Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

1991 ◽  
Vol 28 (4) ◽  
pp. 822-832 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Time Interval ◽  
Reliability Model ◽  
Finite State ◽  
The Laplace Transform ◽  
Semi Markov Process ◽  
Number Of Visits ◽  
Semi Markov Processes ◽  
Finite State Space

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t →∞.

Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

1991 ◽  
Vol 28 (04) ◽  
pp. 822-832 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Time Interval ◽  
Reliability Model ◽  
Finite State ◽  
The Laplace Transform ◽  
Semi Markov Process ◽  
Number Of Visits ◽  
Semi Markov Processes ◽  
Finite State Space

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB (t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB (t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB (t) as t →∞.

Aggregated semi-Markov processes incorporating time interval omission

1991 ◽  
Vol 23 (04) ◽  
pp. 772-797 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Renewal Process ◽  
Stochastic Modelling ◽  
Complete Information ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Finite State ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Finite State Space

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the process is in. We show that complete information concerning the aggregated process is contained in an embedded Markov renewal process, whose parameters, moments and equilibrium behaviour are determined. Such processes have found considerable application in stochastic modelling of single ion channels. In that setting there is time interval omission, i.e. brief sojourns in either class failed to be detected. Complete information on the aggregated process incorporating time interval omission is contained in a Markov renewal process, whose properties are derived, obtained from the above Markov renewal process by a further embedding. The embedded Markov renewal framework is natural, and its invariance to time interval omission leads to considerable economy in the derivation of properties of the observed process. The results are specialised to the case when the underlying process is a continuous-time Markov chain.

Aggregated semi-Markov processes incorporating time interval omission

1991 ◽  
Vol 23 (4) ◽  
pp. 772-797 ◽  
Author(s):  
Frank Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Renewal Process ◽  
Stochastic Modelling ◽  
Complete Information ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Finite State ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Finite State Space

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the process is in. We show that complete information concerning the aggregated process is contained in an embedded Markov renewal process, whose parameters, moments and equilibrium behaviour are determined. Such processes have found considerable application in stochastic modelling of single ion channels. In that setting there is time interval omission, i.e. brief sojourns in either class failed to be detected. Complete information on the aggregated process incorporating time interval omission is contained in a Markov renewal process, whose properties are derived, obtained from the above Markov renewal process by a further embedding. The embedded Markov renewal framework is natural, and its invariance to time interval omission leads to considerable economy in the derivation of properties of the observed process. The results are specialised to the case when the underlying process is a continuous-time Markov chain.

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.

A temporal factorization at the maximum for certain positive self-similar Markov processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1045-1069
Author(s):  
Matija Vidmar
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Markov Processes ◽  
Absorption Time ◽  
The Laplace Transform ◽  
Self Similar

AbstractFor a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

General solidarity theorems for semi-Markov processes

1972 ◽  
Vol 9 (04) ◽  
pp. 789-802
Author(s):  
Choong K. Cheong ◽  
Jozef L. Teugels
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Transition Probabilities ◽  
Large Set ◽  
Semi Markov Process ◽  
Regularly Varying ◽  
Specific Pair ◽  
Semi Markov Processes

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij (t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij (t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij (t). Among others it is shown that f(t) can be taken regularly varying.

Estimation of transition probabilities in a multiparticle semi-Markov system

1976 ◽  
Vol 13 (4) ◽  
pp. 696-706 ◽  
Author(s):  
David Burman
Keyword(s):  
Markov Process ◽  
Time Distribution ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Strong Limit ◽  
Stieltjes Transform ◽  
Markov System ◽  
Sample Paths ◽  
Finite State ◽  
Semi Markov Process

Particles enter a finite-state system and move according to independent sample paths from a semi-Markov process. Strong limit theorems are developed for the ratio of the flow of particles from states i to j and the flow out of When the cumulative arrival of particles into the system up to time t, A (t) ∼ λtα, then a.s. When A (t)∼ λekt, then the flow between states must be normalized by the Laplace–Stieltjes transform of the conditional holding time distribution, in order to make the ratio an unbiased estimator of ρij.

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