On the Invariance of Stationary State Probabilities of a Non-Product-Form Single-Line Queueing System

We consider a single-line queueing system (QS) with Poisson input flow of varying intensity, which depends on the number of demands in the system. The job size (length) distribution for a demand depends on the number of demands in the system at the arrival moment. The service rate also depends on the number of calls in the QS. If the job size for a new arrival is larger than the remaining job size for the currently processed demand, then the arrival is put at the beginning of the queue with a certain probability, which depends on the total number of demands in the system. Otherwise, it occupies the server and displaces the currently processed demand, which is put at the beginning of the queue. The probability distribution of stationary states of the QS 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.