A discrete time Geo/Geo/1 inventory system with modified $N$-policy

M. P. Anilkumar; K. P. Jose

doi:10.26637/MJM0803/0023

Abstract

This paper considers a discrete-time $(s, S)$ inventory system with positive service and $N$-policy. The arrival of customers constitutes a Bernoulli process. The system will be on vacation up to $N$ customers and it starts batch service of size $N$ with geometrically distributed service time. The subsequent customers who arrive during the batch service period are served in a single with geometrically distributed service time. The maximum storage of inventory is $s$. Whenever the on-hand inventory level drops to prefixed level $s$, an order for replenishment is placed. Lead time is also geometrically distributed. The system is analysed and the stability condition is derived using Matrix Analytic Method. Busy period, waiting-time distribution, reorder time distribution and inter-replenishment time are obtained. Numerical experiments are also incorporated to study system variation of system parameters.

Keywords:

Bernoulli process, Discrete time inventory, Geometric distribution, Matrix Analytic Method, N-Policy

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

M. P. Anilkumar Department of Mathematics, T. M. Government College, Vakkad P.O, Tirur, Malappuram-676502, Kerala, India. https://orcid.org/0000-0002-7005-4843
K. P. Jose P. G. & Research Department of Mathematics, St.Peter’s College, Kolenchery-682311 Kerala, India.

Pages: 868-876
Date Published: 01-07-2020
Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)

Torben Meisling. Discrete-time queuing theory. Operations Research, 6(1):96-105, 1958.

Stella C Dafermos and Marcel F Neuts. A single server queue in discrete time. Technical report, Purdue Univ Lafayette Ind Dept of Statistics, 1969.

Zhaotong Lian, Liming Liu, and Marcel F Neuts. A discrete-time model for common life time inventory systems. Mathematics of Operations Research, 30(3):718$732,2005$.

A Krishnamoorthy, B Lakshmy, and R Manikandan. A survey on inventory models with positive service time. Opsearch, 48(2):153-169, 2011.

A Krishnamoorthy and K P Jose. Comparison of inventory systems with service, positive lead-time, loss, and retrial of customers. International Journal of Stochastic Analysis, 2007, 2008.

Micha Yadin and PinhasNaor. Queueing systems with a removable service station. Journal of the Operational Research Society, 14(4):393-405, 1963.

Daniel P Heyman. Optimal operating policies for M/G/1 queuing systems. Operations Research, 16(2):362-382, 1968.

KR Balachandran. Control policies for a single server system. Management Science, 19(9):1013-1018, 1973.

Jesús R Artalejo. A unified cost function for M/G/1 queueing systems with removable server. Trabajos de Investigacionoperativa, 7(1):95, 1992.

KG Gakis, Hyun-Ku Rhee, and BD Sivazlian. Distributions and first moments of the busy and idle periods in controllable M/G/1 queueing models with simple and dyadic policies. Stochastic Analysis and Applications, 13(1):47-81, 1995.

A. Krishnamoorthy, Resmi Varghese, and B. Lakshmy. An (s, q) inventory system with positive lead time and service time under n-policy. Calcutta Statistical Association Bulletin, 66(3-4):241-260, 2014.

A Krishnamoorthy and T G Deepak. Modified n-policy for M/G/1 queues. Computers & Operations Research, 29(12):1611-1620, 2002.

Attahiru Sule Alfa. Discrete time queues and matrixanalytic methods. Top, 10(2): 147-185, 2002.

Attahiru S Alfa. Applied discrete-time queues. Springer, 2016.

Marcel F Neuts. Matrix-geometric solutions in stochastic models: an algorithmic approach. Courier Corporation, 1994.

DH Shi, J Guo, and L Liu. Sph-distributions and the rectangle-iterative algorithm. Lecturer note in Pure and Applied Mathematics, pages 207-224, 1996.