Product-form characterization for a two-dimensional reflecting random walk

Structure-Reversibility of a Two-Dimensional Reflecting Random Walk and Its Application to Queueing Network

“…It is notable that our stationary distribution is closely related to the one which was recently obtained by Latouche and Miyazawa [6]. They derived it by characterizing a class of the reflecting random walks whose stationary distribution has product form.…”

“…Then, we can see that (3.13), (3.14), (3.17) and (3.24)-(3.29) of [6] hold with η 1 = 0.3 and η 2 = 0.2, and therefore, the stationary distribution of this random walk has a product form. For this random…”

Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

“…Two-dimensional semimartingale reflecting Brownian motion (SRBM) in the quarter plane received a lot of attention from the mathematical community. Problems such as SRBM existence [39,40], stationary distribution conditions [19,22], explicit forms of stationary distribution in special cases [7,8,19,23,30], large deviations [1,7,33,34] construction of Lyapunov functions [10], and queueing networks approximations [19,21,31,32,43] have been intensively studied in the literature. References cited above are non-exhaustive, see also [42] for a survey of some of these topics.…”

Modeling the impact of interference on wireless ad hoc network performance

“…This extends results of Bayer and Boucherie [BB02] who consider very specific boundary behavior. A product-form characterization for random walks in the quarter-plane is given by Latouche and Miyazawa in [LM14]. We demonstrate that a coupled queue with forwarding has a geometric product-form distribution if and only if µ 1 = µ * 1 and µ 2 = µ * 2 .…”

“…Also, in the G network we have a 1,1 = b 1,1 = c 1,1 = 0. In [LM14] necessary and sufficient conditions are provided for a more general network in which a 1,1 = 0, b 1,1 = 0 or c 1,1 = 0. In our first results below we derive such conditions from first principles, because it enables us to obtain these conditions in the form that is most suitable for follow-up results in this section.…”