9. Reliability theory

Transcription

1 Material based on original slides by Tuomas Tirronen ELEC-C720 Modeling and analysis of communication networks

2 Contents Introduction Structural system models Reliability of structures of independent repairable components Reliable network topology design 2

3 History As a technological concept, reliability emerged after WW, practical methods were developed during and after WW2 For example Lusser s law, i.e., product probability law of series of components, was formulated by Robert Lusser during V flying bomb tests Arised from the need to improve and control the quality of industrial products with many parts 50s, 60s 70s ballistic missiles, space programs first journal, IEEE Transactions on Reliability 963 safety of nuclear power plants 80s, 90s oil and gas industries, computer programs to evaluate reliability, software reliability,... 00s, new kinds of operation concepts (remote control/maintenance of systems) require reliability analysis, network reliability etc 3

4 Approaches to reliability Hardware reliability Physical approach Strength S of an item is a random variable Load L the item is exposed to is another random variable Reliability R = Pr(S > L) Structural reliability analysis Actuarial approach our approach Time to failure T is studied using its distribution F(t) All information of individual strengths, loads, etc is conveyed in F(t) System reliability analysis Software reliability Human reliability 4

5 Basic concepts () Reliability: The ability of an item to perform a required function, under given environmental and operational conditions and for a stated period of time (ISO 8402) the item can be a single component or a larger entity (system) required function may refer to a single function or many Quality: The totality of features and characteristics of a product or service that bear on its ability to satisfy stated or implied needs (ISO 8402) i.e., conformance to specifications reliability can be seen as an extension of quality into the time domain Availability: The ability of an item to perform its required function at a stated instant of time or over a stated period of time (BS4778) i.e., can the item be used at some time instant or what is the time fraction the item is usable (= average availability) 5

6 Basic concepts (2) Maintainability: The ability of an item, under stated conditions, to be retained in, or restored to, a state in which it can perform its required functions, when maintenance is performed under stated conditions and using prescribed procedures and resources (BS4778) if an item can be repaired, then maintainability determines the availability of the item Dependability: Collective term to describe availability performance and influencing factors: Reliability performance, maintainability performance and maintenance support performance (IEC60300) umbrella term often used when covering reliability issues Safety: Freedom from those conditions that can cause death, injury, occupational illness, or damage to or loss of equipment or property (MIL-STD-882D) Security: Dependability with respect to prevention of deliberate hostile actions 6

7 Basic concepts(3) Fault Error a defect or mistake which leads to error. Reason for an error. a system state which can lead to a failure Failure The termination of its ability to perform a required function (BS 4778) An unacceptable deviation from the design tolerance or in the anticipated delivered service, an incorrect output, the incapacity to perform the desired function (NASA 2002) 7

8 Basic concepts (4) Cause Fault Fault prevention aim is to design a system without faults physical shielding of components, careful manufacturing etc. Error Failure Fault tolerance aim is to be able to provide the service even in the presence of faults main tool: redundancy! hardware software information time 8

9 Repairable and nonrepairable items Can study two types of items Nonrepairable items The item can be single item or larger system We are only interested in the time until first failure whatever happens after this is of no interest to us Interesting measures include: Mean time to failure, reliability (function) and failure rate Our focus Repairable items Single item or larger system Interesting measures include: Availability, mean time between failures, mean down time, number of failures in some time interval In some sources the term dependability is used instead of availability to mean the same thing 9

10 Systems of items We also study systems of many items or components. There are two possibilities for modeling systems: Our focus Systems of independent components Easy analysis: independence of components independence of probabilities Most examples during this course assume independence Systems with dependent components Exact analysis is harder, even impossible, because of the dependencies Analysis of the system as a stochastic process 0

11 Tools and models As function of time, we have state models where systems are modelled as stochastic processes (cf. queueing models in earlier lectures) especially repairable items/systems the failure process, repair times, etc. Structure of systems and its subsystems -> structural models reliability block diagrams, structure function Tools: Basic probability theory Stochastic processes Markov chains/processes (Renewal processes) Statistical methods Main limitations of probabilistic reliability analysis : human errors, human factor

12 Applications Risk analysis Identification of accidental events Causal analysis Consequence analysis Environmental protection Quality Optimization and maintenance Engineering design Verification of quality Research and development... 2

13 Reliability in communications and networking () From user point of view, an interesting quality of service concept is the network availability = Pr (user can access the agreed network services at time ) Average availability tells us the time fraction the system is available A way to understand availability of networks is to study the downtime of a network (or outage of some specific service) per year # nines Avg. availability Downtime / year 2-nines hours 3-nines hours 46 mins 4-nines mins 34 secs 5-nines mins 5 secs 6-nines secs 7-nines secs 3

14 Reliability in communications and networking (2) In addition to availability, network operators, service providers and equipment manufacturers are interested also in reliability of components (mean times to failure, number of failures in some time interval etc.) maintainability security of networks Reliability is an important factor when planning new services, networks or equipment Note that dependability, reliability and availability may have different definitions in different sources. Be careful to understand what are the definitions of the different concepts. 4

15 Aim of the lecture We focus on Repairable systems Systems of independent components Exponential assumptions on mean time to failure and mean down time Thus, we get simple models using Markovian analysis Apply the models to topology design of communication networks where availability is defined as connectivity of the network 5

16 Literature Reliability theory / Dependability System Reliability Theory: Models, Statistical Methods and Applications, 2nd edition, Marvin Rausand and Arnljot Høyland, Wiley, 2004 Mesh-Based Survivable Networks: Options and Strategies for Optical, MPLS, SONET and ATM Networking, Wayne D. Grover, Prentice Hall, 2004 Moniste: Luotettavuus, käytettävyys, huollettavuus (luotettavuusteoria.pdf), Keijo Ruohonen, TTKK, 2002 TKK courses AS-6.380, Mat

17 Contents Introduction Structural system models Reliability of structures of independent repairable components Reliable network topology design 7

18 Reliability block diagrams Reliability block diagrams (RBD) are used to describe the function of a system of components it shows the logical connections between components A system works if there is a path of functioning components from the start point (a) to the end point (b) RBDs give a deterministic model for the structure of a system the whole system works properly if and only if some set of the components function It is important to determine which specific function of the system is modelled: the logical structure may be different for different functions 8

19 Series and parallel structures When a system functions if and only if all of the components function, the logical structure is a series structure b a When a system functions if at least one of all possible n components functions, the logical structure is a parallel structure a b Series and parallel structures can be further combined to model more complex structures 9

20 Structure function () The state vector of a structure is x = (x, x 2,..., x n ), where each state variable x i is either when component i is functioning or 0 when component i is in a failed state The structure function of the system is ε(x) 0 if if thesystemis functioning the systemis in a failed state For a series structure, the structure function is ε( x) x x Κx n x i system works if and only if x i = for all i 2 n i 20

21 Structure function (2) For parallel structure the structure function is n n ) (, x2) Κ(, xn), (, xi ) i i ε( x), (, x Β x If any x i =, then the system functions The last operator (upwards product) is reap ip Example: For structure with 2 components in parallel we have ε( x Β 2, x2) xi, (, x)(, x2) i x x 2, x x 2 i 2

22 Path set and cut set methods For small systems the structure function ( ) can be written down by visually inspecting the system as a combination of series and parallel structures However, for large systems it is not possible! Therefore, we need systematic computational methods for generating the structure function ( ) Path set and cut set methods allow this 22

23 Path/cut sets () Definition: A path set P is set of components which by functioning ensure that the system is functioning. A path set is minimal if it cannot be reduced without losing its status as a path set. Definition: A cut set K is set of components which by failing cause the system to fail. A cut set is minimal if it cannot be reduced. Example: Path sets: {,2} {,3} {, 2,3} Cut sets: {} {2,3} {, 2} {,3} {, 2,3} 2 3 Minimal path sets: Minimal cut sets: P {, 2} P2 {,3} K } K {2,3} { 2 23

24 Path set method Let us denote θ j (x) the structure function of jth minimal path The whole structure functions if and only if at least one minimal path set is functioning, Path set method: θ (x) j x i i. Determine the path sets of the structure j j P j Β ε( x), (, θ ( x)) θ ( x) j j Β j x i P j i 2. Determine minimal path sets P j 3. Calculate the structure functions of minimal path setc as series stuctures 4. Take ip over all functions you get in step Simplify as needed (TIP: Power of binary variable = variable without any power, x ij =x i ) 24

25 Cut set method Let us denote ϕ j (x) the structure function of jth minimal cut Now the structure fails if and only if at least one structure corresponding to the minimal cut sets fail Cut set method: Β i K j i K j ε( x) ϕ ( x). Determine the cut sets of the structure 2. Determine minimal cut sets K j 3. Calculate the structure functions of minimal cuts sets as parallel structures 4. Multiply all functions you get in step Simplify as needed j j i j,, ϕ (x) x ( x ) j Β x i K j i i 25

26 Demo/Exercise Determine the structure function of independent components below a) directly (by using results for series/parallel structures and combining) b) using path set method c) using cut set method

27 Contents Introduction Structural system models Reliability of structures of independent repairable components Reliable network topology design 27

28 Repairable components/systems Now we study systems where components can be repaired or replaced upon failures (or even before), i.e., repairable components We are interested for example in system reliability component/system availability: mean number of failures during a time interval mean time between failures, MTBF mean downtime (or repair time) of systems, MDT (MTTR) For this purpose we can model the systems/failure processes as stochastic processes thus, we have studied the theoretical background already in the beginning of this course 28

29 Reliability of maintained systems () The system is called maintainable, when its components are repaired/restored to working condition using some kind of maintenance Can be preventive, corrective, Let X(t) denote the stochastic process of the system with X(t) = if the system is operational and 0 otherwise The main measure is availability, A(t) also ( ) = Ā(t) = A(t), the unavailability is studied Availability A( t) P{ X ( t) } Average availability A av ( σ ) σ Long run average availability 0 A A( t) dt av lim A σ av t ( σ ) lim σ σ Limiting availability (when exists) A lim A ( t) σ σ 0 A( t) dt 29

30 Availability of single component as on-off process () We can model a single component as an on-off type process X(t) with = if component is operational 0 otherwise Measures related to maintainable systems are Mean time between failures, MTBF Mean downtime, MDT Mean time to failure, MTTF MTBF X(t) MTTF 0 MDT t 30

31 Reliability of single component as on-off process (2) Markov model MTTF is independent and exponentially distributed with mean / MDT is independent and exponentially distributed with mean / ~Exp(κ) ~Exp(λ) Steady state distribution simply: κ U av ο 0 κ λ λ Aav ο κ λ Steady-state distribution holds even when MTTF and MDT have general distributions (but still independent), insensitivity property Then no more a Markovian process but a so-called renewal process 3 0 λ κ MDT MTTF MDT MTTF MTTF MDT

32 Examples () Example : A machine has MTTF = 000 hours and MDT = 5 hours The average availability is A av MTTF MTTF MDT Example 2: Item has independent uptimes with constant failure rate κ Downtimes are IID with mean MDT. Usually we have MDT MTTF, the average unavailability is then approximately A av, A av, κ MDT κ MDT MTTF MTTF MDT κ MDT MDT MTTF MDT 32

33 Systems of independent components () Consider a system consisting of n independent components The state vector of a system is X t) ( X ( t), X ( t),..., X ( )) MTTF and MDT of component i independent and exp. distributed with mean / and /, respectively Let = = = / + That is, is the availability of component i ( 2 n t Then the steady state distribution of state =,, is simply the product of Bernoulli distributions of each component i, = Again distribution holds even under general distributions for MTTF and MDT (insensitivity) 33

34 9. Reliability theory Systems of independent components (2) In general, the average availability of the system is defined as = = The state space Ω can be partitioned into two sets. Up states Ω where the system is working. Note that some components may be in failed state, but the system still provides the intended service. 2. Down states Ω where the system does not perform the required function The (average) availability of the system is given by = ( ) = = ( ) similarly, unavailability is the sum of probabilities of down states 34

35 Systems of independent components (3) As ( ) is a binary-valued function, For series structure: (independence of s!!) 35 )] ( [ } ) ( { X X ε ε E P A av n i i n i i n i i av p X E X E E A ] [ )] ( [ X ε And similarly for parallel structure: Β Β n i i n i i n i i n i i n i i av p p X E X E X E E A ) ( ]) [ ( ) ( )] ( [,,,,,, X ε

36 Systems of independent components (4) However, in general = ( ) ( ) Thus, to calculate availability one can not just write down the structure function ( ) and replace s by the corresponding s! Instead, the function must be first simplified Note that ( ) is a polynomial function All higher exponents of s are equal to, i.e., etc. To the simplified structure function one can then apply the expectation operator 36

37 Demo/exercise Calculate the availability of the system below using the data given in the table 2 3 i MTTF i (hours) MDT i (hours) Hint: Use the structure function derived earlier, use availabilities of components 37

38 Models with state-dependent rates Earlier we assumed components are completely independent from each other Markov models can have state-dependent rates The dynamics (or transition rates) may depend on the state to reflect some physical causes resulting from the given state For example, if there is only one repair man, when there are many faults the repair rates are affected But still we assume that MTTF s and MDT s obey exponential distributions One can construct the associated Markov process and solve steady state via global balance equations 38

39 Example Consider parallel structure of two components. Uptimes are exp. distributed with rates κ 0 and κ. Repair rates are, correspondingly, λ 0 and λ. Also, there is only one person to repair and he spends half of the time repairing component and 2 when both are down. λ 2 0 κ 2 κ λ λ 2 /2 κ 2 κ 2 3 λ /2 System state State of component State of component Now solve equilibrium probabilities ο i. Average availability is the probability that at least one component works: A av ο 0 ο ο 2 39

40 Contents Introduction Structural system models Reliability of structures of independent repairable components Reliable network topology design 40

41 Topology design problem Topology design is the starting point in network design Think of the network as a graph with nodes connected by links Typically network topology is heavily influenced by the set of physical locations that need connectivity, so nodes are often given Also, many of the primary links between nodes are defined by the node locations In practice, design space allows to add some or few additional links and nodes Question is.. Given a network topology (nodes + links), what is a reliable network? By considering the network as a graph, reliability/availability can be formalized by the notion of graph connectivity 4

42 Graphs and k-connectivity Consider the network as a graph G(N,J) consisting of a set of nodes N and set of links J Definition: A graph is said to be connected if there exists a path between every pair of nodes in the graph. Definition: Graph G is k-edge-connected if it remains connected after removal of any k- edges. Remember: edge = link Definition: Graph G is k-vertex-connected if it remains connected after removal of any k- vertices. Remember: vertex = node Removal of node means that all links connected to the node are removed from the graph Efficient algorithms exist to check k-connectivity of the graph 42

43 Examples -edge-connected 2-edge-connected 43

44 Topology design method () Topology design objective: For redundancy, all nodes in the network need to be at least 2-(edge)- connected with probability (i.e., 5 nines ) That is, the network must be resilient to single link failures Consider a given network topology represented by graph G(N,J) Assume that link is operational with probability, but the nodes are perfectly reliable State =,, State space Ω= 0, 44

45 9. Reliability theory Topology design method (2) The structure function is then =, if network in state is 2 connected 0, otherwise And the availability is defined as = is 2 connected = ( ) Note that the size of state space is 2^J (grows exponentially!) If availability is too low new links need to be added Need to define heuristics for identifying most useful locations 45

46 Topology design method (3) Taking into account node failures Node is operational with probability We still require that all nodes must stay 2-connected with 5-nines Thus, all nodes must then be operational and = is 2 connected all nodes on all nodes on =( ) is 2 connected all nodes on The conditional probability of 2-(edge)-connectedness is evaluated as before assuming that nodes do not fail Note! This is just one version of the topology design objective and new ones can be easily defined. 46

47 THE END What you should understand/remember: what kind of things reliability theory studies basic measures, MTTF, MDT how to calculate structure function of simple systems and how to use that to calculate the availability/reliability of a system how to make Markov models of simple maintained systems and calculate the availability how can graph connectivity be used as a measure of reliability in data network topology design 47