Wireless three-hop networks with stealing II: exact solutions through boundary value problems

Abstract

We study the stationary distribution of a random walk in the quarter plane arising in the study of three-hop wireless networks with stealing. Our motivation is to find exact tail asymptotics (beyond logarithmic estimates) for the marginal distributions, which requires an exact solution for the bivariate generating function describing the stationary distribution. This exact solution is determined via the theory of boundary value problems. Although this is a classical approach, the present random walk exhibits some salient features. In fact, to determine the exact tail asymptotics, the random walk presents several unprecedented challenges related to conformal mappings and analytic continuation. We address these challenges by formulating a boundary value problem different from the one usually seen in the literature.