Dept. of Math. Unvesty of Oslo Statstcal Reseach Repot No 3 ISSN 0806 3842 June 2010 Bayesan Assessment of Avalabltes and Unavalabltes of Multstate Monotone Systems Bent Natvg Jøund Gåsemy Tond Retan June 29, 2010 Abstact In the pesent pape we consde a multstate monotone system of multstate components. Followng a Bayesan appoach, the ambton s to ave at the posteo dstbutons of the system avalabltes and unavalabltes to the vaous levels n a fxed tme nteval based on both po nfomaton and data on both the components and the system. We ague that a ealstc appoach s to stat out by descbng ou uncetanty on the component avalabltes and unavalabltes to the vaous levels n a fxed tme nteval, based on both po nfomaton and data on the components, by the moments up tll ode m of the magnal dstbutons. Fom these moments analytc bounds on the coespondng moments of the system avalabltes and unavalabltes to the vaous levels n a fxed tme nteval ae aved at. Applyng these bounds and po system nfomaton we may then ft po dstbutons of the system avalabltes and unavalabltes to the vaous levels n a fxed tme nteval. These can n tun be updated by elevant data on the system. Ths genealzes esults gven n (Natvg and Ede 1987) consdeng a bnay monotone system of bnay components at a fxed pont of tme. Futhemoe, consdeng a smple netwok system, we show that the analytc bounds can be slghtly mpoved by staghtfowad smulaton technques. Keywods avalabltes Bayesan assessment multstate monotone systems unavalabltes AMS 2000 Classfcaton 62NO5, 90B25 B. Natvg J. Gåsemy Depatment of Mathematcs, Unvesty of Oslo, P.O. Box 1053 Blnden, N 0316, Oslo, Noway e-mal: bent@math.uo.no e-mal: gaasemy@math.uo.no T. Retan Depatment of Bology, Unvesty of Oslo, P.O. Box 1066 Blnden, N 0316, Oslo, Noway e-mal: tond.etan@bo.uo.no 1
1. Basc defntons and deas Let S {0, 1,..., M} be the set of states of the system; the M + 1 states epesentng successve levels of pefomance angng fom the pefect functonng level M down to the complete falue level 0. Futhemoe, let C {1,..., n} be the set of components and n geneal S, 1,..., n the set of states of the th component. We eque {0, M} S S. Hence, the states 0 and M ae chosen to epesent the endponts of a pefomance scale that mght be used fo both the system and ts components. Note that n most applcatons thee s no need fo the same detaled descpton of the components as fo the system. Let x, 1,..., n denote the state o pefomance level of the th component at a fxed pont of tme and x (x 1,..., x n ). It s assumed that the state,, of the system at the fxed pont of tme s a detemnstc functon of x;.e. (x). Hee x takes values n S 1 S 2 S n and takes values n S. The functon s called the stuctue functon of the system. We often denote a multstate system by (C, ). We stat by gvng a sees of basc defntons. Defnton 1 A system s a multstate monotone system (MMS) ff ts stuctue functon satsfes: () (x) s non-deceasng n each agument () (0) 0 and (M) M 0 (0,..., 0), M (M,..., M). Defnton 2 The monotone system (A, χ) s a module of the monotone system (C, ) ff (x) [χ(x A ), x Ac ], whee s a monotone stuctue functon and A C. Intutvely, a module s a monotone subsystem that acts as f t wee just a supecomponent. Defnton 3 A modula decomposton of a monotone system (C, ) s a set of dsjont modules {(A k, χ k )} togethe wth an oganzng monotone stuctue functon,.e. () C 1 A whee A A j j, () (x) [χ 1 (x A1 ),..., χ (x A )] [χ(x)]. Makng a modula decomposton of a system s a way of beakng t nto a collecton of subsystems whch can be dealt wth moe easly. In the followng y < x means that y x fo 1,..., n, and y < x fo some. Defnton 4 Let be the stuctue functon of an MMS and let j {1,..., M}. A vecto x s sad to be a path vecto to level j ff (x) j. The coespondng path set s gven by C j (x) { x 1}. A mnmal path vecto to level j s a path vecto x such that (y) < j fo all y < x. The coespondng path set s also sad to be mnmal. 2
Defnton 5 Let be the stuctue functon of an MMS and let j {1,..., M}. A vecto x s sad to be a cut vecto to level j ff (x) < j. The coespondng cut set s gven by D j (x) { x < M}. A mnmal cut vecto to level j s a cut vecto x such that (y) j fo all y > x. The coespondng cut set s also sad to be mnmal. We now consde the elaton between the stochastc pefomance of the system (C, ) and the stochastc pefomances of the components. Let τ be an ndex set contaned n [0, ). Defnton 6 The pefomance pocess of the th component, 1,..., n s a stochastc pocess {X (t), t τ}, whee fo each fxed t τ, X (t) s a andom vaable whch takes values n S. X (t) denotes the state of the th component at tme t. The jont pefomance pocess fo the components {X(t), t τ} {(X 1 (t),..., X n (t)), t τ} s the coespondng vecto stochastc pocess. The pefomance pocess of an MMS wth stuctue functon s a stochastc pocess {(X(t)), t τ}, whee fo each fxed t τ, (X(t)) s a andom vaable whch takes values n S. (X(t)) denotes the system state at tme t. We assume that the sample functons of the pefomance pocess of a component ae contnuous fom the ght on τ. It then follows that the sample functons of {(X(t)), t τ} ae also contnuous fom the ght on τ. Now consde a tme nteval I [t A, t B ] [0, ) and let τ(i) τ I. Defnton 7 The magnal pefomance pocesses {X (t), t τ}, 1,..., n ae ndependent n the tme nteval I ff, fo any ntege m and {t 1,..., t m } τ(i) the andom vectos {X 1 (t 1 ),..., X 1 (t m )},..., {X n (t 1 ),..., X n (t m )} ae ndependent. Defnton 8 The jont pefomance pocess fo the components {X(t), t τ} s assocated n the tme nteval I ff, fo any ntege m and {t 1,..., t m } τ(i) the andom vaables n the aay X 1 (t 1 )... X 1 (t m ). X n (t 1 )... X n (t m ) ae assocated. Fo an ntoducton to the theoy of assocated andom vaables we efe to (Balow and Poschan 1975). Defnton 9 Let 1,..., n, j 1,..., M. The avalablty, p j(i), and the unavalablty, q j(i), to level j n the tme nteval I of the th component ae gven by p j(i) P [X (s) j s τ(i)] q j(i) P [X (s) < j s τ(i)]. The avalablty, p j(i), and the unavalablty, q j(i), to level j n the tme nteval I fo an MMS wth stuctue functon ae gven by p j(i) P [(X(s)) j s τ(i)] q j(i) P [(X(s)) < j s τ(i)]. 3
Let fo 1,..., n, j 0,..., M, j(i) j(i) p j(i) p j(i) p j+1(i) p j+1(i) P [ mn s τ(i) X (s) j] P [ mn (X(s)) j]. s τ(i) Intoduce fo 1,..., n the component avalablty and unavalablty vectos p (I) { p j(i) q }j1,...,m (I) { q j(i) } j1,...,m, the n M component avalablty and unavalablty matces P (I) { p j(i) } 1,...,n j1,...,m Q (I) and the system avalablty and unavalablty vectos p (I) { p j(i) q }j1,...,m (I) { q j(i) } 1,...,n j1,...,m { q j(i) } j1,...,m. Fnally, ntoduce fo 1,..., n the component paamete vectos (I) the n (M + 1) paamete matx R (I) and the system paamete vecto (I) { j(i) } j0,...,m, { j(i) } 1,...,n j0,...,m { j(i) } j0,...,m. When I [t, t], we just dop I fom the notaton and use elablty and unelablty nstead of espectvely avalablty and unavalablty. Note that fo 1,..., n p j(i) + q j(i) 1 p j(i) + q j(i) 1. (1) Suppose now that we un K ndependent expements fo component egsteng x (k) (s) s τ(i) n the kth expement, k 1,..., K, 1,..., n. Let fo j 1,..., M, 1,..., n D 1j(I) K I[x (k) (s) j s τ(i)] D 2j(I) and fo j 0,..., M, 1,..., n K I[x (k) (s) < j s τ(i)], D j(i) K I[ mn s τ(i) x(k) 4 (s) j].
Let fo 1, 2 D (I) (D 1(I),..., D M(I) ), D (I) (D (I) 1,..., Dn (I) ). Futhemoe, let D (I) (D 0(I),..., D M(I) ) and D (I) (D (I) 1,..., D(I) n ). Suppose also that we un K ndependent expements on the system level egsteng (x (k) (s)) s τ(i) n the kth expement, k 1,..., K. Let fo j 1,..., M and fo j 0,..., M Let fo 1, 2 D (I) (D 0(I),..., D M(I) D 1j(I) D 2j(I) K I[(x (k) (s)) j s τ(i)] K I[(x (k) (s)) < j s τ(i)], D j(i) K I[ mn s τ(i) (x(k) (s)) j]. (D 1(I),..., D M(I) ). Futhemoe, let D (I) ). When I [t, t], we also dop I fom the notaton n all these data vaables and data vectos. Assume that the po dstbuton of espectvely the component avalablty and unavalablty matces, befoe unnng any expement on the component level, π ) and π(q(i) ), can be wtten as π n ) n π (p (I) ) π ) 1 whee fo 1,..., n π (p (I) 1 π (q (I) ), ) s the po magnal dstbuton of p (I) and. Hence, we assume that the π (q (I) ) s the po magnal dstbuton of q (I) components have ndependent po component avalablty vectos and ndependent po component unavalablty vectos. Note that befoe the expements ae caed though D 1j(I) s bnomally dstbuted wth paametes K and p j(i), and D 2j(I) bnomally dstbuted wth paametes K and q j(i). We assume that gven P (I), D1(I) 1,..., Dn 1(I) ae ndependent and that gven Q (I), D2(I) 1,..., Dn 2(I) ae ndependent. Hence, snce we have assumed that the components have ndependent po avalablty vectos, usng Bayes theoem the posteo dstbuton of the component avalablty matx, π D1(I) ), can be wtten as π(d 1(I) P (I) (I) )π(p ) π(d 1(I) P (I) (I) )π(p n 1 π (D 1(I) p (I) )π (p (I) ) n n 1 π (D 1(I) p (I) )π (p (I) )dp (I) 1 π D1(I) ) n 1 whee π (p (I) π (p (I) D 1(I) ), D 1(I) )dp (I) π (D 1(I) p (I) )π (p (I) ) π (D 1(I) ) ) s the posteo magnal dstbuton of p (I). Smlaly, the posteo dstbuton of the component unavalablty matx, π D2(I) ), 5
can be wtten as π D2(I) ) n 1 π (q (I) D 2(I) ). Hence, the posteo component avalablty vectos ae ndependent gven D 1(I) and the posteo component unavalablty vectos ae ndependent gven D 2(I). Now specalze I [t, t] and assume that the component states X 1,..., X n ae ndependent gven P. Snce n ths case p s a functon of P, the dstbuton, π(p (P ) D 1 ), can then be aved at. Based on po knowledge on the system level ths may be adjusted to the po dstbuton of the system elablty vecto, π 0 (p (P ) D 1 ). Note that befoe the expements ae caed though D 1j s bnomally dstbuted wth paametes K and pj. Includng the data D 1, we end up wth the posteo dstbuton of the system elablty vecto, π(p (P ) D 1, D 1 ), fo j 1,..., M. When consdeng the case I [t, t], we can nstead of P as well consde the paamete matx R and assume that the components have ndependent po vectos, 1,..., n, each havng a Dchlet dstbuton beng the natual conjugate po. Futhemoe, we assume that gven R, D 1,..., D n ae ndependent. Note that befoe the expements ae caed though D s multnomally dstbuted wth paametes K and. Hence, the posteo magnal dstbuton of gven the data D, π ( D ), s Dchlet. Futhemoe, we have n π(r D) π ( D ). 1 Hence, lfe can be made easy at the component level. Assume that the component states X 1,..., X n ae ndependent gven R. The dstbuton, π( (R ) D), s ted to be aved at. If ths s successful, based on po knowledge on the system level, t s adjusted to π 0 ( (R ) D). Ths may be possble fo smple systems. Note that befoe the expements ae caed though, D s multnomally dstbuted wth paametes K and. Hence, f π 0 ( (R ) D), as n a deam, ended up as a Dchlet dstbuton, the posteo dstbuton, π( (R ) D, D ), also would be a Dchlet dstbuton. Do not foget ths was a deam, also based on ndependent components gven R! So lfe wll at least not be easy at the system level even when I [t, t]. Fo an abtay I p (I) (I) s not a functon of just P, and q(i) not a functon of just Q (I). Hence, the appoach above fo the case I [t, t] can not be extended. In Secton 2 we dscuss two dffeent appoaches to the computaton of posteo moments fo component avalabltes and unavalabltes, the fst one genealzng an appoach gven n (Mastan and Sngpuwalla 1978). In Secton 3 we stat out by descbng ou uncetanty on the component avalabltes and unavalabltes to the vaous levels n a fxed tme nteval, based on both po nfomaton and data on the components, by the moments up tll ode m of the magnal dstbutons. Fom these moments analytc bounds on the coespondng moments of the system avalabltes and unavalabltes to the vaous levels n a fxed tme nteval ae aved at. Applyng these bounds and po system nfomaton we may then ft po dstbutons of the system avalabltes and unavalabltes to the vaous levels n a fxed tme nteval. These can n tun be updated by elevant data on the system. Ths genealzes esults gven n (Natvg and Ede 1987) consdeng a bnay monotone system of 6
bnay components at a fxed pont of tme. In Secton 4 we pesent a staghtfowad smulaton technque fo obtanng bounds that mpove the analytc bounds. Consdeng a smple netwok system, we show that the fome bounds ae slghtly bette than the latte. 2. Moments fo posteo component avalabltes and unavalabltes Based on the expeences of the pevous secton we educe ou ambtons. We stat by specfyng magnal moments ) s } and E{(q j(i) ) s } fo s 1,..., m+k, j 1,..., M of π (p (I) ) and π (q (I) ), 1,..., n. We wll fst llustate how these can be updated to gve posteo moments ) s D 1j(I) } and E{(q j(i) ) s D 2j(I) } fo s 1,..., m, j 1,..., M by usng Lemma 1 n (Mastan and Sngpuwalla 1978). Note that we loose nfomaton by condtonng on D 1j(I) nstead of D 1(I) and D 2j(I) nstead of D 2(I). Howeve, such mpoved condtonng does not wok wth ths appoach. We have 1 0 Hence, ) s D 1j(I) } (p j(i) ) s+d1j(i) 1 0 (p j(i) K D 1j(I) 0 ) s (p j(i) ) D1j(I) ( K D 1j(I) ) (1 p j(i) ( 1) (p j(i) ) K D1j(I) π (p j(i) )dp j(i) ) π (p j(i) )dp j(i). ) s D 1j(I) } K D 1j(I) 0 K D 1j(I) 0 ( K D 1j(I) ( K D 1j(I) A smla expesson s vald fo E{(q j(i) ) ( 1) ) ( 1) ) s+d1j(i) + } ) D1j(I) + } ) s D 2j(I) }. The advantage of usng ) and ths lemma s that t s applcable fo geneal po dstbutons π (p j(i) π (q j(i) ). A seous dawback s, howeve, that to ave at ) s D 1j(I) } ) s D 2j(I) } fo s 1,..., m, j 1,..., M one must specfy magnal and E{(q j(i) moments up tll ode m + K of the coespondng po dstbutons π (p j(i). ) and π (q j(i) ). Ths may be completely unealstc unless K s small. A moe ealstc altenatve s gven n the followng. Assume that the components have ndependent po paamete vectos (I), 1,..., n, each havng a Dchlet dstbuton beng the natual conjugate po. Note that befoe the expements ae caed though D (I) s multnomally dstbuted wth paametes K and (I). Hence, the posteo magnal dstbuton of (I) gven the data D (I), π ( (I) D (I) ), s Dchlet. We now have p j(i) M lj l(i) D 1j(I) M lj D l(i). (2) Hence, the posteo magnal dstbuton of p j(i) gven the data D 1j(I) Accodngly, we loose no nfomaton by condtonng on D 1j(I) 7 s beta. nstead of D 1(I).
We now assume that the po dstbuton π (p j(i) ) s beta wth paametes a j(i) and b j(i). It then follows that π (p j(i) D 1j(I) ) s beta wth paametes a j(i) + D 1j(I) and b j(i) + K D 1j(I). We have ) s D 1j(I) } (p j(i) ) aj(i) Γ(aj(I) Γ(a j(i) 1 0 (p j(i) Γ(a j(i) ) aj(i) Γ(aj(I) Γ(a j(i) +D 1j(I) + b j(i) + D 1j(I) 1 (p j(i) 0 1 (1 p j(i) Γ(a j(i) + D 1j(I) +D 1j(I) + b j(i) + D 1j(I) + K )Γ(a j(i) )Γ(a j(i) ) s Γ(a j(i) + b j(i) + K ) Γ(a j(i) + D 1j(I) )Γ(b j(i) ) bj(i) +K D 1j(I) 1 dp j(i) + D 1j(I) + s) + b j(i) + K + s) + b j(i) + K + s) + s)γ(b j(i) +s 1 (1 p j(i) + K )Γ(a j(i) )Γ(a j(i) + K D 1j(I) ) ) bj(i) +K D 1j(I) 1 dp j(i) + D 1j(I) + s) + b j(i) + K + s), + K D 1j(I) ) the ntegal beng equal to 1 snce we ae ntegatng up a beta densty wth paametes a j(i) + D 1j(I) + s and b j(i) + K D 1j(I). A smla expesson s vald fo E{(q j(i) ) s D 2j(I) }. 3. Bounds fo moments fo system avalabltes and unavalabltes Fom the magnal moments E{(p l(i) ) s D 1l(I) ) s D 2l(I) }, we de- ) s D 1(I) } and uppe bounds ve lowe bounds on the magnal moments } and E{(q l(i) on the magnal moments ) s D 2(I) } fo s 1,..., m, l 1,..., M, j 1,..., M. Smlaly, we deve lowe bounds on the magnal moments E{(q j(i) ) s D 2(I) } and uppe bounds on the magnal moments E{(q j(i) ) s D 1(I) }. Note that we now do not necessaly need the magnal pefomance pocesses of the components to be ndependent n I. Fom these bounds and po knowledge on the system level we may ft π 0 (p (I) Ths may fnally be updated to gve π(p (I) D1(I) ) and π(q (I) ) and π 0(q (I) ). D2(I) ). What we wll concentate on s how to establsh the bounds on the magnal moments of system avalabltes and unavalabltes fom the magnal moments of component avalabltes and unavalabltes. To smplfy notaton we dop the efeence to the data (D 1(I), D 2(I) ) fom expements on the component level. Let us just fo a whle etun to the case I [t, t] and assume that the component states X 1,..., X n ae ndependent gven R. Then we get p j (R ) x I[(x) j] n 1 x. Hence, snce we assume that the components have ndependent po vectos fo 1,..., n, genealzng a esult n (Natvg and Ede 1987), we get 8
E{p j (R )} n I[(x) j] E{ x } pj (E{R }), x 1 whee E{R } { E{ j }}. 1,...,n j0,...,m Accodngly, one can ave at an exact expesson fo E{p j (R )} fo not too lage systems. The pont s, howeve, that thee seems to be no easy way to extend the appoach above to gve exact expessons fo hghe ode moments of p j (R ). Hence, even when I [t, t] and component states ae ndependent gven R, one needs bounds on hghe ode moments of p j (R ). We need the followng theoem poved n (Natvg and Ede 1987). Theoem 1. If Y 1,..., Y n ae assocated andom vaables such that 0 Y 1, 1,..., n, then fo α > 0 1 n E{( Y ) α } 1 n E{(Y ) α } (3) 1 n n E{ Y } E{1 (1 Y )} 1 n E{Y }. (4) In the specal case of ndependent andom vaables Y 1 and Y 2 wth 0 Y 1, 1, 2, we have 1 2 E{( Y ) 2 } 1 2 E{(Y ) 2 }. (5) Poof: Fo the case Y 1,..., Y n bnay and α 1, Equatons (3) and (4) ae poved n Theoem 3.1, page 32 of (Balow and Poschan 1975). The poof, howeve, also woks when 0 Y 1, 1,..., n. Usng ths fact and that non-deceasng functons of assocated andom vaables ae assocated we get 1 n n E{( Y ) α } E{ (Y ) α } 1 1 n E{(Y ) α }, and Equaton (3) s poved. Equaton (4) s poved n the same way. Fnally, Equaton (5) follows snce 1 2 ( Y ) 2 1 2 1 Y 2 + 2(Y 2 1 Y 1 )(Y 2 2 Y 2 ) and that Y 1 and Y 2 ae assumed to be ndependent. Equaton (5) eveals the unpleasant fact that a symmety n Equatons (3) and (4) seems only possble fo α 1, when Y 1,..., Y n ae not bnay. 2 1 Y 2, 9
In the followng, consdeng an MMS (C, ), fo j {1,..., M} let y j k (y j 1k,..., yj nk ), k 1,..., nj be ts mnmal path vectos to level j and zj k (z j 1k,..., zj nk ), k 1,..., mj be ts mnmal cut vectos to level j and C j (yj k ), k 1,..., nj and Dj (zj k ), k 1,..., mj the coespondng mnmal path and cut sets to level j. The thee followng theoems ae taken fom (Natvg 2011). Theoem 2. Let (C, ) be an MMS and let fo j 1,..., M l j(i) ) max l j(i) ) max p y 1 k n j C j (yj k ) j k (I) q z j k +1(I) 1 k m j. D j (zj k ) If the jont pefomance pocess of the system s components s assocated n I, o the magnal pefomance pocesses of the components ae ndependent n I, then l j(i) l j(i) ) pj(i) nf t τ(i) ) qj(i) nf t τ(i) [ 1 l j([t,t]) [ 1 l j([t,t]) (Q ([t,t]) ) ] 1 (P j([t,t]) Theoem 3. Let (C, ) be an MMS and let fo j 1,..., M l j(i) l j(i) t τ(i) j m ) p z D j (zj k ) j n ) j(i) l ) (6) ) ] 1 l j(i) (P j(i) ). (7) j k +1(I) q y j k (I). C j (yj k ) If the magnal pefomance pocesses of the components ae ndependent n I, then l j(i) ) [ pj(i) nf 1 l j([t,t]) (Q ([t,t]) ) ] j(i) 1 l ) (8) l j(i) ) qj(i) nf t τ(i) [ 1 l j([t,t]) (P ([t,t]) ) ] 1 l j(i) ). (9) Theoem 4. Let (C, ) be an MMS wth modula decomposton gven by Defnton 3 and let fo j 1,..., M B j(i) ) max B j(i) ) max 1 k j j k M [max[l k(i) [max[ l k(i) ), l k(i) ), l k(i) )]] )]. Intoduce the followng M module avalablty and unavalablty matces P (I) { p j(i) χ k },..., j1,...,m Q (I) { qχ j(i) k },...,, (10) j1,...,m and coespondngly defne the followng M matces B (I) 10 (I) (P ), B (I) ).
Assume the magnal pefomance pocesses of the components to be ndependent n the tme nteval I. Then fo j 1,..., M B j(i) (B (I) (I) (P nf t τ(i) 1 [ 1 B j([t,t]) j(i) B ( B (I) )) B j(i) ) pj(i) (Q ([t,t]) ) ] 1 j(i) B ) ). (11) B j(i) ( nf t τ(i) (I) B 1 B j(i) [ 1 B j([t,t]) (B (I) )) B j(i) (P ([t,t]) (I) (P ) qj(i) ) ] 1 B j(i) ) )). (12) We ae now eady to establsh the bounds fo the moments of system avalabltes and unavalabltes. Intoduce the m n M aays of component avalablty and unavalablty moments E{ )m } { ) s } } s1,...,m 1,...,n j1,...,m (13) E{ )m } { E{(q j(i) ) s } } s1,...,m. (14) 1,...,n j1,...,m Theoem 5. Let (C, ) be an MMS. Assume that espectvely the component avalablty vectos p (I) 1,..., n and the component unavalablty vectos 1,..., n ae ndependent. Let q (I) l j(i)m )m }) max 1 k n j u j(i)m )m }) mn E{(q z j k +1([I]) ) } D j (zj k ) 1 k m j o l j(i)m )m }) max 1 k m j ū j(i)m )m }) mn E{(p y j k ([I]) ) }. C j (yj k ) 1 k n j o E{(p y j k (I) ) m } C j (yj k ) m ( ) m ( 1) E{(q z j k +1(I) ) m } D j (zj k ) m ( ) m ( 1) If the jont pefomance pocess of the system s components s assocated n I, o the magnal pefomance pocesses of the components ae ndependent n I, then fo m 1, 2,... l j(i)m )m }) ) m } 11
) m } u j(i)m )m }) (15) Poof: l j(i)m E{(q j(i) Fom Equaton (6) we have ) m } E{ max 1 k n j E{(p y j k (I) ) m }, 1 k n j, C j (yj k ) )m }) E{(q j(i) ) m } ) m } ū j(i)m )m }). (16) (p y j k (I) ) m } E{ (p y j k (I) ) m } C j (yj k ) C j (yj k ) havng used the ndependence of the component avalablty vectos. Snce the nequalty holds fo all 1 k n j, the lowe bound of Equaton (15) follows. Smlaly fom Equaton (6) ) m } E{( mn [1 q z j k +1([I]) 1 k m j ]) m } D j (zj k ) mn 1 k m j mn 1 k m j mn E{(1 m E{ m 1 k m j o o ( m D j (zj k ) q z ( m ) ( 1) ) ( 1) j k +1([I]) ) m } (q z j k +1([I]) ) } D j (zj k ) E{(q z j k +1([I]) ) }, D j (zj k ) havng used the ndependence of the component unavalablty vectos. Hence, the uppe bound of Equaton (15) s poved. The bounds of Equaton (16) follow completely smlaly fom Equaton (7). Note that espectvely p j([t1,t1]) and p j([t2,t2]) dependent fo t 1 τ(i), t 2 τ(i), t 1 t 2. Hence,, and q j([t1,t1]) and q j([t2,t2]) ae E(p j D1(I) ) E(p j D1 ) E(q j D2(I) ) E(q j D2 ). Ths means that we cannot apply the best uppe bounds n Equatons (6) and (7). Theoem 6. Let (C, ) be an MMS. Assume that espectvely the component avalablty vectos p (I) 1,..., n and the component unavalablty vectos 1,..., n ae ndependent. Let q (I) m j l j(i)m )m }) m o ( ) m ( 1) 12
D j (zj k ) so u j(i)1 )1 }) ( ) ( 1) s E{(p zj k +1(I) ) s } s n j l j(i)m )m }) C j (yj k ) so (1 E{q y j k ([I]) }) C j (yj k ) n j m o ( ) ( 1) s E{(q yj k (I) ) s } s m j ū j(i)1 )1 }) ( ) m ( 1) (1 E{p z j k +1([I]) }). D j (zj k ) If the magnal pefomance pocesses of the components ae ndependent n I, then fo m 1, 2,... l j(i)m )m }) ) m } (17) E{p j(i) } u j(i)1 )1 }) (18) l j(i)m )m }) E{(q j(i) ) m } (19) Poof: E{q j(i) Fom Equaton (8) we have } ū j(i)1 )1 }). (20) m j m j ) m } E{( E{( p z D j (zj k ) p z j k +1(I) D j (zj k ) ) m }, havng appled Equaton (3). The andom vaables p z j k +1(I) D j (zj k ) j k +1(I) ) m }, k 1,..., m j, ae assocated snce ndependent andom vaables and non-deceasng functons of assocated andom vaables ae asssocated, havng used the ndependence of the component avalablty vectos. Contnung the devaton we get m j m E{ o ( ) m ( 1) D j (zj k ) so ( ) ( 1) s (p zj k +1(I) ) s } s 13
m j m o ( ) m ( 1) D j (zj k ) so ( ) ( 1) s E{(p zj k +1(I) ) s }, s agan havng appled the ndependence of the component avalablty vectos. Hence, Equaton (17) follows. Smlaly fom Equaton (8) } E{ E{p j(i) n j n j (1 q y j k ([I]) )} C j (yj k ) (1 E{q y j k ([I]) }), C j (yj k ) havng used Equaton (4), notng that the andom vaables (1 q y j k ([I]) ), k 1,..., n j, C j (yj k ) ae assocated by the same agument as above, and the ndependence of the component unavalablty vectos. Hence, Equaton (18) s poved. The bounds of Equatons (19) and (20) follow completely smlaly fom Equaton (9). Due to the lack of symmety n Theoem 1, we have not been able to obtan coespondng uppe bounds fo ) m } and E{(q j(i) ) m }, m 2, 3,... n ths theoem. As fo Theoem 5 we cannot apply the best uppe bounds n Equatons (8) and (9). Coollay 7. Make the same assumptons as n Theoem 6 and let L j(i)m )m }) max[l j(i)m U j(i)1 )1 }) mn[u j(i)1 L j(i)m )m }) max[ l j(i)m Ū j(i)1 )1 }) mn[ū j(i)1 Then fo m 1, 2,... )m }), )m }), l j(i)m )m })] )1 }), u j(i)1 )1 })] j(i)m l )m })] )1 }), ū j(i)1 )1 })]. L j(i)m E{p j(i) L j(i)m E{q j(i) )m }) ) m } (21) } U j(i)1 )1 }) (22) )m }) E{(q j(i) ) m } (23) } Ū j(i)1 )1 }). (24) 14
Coollay 8. Make the same assumptons as n Theoem 6 and let B j(i)m )m }) max C j(i)1 D j(i)m B j(i)m Then fo m 1, 2,... )1 }) mn 1 k j )m }) mn )m }) max 1 k j j k M [L k(i)m k(i)1 [U 1 k j [u k(i)m k(i)m [ L )m })] )1 })] )m })] )m })] C j(i)1 )1 k(i)1 }) mn [Ū j k M )1 })] D j(i)m )m }) mn j k M [ū k(i)m )m })]. L j(i)m )m }) B j(i)m )m }) ) m } (25) E{p j(i) } C j(i)1 )1 }) U j(i)1 )1 }) (26) ) m } D j(i)m )m }) u j(i)m )m }) (27) L j(i)m )m }) B j(i)m )m }) E{(q j(i) ) m } (28) E{q j(i) } C j(i)1 )1 }) Ū j(i)1 )1 }) (29) E{(q j(i) ) m } D j(i)m )m }) ū j(i)m )m }). (30) It s mpotant to note that the bounds fo )} and E{(q j(i) )} gven n ths secton equal the bounds n Secton 3.2 of (Natvg 2011) by eplacng P (I) by E{ )1 } and Q (I) by E{ )1 }. Howeve, ths s not tue fo hghe ode moments. 4. A smulaton appoach and a case study An objecton aganst the bounds n Theoems 5 and 6 and Coollaes 7 and 8 s that they ae based on knowng all mnmal path and cut vectos of the system. It s natual to ty to both mpove the bounds and educe the computatonal complexty by ntoducng modula decompostons. Lookng at the bounds fo ) m } and E{(q j(i) ) m } fo m 2, 3,..., t seems that only the lowe bounds of Theoem 5 ae of the fom that fts nto the machney of Secton 3.3 of (Natvg 2011). We now get the followng theoem Theoem 9. Let (C, ) be an MMS wth modula decomposton gven by Defnton 3. Make the same assumptons as n Theoem 6. Then fo j 1,..., M, m 1, 2,... 15
l j(i) )m }) l j(i) (l (I) (E{(P (I) )m }) ) m } (31) l j(i) )m }) l j(i) ( l (I) )m }) E{(q j(i) ) m }. (32) Poof: Equaton (31) follows fom the lowe bound of Equaton (15) and fom Equaton (3.45) of (Natvg 2011), by n the last expesson eplacng P (I) by E{ )m }. Note that n ths case the latte aay gven by Equaton (13) can be eplaced by an n M matx by fxng s at m. Hence, an n M matx s eplaced by an n M matx. Equaton (32) follows by a dualty agument. Hence, ou analytcal bounds ae not mpoved by usng a modula decomposton. On the othe hand the computatonal complexty s educed snce we have to fnd mnmal path and cut vectos only fo each module and fo the oganzng stuctue. All analytcal bounds on the magnal moments ) m D 1(I) } and E{(q j(i) ) m D 2(I) } fo m 1,..., j 1,..., M gven n Secton 2 can be mpoved by staghtfowad smulaton technques. Let us llustate ths on the lowe bounds n Equaton (15). As n the poof of Theoem 5, wth full notaton, we have fom Equaton (6) ) m D 1(I) } E{ max 1 k n j (p y j k (I) ) m D 1(I) }. (33) C j (yj k ) Fo 1,..., n we smulate fom the posteo magnal dstbuton of (I) gven the data D (I), π ( (I) D (I) ), assumed beng Dchlet. We then calculate p j(i) fom Equaton (2) fo 1,..., n, j 1,..., M. Fo each ound of n smulatons the quantty C j k (I) (yjk )(pyj ) m s calculated, and the max 1 k n j ght hand sde of Equaton (33) s estmated by the aveage of the smulated quanttes. Theoetcally, as seen fom the poof, ths mpoves the lowe bound of Equaton (15) of Theoem 5. Smlaly, we obtan a smulated lowe bound whch mpoves the lowe bound of Equaton (17) of Theoem 6. In pactce, the analytc bounds may be magnally bette due to smulaton uncetanty. Ths smulaton technque can also be appled to ave at mpoved bounds usng modula decompostons. Fom Theoem 4 we fo nstance get the followng nequaltes as statng ponts fo the smulatons. Coollay 10. Let (C, ) be an MMS wth modula decomposton gven by Defnton 3. Assume the magnal pefomance pocesses of the components to be ndependent n the tme nteval I. Then fo j 1,..., M E{(B j(i) (B (I) (P (I) E{(p j )m D 2(I) } E{(1 E{( j(i) B ( )))m D 1(I) } E{(p j )m D 1(I) } j(i) B ( (I) B )))m D 2(I) } (34) (I) B )))m D 2(I) } E{(q j )m D 2(I) } E{(q j )m D 1(I) } E{(1 B j(i) (B (I) (P (I) )))m D 1(I) }. (35) 16
Fgue 1 A smple netwok. To llustate the theoy consde the smple netwok system depcted n Fgue 1. Hee module 1 s the paallel system of the components a 1 and b 1 and module 2 the paallel system of the components a 2 and b 2. We assume that the set of states of the th component s gven by S {0, 3}, 1, 2, 3, 4,.e. we have a bnay descpton at the component level. Let fo each module the state be 0 f nethe of the components wok, 1 f one component woks and 3 f two components wok. The states of the system ae gven n Table 1. Table 1 States of the smple netwok system of Fgue 1. 3 0 2 3 Module 2 1 0 1 2 0 0 0 0 0 1 3 Module 1 Note fo nstance that the state 1 s ctcal both fo each module and the system as a whole n the sense that the falng of a component leads to the 0 state. The mnmal path and cut vectos fo the system of Fgue 1 ae gven n espectvely Tables 2 and 3. Table 2 Mnmal path vectos fo the system of Fgue 1. Level Component 1 Component 2 Component 3 Component 4 1 0 3 0 3 1 0 3 3 0 1 3 0 0 3 1 3 0 3 0 2 0 3 3 3 2 3 0 3 3 2 3 3 0 3 2 3 3 3 0 3 3 3 3 3 Table 3. Mnmal cut vectos fo the system of Fgue 1. Level Component 1 Component 2 Component 3 Component 4 1, 2 0 0 3 3 1, 2 3 3 0 0 2 0 3 0 3 2 0 3 3 0 2 3 0 0 3 2 3 0 3 0 3 0 3 3 3 3 3 0 3 3 3 3 3 0 3 3 3 3 3 0 17
Followng the agument leadng to Equaton (2) we assume that the posteo magnal dstbuton of p j(i) gven the data D 1j(I) s beta wth paametes αe{p j(i) } and α(1 E{p j(i) }). Hence, V a{p j(i) and the second ode moment s gven by } E{p j(i) }(1 E{p j(i) })/(α + 1), ) 2 } E{p j(i) }(1 + E{p j(i) }α)/(α + 1). E{p j(i) } s chosen to be equal to the value of p j(i) calculated by a standad detemnstc analyss as gven n (Natvg 2011). In these calculatons the magnal pefomance pocesses of the two modules ae assumed ndependent n the tme nteval I and also that the two components of each module fal and ae epaed/eplaced ndependently of each othe. All components have the same nstantaneous falue ate λ 0.001 and epa/eplacement ate µ 0.01. In Table 4 the analytcal lowe bounds fom Coollay 8, B j(i)m )m }), fo ) m } fo m 1, 2 and the coespondng smulated lowe bounds, both not usng and usng modula decompostons, ae calculated fo the tme nteval I equal to [100, 110], [100, 200], [1000, 1100], the paamete α equal to 1, 10, 1000 and fo system level j equal to 1, 2, 3. Lookng at the lowe bounds fo )} thee ae just mno dffeences between the analytcal and the smulated bounds that ae not based on modula decompostons except fo α 1, j 2 and the two longest ntevals [100, 200] and [1000, 1100], whee the mpovements ae qute small. Coespondngly, thee ae just mno mpovements of the smulated bounds that ae based on modula decompostons compaed to the ones that ae not except fo α 10, 1000, j 2 and the two longest ntevals [100, 200] and [1000, 1100], whee the mpovements ae qute small. Futhemoe, the analytcal bounds do not not depend on the α paamete whch s natual snce E{p j(i) } s ndependent of ths paamete. That these lowe bounds ae deceasng n the length of the nteval I and the system state j s just a eflecton of the fact that these popetes hold fo )}. Tunng to the lowe bounds fo ) 2 } thee ae agan just mno dffeences between the analytcal and the smulated bounds that ae not based on modula decompostons agan except fo α 1, j 2 and the two longest ntevals [100, 200] and [1000, 1100], whee the mpovements ae qute small. Coespondngly, thee ae just mno mpovements of the smulated bounds that ae based on modula decompostons compaed to the ones that ae not except fo α 10, 1000, j 2 and the two longest ntevals [100, 200] and [1000, 1100], whee the mpovements ae qute small. Futhemoe, these bounds ae deceasng n the α paamete whch s natual snce ) 2 } s deceasng n ths paamete. That these lowe bounds ae deceasng n the length of the nteval I and the system state j s just a eflecton of the fact that these popetes also hold fo ) 2 }. It should also be noted that combnng the lowe bounds fo fo V a{(p j(i) )} and ) 2 } does not lead to a lowe bound )}. Howeve, fo the analytcal lowe bounds t s evealng that ths leads to postve vaances. Fo the coespondng smulated lowe bounds ths s obvously the case. Fnally, t should be emaked that n contast to the analytcal bounds, the smulated bounds ae mpoved by usng modula decompostons. 18
Fo compute code used n ths secton we efe to http://folk.uo.no/tond/system/. Refeences Balow RE, Poschan, F (1975) Statstcal theoy of elablty and lfe testng. Pobablty models. Holt, Rnehat and Wnston, New Yok. Mastan DV, Sngpuwalla ND (1978) A Bayesan estmaton of the elablty of coheent stuctues. Opeat Res 26:663-672 Natvg B (2011) Multstate systems elablty theoy wth applcatons. Wley, New Yok. Natvg B, Ede H (1987) Bayesan estmaton of system elablty. Scand J Statst 14:319-327 Table 4. Lowe bounds of the smple netwok system of Fgue 1. Analytcal Sm. mnus m.d. Sm. plus m.d. I α j 1. m. 2. m. 1. m. 2. m. 1. m. 2. m. [100, 110] 1 1 0.9902 0.9833 0.9902 0.9833 0.9902 0.9833 [100, 110] 1 2 0.9710 0.9507 0.9726 0.9546 0.9730 0.9551 [100, 110] 1 3 0.7481 0.6487 0.7481 0.6487 0.7481 0.6487 [100, 110] 10 1 0.9902 0.9807 0.9902 0.9807 0.9902 0.9807 [100, 110] 10 2 0.9710 0.9433 0.9713 0.9446 0.9724 0.9465 [100, 110] 10 3 0.7481 0.5751 0.7481 0.5752 0.7481 0.5752 [100, 110] 1000 1 0.9902 0.9805 0.9902 0.9805 0.9902 0.9805 [100, 110] 1000 2 0.9710 0.9428 0.9710 0.9428 0.9722 0.9451 [100, 110] 1000 3 0.7481 0.5598 0.7481 0.5598 0.7481 05598 [100, 200] 1 1 0.9555 0.9263 0.9555 0.9262 0.9555 0.9262 [100, 200] 1 2 0.8723 0.7947 0.8857 0.8226 0.8884 0.8256 [100, 200] 1 3 0.5219 0.3821 0.5217 0.3819 0.5217 0.3819 [100, 200] 10 1 0.9555 0.9142 0.9555 0.9142 0.9555 0.9142 [100, 200] 10 2 0.8723 0.7641 0.8751 0.7731 0.8834 0.7865 [100, 200] 10 3 0.5219 0.2903 0.5219 0.2903 0.5219 0.2903 [100, 200] 1000 1 0.9555 0.9130 0.9555 0.9130 0.9555 0.9130 [100, 200] 1000 2 0.8723 0.7610 0.8723 0.7611 0.8819 0.7778 [100, 200] 1000 3 0.5219 0.2726 0.5219 0.2726 0.5219 0.2726 [1000, 1100] 1 1 0.9380 0.8986 0.9381 0.8987 0.9381 0.8987 [1000, 1100] 1 2 0.8254 0.7256 0.8460 0.7663 0.8500 0.7705 [1000, 1100] 1 3 0.4578 0.3157 0.4580 0.3159 0.4580 0.3159 [1000, 1100] 10 1 0.9380 0.8818 0.9380 0.8818 0.9380 0.8818 [1000, 1100] 10 2 0.8254 0.6857 0.8296 0.6987 0.8422 0.7178 [1000, 1100] 10 3 0.4578 0.2265 0.4578 0.2266 0.4578 0.2266 [1000, 1100] 1000 1 0.9380 0.8800 0.9380 0.8799 0.9380 0.8799 [1000, 1100] 1000 2 0.8254 0.6813 0.8254 0.6815 0.8400 0.7057 [1000, 1100] 1000 3 0.4578 0.2098 0.4578 0.2098 0.4578 0.2098 19