Alternative Analysis of Finite-Time Probability Distributions of Renewal Theory

Abstract

We introduce a level-crossing analysis of the finite time-t probability distributions of the excess life, age, total life, and related quantities of renewal processes. The technique embeds the renewal process as one cycle of a regenerative process with a barrier at level t, whose limiting probability density function leads directly to the time-t quantities. The new method connects the analysis of renewal processes with the analysis of a large class of stochastic models of Operations Research. Examples are given.

Note on the service time in an M/G/1 queue with bounded workload

“…Sample path of workload {W K (t)} t≥0 bounded at level K shows a truncated service time S K , busy cycle, busy period, a state-space level x ∈ (0, K ). (Brill, 2014); processes in the natural sciences where there is a natural finite barrier, e.g., the concentration of a solution when solute is added in random amounts at a fixed temperature and pressure-assuming that the solute precipitates gradually out of solution over time, and the excess over the concentration barrier is lost.…”