$Geo/G/1/\infty$-queue with one “nonstandard” discipline of service

The object of consideration is s queueing system $Geo/G/1/\infty$ with the service discipline according to which upon the arrival of a new customer, its length is compared to the length of (remaining) length of the customer on the server. The customer with the minimum length occupies the server whereas the other becomes the first in the queue thus shifting the remaining queue for one place. For this system, main nonstationary characteristics are found. In particular, it is demonstrated that, unlike the continuous time case, for the discrete time case, the stationary distribution of the number of customers in the system is not invariant with respect to the loading.