![— Dependability:
— The concept of dependable computing first appears in the 1830’s
in the context of Babbage’s Calculating Engine [1,2].
— The ability to deliver service that can be justifiably be trusted [3].
— The ability to avoid service failures that are more frequent and
more severe than acceptable [3].
What is Dependability?](https://image.slidesharecdn.com/state-model-based-140626140700-phpapp01/85/State-model-based-4-320.jpg)
![According to different properties, which may be more or less
emphasized depending on the application intended for the computer
system under consideration:
— Availability is always required, although to a varying degree
depending on the application;
— reliability, safety, confidentiality may or may not be required
according to the application [3].
Dependability
and
its
a4ributes](https://image.slidesharecdn.com/state-model-based-140626140700-phpapp01/85/State-model-based-8-320.jpg)
![State-space methods are much more comprehensive. They
allow explicit modeling of complex relationships (e.g., [5]),
and their transition structure encodes important sequencing
information. Historically, state-space methods have been
explored in the context of mathematical models that specify
probabilistic assumptions about time durations and
transition behavior. We now review those models and
comment on how they are being applied in the security
context.
What is Model State?](https://image.slidesharecdn.com/state-model-based-140626140700-phpapp01/85/State-model-based-16-320.jpg)
![• [1] D. Lardner, Babbage's calculating engine. Edinburgh Review, July 1834. Reprinted in P.
Morrison and E. Morrison, editors, Charles Babbage and His Calculating Engines. Dover, 1961.
• [2] C. Babbage. On the mathematical powers of the calculating engine (December 1837).
Unpublished Manuscript. Buxton MS7, Museum of the History of Science. In B. Randell, editor,
The Origins of Digital Computers: Selected papers, pages 17-52. Springer, 1974.
• [3]Fundamental Concepts of Dependability by A Avizienis, J C Laprie, B Randell, Brian Randell
• K. Goseva-Popstojanov, K. S. Trivedi, Stochastic Modeling Formalisms for Dependability,
Performance and Performability, LNCS 1769, 2000
• [5] David M. Nicol, Fellow, IEEE, William H. Sanders, Fellow, IEEE, and Kishor S. Trivedi, Fellow,
IEEE, Dependability to Security Model-Based Evaluation
• [6] . K. Muppala, M. Malhotra, and K. S. Trivedi, “Markov dependability models of complex
systems: Analysis techniques,” in Reliability and Maintenance of Complex Systems, S. Ozekici,
Ed. Berlin, Germany: Springer, 1996, pp. 442–486.
• [7] Vedran Kordic, Petri Net Theory and Applications
• [8] Peter J. Haas, Stochastic Petri Nets- Modelling, Stability, Simulation, Springer.
• [9] Ebeling, C. E., An Introduction to Reliability and Maintainability Engineering. Illinois, Waveland
Press, 1997
References](https://image.slidesharecdn.com/state-model-based-140626140700-phpapp01/85/State-model-based-46-320.jpg)
State model based
- 1. State Model Based Disciplina: Tópicos Avançado em Avaliação de Desempenho de Sistemas Aluno: Rafael Roque de Souza Professores: Eduardo Tavares, Ricardo Massa
- 2. Agenda — Dependability: — Motivation — Dependability — Attributes — Model Base State — Modelos Formais — Makov chain — Petri Net — Tools — Conclusão 2
- 3. 3/81DRCN2009 Copyright © by Kishor S. Trivedi Health & Medicine Communication Avionics Entertainment Banking Motivation: Dependence on Computer Systems Motivation
- 4. — Dependability: — The concept of dependable computing first appears in the 1830’s in the context of Babbage’s Calculating Engine [1,2]. — The ability to deliver service that can be justifiably be trusted [3]. — The ability to avoid service failures that are more frequent and more severe than acceptable [3]. What is Dependability?
- 5. More… Default approach: Utilize a formalism to model system dependability • Quantify the availability of components, calculate system availability based on this data and a set of assumptions - the availability model • Most models expose the same expressiveness • Each formalism allows to focus on certain aspects • Component-based models: Reliability block diagram, fault tree • State-based models: Markov chains, petri nets • System understanding evolved from hardware to software to IT infrastructures What is Dependability?
- 6. • Some assumptions – All failure and repair events are exponentially distributed – Components are either fully working or completely failed – All failure and repair events are pair-wisely stochastically independent – Correct functioning at t can be treated as event, with event probability derived from the availability value computed by failure rate and repair rate Exemple Dependable Systems Course PT 2011 Dependability Modeling 3 Up Down µ = 1 MT T R = 1 MT T F • Some assumptions • All failure and repair events are exponentially distributed • Components are either fully working or completely failed • All failure and repair events are pair-wisely stochastically independent • Correct functioning at t can be treated as event, with event probability derived from the availability value computed by failure rate and repair rate
- 7. Schedule — Dependability: — Dependability — Attributes — Model Base State — Modelos Formais — Makov chain — Petri Net — Tools — Conclusão 7
- 8. According to different properties, which may be more or less emphasized depending on the application intended for the computer system under consideration: — Availability is always required, although to a varying degree depending on the application; — reliability, safety, confidentiality may or may not be required according to the application [3]. Dependability and its a4ributes
- 9. Dependability and its a4ributes — Trustworthiness of a computer system such that reliance can justifiably be placed on the service it delivers, It encompasses the following attributes: — Availability: readiness for correct service — Reliability: continuity of correct service — Safety: absence of catastrophic consequences — Integrity: absence of improper system alterations — Maintainability: ability to undergo modifications and repairs 9
- 10. What is Availability? Availability is the probability that the system will still be operating to requirements at a given time. — Availability is a function not only of how rarely a system fails (reliability) but also of how quickly it can be repaired (time to repair) — Availability of 0.998 means software is available for 998 out of 1000 time units 10
- 11. 11 What is Reliability? — Reliability is the probability that the system will deliver a set of services for a given period of time, whereas a system is fault tolerant when it does not fail even when there are faulty components.
- 12. Example • Reliability x Availability? • Look at the example: – Consider a site online that negotiators get down to 1 minute every 4 hours, ie every 240 (4x60) minutes. The availability and 239/240 = 99 583% (relative high availability) – Reliability: can it be low down periods occur at critical times when the market isfluctuating and clients who trade their shares!
- 13. Safety is an extension of reliability. When the state of correct service and the states of incorrect service due to non-catastrophic failure are grouped into a safe state (in the sense of being free from catastrophic damage, not from danger), safety is a measure of continuous safeness, or equivalently, of the time to catastrophic failure. Safety is thus reliability with respect to catastrophic failures. What is Safety
- 14. • A measure of the time to service restoration since the last failure occurrence, or equivalently, measure of the continuous delivery of incorrect service; • The maintainability model situations where the system fails and the return to proper functioning of the state requires any maintenance. The maintainability and defined as the probability of a repair system was successfully completed in given time. What is Maintainability
- 15. Agenda — Dependability: — Dependability — Attributs — Model Base State — Model — Makov chain — Petri Net — Tools — Conclusão 15
- 16. State-space methods are much more comprehensive. They allow explicit modeling of complex relationships (e.g., [5]), and their transition structure encodes important sequencing information. Historically, state-space methods have been explored in the context of mathematical models that specify probabilistic assumptions about time durations and transition behavior. We now review those models and comment on how they are being applied in the security context. What is Model State?
- 17. • In contrast with state-space models, combinatorial models do not enumerate all possible system states to obtain a solu- tion. Instead, simpler approaches are used to compute system dependability measures. Despite several extensions that have been made to combinatorial models, they do not easily capture certain features, such as stochastic dependence and imperfect fault coverage. We present a brief overview of combinatorial models. What is Model State?
- 18. • Reliability: “The ability of a system to perform a required function under given conditions for a given time interval.” No recovery is assumed after system fails (there can be recovery after a component failure) • • Availability: “The ability of a system to be in a state to perform a required function at a given instant of time or at any instant of time within a given time interval.“ Dependability Attributes or Measures 7/81DRCN2009 Copyright © by Kishor S. Trivedi Reliability Availability Dependability Measures Dependability Attributes or Measures • Reliability: “The ability of a system to perform a required function under given conditions for a given time interval.” No recovery is assumed after system fails (there can be recovery after a component failure) • Availability: “The ability of a system to be in a state to perform a required function at a given instant of time or at any instant of time within a given time interval.“
- 19. 8/81DRCN2009 Copyright © by Kishor S. Trivedi Numerical solution tool Close-form solution Dependability Evaluation Methods Model-based Discrete-event simulation Hybrid Analytic Models Quantitative evaluation Measurement-based Numerical solution Of analytic models Not as well utilized; Unnecessarily excessive Use of simulation Dependability Evaluation Methods
- 20. Modeling Taxonomy 23/81DRCN2009 Copyright © by Kishor S. Trivedi Modeling Taxonomy Hierarchical models largeness Abstract models Discrete-event simulation Hybrid Analytic models Combinatorial models Efficiency, simplicity State-space models Dependency capture
- 21. • 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 Models
- 22. • State space explosion problem or the largeness problem • Stochastic Petri nets and related formalisms for easy specification and automated generation/solution of underlying Markov model • Or use hierarchical (Multilevel) model composition. – e.g. Upper level : FT or RBD, lower level: Markov chains – Many practical examples of the use of hierarchical models exist Problem with State Space Models
- 23. • Can relax the assumption of exponential distributions State-Space model taxonomy 21/81DRCN2009 Copyright © by Kishor S. Trivedi State-Space model taxonomy (discrete) State space models Markovian models non-Markovian models discrete-time Markov chains (DTMC) continuous-time Markov chains (CTMC) Markov reward models (MRM) Semi-Markov process (SMP) Markov regenerative process Non-Homogeneous Markov Can relax the assumption of exponential distributions
- 24. Agenda — Dependability: — Dependability — Attributs — Model Base State — Modelos Formais — Makov chain — Petri Net — Tools — Conclusão 24
- 25. • Discrete random process, usually drawn as state transition diagram • Markov property - Next step depends only on the current step • Impossible to predict future states, but useful for statistical properties • Finite state space (chain), transitions with probabilities, initial state probabilities • Transient state - Probability to not return to this state (finite number of visits) • Recurrent state - Probability of 1 to return to this state after unspecified time t • Mean recurrence time can be used as MTTF metric • Time-homogeneous Markov chains - Transition probabilities do not change in time. What is Markov Chain Dependable Systems Course • Discrete random process, usually drawn as state transition • Markov property - Next step depends only on the curr • Impossible to predict future states, but useful for statist • Finite state space (chain), transitions with probabilities, • Transient state - Probability to not return to this state (fini • Recurrent state - Probability of 1 to return to this state af • Mean recurrence time can be used as MTTF metric • Time-homogeneous Markov chains - Transition probabi 8 • Discrete random process, usually drawn as state transitio • Markov property - Next step depends only on the cur • Impossible to predict future states, but useful for statis • Finite state space (chain), transitions with probabilities • Transient state - Probability to not return to this state (fin • Recurrent state - Probability of 1 to return to this state a • Mean recurrence time can be used as MTTF metric • Time-homogeneous Markov chains - Transition probab
- 26. • Discrete-time Markov chain (DTMC) • System state only changes after a fixed time interval, system is in exactly one state • Transition to next state depends on transition probability (non- negative) at t • Each row of the probability transition matrix represents flow out of that state, the columns the transition flow into the state, row sum is one • Continuous-time Markov chain (CTMC) • Allows state changes at any instance of time - continous parameter space, still discrete state space • Transition to next state after spending some time in a state - holding time • Generator matrix Q therefore expresses transition rates instead of probabilities • By definition, the diagonal entries are equal to minus the total rate out of that state – Rates with which no state change takes place What is Markov Chain
- 27. • Initial distribution vector can be combined with transition matrix to find probabilities for being in one of the states after one step • Each row sum of the transition matrix is 1 Markov Chains - DTMC Example Dependable Systems Course PT 2011 Markov Chains - DTMC Example 10 Transition Matrix • Each row sum of the transition matrix is 1 • Initial distribution vector can be combined with transition matrix to find probabilities for being in one of the states after one step Probability Matrix after 2 steps (C) Tamara Lynn Anthony ns - DTMC Example Transition Matrix he transition matrix is 1 Probability Matrix after 2 stepsPT 20110 Transition Matrix with transition matrix to find probabilities p Probability Matrix after 2 steps
- 28. • Each state represents a particular error state, transition with component failure rate • States expresses number of failed components at any given time • Time-homogeneous process - Failure / repair rates do not change over time • Components have identical failure rates and identical repair rates – Failure and repair events are independent, process is memory-less • Row sum is zero: Probability mass flowing out of state i will go to some other state – Example: Markov Chains - CTMC Exampleexpresses number of failed components at any given time homogeneous process - Failure / repair rates do not change o onents have identical failure rates and identical repair rates and repair events are independent, process is memory-less m is zero: Probability mass flowing out of state i will go to some :
- 29. 3 s0 = µs1 3 s0 + 2µs2 = µs1 + 2 s1 2 s1 + 3µs3 = 2µs2 + s2 s2 = 3µs3 s0 + s1 + s2 + s3 = 1 s0 = µ 3 s1 s2 = µ s1 s3 = 2 3µ2 s1 s1 = 3µ2 (µ + )3 s0 = µ3 (µ + )3 ; s1 = 3µ2 (µ + )3 ; s2 = 3µ 2 (µ + )3 ; s3 = 3 (µ + )3 A = s0 + s1 = µ2 (µ + 3 ) (µ + )3 = 3a2 + 2a3 a = µ (µ + ) Dependable Systems Course Example: 2-of-3 System 13 Example: 2-of-3 System
- 30. • Interested in steady-state availability of the system • Interpretation as steady-state probability for the system being operational at – Derived from probability vector -> contains steady-state probabilities for the system being in one of the failure states after a number of steps – ,Static‘ steady-state availability computable if probabilities are in equilibrium • Probability for leaving state is similar to probability for going into that state - probability mass is evenly distributed – Typically achieved after a high number of steps Example: 2-of-3 System Example: 2-of-3 System • Interested in steady-state availability of the system
- 31. • Resulting formula equals to result from Boolean investigation, but Markov chains also support non-independent events - common cause failure • Markov chains grow exponentially with their number of components - which is bad – Divide-and-conquer - Decompose and aggregate chain parts – Structural decomposition - Consider a system as set of independent subsystems – Behavioral decomposition - Assume time constants for some fault occurences and handling processes based on criticality - e.g. fault in parked airplane Markov ChainsMarkov Chains • Resulting formula equals to result from Boolean investigation, but Markov chains also support non-independent events - common cause failure • Markov chains grow exponentially with their number of components - which is b • Divide-and-conquer - Decompose and aggregate chain parts • Structural decomposition - Consider a system as set of independent subsyste • Behavioral decomposition - Assume time constants for some fault occurences handling processes based on criticality - e.g. fault in parked airplane
- 32. • Mathematical model for concurrent systems with many components (Carl Adam Petri) • Bipartit directed graph (places vs. transitions) • Each place has a capacity for tokens, default is unlimited or one • Each arc has a weight expressing a cost factor, default is one • Places are pre- / postconditions for transitions • Distribution of tokens is called a marking • Every net has an initial marking What is Petri Nets Dependable Systems Course PT 2011 Stochatic Petri Nets • Mathematical model for concurrent systems with many components (Carl Adam Petri) • Bipartit directed graph (places vs. transitions) • Each place has a capacity for tokens, default is unlimited or one • Each arc has a weight expressing a cost factor, default is one • Places are pre- / postconditions for transitions • Distribution of tokens is called a marking • Every net has an initial marking 16 Place / State Transition Token Input place of the transition Output place of the transition
- 33. Extension SPN
- 34. • Transition is activated (may fire) when – All input places contain enough tokens for the transition costs – All output places have enough capacity to take the new tokens • Tokens are consumed and placed in output places, considering the arc weights • Atomic nondeterminstic operation - any activated transition may fire • Firing happens with given delay • More complex Petri net versions can • distinguish different token types – Colored tokens (data values) – Activation times for tokens • Petri nets allow both formal analysis (for exponential distribution) and simulation Stochastic Petri Nets Dependable Systems Course PT 2011 • Transition is activated (may fire) when • All input places contain enough tokens for the transition costs • All output places have enough capacity to take the new tokens • Tokens are consumed and placed in output places, considering the arc weights • Atomic nondeterminstic operation - any activated transition may fire • Firing happens with given delay • More complex Petri net versions can distinguish different token types • Colored tokens (data values) • Activation times for tokens • Petri nets allow both formal analysis (for exponential distribution) and simulation 17
- 35. • A stochastic process and a sequence of random variables indexed on time witha well-defined correlation structure • Have probability distributions associated with them – Arrival of customers in a bank queue – Number of requests in a Web Server • Why stochastic modeling? – In many systems, you need to join in time to events • How to model stochastic processes? – Analytical queuing – theory models – Petri Nets Stochastic Petri Nets
- 36. • Probabilistic behavior model • Distributions: – Exponential - SPN (stochastic Petri net) – exponential or immediate GSPN(Generalized) – There are other models with arbitrary functions Stochastic Petri Nets
- 37. • Reachability set – All possible markings reachable from an initial marking – Possible analysis questions • Can some system state (e.g. an error state) be reached at all ? • Exists a firing sequence that transforms M0 to M ? • Boundedness – Marking is bounded if there is a k so that for every reachable marking the number of tokens in each place is bounded by k – Useful for modeling limited (bounded) resources Typical Petri Net Properties
- 38. • Complexity of the petri net does not depend on the number of components ! Example: 2-of-3 System Dependable Systems Course PT 2011 Example: 2-of-3 System 23 Complexity of the petri net does not depend on the number of components !
- 39. • Modeling of cold standby components (inhibitor arc) • Limited repair capacities - at most R repairmen available at a time • Dependability analysis - prove that there is no state where some property is violated Example: K-of-N With Standby and Repairmen Dependable Systems Course PT 2011 Example: K-of-N With Standby and Repairmen • Modeling of cold standby components (inhibitor arc) • Limited repair capacities - at most R repairmen available at a time • Dependability analysis - prove that there is no state where some property is violated
- 40. • buffer size = #token_capacity(p1 + p2) • unit count = #token_capacity(p3 + p4) • Firing rate of t1 is arrival rate t2 is an immediate transition • Firing rate of t3 is the service rate, depends on token count in p4 Example: Parallel System with Input Buffer Dependable Systems Course PT 2011 Example: Parallel System with Input Buffer 25 Input buffer with positions Identical units • buffer size = #token_capacity(p1 + p2) • unit count = #token_capacity(p3 + p4) • Firing rate of t1 is arrival rate • t2 is an immediate transition • Firing rate of t3 is the service rate, depends on token count in p4 Free buffer positions Filled buffer positions Free units (C)AndreaBobbio Active units
- 41. Dependable Systems Course PT 2011 Example: Parallel System with Input Buffer 26 (C) Andrea Bobbio • Light lines - Fault free operation • Heavy lines - Failures • Dotted lines - repairs • Rate computation demands exponential distribution Example: Parallel System with Input Buffer
- 42. • In many cases, simulation is the only way to solve the net • More than one outgoing non-exponential distribution • Special guard functions • Complexity issues • ... • Typical simulation problems • Modeled failure rates might be small, so many runs needed for valid result • Random number generation • Confidence intervals Petri Net Simulation
- 43. • Petri net has according reachability graph • Combines to Markov chain when transition probabilities are given Petri Net ->Markov Chain able Systems Course PT 2011 tri Net -> Markov Chain • Petri net has according reachability graph • Combines to Markov chain when transition probabilities are given 21 Petri Net -> Markov Chain • Petri net has according reachabi • Combines to Markov chain wh probabilities are given • Petri net has according reachability • Combines to Markov chain when probabilities are given • Petri net has according reachability graph • Combines to Markov chain when transition probabilities are given
- 44. Agenda — Dependability: — Dependability — Attributs — Model Base State — Modelos Formais — Makov chain — Petri Net — Tools — Conclusão 44
- 45. • Astro • Mercury • TimeNet • Sharp • CPN Tool • INA • ... Tools
- 46. • [1] D. Lardner, Babbage's calculating engine. Edinburgh Review, July 1834. Reprinted in P. Morrison and E. Morrison, editors, Charles Babbage and His Calculating Engines. Dover, 1961. • [2] C. Babbage. On the mathematical powers of the calculating engine (December 1837). Unpublished Manuscript. Buxton MS7, Museum of the History of Science. In B. Randell, editor, The Origins of Digital Computers: Selected papers, pages 17-52. Springer, 1974. • [3]Fundamental Concepts of Dependability by A Avizienis, J C Laprie, B Randell, Brian Randell • K. Goseva-Popstojanov, K. S. Trivedi, Stochastic Modeling Formalisms for Dependability, Performance and Performability, LNCS 1769, 2000 • [5] David M. Nicol, Fellow, IEEE, William H. Sanders, Fellow, IEEE, and Kishor S. Trivedi, Fellow, IEEE, Dependability to Security Model-Based Evaluation • [6] . K. Muppala, M. Malhotra, and K. S. Trivedi, “Markov dependability models of complex systems: Analysis techniques,” in Reliability and Maintenance of Complex Systems, S. Ozekici, Ed. Berlin, Germany: Springer, 1996, pp. 442–486. • [7] Vedran Kordic, Petri Net Theory and Applications • [8] Peter J. Haas, Stochastic Petri Nets- Modelling, Stability, Simulation, Springer. • [9] Ebeling, C. E., An Introduction to Reliability and Maintainability Engineering. Illinois, Waveland Press, 1997 References
- 47. • http://www.ee.duke.edu/~kst/ • http://www.modcs.org • http://www.informatik.uni-hamburg.de/TGI/PetriNets • http://www.informatik.uni-hamburg.de/TGI/PetriNets/tools/quick.html • http://tandem.bu.edu/rsg.html Links