Uniform limit theorems for non-singular renewal and Markov renewal processes

1978 ◽  
Vol 15 (01) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

1978 ◽  
Vol 15 (1) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.


2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.


2007 ◽  
Vol 44 (2) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.


2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.


1981 ◽  
Vol 18 (03) ◽  
pp. 752-756
Author(s):  
Per Kragh Andersen

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.


1968 ◽  
Vol 5 (2) ◽  
pp. 387-400 ◽  
Author(s):  
Jozef L. Teugels

In [3], Kendall proved a solidarity theorem for irreducible denumerable discrete time Markov chains. Vere-Jones refined Kendall's theorem by obtaining uniform estimates [14], while Kingman proved analogous results for an irreducible continuous time Markov chain [4], [5].We derive similar solidarity theorems for an irreducible Markov renewal process. The transient case is discussed in Section 3, and Section 4 deals with the positive recurrent case. Recently Cheong also proved solidarity theorems for Semi-Markov processes [1]. His theorems use the Markovian structure, while our emphasis is on the renewal aspects of Markov renewal processes.An application to the M/G/1 queue is included in the last section.


1991 ◽  
Vol 23 (01) ◽  
pp. 64-85 ◽  
Author(s):  
C. Y. Teresalam ◽  
John P. Lehoczky

This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of queueing systems.


1975 ◽  
Vol 12 (1) ◽  
pp. 167-169 ◽  
Author(s):  
Mats Rudemo

Examples are given of point processes that are non-stationary but have stationary forward recurrence time distributions. They are obtained by modification of stationary Poisson and renewal processes.


1991 ◽  
Vol 23 (1) ◽  
pp. 64-85 ◽  
Author(s):  
C. Y. Teresalam ◽  
John P. Lehoczky

This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of queueing systems.


1975 ◽  
Vol 12 (01) ◽  
pp. 167-169
Author(s):  
Mats Rudemo

Examples are given of point processes that are non-stationary but have stationary forward recurrence time distributions. They are obtained by modification of stationary Poisson and renewal processes.


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