Intuitive approximations for a residual waiting time process

The residual waiting time process, also known as the residual life process, represents the remaining time until the next renewal event, observed at an arbitrary moment. This process arises naturally in diverse areas such as queueing systems, reliability analysis, and inventory modeling. However, obtaining exact expressions for the expected residual waiting time is often analytically challenging, especially when the interarrival time distribution deviates from the Erlang distribution case. In this study, we propose intuitive approximations for the expected value of residual waiting time process, based on intuitive approximation of the renewal function. Two classes of interarrival distributions are examined: heavy-tailed distributions with regularly varying tails, and light-tailed distributions belonging to the special class of distributions denoted by $\mathrm{\Gamma }(g)$, which naturally arises in extreme value theory. Using theoretical results from renewal theory and equilibrium distributions, intuitive approximation formulas are derived for both distributional settings. In particular, we investigate the Erlang distribution as a case study, comparing expected value of the residual waiting time computed via the exact renewal function with that obtained from the intuitive approximation. Moreover, for the Pareto and Burr XII distributions, we conduct case studies demonstrating how intuitive approximation closely matches asymptotic results for the expected value of the residual waiting time in the absence of exact formulas. This work provides a practical and mathematically grounded framework for analyzing systems involving stochastic arrivals, with potential extensions to higher-order moments.