In this paper we consider a single server queueing model with under general bulk service rule with infinite upper bound on the batch size which we call group clearance. The arrivals occur according to a batch Markovian point process and the services are generally distributed. The customers arriving after the service initiation cannot enter the ongoing service. The service time is independent on the batch size. First, we employ the classical embedded Markov renewal process approach to study the model. Secondly, under the assumption that the services are of phase type, we study the model as a continuous-time Markov chain whose generator has a very special structure. Using matrix-analytic methods we study the model in steady-state and discuss some special cases of the model as well as representative numerical examples covering a wide range of service time distributions such as constant, uniform, Weibull, and phase type.

In this paper, we consider a queuing inventory system with heterogeneous customers of K types arriving according to a marked Markovian arrival process. Each class of customers differs by nature of the service they seek and different priorities are assigned for each class resulting in different levels of inventory admitted to exhaust for customers of each class. A single service node is provided for each class with exponential services having class-dependent service rates. All classes of customers are served from a single source of inventory replenished according to $(s,S)$ policy with exponentially distributed lead time. Stability condition and steady state probabilities are obtained by matrix-analytic method. Some important performance measures are also derived. Inventory recycle time was analyzed in detail. Useful cost function and numerical illustrations are also given. The optimization problem is interesting and can be solved in similar real scenario.