distribution. The objective function consists of a weighted sum of the waiting times, idle time and overtime. We prove that this objective function is multimodular, and then use a local search algorithm which in that case is guaranteed to find the optimal solution. Numerical examples show that this method gives considerable improvements over the standard even-spaced schedule, and that the schedules for different service time distributions can look quite different.

Optimal Control of Replenishment and Substitution in an Inventory System with Nonstationary Batch Demand

We study an inventory system in which a supplier supplies demand using two mutually substitutable products over a selling season of T periods, with a single replenishment opportunity at the beginning of the season. As the season starts, customer orders arrive in each period, for either type of products, following a nonstationary Poisson process with random batch sizes. The substitution model we consider combines the usual supplier‐driven and customer‐driven schemes, in that the supplier may choose to offer substitution, at a discount price, or may choose not to; whereas the customer may or may not accept the substitution when it is offered. The supplier's decisions are the supply and substitution rules in each period throughout the season, and the replenishment quantities for both products at the beginning of the season. With a stochastic dynamic programming formulation, we first prove the concavity of the value function, which facilitates the solution to the optimal replenishment quantities. We then show that the optimal substitution follows a threshold rule, and establish the monotonicity of the thresholds over time and with respect to key cost parameters. We also propose a heuristic exhaustive policy, and illustrate its performance through numerical examples.

Load distribution performance of super-node based peer-to-peer communication networks: A nonstationary Markov chain approach

Voice over Internet protocol (VoIP) services using peer-to-peer (P2P) technology have become popular in recent years. In P2P-based VoIP networks such as Skype and P2P session initiation protocol (P2PSIP), super nodes are chosen from among all ordinary end-user nodes and handle particular tasks such as the management of user information, call establishment, and traffic relay. Future communication networks based on P2P technology must support a huge number of user nodes. A fundamental analysis of the load distribution in decentralized user-information management is needed to develop efficient and robust communication networks. In this paper, we analyze the performance of the P2P-based dynamic load distribution. In our analytical model, new nodes join the network according to a nonstationary Poisson process, and the stochastic behavior of the number of online nodes is analyzed approximately with an M($t$)/M/$\infty$ queue. We focus on two performance measures that significantly affect the quality of service (QoS) provided to the users: the churn rate and the load of super nodes. Numerical examples show that the performance of the P2P-based VoIP networks is sensitive to the sojourn time of super nodes and the maximum number of nodes managed by a super node.

RECONCILIATION OF WAITING TIME STATISTICS OF SOLAR FLARES OBSERVED IN HARD X-RAYS

We study the waiting time distributions of solar flares observed in hard X-rays with ISEE-3/ICE, HXRBS/SMM, WATCH/GRANAT, BATSE/CGRO, and RHESSI. Although discordant results and interpretations have been published earlier, based on relatively small ranges ($< 2$ decades) of waiting times, we find that all observed distributions, spanning over 6 decades of waiting times ($\Delta t \approx 10^{-3}- 10^3$ hrs), can be reconciled with a single distribution function, $N(\Delta t) \propto \lambda_0 (1 + \lambda_0 \Delta t)^{-2}$, which has a powerlaw slope of $p \approx 2.0$ at large waiting times ($\Delta t \approx 1-1000$ hrs) and flattens out at short waiting times $\Delta t \lapprox \Delta t_0 = 1/\lambda_0$. We find a consistent breakpoint at $\Delta t_0 = 1/\lambda_0 = 0.80\pm0.14$ hours from the WATCH, HXRBS, BATSE, and RHESSI data. The distribution of waiting times is invariant for sampling with different flux thresholds, while the mean waiting time scales reciprocically with the number of detected events, $\Delta t_0 \propto 1/n_{det}$. This waiting time distribution can be modeled with a nonstationary Poisson process with a flare rate $\lambda=1/\Delta t$ that varies as $f(\lambda) \propto \lambda^{-1} \exp{-(\lambda/\lambda_0)}$. This flare rate distribution represents a highly intermittent flaring productivity in short clusters with high flare rates, separated by quiescent intervals with very low flare rates.

Multiple allocation algorithm based on machine vision for elevator group control system and simulation system design

To increase the efficiency of elevator group system, the paper introduced a new designed simulation system based on non-stationary Poisson process and Model/View/Controller (MVC) pattern. At the same time, a new dispatch algorithm was proposed based on machine vision to inspect the hall traffic. The proposed algorithm can serve one hall-call by allocating more than one elevator. Finally simulation results show that the new method outperforms the minimum long-waiting method in terms of average waiting time and left-waiting people.

Fluctuations and pseudo long range dependence in network flows: A non-stationary Poisson process model

In the study of complex networks (systems), the scaling phenomenon of flow fluctuations refers to a certain power-law between the mean flux (activity) ⟨Fi⟩ of the i-th node and its variance σi as σi α ⟨Fi⟩α. Such scaling laws are found to be prevalent both in natural and man-made network systems, but the understanding of their origins still remains limited. This paper proposes a non-stationary Poisson process model to give an analytical explanation of the non-universal scaling phenomenon: the exponent α varies between 1/2 and 1 depending on the size of sampling time window and the relative strength of the external/internal driven forces of the systems. The crossover behaviour and the relation of fluctuation scaling with pseudo long range dependence are also accounted for by the model. Numerical experiments show that the proposed model can recover the multi-scaling phenomenon.

Stochastic technical change, non-renewable resource and optimal sustainable growth

We develop a stochastic endogenous growth model involving a non-renewable resource, in which innovation arrivals are governed by a non-stationary Poisson process. Using a CRRA analytical example, we characterize the optimal trajectories of the model and analyze the effects of uncertainty in the sense of Rothschild and Stiglitz by computing a mean-preserving spread. We show that increased variability in the innovation process always implies a smaller optimal R&D effort, since this leads to a reduced marginal rate of return. Effects on the other variables of the model may also be unambiguously identified depending upon the relative risk aversion of agents, the social discount rate and the marginal arrival rate of innovations. Finally, we investigate the conditions under which, on average, the economy reaches a sustainable growth path.