  In this paper is described the general aspect of a numerical method for
piecewise determin-istic Markov processes with boundary. Under very natural
hypotheses, a crucial result about uniqueness of solution of a generalized
Kolmogorov equation with respect to a test function space is proved. Next we
prove the existence and uniqueness of a positive solution to the finite volume
scheme without result about convergence. Finally different models of
transmission control protocol window-size processes are simulated to illustrate
the efficiency of the numerical method for describing the evolution of the
density of a piecewise deterministic Markov process with boundary. Obviously
some technical aspects have been skipped for reader convenience but the full
theory will be exposed in a forthcoming paper in collaboration with C.
