Time dependent analysis of multivariate marked renewal processes

2001 ◽  
Vol 38 (3) ◽  
pp. 707-721 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Compound Poisson Process ◽  
Passage Time ◽  
Time Dependent ◽  
Renewal Processes ◽  
First Passage ◽  
Time Dependent Analysis

The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.

Time dependent analysis of multivariate marked renewal processes

2001 ◽  
Vol 38 (03) ◽  
pp. 707-721 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Compound Poisson Process ◽  
Passage Time ◽  
Time Dependent ◽  
Renewal Processes ◽  
First Passage ◽  
Time Dependent Analysis

The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.

scholarly journals On functionals of a marked Poisson process observed by a renewal process

2001 ◽  
Vol 26 (7) ◽  
pp. 427-436 ◽  
Author(s):  
Jewgeni H. Dshalalow ◽  
Jean-Baptiste Bacot
Keyword(s):  
Laplace Transform ◽  
Poisson Process ◽  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
Time Dependent ◽  
First Passage ◽  
Time Dependent Solution ◽  
Time Dependent Analysis

We study the functionals of a Poisson marked processΠobserved by a renewal process. A sequence of observations continues untilΠcrosses some fixed level at one of the observation epochs (the first passage time). In various stochastic models applications (such as queueing withN-policy combined with multiple vacations), it is necessary to operate with the value ofΠprior to the first passage time, or prior to the first passage time plus some random time. We obtain a time-dependent solution to this problem in a closed form, in terms of its Laplace transform. Many results are directly applicable to the time-dependent analysis of queues and other stochastic models via semi-regenerative techniques.

scholarly journals First excess levels of vector processes

1994 ◽  
Vol 7 (3) ◽  
pp. 457-464 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Point Process ◽  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Compound Process ◽  
First Time

This paper analyzes the behavior of a point process marked by a two-dimensional renewal process with dependent components about some fixed (two-dimensional) level. The compound process evolves until one of its marks hits (i.e. reaches or exceeds) its associated level for the first time. The author targets a joint transformation of the first excess level, first passage time, and the index of the point process which labels the first passage time. The cases when both marks are either discrete or continuous or mixed are treated. For each of them, an explicit and compact formula is derived. Various applications to stochastic models are discussed.

scholarly journals On the level crossing of multi-dimensional delayed renewal processes

1997 ◽  
Vol 10 (4) ◽  
pp. 355-361 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
The Other ◽  
Renewal Processes ◽  
Active Components ◽  
Level Crossing ◽  
First Passage ◽  
Informal Discussion

The paper studies the behavior of an (l+3)th-dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the “crossing level” and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented.

First-passage times with PFr densities

1985 ◽  
Vol 22 (1) ◽  
pp. 185-196 ◽  
Author(s):  
David Assaf ◽  
Moshe Shared ◽  
J. George shanthikumar
Keyword(s):  
First Passage Time ◽  
Compound Poisson Process ◽  
Passage Time ◽  
First Passage Times ◽  
Embedded Markov Chain ◽  
First Passage ◽  
Completely Monotone ◽  
Totally Positive ◽  
Finite State ◽  
Passage Times

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

First passage time of filtered Poisson process with exponential shape function

2005 ◽  
Vol 20 (1) ◽  
pp. 57-65 ◽  
Author(s):  
A. Novikov ◽  
R.E. Melchers ◽  
E. Shinjikashvili ◽  
N. Kordzakhia
Keyword(s):  
Poisson Process ◽  
Shape Function ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

On the evaluation of first-passage-time probability densities via non-singular integral equations

1989 ◽  
Vol 21 (1) ◽  
pp. 20-36 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
S. Sato
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Singular Integral Equations ◽  
Passage Time ◽  
Time Dependent ◽  
Computational Results ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

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