The equilibrium distribution for a class of multi-dimensional random walks

In previous papers, it has been proved that the equilibrium distribution of homogeneous, nearest-neighboring random walks on a two-dimensional grid can be constructed explicitly through a compensation procedure if and only if there are no transitions to the North, North-East and East for points in the interior. In the present paper the extension to N-dimensional random walks is investigated. It appears that for higher dimensions the same condition should be satisfied for each plane in the grid space. Since induction with respect to the dimension is applied. the step from dimension 2 to dimension 3 is worked out in detail. For the proof of the if-part the condition is added that the random walk satisfies the so-called projection property on the boundaries. For 3-dimensional random walks, the eqUilibrium distribution appears to be the sum of six alternating series of binary trees of product forms. These analytic results make it possible to develop efficient numerical procedures. Such procedures are sketched in the paper. As a numerical illustration, the procedures are applied to the model of a 2 x 3 switch.