Joint stationary distribution in the two-channel queueing system with ordered entry, governed by one queue skipping policy

Consideration is given to the queueing system with the ordered entry, which is governed by one special queue skipping policy. The system consists of two single-server queues, say~Q1 and~Q2, each with infinite capacity. New customers arrive only in batches and only to~Q1. Upon arrival of a batch its size is compared with the current total number of customers in~Q1. 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 to~Q2 and the arrived batch enters the system; otherwise the new batch goes to~Q2. Whenever a~batch arrives to~Q2 the same comparison is performed. The batch pushed-out from~Q2 is considered as lost. Under the assumption that the service times are exponential and the batch inter-arrival times are i.i.d. we sketch the procedure for the computation of the joint stationary distribution of the queues' content. Obtaining stability criteria as well as closed-form expressions remain the open issue.