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.

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.

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 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.

A conservation property for general GI/G/1 queues with an application to tandem queues

1979 ◽  
Vol 11 (03) ◽  
pp. 660-672 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Limit Theorem ◽  
Total Variation ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Output Process ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Conservation Property ◽  
Positive Recurrent

We show that, if the input process of a generalGI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

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 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.

A conservation property for general GI/G/1 queues with an application to tandem queues

1979 ◽  
Vol 11 (3) ◽  
pp. 660-672 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Limit Theorem ◽  
Total Variation ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Output Process ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Conservation Property ◽  
Positive Recurrent

We show that, if the input process of a general GI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

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