Queueing-inventory models with batch demands and positive service times

We consider queueing-inventory models in which customers arrive according to a point process and each customer demands for one or more items but not exceeding a pre-determined (finite) value, say, $N$. The demands of the customers require positive service times. Replenishments are based on $(s, S)$-type policy and the lead times are assumed to be random. We consider two models. In Model 1, any arriving customer finding the inventory level to be zero will be lost. In Model 2, the loss of customers occur in two ways. First, an arriving customer finding the inventory level to be zero with the server being idle will be lost, and secondly, the customers, if any, present at a service completion with zero inventory will all be lost. We assume that in both the models the demands may be met partially based on the requests and the available inventory levels at that time. The inventory level is reduced by the amount to meet (fully or partially) the requisite demand of the customer at the beginning of the service. Under the assumption that all underlying random variables are exponential, we perform the steady-state analysis of the models using the classical matrix-analytic methods. Illustrative examples comparing the two models are presented.