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Andrea Bobbio Dipartimento di Informatica Universit à del Piemonte Orientale, “ A. Avogadro ” 15100 Alessandria (Italy) bobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio. Dependability Theory and Methods 5. Markov Models. Bertinoro, March 10-14, 2003.
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Andrea Bobbio Dipartimento di Informatica Università del Piemonte Orientale, “A. Avogadro” 15100 Alessandria (Italy) bobbio@unipmn.it - http://www.mfn.unipmn.it/~bobbio Dependability Theory and Methods5. Markov Models Bertinoro, March 10-14, 2003 Bertinoro, March 10-14, 2003
States and labeled state transitions State can keep track of: Number of functioning resources of each type States of recovery for each failed resource Number of tasks of each type waiting at each resource Allocation of resources to tasks A transition: Can occur from any state to any other state Can represent a simple or a compound event State-Space-Based Models Bertinoro, March 10-14, 2003
Transitions between states represent the change of the system state due to the occurrence of an event Drawn as a directed graph Transition label: Probability: homogeneous discrete-time Markov chain (DTMC) Rate: homogeneous continuous-time Markov chain (CTMC) Time-dependent rate: non-homogeneous CTMC Distribution function: semi-Markov process (SMP) State-Space-Based Models (Continued) Bertinoro, March 10-14, 2003
Should I Use Markov Models? State-Space-Based Methods + Model Dependencies +Model Fault-Tolerance and Recovery/Repair + Model Contention for Resources + Model Concurrency and Timeliness + Generalize to Markov Reward Models for Modeling Degradable Performance Modeler’sOptions Bertinoro, March 10-14, 2003
Should I Use Markov Models? + Generalize to Markov Regenerative Models for Allowing Generally Distributed Event Times + Generalize to Non-Homogeneous Markov Chains for Allowing Weibull Failure Distributions + Performance, Availability and Performability Modeling Possible - Large (Exponential) State Space Modeler’sOptions Bertinoro, March 10-14, 2003
Modeling Performance, Availability and Performability Modeling Complex Systems We Need Automatic Generation and Solution of Large Markov Reward Models In order to fulfill our goals Bertinoro, March 10-14, 2003
Choice of the model type is dictated by: Measures of interest Level of detailed system behavior to be represented Ease of model specification and solution Representation power of the model type Access to suitable tools or toolkits Model-based evaluation Bertinoro, March 10-14, 2003
x i State space models s s’ A transition represents the change of state of a single component Z(t)is the stochastic process Pr {Z(t) = s}is the probability of finding Z(t)in state sat timet. Pr {s s’, t} = Pr {Z(t+ t) = s’| Z(t) = s} Bertinoro, March 10-14, 2003
x i State space models s s’ If s s’ represents a failure event: Pr {s s’, t} = = Pr {Z(t+ t) = s’| Z(t) = s} = it If s s’ represents a repair event: Pr {s s’, t} = = Pr {Z(t+ t) = s’| Z(t) = s} = it Bertinoro, March 10-14, 2003
Markov Process: definition Bertinoro, March 10-14, 2003
Transition Probability Matrix initial
Transient analysis Given that the initial state of the Markov chain, then the system of differential Equations is written based on: rate of buildup = rate of flow in - rate of flow out for each state (continuity equation).
Steady-state condition If the process reaches a steady state condition, then:
Steady-state analysis (balance equation) The steady-state equation can be written as a flow balance equation with a normalization condition on the state probabilities. (rate of buildup) = rate of flow in - rate of flow out rate of flow in = rate of flow out for each state (balance equation).
2-component system Bertinoro, March 10-14, 2003
2-component system Bertinoro, March 10-14, 2003
2-component system Bertinoro, March 10-14, 2003
2-component series system A1 A2 A1 A2 2-component parallel system Bertinoro, March 10-14, 2003
2-component stand-by system A B Bertinoro, March 10-14, 2003
Repairable system: Availability Bertinoro, March 10-14, 2003
Repairable system: 2 identical components Bertinoro, March 10-14, 2003
Repairable system: 2 identical components Bertinoro, March 10-14, 2003
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. 2-component Markov availability model Bertinoro, March 10-14, 2003
Markov availability model • 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. Bertinoro, March 10-14, 2003
Markov availability model 2 1 0 Non-shared (independent) repair 2 1 0 Shared repair Bertinoro, March 10-14, 2003
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 Bertinoro, March 10-14, 2003
For any state: Rate of flow in = Rate of flow out Considering the shared case i: steady state probability that system is in state i Steady-state balance equations Bertinoro, March 10-14, 2003
Hence Since We have Or Steady-state balance equations
Steady-state Unavailability: For the Shared Case = 0= 1 - Ashared Similarly, for the Non-Shared Case, Steady-state Unavailability =1 - Anon-shared Downtime in minutes per year = (1 - A)* 8760*60 Steady-state balance equations(Continued) Bertinoro, March 10-14, 2003
Steady-state balance equations Bertinoro, March 10-14, 2003
Absorbing states MTTF Bertinoro, March 10-14, 2003
Absorbing states - MTTF Bertinoro, March 10-14, 2003
Next consider a modification of the 2-component parallel system 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 the conditional probability that the system recovers given that a fault has occurred. The state diagram is now given by the following picture: Markov model with imperfect coverage Bertinoro, March 10-14, 2003
Now allow for Imperfect coverage c Bertinoro, March 10-14, 2003
Assume that the initial state is 2 so that: Then the system of differential equations are: Markov modelwith imperfect coverage Bertinoro, March 10-14, 2003
After solving the differential equations we obtain: R(t)=P2(t) + P1(t) From R(t), we can obtain system MTTF: It should be clear that the system MTTF and system reliability are critically dependent on the coverage factor. Markov model with imperfect coverage Bertinoro, March 10-14, 2003
Measurement data from an operational system Large amount of data needed Improved instrumentation needed Fault-injection experiments Expensive but badly needed Tools from CMU,Illinois, LAAS (Toulouse) A fault/error handling submodel (FEHM) Phases: detection, location, retry, reconfig, reboot Estimate duration and probability of success of each phase Source of fault coverage data Bertinoro, March 10-14, 2003
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 investigate the effects of detection delay on system reliability and mean time to failure. Redundant System with Finite Detection Switchover Time Bertinoro, March 10-14, 2003
Assumptions: Two units have the same MTTF and MTTR; Single shared repair person; Average detection/switchover time tsw=1/; We need to use a Markov model. Redundant System with Finite Detection Switchover Time Bertinoro, March 10-14, 2003
Redundant System with Finite Detection Switchover Time 2 1D 1 0 Bertinoro, March 10-14, 2003
After solving the Markov model, we obtain steady-state probabilities: Redundant System with Finite Detection Switchover Time Bertinoro, March 10-14, 2003
Closed-form Bertinoro, March 10-14, 2003