A waveguide occupies a d+1-dimensional domain with several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem with spectral parameter μ, selfadjoint with respect to the Green formula. At infinity, the coefficients of the problem stabilize at an exponential rate to functions independent of the axial variable in the corresponding cylinder. On every interval of the continuous spectrum between neighboring "thresholds", a unitary scattering matrix S(μ) is defined; the size of S(μ) is finite for any μ, remains to be constant on any such interval, and varies from an interval to an interval. The basic result claims the existence of finite one-sided limits of the scattering matrix S(μ) at every threshold.