5.7.2 __Availability Modeling
(Markov Process Approach)__

A Markov process (Ref. [2]) is a mathematical
model that is useful in the study of the availability of complex systems. The
basic concepts of the Markov process are
those of “state” of the system (e.g., operating, nonoperating) and state
“transition” (from operating to nonoperating due to failure, or from
nonoperating to operating due to repair).

A graphic example of a Markov
process is presented by a frog in a lily pond. As time goes by, the frog jumps
from one lily pad to another according to his whim of the moment. The state of
the system is the number of the pad currently occupied by the frog; the state
transition is, of course, his leap.

Any Markov process is defined by a
set of probabilities pij which define the probability of transition from any
state i to any state j. One of the most important features of any Markov model
is that the transition probability p_{ij} depends only on states i and j and is completely
independent of all past states except the last one, state i; also
p_{ij} does not change with
time.

In system availability modeling utilizing the Markov process
approach, the following additional assumptions are made:

(1) |
The conditional probability of a failure
occurring in time (t, t + dt) is l dt. |

(2) |
The conditional probability of a repair
occurring in time (t, t + dt) is m dt. |

(3) |
The probability of two or more failures
or repairs occurring simultaneously is zero. |

(4) |
Each failure or repair occurrence is
independent of all other occurrences. |

(5) |
l (failure rate) and m
(repair rate) are constant (e.g.,
exponentially distributed). |

Let us now apply the Markov
process approach to the availability analysis of a single unit with failure
rate l and repair
rate m.