Poisson approximation for some point processes in reliability

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

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Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes

Journal of Applied Probability ◽

10.1017/s002190020010806x ◽

1985 ◽

Vol 22 (04) ◽

pp. 828-835 ◽

Cited By ~ 2

Author(s):

J. Chandramohan ◽

Lung-Kuang Liang

Keyword(s):

Markov Chain ◽

Poisson Process ◽

Renewal Process ◽

Point Processes ◽

Renewal Processes ◽

Single Pair ◽

Doubly Stochastic ◽

Doubly Stochastic Poisson Process ◽

Non Homogeneous Poisson Process ◽

Dependent Point

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.

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Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes

Journal of Applied Probability ◽

10.2307/3213950 ◽

1985 ◽

Vol 22 (4) ◽

pp. 828-835 ◽

Cited By ~ 6

Author(s):

J. Chandramohan ◽

Lung-Kuang Liang

Keyword(s):

Markov Chain ◽

Poisson Process ◽

Renewal Process ◽

Point Processes ◽

Renewal Processes ◽

Single Pair ◽

Doubly Stochastic ◽

Doubly Stochastic Poisson Process ◽

Non Homogeneous Poisson Process ◽

Dependent Point

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.

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On doubly stochastic Poisson processes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410003838x ◽

1964 ◽

Vol 60 (4) ◽

pp. 923-930 ◽

Cited By ~ 75

Author(s):

J. F. C. Kingman

Keyword(s):

Poisson Process ◽

Stationary Point ◽

Point Processes ◽

Poisson Processes ◽

Doubly Stochastic ◽

Doubly Stochastic Poisson Process ◽

Stationary Point Process ◽

Special Case ◽

Stationary Point Processes ◽

Made In

The class of stationary point processes known as ‘doubly stochastic Poisson processes’ was introduced by Cox (2) and has been studied in detail by Bartlett (1). It is not clear just how large this class is, and indeed it seems to be a problem of some difficulty to decide of a general stationary point process whether or not it can be represented as a doubly stochastic Poisson process. (A few simple necessary conditions are known. For instance, Cox pointed out in the discussion to (1) that a double stochastic Poisson process must show more ‘dispersion’ than the Poisson process. Such conditions are very far from being sufficient.) The main result of the present paper is a solution of the problem for the special case of a renewal process, justifying an assertion made in the discussion to (1).

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SEGMENTED POINT PROCESS MODELS FOR MAINTAINED SYSTEMS

International Journal of Reliability Quality and Safety Engineering ◽

10.1142/s0218539307002738 ◽

2007 ◽

Vol 14 (05) ◽

pp. 431-458 ◽

Cited By ~ 5

Author(s):

A. SYAMSUNDAR ◽

V. N. A. NAIKAN

Keyword(s):

Poisson Process ◽

Point Process ◽

Point Processes ◽

Single Point ◽

Process Models ◽

Parameter Estimates ◽

Minimal Repair ◽

Homogeneous Poisson Process ◽

Stable Conditions ◽

Point Process Models

A maintained system is generally modeled using point processes. The most common processes used are the renewal process and the non homogeneous Poisson process corresponding to maximal and minimal repair situations with homogeneous Poisson process being a special case of both. A general repair formulation with a factor indicating the degree of repair is introduced into the minimal repair model to form an Arithmetic Reduction of Intensity model. These processes are generally able to model maintained systems with a fair degree of accuracy when the system is operating under stable conditions. However whenever there is a change in the environment these models which are monotonic in nature are not able to accommodate this change. Such systems operating under different environments need to be modelled by segmented models with the system domain divided into segments at the points of changes in the environment. The individual segments can then be modeled by any of the above point process models and these can be combined to form a composite model. This paper proposes a statistical model of such an operating/maintenance environment. Its purpose is to quantify the impacts of changes in the environment on the failure intensities. Field data from an industrial-setting demonstrate that appropriate parameter estimates for such phenomena can be obtained and such models are shown to more accurately describe the maintained system in a changing environment than the single point process models usually used.

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Functional Principal Component Modelling of the Intensity of a Doubly Stochastic Poisson Process

Compstat ◽

10.1007/978-3-642-57489-4_55 ◽

2002 ◽

pp. 373-376

Author(s):

Ana M. Aguilera ◽

Paula R. Bouzas ◽

Nuria Ruiz-Fuentes

Keyword(s):

Poisson Process ◽

Principal Component ◽

Doubly Stochastic ◽

Doubly Stochastic Poisson Process ◽

Functional Principal Component

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Rarefactions of compound point processes

Journal of Applied Probability ◽

10.1017/s0021900200037426 ◽

1984 ◽

Vol 21 (04) ◽

pp. 710-719

Author(s):

Richard F. Serfozo

Keyword(s):

Poisson Process ◽

Point Process ◽

Classical Result ◽

Point Processes ◽

Rare Events ◽

Random Measure ◽

Bernoulli Trials ◽

Random Measures ◽

Infinitely Divisible ◽

Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

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A limit theorem for spatial point processes

Advances in Applied Probability ◽

10.1017/s0001867800016001 ◽

1986 ◽

Vol 18 (03) ◽

pp. 646-659 ◽

Cited By ~ 1

Author(s):

Steven P. Ellis

Keyword(s):

Limit Theorem ◽

Poisson Process ◽

Asymptotic Distribution ◽

Point Process ◽

Point Processes ◽

Affine Transformations ◽

Density Estimates ◽

Spatial Point Processes ◽

Spatial Point ◽

As If

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

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On Laslett's transform for the Boolean model

Advances in Applied Probability ◽

10.1017/s0001867800010594 ◽

2001 ◽

Vol 33 (1) ◽

pp. 1-5 ◽

Cited By ~ 1

Author(s):

A. D. Barbour ◽

V. Schmidt

Keyword(s):

Poisson Process ◽

Point Process ◽

Simple Proof ◽

Poisson Point Process ◽

Boolean Model ◽

Random Sets ◽

Convex Compact ◽

Homogeneous Poisson Process ◽

Vertical Strip ◽

Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

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State aggregation and discrete-state Markov chains embedded in a class of point processes

Journal of Applied Probability ◽

10.1017/s0021900200102554 ◽

1995 ◽

Vol 32 (01) ◽

pp. 39-51

Author(s):

Xi-Ren Cao

Keyword(s):

Markov Chains ◽

Poisson Process ◽

Point Process ◽

Point Processes ◽

Marked Point ◽

Practical Importance ◽

Marked Point Process ◽

Interarrival Time ◽

Discrete State ◽

State Aggregation

One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.

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