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.

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.

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.

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.

Hitting-Time Densities of a Two-Dimensional Markov Process

1992 ◽  
Vol 6 (4) ◽  
pp. 561-580
Author(s):  
C. H. Hesse
Keyword(s):  
Global Analysis ◽  
First Passage Time ◽  
Type Equation ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Wide Range ◽  
The Laplace Transform ◽  
Mean And Variance

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

First-passage time for a particular stationary periodic Gaussian process

1976 ◽  
Vol 13 (01) ◽  
pp. 27-38 ◽  
Author(s):  
L. A. Shepp ◽  
D. Slepian
Keyword(s):  
Gaussian Process ◽  
Infinite Series ◽  
First Passage Time ◽  
Passage Time ◽  
Numerical Calculations ◽  
Time Interval ◽  
Two Dimensional ◽  
First Passage ◽  
Gaussian Density ◽  
First Passage Probability

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x 0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

Remarks on renewal theory for percolation processes

1976 ◽  
Vol 13 (02) ◽  
pp. 290-300 ◽  
Author(s):  
R. T. Smythe
Keyword(s):  
Ergodic Theorem ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Integer Lattice ◽  
Two Dimensional ◽  
First Passage Percolation ◽  
First Passage ◽  
Underlying Distribution ◽  
Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L 1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

First Passage Time Distribution of a Two-Dimensional Wiener Process with Drift

1993 ◽  
Vol 7 (4) ◽  
pp. 545-555 ◽  
Author(s):  
Marco Dominé ◽  
Volkmar Pieper
Keyword(s):  
Wiener Process ◽  
First Passage Time ◽  
Exit Time ◽  
Passage Time ◽  
Absorbing Boundary Conditions ◽  
Two Dimensional ◽  
First Exit Time ◽  
First Passage ◽  
Absorbing Boundary ◽  
Starting Point

The two-dimensional correlated Wiener process (or Brownian motion) with drift is considered. The Fokker-Planck (or Kolmogorov forward) equation for the Wiener process (X1(t), X2(t)) is solved under absorbing boundary conditions on the lines x1= h1 and x2 = h2 and a fixed starting point (x0,1, x0,2). The first passage (or first exit) time when the process leaves the domain G = ( −∞, h1) × ( −∞, h2) is derived.

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