Estimation of the stationary distribution of semi-Markov processes with Borel state space

2006 ◽  
Vol 76 (14) ◽  
pp. 1536-1542 ◽  
Author(s):  
N. Limnios
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Borel State Space ◽  
Semi Markov Processes

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.

Stochastic Petri Nets: Modeling Power and Limit Theorems

1991 ◽  
Vol 5 (4) ◽  
pp. 477-498 ◽  
Author(s):  
Peter J. Haas ◽  
Gerald S. Shedler
Keyword(s):  
Petri Nets ◽  
State Space ◽  
Markov Processes ◽  
Discrete Event ◽  
Building Blocks ◽  
The State ◽  
Stochastic Petri Nets ◽  
Transition Mechanism ◽  
Modeling Power ◽  
Semi Markov Processes

Generalized semi-Markov processes and stochastic Petri nets provide building blocks for specification of discrete event system simulations on a finite or countable state space. The two formal systems differ, however, in the event scheduling (clock-setting) mechanism, the state transition mechanism, and the form of the state space. We have shown previously that stochastic Petri nets have at least the modeling power of generalized semi-Markov processes. In this paper we show that stochastic Petri nets and generalized semi-Markov processes, in fact, have the same modeling power. Combining this result with known results for generalized semi-Markov processes, we also obtain conditions for time-average convergence and convergence in distribution along with a central limit theorem for the marking process of a stochastic Petri net.

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.

Markov-renewal fluid queues

2004 ◽  
Vol 41 (03) ◽  
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.

A SUCCESSIVE LUMPING PROCEDURE FOR A CLASS OF MARKOV CHAINS

2012 ◽  
Vol 26 (4) ◽  
pp. 483-508 ◽  
Author(s):  
Michael N. Katehakis ◽  
Laurens C. Smit
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Queueing Models ◽  
Semi Markov Processes ◽  
Stationary Probabilities

A class of Markov chains we call successively lumbaple is specified for which it is shown that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a(typically much) smaller state space and this yields significant computational improvements. We discuss how the results for discrete time Markov chains extend to semi-Markov processes and continuous time Markov processes. Finally, we will study applications of successively lumbaple Markov chains to classical reliability and queueing models.

Quasi-stationary distributions in semi-Markov processes

1970 ◽  
Vol 7 (2) ◽  
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.

Convergence of quasi-stationary to stationary distributions for stochastically monotone Markov processes

1986 ◽  
Vol 23 (1) ◽  
pp. 215-220 ◽  
Author(s):  
Moshe Pollak ◽  
David Siegmund
Keyword(s):  
Markov Process ◽  
State Space ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Stationary Distributions ◽  
New Process ◽  
Quasi Stationary Distribution

It is shown that if a stochastically monotone Markov process on [0,∞) with stationary distribution H has its state space truncated by making all states in [B,∞) absorbing, then the quasi-stationary distribution of the new process converges to H as B →∞.

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

1989 ◽  
Vol 3 (3) ◽  
pp. 405-415 ◽  
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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