On the distribution of the number of calls in the \(D_\eta/D_{\xi}^{k}/1\) queueing system (Q2722139)

Let \(\eta,\xi\), \(k\in N_{+}=\{1,2,\dots \},\) be independent r.v.s with finite mean values. Let us consider a homogeneous Markov chain \(\{Y_{n}; n\geq 0\}\) with phase space of states \(N\cup N^{3},\) \(N=\{0, 1, \dots \},\) and some special transition probabilities for one step which depend on \(\eta\) and \(\xi.\) Such a random sequence \(\{Y_{n}; n\geq 0\}\) describes an evolution of single-channel queueing system with the following properties: 1) claims on serving device arrive by one, over i.i.d. with \(\eta\) time interval; 2) claims are served by groups; 3) queue length is infinite. This queueing system is called \(D_{\eta}/D_{\xi}^{k}/1\)-queueing system. The distribution of the number of claims which contain this queueing system in transient and stationary regimes is found. Factorization methods are used to solve this problem. It is known that the idea to use factorization for the solution of boundary problems for a wide class of random processes belongs to V. S. Korolyuk.