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The DFR Property for Counting Processes Stopped at an Independent Random Time

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

F. G. Badía  and
C. Sangüesa
F. G. Badía*
Affiliation:
University of Zaragoza
C. Sangüesa*
Affiliation:
University of Zaragoza
*
∗ Postal address: Maria de Luna 3, Zaragoza, 50018, Spain. Email address: gbadia@unizar.es
∗∗ Postal address: Pedro Cerbuna 11, Zaragoza, 50009, Spain. Email address: csangues@unizar.es
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Abstract

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In this paper we consider general counting processes stopped at a random time T, independent of the process. Provided that T has the decreasing failure rate (DFR) property, we present sufficient conditions on the arrival times so that the number of events occurring before T preserves the DFR property of T. In particular, when the interarrival times are independent, we consider applications concerning the DFR property of the stationary number of customers waiting in queue for specific queueing models.

Keywords

Decreasing failure ratecounting processrenewal processstochastic orderassociationLittle's law

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 574 - 585
Copyright
© Applied Probability Trust 

References

Asmussen, S., Klüppelberg, C. and Sigman, K. (1999). Sampling at subexponential times, with queueing applications. Stoch. Process. Appl. 79, 265–286.CrossRefGoogle Scholar
Badía, F. G. (2011). Hazard rate properties of a general counting process stopped at an independent random time J. Appl. Prob. 48, 56–67.CrossRefGoogle Scholar
Badía, F. G. and Salehi, E. T. (2012). Preservation of reliability classes associated with the mean residual life by a renewal process stopped at a random time Appl. Stoch. Models Business Industry. 28, 381–394.CrossRefGoogle Scholar
Badía, F. G. and Sangüesa, C. (2008). Preservation of reliability classes under mixtures of renewal processes. Prob. Eng. Inf. Sci. 22, 1–17.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Reinhart and Winston, New York.Google Scholar
Cox, D. R. (1962). Renewal Theory. Methuen, London.Google Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1, 627–649.CrossRefGoogle Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
Finkelstein, M. S. and Esaulova, V. (2001). Why the mixture failure rate decreases. Reliab. Eng. Syst. Safe. 71, 173–177.CrossRefGoogle Scholar
Grandell, J. (1997). Mixed Poisson Processes. Chapman & Hall, London.CrossRefGoogle Scholar
Gurland, J. and Sethuraman, J. (1995). How pooling failure data may reverse increasing failure rates. J. Amer. Statist. Assoc. 90, 1416–1423.CrossRefGoogle Scholar
Haji, R. and Newell, G. F. (1971). A relation between stationary queue and waiting time distributions. J. Appl. Prob. 8, 617–620.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics 5, 375–383.CrossRefGoogle Scholar
Ross, S. M., Shanthikumar, J. G. and Zhu, Z. (2005). On increasing-failure-rate random variables. J. Appl. Prob. 42, 797–809.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1988). Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446.CrossRefGoogle Scholar
Shanthikumar, J. G. (1988). DFR property of first-passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.CrossRefGoogle Scholar
Song, R., Karon, J. M., White, E. and Goldbaum, G. (2006). Estimating the distribution of a renewal process from times at which events from an independent process are detected. Biometrics 62, 838–846.CrossRefGoogle ScholarPubMed