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Compound geometric approximation under a failure rate constraint

Part of: Distribution theory Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  24 October 2016

Fraser Daly
Fraser Daly*
Affiliation:
Heriot-Watt University
*
*Postal address: Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK. Email address: f.daly@hw.ac.uk
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Abstract

We consider compound geometric approximation for a nonnegative, integer-valued random variable W. The bound we give is straightforward but relies on having a lower bound on the failure rate of W. Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth–death processes and Poisson processes.

Keywords

Compound geometric distributionfailure ratehazard rate orderingM/G/1 queuebirth–death processPoisson process

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60E15: Inequalities; stochastic orderings 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62E10: Characterization and structure theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 700 - 714
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Asmussen, S. (2003).Applied Probability and Queues, 2nd edn.Springer,New York.Google Scholar
[2] Barbour, A. D. and Chen, L. H. Y. (2015).An Introduction to Stein's Method,Singapore, University Press.Google Scholar
[3] Barbour, A. D. and Grübel, R. (1995).The first divisible sum.J. Theoret. Prob. 8,39–47.CrossRefGoogle Scholar
[4] Bon, J.-L. (2006).Error bounds for exponential approximation of large-system reliability.J. Math. Sci. (N.Y.) 138,5366–5376.CrossRefGoogle Scholar
[5] Brown, M. (1990).Error bounds for exponential approximations of geometric convolutions.Ann. Prob. 18,1388–1402.CrossRefGoogle Scholar
[6] Brown, M. (2014).Variance bounds under a hazard rate constraint.Prob. Eng. Inf. Sci. 28,203–208.Google Scholar
[7] Brown, M. (2015).Sharp bounds for exponential approximations under a hazard rate upper bound.J. Appl. Prob. 52,841–850.Google Scholar
[8] Brown, M. and Kemperman, J. H. B. (2009).Sharp two-sided bounds for distributions under a hazard rate constraint.Prob. Eng. Inf. Sci. 23,31–36.Google Scholar
[9] Daly, F. (2010).Stein's method for compound geometric approximation.J. Appl. Prob. 47,146–156.Google Scholar
[10] Fallat, S. M. and Johnson, C. R. (2011).Totally Nonnegative Matrices,Princeton University Press.Google Scholar
[11] Kalashnikov, V. (1997).Geometric Sums: Bounds for Rare Events with Applications.Kluwer,Dordrecht.Google Scholar
[12] Karlin, S. and McGregor, J. (1959).Random walks.Illinois J. Math. 3,66–81.Google Scholar
[13] Kleinrock, L. (1975).Queueing Systems, Vol. 1.John Wiley,New York.Google Scholar
[14] Kyprianou, E. K. (1972).The quasi-stationary distributions of queues in heavy traffic.J. Appl. Prob. 9,821–831.Google Scholar
[15] Müller, A. and Stoyan, D. (2002).Comparison Methods for Stochastic Models and Risks,John Wiley,Chichester.Google Scholar
[16] Obretenov, A. (1981).Upper bound of the absolute difference between a discrete IFR distribution and a geometric one. In Proceedings of the Sixth Conference on Probability Theory,Editura Academiei Republicii Socialiste România,Bucharest, pp.177–184.Google Scholar
[17] Peköz, E. A. (1996).Stein's method for geometric approximation.J. Appl. Prob. 33,707–713.CrossRefGoogle Scholar
[18] Peköz, E. A. and Röllin, A. (2011R).New rates for exponential approximation and the theorems of Rényi and Yaglom.Ann Prob. 39,587–608.Google Scholar
[19] Peköz, E. A.,Röllin, A. and Ross, N. (2013).Total variation error bounds for geometric approximation.Bernoulli 19,610–632.Google Scholar
[20] Phillips, M. J. and Weinberg, G. V. (2000).Non-uniform bounds for geometric approximation.Statist. Prob. Lett. 49,305–311.Google Scholar
[21] Ross, S. M. (2006).Bounding the stationary distribution of the M/G/1 queue size.Prob. Eng. Inf. Sci. 20,571–574.CrossRefGoogle Scholar
[22] Ross, S. M.,Shanthikumar, J. G. and Zhu, Z. (2005).On increasing-failure-rate random variables.J. Appl. Prob. 42,797–809.CrossRefGoogle Scholar
[23] Shaked, M. and Shanthikumar, J. G. (2007).Stochastic Orders.Springer,New York.Google Scholar
[24] Shanthikumar, J. G. (1988).DFR property of first-passage times and its preservation under geometric compounding.Ann. Prob. 16,397–406.Google Scholar
[25] Van Doorn, E. A. and Schrijner, P. (1995).Ratio limits and limiting conditional distributions for discrete-time birth–death processes.J. Math. Anal. Appl. 190,263–284.Google Scholar
[26] Vellaisamy, P. and Chaudhuri, B. (1996).Poisson and compound Poisson approximations for random sums of random variables.J. Appl. Prob. 33,127–137.Google Scholar