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Ergodicity properties of stress release, repairable system and workload models

Published online by Cambridge University Press:  01 July 2016

Günter Last*
Affiliation:
Universität Karlsruhe
*
Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe (TH), D-76128 Karlsruhe, Germany. Email address: [email protected]

Abstract

In this paper we derive some of the main ergodicity properties of a class of Markov renewal processes and the associated marked point processes. This class represents a generic model of applied probability and is of importance in earthquake modeling, reliability theory and queueing.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Alsmeyer, G. (1994). On the Markov renewal theorem. Stoch. Process. Appl. 50, 3756.CrossRefGoogle Scholar
[2] Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.Google Scholar
[3] Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, Berlin.Google Scholar
[4] Block, H. W., Borges, W. S. and Savits, T. H. (1985). Age-dependent minimal repair. J. Appl. Prob. 22, 370385.Google Scholar
[5] Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. John Wiley, Chichester.Google Scholar
[6] Borovkov, K. and Vere-Jones, D. (2000). Explicit formulae for stationary distributions of stress release processes. J. Appl. Prob. 37, 315321.Google Scholar
[7] Browne, S. and Sigman, K. (1992). Work-modulated queues with applications to storage processes. J. Appl. Prob. 29, 699712.Google Scholar
[8] Dai, J. and Meyn, S. P. (1995). Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans. Automatic Control 40, 18891904.Google Scholar
[9] Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.CrossRefGoogle Scholar
[10] Dorado, C., Hollander, M. and Sethuraman, J. (1997). Nonparametric estimation for a general repair model. Ann. Statist. 25, 11401160.CrossRefGoogle Scholar
[11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
[12] Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press.CrossRefGoogle Scholar
[13] Harrison, J. M. and Resnick, S. I. (1976). The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 1, 347358.Google Scholar
[14] Kijima, M. (1989). Some results for repairable systems. J. Appl. Prob. 26, 89102.CrossRefGoogle Scholar
[15] Konstantopoulos, T. and Last, G. (1999). On the use of Lyapunov function methods in renewal theory. Stoch. Process. Appl. 79, 165178.Google Scholar
[16] Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line. Springer, New York.Google Scholar
[17] Last, G. and Szekli, R. (1998). Stochastic comparison of repairable systems. J. Appl. Prob. 35, 348370.CrossRefGoogle Scholar
[18] Last, G. and Szekli, R. (1998). Asymptotic and monotonicity properties of some repairable systems. Adv. Appl. Prob. 30, 10891110.Google Scholar
[19] Last, G. and Szekli, R. (1999). On Palm and time stationarity of repairable systems. Stoch. Process. Appl. 79, 1743.CrossRefGoogle Scholar
[20] Last, G. and Szekli, R. (2001). Moments and Blackwell's convergence for repairable systems with heavy tailed lifetimes. Markov Process. Relat. Fields 7, 469490.Google Scholar
[21] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
[22] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487517.Google Scholar
[23] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes Adv. Appl. Prob. 25, 518548.Google Scholar
[24] Tuominen, P. and Tweedie, R. L. (1993). Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Prob. 26, 775–598.Google Scholar
[25] Vere-Jones, D. (1978). Earthquake predicition—a statistician's view. J. Phys. Earth 26, 129146.CrossRefGoogle Scholar
[26] Vere-Jones, D. (1988). On the variance properties of stress release models. Austral. J. Statist. 30A, 123135.CrossRefGoogle Scholar
[27] Vere-Jones, D. and Ogata, Y. (1984). On the moments of a self-correcting process. J. Appl. Prob. 21, 335342.Google Scholar
[28] Zheng, X. (1991). Ergodic theorems for stress release processes. Stoch. Process. Appl. 37, 239258.Google Scholar