Numerical Solution of Renewal-Type Integral Equations

The integral equation of renewal type has numerous applications in applied probability. However, it is usually not solvable in closed form. Therefore, there is considerable interest in finding approximate solutions that are good for stochastic modeling. In this paper, we describe a numerical method for approximate solution of integral equations of renewal type based on quadrature rules for Stieltjes integrals previously developed by the author. We review a history of methods for approximate solution of integral equations of renewal type. We provide an error analysis 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. The proposed numerical method is simple, amenable to direct error analysis, and performs better than previously proposed approximations in many cases of significant practical interest.