scholarly journals Theory of semiregenerative phenomena

1988 ◽  
Vol 25 (A) ◽  
pp. 257-274
Author(s):  
N. U. Prabhu
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Renewal Process ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Discrete Time Case ◽  
Markov Renewal Theory

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

scholarly journals Theory of semiregenerative phenomena

1988 ◽  
Vol 25 (A) ◽  
pp. 257-274 ◽  
Author(s):  
N. U. Prabhu
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Renewal Process ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Discrete Time Case ◽  
Markov Renewal Theory

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

Periodicity in Markov renewal theory

1974 ◽  
Vol 6 (1) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Regenerative Processes ◽  
Markov Kernel ◽  
Transient States ◽  
Initial States ◽  
Markov Renewal Theory ◽  
Direct Inspection

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

Periodicity in Markov renewal theory

1974 ◽  
Vol 6 (01) ◽  
pp. 61-78 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Renewal Process ◽  
Limit Theorems ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Regenerative Processes ◽  
Markov Kernel ◽  
Transient States ◽  
Initial States ◽  
Markov Renewal Theory ◽  
Direct Inspection

In an irreducible Markov renewal process either all states are periodic or none are. In the former case they all have the same period. Periodicity and the period can be determined by direct inspection from the semi-Markov kernel defining the process. The periodicity considerably increases the complexity of the limits in Markov renewal theory especially for transient initial states. Two Markov renewal limit theorems will be given with particular attention to the roles of periodicity and transient states. The results are applied to semi-Markov and semi-regenerative processes.

scholarly journals A note on repeated sequences in Markov chains

1987 ◽  
Vol 19 (03) ◽  
pp. 739-742 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Transition Matrix ◽  
Repeated Sequences ◽  
Markov Renewal Process ◽  
Conditional Mean ◽  
Sojourn Times

If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

scholarly journals Relative stability and weak convergence in non-decreasing stochastically monotone Markov chains

1991 ◽  
Vol 4 (4) ◽  
pp. 293-303
Author(s):  
P. Todorovic
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Renewal Process ◽  
Relative Stability ◽  
Transition Probability ◽  
Markov Renewal Process

Let {ξn} be a non-decreasing stochastically monotone Markov chain whose transition probability Q(.,.) has Q(x,{x})=β(x)>0 for some function β(.) that is non-decreasing with β(x)↑1 as x→+∞, and each Q(x,.) is non-atomic otherwise. A typical realization of {ξn} is a Markov renewal process {(Xn,Tn)}, where ξj=Xn, for Tn consecutive values of j, Tn geometric on {1,2,…} with parameter β(Xn). Conditions are given for Xn, to be relatively stable and for Tn to be weakly convergent.

scholarly journals On the Dual Relationship Between Markov Chains of GI/M/1 and M/G/1 Type

2010 ◽  
Vol 42 (1) ◽  
pp. 210-225 ◽  
Author(s):  
P. G. Taylor ◽  
B. Van Houdt
Keyword(s):  
Markov Chains ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Dual Processes ◽  
Design Of Algorithms ◽  
Type Process ◽  
The Matrix ◽  
Positive Recurrent ◽  
Matrix Transforms ◽  
Phd Thesis

In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G(z, s) for the M/G/1-type process and R(z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

MONOTONICITY AND CONVEXITY OF SOME FUNCTIONS ASSOCIATED WITH DENUMERABLE MARKOV CHAINS AND THEIR APPLICATIONS TO QUEUING SYSTEMS

2005 ◽  
Vol 20 (1) ◽  
pp. 67-86 ◽  
Author(s):  
Hai-Bo Yu ◽  
Qi-Ming He ◽  
Hanqin Zhang
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Stochastic Dominance ◽  
Continuous Time ◽  
Queue Length ◽  
Queuing Theory ◽  
Queuing Systems ◽  
Discrete Time Case ◽  
Monotone Matrix

Motivated by various applications in queuing theory, this article is devoted to the monotonicity and convexity of some functions associated with discrete-time or continuous-time denumerable Markov chains. For the discrete-time case, conditions for the monotonicity and convexity of the functions are obtained by using the properties of stochastic dominance and monotone matrix. For the continuous-time case, by using the uniformization technique, similar results are obtained. As an application, the results are applied to analyze the monotonicity and convexity of functions associated with the queue length of some queuing systems.

scholarly journals A note on repeated sequences in Markov chains

1987 ◽  
Vol 19 (3) ◽  
pp. 739-742 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Transition Matrix ◽  
Repeated Sequences ◽  
Markov Renewal Process ◽  
Conditional Mean ◽  
Sojourn Times

If (non-overlapping) repeats of specified sequences of states in a Markov chain are considered, the result is a Markov renewal process. Formulae somewhat simpler than those given in Biggins and Cannings (1987) are derived which can be used to obtain the transition matrix and conditional mean sojourn times in this process.

scholarly journals On the Dual Relationship Between Markov Chains of GI/M/1 and M/G/1 Type

2010 ◽  
Vol 42 (01) ◽  
pp. 210-225 ◽  
Author(s):  
P. G. Taylor ◽  
B. Van Houdt
Keyword(s):  
Markov Chains ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Dual Processes ◽  
Design Of Algorithms ◽  
Type Process ◽  
The Matrix ◽  
Positive Recurrent ◽  
Matrix Transforms ◽  
Phd Thesis

In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G (z, s) for the M/G/1-type process and R (z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

Discrete-time Markov chains

2011 ◽  
pp. 1-184
Author(s):  
Yuri Suhov ◽  
Mark Kelbert
Keyword(s):  
Markov Chains ◽  
Discrete Time
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