Two-commodity queueing-inventory system with two classes of customers

Abstract

We analyze a two-commodity queueing-inventory system with an individual ordering policy. The maximum storage capacity for the ith commodity is $$S_i$$ $$(i=1,2)$$ . The reorder level for ith commodity is fixed as $$s_i$$ and whenever the inventory level of ith commodity falls on $$s_i$$ an order for $$Q_i\,(=S_i-s_i)$$ 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.