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

We consider a two-dimensional skip-free reflecting random walk on the non-negative integers, which is referred to as a 2-d reflecting random walk. We give necessary and sufficient conditions for the stationary distribution to have a product-form. We also derive simpler sufficient conditions for the product-form for a restricted class of 2-d reflecting random walks. We apply these results and obtain a product-form approximation of the stationary distribution through a suitable modification of the parameters of the random walk.