Markov-renewal fluid queues

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.

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A Quasi-Reversibility Approach to the Insensitivity of Generalized Semi-Markov Processes

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800001273 ◽

1989 ◽

Vol 3 (3) ◽

pp. 405-415 ◽

Cited By ~ 1

Author(s):

Panagiotis Konstantopoulos ◽

Jean Walrand

Keyword(s):

Markov Process ◽

Stationary Distribution ◽

Markov Processes ◽

Simple Proof ◽

Queueing Network ◽

Simple Criterion ◽

Algebraic Criterion ◽

Semi Markov Process ◽

Semi Markov Processes ◽

Necessary And Sufficient

This paper is concerned with a certain property of the stationary distribution of a generalized semi-Markov process (GSMP) known as insensitivity. It is well-known that the so-called Matthes' conditions form a necessary and sufficient algebraic criterion for insensitivity. Most proofs of these conditions are basically algebraic. By interpreting a GSMP as a simple queueing network, we are able to show that Matthes' conditions are equivalent to the quasi-reversibility of the network, thus obtaining another simple proof of the sufficiency of these conditions. Furthermore, we apply our method to find a simple criterion for the insensitivity of GSMP's with generalized routing (in a sense that is introduced in the paper).

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General solidarity theorems for semi-Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200036159 ◽

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.

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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

Journal of Konbin ◽

10.2478/jok-2013-0052 ◽

2012 ◽

Vol 24 (1) ◽

pp. 49-58 ◽

Cited By ~ 1

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.

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

Advances in Decision Sciences ◽

10.1155/2012/123635 ◽

2012 ◽

Vol 2012 ◽

pp. 1-12 ◽

Cited By ~ 3

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.

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Quasi-stationary distributions in semi-Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200034951 ◽

1970 ◽

Vol 7 (02) ◽

pp. 388-399 ◽

Cited By ~ 7

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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General solidarity theorems for semi-Markov processes

Journal of Applied Probability ◽

10.2307/3212615 ◽

1972 ◽

Vol 9 (4) ◽

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.

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Quasi-stationary distributions in semi-Markov processes

Journal of Applied Probability ◽

10.2307/3211972 ◽

1970 ◽

Vol 7 (2) ◽

pp. 388-399 ◽

Cited By ~ 9

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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On the entropy for semi-Markov processes

Journal of Applied Probability ◽

10.1239/jap/1067436100 ◽

2003 ◽

Vol 40 (4) ◽

pp. 1060-1068 ◽

Cited By ~ 4

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.

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Additional quasi-stationary distributions for semi-Markov processes

Journal of Applied Probability ◽

10.2307/3212336 ◽

1972 ◽

Vol 9 (3) ◽

pp. 671-676 ◽

Cited By ~ 3

Author(s):

David C. Flaspohler ◽

Paul T. Holmes

Keyword(s):

Markov Process ◽

Markov Processes ◽

Sufficient Conditions ◽

Initial State ◽

Stationary Distributions ◽

Absorbing State ◽

Semi Markov Process ◽

Image Position ◽

Semi Markov Processes

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits: where Xj(t) is the amount of time prior to time t spent in state j.The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.

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