On the asymmetric clocked buffered switch

A $2 times 2$ clocked buffered switch is a device used in data-processing networks for routing messages from one node to another. The message handling process of this switch can be modelled as a two-server, time slotted, queueing process with state space the number of messages $( {bf x_n , {bf y_n )$ present at the servers at the end of a time slot. The $({bf x_n , {bf y_n )$-process is a two-dimensional nearest-neighbour random walk. In the present study the bivariate generating function $Phi (p , q)$ of the stationary distribution of this random walk is determined, assuming that this distribution exists. $Phi (p, q)$ is known, whenever $Phi (p,0)$ and $Phi ( 0 , q)$ are known. The essential points of the present study are the construction of these two functions from the knowledge of their poles and zeros and the simple determination of these poles and zeros.