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7. Markov Models
Reliable System Design 2011by: Amir M. Rahmani
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Markov Models
The primary difficulty with the combinatorial models is that many complex systems cannot be modeled easily in a combinatorial fashion.
The fault coverage is sometimes difficult to incorporate into the reliability expression in a combinatorial model.
The process of repair is very difficult to model in a combinatorial model.
Alternative: Markov models
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Markov Process
In 1907 A.A. Markov published a paper in which he defined and investigated the properties of what are now known as Markov processes.
A Markov process with a discrete state space is referred to as a Markov Chain.
A set of random variables forms a Markov chain if the probability that the next state is Sn+1 depends only on the current state Sn, and not on any previous states
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Markov Process
A stochastic process is a function whose values are random variables
The classification of a random process depends on different quantities
• – state space• – index (time) parameter• – statistical dependencies among the random
variables X(t) for different values of the index parameter t.
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Markov Process
Categories of Markov state-space models:• 1. Discrete space and discrete time• 2. Discrete space and continuous time• 3. Continuous space and discrete time• 4. Continuous space and continuous time
The first two categories involve a discrete space; that is, the states of the system can be numbered with an integer.
In the first and the third categories, the system changes by discrete time steps.
The second category is the one most useful for modeling fault-tolerant systems.
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Markov Process
States must be• – mutually exclusive• – collectively exhaustive
Let Pi(t)= Probability of outgoing in the state Si at time t.
Markov Properties• – future state probability depends only on current state
• independent of time in state• path to state
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State Transition Diagrams
A Markov state transition diagram can graphically represent all:
• 1- System states and their initial conditions. • 2- Transitions between system states and corresponding
transition rates
The transition rates are replaced with equivalent transition probabilities considering that the state transition time is very small (Δt ) this leads to
• 1- A situation where the system can remain in the current state after time t with some probability.
• 2- Thus, in the above case, a situation where the system can go to the next state(s) (transition rates) after time t with some probability.
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Construction of State Transition Diagram
The basic steps in constructing state transition diagrams are:
• 1- Define the failure criteria of the system. • 2- Enumerate all of the possible states of the
system and classify them into good or failed states.
• 3- Determine the transition rates between various states and draw the state transition diagram
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Example
State diagram for one component Let X denote the lifetime for a component. The Markov property is defined as follows:
The probability that a component fails in the small interval Λt is proportional to the length of the interval.
λ is the proportional constant. The probability above does not depend on the
time t.
0
)/(
t
ttXttXP
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Markov Process Assume exponential failure law with failure rate λ. Probability that system failed at t+Δt, given that is was
working at time t is given by
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Reliability for one component
The probability that the component works at the time t+ Δt is
We divide with Δt
Let Δt →0 , and we get
)()1()( 11 tPtttP
)())()(
111 tP
t
t
t
tPttP
)()( 11 tPtP
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Reliability for one component
The solution to this differential equation is
Assuming that the component works at the time t = 0, so
The reliability of the component is:
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Failure probability for one component
The probability that the component does not work at the time t+ Δt is
We divide with Δt
Let Δt →0 , and we get
)()()( 010 tPttPttP
)()()(
100 tP
t
tPttP
)()( 10 tPtP
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Failure probability for one component
Solving the differential equation yields
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Markov chain modelThe equation system can be written using matrices
where
and
Q is called the transition rate matrix.
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Cold stand-by system with one spare
State diagram
State labeling• 2 Primary module works• 1 Spare module works (Primary module does not work)• 0 No module works, system failure
Assumption: The failure rate for the spare is zero.
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Cold stand-by system with one spare
We calculate the reliability of the system by solving the equation system
Where
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The Equation System
We solve this by Laplace transform using the following relation
Laplace transforms: Time function Laplace transform
2
2
)(
1
1
1
11
ste
se
st
s
t
t
)0()()(~
PsPstP
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Solving the Equation System
The Laplace transform get
where
which give us
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Solving the Equation System
1- We compute
which gives the following time function
2- We compute
The reliability of the system can be written as:
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Calculating MTTF
Let X1 and X2 denote the time spent in state 2 and state 1, respectively. MTTF for the system can then be written as
Alternatively, the MTTF can be computed as
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Reliability
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Coverage
Designing a fault-tolerant system that will correctly detect, mask or recover from every conceivable fault, or error, is not possible in practice.
Even if a system can be designed to tolerate a very large number of faults, or errors, there are for most systems a non-zero probability that a single fault will be remained. such faults are known as “non-covered” faults.
The probability that a fault is covered (i.e., correctly handled by the fault-tolerance mechanisms) is known as the coverage factor, and denoted c.
The probability that a fault is non-covered can then be written as 1 - c.
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Cold Stand-by system with Coverage factor
State diagram
We can write-up the Q-matrix directly by inspecting the state diagram.
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Solving the Equation System
We have the following equation system
After applying the Laplace transform, we get
We then compute
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Solving the Equation System
can we compute directly from the first equation
We then compute
Reliability for the system is
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The Reliability with Coverage factor
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Calculating MTTF
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Availability
Definition: the probability that a system is functioning properly at a given time t.
When calculating the availability we consider both failures and repairs. We must make assumptions about the function time (up time) and the repair time (down time).
The repair time consists of the time it takes to perform the repair, the time between the system failure and the repair is started, and the time it takes to restart the system after the repair is completed.
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Steady-state Availability
E [X0] = MTTFF (Mean Time To First Failure)
E [Xi] = MTTF (Mean Time To Failure)
E [Yi] = MTTR (Mean Time To Repair)
MTTR + MTTF = MTBF (Mean Time Between Failures)
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Design Tradeoffs
How to make availability approach 100%?
MTTF → infinity (high reliability) MTTR → zero (fast recovery)
MTTRMTTF
MTTF tyAvailabili
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Availability vs. Reliability
– Reliability is measured by mean time To failure (MTTF)
- There is no repair in the state of system failure for modeling reliability.
– Availability is a function of MTTF and mean time to repair (MTTR) MTTF/(MTTF+MTTR)
– A system may have a high MTBF, but low availability
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Markov chain model for a simplex system
State0: System OK Failure rate: λ1: System failure Repair rate: μ
Availability: A(t) = P0 (t)Reliability: R(t) = e-λt
Maintainability: M(t) = 1 – e-μt
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The availability for a simplex system
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The availability for a simplex system
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Steady-state Availability
Assuming exponentially distributed function times and repair times, we get
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Markov chain for a hot stand-by system
State0,1: System OK Failure rate: λ2: System failure Repair rate: μ
Availability: A(t) = P0 (t) + P1 (t)
Assumption: Only one repair-person works with the system when a failure has occurred.
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Safety Definition: The probability that a system is either
functioning properly, or is in safe failed state.
Calculating safety is similar to calculating reliability.
In a reliability model there is usually only one absorbing state, while in a safety model there are at least two absorbing states.
Among the absorbing states in a safety model, at least one represents that system is in a safe shut-down state, and at least one represents that a catastrophic failure has occurred.
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Safety for a simplex system with coverage factorWe obtain the following markov chain model
and the corresponding transition-rate matrix
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Safety for a simplex system with coverage factor
The solutions of the differential equations are:
The safety of the system is:
The steady-state safety is:
