In this paper we deal with the effects of the electron absorption at the physical boundaries like walls, electrodes, diagnostic probes and divertors on the plasma and sheath parameters. The effects of the electron absorption manifest themselves via lack of reflected electrons in their velocity distribution in approaching to the boundary. The rate of this lack toward the boundary determines the electron density profile. It turns out that the ion velocity distribution in laboratory and fusion plasmas is strongly (and in some cases entirely) determined by the electron density profile. This is apparently striking but, nevertheless, well established (and physically clear) why and how a kinetic picture, i.e., velocity distribution emerges from a hydrodynamic quantity. Whereas, in mathematical sense, we always prefer to calculate hydrodynamic quantities from the kinetic ones, in the nature this approach works well only when the thermodynamic equilibrium at least for the electron velocity population is obeyed. Otherwise, statistical systems became strongly influenced by the sources of non-equilibrium which, under consideration here, is represented by an electron sink, i.e., by electron absorbing boundaries. Our general task is to find how to best correlate the hydrodynamic and kinetic parameters up to such surfaces which terminate the plasma locally (i.e., electrodes) or globally (i.e. divertors) with special attention to the boundary conditions which arise in the approximate two scale-approach, i.e., at the plasma-sheath boundary. Here in particular we present a complete set of boundary conditions and consequent solutions on a basic and very important, but surprisingly rarely tackled plasma-sheath model when the electron velocity distribution is cut-off maxwellian. Basically we follow the methodology first applied by Tonks and Langmuir [1] and further elaborated for various cases during the previous century (see e.g., [2] and references therein). While the sheath boundary conditions and solutions for the case of strong electron absorption were already discussed by Andrews and Varey [3] the plasma region is here treated for the first time. Consequences of the results obtained from the sheath and plasma sides for this particular case to the general bounded plasma theory and its further possible developments and applications to laboratory and fusion devices are discussed. [1] L. Tonks and I. Langmuir: "A general theory of the plasma of an arc" Phys Rev. 876, 34 (1929) [2] N. Jelic, M. Cercek, M. Stanojevic and T. Gyergyek: "An investigation of the collisinonless discharge in the presence of an electron beam", J. Phys. D, 2487, 27 (1994) [3] J.G.Andrews and R,H. Varey: "The sheath at an electrode close to plasma potential", J. Phys. A."Gen. Phys, 413, 5 (1970)