Unreliable production–inventory systems with hyper-exponential renewal demand processes

We investigate a production–inventory system in which an unreliable machine is susceptible to failure following which it must be repaired to make it operative again. The inter-arrival times of demand arrival processes are assumed to be hyper-exponentially distributed, demand sizes are i.i.d. random variables. An (m,M) policy is used to control the machine's setups and shutdowns. We give a stability condition of the inventory process, and then obtain its steady-state distribution which is used to calculate some performance measures of the system.