The difference of two renewal processes level crossing and the infimum

Abstract

We consider the difference process N of two independent renewal (counting) processes. Second-order approximations to the distribution function of the level crossing time are given. Direct application of the second-order approximation is complicated by the occurrence of an (in general) unknown term E[Mtilde], which denotes the expected minimum of the stationary version of N. However, this number is obtained for a wide class of processes N, using matrix-geometric techniques. Numerical experiments have been carried out, in which the new approximations were compared to simulation, first-order and/or exact results. These results confirm that the second-order approximations are considerably better than the (known) first-order ones.

Keywords:

• Difference process of independent renewal processes,
• boundary crossing,
• second-order approximation,
• PH-renewal processes,
• matrix-geometric solutions.