Abstract
This paper studies an M/M/1 queueing-inventory system with batch demands. Customers arrive in the system according to a compound Poisson process, where the size of the batch demands for each arrival is a random variable that follows a geometric distribution. The inventory is replenished according to the standard (s,S) policy. The replenishment time follows an exponential distribution. Two models are considered. In the first model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer takes away all the items in the inventory, and a part of the customer’s batch demands is lost. In the second model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer leaves without taking any item from the inventory, and all of the customer’s batch demands are lost. For these two models, the authors derive the stationary conditions of the system. Then, the authors derive the stationary distributions of the product-form of the joint queue length and the on-hand inventory process. Besides this, the authors obtain some important performance measures and the average cost functions by using these stationary distributions. The results are illustrated by numerical examples.
Similar content being viewed by others
References
Schwarz M, Sauer C, Daduna H, et al., M/M/1 Queueing systems with inventory, Queueing Systems, 2006, 54(1): 55–78.
Simchi L D and Sigman K, Light traffc heuristic for an M/G/1 queue with limited inventory, Annals of Operations Research, 1992, 40(1): 371–380.
Berman O, Kaplan E H, and Shimshak D G, Deterministic approximations for inventory management at service facilities, IIE Transactions, 1993, 25(5): 98–104.
Berman O and Kim E, Stochastic models for inventory management at service facilities, Stochastic Models, 1999, 15(4): 695–718.
Berman O and Kim E, Dynamic inventory strategies for profit maximization in a service facility with stochastic service, demand and lead time, Mathematical Methods of Operations Research, 2004, 60(3): 497–521.
Saffari M, Asmussen S, and Haji R, The M/M/1 queue with inventory, lost sale and general lead times, Queueing Systems, 2013, 75(1): 65–77.
Krenzler R and Daduna H, Loss systems in a random environment steady-state analysis, Queueing Systems, 2015, 80(1): 127–153.
Xu J and Chen S, Optimal production and rationing policies of a make-to-stock production system with batch demand and backordering, Operations Research Letters, 2010, 38: 231–235.
Huang B and Wu A, EOQ model with batch demand and planned backorders, Applied Mathematical Modelling, 2016, 40: 5482–5496.
Saidane S, Babai M, Aguir M, et al., On the performance of the base-stock inventory system under a compound erlang demand distribution, Computers and Industrial Engineering, 2013, 66(3): 548–554.
Mohebbi S, Supply interruptions in a lost-sales inventory system with random lead time, Operations Research, 2001, 30(3): 411–426.
Axsater S, Initiation of an inventory control system when the demand starts at a given time, International Journal of Production Economics, 2013, 143(2): 553–556.
Goh C H, Greenberg B S, and Matsuo H, Perishable inventory systems with demand and arrivals, Operations Research Letters, 1993, 13: 1–8.
Tang O, Application of transforms in a compound demands process, Scientific Journal on Traffic and Transportation Technology, 2001, 13(6): 355–364.
Anily S, Beja A, and Mendel A, Optimal lot sizes with geometric production yield and rigid demand, Operations Research, 2002, 50(3): 424–432.
Neuts M F, Matrix Geometric Solutions in Stochastic Models — An Algorithmic Approach, John Hopkins University Press, Baltimore, 1981.
Krishnamoorthy A, Manikandan R, and Lakshmy B, A revisit to queueing-inventory system with positive service time, Annals of Operations Research, 2013, 233(1): 1–16.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported in part by the Natural Science Foundation of Hebei Province, China under Grant No. A2017203078, and Natural Science Research Project of the Education Department of Henan Province, China under Grant No. 2011C110002.
This paper was recommended for publication by Editor WANG Shouyang.
Rights and permissions
About this article
Cite this article
Yue, D., Zhao, G. & Qin, Y. An M/M/1 Queueing-Inventory System with Geometric Batch Demands and Lost Sales. J Syst Sci Complex 31, 1024–1041 (2018). https://doi.org/10.1007/s11424-018-6277-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-018-6277-y