Analytical and simulation studies of queueing-inventory models with MAP demands in batches and positive phase type services

Queueing-inventory models have many practical applications and have been studied extensively in the literature. Most of the studies focus on models in which the demands occur singly. Only a very few papers analyze models wherein the demands occur in batches. In this paper we consider batch demands in the context of two models, both of which assume that the demands occur according to a versatile Markovian point process. The demands need to be serviced with items from the inventory, and the service times are assumed to be of phase type. The replenishment of the inventory is based on the (s, S)-type policy and the lead times are assumed to be random. These two models are such that in the first model an arriving customer finding the inventory level to be zero will be lost; and in the second model a customer can be lost either at the time of an arrival (wherein the server is idle due to zero inventory) or at the time of a service completion (at which time the inventory level becomes zero). In the second model, all waiting customers are removed from the system due to zero inventory. These two models are studied in steady-state using the classical matrix-analytic methods in single server case, and in the case of multi-server systems we resort to simulation using ARENA. Illustrative examples, including an optimization problem, comparing the two models are presented.