scholarly journals Passage time distributions in large Markov chains

2002 ◽  
Author(s):  
Peter G. Harrison ◽  
William J. Knottenbelt
Keyword(s):  
Markov Chains ◽  
Passage Time

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.

Bounds and Approximations for the Transient Behavior of Continuous-Time Markov Chains

1989 ◽  
Vol 3 (2) ◽  
pp. 175-198 ◽  
Author(s):  
Bok Sik Yoon ◽  
J. George Shanthikumar
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
First Passage Time ◽  
Transient Behavior ◽  
Passage Time ◽  
Lower And Upper Bounds ◽  
Simple Method ◽  
Continuous Time Markov Chains ◽  
Recent Addition ◽  
Dependent Probability

Discretization is a simple, yet powerful tool in obtaining time-dependent probability distribution of continuous-time Markov chains. One of the most commonly used approaches is uniformization. A recent addition to such approaches is an external uniformization technique. In this paper, we briefly review these different approaches, propose some new approaches, and discuss their performances based on theoretical bounds and empirical computational results. A simple method to get lower and upper bounds for first passage time distribution is also proposed.

scholarly journals Sojourn time distributions in a Markovian G-queue with batch arrival and batch removal

1999 ◽  
Vol 12 (4) ◽  
pp. 339-356 ◽  
Author(s):  
Yang Woo Shin
Keyword(s):  
Markov Chains ◽  
Sojourn Time ◽  
First Passage Time ◽  
Passage Time ◽  
Batch Arrival ◽  
Single Server ◽  
Negative Customers ◽  
First Passage ◽  
Markovian Queue ◽  
Sojourn Time Distributions

We consider a single server Markovian queue with two types of customers; positive and negative, where positive customers arrive in batches and arrivals of negative customers remove positive customers in batches. Only positive customers form a queue and negative customers just reduce the system congestion by removing positive ones upon their arrivals. We derive the LSTs of sojourn time distributions for a single server Markovian queue with positive customers and negative customers by using the first passage time arguments for Markov chains.

scholarly journals On the First Passage Time Distribution for a Class of Markov Chains

1983 ◽  
Vol 11 (4) ◽  
pp. 1000-1008 ◽  
Author(s):  
Mark Brown ◽  
Narasinga R. Chaganty
Keyword(s):  
Markov Chains ◽  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

Spectral structure of the first-passage-time densities for classes of Markov chains

1987 ◽  
Vol 24 (03) ◽  
pp. 631-643 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
Spectral Structure ◽  
First Passage ◽  
Completely Monotone ◽  
Spectral Representations ◽  
Absorbing States ◽  
Birth Death

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

Spectral structure of the first-passage-time densities for classes of Markov chains

1987 ◽  
Vol 24 (3) ◽  
pp. 631-643 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
Spectral Structure ◽  
First Passage ◽  
Completely Monotone ◽  
Spectral Representations ◽  
Absorbing States ◽  
Birth Death

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

Passage-time generating functions for continuous-time finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 414-426 ◽  
Author(s):  
J. N. Darroch ◽  
K. W. Morris
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Functions ◽  
Continuous Time ◽  
Passage Time ◽  
Weighted Sum ◽  
Continuous Time Markov Chain ◽  
Finite Markov Chains

Let S denote a subset of the states of a finite continuous-time Markov chain and let Y(a) denote the time that elapses until a weighted sum of a time units have been spent in S. Formulae are derived for the generating functions of Y(a) and of Y(a + b) – Y(b).

Spectral Analysis, without Eigenvectors, for Markov Chains

1991 ◽  
Vol 5 (2) ◽  
pp. 131-144 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Spectral Analysis ◽  
Markov Chains ◽  
Transition Probabilities ◽  
First Passage Time ◽  
Passage Time ◽  
Interpolation Polynomial ◽  
Simple Eigenvalue ◽  
First Passage ◽  
Free Representation ◽  
The Right

The Lagrange-Sylvester interpolation polynomial approach provides a simple, eigenvector-free representation for finite diagonalizable matrices. This paper discusses the Lagrange-Sylvester methodology and applies it to skip free to the right Markov chains. It leads to relatively simple, eigenvalue-based expressions for first passage time distributions and transition probabilities.

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