A Blood Bank Model with Perishable Blood and Demand Impatience

We consider a stochastic model for a blood bank, in which amounts of blood are offered and demanded according to independent compound Poisson processes. Blood is perishable, i.e., blood can only be kept in storage for a limited amount of time. Furthermore, demand for blood is impatient, i.e., a demand for blood may be canceled if it cannot be satisfied soon enough. For a range of perishability functions and demand impatience functions, we derive the steady-state distributions of the amount of blood kept in storage, and of the amount of demand for blood (at any point in time, at most one of these quantities is positive). Under certain conditions we also obtain the fluid and diffusion limits of the blood inventory process, showing in particular that the diffusion limit process is an Ornstein-Uhlenbeck process.