A production inventory model with self-generation of priorities and retrial of customers

In this article we consider a model in which arrival of customers to the production inventory follows a Poisson process. The production starts when the inventory level depletes to s due to the service provided to the arriving customers. The time between two successive items added to the inventory exponentially distributed with parameter β. The production process shuts down when the inventory level reaches S. An arriving customer who finds the inventory dry or server busy proceeds to an orbit of infinite capacity with probability σ and is lost forever with probability 1 — σ. The inter-retrial time follows an exponential distribution with a linear rate iθ, if there are i customers in orbit. A retrial customer who finds inventory dry or server busy returns to the orbit with probability δ and lost forever with probability 1 — δ. The model is studied as a level dependent quasi birth-death(LDQBD) process and matrix geometric solution obtained.