We consider the initial-boundary value problem for the linear one-dimensional Jin-Xin relaxation model in a quarter plane. Our main interest is to understand the boundary layer behavior of the solution and its asymptotic convergence to the corresponding equilibrium system of hyperbolic conservation laws. We identify and rigorously justify a necessary and sufficient condition (which we refer to as Stiff Kreiss Condition) on the boundary condition to guarantee the uniform well-posedness of the initial-boundary value problem for the relaxation system independent of the relaxation parameter. The Stiff Kreiss Condition is derived by using a normal mode analysis. The asymptotic convergence and boundary layer behavior are studied by the Laplace transform and a matched asymptotic analysis. An optimal rate of convergence is obtained.