A Semi-Markovian Analysis of an Inventory Model with Inventory-Level Dependent Arrival and Service Processes

Two Semi Markov processes are defined to describe the service and inter-arrival times of an \(s-S\) inventory model with zero lead time, in which both inter-arrival time and service time depend upon the inventory level. It is assumed that both the service time and the inter-arrival time follow Phase-Type distributions, which are determined by the current inventory level. The marginal distributions of both the service time and the inter-arrival time are obtained. A continuous parameter Markov chain is used to model the queue size. Condition for stability and the steady state characteristics of the system are derived. The impact of interdependence between service and arrival processes, along with inventory level on the system is examined. Furthermore, a numerical analysis is also done to explain the consequences of this dependency on steady-state system characteristics.