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

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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$a\in (0,1)$$</EquationSource> </InlineEquation>. The result is used to obtain the overflow distribution of this controlled workload process; we derive <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathbb{E}e^{-\alpha} W(T)$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathbb{E}[e^{{-\alpha}^{W(T)}} | W(T) \geq 1]$$</EquationSource> </InlineEquation>, 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. Copyright Springer-Verlag 2006