A Markov-Modulated (s, S) Inventory System with Repeated Calls and Blocked Demands

TL;DR

This paper models a Markov-modulated (s, S) inventory system with demand failures, analyzing its stability and providing a closed-form formula for the steady-state minimum cost.

Contribution

It introduces a novel application of ergodicity criteria to a Markov-modulated inventory system with retrial queues and derives a closed-form cost formula.

Findings

Derived a traffic-intensity formula for the system.

Established conditions for system ergodicity.

Provided an analytic solution for the minimum cost.

Abstract

In this article, we consider a continuous review (s, S) inventory system with failures of demand fulfillment (service) modeled as a Markov-modulated retrial queueing system. The inventory system features a single product that experiences Markovian inter-demand and service intervals with random service interruptions and instantaneous replenishments. A recently developed criterion for the ergodicity of a class of discrete-time level-dependent-quasi-birth-and-death (LDQBD) processes with convergent transition matrix rows is applied to the jump chain of the process in order to elicit a closed-form traffic-intensity formula. An analytic solution for the steady-state average minimum cost is provided.