On the invariance of stationary state probabilities of a non-product-form single-line queueing system (Q1812374)

The paper considers a single-line queueing system with losses and with \(N\) waiting places. The intensity of Poisson input flow depends on the number of demands in the system. The job size for a demand that enters the system when it contains \(k\) demands has an arbitrary distribution \(H_k(x)\) with finite expectation. The first demand from the queue is processed with rate \(\alpha_k\) provided that there are \(k\) demands in the queue. If, for a new demand, its job size is less than the remaining job size for the demand currently processed at the arrival moment, the new arrival displaces the demand at the server, which is put at the beginning of the queue, and its processing starts instantly. Otherwise, with probability \(1-a_k\), the new arrival also displaces the demand at the server, which is put at the beginning of the queue, and its processing starts instantly; with probability \(a_k\), the new arrival becomes the second in the queue (the first in the queue is the demand currently processed). For a piecewise linear Markov process which describes the system, the stationary distribution is found and necessary and sufficient conditions for this distribution to be invariant with respect to the job size distribution with a fixed mean are obtained.