6.4.4.2.4.1 Markov Modeling (Ref. [9])
Markov modeling processes are stochastic processes using random
variables to describe the states of the process, transition probabilities for
changes of state and time or event parameters for measuring the process. A
stochastic process is said to be a Markov property if the conditional
probability of any future event, given any past events and the present state,
is independent of the past events and depends only on the present state of the
process (Ref.
[10]).
The advantages for using Markov modeling methods include the
flexibility in expressing dynamic system behavior. These types of behavior
include:
(1) |
Complex repair. Situations consisting of repairs
of either individual components or groups of components or partial
repair of components. |
(2) |
Standby spares. Standby conditions
include hot, warm and cold spares. Hot spares are power-on units with
identical stresses as apply to the active units, where warm spares
have power-on but have lower stresses. Cold spares are power-off
units. |
(3) |
Sequence dependent. This behavior includes:
functional dependency in which the failure of one component can cause
the unavailability of other components; priority dependency in which
behavior will differ depending on whether an event occurs before or
after another; and sequence enforcement in which it is impossible for
certain events to occur before others have occurred. |
(4) |
Imperfect fault coverage.
Imperfect fault coverage conditions arise when a dynamic
reconfiguration process that is invoked in response to a fault or
component failure has a chance of not being successful leading to
system failure. |
The disadvantages of using Markov
modeling techniques include state size and model construction. Solving models
with thousands of states can challenge the computer resources available. Also,
the problem of identifying all the states and transitions correctly can be a
difficult assignment.
Software tools for performing
dependability analysis, such as Markov modeling include (see the RAC Web Site;
the URL is http://rac.iitri.org/DATA/RMST):
(1) |
HARP, Hybrid-Automated Reliability Predictor was
developed to input system conditions directly in the form of a Markov
model or in the form of a dynamic fault tree. |
(2) |
SHARPE, Symbolic Hierarchical Automated
Reliability and Performance Evaluation, is an integrated tool that
allows models to be solved either individually or combined
hierarchically. In addition to Markov models, SHARPE can solve
reliability block diagrams, fault trees and generalized stochastic
Petri nets. |
(3) |
CARMS, Computer-Aided Rate
Modeling and Simulation, is an interactive Markov modeling tool
designed for reliability analysis of redundant systems. |
(4) |
CARSA, Computer-Aided Redundant
System Reliability Analysis, utilizes Markov modeling for failure
effect coverage. CARSA by-passes disadvantages of Markov modeling
(larger number of states) by partitioning the system so that the model
is a lower
dimension. |