Unreliable production–inventory system with superposition of k Poisson demand arrival processes

A production–inventory system is studied in which an unreliable machine is susceptible to failure following which it must be repaired to make it operative again and the demands for the product are the superposition of k Poisson demand arrival processes, demand sizes are independent and identically distributed random variables. A two-critical-number policy is used to control a machine's setups and shutdowns. The sufficient conditions to ensure the existence of the stationary distribution of inventory level and explicit expression of this stationary distribution are obtained. Moreover, some performance measures of the system under steady-state condition are also obtained.