First passage times and lumpability of semi-Markov processes

1988 ◽  
Vol 25 (04) ◽  
pp. 675-687 ◽  
Author(s):  
Ushio Sumita ◽  
Maria Rieders
Keyword(s):  
Markov Processes ◽  
First Passage Times ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Passage ◽  
The Laplace Transform ◽  
First Exit Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient ◽  
Passage Times

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

First passage times and lumpability of semi-Markov processes

1988 ◽  
Vol 25 (4) ◽  
pp. 675-687 ◽  
Author(s):  
Ushio Sumita ◽  
Maria Rieders
Keyword(s):  
Markov Processes ◽  
First Passage Times ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Passage ◽  
The Laplace Transform ◽  
First Exit Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient ◽  
Passage Times

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

On first-passage times in increasing Markov processes

1996 ◽  
Vol 26 (3) ◽  
pp. 199-203 ◽  
Author(s):  
Rafael Pérez-Ocón ◽  
M.Luz Gámiz-Pérez
Keyword(s):  
Markov Processes ◽  
First Passage Times ◽  
First Passage ◽  
Passage Times

Hidden Markov Processes: The Complete Realization Problem

2014 ◽  
Author(s):  
M. Vidyasagar
Keyword(s):  
Markov Processes ◽  
Hidden Markov ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Rank Condition ◽  
Mixing Processes ◽  
Realization Problem ◽  
Hidden Markov Processes ◽  
Necessary And Sufficient

This chapter considers hidden Markov processes (HMPs), focusing on the so-called complete realization problem. It is quite easy to prove a universal necessary condition for the given process to have a hidden Markov model (HMM). However, this condition is not sufficient in general. In principle, one can derive a “necessary and sufficient condition,” but the “necessary and sufficient condition” is virtually a restatement of the problem to be solved and does not shed any insight into the solution. The chapter first introduces a very useful matrix known as the “Hankel” matrix before discussing the nonsufficiency of the finite Hankel rank condition, an abstract necessary and sufficient condition, and the existence of regular quasi-realizations. It also describes the spectral properties of alpha-mixing processes and goes on to analyze ultra-mixing processes and a sufficient condition for the existence of HMMs.

Ageing first-passage times of Markov processes: a matrix approach

1997 ◽  
Vol 34 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Haijun Li ◽  
Moshe Shaked
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Critical Value ◽  
First Passage Times ◽  
Matrix Approach ◽  
First Passage ◽  
Passage Times

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

Processes with new better than used first-passage times

1984 ◽  
Vol 16 (03) ◽  
pp. 667-686 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Point Process ◽  
Markov Processes ◽  
Probability Space ◽  
Sufficient Conditions ◽  
General Point ◽  
First Passage ◽  
Semi Markov Processes ◽  
New Better Than Used ◽  
Better Than ◽  
Passage Times

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R 0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.

First-passage percolation on the square lattice, II

1977 ◽  
Vol 9 (2) ◽  
pp. 283-295 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Sufficient Conditions ◽  
Initial Result ◽  
First Passage Times ◽  
Square Lattice ◽  
Maximum Height ◽  
First Passage Percolation ◽  
First Passage ◽  
Necessary And Sufficient ◽  
Passage Times

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.

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