Production inventory with service time and interruptions

Abstract

This paper considers an (s,S) production inventory system with positive service time, with time for producing each item following Erlang distribution. Customers arrive according to a Poisson process. A customer who arrives when there is no inventory in the system is considered lost. On the other hand, a customer who finds a busy server with at least one inventory in the system joins a queue of infinite capacity. When the inventory level falls to s, production process is switched on, and it is switched off when the inventory level reaches back to S. Service time to each customer also follows an Erlang distribution. The service of a customer may be interrupted, where the time for such a phenomenon follows an exponential distribution, whenever it occurs. An interrupted service, after repair, resumes from where it was stopped. The correction/repair time follows an exponential distribution. We assume that the service of a single customer may encounter any number of interruptions and that the customer being served waits there until his service is completed. Moreover, at a time the server is subject to at most one interruption. We also assume that no inventory is lost due to a service interruption. Like the service process, the production process also is subject to interruptions, where the duration to an interruption follows an exponential distribution. However, in contrast to the service interruption, in the case of interruption to production process, we assume that the item being processed is lost because of interruption. That is, the production process, on being interrupted, restarts from the beginning, after repair. The repair time of an interrupted production process follows exponential distribution. Few of the last service phases are assumed to be protected in the sense that the service will not be interrupted while being in these phases. The same is assumed for the production process also.

The model is analysed as a level-independent quasi-birth–death process. We apply a novel method to obtain an explicit expression for the necessary and sufficient condition for the stability of the system under study. This method works even if we assume general phase-type distributions for the production as well as the service processes, and hence can be used to characterise the stability of inventory systems where the assumption of disallowing the customers to join the system, when there is a shortage of inventory has been made. Under stability, we apply matrix analytic methods to compute the system state distribution. In consequence to that, several system performance measures have been derived, and their dependence on the system parameters has been studied numerically.

Keywords:

• (s,S) production inventory
• positive service time
• interruption
• Erlang distribution
• phase-type distribution
• LIQBD
• matrix analytic methods

Acknowledgements

The authors gratefully acknowledge the suggestions and constructive criticisms of their reviewers, the associate editor and the editor on earlier versions of this paper who helped to improve the presentation.

Additional information

Notes on contributors

A. Krishnamoorthy

A. Krishnamoorthy is emeritus professor in the Department of Mathematics at Cochin University of Science & Technology. He has guided 30 research students and has about 110 research papers in stochastic modelling, with particular reference to queues, inventory and reliability.

Sajeev S. Nair

Sajeev S. Nair is currently assistant professor in the Department of Mathematics at Government Engineering College, Thrissur, Kerala. He received his doctoral degree from the Cochin University of Science & Technology in December 2012. His research interests include stochastic modelling in queues and inventory.

Viswanath C. Narayanan

Viswanath C. Naryanan is currently assistant professor in the Department of Mathematics at Government Engineering College, Thrissur, Kerala. His research interests include stochastic modelling in queues, inventory, reliability, biology and immunology. He has about 30 published research papers to his credit .