We study the global behavior of positive solutions of the system of rational difference equations x n + 1 = f ( y n − q , x n − s ) , y n + 1 = g ( x n − t , y n − p ) , n = 0 , 1 , 2 , … , where p , q , s , t ∈ { 0 , 1 , 2 , … } with s ≥ t and p ≥ q , the initial values x − s , x − s + 1 , … , x 0 , y − p , y − p + 1 , … y 0 ∈ ( 0 , + ∞ ) . We give sufficient conditions under which every positive solution of this system converges to the unique positive equilibrium.