scholarly journals Nonlinear Markov Renewal Theory with Statistical Applications

1992 ◽  
Vol 20 (2) ◽  
pp. 753-771 ◽  
Author(s):  
Vincent F. Melfi
Keyword(s):  
Renewal Theory ◽  
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.

ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

2012 ◽  
Vol 26 (2) ◽  
pp. 221-244 ◽  
Author(s):  
M. L. Chaudhry ◽  
S. K. Samanta ◽  
A. Pacheco
Keyword(s):  
Length Distribution ◽  
Renewal Theory ◽  
Computationally Efficient ◽  
Form Analysis ◽  
Matrix Geometric Method ◽  
Alternative Approach ◽  
The Matrix ◽  
Sojourn Time Distribution ◽  
System Length ◽  
Markov Renewal Theory

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

Markov renewal theory

1969 ◽  
Vol 1 (02) ◽  
pp. 123-187 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
State Space ◽  
Renewal Theory ◽  
The State ◽  
Countable State Space ◽  
Countable State ◽  
Markov Renewal Theory ◽  
Random Amount

Consider a stochastic process X(t) (t ≧ 0) taking values in a countable state space, say, {1, 2,3, …}. To be picturesque we think of X(t) as the state which a particle is in at epoch t. Suppose the particle moves from state to state in such a way that the successive states visited form a Markov chain, and that the particle stays in a given state a random amount of time depending on the state it is in as well as on the state to be visited next. Below is a possible realization of such a process.

Renewal theorems for processes with dependent interarrival times

2018 ◽  
Vol 50 (4) ◽  
pp. 1193-1216
Author(s):  
Sabrina Kombrink
Keyword(s):  
Point Processes ◽  
Renewal Theory ◽  
Renewal Theorem ◽  
Hölder Continuous ◽  
Finite State ◽  
Key Renewal Theorem ◽  
Interarrival Times ◽  
Markov Renewal Theory ◽  
Discrete Measures ◽  
Finite State Space

Abstract In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry, for instance to the problem of Minkowski measurability of self-conformal sets.

A BAYESIAN APPROACH TO FIND RANDOM-TIME PROBABILITIES FROM EMBEDDED MARKOV CHAIN PROBABILITIES

2007 ◽  
Vol 21 (4) ◽  
pp. 551-556 ◽  
Author(s):  
Winfried K. Grassmann ◽  
Javad Tavakoli
Keyword(s):  
Markov Chain ◽  
Bayesian Approach ◽  
Queuing Theory ◽  
Classical Method ◽  
Renewal Theory ◽  
Random Time ◽  
Embedded Markov Chain ◽  
Markov Chain Approach ◽  
Complex Approach ◽  
Markov Renewal Theory

The embedded Markov chain approach is widely used in queuing theory, in particular in M/G/1 and GI/M/c queues. In these cases, one has to relate the embedded equilibrium probablities to the corresponding random-time probabilities. The classical method to do this is based on Markov renewal theory, a rather complex approach, especially if the population is finite or if there is balking. In this article we present a much simpler method to derive the random-time probabilities from the embedded Markov chain probabilities. The method is based on conditional probability. Our approach might also be applicable in such situations.

The N/G/1 queue and its detailed analysis

1980 ◽  
Vol 12 (01) ◽  
pp. 222-261 ◽  
Author(s):  
V. Ramaswami
Keyword(s):  
Poisson Process ◽  
Waiting Time ◽  
Queue Length ◽  
Renewal Theory ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Special Cases ◽  
Complex Models ◽  
Markov Modulated ◽  
Markov Renewal Theory

We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.

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