Two-commodity queueing-inventory system with two classes of customers
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Two‑commodity queueing‑inventory system with two classes of customers Serife Ozkar1 · Umay Uzunoglu Kocer2 Accepted: 1 September 2020 © Operational Research Society of India 2020
Abstract We analyze a two-commodity queueing-inventory system with an individual ordering policy. The maximum storage capacity for the ith commodity is Si (i = 1, 2) . The reorder level for ith commodity is fixed as si and whenever the inventory level of ith commodity falls on si an order for Qi (= Si − si ) items ( i = 1, 2 ) is placed for that commodity irrespective of the inventory level of the other commodity. There are two types of customers, which are classified as a priority (Type-1) and ordinary (Type-2) customers. Priority customers demand only commodity-1, whereas ordinary customers demand only commodity-2. Each customer class arrives according to an independent Poisson process with different rates. Service time for each customer class is also independent of each other and follows an exponential distribution. Type-1 customers have non-preemptive priority over Type-2 customers. When the server is idle and there are both types of customers, then the service is offered to Type-1 customers. Type-2 customers have got service when there are no Type-1 customers waiting. It is assumed that waiting space for the priority customers is finite whereas there is no queue capacity for ordinary customers. The system is formulated by a five-dimensional continuous-time Markov chain. The structure of the infinitesimal generator matrix is shown to be of the QBD type. Steady-state distribution is obtained using the matrix-geometric method. The system load is formulated in a closed-form. A comprehensive numerical study is performed on the performance measures. Finally, an optimization study is presented. Keywords Quasi-birth-and-death process · Matrix geometric method · Queueinginventory system · Two-commodity system · Two customer classes Mathematics Subject Classification 60K25 · 90B05 · 90B22
* Serife Ozkar [email protected] 1
Department of International Trade and Logistics, Balikesir University, Balikesir, Turkey
2
Department of Statistics, Dokuz Eylul University, Izmir, Turkey
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1 Introduction Multi-commodity inventory systems are commonly encountered in many real-life situations. These systems are more complex than single commodity inventory systems due to the multitude of items stocked and their coordinated actions. For multi-product inventory systems, the main problem is the interaction of the reorder points and reorder times for each individual item. In the earliest studies, inventory models were proposed with independently established reorder points. The individual ordering policy consists of the calculation of optimum order quantity and/or reordering periods for each item. Implementation of an individual policy gives considerable flexibility to the system in selecting the individually best inventory models for every single item, and in modifying the policy independently. However, when t
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