ABSTRACT
The piecewise deterministic processes are jump type stochastic processes whose trajectories between the jumps are deterministic. They may arise when approximating the solution of a dynamical system via time discretization. They also arise naturally when the constant evolution model between jumps of a pure jump process is replaced by some deterministic flow of a dynamical system. Application areas of such processes include ruin theory, communication networks, queuing theory, biochemistry and bacterial population growth. This chapter describes the evolution semigroups and the infinitesimal generators, as well as the Doeblin-Ito formulae for these processes. It provides a thorough discussion of applications in the transmission control protocol (TCP) used to control the information transmission in internet applications. The transmission control protocol is a sophisticated reliability protocol used by most Internet applications to control the transmission of information. More than 50% of Internet traffic uses TCP.
