|Abstract or Summary
- Occasions arise in engineering for conducting probabilistic analyses concerned
with the "state" character of systems which range in scale from the circuit level (and even
below) through the black box level and on up through the system level.
These stochastic systems are systems with random as well as deterministic features.
Such a system often has the following character: it has a finite number of states (or
modes) and at any time is in one of these states; it is subject to external events "input"
events which tend to induce changes of state; it is the source of events, the occurrence
of these "output" events being dependent on the current state and the current input
A necessary background in finite-state, discrete-time Markov chains is first presented.
Then the stochastic chain is formulated and, as an illustration, applied to a rather
complex though not untypical problem, that of evaluating the worth of two alternative
aircraft navigation systems.
Following this practically based example, an entropy analysis is worked out. The
information theory notion of entropy is used to measure the randomness of the sets of
events constituting inputs and outputs of stochastic chains. Also, methods of matrix
analysis are developed; these methods provide tools for framing solutions to the canonical
equations of discrete-time variable and continuous-time variable stochastic chains.
Solutions are developed not only for the general case, but also for important special
cases. Digital computer solution methods are outlined too, and are illustrated by
The formulation of continuous-variable stochastic chains, and the derivation of
solution forms and outlining of computer solution techniques completes the treatment of
single stochastic chains. The rest of the thesis extends the model: networks of stochastic
chains are formulated and applied.
Such networks can be applied to stochastic analysis problems of potentially arbitrary
complexity; indeed, without such networks, stochastic chain analysis would be limited
to systems possessing only a relatively few states (or modes) of operation. Finally,
three particular application areas are explored: reliability of electronic equipments
(briefly); and simulation analysis and probabilistic logics (both of these latter in considerable
detail). In each area, stochastic chains used in the network context can provide
potent means for analyzing complex and, in many cases, previously intractable problems.