Poisson Time Research Articles

Causal Behavior of Switched Delay Systems as Multi-Mode Multi-Dimensional Systems*

Abstract This paper is about minimal causal models for delay systems, where the delay is switched between fixed values. Deterministic and random switching are discussed. General properties of multi-mode multidimensional (M 3 D) systems are described. We digress about the constraint “delay derivative > 1” that needs to be imposed on any delay system. Its relevance stems from the fact that a causal minimal representation of a switched delay system must be cast as an M 3 D system. The backwards Kolmogorov equation for a delay system where switching occurs at Poisson times (PM 3 D) is derived. The multidimensional character adds a new twist to this problem: the backward equation is given as a set of coupled PDE's involving unknown functions with different number of arguments.

Shear-induced chaos

Guided by a geometric understanding developed in earlier works of Wang and Young, we carry out numerical studies of shear-induced chaos in several parallel but different situations. The settings considered include periodic kicking of limit cycles, random kicks at Poisson times and continuous-time driving by white noise. The forcing of a quasi-periodic model describing two coupled oscillators is also investigated. In all cases, positive Lyapunov exponents are found in suitable parameter ranges when the forcing is suitably directed.

Transient Moments of the TCP Window Size Process

The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.

Transient Moments of the TCP Window Size Process

The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.

The influence of dependence on data network models

Consider an infinite-source marked Poisson process to model end user inputs to a data network. At Poisson times, connections are initated. The connection is characterized by a triple (F, L, R) denoting the total quantity of transmitted data in a connection, the length or duration of the connection, and the transmission rate; the three quantities are related by F = LR. How critical is the dependence structure of the mark for network characteristics such as burstiness, distribution tails of cumulative input, and long-range dependence properties of traffic measured in consecutive time slots? In a previous publication (D'Auria and Resnick (2006)) we assumed that F and R were independent. Here we assume that L and R are independent. The change in dependence assumptions means that the model properties change dramatically: tails of cumulative input per time slot are dramatically heavier, traffic cannot be approximated by a Gaussian distribution, and the decay of dependence cannot be measured in the traditional way using correlation functions. Different network applications are likely to have different mark dependence structure. We argue that the present independence assumption on L and R is likely to be appropriate for network applications such as streaming media or peer-to-peer networks. Our conclusion is that it is desirable to separate network traffic by application and to model each application with its own appropriate dependence structure.

The influence of dependence on data network models

Consider an infinite-source marked Poisson process to model end user inputs to a data network. At Poisson times, connections are initated. The connection is characterized by a triple (F,L,R) denoting the total quantity of transmitted data in a connection, the length or duration of the connection, and the transmission rate; the three quantities are related byF=LR. How critical is the dependence structure of the mark for network characteristics such as burstiness, distribution tails of cumulative input, and long-range dependence properties of traffic measured in consecutive time slots? In a previous publication (D'Auria and Resnick (2006)) we assumed thatFandRwere independent. Here we assume thatLandRare independent. The change in dependence assumptions means that the model properties change dramatically: tails of cumulative input per time slot are dramatically heavier, traffic cannot be approximated by a Gaussian distribution, and the decay of dependence cannot be measured in the traditional way using correlation functions. Different network applications are likely to have different mark dependence structure. We argue that the present independence assumption onLandRis likely to be appropriate for network applications such as streaming media or peer-to-peer networks. Our conclusion is that it is desirable to separate network traffic by application and to model each application with its own appropriate dependence structure.

The invariance principle for the Ornstein–Uhlenbeck process with fast Poisson time: An estimate for the rate of convergence

The invariance principle for the Ornstein–Uhlenbeck process with fast Poisson time: An estimate for the rate of convergence

The influence of media reporting of the suicide of a celebrity on suicide rates: a population-based study

The impact of media reporting of suicides of entertainment celebrities may affect suicide rates due to an imitation effect. We investigated the impact on suicides of the media reporting of the suicide of a male television celebrity. All suicides during 2003-2005 in Taiwan (n = 10,945) were included in this study. A Poisson time series autoregression analysis was conducted to examine whether there was an increase in suicides during the 4-week period after extensive media reporting of the celebrity suicide. After controlling for seasonal variation, calendar year, temperature, humidity and unemployment rate, there was a marked increase in the number of suicides during the 4-week period after media reporting (relative risk = 1.17, 95% CI 1.04-1.31). The increase was in men (relative risk = 1.30, 95% CI 1.14-1.50) and for the individuals using the same highly lethal method (hanging) as the TV actor did (relative risk = 1.51, 95% CI 1.25-1.83). However, the age groups in which the increase occurred were younger than the age of the celebrity. The extensive media reporting of the celebrity suicide was followed by an increase in suicides with a strong implication of a modelling effect. The results provide further support for the need for more restrained reporting of suicides as part of suicide prevention strategies to decrease the imitation effect.

The Influence of Media Coverage of a Celebrity Suicide on Subsequent Suicide Attempts

To investigate the impact of media reporting of a celebrity suicide on subsequent suicide attempts. A Poisson time series autoregression analysis was conducted to examine whether there was a significant increase in suicide attempts during the 3-week period after the start of extensive media reporting of a celebrity suicide. The reporting began on May 2, 2005, and lasted about 17 days. To investigate the influence of media reporting on suicide attempts, a structured interview was conducted with 124 suicide attempters identified from 2 counties in Mid Taiwan who had exposure to the media reporting. After controlling for seasonal variation, calendar year, temperature, and humidity, there was a marked increase in the number of suicide attempts during the 3-week period after media reporting began (adjusted relative risk = 1.55, 95% CI = 1.26 to 1.91). Among 124 suicide attempters exposed to the media reports, 23.4% reported an influence from them. There was no relationship between the attempters' ages and the age of the celebrity or the method, but male attempters had a significantly higher risk for such influence. A considerably higher risk for such influence was found among subjects with a history of suicide attempt(s) in the previous year (odds ratio = 52.3, 95% CI = 5.96 to 459.1). The extensive media reporting of the suicide of a celebrity was followed by an increase in suicide attempts. The effect was particularly marked in individuals with a recent history of a suicide attempt. The results provide further support for the need for more restrained reporting of suicides as part of suicide prevention strategies and for special vigilance for contagious effects of such reporting on people who have carried out recent suicidal acts.

Random Flights in Higher Spaces

We consider in this paper random flights in ℝd performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S1d. We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ3 are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2.

Poisson's ratio as a sensitive indicator of (fatigue) damage in fibre‐reinforced plastics

ABSTRACTEven if the extent of damage in fibre‐reinforced plastics is limited, it already affects the elastic properties. Therefore, the damage initiation and propagation in composite structures is monitored very carefully. Beside the use of nondestructive testing methods (ultrasonic inspection, optical fibre sensing), the follow‐up of the degradation of engineering properties such as the stiffness is a common approach.In this paper, it is investigated if the Poisson's ratio can be used as a sensitive indicator of (fatigue) damage in fibre‐reinforced plastics. Static, cyclic and fatigue tests have been performed on [0°/90°]2s glass/epoxy laminates, and axial and transverse strain were measured continuously. The evolution of the Poisson's ratio νxy versus time and axial strain ɛxx is studied. It is concluded that the degradation of the Poisson's ratio can be a valuable indicator of damage, in combination with the stiffness degradation.

A Comprehensive Standardized Model for Ultrawideband Propagation Channels

A comprehensive statistical model is described for ultrawideband (UWB) propagation channels that is valid for a frequency range from 3-10 GHz. It is based on measurements and simulations in the following environments: residential indoor, office indoor, builtup outdoor, industrial indoor, farm environments, and body area networks. The model is independent of the used antennas. It includes the frequency dependence of the path gain as well as several generalizations of the Saleh-Valenzuela model, like mixed Poisson times of arrival and delay-dependent cluster decay constants. A separate model is specified for the frequency range below 1 GHz. The model can thus be used for realistic performance assessment of UWB systems. It was accepted by the IEEE 802.15.4a Task Group as standard model for evaluation of UWB system proposals. This paper also presents a critical assessment of the applicability of the model and possible generalizations and improvements

Modelling nonlinear count time series with local mixtures of Poisson autoregressions

A novel class of nonlinear models is studied based on local mixtures of autoregressive Poisson time series. The proposed model has the following construction: at any given time period, there exist a certain number of Poisson regression models, denoted as experts, where the vector of covariates may include lags of the dependent variable. Additionally, the existence of a latent multinomial variable is assumed, whose distribution depends on the same covariates as the experts. The latent variable determines which Poisson regression is observed. This structure is a special case of the mixtures-of-experts class of models, which is considerably flexible in modelling the conditional mean function. A formal treatment of conditions to guarantee the asymptotic normality of the maximum likelihood estimator is presented, under stationarity and nonstationarity. The performance of common model selection criteria in selecting the number of experts is explored via Monte Carlo simulations. Finally, an application to a real data set is presented, in order to illustrate the ability of the proposed structure to flexibly model the conditional distribution function.

Invariance principle for one class of Markov chains with fast Poisson time. Estimate for the rate of convergence

We obtain an estimate for the rate of convergence of normalized Poisson sums of random variables determined by the first-order autoregression procedure to a family of Wiener processes.

The role of PASTA in network measurement

Poisson Arrivals See Time Averages (PASTA) is a well known property applicable to many stochastic systems. In active probing, PASTA is invoked to justify the sending of probe packets (or trains) at Poisson times in a variety of contexts. However, due to the diversity of aims and analysis techniques used in active probing, the benefits of Poisson based measurement, and the utility and role of PASTA, are unclear. Using a combination of rigorous results and carefully constructed examples and counter-examples, we map out the issues involved, and argue that PASTA is of very limited use in active probing. In particular, Poisson probes are not unique in their ability to sample without bias. Furthermore, PASTA ignores the issue of estimation variance, and the central need for an inversion phase to estimate the quantity of interest ased on what is directly observable. We give concrete examples of when Poisson probes should not be used, and explain why, and offer initial guidelines on suitable alternative sending processes.

Climatic, high tide and vector variables and the transmission of Ross River virus

This report assesses the impact of the variability in environmental and vector factors on the transmission of Ross River virus (RRV) in Brisbane, Australia. Poisson time series regression analyses were conducted using monthly data on the counts of RRV cases, climate variables (Southern Oscillation Index and rainfall), high tides and mosquito density for the period of 1998-2001. The results indicate that increases in the high tide (relative risk (RR): 1.65; 95% confidence interval (CI): 1.20-2.26), rainfall (RR: 1.45; 95% CI: 1.21-1.73), mosquito density (RR: 1.17; 95% CI: 1.09-1.27), the density of Culex annulirostris (RR: 1.25; 95% CI: 1.13-1.37) and the density of Ochlerotatus vigilax (RR: 2.39; 95% CI: 2.30-2.48), each at a lag of 1 month, were statistically significantly associated with the rise of monthly RRV incidence. The results of the present study might facilitate the development of early warning systems for reducing the incidence of this wide-spread disease in Australia and other Pacific island nations.

Occupancy importance factor in earthquake engineering

Seismic codes require that important structures be designed by increasing the seismic design coefficients of the ordinary structures by a factor called the importance factor. At the present time, shear base coefficients are established such that the probability that their value may be exceeded over a period of time does not exceed a small number. On the other hand, at sites near seismic sources, there is a saturation of maximum ground accelerations, as the magnitude of the earthquake increases. Therefore, it would seem that the importance factors at these sites should be smaller than at sites away from the source. To study this, values of the importance factor are calculated for a site in the near-field. We consider that the design is such that it minimizes the present value of the total cost including initial cost as well as losses due to damage and failure. The study considers Poisson and non-Poisson interarrival times for earthquake occurrence. The results show that the importance factor for a site in the near-field is smaller than that of a site away from the source.

Sporadic and Continuous Clearing Policies for a Production/Inventory System Under an M/G Demand Process

A production/inventory system is filled continuously at rate 1 and satisfies demands of i.i.d. random sizes that arrive at Poisson times. For this system we consider two clearing policies. Under sporadic review, clearing takes place after a random time independent of the content process. Under continuous review, the system is cleared as soon as the inventory level reaches some pre-specified threshold. We derive explicit results for the associated expected discounted cost functionals under both types of clearing.

Batch arrival queues with vacations and server setup

We study a single server queueing system whose arrival stream is compound Poisson and service times are generally distributed. Three types of idle period are considered: threshold, multiple vacations, and single vacation. For each model, we assume after the idle period, the server needs a random amount of setup time before serving. We obtain the steady-state distributions of system size and waiting time and expected values of the cycle for each model. We also show that the distributions of system size and waiting time of each model are decomposed into two parts, whose interpretations are provided. As for the threshold model, we propose a method to find the optimal value of threshold to minimize the total expected operating cost.

A Two-Sided First-Exit Problem for a Compound Poisson Process with a Random Upper Boundary

We consider the first-exit time of a compound Poisson process from a region that is bounded from below by an increasing straight line, while its upper boundary has positive jumps of i.i.d. sizes at Poisson times and increases linearly between jumps. An integral equation for the corresponding Laplace-Stieltjes transforms is derived and solved. The case of exponential jumps is treated separately. The problem has applications in queueing and risk theory.