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Improved bounds for the solutions of renewal equations
Published online by Cambridge University Press: 06 June 2022
Abstract
Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time.
Keywords
- Type
- Research Article
- Information
- Probability in the Engineering and Informational Sciences , Volume 37 , Issue 3 , July 2023 , pp. 740 - 777
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press
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