Characterization theorem on losses in GIX/GIY/1/nGI^X/GI^Y/1/n queues

In this paper, we prove a characterization theorem on the number of losses during a busy period in $GI^X/GI^Y/1/n$ queueing systems, in which interarrival time distribution belongs to the class NWUE.Comment: to appear in: Operations Research Letter

Crossings states and sets of states in P\'olya random walks

We consider the P\'olya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up- and down-directed state-crossings and different sets of states crossings. One of the most important results of this paper is that the expected number of undirected state-crossings $\mathbf{n}$ is equal to 1 for any state $\mathbf{n}\in\mathbb{Z}^2\setminus\{\mathbf{0}\}$. As well, the results of the paper are extended to $d$-dimensional random walks, $d\geq2$, in bounded areas.Comment: Dear readers. I made a tremendous work to revise this paper after referee report. There are 30 pages of 11pt format, 4 figures and 1 tabl