Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.Comment: 42 pages, preprint, CNRS UMR 808

A load-balanced network with two servers

A load-balanced network with two queues Q 1 and Q 2 is considered. Each queue receives a Poisson stream of customers at rate i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p i *, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system

A load-balanced network with two servers

A load-balanced network with two queues Q 1 and Q 2 is considered. Each queue receives a Poisson stream of customers at rate i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p i *, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system