Thesis (BS Applied Mathematics) -- University of the Philippines Mindanao, 2005

Renewal functions represent the long-run behavior of a renewal process which plays a significant role in applied probability studies such as inventories, queuing, reliability and warranty. For large values of time t, the asymptotic approximation of the renewal function is suitably matched. To investigate the limitations of the asymptotic approximation, in particular its lower bound for time t, its performance was observed relative to the results of a numerical approximation to the renewal function based on the concept of the Riemann integration. By using the Exponential and Erlang-2 distribution as the density of the renewal function, the individual behaviors of the approximations were the first compared when an existing renewal function was available. Due to the asymptotic linear behavior of the densities, the lower bound of time ts of the asymptotic approximation was easy to obtain when the gamma and Weibull distribution has its variability at c2x <1 and the Lognormal distribution has its variability c2x <1. On the other hand, the identification of the lower bound of time ts for the Gamma and Weibull distribution when its variability is at c2x <1 and the Lognormal distribution when its variability is at c2x <1 was difficult to obtain owing to the resulting large run-time required by the numerical part of the method.