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 the entropy for semi-Markov processes

2003 ◽  
Vol 40 (4) ◽  
pp. 1060-1068 ◽  
Author(s):  
Valerie Girardin ◽  
Nikolaos Limnios
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Relative Entropy ◽  
Continuous Time ◽  
Entropy Rate ◽  
Renewal Processes ◽  
Semi Markov Process ◽  
Continuous Time Processes ◽  
Semi Markov Processes ◽  
Jump Markov Processes

The aim of this paper is to define the entropy of a finite semi-Markov process. We define the entropy of the finite distributions of the process, and obtain explicitly its entropy rate by extending the Shannon–McMillan–Breiman theorem to this class of nonstationary continuous-time processes. The particular cases of pure jump Markov processes and renewal processes are considered. The relative entropy rate between two semi-Markov processes is also defined.

On a fundamental identity in the theory of semi-Markov processes

1972 ◽  
Vol 4 (02) ◽  
pp. 258-270 ◽  
Author(s):  
E. Arjas
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Markov Processes ◽  
Partial Sums ◽  
Discrete State ◽  
Fundamental Identity ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Wald's Identity ◽  
Discrete State Space

A fundamental identity, due to Miller (1961a), (1962a, b) and Kemperman (1961), is generalized to semi-Markov processes. Thus the identity applies to processes defined on a Markov chain with discrete state space and random walks with Markov dependent steps (Section 2). Wald's identity is discussed briefly in Section 3. Section 4 is a study of the maxima of partial sums, and Section 5 of maxima in a semi-Markov process.

On the entropy for semi-Markov processes

2003 ◽  
Vol 40 (04) ◽  
pp. 1060-1068 ◽  
Author(s):  
Valerie Girardin ◽  
Nikolaos Limnios
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Relative Entropy ◽  
Continuous Time ◽  
Entropy Rate ◽  
Renewal Processes ◽  
Semi Markov Process ◽  
Continuous Time Processes ◽  
Semi Markov Processes ◽  
Jump Markov Processes

The aim of this paper is to define the entropy of a finite semi-Markov process. We define the entropy of the finite distributions of the process, and obtain explicitly its entropy rate by extending the Shannon–McMillan–Breiman theorem to this class of nonstationary continuous-time processes. The particular cases of pure jump Markov processes and renewal processes are considered. The relative entropy rate between two semi-Markov processes is also defined.

On a fundamental identity in the theory of semi-Markov processes

1972 ◽  
Vol 4 (2) ◽  
pp. 258-270 ◽  
Author(s):  
E. Arjas
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Markov Processes ◽  
Partial Sums ◽  
Discrete State ◽  
Fundamental Identity ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Wald's Identity ◽  
Discrete State Space

A fundamental identity, due to Miller (1961a), (1962a, b) and Kemperman (1961), is generalized to semi-Markov processes. Thus the identity applies to processes defined on a Markov chain with discrete state space and random walks with Markov dependent steps (Section 2). Wald's identity is discussed briefly in Section 3. Section 4 is a study of the maxima of partial sums, and Section 5 of maxima in a semi-Markov process.

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.

Markov-renewal fluid queues

2004 ◽  
Vol 41 (3) ◽  
pp. 746-757 ◽  
Author(s):  
Guy Latouche ◽  
Tetsuya Takine
Keyword(s):  
Markov Process ◽  
Exponential Distribution ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Input Rate ◽  
Fluid Queue ◽  
Birth And Death Processes ◽  
Fluid Queues ◽  
Semi Markov Process ◽  
Semi Markov Processes

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov-renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (03) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij (t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

Quasi-stationary distributions in semi-Markov processes

1970 ◽  
Vol 7 (02) ◽  
pp. 388-399 ◽  
Author(s):  
C. K. Cheong
Keyword(s):  
Markov Process ◽  
State Space ◽  
Markov Processes ◽  
Transition Probability ◽  
Main Concern ◽  
Initial State ◽  
Stationary Distributions ◽  
Semi Markov Process ◽  
Image Position ◽  
Semi Markov Processes

Our main concern in this paper is the convergence, as t → ∞, of the quantities i, j ∈ E; where Pij (t) is the transition probability of a semi-Markov process whose state space E is irreducible but not closed (i.e., escape from E is possible), and rj is the probability of eventual escape from E conditional on the initial state being i. The theorems proved here generalize some results of Seneta and Vere-Jones ([8] and [11]) for Markov processes.

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