A controlled M / G / 1 workload process with an application to perishable inventory systems

Abstract

We derive the stationary distribution of the regenerative process W(t), t ≥ 0, whose cycles behave like an M / G / 1 workload process terminating at the end of its first busy period or when it reaches or exceeds level 1, and restarting with some fixed workload $a\in (0,1)$. The result is used to obtain the overflow distribution of this controlled workload process; we derive $\mathbb{E}e^{-\alpha} W(T)$ and $\mathbb{E}[e^{{-\alpha}^{W(T)}} | W(T) \geq 1]$, where T is the duration of the first cycle. W(t) can be linked to a certain perishable inventory model, and we use our results to determine the distribution of the duration of an empty period.