On approximations for the distribution of first level crossing time

TL;DR

This paper evaluates the accuracy of existing approximations for the distribution of the first crossing time of a compound renewal process, comparing them with exact and simulation results across various inter-renewal and jump size distributions.

Contribution

It assesses the performance of specific approximation methods for the first crossing time distribution, including cases lacking exact solutions, using simulations and exact calculations.

Findings

Approximations perform well for exponential distributions.

Comparison shows varying accuracy across different distributions.

Simulations validate the effectiveness of certain approximation methods.

Abstract

We investigate performance of approximations put forth in \citeNP{[Malinovskii 2017a]} and \citeNP{[Malinovskii 2017b]} for the distribution of the time of first level $u$ crossing by the random process $\homV{s}-cs$, $s>0$, where $\homV{s}$ is compound renewal process. In the case of Exponential inter-renewal and jump size random variables, we compare the approximations with exact and with simulation results. In a few other cases including Erlang and Pareto inter-renewal and jump size random variables, where exact results are absent, we compare the approximations with simulation results.