Numerical solutions of renewal-type integral equations

The integral equation of renewal type has many applications in applied probability. However, it is rarely solvable in closed form. In this paper, we describe a numerical method for finding approximate solutions to integral equations of renewal type based on quadrature rules for Stieltjes integrals previously developed by the author. We provide error analyses for our numerical solution based on the trapezoid-like rule for Stieltjes integrals and describe computational experience with two solution algorithms based on the trapezoid-like and the Simpson-like rules for Stieltjes integrals. We discuss a link between discrete and continuous renewal theory that is illuminated by the numerical solution method. The proposed numerical method is simple, amenable to direct error analysis, performs better than previously proposed approximations in many cases of significant practical interest, and is capable of handling some cases on which other methods fail.