Section: New Results

Properties of random walks in orthants

Participant : Guy Fayolle.

We pursued works initiated these last years in several directions.

Explicit criterion for the finiteness of the group in the quarter plane

In the book [3] , original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps, the general solution being obtained via reduction to boundary value problems. Among other things, an important quantity, the so-called group of the walk, allows to deduce theoretical features about the nature of the solutions. In particular, when the order of the group is finite, necessary and sufficient conditions have been given in [3] for the solution to be rational or algebraic. When the underlying algebraic curve is of genus 1, we propose, in collaboration with R. Iasnogorodski (St-Petersbourg, Russia), a concrete criterion ensuring the finiteness of the group. It turns out that this criterion is always tantamount to the cancellation of a single constant, which can be expressed as the determinant of a matrix of order 3 or 4, and depends in a polynomial way on the coefficients of the walk [55] .

About a possible analytic approach for walks with arbitrary big jumps in ${\mathbb{Z}}_{+}^2$

The article [21] , achieved in collaboration with K. Raschel (CNRS and University F. Rabelais, Tours) considers random walks with arbitrary big jumps. For that class of models, we announce a possible extension of the analytic approach proposed in [3] , initially valid for walks with small steps in the quarter plane. New technical challenges arise, most of them being tackled in the framework of generalized boundary value problems on compact Riemann surfaces.

Correction of papers

Guy Fayolle found important errors in several articles dealing with models involving random walks in ${\mathbb{Z}}_{+}^2$. This is the object of the letter to the editors [19] . The concerned authors have provided new correct versions of their studies.

Communication networks with harvesting energy supply

In collaboration with S. Foss (Heriot-Watt University, Edinburgh), we started to analyze stability and performance of a number of models of parallel queues with multiple access and individual energy supplies. Energy limitation in general decreases the stability region, but also may increase it for specific parameter regions. The most difficult and intriguing cases arise when the input rates of requests and of energy items are close. Preliminary models of physical interest involve random walks in ${\mathbb{Z}}_{+}^4$.