On (s,S) inventory system with random lead time and repeated demands

P V Ushakumari

doi:10.1155/jamsa/2006/81508

Abstract

We consider an (s,S) inventory system with random lead time and repeated demands of unsatisfied demands from the orbit. Whenever the inventory level falls to the level s, an order is placed to bring the level to S. The quantity ordered is M=S−s. Demands to the system are served immediately if there is a positive inventory. Otherwise it will go to a pool of unsatisfied customers called orbit. After a random amount of time, that demand is retried for service. We assume a Markovian setup for the time between consecutive arrivals, replenishments, and retrials. We obtained the condition for ergodicity of the system, steady state system size probabilities, expected length of the busy period of the system, expected inventory level, expected number of customers waiting in the orbit, expected waiting times, and so forth. A control problem is studied and some numerical illusrtations are provided.

Highlights

The (s,S) inventory system with positive lead time has been studied by several researchers
The demands occurring during the stock-out period are either lost or satisfied only after the arrival of ordered items. In the latter case it is assumed that either all or a prefixed number of demands occurring during the stock-out period are satisfied
In some applications, the demands during the stock-out period go to an orbit of unsatisfied customers and after a random amount of time, retry for service

Summary

Introduction

The (s,S) inventory system with positive lead time has been studied by several researchers. In some applications, the demands during the stock-out period go to an orbit of unsatisfied customers and after a random amount of time, retry for service. We can see such situations in production inventory systems with positive lead times. The time between successive retrials is random and follows exponential distribution with rate αj(> 0) when j demands in the orbit. We obtained the steady state system size probabilities, expected length of busy period of the system, expected waiting time in the orbit, expected inventory level, expected number of waiting customers, and so forth.

Model description

Ergodicity

Steady state results

Busy period analysis

Waiting time

Control problem