Existence of stationary queue-size distributions in the systems that work only on the biggest batches of customers

Consideration is given to the infinite capacity FIFO queueing systems with a single server, that work only on the biggest batches of customers. New customers arrive only in batches. Upon arrival of a batch its size is compared with the current total number of customers the system. If the size of the batch is larger than that number, all customers residing in the system (including the one in server) are pushed-out and the arrived batch enters the system; otherwise the new batch is considered as lost. Conditions of the existence of the stationary queue-size distributions of such systems are of interest. It is shown that in the classical cases they follow the intuition. Yet the preliminary analysis shows, that finiteness of the mean batch size may not be necessary for the stability of such queues.