Post on 17-Dec-2015
Chapter 8
Continuous Time Markov Chains
Markov Availability Model
2-State Markov Availability Model
1) Steady-state balance equations for each state:– Rate of flow IN = rate of flow OUT
• State1:• State0:
2 unknowns, 2 equations, but there is only one independent equation.
UP1
DN0
MTTR
MTTF
1
1
10 01
2-State Markov Availability Model(Continued)
1) Need an additional equation: 110
Downtime in minutes per year = * 8760*60
1
11 111
min356.510199999.0 5 DTMYAA ssss
MTTRMTTF
MTTR
MTTRMTTF
MTTFAss
11
1
1
11
MTTRMTTF
MTTRAss
1
2-State Markov Availability Model(Continued)
2) Transient Availability for each state:– Rate of buildup = rate of flow IN - rate of flow OUT
This equation can be solved to obtain assuming P1(0)=1
)()( 101 tPtP
dt
dP
havewetPtP 1)()(since 10 )())(1( 111 tPtP
dt
dP
)()( 11 tP
dt
dP
tetAtP )(1 )()(
2-State Markov Availability Model(Continued)
3)
4) Steady State Availability:
tetR )(
ss
tAtA )(lim
• Assume we have a two-component parallel
redundant system with repair rate .
• Assume that the failure rate of both the components
is .
• When both the components have failed, the system
is considered to have failed.
Markov availability model
Markov availability model (Continued)
• Let the number of properly functioning components be the state
of the system. The state space is {0,1,2} where 0 is the system
down state.
• We wish to examine effects of shared vs. non-shared repair.
2 1 0
2
2
2 1 0
2
Non-shared (independent) repair
Shared repair
Markov availability model (Continued)
• Note: Non-shared case can be modeled & solved using a RBD
or a FTREE but shared case needs the use of Markov chains.
Markov availability model (Continued)
Steady-state balance equations
• For any state:Rate of flow in = Rate of flow outConsider the shared case
i: steady state probability that system is in state i
122 021 2)(
01
Steady-state balance equations (Continued)
• Hence
Since
We have
or
12 2
1210
01
12 000
2
20
21
1
Steady-state balance equations (Continued)
• Steady-state unavailability = 0= 1 - Ashared
Similarly for non-shared case,
steady-state unavailability = 1 - Anon-shared
• Downtime in minutes per year = (1 - A)* 8760*60
2
221
11
sharednonA
Steady-state balance equations
Homework 5:
• Return to the 2 control and 3 voice channels example and
assume that the control channel failure rate is c, voice channel
failure rate is v.
• Repair rates are c and v, respectively. Assuming a single
shared repair facility and control channel having preemptive
repair priority over voice channels, draw the state diagram of a
Markov availability model. Using SHARPE GUI, solve the
Markov chain for steady-state and instantaneous availability.
Markov Reliability Model
• Consider the 2-component parallel system but disallow repair
from system down state
• Note that state 0 is now an absorbing state. The state diagram
is given in the following figure.
• This reliability model with repair cannot be modeled using a
reliability block diagram or a fault tree. We need to resort to
Markov chains.
(This is a form of dependency since in order to repair a
component you need to know the status of the other
component).
Markov reliability model with repair
• Markov chain has an absorbing state. In the steady-state, system will be in state 0 with probability 1. Hence transient analysis is of interest. States 1 and 2 are transient states.
Markov reliability model with repair (Continued)
Absorbing state
Assume that the initial state of the Markov chain
is 2, that is, P2(0) = 1, Pk (0) = 0 for k = 0, 1.
Then the system of differential Equations is written
based on:
rate of buildup = rate of flow in - rate of flow out
for each state
Markov reliability model with repair (Continued)
Markov reliability model with repair (Continued)
)()()(2)(
121 tPtPdt
tdP
)()(2)(
122 tPtPdt
tdP
)()(
10 tPdt
tdP
After solving these equations, we get
R(t) = P2(t) +P1(t)
Recalling that , we get:
Markov reliability model with repair (Continued)
0
)( dttRMTTF
222
3
MTTF
Note that the MTTF of the two component parallel redundant system, in the absence
of a repair facility (i.e., = 0), would have
been equal to the first term,
3 / ( 2* ), in the above expression.
Therefore, the effect of a repair facility is to
increase the mean life by / (2*2), or by a
factor
Markov reliability model with repair (Continued)
13
)
2321(
2
Markov Reliability Model with Imperfect Coverage
Markov model with imperfect coverage
Next consider a modification of the above example proposed by Arnold as a model of duplex processors of an electronic switching system. We assume that not all faults are recoverable and that c is the coverage factor which denotes theconditional probability that the system recovers given that a fault has occurred. The state diagram is now given by the following picture:
Now allow for Imperfect coverage
c
Markov modelwith imperfect coverage (Continued)
Assume that the initial state is 2 so that:
Then the system of differential equations are:
0)0()0(,1)0( 102 PPP
)()()1(2)(
)()()(2)(
)()()1(2)(2)(
120
121
1222
tPtPcdt
tdP
tPtcPdt
tdP
tPtPctcPdt
tdP
Markov model with imperfect coverage (Continued)
After solving the differential equations we obtain:
R(t)=P2(t) + P1(t)
From R(t), we can system MTTF:
It should be clear that the system MTTF and system reliability are
critically dependent on the coverage factor.
)]1([2
)21(
c
cMTTF
SOURCES OF COVERAGE DATA
• Measurement Data from an Operational system: Large
amount of data needed;
Improved Instrumentation Needed
• Fault/Error Injection Experiments
Costly yet badly needed: tools from
CMU, Illinois, Toulouse
SOURCES OF COVERAGE DATA (Continued)
• A Fault/Error Handling Submodel
Phases of FEHM:
Detection, Location, Retry, Reconfig, Reboot
Estimate Duration & Prob. of success of each phase
IBM(EDFI), HARP(FEHM), Draper(FDIR)
Homework 6:
• Modify the Markov model with imperfect coverage to allow
for finite time to detect as well as imperfect detection. You
will need to add an extra state, say D. The rate at which
detection occurs is . Draw the state diagram and using
SHARPE GUI investigate the effects of detection delay on
system reliability and mean time to failure.