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 a ε (0,1). The result is used to obtain the overflow distribution of this controlled workload process; we derive double struck E sign^-αW(T) and double struck E sign[e ^αW(T) | W(T) ≥ 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.