tailieunhanh - Asymptotic results for the first and second moments and numerical computations in discrete-time bulk-renewal process

This paper introduces a simplified solution to determine the asymptotic results for the renewal density. It also offers the asymptotic results for the first and second moments of the number of renewals for the discrete-time bulk-renewal process. The methodology adopted makes this study distinguishable compared to those previously published where the constant term in the second moment is generated. In similar studies published in the literature, the constant term is either missing or not clear how it was obtained. | Yugoslav Journal of Operations Research xx (201x), Number nn, zzz–zzz DOI: ASYMPTOTIC RESULTS FOR THE FIRST AND SECOND MOMENTS AND NUMERICAL COMPUTATIONS IN DISCRETE-TIME BULK-RENEWAL PROCESS James J. KIM Royal Canadian Air Force (RCAF), Ottawa ON., Canada Mohan L. CHAUDHRY Royal Military College of Canada, Kingston ON., Canada chaudhry-ml@ Abdalla MANSUR Abu Dhabi Mens College, Higher Colleges of Technology, Abu Dhabi, United Arab Emirates amansur@ Received: April 2018 / Accepted: October 2018 Abstract: This paper introduces a simplified solution to determine the asymptotic results for the renewal density. It also offers the asymptotic results for the first and second moments of the number of renewals for the discrete-time bulk-renewal process. The methodology adopted makes this study distinguishable compared to those previously published where the constant term in the second moment is generated. In similar studies published in the literature, the constant term is either missing or not clear how it was obtained. The problem was partially solved in the study by Chaudhry and Fisher where they provided a asymptotic results for the non-bulk renewal density and for both the first and second moments using the generating functions. The objective of this work is to extend their results to the bulk-renewal process in discrete-time, including some numerical results, give an elegant derivation of the asymptotic results and derive continuous-time results as a limit of the discrete-time results. Keywords: Renewal Theory, Discrete-time, Bulk-renewal Process, Generating Function, Asymptotic Results. MSC: 60K05, 62E20, 60K25. 2 , and al. / Asymptotic Results for the First and Second Moments 1. INTRODUCTION Renewal theory and its applications have a significant role in many different areas such as failure and replacement of equipment, risk-based asset management models and queues [8]. The .