Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 261

Appendix A
Stochastic process
fundamentals

In this appendix we briefly summarize the basic results and definitions con-
cerning the classes of stochastic processes produced by the execution of
GSPNs.
The stochastic processes on which we mostly focus our attention are Marko-
vian processes with finite state spaces (finite Markov chains), and their gen-
eralizations (Markov reward processes, and semi-Markov processes).
The goal of this appendix is not to provide a complete, rigorous, and formal
treatment of the whole subject, which well deserves a topic by itself, rather
to introduce the notation used in the topic, and to develop in the reader some
intuition about the structure and the dynamics of the stochastic processes
of interest.
Readers who desire a more formal and complete treatment of the theory
of stochastic processes are referred to the vast literature that exists on this
subject.

A.1 Basic Definitions

Stochastic processes are mathematical models useful for the description of
random phenomena as functions of a parameter which usually has the mean-
ing of time.
From a mathematical point of view, a stochastic process is a family of ran-
dom variables { X(t),t ∈ T } defined over the same probability space, indexed
by the parameter t, and taking values in the set S. The values assumed by
the stochastic process are called states, so that the set S is called the state
space of the process.
Alternatively, we can view the stochastic process as a set of time functions,
one for each element of the probability space. These time functions are called
either sample functions, or sample paths, or realizations of the stochastic
process.

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