A Markovian inventory system with random shelf time and back orders

In this paper we consider an inventory system in which the demands occur according to a Markovian arrival process (MAP). The items in the inventory have shelf times that are assumed to follow an exponential distribution. The inventory is replenished according to an (s, S) policy and the replenishing times are assumed to follow a phase type distribution. Back orders of demands are allowed up to a certain level. Using matrix analytic methods we perform the steady state analysis of the inventory model. We also show that the reorder process can be modelled as a MAP. An optimization problem is proposed and several interesting numerical examples are presented.