Distribution of the Non-Stationary Number of Free Places of the Queuing System \(\boldsymbol{GJ|M|1|N}\)

In this paper, the Laplace transform of the generating function of the distribution of a nonstationary queue length system with a limited queue \(GJ|M|1|N\) is found. Based on this, one duality relation is established between the distributions of non-stationary queue lengths of the dual systems \(M|G|1|N+1\) and \(GJ|M|1|N\) . The Laplace–Stieltjes transform of the distribution function of the time interval until the moment of the first failure in the system \(GJ|M|1|N\) is found. A formula is derived that determines the Laplace transform of the distribution of the nonstationary queue length of this system.