COMPARISONS OF DEPENDENCE FOR STATIONARY MARKOV PROCESSES

2000 ◽  
Vol 14 (3) ◽  
pp. 299-315 ◽  
Author(s):  
Taizhong Hu ◽  
Xiaoming Pan
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Waiting Time ◽  
Continuous Time ◽  
Queueing Theory ◽  
Stationary Markov Process ◽  
Stationary Waiting Time ◽  
Markov Modulated ◽  
Stationary Workload ◽  
Monotonicity Results

Results and conditions which quantify the decrease in dependence with lag for a stationary Markov process and enable one to compare the dependence for two stationary Markov processes are obtained. The notions of dependence used in this article are the supermodular ordering and the concordance ordering. Both discrete-time and continuous-time Markov processes are considered. Some applications of the main results are given. In queueing theory, the monotonicity results of the waiting time of the nth customer as well as the stationary waiting time in an MR/GI/1 queue and the stationary workload in a Markov-modulated queue are established, thus strengthening previous results while simplifying their derivation. This article is a continuation of those by Fang et al. [7] and Hu and Joe [10].

Proportional intensities and strong ergodicity for Markov processes

1983 ◽  
Vol 20 (01) ◽  
pp. 185-190 ◽  
Author(s):  
Mark Scott ◽  
Dean L. Isaacson
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Strong Ergodicity ◽  
Transition Functions ◽  
The Matrix ◽  
Intensity Functions ◽  
Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

Homecomings of Markov processes

1973 ◽  
Vol 5 (01) ◽  
pp. 66-102 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Continuous Time ◽  
Random Time ◽  
Random Set ◽  
Theoretical Interest ◽  
Different Types ◽  
Treat All ◽  
Image Position

Ifx0is a particular state for a continuous-time Markov processX, the random time setis often of both practical and theoretical interest. Ignoring trivial or pathological cases, there are four different types of structure which this random set can display. To some extent, it is possible to treat all four cases in a unified way, but they raise different questions and require different modes of description. The distributions of various random quantities associated withcan be related to one another by simple and useful formulae.

Monotonicity results for MR/GI/1 queues

1997 ◽  
Vol 34 (02) ◽  
pp. 514-524 ◽  
Author(s):  
Nicole Bäuerle
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Transition Matrix ◽  
Stochastic Ordering ◽  
Arrival Process ◽  
Stationary Waiting Time ◽  
Monotonicity Results

This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the nth customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.

scholarly journals Possibility of Application of the Theory of Semi-Markov Processes to Determine Reliability of Diagnosing Systems / MOŻLIWOŚĆ ZASTOSOWANIA TEORII PROCESÓW SEMI-MARKOWA DO OKREŚLENIA NIEZAWODNOŚCI SYSTEMÓW DIAGNOZUJĄCYCH

2012 ◽  
Vol 24 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Jerzy Girtler
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Continuous Time ◽  
Diagnostic Tests ◽  
The State ◽  
Actual State ◽  
Reliability Model ◽  
Discrete State ◽  
Semi Markov Process ◽  
Semi Markov Processes

Abstract The paper provides justification for the necessity to define reliability of diagnosing systems (SDG) in order to develop a diagnosis on state of any technical mechanism being a diagnosed system (SDN). It has been shown that the knowledge of SDG reliability enables defining diagnosis reliability. It has been assumed that the diagnosis reliability can be defined as a diagnosis property which specifies the degree of recognizing by a diagnosing system (SDG) the actual state of the diagnosed system (SDN) which may be any mechanism, and the conditional probability p(S*/K*) of occurrence (existence) of state S* of the mechanism (SDN) as a diagnosis measure provided that at a specified reliability of SDG, the vector K* of values of diagnostic parameters implied by the state, is observed. The probability that SDG is in the state of ability during diagnostic tests and the following diagnostic inferences leading to development of a diagnosis about the SDN state, has been accepted as a measure of SDG reliability. The theory of semi-Markov processes has been used for defining the SDG reliability, that enabled to develop a SDG reliability model in the form of a seven-state (continuous-time discrete-state) semi-Markov process of changes of SDG states.

scholarly journals Monounireducible Nonhomogeneous Continuous Time Semi-Markov Processes Applied to Rating Migration Models

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Guglielmo D'Amico ◽  
Jacques Janssen ◽  
Raimondo Manca
Keyword(s):  
Distribution Function ◽  
Markov Process ◽  
Markov Processes ◽  
Continuous Time ◽  
Topological Structure ◽  
Credit Rating ◽  
Fundamental Importance ◽  
Sufficient Condition ◽  
Semi Markov Process ◽  
Semi Markov Processes

Monounireducible nonhomogeneous semi- Markov processes are defined and investigated. The mono- unireducible topological structure is a sufficient condition that guarantees the absorption of the semi-Markov process in a state of the process. This situation is of fundamental importance in the modelling of credit rating migrations because permits the derivation of the distribution function of the time of default. An application in credit rating modelling is given in order to illustrate the results.

On some waiting time problems

2000 ◽  
Vol 37 (03) ◽  
pp. 756-764 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Survival Analysis ◽  
Finite Number ◽  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Waiting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Semi Markov Processes

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

scholarly journals On Markov Modulated Mean-Reverting Price-Difference Models

2008 ◽  
Vol 13 ◽  
pp. 55
Author(s):  
W. P. Malcom ◽  
Lakhdar Aggoun ◽  
Mohamed Al-Lawati
Keyword(s):  
Stochastic Model ◽  
Markov Processes ◽  
Continuous Time ◽  
Price Difference ◽  
Time Dynamics ◽  
Price Process ◽  
Finite State ◽  
The Mean ◽  
New Feature ◽  
Markov Modulated

In this paper we develop a stochastic model incorporating a double-Markov modulated mean-reversion model. Unlike a price process the basis process X can take positive or negative values. This model is based on an explicit discretisation of the corresponding continuous time dynamics. The new feature in our model is that we suppose the mean reverting level in our dynamics as well as the noise coefficient can change according to the states of some finite-state Markov processes which could be the economy and some other unseen random phenomenon.  

scholarly journals The relationship between intensity and stochastic matrices for continuous-time discrete value stochastic non-homogeneous processes with Markov property

2017 ◽  
Vol 13 (3) ◽  
pp. 7244-7256
Author(s):  
Mi los lawa Sokol
Keyword(s):  
Markov Process ◽  
Stochastic Processes ◽  
Markov Processes ◽  
Continuous Time ◽  
Markov Property ◽  
Time Dependent ◽  
Stochastic Matrices ◽  
Homogeneous Markov Process ◽  
The Relationship ◽  
Time Form

The matrices of non-homogeneous Markov processes consist of time-dependent functions whose values at time form typical intensity matrices. For solvingsome problems they must be changed into stochastic matrices. A stochas-tic matrix for non-homogeneous Markov process consists of time-dependent functions, whose values are probabilities and it depend on assumed time pe- riod. In this paper formulas for these functions are derived. Although the formula is not simple, it allows proving some theorems for Markov stochastic processes, well known for homogeneous processes, but for non-homogeneous ones the proofs of them turned out shorter.

The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach

2003 ◽  
Vol 40 (4) ◽  
pp. 881-892 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Finite Alphabet ◽  
Algorithmic Approach ◽  
First Occurrence ◽  
Semi Markov Processes ◽  
Pattern Occurrences ◽  
Continuous Time Models

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.

On some waiting time problems

2000 ◽  
Vol 37 (3) ◽  
pp. 756-764 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Survival Analysis ◽  
Finite Number ◽  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Waiting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Semi Markov Processes

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

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