
Markov Model
Markov models are some of the most powerful tools available
to engineers and scientists for analyzing complex systems.
This analysis yields results for both the time dependent
evolution of the system and the steady state of the system.
For example, in Reliability Engineering, the operation
of the system may be represented by a state diagram, which
represents the states and rates of a dynamic system. This
diagram consists of nodes (representing a possible state
of the system, which is determined by the states of the
individual components & subcomponents) connected by
arrows (representing the rate at which the system operation
transitions from one state to the other state). Transitions
may be determined by a variety of possible events, for example
the failure or repair of an individual component. A statetostate
transition is characterized by a probability distribution.
Under reasonable assumptions, the system operation may be
analyzed using a Markov model.
A Markov model analysis can yield a variety of useful performance
measures describing the operation of the system. These performance
measures include the following:
* system reliability
* availability
* mean time to failure (MTTF)
* mean time between failures (MTBF)
* the probability of being in a given state at a given time
* the probability of repairing the system within a given
time period (maintainability)
* the average number of visits to a given state within a
given time periodand many other measures.
The name Markov model is derived from one of the assumptions
which allows this system to be analyzed; namely the Markov
property. The Markov property states: given the current
state of the system, the future evolution of the system
is independent of its history. The Markov property is assured
if the transition probabilities are given by exponential
distributions with constant failure or repair rates. In
this case, we have a stationary, or time homogeneous, Markov
process. This model is useful for describing electronic
systems with repairable components, which either function
or fail. As an example, this Markov model could describe
a computer system with components consisting of CPUs, RAM,
network card and hard disk controllers and hard disks.
The assumptions on the Markov model may be relaxed, and
the model may be adapted, in order to analyze more complicated
systems. Markov models are applicable to systems with common
cause failures, such as an electrical lightning storm shock
to a computer system. Markov models can handle degradation,
as may be the case with a mechanical system. For example,
the mechanical wear of an aging automobile leads to a nonstationary,
or nonhomogeneous, Markov process, with the transition
rates being time dependent. Markov models can also address
imperfect fault coverage, complex repair policies, multioperationalstate
components, induced failures, dependent failures, and other
sequence dependent events
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