In this paper, we consider a generalized two-demand queueing model, the same model studied in Wright (Adv. Appl. Prob., 24, 986–1007, 1992). Using this model, we show how the kernel method can be applied to a two-dimensional queueing system for exact tail asymptotics in the stationary joint distribution and also in the two marginal distributions. We demonstrate in detail how to locate the dominant singularity and how to determine the detailed behavior of the unknown generating function around the dominant singularity for a bivariate kernel, which is much more challenging than the analysis for a one-dimensional kernel. This information is the key for characterizing exact tail asymptotics in terms of asymptotic analysis theory. This approach does not require a determination or presentation of the unknown generating function(s).