Another characterization of homogeneous Poisson processes

To ensure the effective and efficient operation of Defence aviation equipment there is a clear need for a component life model that is representative of the true life of a component. However, the large and often sophisticated RAM models used to manage defence aviation platforms, through various engineering and logistics activities, use models that cannot accurately represent this life. The main difference is in the underlying repair assumption. Specifically, the Ordinary Renewal Process (ORP) uses an as-good-as-new repair assumption while the Non-Homogenous Poisson Process (NHPP) uses an as-bad-as-old repair assumption. However, it is highly unlikely that any component, typically referred to as a Repairable Item (RI), will readily fit into either repair assumption. Therefore, despite the best endeavours of both engineering and logistics staff given the underlying repair assumption and the limitation these impose on the model, any solution will be suboptimal. Accordingly, there is a need for a RI life model that can contend with imperfect maintenance, imperfect inspection and can adapt to the limitations in data and include a number of additional factors including aging of the component, number of repairs, effectiveness of the repair, skill of the technicians, etc. Eight cases were developed as part of the overall modelling scheme. These eight cases are further divided into 2 main types; the first type representing cases where failure times are known and the second type where failure times are unknown. The cases incrementally modify these types through the addition of factors including multiple failure modes and their inter-dependence, and imperfect inspection and maintenance, in order to achieve a more realistic representation. Each of these cases were then solved using utilising a Markov Chain Monte Carlo (MCMC) sampling procedure, concentrating only on the analysis of the KIJIMA Type I GRP model with an underlying Weibull Time-To- Failure (TTF) distribution. The MCMC was made possible through the use of a Slice Sampling and Auxiliary Variable techniques. The resulting models have the ability to accurately model, and specifically predict, the future failure trends. Furthermore, the model allows the analyst to compare the maintenance effectiveness either in isolation, or in comparison (benchmarking) of various maintenance activities/facilities

In this paper, Markov Renewal processes having a finite number of states are studied. Explicit expressions are derived for the distribution functions of first passage times, as well as for the marginal distribution function of the corresponding Semi-Markov process. Double generating functions are obtained for the distribution functions of the $N_j$-processes. The limiting behavior of a Markov Renewal process is discussed, the stationary probabilities being derived completely. General Markov Renewal processes are introduced, and a related stationary process is determined. Several examples are given.

We consider simple models of congestion control in high-speed networks, and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ON-OFF type with exponential ON and OFF times, then, under a certain scaling, the steady-state distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions. For the case where the source arrival process is Poisson and the service times are exponential, the steady-state distribution consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and service-time distributions that are phase type, and show the impact of these distributions on the steady-state distribution of the number of sources in the system. We also establish an insensitivity to service-time distribution when the arrival process is Poisson. We use these results to relate the capacity of a bottleneck node to performance measures of interest for best effort traffic.

In this paper, an analytic approximation is derived for the end-to-end delay-jitter incurred by a periodic traffic with constant packet size. The single node case is considered first. It is assumed that the periodic traffic is multiplexed with a background packet stream under the FCFS service discipline. The processes governing the packet arrivals and the packet sizes of the background traffic are assumed to be general renewal processes. A very simple analytical approximation is derived and its accuracy is assessed by means of event-driven simulations. This approximation is then extended to the multiple node case yielding a simple analytical approximation of the end-to-end jitter. This approximation is shown to be fairly accurate in the light to moderate traffic conditions typically encountered in IP core networks.

In this paper, a multiserver retrial system with constant retrial policy is considered. The input is assumed to be a general renewal process, service times are iid with a general distribution and the retrial attempts follow an exponential distribution. The system is described with a regenerative process, and we focus on the logarithmic asymptotics of the large deviation (overflow) probability that the orbit size reaches a high level N within regeneration cycle. To analyze the upper bound of this probability, we interpret the retrial system as a buffered system with service times of a special type. Simulation results show the accuracy of the obtained bound for Weibull/M/2 retrial system.

The need for persistent patrol and detection arises in many contexts such as crime prevention, search and rescue, post-conflict stability operations, and peace keeping. In these situations, military or police units are not only effective deterrents to would-be adversaries but also a speedy task force to intercept any trespassers or provide swift security and assistance. With recent advances in technology, unmanned aerial vehicles (UAVs) are well-suited for these tasks because they possess a large bird’s-eye view and are unhindered by ground obstacles. The path planning algorithms used in such missions play a critical role in minimizing the required resources, and maximizing the quality of provided service. Moreover, in many of these mission scenarios, if a patrol pattern is regular or predictable, adversaries can plan a counter strategy to the patrolling effort. In other words, unpredictability is one of the key features of planning in these circumstances. In this work, we propose and analyze the Persistent Patrol and Detection Problem (PPDP), a generic mathematical model for UAVs with limited sensors performing such a mission in a stochastic environment. Incidents occur dynamically and stochastically according to a general renewal process with known time intensity and spatial distribution in a planar region. The UAV is modeled as a point mass traveling at a constant speed with unbounded acceleration. The UAV detects incidents within the footprint of its onboard sensors, i.e., within its visibility range. We want to minimize the expected waiting time between the occurrence of an incident, and its detection epoch. Furthermore, we prefer stochastic trajectories so that would-be adversaries cannot predict the UAV’s motion based on past observations to plan for evasion. Related research focuses on search and rescue missions in which the number of searched objects is known at the beginning of the missions.1,2, 3 In other words, these works present static problems in terms of the number of objects to be found. In contrast, incidents of interest in the PPDP arrive continuously with unknown arrival times, and therefore the search effort must be persistent and preventive. This inherent difference between the PPDP and previous works has made the PPDP a dynamic problem in term of the number of incidents. ∗V. A. Huynh is with the Laboratory of Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139. vuhuynh@mit.edu †J. J. Enright is with Kiva Systems, 225 Wildwood Ave, Woburn, MA 01801. jenright@kivasystems.com ‡E. Frazzoli is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139. frazzoli@mit.edu

A major objective of plant ecology research is to determine the underlying processes responsible for the observed spatial distribution patterns of plant species. Plants can be approximated as points in space for this purpose, and thus, spatial point pattern analysis has become increasingly popular in ecological research. The basic piece of data for point pattern analysis is a point location of an ecological object in some study region. Therefore, point pattern analysis can only be performed if data can be collected. However, due to the lack of a convenient sampling method, a few previous studies have used point pattern analysis to examine the spatial patterns of grassland species. This is unfortunate because being able to explore point patterns in grassland systems has widespread implications for population dynamics, community‐level patterns, and ecological processes. In this study, we developed a new method to measure individual coordinates of species in grassland communities. This method records plant growing positions via digital picture samples that have been sub‐blocked within a geographical information system (GIS). Here, we tested out the new method by measuring the individual coordinates of Stipa grandis in grazed and ungrazed S. grandis communities in a temperate steppe ecosystem in China. Furthermore, we analyzed the pattern of S. grandis by using the pair correlation function g(r) with both a homogeneous Poisson process and a heterogeneous Poisson process. Our results showed that individuals of S. grandis were overdispersed according to the homogeneous Poisson process at 0–0.16 m in the ungrazed community, while they were clustered at 0.19 m according to the homogeneous and heterogeneous Poisson processes in the grazed community. These results suggest that competitive interactions dominated the ungrazed community, while facilitative interactions dominated the grazed community. In sum, we successfully executed a new sampling method, using digital photography and a geographical information system, to collect experimental data on the spatial point patterns for the populations in this grassland community.

This paper describes a method for planning the duration of a randomized parallel group study in which the response of interest is a potentially recurrent event. At the design stage we assume patients accrue at a constant rate, we model events via a homogeneous Poisson process, and we utilize an independent exponential censoring mechanism to reflect loss to follow-up. We derive the appropriate study duration to ensure satisfaction of power requirements for the effect size of interest under a Poisson regression model. An application to a kidney transplant study illustrates the potential savings of the Poisson-based design relative to a design based on the time to the first event. Revised design criteria are also derived to accommodate overdispersed Poisson count data. We examine the frequency properties of two non-parametric tests recently proposed by Lawless and Nadeau for trials based on the above design criteria. In simulation studies involving homogeneous and non-homogeneous Poisson processes they performed well with respect to their type I error rate and power. Results from supplementary simulation studies indicate that these tests are also robust to extra-Poisson variation and to clustering in the event times, making these tests attractive in their generality. We illustrate both tests by application to data from a completed kidney transplant study.

The Bayesian approach to analyzing data from repairable systems is presented in this chapter. Specifically, the homogeneous Poisson process and the power law process, a nonhomogeneous Poisson process with intensity equal to λ( t ) = (β/θ)( t /θ) β−1 , t > 0, are studied. Other models, such as the renewal process, follow a similar pattern. Also, measures of system performance, such as availability, are discussed, where the Bayesian approach provides a convenient way to combine uncertainties. For multiple copies of a repairable system, we discuss empirical Bayes and hierarchical Bayes methods.

Molecular loci that fail relative-rate tests are said to be "overdispersed." Traditional molecular-clock approaches to estimating divergence times do not take this into account. In this study, a method was developed to estimate divergence times using loci that may be overdispersed. The approach was to replace the traditional Poisson process assumption with a more general stationary process assumption. A probability model was developed, and an accompanying computer program was written to find maximum-likelihood estimates of divergence times under both the Poisson process and the stationary process assumptions. In simulation, it was shown that confidence intervals under the traditional Poisson assumptions often vastly underestimate the true confidence limits for overdispersed loci. Both models were applied to two data sets: one from land plants, the other from the higher metazoans. In both cases, the traditional Poisson process model could be rejected with high confidence. Maximum-likelihood analysis of the metazoan data set under the more general stationary process suggested that their radiation occurred well over a billion years ago, but confidence intervals were extremely wide. It was also shown that a model consistent with a Cambrian (or nearly Cambrian) origination of the animal phyla, although significantly less likely than a much older divergence, fitted the data well. It is argued that without an a priori understanding of the variance in the time between substitutions, molecular data sets may be incapable of ever establishing the age of the metazoan radiation.

The wetland plants are very sensitive to hydrological regimes. In this study, the individual sizes of a widely distributed species (i.e., Deyeuxia angustifolia) at three typical marshes with different water table depths (i.e., wet meadow (WM) marsh; seasonal inundated (SI) marsh; perennial inundated (PI) marsh) were investigated in the Sanjiang Plain of Northeast China. Concurrently, three primary point pattern processes (homogeneous Poisson (HP) process, homogeneous Thomas (HT) process, and inhomogeneous Thomas (IT) process) were used to model spatial patterns in the distribution at 0-50cm scale for this tillering-cloning species. The plant height, diameter at breast height (d.b.h), internode number, branches number, and individual aboveground biomass of D. angustifolia decreased sharply with rising water level; however, its density and coverage increased first and then decreased as water level increases. The distribution of D. angustifolia totally diverged from the complete spatial randomness (CSR) model (i.e., HP process) suggesting strong aggregation at 0-50cm scale in all marshes, and aggregated intensity enhanced with increasing water level. Interestingly, the spatial distribution of D. angustifolia fits better with the nested double-cluster model (i.e., IT process) at all scales in WM and SI marshes, indicating that there is a series of clustered patterns under the slight flood stress. However, the spatial pattern fits well with the Poisson cluster model (i.e., HT process) at all scales in PI marsh, implying the small-scale clustering disappeared with the intensification of flooding stress. Our results highlight that the D. angustifolia population could adapt to flooding stress in a certain degree via individual miniaturization strategies and multi-aggregation mechanisms in the freshwater wetlands.

We consider an incomplete repair model; that is, the impact of repair is not minimal as in the homogeneous Poisson process and not “as good as new” as in renewal processes but lies between these boundary cases. The repairs are assumed to affect the failure intensity following a virtual age process affect of the general form proposed by Kijima. In previous works field data from an industrial setting were used to fit several models. In most cases the estimated rate of occurrence of failures was that of an underlying exponential distribution of the time between failures. In this chapter it is shown that there exist maintenance schedules under which the failure behavior of the failure–repair process becomes a homogeneous Poisson process.

<p>The renewal process defines as a counting process where the times between the count renewals is a random variables and their distribution is identical. In the electricity generation machines there are spare parts replaced due to damage or expired and replacement process occur repeatedly and the renewal process of here assume that times between replacements are independent random variables and it has identical probability distribution. In this paper, renewal process model has applied on the time of fault for machine in Bahri Thermal Station for electricity generation, which is belong to the National Electricity Authority in Sudan during the period (2011-2015). Through the renewal process model estimation is clear that, the failure time (renewal) for the machines follow Weibull distribution with 2-parameters and when the time trend has been tested it is clear that no trend exist which mean that the renewal process represent Homogeneous Poisson Process (HPP), and the repair rate (renewal) is occurred constantly. In addition, the findings approve that whenever the repair rate (renewal) increase the mean time between failures (MTBF) increases too and this clear in machine no (6).</p>

Ruin probability in the dual risk model with two revenue streams

LetY0, Y1, Y2, … be an i.i.d. sequence of random variables with continuous distribution function, and letPbe a simple point process on 0≦t≦∞, independent of theYj's.We assume thatPhas a point att= 0; we associateYjwith thejth point ofj≧0, and we say that theYj'soccur at the arrival times ofP. Y0is considered a ‘reference value'. The firstYj(j≧1) to exceed all previous ones is called the first ‘record value', and the time of its occurrence is the first ‘record time'. Subsequent record values and times are defined analogously. We give an infinite series representation for the joint characteristic function of the firstnrecord times, for generalP; in some cases the series can be summed. We find the intensity of the record process whenPis a general birth process, and whenPis a linear birth process withmimmigration sources we find the distribution of the number of records in (0,t]. Form =0 (the Yule process) we give moments of record times and a compact form for the record process intensity. We show that the records occur according to a homogeneous Poisson process whenm =1, and we display a different model with the same behavior, leading to statistical non-identifiability if only the record times are observed. Form =2, the records occur according to a semi-Markov process; again we display a different model with the same behavior. Finally we give a new derivation of the joint distribution of the interrecord times whenPis an arbitrary Poisson process. We relate this result to existing work and to the classical record model. We also obtain a new characterization of the exponential distribution.