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A threshold policy in a Markov-modulated production system with server vacation: the case of continuous and batch supplies

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  29 November 2018

Yonit Barron
Yonit Barron*
Affiliation:
Ariel University
*
* Postal address: Department of Industrial Engineering and Management, Ariel University, Ariel 40700, Israel. Email address: barron@ariel.ac.il
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Abstract

We consider a Markov-modulated fluid flow production model under the D-policy, that is, as soon as the storage reaches level 0, the machine becomes idle until the total storage exceeds a predetermined threshold D. Thus, the production process alternates between a busy and an idle machine. During the busy period, the storage decreases linearly due to continuous production and increases due to supply; during the idle period, no production is rendered by the machine and the storage level increases by only supply arrivals. We consider two types of model with different supply process patterns: continuous inflows with linear rates (fluid type), and batch inflows, where the supplies arrive according to a Markov additive process (MAP) and their sizes are independent and have phase-type distributions depending on the type of arrival (MAP type). Four types of cost are considered: a setup cost, a production cost, a penalty cost for an idle machine, and a storage cost. Using tools from multidimensional martingale and hitting time theory, we derive explicit formulae for these cost functionals in the discounted case. Numerical examples, a sensitivity analysis, and insights are provided.

Keywords

Vacation𝐷-policyMAPMarkov-modulated fluid flow processphase-type distribution

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 90B05: Inventory, storage, reservoirs 90B30: Production models 90B22: Queues and service
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1246 - 1274
Copyright
Copyright © Applied Probability Trust 2018 

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