CODETTA RAITERI D

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CODETTA RAITERI, Daniele

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Parametric Dynamic Fault Tree and its Solution through ModularizationDaniele Codetta RaiteriDipartimento di Informatica, Universita di TorinoCorso Svizzera 185, 10149 Torino, Italycodetta@di.unito.it1. IntroductionThe Fault Tree (FT) [1] is a widespread model for thedependability analysis of complex systems and allows tomodel how combinations of component failures can deter-mine the failure of subsystems or of the whole system; anexample is reported in Fig. 1; the nodes can be failure eventsor gates: failure events are represented as a bar and areequivalent to a boolean variable whose value is 0 until thefailure event has not yet occured, or 1 when the event hasoccured; gates are connected to events by means of arcs andhave several input events and a unique output event, con-nected respectively below and above the gate. The eventsthat are represented as a bar with a circle, are called basicevents (BE) and correspond to the failure events of physi-cal components of the system; the occurence time of suchevents is ruled by a probability distribution associated toeach BE, typically an exponential distribution whose pa-rameter is called failure rate. The internal events, repre-sented as an empty bar, correspond to subsystems failures;an internal event is the output of a gate and occurs when aparticular combination (determined by the type of the gate)of the gate input events occurs. While BEs can not be theoutput of any gate, there is a unique event called Top Event(TE), indicated by a black bar, that can only be the output ofa gate; TE represents the failure of the whole system. A FTmay contain several kinds of gates; in the standard versionof this model, BEs are considered as statistically indipen-dent and three gates corresponding to the AND, OR, K of Nboolean functions are defined; such a model can be easilyanalyzed in a combinatorial way [1], but it suffers from theinability to represent dependencies among failure events; inorder to overcome this limitation, some new gates called dy-namic gates were introduced and they are: Priority AND (PAND) - it fails if all of its input eventsfail and in a specified order (from left to right); Functional Dependency Gate (FDEP) - it forces a setof dependent events to occur when a particular eventcalled Trigger occurs; Sequence Enforcing Gate (SEQ) - it forces a set ofevents to occur in a specified order (from left to right); Warm Spare Gate (WSP) - it models the presence of aset of spare components able to replace one or moremain components when one of them fails; the spareschange their failure rate when turning from the dor-mant state (stand-by) to the working state.The resulting model is called Dynamic Fault Tree (DFT)[2]. The introduction of dynamic gates changes the wayto perform the system dependability analysis: while stan-dard FTs are solved in a combinatorial way, DFTs requirethe state space analysis. Another evolution of the modelconcerns the way to represent the system redundanciesand symmetries compactly; this purpose is achieved by theParametric Fault Tree (PFT) [3]; using PFT, identical sub-systems are represented by a unique parameterized sub-tree whose root is a (Basic) Replicator Event (RE), indi-cated by a dotted bar; a parameter is associated to the REand its variation range (for instance, from 1 to 3) indicateshow many identical subtrees are represented in a compactway; such parameter will be associated even to the eventsinside the replicated subtree; a replicated subtree may fur-therly contain other REs using combinations of several pa-rameters. DFT and PFT formalisms can be integrated gen-erating the Parametric Dynamic Fault Tree (PDFT) [4] sup-porting both parameterization and dependencies in the fail-ure mode. As DFT, PDFT needs the state space solution.2. The solution techniqueFig. 1 is an example of DPFT: this system is composedby three subsystems called  , 	 ,  , and itfails when two of them are failed (  is the output of a Kof N gate with 	 and  );  (output of an ORgate) fails if at least one among ff , fi	 and (represented as ffifl "! with  varying from  to  ) fails;ffifl "! fails when all of its components ( #fl "! and $fl &%(')!with ' varying from  to 	 ) fail ( ffifl "! is the output of anAND gate). 	 is the output of a PAND gate, so it failsif both C and D F are failed and * failed before + , ; + ,fails if at least one +-fl./! is failed;  fails if  or 0 ,fails; 0 , (output of a WSP gate) fails when the main com-ponent 0 is failed and there are no spares 12fl435! availableto replace it.As mentioned in section 1, PDFT requires the state spacesolution because it can contain dependencies in the failuremode due to dynamic gates; state space solution may becomputationally very expensive, so in our approach we tryto apply such technique only to those subtrees that speci-fically require it. So, the first step of the adopted solutionprocess is called modularization and consists of detectingmodules, i. e. indipendent subtrees; a subtree is a moduleif both it does not share any node with other subtrees andit does not descend from a dynamic gate; modules are de-tected applying a previously realized linear time algorithm[5] that has been adapted to consider the presence of depen-dencies due to dynamic gates [6] or shared parameterizednodes [4]. Once modules are detected, they are classified asdynamic or static if they contain or not at least one dynamicgate (classication step). Dynamic modules are then classi-fied as minimal if they do not contain other modules of anykind.Only dynamic modules need a state space solution; away to perform such analysis consists of translating themin a High Level Stochastic Petri Net in the form of Stocha-stic Well-formed Net (SWN) [7]; this is done by means of aspecific translator (Fig. 3, Fig. 4). Suppose we have to per-form a transient analysis of the PDFT, in other words wehave to calculate the probability that the system is failed ata given mission time: we detach each minimal dynamic mo-dule (MDM) and we translate it in a SWN. Instead of theordinary state space, a symbolic state space, whose dimen-sions are smaller, can be generated from a SWN and ana-lized; in this way, we solve in isolation every MDM, calcu-lating its probability of failure at the given mission time (de-composition step). The combined use of parameterization,modularization, decomposition and translation of MDMs inSWN can lead to a relevant reduction of the state space di-mensions [4].The translation of a MDM in SWN is performed in thisway: for each BE and for each gate inside the MDM, a SWNis created separately; considering the MDM in Fig. 1 whoseroot is 	 , we create a distinct SWN for the BEs * (Fig.2.a) and +-fl46! (Fig. 2.b), and for the gates PAND (Fig. 2.c)and OR (Fig. 2.d); then, such SWNs are composed togetherperforming a superposition over the common places corre-sponding to MDM events; in this case, * , +7fl4/! and + , .The resulting net is shown in Fig. 3; SWN compositionalityis very flexible allowing the generation of the correspon-ding SWN even for complicated combinations of gates.Now we replace in the PDFT each detached MDM witha BE which has not a failure rate but a probability of failurethat is equal to the probability calculated on the correspon-ding module (substitution step); at this point we obtain aPFT that is no more dynamic because it does not containany dynamic gate; we call it Reduced PFT (Fig. 5) and itcan be solved in a combinatorial way after having been un-folded, i. e. converted in the equivalent FT.The whole process has been implemented following amulti-solution multi-formalism approach [8] adapting thegraphical tool called DrawNET++ [9] to the PDFT forma-lism, using the GreatSPN tool [10] as SWN solver and theSHARPE package [11] to solve the reduced PFT, once un-folded.TESYS1 SYS2 SYS3SUB(i)i: 1, 2, 3A(i) B(i, j)j: 1, 2CD_FD(k)k: 1, 2, 3EMSP(h)h: 1, 2M_FMDMMDMk = 2n = 3Figure 1. An example of DPFTDD_fail< k >< k >CC_failCD_FSYS2NO_FAILPANDNO_PANDD D_FOR< k >a)b)c)d)Figure 2. SWNs for each BE and gate in SYS2CC_failDD_failD_FSYS2NO_FAILORPANDNO_PAND< k >< k >< k >Figure 3. SWN for the module SYS2SPSP_fail_OFFMM_fail< h >< h >< h >< h >< h >< S >< S > SP_fail_ONSP_current< h >M_F< S >Figure 4. SWN for the module M FTESYS1 SYS3SUB(i)i: 1, 2, 3A(i) B(i, j)j: 1, 2ESYS2_modM_F_modk = 2n = 3Figure 5. The reduced PFT3. Future worksFuture developing on PDFT will regard mainly these as-pects: the integration of the PDFT formalism with the Re-pairable Fault Tree (RFT) [12] formalism; this issuehas alreay been partially studied in [4], but only regar-ding the WSP gate; failure and repair semantics mustbe defined in order to integrate dynamic gates and Re-pair Boxes (RB) [4][12]; for instance, if one of the in-put events of a PAND gate is repairable, such event isrepeatable: which is the failure order causing the fail-ure of the PAND gate? Currently only the quantitative analysis is available forPDFT, so a way to perform a qualitative analysis ofPDFT must be studied in order to detect minimal cutsets (MCS) or sequences, i. e. the minimal sets or se-quences of BEs leading to the failure of the whole sys-tem; this issue was already partially faced in the caseof PFT [3] and DFT [13]. The solution of the reduced PFT requires an unfoldingstep; we would like to be able to perform a combi-natorial analysis of a PFT directly, without unfoldingit; this might be done through a previous qualitativeanalysys step.References[1] W.G. Schneeweiss. The Fault Tree Method. LiLoLe Verlag,1999.[2] R. Manian, D.W. Coppit, K.J. Sullivan, and J.B. Dugan.Bridging the gap between systems and dynamic fault treemodels. Proceedings IEEE Annual Reliability and Maintain-ability Symposium, pages 105–111, 1999.[3] A. Bobbio, G. Franceschinis, R. Gaeta, and L. Portinale.Parametric fault-tree for the dependability analysis of redun-dant systems and its high level Petri net semantics. IEEETransactions Software Engineering, 29:270–287, 2003.[4] A. Bobbio and D. Codetta Raiteri. Parametric fault-trees withdynamic gates and repair boxes. In Proceedings of the An-nual Reliability and Maintainability Symposium, pages 459–465, Los Angeles, CA USA, January 2004.[5] Y. Dutuit and A. Rauzy. A linear-time algorithm to find mod-ules of fault tree. IEEE Transactions on Reliability, 45:422–425, 1996.[6] A. Anand and A. Somani. Hierarchical analysis of fault treeswith dependencies, using decomposition. Proceedings IEEEAnnual Reliability and Maintainability Symposium, pages69–75, 1998.[7] G. Chiola, C. Dutheillet, G. Franceschinis, and S. Had-dad. Stochastic Well-Formed coloured nets and multipro-cessor modelling applications. In K. Jensen and G. Rozen-berg, editors, High-Level Petri Nets. Theory and Application.Springer Verlag, 1991.[8] G.Franceschinis M.Gribaudo M.Iacono N.MazzoccaV.Vittorini. Towards An Object Based Multiformal-ism Multi-Solution Modeling Approach. Proceedings Sec-ond Workshop on Modeling of Objects, Components andAgents, 2002.[9] V.Vittorini G.Franceschinis M.Gribaudo M.IaconoC.Bertoncello. DrawNet++: a Flexible Framework forBuilding Dependability Models. Proceedings Interna-tional Conference on Dependable Systems and Networks,2002.[10] G. Chiola, G. Franceschinis, R. Gaeta, and M. Ribaudo.GreatSPN 1.7: Graphical Editor and Analyzer for Timed andStochastic Petri Nets. Performance Evaluation, 24:47–68,November 1995.[11] R.A.Sahner K.S.Trivedi A.Puliafito. Performance And Relia-bility Analysis Of Computer Systems; An Example-Based Ap-proach Using the SHARPE Software Package. Kluwer Aca-demic Publishers, 1996.[12] D. Codetta Raiteri, G. Franceschinis, M. Iacono, and V. Vit-torini. Repairable fault tree for the automatic evaluation ofrepair policies. DSN Performance and Dependability Sym-posium (to appear), July 2004.[13] Z. Tang and J.B. Dugan. Minimal cut set/sequence genera-tion for dynamic fault trees. In Procedings of the Annual Re-liability and Maintainability Symposium, Los Angeles, CAUSA, January 2004.

Parametric Dynamic Fault Tree and its Solution through Modularization

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Parametric Dynamic Fault Tree and its Solution through Modularization

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