Some queueing problems (symmetric shortest queue one, for instance) may be treated as random walks on multidimensional grids. The main purpose of the paper is to explore the basic conditions for two-dimensional random walks to have stationary distribution as infinite series of some product form. It is considered continuous time Markov process on the pairs of nonnegative integers, and it is assumed that transitions can only take place to adjacent points and rates for transitions do not depend on the starting points (both in inner area and on boundaries). It is assumed that stationary probabilities exist and satisfy usual balance linear equations.
The essence of compensation approach is to choose the product form coefficients in such way that the “initial” solution for inner area turns to be one for boundaries also. This compensation approach generates an infinite, in general, sequence, so-called product form compensation terms to remove the errors of “initial” solution on horizontal and vertical boundaries. The result compensation is expressed as infinite sum of compensation terms (here the linear structure of the equations is used). The recursion relations for compensation terms are found and the problem of absolutely convergence of compensation series is solved (in the terms of solutions of equation for inner area). The convergence criterion is obtained as the restriction on the set of “initial” solutions (feasible ones) and the mean boundary drifts. Eventually there are obtained the conditions which guarantee the product form solution for probabilities having sufficiently large sum of (both) coordinates. It is mentioned that compensation method is a numerically oriented one and leads to efficient numerical procedures.