به طور کلی در فرآیندهای مارکوف ارگودیک دو بعدی یافتن فرم بسته توزیع ایستا، تنها برای حالات خیلی خاص امکان پذیر است. با توجه به این مشکل و نیز با توجه به اهمیت توزیع ایستا، بررسی و مطالعه رفتار مجانبی توزیع ایستای این فرآیندها مورد توجه قرار گرفته است. زنجیر قدم زدن تصادفی دو بعدی که در برخی متون به آن، فرآیند qbd دو طرفه نیز می گویند، یکی از این فرآیندها است. یک فرآیند qbd زمان گسسته یک زنجیر مارکوف دو بعدی است. اولین متغیر را طبقه و دومین متغیر را فاز می نامند و ماتریس احتمال انتقال آن دارای ساختار سه قطری بلوکی است (انتقال در جهت طبقه پرش آزاد است). در صورتی که متغیر فاز نیز دارای خاصیت پرش آزاد باشد، زنجیر یک فرآیند qbd دو طرفه است. هدف ما در این پایان نامه بررسی روش های به دست آوردن مجانب های دقیق و نرخ های نزول سخت توزیع ایستای این زنجیر در جهت های مختصات همچنین توزیع های کناری آن بر اساس مقاله میازاوا (2011) است. برای این منظور با استفاده از روش های تحلیل ماتریسی، نمایش هندسی-ماتریسی توزیع ایستای زنجیر نشان داده شده و با استفاده از توابع مولد گشتاور و احتمالات انتقال یک مرحله ای نرخ های نزول به دست آمده است. در نهایت از این نتایج برای به دست آوردن نرخ های نزول شبکه جکسون دو گره ای و اصلاح یافته آن به عنوان مثال های مورد توجه در نظریه صف استفاده می شود. 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 process 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. in general, a function f(n) for n ? 0 is said to have exact asymptotics g(n) if there is a positive constant c such that lim?(n ? ?)??(f(n))/(g(n))?=c and to have a rough decay rate ? if lim?(n ? ?)??1/n? log?(f(n) )= ? and r = e^?, is referred to as a rough geometric decay rate. decay rate problems have been studied for a much more general two dimensional reflected random walk, so called, n-partially homogeneous chain by borokov (2001), where the n is a positive real number and specifies the depth of the boundary faces. both the rough and exact asymptotics have been considered in all directions for the 0-partially homogeneous chain, which includes the double qbd process as a special case. however, the decay rate is not made explicit even for this special case, in particular, for the exact asymptotics, and there is no answer for the marginal distributions. in this thesis we consider this problem using the matrix analytic methods. we are only concerned with the double qbd process, i.e., a skip free 0-partially homogeneous chain and consider asymptotic behavior of its stationary distribution in the directions of coordinates. this allows us to use the nice geometric structure of the stationary distribution, which is called a matrix geometric form in the matrix analytic literature. thus, we directly will be considered itself. we will explain how to find full spectra of a rate matrix for the matrix geometric form, and characterize the rough decay rates as solutions of an optimization problem. we then get the rough and exact asymptotics with help of graphical interpretation of the spectra, which can be done using only primitive modeling data, i.e., one step transition probabilities. this approach not only sharpens existing results under weaker assumptions for the decay rate problem on the double qbd process, but also enables us to find the rough and exact asymptotics of the marginal distributions. in the last chapter, beside a simple example the decay rates for jackson networks and their modifications will be investigated.