Two-Priority Queueing System with LCFS service, Probabilistic Priority and Batch Arrivals

Consideration is given to the single server queueing system with two independent flows of customers: a batch Poisson flow and a flow of saturating customers. The size of a batch and the service times of the customers in a batch are governed by a joint probability distribution. Whenever the server becomes free and there are no customers in the queue, a customer from the saturating flow occupies the server. The service times of saturating customers are general i.i.d. Customers from the Poisson flow have relative priority over the saturating customers. The service policy is Last-Come-First-Served with probabilistic priority. It is being discussed that the joint probability distribution of the total number of customers in the system and the remaining service time of the customer in service can be obtained in a recursive way. For the special case, when the size of the arriving batch and the service times in the batch are sampled independently, explicit expressions are provided.