Two-commodity queueing-inventory system with phase-type distribution of service times

A two-commodity queueing-inventory system with phase-type service times and exponential lead times is considered. There are two types of customers; Type 1 and Type 2. Demands from each customer type occur independently according to a Poisson process with different rates whereas the service times follow a phase-type distribution. Type 1 customers have a non-preemptive priority over Type 2 customers. We assume a finite waiting space for Type 1 customers whereas there is no limit on the waiting room for Type 2 customers. Type i customers demand only commodity i, $$i=1,2$$ i = 1 , 2 . For the ith commodity, $$S_i$$ S i and $$s_i$$ s i represent, respectively, the maximum inventory level and the reorder level. Whenever the inventory level of ith commodity drops to $$s_i$$ s i , an order is placed from retailer-i to make the inventory level $$S_i$$ S i . The lead times of the commodities are exponentially distributed with different parameters. When there is a Type i customer waiting in the queue, if the inventory level of ith commodity is zero (or reaches zero), a decision of immediate purchase is made so as not to lose the waiting customer. The queueing-inventory model in the steady-state is analyzed using the matrix-geometric method. The system performance is examined for different values of parameters. Besides, an optimization study is performed for some system parameters.