Nonlinearly perturbed renewal equations: The nonpolynomial case
Author:
Ying Ni
Journal:
Theor. Probability and Math. Statist. 84 (2012), 117-129
MSC (2010):
Primary 60K05, 34E10; Secondary 60K25
DOI:
https://doi.org/10.1090/S0094-9000-2012-00865-X
Published electronically:
July 31, 2012
MathSciNet review:
2857422
Full-text PDF Free Access
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Abstract: Models of nonlinearly perturbed renewal equations with nonpolynomial perturbations are studied. Exponential asymptotic expansions are given for the solutions to the perturbed renewal equations under consideration. An application to perturbed M/G/1/ queues is presented.
References
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- William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- Mats Gyllenberg and Dmitrii S. Silvestrov, Quasi-stationary phenomena in nonlinearly perturbed stochastic systems, De Gruyter Expositions in Mathematics, vol. 44, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. MR 2456816
- Y. Ni, D. Silvestrov, and A. Malyarenko, Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations, J. Numer. Appl. Math. 1(96) (2008), 173–197.
- Y. Ni, Analytical and numerical studies of perturbed renewal equations with multivariate non-polynomial perturbations, Journal of Applied Quantitative Methods 5(3) (2010a), 498–515.
- Y. Ni, Perturbed Renewal Equations with Non-polynomial Perturbations, Licentiate Thesis, Mälardalen University, 2010b.
- D. S. Sīl′vestrov, A generalization of the renewal theorem, Dokl. Akad. Nauk Ukrain. SSR Ser. A 11 (1976), 978–982, 1052 (Russian, with English summary). MR 0483057
- D. S. Sīl′vestrov, The renewal theorem in the scheme of series. I, Teor. Verojatnost. i Mat. Statist. 18 (1978), 144–161, 183 (Russian, with English summary). MR 0488350
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- D. S. Silvestrov, Exponential asymptotics for a perturbed renewal equation, Teor. Ĭmovīr. Mat. Stat. 52 (1995), 143–153 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 52 (1996), 153–162. MR 1445549
References
- S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000. MR 1794582 (2001m:62119)
- W. Feller, An Introduction to Probability Theory and its Applications, vol. II, Wiley, New York, 1971. MR 0270403 (42:5292)
- M. Gyllenberg and D. S. Silvestrov, Quasi-stationary Phenomena in Nonlinearly Perturbed Stochastic Systems, De Gruyter Expositions in Mathematics, vol. 44, Walter de Gruyter, Berlin, 2008. MR 2456816 (2009k:60005)
- Y. Ni, D. Silvestrov, and A. Malyarenko, Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations, J. Numer. Appl. Math. 1(96) (2008), 173–197.
- Y. Ni, Analytical and numerical studies of perturbed renewal equations with multivariate non-polynomial perturbations, Journal of Applied Quantitative Methods 5(3) (2010a), 498–515.
- Y. Ni, Perturbed Renewal Equations with Non-polynomial Perturbations, Licentiate Thesis, Mälardalen University, 2010b.
- D. S. Silvestrov, A generalization of the renewal theorem, Dokl. Akad. Nauk. Ukr. SSR, Ser. A, 11 (1976), 978–982. MR 0483057 (58:3086)
- D. S. Silvestrov, The renewal theorem in a series scheme. I, Teor. Veroyatn. Mat. Stat. 18 (1978), 144–161; English transl. in Theory Probab. Math. Statist. 18, 155–172. MR 0488350 (58:7899)
- D. S. Silvestrov, The renewal theorem in a series scheme. II, Teor. Veroyatn. Mat. Stat. 20 (1979), 97–116; English transl. in Theory Probab. Math. Statist. 20, 113–130. MR 529265 (80g:60092)
- D. S. Silvestrov, Exponential asymptotic for perturbed renewal equations, Teor. Ǐmovirn. Mat. Stat. 52 (1995), 143–153; English transl. in Theory Probab. Math. Statist. 52, 153–162. MR 1445549 (97m:60127)
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Additional Information
Ying Ni
Affiliation:
Division of Applied Mathematics, School of Education, Culture, and Communication, Mälardalen University, Västerås 721 23, Sweden
Email:
ying.ni@mdh.se
Keywords:
Perturbed renewal equation,
nonpolynomial perturbation
Received by editor(s):
October 30, 2010
Published electronically:
July 31, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
