A Fault Tree Analysis Strategy Using Binary Decision Diagrams.


 Beverly Porter
 2 years ago
 Views:
Transcription
1 A Fault Tree Analyss Strategy Usng Bnary Decson Dagrams. Karen A. Reay and John D. Andrews Loughborough Unversty, Loughborough, Lecestershre, LE 3TU. Abstract The use of Bnary Decson Dagrams (BDDs) n fault tree analyss provdes both an accurate and effcent means of analysng a system. There s a problem however, wth the converson process of the fault tree to the BDD. The varable orderng scheme chosen for the constructon of the BDD has a crucal effect on ts resultng sze and prevous research has faled to dentfy any scheme that s capable of producng BDDs for all fault trees. Ths paper proposes an analyss strategy amed at ncreasng the lkelhood of obtanng a BDD for any gven fault tree, by ensurng the assocated calculatons are as effcent as possble. The method mplements smplfcaton technques, whch are appled to the fault tree to obtan a set of 'mnmal' subtrees, equvalent to the orgnal fault tree structure. BDDs are constructed for each, usng orderng schemes most suted to ther partcular characterstcs. Quanttatve analyss s performed smultaneously on the set of BDDs to obtan the top event probablty, the system uncondtonal falure ntensty and the crtcalty of the basc events.. Introducton The Bnary Decson Dagram (BDD) method () has emerged as an alternatve to conventonal technques for performng both qualtatve and quanttatve analyss of fault trees. BDD's are already provng to be of consderable use n relablty analyss, provdng a more effcent means of analysng a system, wthout the need for the approxmatons prevously used n the tradtonal approach of Knetc Tree Theory (2). The BDD method does not analyse the fault tree drectly, but converts the tree to a bnary decson dagram, whch represents the Boolean equaton for the top event. The dffculty, however, les wth the converson of the tree to the BDD. An orderng of the fault tree varables (basc events) must be chosen and ths orderng can have a crucal effect on the sze of the resultng BDD; t can mean the dfference between a mnmal BDD wth few nodes, provdng an effcent analyss and beng able to produce any BDD at all. There s no unversal orderng scheme that can be successfully used to produce a mnmal BDD for all fault trees; ndeed no scheme has been found that wll produce a BDD (mnmal or otherwse) for some large fault trees. Emphass n the research has now turned to applyng alternatve technques that wll ncrease the lkelhood of obtanng a BDD for any gven fault tree, by ensurng the assocated calculatons are as effcent as possble.
2 In ths paper, an analyss strategy s proposed whch mplements these requrements. The ntal stage combnes two smplfcaton technques that have been shown to be advantageous n the constructon of BDDs: Faunet reducton (3), and lneartme modularsaton (4). The reducton technque reduces the fault tree to ts mnmal logc form, whlst modularsaton dentfes ndependent subtrees (modules) exstng wthn the tree that can be analysed separately. Ths results n a set of 'mnmal' fault trees, equvalent to the orgnal fault tree structure. A neural network s used to select the most approprate orderng scheme (5,6) for each ndependent module of the fault tree, based upon ts ndvdual characterstcs. BDDs are obtaned for each module n separate computatons, culmnatng n a set of BDDs, whch together represent the orgnal system. Quanttatve analyss s performed smultaneously on the set of BDDs to obtan the top event probablty, the system uncondtonal falure ntensty and the crtcalty of the basc events. Each of these stages s descrbed n more detal n the followng sectons and demonstrated throughout wth the use of an example fault tree. 2. Smplfcaton of the Fault Tree Structure Two preprocessng technques are appled to the fault tree n order to obtan the smallest possble subtrees, so that the process of constructng the BDDs becomes smple and effcent. The frst stage of preprocessng s Faunet reducton, a technque that s used to restructure the fault tree to ts most concse form. Once ths has been appled however, t s possble to smplfy the analyss further by dentfyng ndependent subtrees (modules) wthn the fault tree that can be treated separately. The lneartme algorthm s an extremely effcent method of modularsaton and forms the second stage of the fault tree preprocessng. Ths results n a set of ndependent fault trees each wth the smplest possble structure, whch together descrbe the orgnal system. 2. Faunet Reducton FAUNET reducton s a technque that s used to reduce the fault tree to ts mnmal form, so elmnatng any nose from the system, wthout alterng the underlyng logc. Its effectveness has been demonstrated wth ts applcaton to a large set of fault trees, where t decreased the sze of the resultng BDDs by approxmately 5%. The method conssts of three stages: Contracton Subsequent gates of the same type are contracted to form a sngle gate. Ths gves a fault tree wth an alternatng sequence of AND gates and OR gates. 2
3 Factorsaton Pars of events that always occur together n the same gate type are dentfed. They are combned to form a sngle complex event, whch are gven a numercal label from 2 upwards. Extracton The followng two structures are dentfed and replaced: restructure X X X2 X X3 X2 X3 restructure X X X2 X X3 X2 X3 Fgure : The extracton procedure The above three steps are repeated untl no further changes are possble n the fault tree, resultng n a more compact representaton of the system. For example, consder the fault tree llustrated n Fgure 2. 3
4 Top G G2 G3 G4 G5 G6 a G7 G8 b G9 c d G G e G2 a f g h d G3 G4 j k m G5 n p q n k e G6 r s Fgure 2: Example fault tree Upon applcaton of the Faunet reducton technque to ths tree, we obtan the much smaller fault tree shown n Fgure 3. The correspondng complex event data s shown n Table. Complex Event, X c Value of the gate Event Event 2 2 AND g h 2 OR p q 22 OR r s 23 OR 2 b 24 OR j 2 25 AND 24 k 26 OR 25 n Table : The complex event data after Faunet reducton. 4
5 Top G2 26 G3 G4 G6 a G7 23 G9 c d e G2 a f d m G5 22 e Fgure 3: The resultng fault tree after the applcaton of Faunet reducton Havng reduced the fault tree to a more concse form, we now consder the second preprocessng technque of modularsaton. 2.2 Modularsaton The modularsaton procedure does not alter the structure of the tree, but detects modules. A module of a fault tree s a subtree that s completely ndependent from the rest of the tree. That s, t contans no basc events that appear elsewhere n the fault tree. The advantage of dentfyng these modules s that each one can be analysed separately from the rest of the tree. The results from subtrees dentfed as modules are substtuted nto the hgherlevel fault trees where the modules occur. Usng the lneartme algorthm, the modules can be dentfed after two depthfrst traversals of the fault tree. The frst of these performs a stepbystep traversal recordng for each gate and event, the step number at the frst, second and fnal vsts to that node. The step number of the second vst to each event s always equvalent to the step number of the frst vst to that event. To demonstrate ths, refer to the fault tree n Fgure 3. Startng at the top event and progressng through the tree n a depthfrst manner (and consderng the event nputs to a gate before any gate nputs), the gates and events are vsted n the order shown n Table 2. Each gate s vsted at least twce: once on the way down the tree and agan on the way back 5
6 up the tree. Once a gate has been vsted, t can be vsted agan, but the depthfrst traversal beneath that gate s not repeated. The step numbers of the vsts to the gates and events are shown n Tables 3 and 4. Step number Node Top 26 G2 a G6 e G2 m G5 22 e Step number Node G5 G2 G6 G7 a f G7 G2 G3 23 G9 Step number Node d G9 G3 G4 c d G4 Top Table 2: Order n whch the gates and events are vsted n the depthfrst traversal of the fault tree n Fgure 3. The second pass through the tree fnds the maxmum (Max) of the last vsts and the mnmum (Mn) of the frst vsts of the descendants (any gates and events appearng below that gate n the tree) of each gate; these values are shown n Table 3. Gate Top G2 G3 G4 G6 G7 G9 G2 G5 Vst Vst Last vst Mn Max Table 3: Data for the fault tree gates. Event a c d e f m Vst Vst Last vst Table 4: Data for the fault tree events. The prncple of the algorthm s that f any descendant of a gate has a frst vst step number smaller than the frst vst step number of the gate, then t must also occur beneath another gate. Conversely, f any descendant has a last vst step number greater than the second vst step number of the gate, then agan t must occur elsewhere n the tree. Therefore, a gate can be dentfed as headng a module only f: 6
7 The frst vst to each descendant s after the frst vst to the gate and The last vst to each descendant s before the second vst to the gate Therefore, the followng gates can be dentfed as modules: Top, G2 and G6 For completeness, the top event (Top) s ncluded n ths lst, even though t wll always be a module of the fault tree. The occurrences of these subtrees are replaced by the sngle modular events, whch are named n the same way as complex events (.e. they take on the next avalable value above 2). Top  27, G228, G629 Three separate fault trees as shown n Fgure 4 now replace the fault tree n Fgure 3. Top G2 G6 28 G3 26 G4 29 a G7 e G2 23 G9 c d a f m G5 d e 22 (a)  Module 27 (b)  Module 28 (c)  Module 29 Fgure 4: The three modules obtaned from the fault tree shown n Fgure 3 Havng reduced the fault tree to ts mnmal form and dentfed all the ndependent modules, the preprocessng of the fault tree s complete and the next step s to obtan the assocated BDDs. 3. Obtanng the Assocated Bnary Decson Dagrams A BDD must be constructed for each of the modules. As they all have dfferent propertes, usng the same varable orderng scheme for each may not be approprate. Therefore an orderng scheme s selected for each module based on ts unque characterstcs through the 7
8 use of a preprogrammed neural network. The neural network selects the best orderng scheme from eght possble alternatves, whch nclude both structural and weghted schemes. The BDD for each module s then obtaned usng the varable orderng determned by the approprate scheme. Consderng the module '27' n Fgure 4(a), the modfed prorty depthfrst scheme (a depthfrst leftrght exploraton, consderng the most repeated events under any gate frst and consderng gates wth only event nputs before any others) was dentfed as the most sutable by the neural network. Ths gves the followng orderng: 28 < 26 < d < c < 23 < The BDD obtaned usng ths orderng s shown n Fgure 5. It s known as the 'prmary' BDD, as t represents the top event and s used to calculate the system unavalablty. F 28 F2 26 F3 d F4 23 c F5 F6 F7 23 Fgure 5: The prmary BDD (module '27') obtaned from the orderng 28<26<d<c<23< Each module s treated n the same manner, wth ts BDD nodes labelled consecutvely from the one prevously constructed n order to avod confuson. The BDDs were constructed for the two remanng modules n the example. The modfed depthfrst scheme (a depthfrst leftrght exploraton, consderng the most repeated events under any gate frst) was used for module '28', producng the orderng: a<29<f 8
9 The modfed topdown scheme (a leftrght, topdown exploraton of the tree, consderng repeated events frst) was used for module '29' gvng: e<m<22 The resultng BDDs, whch also llustrate the node labellng, are shown n Fgure 6. F F8 a F9 F m e F2 (a)  BDD for module '28' (b) BDD for module '29' Fgure 6: The BDDs for modules '28' and '29', demonstratng node labellng Once the complete set of BDD data has been computed, the quanttatve analyss can begn. 4. Quanttatve Analyss Quanttatve analyss performed on BDDs s an exact and effcent procedure (7), whch allows us to determne many propertes of the system under consderaton. To date, the methods have only been used on BDDs consstng entrely of basc events. As the technques of reducton and modularsaton produce both complex and modular events, the methods need to be extended to consder these extra factors. The followng sectons descrbe the extenson of the current methods for dealng wth BDDs to those contanng complex events and/or modular events. The am of the analyss s to obtan not only the top event probablty and uncondtonal falure ntensty, but to be able to extract the crtcalty functon for the basc events that contrbute to the complex events and modules. Ths s essental, as although we may use reducton and modularsaton to help construct the BDDs, we must be able to analyse the system n terms of ts orgnal components. 4. System Unavalablty The probablty of occurrence of the top event (Q sys ) s calculated by summng the probabltes of the dsjont paths through the prmary BDD. A depthfrst algorthm can perform ths calculaton very effcently; further dscusson of ths procedure can be found n reference 7. 9
10 The unavalablty of each encoded event s requred for ths calculaton. Therefore, the probabltes of the complex and modular events must be obtaned from the basc event data. Determnng the unavalablty of complex events s straghtforward, as they are only a combnaton of two component events. The calculaton depends on whether the events were combned under an AND gate or an OR gate, but f we call the complex event x c and ts consttuent events x and x 2, then we can say: AND Gate: q c = q q 2 (a) OR Gate: q c = q + q 2  q q 2 (b) The probabltes of the complex events are calculated as they are formed, makng the process as effcent as possble. The calculaton of the modular events' probabltes s effectvely that of fndng the probablty of occurrence of the 'top event' of each of the modules. Agan, a depthfrst algorthm s used (as shown n Fgure 7), whch can repeatedly call tself should further modular events be located wthn the module tself. Thus, the unavalablty of modules encodng only basc and complex events wll necessarly be evaluated frst. module_prob(f) { F = te(x, J, K) Consder '' branch: f (J = ) then po [F] = else po [F] = module_prob(j) Consder '' branch: f (K = ) then po [F] = else po [F] = module_prob(k) Calculate and return probablty value of node: f (x s a modular event wth unknown probablty and module root node R) then q = module_prob(r) probablty[f] = q.po [F] + (q ).po [F] } return(probablty[f]) Fgure 7: The algorthm for calculatng the probablty of a module. Havng obtaned the probabltes of all complex and modular events, the system unavalablty can easly be determned.
11 4.2 System Uncondtonal Falure Intensty The system uncondtonal falure ntensty, w sys (t), defned as the probablty that the top event occurs at t per unt tme, s gven by: w sys (t) = G ( q (t)).w(t) (2) where G (q(t)) s the crtcalty functon for each component and w (t) s the component uncondtonal falure ntensty The crtcalty functon s defned as the probablty that the system s n a crtcal state wth respect to component and that the falure of component would then cause the system to go from a workng to a faled state. Therefore: G ( q(t)) = Q(, q(t)) Q(, q(t)) (3) where Q(,q(t)) s the probablty of system falure wth q (t)=and Q(,q(t)) s the probablty of system falure wth q (t)=. An effcent method of calculatng the crtcalty functon from the BDD (7) consders the probabltes of the path sectons of the BDD up to and after the nodes n queston, resultng n the followng expresson: G ( q (t)) = pr x ( q(t))[po x ( q(t))  po ( q(t))] x (4) n where: prx ( q (t))  the probablty of the path secton from the root vertex to the node x (set to one for the root vertex) po ( q(t)) x  the probablty of the path secton from the '' branch of node x to a termnal node po ( q(t)) x  the probablty of the path secton from the '' branch of node x to a termnal node n  all nodes for varable x n the BDD. For a sngle BDD encodng only basc events, one pass of the BDD s requred to calculate prx (q), po x (q) and po x (q) for each node (subsequently referred to as the 'path probabltes' of a node), from whch the crtcalty functon of each basc event can be determned, leadng to the evaluaton of the system falure ntensty. However, ths method does not take account of complex and modular events. It s possble to calculate w sys by consderng only the events encoded n the prmary BDD, but ths requres not only the crtcalty of the modular and complex events but also ther falure ntenstes. Although these are relatvely smple to calculate, they are values that have no further use n the analyss. Instead, we calculate the crtcalty functons of each of the basc events and use these together wth ther uncondtonal
12 falure ntenstes to calculate w sys. Ths also allows analyss of the contrbutons to system falure through component or basc event mportance measures. G (q) s Brnbaum's measure of component mportance. It s also a major element requred to evaluate the crtcalty measure of component mportance. The crtcalty functons of the basc events wthn the prmary BDD are stll calculated at the end of the analyss once the path probabltes have been found for the nodes of the prmary BDD. The calculaton of the crtcalty functons of the basc events ncorporated wthn complex events and modules are descrbed n the followng sectons. 4.3 Crtcalty of Basc Events wthn Complex Events Once the path probabltes are known for a complex event node, the complex event must be further analysed by assgnng approprate values of prx (q), po x (q) and po x (q) to ts component events. These are requred so that the crtcalty functons of the basc events can be evaluated. Consder the complex event X c, shown n Fgure 8. pr c Xc po c po c Fgure 8: A complex event node wthn a BDD The two events that make up ths complex event are ether joned by an AND gate or an OR gate, whch gves the possble te (fthenelse () ) structures and correspondng BDDs as shown n Fgure 9. AND: X c = X.X 2 OR: X c = X + X 2 X c = te(x, te(x 2,, ), ) X c = te(x,, te(x 2,, )) X X X 2 X 2 Fgure 9: The possble BDD structures of a complex event 2
13 The complex event node effectvely replaces one of these structures n the orgnal BDD  ths could be ether the prmary BDD or the BDD of a module. In order to evaluate the path probabltes of the nodes encodng these component events, we smply replace any termnal '' branches wth the probablty of the paths below the '' branch of the complex node and the termnal '' branches wth the probablty of the paths below the '' branch of the complex node. The probablty of the paths before the root vertex does not have the usual value of, but takes on the value of prx (q) of the complex event node. Ths s shown n Fgure. pr c pr c X X X 2 po c po c X 2 po c po c po c po c (a) X c = X. X 2 (b) X c = X + X 2 Fgure : The complex event structure Usng Fgure, we can calculate the values of varables X and X 2 : prx (q), po x (q) and po (q) x for the X : AND: OR: pr = prc (5) X : prc po + c po = po (7) pr = () c po = po (2) po + X 2 : 2 prc. q = q 2.poc ( q 2 ). poc (6) = q2.poc ( q2 ). poc (3) pr = (8) X 2 : pr prc.( q) c po 2 = po (9) c po 2 = po () 2 = (4) c po 2 = po (5) c po 2 = po (6) As the events X and X 2 may be ether basc events or other complex events, ths process s repeated untl values have been calculated for all contrbutng basc events. The crtcalty functons of the basc events are calculated as they are encountered, usng Equaton 4. The algorthm mplementng ths method s shown n Fgure. 3
14 complex_calc(x c) f (<op> = OR) { { x c = x <op> x 2 po [x ] = po [x c] po [x Calculate probabltes: ] = q 2.po [x c] + (q 2).po [x c] pr[x 2] = pr[x c].(q ) pr[x ] = pr[x c] } po [x 2] = po [x c] po [x 2] = po If contrbutng events are basc then calculate crtcalty, [x c] otherwse call functon agan: f (<op> = AND) f (x { s a basc event) then G = G + pr[x ].(po [x ]  po [x ]) po [x ] = q 2.po [x c] + (q 2).po else complex_calc(x [x ) c] po [x ] = po [x c] f (x 2 s a basc event) then G 2 = G 2 + pr[x 2].(po [x 2]  po [x 2]) pr[x 2] = pr[x c].q else complex_calc(x 2) } } Fgure : The calculaton of the crtcalty functons of basc events wthn complex events. Any complex event may appear more than once n the BDD, resultng n new values of prx (q), po x (q) and po x (q) beng calculated for ts component events on each occason. The crtcalty functon for each of the contrbutng basc events must be n stages, usng the newly assgned values each tme. Once ths addtonal crtcalty value has been calculated for each of the contrbutng basc events, t s added to the current value so that t s calculated as the analyss proceeds, rather than as a separate procedure at the end of the analyss as s the case for the basc events n the prmary BDD. 4.4 Crtcalty of Basc Events wthn Modules Modular events are dealt wth n a smlar way to complex events. Once the path probabltes of the modular event node are known, the module s further analysed to determne the path probabltes of ts component nodes. These probabltes must be calculated as they would have been, had the module not been replaced by the sngle modular event. In order to do ths, the values of po x (q) and po x (q) of the modular event replace the termnal '' and '' branches, and the probablty of the paths before the root vertex of the module s assgned the value of prx (q) of the modular event. Ths s demonstrated n Fgure 2. 4
15 Module X m : pr m X pr m X X m + X 2 X 2 po m X 3 po m po m po m X 3 po m po m Fgure 2: Replacng a modular event wth the entre module structure. Unlke complex events, the structure of modules s not fxed. They can contan any number of events (basc, complex, or ndeed other modular events), connected by any number of gates. Therefore, the probabltes are assgned by performng a pass through the whole BDD, a process that s capable of dealng wth any structure. The crtcalty functons of the basc events are then calculated accordng to equaton 4. As wth the complex events, the calculatons requred to obtan the path probabltes for the nodes wthn the module must be repeated for each occurrence of the modular event n the BDD. The values are then used to calculate the addtonal contrbutons to the crtcalty functons of the basc events that arse due to the further occurrences of the modular event. Ths can be seen n the followng example. Havng determned the crtcalty functon of each basc event, the system falure ntensty can be evaluated usng equaton 2 and any further mportance measure analyss undertaken. 4.5 Quanttatve Analyss Example Ths quanttatve analyss can be demonstrated usng the set of example BDDs obtaned n Secton 3. The basc event data s shown n Table 5. Event a b c d e f g h q w.94 x x x x x x x x x 6 Event j k m n p q r s q w 3.92 x x x x 6.2 x x x x 6 Table 5: Basc event data for the example fault tree. 5
16 System Unavalablty The probabltes of the complex events are calculated accordng to equatons a and b, as the complex events are formed. These are shown n Table 6. Complex Event, X c Unavalablty of the complex event, q c.8 x x x x x x x 3 Table 6: Complex event data. The probabltes of occurrence of modules '28' and '29' are also needed and are evaluated by calculatng the probablty of the 'top event' of each module. Consderng module '29', the dsjont paths through the BDD are:. e.m 2. e.m.22 Therefore the probablty of the module s gven by: q 29 = q e. q m + q e.(  q m ).q 22 =.4 x 4 Smlarly for module '28', the dsjont paths through the BDD are. a 2. a.29 whch gves: q 28 = q a + (  q a ).q 29 = 3. x 3 Havng obtaned the probabltes of each of the events wthn the prmary BDD, the top event probablty can be calculated. The dsjont paths through the prmary BDD are: d d d.c.23 from whch we can calculate the system unavalablty as: Q sys = q 28.q 26.q d.q 23 + q 28.q 26.q d.(  q 23 ).q + q 28.q 26.(  q d ).q c.q 23 = 2.77 x 9 6
17 System Uncondtonal Falure Intensty The calculatons for the system falure ntensty start by determnng the path probabltes prx (q), po x (q) and po (q) x n Table 7. for the nodes of the prmary BDD. The calculatons are shown Node Varable One branch Zero branch prx (q) (q) po x (q) po x F 28 F2. F2 26 F3 F3 d F4 F5 F4 23 F6 F5 c F7 F6 F7 23 q 26.po [F2] + (q 26). po 7. [F2] = 8.89 x pr[f]*q 28 = q d.po [F3] + (q d). 3. x 3 po 4. [F3] =.73 x pr[f2]*q 26 = q 23.po [F4] + (q 23)..6 x 5 po [F4] =.36 x 2 q c.po [F5] + (q c). po [F5] = 3.74 x 5 pr[f3]*q d =.6 x 7. q.po [F6] + (q ). po [F6] = 9. x 3 pr[f3].(q d) = q 23.po [F7] + (q 23)..58 x 5 po 3. [F7] = 4.68 x prf4*(q 23) 7.. =.59 x pr[f5].q c = x Table 7: Results of the quanttatve analyss appled to the prmary BDD. The values of prx (q), po x (q) and po x (q) for the basc events wthn the complex events can be calculated accordng equatons 56. Dealng wth the frst occurrence of the complex event '23' at node F4, t can be expanded n terms of ts basc events to gve the values shown n Table 8. The crtcalty functons of the basc events 'b', 'g' and 'h' can be evaluated at ths stage and are also shown n Table 8. Complex event, X C Gate type Component event prx (q) po x (q) po x (q) of the component event Crtcalty 23 OR X = 2 pr 23 =.6 x 7 po 23 =. X 2 = b q b.po 23 + (q b). po 23 =.35 x 2  pr 22.(q 2) =.6 x 7 po 23 =. po 23 = 9. x x 7 2 AND X = g pr 2 =.6 x 7 q h.po 2 + (q h). po 2 = 2.53 x 2 po 2 =.35 x x 9 X 2 = h pr 2.q g = 2.4 x 9 po 2 =. po 2 =.35 x x 9 Table 8: Calculatng the crtcalty functons of the basc events wthn event '23'. 7
18 The calculatons are repeated for the second occurrence of ths complex event at node F7. Ths results n addtonal crtcalty values for the basc events whch are added together to gve the total crtcalty functon: G b =.58 x x 7 = 2.85 x 7 G g =.89 x x 9 = 3.4 x 9 G h = 2.36 x x 9 = 4.25 x 9 The complex event 26 appears only once n the prmary BDD, and expandng t out n terms of ts basc events gves the followng crtcalty functons: G j = 3.7 x 9, G k = 9.88 x 9, G n = 5.4 x 7, G p = 3.72 x 9, G q = 3.72 x 9 Module 28, whch s encoded n node F, s analysed to obtan the path probabltes of ts component nodes. The probabltes po x (q) and po x (q) of the modular event (8.89 x 7 and. respectvely), replace the termnal '' and '' branches and the value prx (q) of the modular event s assgned to the module's root vertex. The resultant calculatons are shown n Table 9. Node Varable One branch Zero branch prx (q) (q) po x (q) po x Crtcalty F8 a F9 pr[f] =. F9 29 po q [F] = 29.po [F9] x 7 (q 29).po [F9] 8.89 x 7 =.2 x  pr[f8]*q a = 3. x 3 po [F] = 8.89 x 7 po [F] =.  Table 9: Results of the quanttatve analyss appled to module '28'. Node F9 encodes another module '29', whch must also be analysed n terms of ts basc events. The path probabltes are calculated for each node gvng the results shown n Table. Node Varable One branch Zero branch prx (q) (q) po x (q) po x Crtcalty F e F F m F2 F2 22 q pr[f9] = m.po [F] + 3. x 3 (q m).po [F2] po [F9] =. 8.7 x  = 2.9 x 8 pr[f].q e = po [F9] = x 5 x 7 q 22.po [F2] + (q 22).po [F2] 9.7 x 2 =.59 x 8 pr[f].q m =.58 x 7 po [F9] = 8.89 x 7 po [F9] =.  Table : Results of the quanttatve analyss appled to module '29'. 8
19 The complex event 22 s expanded out n terms of ts basc events to obtan the crtcalty functons: G r =.39 x 3, G s =.38 x 3 The only crtcalty functons that reman to be calculated are those for the basc events wthn the prmary BDD: G c = 7.4 x 8, G d = 2.7 x 7, G =.59 x 7 The system falure ntensty s calculated accordng to Equaton 2 usng the basc events' falure ntenstes and crtcalty functons to gve: W sys =.8 x  5. Conclusons Ths paper has ntroduced an analyss strategy for dealng wth the effcent constructon of BDDs from fault trees. The resultng BDDs can encode both complex and modular events, for whch the necessary quanttatve analyss has been developed. It has also been shown how the analyss proceeds to enable the calculaton of the top event probablty and the system uncondtonal falure ntensty. In addton, a method to extract the crtcalty functons for the basc events, whch are consttuents of both complex events and modules, has been developed. Ths enables the system to be analysed n terms of ts orgnal components. Further quanttatve analyss s possble; the methods could be extended to nclude the calculaton of other mportance measures for the basc events. 6. References. Rauzy, A. New Algorthms for Fault Tree Analyss, Relab. Engng. Syst. Safety, 4, pp232, Vesely, W. E., "A Tme Dependent Methodology for Fault Tree Evaluaton", Nuclear Eng and Des, 3, pp33736, Platz, O. and Olsen J. V. FAUNET: A Program Package for Evaluaton of Fault Trees and Networks, Research Establshment Rs! Report No 348, DK4 Rosklde, Denmark, Sept Dutut, Y. and Rauzy, A. A LnearTme Algorthm to fnd Modules of Fault Trees, IEEE Trans. Relablty, 45, No. 3, Bartlett, L. M. Varable Orderng Heurstcs for Bnary Decson Dagrams, Doctoral Thess, Loughborough Unversty, 2 9
20 6. Bartlett, L. M and Andrews, J. D. "Selectng an Orderng Heurstc for the Fault Tree Bnary Decson Dagram Converson Process usng Neural Networks", accepted for Publcaton n IEEE Trans. Relablty. 7. Snnamon, R. M. and Andrews, J. D. Quanttatve Fault Tree Analyss usng Bnary Decson Dagrams, Jour. Europ en des Systemes Automats s, 3, 996 2
Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More information1.1 The University may award Higher Doctorate degrees as specified from timetotime in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages  n "Machnes, Logc and Quantum Physcs"
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationSolutions to the exam in SF2862, June 2009
Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodcrevew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCullochPtts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationSingle and multiple stage classifiers implementing logistic discrimination
Sngle and multple stage classfers mplementng logstc dscrmnaton Hélo Radke Bttencourt 1 Dens Alter de Olvera Moraes 2 Vctor Haertel 2 1 Pontfíca Unversdade Católca do Ro Grande do Sul  PUCRS Av. Ipranga,
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationFormulating & Solving Integer Problems Chapter 11 289
Formulatng & Solvng Integer Problems Chapter 11 289 The Optonal Stop TSP If we drop the requrement that every stop must be vsted, we then get the optonal stop TSP. Ths mght correspond to a ob sequencng
More informationStudy on CET4 Marks in China s Graded English Teaching
Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More informationLevel Annuities with Payments Less Frequent than Each Interest Period
Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annutymmedate 2 Annutydue Symoblc approach
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 200502 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 148537801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationLogical Development Of Vogel s Approximation Method (LDVAM): An Approach To Find Basic Feasible Solution Of Transportation Problem
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77866 Logcal Development Of Vogel s Approxmaton Method (LD An Approach To Fnd Basc Feasble Soluton Of Transportaton
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationTime Value of Money Module
Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationRELIABILITY, RISK AND AVAILABILITY ANLYSIS OF A CONTAINER GANTRY CRANE ABSTRACT
Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March RELIABILITY RIK AND AVAILABILITY ANLYI OF A CONTAINER GANTRY CRANE
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationSample Design in TIMSS and PIRLS
Sample Desgn n TIMSS and PIRLS Introducton Marc Joncas Perre Foy TIMSS and PIRLS are desgned to provde vald and relable measurement of trends n student achevement n countres around the world, whle keepng
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More informationTrafficlight a stress test for life insurance provisions
MEMORANDUM Date 006097 Authors Bengt von Bahr, Göran Ronge Traffclght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationAn MILP model for planning of batch plants operating in a campaignmode
An MILP model for plannng of batch plants operatng n a campagnmode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafeconcet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño
More informationAvailabilityBased Path Selection and Network Vulnerability Assessment
AvalabltyBased Path Selecton and Network Vulnerablty Assessment Song Yang, Stojan Trajanovsk and Fernando A. Kupers Delft Unversty of Technology, The Netherlands {S.Yang, S.Trajanovsk, F.A.Kupers}@tudelft.nl
More informationFORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER
FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 276957909 Introducton he convectve heat
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationEnterprise Master Patient Index
Enterprse Master Patent Index Healthcare data are captured n many dfferent settngs such as hosptals, clncs, labs, and physcan offces. Accordng to a report by the CDC, patents n the Unted States made an
More informationExamensarbete. Rotating Workforce Scheduling. Caroline Granfeldt
Examensarbete Rotatng Workforce Schedulng Carolne Granfeldt LTH  MAT  EX   2015 / 08   SE Rotatng Workforce Schedulng Optmerngslära, Lnköpngs Unverstet Carolne Granfeldt LTH  MAT  EX   2015
More informationNew bounds in BalogSzemerédiGowers theorem
New bounds n BalogSzemerédGowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A
More informationThe circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:
polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationInterleaved Power Factor Correction (IPFC)
Interleaved Power Factor Correcton (IPFC) 2009 Mcrochp Technology Incorporated. All Rghts Reserved. Interleaved Power Factor Correcton Slde 1 Welcome to the Interleaved Power Factor Correcton Reference
More informationIDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM
Abstract IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Alca Esparza Pedro Dept. Sstemas y Automátca, Unversdad Poltécnca de Valenca, Span alespe@sa.upv.es The dentfcaton and control of a
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationPeriod and Deadline Selection for Schedulability in RealTime Systems
Perod and Deadlne Selecton for Schedulablty n RealTme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng
More informationControl Charts for Means (Simulation)
Chapter 290 Control Charts for Means (Smulaton) Introducton Ths procedure allows you to study the run length dstrbuton of Shewhart (Xbar), Cusum, FIR Cusum, and EWMA process control charts for means usng
More informationCS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements
Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there
More informationSUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.
SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976761000
More informationStochastic Protocol Modeling for Anomaly Based Network Intrusion Detection
Stochastc Protocol Modelng for Anomaly Based Network Intruson Detecton Juan M. EstevezTapador, Pedro GarcaTeodoro, and Jesus E. DazVerdejo Department of Electroncs and Computer Technology Unversty of
More informationSketching Sampled Data Streams
Sketchng Sampled Data Streams Florn Rusu, Aln Dobra CISE Department Unversty of Florda Ganesvlle, FL, USA frusu@cse.ufl.edu adobra@cse.ufl.edu Abstract Samplng s used as a unversal method to reduce the
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationA ReplicationBased and Fault Tolerant Allocation Algorithm for Cloud Computing
A ReplcatonBased and Fault Tolerant Allocaton Algorthm for Cloud Computng Tork Altameem Dept of Computer Scence, RCC, Kng Saud Unversty, PO Box: 28095 11437 RyadhSaud Araba Abstract The very large nfrastructure
More informationMultiplePeriod Attribution: Residuals and Compounding
MultplePerod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens
More informationExhaustive Regression. An Exploration of RegressionBased Data Mining Techniques Using Super Computation
Exhaustve Regresson An Exploraton of RegressonBased Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationDamage detection in composite laminates using cointap method
Damage detecton n composte lamnates usng contap method S.J. Km Korea Aerospace Research Insttute, 45 EoeunDong, YouseongGu, 35333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The contap test has the
More informationFuzzy TOPSIS Method in the Selection of Investment Boards by Incorporating Operational Risks
, July 68, 2011, London, U.K. Fuzzy TOPSIS Method n the Selecton of Investment Boards by Incorporatng Operatonal Rsks Elssa Nada Mad, and Abu Osman Md Tap Abstract Mult Crtera Decson Makng (MCDM) nvolves
More informationCS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering
Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that
More informationAdaptive Fractal Image Coding in the Frequency Domain
PROCEEDINGS OF INTERNATIONAL WORKSHOP ON IMAGE PROCESSING: THEORY, METHODOLOGY, SYSTEMS AND APPLICATIONS 222 JUNE,1994 BUDAPEST,HUNGARY Adaptve Fractal Image Codng n the Frequency Doman K AI UWE BARTHEL
More informationRESEARCH ON DUALSHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.
ICSV4 Carns Australa 9 July, 007 RESEARCH ON DUALSHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract
More informationNEUROFUZZY INFERENCE SYSTEM FOR ECOMMERCE WEBSITE EVALUATION
NEUROFUZZY INFERENE SYSTEM FOR EOMMERE WEBSITE EVALUATION Huan Lu, School of Software, Harbn Unversty of Scence and Technology, Harbn, hna Faculty of Appled Mathematcs and omputer Scence, Belarusan State
More informationStochastic epidemic models revisited: Analysis of some continuous performance measures
Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,
More information