This thesis considers asymptotic behavior of a skip free random walk in the two dimensional positive quadrant of the lattice with homogeneous reflecting transitions at each boundary face, using the matrix analytic methods and following Miyazawa (2011). Take one of the coordinates of proce as the level and the other coordinate as the phase, a background state. Then, this random walk can be considered as a continuous-time quasi -birth and- death process, a QBD process in short, with infinitely many phases through uniformization due to the homogeneous transition structure. Since the chain is skip free in both dimensions, this reflected random walk is called a double QBD process, DQBD in short. This process is rather simple, but has flexibility to accommodate a wide range of queueing models, including two node networks. It is also amenable to analysis by matrix analytic methods. In general for two-dimensional models with both infinitely many levels and phases, the computation of the exact stationary probability distributions is usually very difficult. Not only because of this but also for its own importance, researchers have studied the tail asymptotics of its stationary distribution