Algebraic reliability of multi-state k -out-of- n systems
Patricia Pascual-Ortigosa, Eduardo Sáenz-de-Cabezón, Henry P. Wynn
AALGEBRAIC RELIABILITY OF MULTI-STATE k -OUT-OF- n SYSTEMS
PATRICIA PASCUAL-ORTIGOSA, EDUARDO S ´AENZ-DE-CABEZ ´ON,AND HENRY P. WYNN
Abstract.
In this paper we review different definitions that multi-state k -out-of- n systems have received along the literature and study them in a unified wayusing the algebra of monomial ideals. We thus obtain formulas and algorithms tocompute their reliability and bounds for it. We provide formulas and computerexperiments for simple and generalized multi-state k -out-of- n systems and forbinary k -out-of- n systems with multi-state components. Introduction
We say that a system is a k -out-of- n :G system (G for good) if it works whenever k of its n components work, and that it is a k -out-of- n :F (F for fail) if it fails whenever k of its n components fail. k -out-of- n systems are one of the most relevant types ofsystems studied in reliability theory due to their theoretical interest and wide rangeof applications, cf. [23, 26, 12]. The multi-state version, which can model moregeneral situations, has been object of intense research in the last decades and is alsoapplied in a variety of situations [21, 22, 5, 14, 39]. Since the first definition of multi-state k -out-of- n systems [16] several authors have proposed different definitions andgeneralizations, together with particular methods to evaluate the reliability of thesesystems, see for instance [6, 21, 2, 9, 10, 3, 5, 35] and references therein.We list a number of examples of this kind of sytems.(1) Power generation . The safety and reliability of power systems is an essentialcomponent of energy security and is increasing its importance in a period inwhich there are likely to be radical changes in energy supply as governmentsadopt zero net carbon strategies and use more renewable sources, such aswind power, which may be more volatile. There are four standard statesof generation for an energy unit: (i) available and in service, (ii) availableand not in service (iii) planned outage, (iv) unplanned outage. Consideringthat a national electricity grid will have many sources of supply and differentcomponents will be in different states, this represents a challenging multi-component multi-state network. There is also a strong time aspect leadingto strict definitions such as FOR: Forced Outage Rate and AV: Availability,which form part of supply contracts and regulation. Recent books are [37,47, 30] for a k -out-of- n approach.(2) DNA repair . DNA damage is a biological process that can upset importantfunctions such as replication. DNA damage is different from mutation, al-though both occur. The system can be in very many states, depending on a r X i v : . [ m a t h . A C ] A p r he amount of cell loss of different types. Areas of study include the funda-mental equilibria between repair and damage, needed to sustain the systems.Initial models make assumptions, similar to those in reliability, for example,that occurrence at break sites happen independently [11].(3) Software reliability and Bayes nets . It is natural in several areas of reliabil-ity to take a probabilistic state-space approach. This is particularly true ofone of the main traditions of software reliability and provides an alternativeto rule based formal methods. An advantage of this approach is that it canmodel systems as a Bayes net and link up with modern theories of causation.Also important in such systems is the idea of degradation which automati-cally implies different levels of reliability and is particularly important in theanalysis of safety critical systems; see [17] for a comprehensive approach.In the failure of k -out-of- n components, the number k is a simple metric to de-scribe degradation (mentioned above) and this extends to the multi-state methodsaddressed here. A useful way to think of the latter is that there is a damage “fron-tier” beyond which the system is deamed to have failed or to have reached a level,for example, at which the unit may be switched off for maintenance. This may beplanned or unplanned (as mentioned above for power generation). Another way ofconceptualizing these issues is that the metric k is simply a way of counting some(bad) aspect of the system and counting is surely a basic combinatorial and alge-braic activity. Broadly, research on the theory of k -out-of- n methods divides into(i) combinatorial and algebraic theory, as in this paper, and (ii) simulations studies,which are typically of a Markovian type. For the combinatorial methods generatingfunctions play an important role [51]. In our work this is reflected in the use ofHilbert series, which are essentially a type of generating function. For sequential k -out-of- n problems one often converts the system into a Markov chain, inspect theergodic behaviour and benchmark against probabilistic asymptotics from large de-viation theory and boundary crossing methods. A main tool is that of de Bruijngraphs which track the change of a moving window between time steps [27] . Signa-ture analysis has also been applied to k -out-of- n systems [33]. The methods employthe inherent symmetries in the order statistics of failure events to simplify reliabil-ity bounds, [25]. Genome analysis is one science that makes much use of a typeof k -out-of- n analysis under a heading of k-mer : the detection of special genomesequence of length k out of a much longer sequence, [38]. There is a dominance ofcomputer base search methods in the area and some also use the de Bruijn graphmethods. The idea of a “special sequence” makes the field quite close to percolationtheory where the sequence is a percolation through a lattice structure of some kind.The algebraic method for the analysis of system reliability associates a monomialideal to a coherent system and by studying algebraic properties of this ideal obtainsinformation about the system and its reliability [43, 44, 45, 46], see Appendix A fora basic introduction to this method. The principal objective is to obtain generalextensions of classical Bonferroni bounds in multi-state system reliability. It is ageneral method that can be adapted to different kind of systems, both binary andmulti-state. In this paper we review the different definitions of multi-state k -out-of- n ystems, study them in an algebraic way, and apply the algebraic method as a unifiedway to compute their reliability. The foundation has two parts: a description of thesystem, including the idea of a state, and the stochastic model which defines theoccupancy of the state. The next step is to map the system into an algebraic objectcalled a monomial ideal, which can be handled via combinatorial algebra, includingthe use of computer algebra (already well developed for this purpose). The compactinclusion-exclusion formulae needed for the bounds start by being distribution-freeand require special Betti numbers which are attached to the ”live” terms in theformulae. For simple probability models it is then straightforward to obtain theactual probability bounds.A problem for the reliability computation of these systems is the computationalburden when complexity increases. Several algorithms have been proposed to com-pute the exact reliability of these systems, see [4, 7, 53, 50, 32]; also, Ding et al.propose in [9] a framework for reliability approximation. Our approach, while enu-merative, shows good performance and can provide both exact reliability and boundsin the case of i.i.d components and in the case of independent non-identical compo-nents.The outline of the paper is the following: in Section 2 we give a quick overview ofthe algebraic method for system reliability analysis, in particular when applied tomulti-state systems. In Section 3 we show the first definitions of multi-state k -out-of- n systems, give an algebraic version of them and use it to analyse the reliabilityof this kind of systems. In Section 4 we study generalized multi-state k -out-of- n systems and in Section 5 we focus on a type of binary k -out-of- n systems withmulti-state components and give an example of application of these systems. Asimple storage problem is used for illustration. Nomenclature S : A coherent system n : Number of components of the system Sm : Maximum level of performance of the system S S = { , . . . , m } : Possible states of the system Sc i : Component i of the system, i ∈ { , . . . , n } m i : Maximum level of performance of the component c i , i ∈ { , . . . , n }S i = { , . . . , m i } : Possible states of the component c i , i ∈ { , . . . , n } φ : S × · · · × S n → S : Structure function of the system S x = ( x , . . . , x n ) : Vector of components’ states F S,j : Set of j -working states of S F S,j : Set of minimal j -working states of SI S,j : j -reliability ideal of SG ( I S,j ) : Unique minimal monomial generating set of I S,j H I S,j : Numerator of the Hilbert series of I S,j β i ( I ) , β i,j ( I ) : Betti numbers and graded Betti numbers of II ( k,n ) ,j : j -reliability ideal of a simple multi-state k -out-of- n system N j : Number of components in state j or above, j ∈ { , . . . , M } n, ( k ,...,k M ) : Generalized multi-state k -out-of- n system I n, ( k ,...,k M ) : j -reliability ideal of a generalized multi-state k -out-of- n system p i,j : Probability that the component i is in level greater than or equal to jR S,j : Probability that the system S is performing at level greater than or equal to jr S,j : Probability that the system S is performing at level jS m,n,k : m -multi-state k -out-of- n :G system J m [ n,k ] : j -reliability ideal of the system S m,n,k N m [ n,k ] : Number of generators of the ideal J m [ n,k ] Algebraic reliability of multi-state systems
Let S be a system with n components that can be in any of a set of m + 1 possiblestates S = { , . . . , m } . Each component c i of S can be in a discrete number ofordered states S i = { , . . . , m i } . The states of the system are also ordered andmeasure the overall performance of the system. We assume that state j representsbetter performance than state i whenever j > i . We define a structure function φ that for each n -tuple of component states outputs the state of the system i.e. φ : S × · · · × S n → S . We say that the system is coherent if φ ( x ) ≥ φ ( y ) whenever x > y , which means that the component states given by x are greater or equal thanthose given by y and there is at least one improvement. Conversely, φ ( x ) ≤ φ ( y )whenever x < y . If m = · · · = m n = 1, then we say that the system has binarycomponents . If m = 1, then we say that the system is itself binary . We havetherefore the following types of systems with respect to their number of states:- If m = 1 and m i = 1 for all i , we have a binary system with binary compo-nents. These are usually simply referred to as binary systems .- If m > m i = 1 for all i , we have a multi-state system with binarycomponents.- If m = 1 and there is at least one component i such that m i >
1, we have abinary system with multi-state components.- If m > i such that m i >
1, we have amulti-state system with multi-state components.We basically follow here the notation in [19] and [36] but we allow a more generalkind of systems, since we do not restrict to the case that max( S ) ≤ max( S i ) ∀ i .For other definitions of multi-state systems and a review of multi-state reliabilityanalysis, we refer to [16, 29, 52, 28] and the references therein.Let S be a coherent system with n components and let F S,j be the set of tuples ofcomponents’ states x such that φ ( x ) ≥ j for some 0 < j ≤ m . The elements of F S,j are called j -working states of S . Let F S,j be the set of minimal j -working states,i.e. states in F S,j such that the degradation of the performance of any componentprovokes that the overall performance of the system is degraded to some j (cid:48) < j . Letnow R = k [ x , . . . , x n ] be a polynomial ring over a field k . Each tuple of components’states ( s , . . . , s n ) ∈ S × · · · × S n corresponds to the monomial x s · · · x s n n in R . The coherence property of the system is equivalent to saying that the elements of F S,j correspond to the monomials in an ideal, denoted by I S,j and called the j -reliabilityideal of S . The unique minimal monomial generating set of I S,j , denoted G ( I S,j ), s formed by the monomials corresponding to the elements of F S,j (see [42, §
2] formore details). Hence, obtaining the set of minimal cuts of S amounts to computethe minimal generating set of I S,j .In order to compute the j -reliability of S (i.e. the probability that the system isperforming at least at level j ) we can use the numerator of the Hilbert series of I S,j ,denoted by H I S,j . The polynomial H I S,j gives a formula, in terms of x , . . . , x n thatenumerates all the monomials in I S,j , i.e. the monomials corresponding to the statesin F S,j . Hence, computing the (numerator of) the Hilbert series of I S,j provides away to compute the j -reliability of S by substituting x ai by p i,a , the probability thatthe component i is at least performing at level a , as explored in [42, §
2] (for thebinary case).Often in practice it is more useful to have bounds on the j -reliability of S ratherthan the exact formula. In order to have a formula that can be truncated at differentsummands to obtain bounds for the j -reliability in the same way that we truncatethe inclusion-exclusion formula to obtain the so-called Bonferroni bounds, we needa special way to write the numerator of the Hilbert series of I S,j . This convenientform is given by the alternating sum of the ranks in any free resolution of the ideal I S,j . Every monomial ideal I has a minimal free resolution, which provides thetightest bounds among the aforementioned ones. The ranks of the free modules inthe minimal free resolution are called the Betti numbers of the ideal and are denotedby β i ( I ), or by β i,j ( I ) in the graded case. In general, the closer the resolution is tothe minimal one, the tighter the bounds obtained, see e.g. [42, § j -reliability of a coherentsystem S works as follows:(1) Associate to the system S its j -reliability ideal I S,j .(2) Obtain the minimal generating set of I S,j to get the set F S,j .(3) Compute the Hilbert series of I S,j to have the j -reliability of S .(3’) Compute any free resolution of I S,j . The alternating sum of the ranks of thisresolution gives a formula for the Hilbert series of I S,j i.e., the unreliabilityof S , which provides bounds by truncation at each summand.The choice between steps (3) or (3’) depends on our needs. If we are only in-terested in computing the full reliability formula, then we can use any algorithmthat computes Hilbert series in step (3). However, if we need bounds for our systemreliability, then we can compute any free resolution of I S,j and thus perform step(3’). If the performing probabilities of the different components are independent andidentically distributed (i.i.d), then in points (3) and (3’) of this procedure we onlyneed the graded version of Hilbert series and free resolutions. Otherwise, we needtheir multigraded version. For more details and the proofs of the results describedhere, we refer to [42, 45]. To see more applications of this method in reliabilityanalysis we refer to [43, 44, 46]. . Simple multi-state k -out-of- n systems The first definition of multi-state k -out-of- n systems was given by El-Neweihi etal. in the seminal work [16]. They define multi-state systems as follows: Definition 3.1 (El-Neweihi et al., 1978) . A system of n components is said to bea multi-state coherent system (MCS) if its structure function φ satisfies:(1) φ is increasing.(2) For level j of component i , there exists a vector ( · i , x ) such that φ ( j i , x ) = j while φ ( l i , x ) (cid:54) = j for l (cid:54) = j for i = 1 , . . . , n and j = 0 , . . . , M .(3) φ ( j ) = j for j = 0 , . . . , M , where j = ( j, . . . , j ).Where ( j i , x ) means that the state of the i ’th component in x is j . Observe thatthis definition is more restrictive than ours in the sense that they assume everycomponent has the same number of states, which is in turn the number of states ofthe system, i.e. M .The definiton of multi-state k -out-of- n systems in [16] is: Definition 3.2 (El-Neweihi, 1978) . A system is a multi-state k -out-of- n system ifits structure function satisfies(3.1) φ ( x ) = x ( n − k +1) where x (1) ≤ x (2) ≤ · · · ≤ x ( n ) is a non decreasing arrangement of x , . . . , x n .Observe that this definition satisfies the conditions given in Definition 3.1. It iseasy to check that φ is an increasing function and φ ( j ) = j for all j = 0 , . . . , M . Tosee condition (2) just observe that there always exists a non decreasing arrangementof x , . . . , x n in which φ ( j i , x ) = j while φ ( l i , x ) (cid:54) = j for l (cid:54) = j for i = 1 , . . . , n and j = 0 , . . . , M . Taking the vector in which the first n − k + 1 components are lowerthan j and the rest of the are greater than j , we have that condition (2) is satisfied. Remark . This kind of systems are called simple multi-state k -out-of- n systems in [26].We describe now the j -reliability ideal of these multi-state k -out-of- n systems: Proposition 3.4.
The ideal I ( k,n ) ,j = (cid:104) (cid:89) σ ⊆{ ,...,n }| σ | = k x ji | i ∈ σ (cid:105) is the j -reliability ideal of a multi-state k -out-of- n system as defined in Definition3.2.Proof. First of all we need to check that all µ ∈ G ( I ( k,n ) ,j ) satisfy φ ( µ ) = j . Let x µ = x ji x ji . . . x ji k be a generator of I ( k,n ) ,j , with { i , . . . , i k } ⊆ { , . . . , n } . If we makea non decreasing arrangement of x i , . . . , x i k we obtain the vector (0 , ..., , j, ..., j ) inwhich the first n − k components are in state 0 and the other components are in state j . Applying the structure function φ to this vector we have that φ (0 , ..., , j, ..., j ) = j . ow, if x ν ∈ I ( k,n ) ,j , there exists x µ ∈ G ( I ( k,n ) ,j ) such that µ ≤ ν . This implies φ ( µ ) ≤ φ ( ν ) and since φ ( µ ) = j and φ is an increasing function, we obtain φ ( ν ) ≥ j .Finally if l < j and φ ( ν ) = l we must have x ν (cid:54)∈ I ( k,n ) ,j . Since φ ( ν ) = l < j we have that there are at most, k − j .This implies that there does not exist any σ ∈ { , . . . , n } with | σ | = k such that (cid:81) x i ∈ σ x ji | x ν , hence x ν / ∈ I ( k,n ) ,j . (cid:3) In [6] Boedigheimer and Kapur define customer-driven reliability models for multi-state systems. They consider systems with M states in which component i can bein M i states. They describe such systems using upper and lower boundary points ,which are enough to describe the system completely and are defined as follows Definition 3.5.
We say x is a lower boundary point (l.b.p.) to level j iff φ ( x ) ≥ j and y < x implies that φ ( y ) < j , for j = 1 , . . . , M . An upper boundary point (u.b.p) to level j is an n -tuple x such that φ ( x ) ≤ j and y > x implies that φ ( y ) > j , for j = 0 , . . . , M − j are the minimal monomialgenerators of the j -reliability ideal of the system. To describe upper boundarypoints algebraically we need the concept of maximal standard pairs [48]. Definition 3.6.
Let I a monomial ideal in R = k [ x , . . . , x n ] and σ ⊆ { , . . . , n } .The pair ( x µ , σ ) is a standard pair for I if it satisfies:- supp( x µ ) ∩ σ = ∅ , where supp( x µ ) is the set of indices i ∈ { , . . . , n } suchthat x i divides x µ .- for all monomials x ν such that supp( x ν ) ⊆ σ we have that x µ x ν / ∈ I .- ( x µ , σ ) (cid:54)⊆ ( x ν , τ ) for any other ( x ν , τ ) satisfying the two previous conditions.We say that ( x µ , σ ) is a maximal standard pair if there is no other standard pair( x ν , σ ) such that x µ divides x ν .Maximal standard pairs are in one-to-one correspondence with upper boundarypoints. Theorem 3.7.
Let I S,j be the j -reliability ideal of a coherent system S for whichcomponent i can be in states (0 , . . . , M i ) . Then µ + (cid:80) i ∈ σ M i is an upper boundarypoint of S for level j − if and only if ( x µ , σ ) is a maximal standard pair of I S,j .Proof. ⇒ ) Let α be an upper boundary point of S for level j −
1. Let σ ⊆ { , . . . , n } be the set of components of S such that α i = M i . We have that σ (cid:54) = { , . . . , n } i.e. there exists at least one component i such that α i (cid:54) = M i hence α is of the form α = µ + (cid:80) i ∈ σ M i . φ ( α ) < j implies x α / ∈ I S,j , and we claim that ( µ, σ ) is a standardpair for I S,j . To see this, let x µ x ν such that supp( x ν ) ⊆ σ . If ν i ≤ M i then clearly x µ x ν / ∈ I S,j because µ + ν ≤ α and φ ( α ) < j . Now, since x α / ∈ I S,j we know thereis no minimal generator of I S,j that divides x α and since M i = α i is the maximalpower to which variable i can possibly be raised to in any generator of I S,j thenno generator will divide x α x ν for any ν such that supp( x ν ) ⊆ σ hence ( µ, σ ) is astandard pair. Assume now that ( µ, σ ) is not maximal. Then there is some i (cid:48) / ∈ σ such that ( µ + 1 i (cid:48) , σ ) is a standard pair for I S,j . Then x µ x i (cid:48) (cid:81) i ∈ σ x M i i / ∈ I S,j i.e. ( α + 1 i ) < j which contradicts the assumption that α is an upper boundary pointof S for level j − ⇐ ) Let ( x µ , σ ) be a maximal standard monomial of I S,j , i.e. x µ / ∈ I S,j and x µ x ν / ∈ I S,j for all x ν such that supp( x ν ) ⊆ σ . Let x α = x µ (cid:81) i ∈ σ x M i i . Since x α / ∈ I S,j we knowthat φ ( α ) < j . Let now β > α , we can assume without loss of generality that β = α + 1 i for some i / ∈ σ . Suppose x β / ∈ I S,j . Then there is no minimal generatorof I S,j that divides x β but since M i is the maximal state of component i , then thereis no minimal generator of I S,j that divides x β x ν for any ν such that its supportis a subset of σ . Finally since the difference between x µ x i and x β is a monomialwhose support is in σ , we have that ( x µ x i , σ ) is a standard pair for I S,j , which isin contradiction with the fact that ( x µ , σ ) is maximal, hence x β ∈ I S,j and α is anupper boundary point of S for level j − (cid:3) Using upper and lower boundary points, Boedigheimer and Kapur define multi-state k -out-of- n systems as follows. Definition 3.8 (Boedigheimer and Kapur, 1994) . φ is a multi-state k -out-of- n : G structure function if, and only if, φ has (cid:0) nk (cid:1) lower boundary points to level j ( j =1 , . . . , M ) and (cid:0) nk − (cid:1) upper boundary points to level j ( j = 0 , . . . , M − I ( k,n ) ,j in Proposition 3.4 has (cid:0) nk (cid:1) elements,i.e. this system has (cid:0) nk (cid:1) lower boundary points. The maximal standard pairs of I ( n,k ) ,j are ( (cid:81) i ∈ σ x j − i , { , . . . , n } − σ ) for all σ ⊆ { , . . . , n } such that | σ | = n − k + 1, i.e.the number of upper boundary points of S for j − (cid:0) nn − k +1 (cid:1) = (cid:0) nk − (cid:1) . Hence,Proposition 3.4 is a proof of the equivalence of definitions 3.2 and 3.8 in the casethat M i = M for all i .If we allow that the number of states of each of the components can be different,then the situation is more complicated. Let n j be the number of components suchthat their maximum performance level M i is bigger than or equal to j . If n j ≥ k thenthe system behaves as a multi-state k -out-of- n system by setting φ as in Definition3.2. The number of lower and upper boundary points does however vary. The lowerboundary points are given by the tuples that have k components at level j and n − k components at level 0, and there are (cid:0) n j k (cid:1) such tuples. And if n j ≥ k then the upperboundary points for level j are given by the tuples in which k − j ), the other component such that itsmaximum level is bigger than j is exactly at level j and the rest of the componentsare at level min { M i , j } . The number of such tuples is (cid:0) n j +1 k (cid:1) . Hence the systembehaves at level j as a k -out-of- n j system according to definition 3.8. In fact, if weonly consider those components whose maximum performance level is bigger than j then the system behaves at level j as a k -out-of- n j system according to bothdefinitions.We can then generalize the ideal in Proposition 3.4 allowing different number oflevels for each component: Definition 3.9.
Let S be a multi-state system with levels { , . . . , M } and suchthat each component i has M i +1 levels of performance { , . . . , M i } . Let n j ≤ n evel Lower boundary points Upper boundary points0 (0 , , , , , (0 , , , , , (0 , , , , , (0 , , , , , (4 , , , , , , , , , (0 , , , , , (0 , , , , , (1 , , , , , (1 , , , , , (1 , , , , , (1 , , , , , (0 , , , , , (0 , , , , , (1 , , , , , (0 , , , ,
0) (4 , , , , , , , , , (1 , , , , , , , , , (0 , , , , , (2 , , , , , (0 , , , , , (2 , , , , , (4 , , , , , , , , , (2 , , , , , , , , Table 1.
Upper and lower boundary points for the system in Exam-ple 3.10be the number of components such that M i ≥ j for each j ∈ { , . . . , M } (for easeof notation we consider that these are components 1 , . . . , n j ). S is a multi-state k -out-of- n system if for every j ∈ { , . . . , M } the j -reliability ideal of S , I S,j , is ofthe form I S,j = (cid:104) (cid:89) σ ⊆{ ,...,n j }| σ | = k x ji | i ∈ σ (cid:105) . Example 3.10.
Let S be a system such that S = { , , , , } , S = { , , , } , S = S = { , , } and S = { , } and let φ ( x ) = x (4) . Observe that n = 5, n = 4, n = 2, n = 1. The system behaves as a 2-out-of-5 for levels j = 1 , , n j system for levels j = 1 , , I S, = (cid:104) x x , x x , x x , x x , x x , x x , x x , x x , x x , x x (cid:105) I S, = (cid:104) x x , x x , x x , x x , x x , x x (cid:105) I S, = (cid:104) x x (cid:105) . Generalized multi-state k -out-of- n systems In [22] Huang, Zuo and Wu introduced generalized multi-state k -out-of- n systems allowing different number of components for a system to perform at each level j naturally extending the capabilities of the systems studied in the previous sectionand providing more flexibility to describe practical situations. The definition in [22]is the following Definition 4.1 (Huang, Zuo and Wu, 2000) . An n -component system is called a generalized multi-state k -out-of- n :G system if φ ( x ) > j, ≤ j ≤ M whenever thereexists an integer value l ( j ≤ l ≤ M ) such that at least k l components are in state l or above. f we denote by φ the structure function of the system S and by N j the numberof components in state j or above, then this definition can be rephrased by sayingthat φ ( S ) ≥ j if N j ≥ k j N j +1 ≥ k j +1 ... N M ≥ k M Hence we can denote a generalized multi-state k -out-of- n system by S n, ( k ,...,k M ) .When k ≤ · · · ≤ k m the system is called an increasing generalized multi-state k -out-of- n :G system, and if k ≥ · · · ≥ k m the system is said to be decreasing . Huang etal. provide formulas for both cases and an enumerative algorithm for the evaluationof the reliability of generalized multi-state k -out-of- n systems when the sequence( k , . . . , k M ) is monotone.Continuing this line M. J. Zuo and Z. Tian defined in [54] generalized multi-state k -out-of- n :F systems. Definition 4.2 (Zuo and Tian, 2006) . An n -component system is called generalizedmulti-state k -out-of- n : F system if φ ( x ) < j, ≤ j ≤ M whenever the states of atleast k l components are below l for all l such that j ≤ l ≤ M .Using this definition they provide a correspondence between generalized multi-state k -out-of- n :G systems and generalized multi-state k -out-of- n :F systems. Theystudy these systems when the sequence ( k , . . . , k M ) is not necessarily monotone andprovide an efficient algorithm that is recursive on M , the number of performancelevels. This algorithm outperforms the one in [22] which is recursive in n .Using the ideals in Proposition 3.4 we can immediately describe the reliabilityideal of a generalized multi-state k -out-of- n :G system given by ( k , . . . , k M ). Proposition 4.3.
The j -reliability ideal of a generalized multi-state k -out-of- n sys-tem S = S n, ( k ,...,k M ) is given by I S,j = I n, ( k j ,...,k M ) = M (cid:88) i = j I ( k i ,n ) ,i . Example 4.4.
We study here Example 8 in [22] with the algebraic method and re-cover the exact same results given there. The system in this example is a generalizedmulti-state k -out-of-3:G system with four states (0 , , ,
3) such that k = 3, k = 2and k = 2, hence it is a decreasing generalized multi-state k -out-of- n :G system.The probabilities of the different components are given by p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 . p , = 0 .
3, where p i,j is the probability that component i isperforming at level j .- For the system to be in state 3 there must be at least 2 components in state 3or above ( k = 2). Hence the corresponding ideal is I S, = (cid:104) x y , x z , y z (cid:105) . he numerator of the Hilbert series is H I S, = x y + x z + y z − x y z )and when plugging the probabilities in, we have that the probability thatthe system is in state 3 or above, denoted R S, , is 0 . r S, .- The system is in state 2 or above if at least 2 components are in state 2or above, hence I S, = I (2 , , + I (2 , , = I (2 , , = (cid:104) x y , x z , y z (cid:105) . Thenumerator of the Hilbert series is H I S, = x y + x z + y z − x y z ) andwe obtain R S, = 0 .
826 and r S, = R S, − R S, = 0 . − .
396 = 0 . k = 3 the system is in state 1 or above if all 3 components are instate 1 or above or if at least 2 components are in state 2 or above or if atleast 2 components are in state 3 or above. The corresponding ideal is then I S, = I (3 , , + I (2 , , + I (2 , , = I (3 , , + I (2 , , = (cid:104) xyz, x y , x z , y z (cid:105) , H I S, = xyz + x y + x z + y z − ( xy z + x yz + x y z ) and we obtain R S, = 0 .
89 and r S, = R S, − R S, = 0 . − .
826 = 0 . r S, = R S, − R S, = 1 − .
89 = 0 . k -out-of- n :G systems given inProposition 4.3 we can develop a recursive method to compute their reliability. Themethod is recursive on M , the number of performance levels and can be used for anysequence ( k , . . . , k M ) describing the system, not necessarily monotone. This methodis an enumerative one that can be used even when the component’s probabilities arenot i.i.d. For the i.i.d. case our method is equivalent to the one in [54] in terms ofcomputational complexity. We will use the technique of Mayer-Vietoris trees, whichwere introduced in [40, 41], see Appendix B for an explanation of the method. Forease of the notation we assume that the sequence ( k , . . . , k M ) is strictly decreasing.In any other case, the only difference is that some of the summands that composethe ideal I n, ( k j ,...,k M ) will be missing, as we saw in Example 4.4 but this fact does notaffect the algorithm description or its performance.Let 1 ≤ j ≤ M and I n, ( k j ,...,k M ) = (cid:80) Mi = j I ( k i ,n ) ,i the j -reliability ideal of the system.We sort the generators of I n, ( k j ,...,k M ) in ascending degree and lexicographically withineach degree. For constructing the Mayer-Vietoris tree we will use as pivot alwaysthe last generator. First, we use as pivots the generators of I ( k M ,n ) ,M . We denoteeach of them by x Mσ = (cid:81) x i ∈ σ x Mi for σ ⊆ { , . . . , n } and | σ | = k M . For each of thesegenerators we obtain as left child in the Mayer-Vietoris tree the ideal denoted by I σ,M given by I σ,M = I n − k M , ( k j − k M ,...,k M − − k M ) + (cid:88) x i / ∈ σ,x i < max( σ ) (cid:104) x Mi (cid:105) , where I n − k M , ( k j − k M ,...,k M − − k M ) ⊆ k [[ n ] − σ ]. On each of the nodes of the tree we useas pivots the monomials in (cid:80) x i / ∈ σ,x i < max( σ ) (cid:104) x Mi (cid:105) and proceed in the same way whenthe node is I σ,M = I n − k M , ( k j − k M ,...,k M − − k M ) . Finally, after using all the generators of I n, ( k j ,...,k M ) as pivots, we are left with the ideal I n, ( k j ,...,k M − ) . This procedure leadsto the following recursive formula for the Betti number of I n, ( k j ,...,k M ) (we give herethe version for i.i.d. components) α ( I n, ( k j ,...,k M ) ) = β α ( I n, ( k j ,...,k M − ) )+ n − k M − (cid:88) i =0 (cid:18) nk M + i (cid:19)(cid:18) i + k M − k M − (cid:19) p k M + i ≥ M β α − i +1 ( I n − k M − i, ( k j − k M − i,...,k M − − k M − i ) )+ (cid:18) nk M + α − (cid:19)(cid:18) α + k M − k M − (cid:19) p k M + α − ≥ M (cid:32) M − (cid:88) i = j (cid:18) n − k M − ( α − k i − k M − ( α − (cid:19) p k i − k M − ( α − ≥ i (cid:33) + p k M + α ≥ M n − k M (cid:88) i =1 ( i + 1) (cid:18) ı α (cid:19) . (4.1)The complete derivation of this formula is straightforward but somewhat tedious.It is based on the analysis of the branches of the Mayer-Vietoris tree, as describedin Appendix B. Observe that the computation for ( k , . . . , k M ) is done in terms ofcases with strictly less than M levels, and hence the recursion is on the numberof performance levels, and not on the number of variables. The efficiency of thismethod is equivalent to the one in [54]. Remark . There are several algorithms to compute the reliability of generalizedmulti-state k -out-of- n systems. Some of them are restricted to identical indepen-dent components. Among these, the algorithm in [22] is as we have seen enumerative(hence of low efficiency) and applicable to monotonic patterns, the one in [54] is alsoenumerative but more efficient and is applicable to monotonic and non-monotonicpatterns. The algorithm in [7] is non enumerative and more efficient than the pre-vious ones. For the case of independent but not necessarily identical componentsthe algorithm by [53] uses a finite Markov chain imbedding (FMCI) approach andis adequate for small size systems, as is the algorithm in [50]. Other more efficientalgorithms include [7], based on conditional probabilities, or [32] using multi-valueddecision diagrams. Our algebraic approach is enumerative and applicable to bothkind of systems (with independent and identical components and with independentnon identical components) and produces not only the full reliability formulas butalso bounds.4.1. Quality of the algebraic bounds.
For a polynomial ring R = k [ x , . . . , x n ]Hilbert’s syzygy theorem (cf. [15] for instance) states that the length of any reso-lution of an ideal in R is bounded above by n + 1. In our context this means thatthe algebraic method using the Betti numbers of reliability ideals produces a com-pact version of the inclusion-exclusion identity and thus a series of Bonferroni-likebounds for the system’s reliability such that if the system S has n components thenthe reliability formula, given by the Hilbert series numerator of I S , has at most n + 1 summands. Every truncation of this formula provides a bound for the relia-bility. We compare these bounds with the following ones considered in [19] for somegeneralized k -out-of- n multi-state systems. f we denote by y m , m = 1 , . . . , M p the minimal path vectors of a given multi-state system S ,with structure function φ then a lower bound for the reliability of S is given (assuming independent components) by l (cid:48) φ ( p ) = max ≤ m ≤ M p ( n (cid:89) i =1 P ( x i ≥ y mi )) = max ≤ m ≤ M p ( n (cid:89) i =1 p y mi i ) . On the other hand, if the minimal cuts of S are given by z m , m = 1 , . . . , M c thenwe have the lower bound l ∗∗ φ ( p ) = M c (cid:89) m =1 n (cid:97) i =1 P ( x i ≥ z mi )) = M c (cid:89) m =1 n (cid:97) i =1 p z mi +1 i where for real numbers p ∈ [0 ,
1] we define (cid:96) ni =1 p i = 1 − (cid:81) ni =1 (1 − p i ). Example 4.6.
Let k = 4 , k = 2 , k = 1 and let n = 8 , ,
14. Let us consider themulti-state generalized k -out-of- n :G systems I n, (4 , , for the following probabilities,independent but not identical: level c c c c c c c c c c c c c c Table 2.
Probabilities p i,j , i.e. P ( c i ≥ j ) for the components ofseveral generalized multistate k -out-of- n systemsThe number of generators (i.e. number of minimal paths) of each of the systemsconsidered are given in Table 3 we also give the number of minimal cuts. Sytem level S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , Table 3.
Number of minimal paths and cuts for several generalizedmultistate k -out-of- n systems ystem Level l l l l l l l S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , Table 4.
Lower bounds for several generalized multi-state k -out-of- n systems. System Lvl. u u u u u u u u S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , Table 5.
Upper bounds for several generalized multi-state k -out-of- n systems.The results are summarized in tables 4 and 5 in which we consider the probabilityof the system performing at levels 1 to 3. In the tables, column l i indicates alower bound given by the first i summands of the Hilbert series numerator of thecorresponding j -reliability ideal, while column u i denotes an upper bound given bythe first i summands. An asterisk indicates that the bound is sharp. Cells with aminus sign − indicate that the bound is meaningless (i.e. upper bounds above 1 orlower bounds below 0).The results in tables 4 and 5 allow us to discuss the strengths and weaknesses ofour method. First of all, for systems with big number of generators, the first boundsare useless due to the fact that each of the first summands of the compact inclusion-exclusion formula consists of a large number of inner summands. As the number ofvariables increases, we obtain a collection of useful bounds, that compare well withthe bounds considered in [19] as we can see in Table 6. Observe that l ∗∗ φ ( p ) behavesvery well in case we have a multistate parallel system, as is the case in level 3 ofour systems. This is because the minimal cuts are unique in these cases. We haveconsidered low working probabilities in our system, since our bounds are sharper inthis case. In case our probabilities are high we can consider the unreliability of the ual systems and thus obtain close bounds. All our bounds were computed in lessthan one second on a laptop . It is worth noting that the performance of our methoddoes not depend on having identical or non-identical probability distributions in thecomponents of the system.System Lvl. l (cid:48) φ ( p ) l ∗∗ φ ( p ) S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , S , (4 , , Table 6.
Lower bounds considered in [19] for some generalized multi-state k -out-of- n systems5. Binary k -out-of- n system with multi-state components The following multi-state generalization of k -out-of- n systems was introduced in[40]. Let S m,n,k be a system with k components, each of which can be in a set ofstates { , , . . . , m } . S m,n,k is called an m -multi-state k -out-of- n :G system if thesystem works whenever the sum of the states of the n components is bigger than orequal to k . Note that this kind of systems allows k to be bigger than n . This is anexample of a binary system with multi-state components. This kind of systems areuseful to model different situations like the following examples:- A storehouse has n storage facilities each of which has a capacity of m units.At any given time each of the facilities is partially full, leaving a real capacitysmaller than or equal to m units. The system is said to work if it is capableto store a new arriving lot that consists of k storage units.- A set of n pumps and pipes contributes to a global pipe that covers theneeds of a power plant. Each individual pipe may supply water at differentlevels { , . . . , m } and we consider that the system is working if the combinedsupply (sum of all the individual supplies) is above level k .The reliability ideal of S m,n,k , denoted by J m [ n,k ] is generated by all monomials x µ in n variables such that the degree of x µ is k and µ j ≤ m for all 1 ≤ j ≤ n . To obtainthe number of generators of the system (i.e. the minimal working states) and theBetti numbers, needed to compute the reliability function and bounds for it in thealgebraic approach, we can proceed as follows.First, we list all the generators in a precise ordering, following Proposition 3.2.14in [40]: For each i from m descending to 0 and for each variable x j for j from 1 to CPU: intel i7-4810MQ, 2.80 GHz. RAM: 16Gb (we call x j the distinguished variable in each step) we form all monomials x µ suchthat - the first j − i - the variable x j has an exponent equal to i - the remaining last n − j variables have an exponent smaller than or equal to i - the degree of x µ equals k Using this ordering and Corollary 3.2.25 in [40] we can obtain the Betti numbers of J m [ n,k ] using only one more piece of information, namely, for each generator x µ of J m [ n,k ] we need to know the number of variables before x j that have a nonzero exponent in x µ . So when we list the generators of J m [ n,k ] we keep track of how many of the first j − j be the distinguished variable and i ≤ m fixed, the exponent of x j in x µ . Now, for each p between 0 and k − i , which represents the sum of the exponentsof the first j − x µ , and for each l between 0 and j −
1, which representsthe number of variables among the first j − p using l summands eachof which is between 1 and i −
1. This number is called the number of restrictedcompositions of p in l summands between and i − C ( p, l, , i − l nonzero summands among the first j − (cid:0) j − l (cid:1) ways. For each of these choices we have that the exponents ofthe last n − j variables sum up to k − i − p and each of these exponents is between0 and i . The number of such compositions is C ( k − i − p, n − j, , i ). Hence, puttingall these considerations together we have the following result. Lemma 5.1.
The number of generators of J m [ n,k ] is (5.1) N m [ n,k ] = k (cid:88) i =0 n (cid:88) j =1 k − i (cid:88) p =0 j − (cid:88) l =0 C ( p, l, , i − (cid:18) j − l (cid:19) C ( k − i − p, n − j, , i ) . All these generators have degree k , hence β ,k ( J m [ n,k ] ) = N m [ n,k ] and β ,j ( J m [ n,k ] ) = 0 forall j (cid:54) = k . Each generator contributes to β i,k + i ( J m [ n,k ] ) with (cid:0) n − l − i (cid:1) elements, hencethe formula for the Betti numbers of J m [ n,k ] is (5.2) β i,k + i ( J m [ n,k ] ) = k (cid:88) i =0 n (cid:88) j =1 k − i (cid:88) p =0 C ( p, l, , i − (cid:18) j − l (cid:19) C ( k − i − p, n − j, , i ) (cid:18) n − l − i (cid:19) and β i,j ( J m [ n,k ] ) = 0 if j (cid:54) = k + i .Remark . The number of restricted compositions of an integer with a given num-ber of bounded summands can be obtained using a certain generating function,as shown in [1, 13, 18]. The following closed formula for some types of restricted ompositions can be found in Theorem 2.1 in [24] which can be used to explic-itly compute the numbers in Lemma 5.1 using that C ( k − i − p, n − j, , i ) = C ( k − i − p + n − j, n − j, , i + n + j ): C ( n, k, , b ) = (cid:88) i = α ,i ,...,i b max { ,α j }≤ i j ≤ min { β j ,γ j } b (cid:89) l =2 (cid:18) k − (cid:80) l − j =2 i j i l (cid:19) , where α j = n − k ( j − − b (cid:88) l = j +1 ( l − j + 1) i l β j = k − b (cid:88) l = j +1 i l γ j = (cid:98) n − k − (cid:80) bl = j +1 ( l − i l j − (cid:99) . In order to obtain the necessary information to construct the reliability polyno-mial and bounds from the Betti numbers of J m [ n,k ] we need their multigraded version.For this, let x µ a minimal generator of J m [ n,k ] and x j its distinguished variable. Let( x i , . . . , x i l ) be the l variables among the first j − x µ . Let P x µ = { x , . . . , ˆ x j , . . . , x n } \ { x i , . . . , x i l } . Then the multidegreesof the contribution of x µ to β i,k + i ( J m [ n,k ] ) are x µ (cid:81) x i ∈ σ x i for each subset σ of P x µ ofcardinality i . Observe that the resolution of J m [ n,k ] is k -linear, i.e. β i,j J m [ n,k ] = 0 forall j (cid:54) = k + i . Example 5.3.
Let S be a system with 4 components, each of which has possiblestates { , , , } such that the system is working whenever the sum of the statesof the components is bigger than or equal 5. The ideal of this system is J , ⊆ R = k [ x, y, z, t ] and is minimally generated by the following 40 monomials, sortedas described before. i = 3 i = 2 x x yt, x zt, x yz, x y , x z , x t x y z, x y t, x yz , x yt , x z t, x zt , x yzty y zt, y z , y t , xy z, xy t, x y y z t, y zt , xy zt, xy z , xy t z z t , xz t, yz t, xyz , x z , y z xz t , yz t , xyz tt xyt , xzt , yzt , x t , y t , z t xyzt And from this we have that β , ( J , ) = 40, β , ( J , ) = 92, β , ( J , ) = 72, β , ( J , ) = 19 and β i,j ( J , ) = 0 otherwise. Observe that, for instance, the multi-degrees of the two contributions of xz t to β , ( J , ) are xyz t and xz t , and themultidegree of its contribution to β , ( J , ) is xyz t since P xz t = { y, t } .We finish with an example of application of these systems. .1. Storage problem using binary k -out-of- n systems with multi-statecomponents. Binary k -out-of- n systems with multi-state components can be usedto model storage problems in which the storage capacity is distributed among sev-eral containers. To illustrate this, let S be the set of n tanks in a wine cellar wheregrape is received in the harvesting season. Each of the tanks T i , i = 1 , . . . , n has atotal capacity of C i tons and when a tractor arrives at the cellar, the staff distributesthe the new coming grapes among different tanks so that the wine produced in thetanks is sufficiently homogeneous in terms of the origin of the grapes.The filling procedure is the following: let G be the number of loads of grapesin the incoming tractor (a load consists of 100Kg). We use a discrete measure oftime, namely time t means that we have already stored in the tanks the grapes of t tractors. We denote by l t a measure of the level of the set of tanks after time t .We can consider l t as the average of the levels of each of the tanks, the minimumor the maximum among them. We choose a level l ≤ min { C , . . . , C n } that we donot want to pass after storing the new coming grapes. Let m = l − l t and observethat in principle l is chosen so that m < G . Among all the possibilities to performthe required load, we choose one randomly. Let us denote by p ti,j the probabilitythat at time t the empty space in tank T i is at least j . We have that p ti, = 1for all i and p ti,j ≥ ≤ j ≤ m . If one or more of the tanks is full attime t we continue with the same procedure on the remaining tanks. Our goal is tostudy the probability p ( l ) , l > l t that we can store the G new coming grape loadsin the n tanks so that no tank is filled beyond l and assuming all tanks are alreadyfilled to level l t . This situation can be modeled by a binary G -out-of- n system withmulti-state components, in which each component can be in states { , . . . , m } . Example 5.4.
Consider a cellar with n = 5 tanks with a capacity of 15 tons each.After a certain time t the maximum level on any of the tanks is 12 . p ( l )behaves for l > p i,j as p i,j = 1 − ( j ) / for all i , and 0 ≤ j ≤
15, and p i,j = 0 if j >
15, i.e. in our case all tanks have thesame probability distribution. Under these conditions we have a binary 15-out-of-5 system with multi-state components such that each component can be in states { , . . . , m = l − } for each l . Using the results in Section 5 we have that theideal of this system is J m [5 , . The number of generators of this ideal, according tothe formula given in Lemma 5.1, gives the number of different ways to allocate thegrapes meeting the requirements of the described procedure. Taking into accountthe probabilities of each of the tanks, we can compute the probability that we canmeet the requirements using the multigraded Betti numbers as computed in Lemma5.1. We used an implementation of the formulas (5.1) and (5.2) and algorithms toobtain the set of generators and Hilbert series of the corresponding ideals within thecomputer algebra system Macaulay2 [31]. The results are shown in Figure 1 andTable 7, in which we also show the time (in seconds) taken for the computation ofthe full list of multigraded Betti numbers, from which we compute the probabilityin each case.
26 128 130 132 134 136 138 140max level l0.00.20.40.60.81.0 p r o b a b ili t y p ( l ) Figure 1.
Probability that we can fill the 5 tanks in Example 5.4 upto level at most l for l from 125 to 140.6. Conclusions and further work
The paper shows how to apply the authors’ work on algebraic reliability to multi-state problems. The key to the extension is to find the right monomial ideal for asuitable generalization of a k -out-of- n system. From this the main technical prob-lem is to find the Betti numbers which give tight reliability bounds: generalizedextensions of Bonferroni bounds. Multi-unit storage, an increasingly important ap-plication, has a natural multi-state description and results are given for some simpleexamples.The methods of this paper should be extendable to any multi-state systems inwhich there is an identifiable state, or collections of states, which indicates a level ofdegradation of the system and for which extremal state may lead to the failure of thesystem. There are two parts of the theory, one based the algebra and combinatoricsof the system and its degradation and the other the stochastic behaviour of thesystem.Future work, therefore, will concentrate on both parts of the theory: algebraicand stochastic and, of course the interplay between the two. We are aware thatstochastic processes are indexed by time and that therefore the works should givegreater priority to the time behaviour bringing in, at least, the standard models offailure. For the algebraic side each ”special” state or pattern is likely to lead todifferent algebra, that is a different ideal or collection of ideals. On the stochasticside we are eager to allow the behaviour systems to be controlled by causal graph(network) based stochastic models, partly because they too are increasingly coveredby algebraic theory, [49]. Multi-state modeling has become increasingly part of areassuch as disease modeling and emergency planning, often under a heading of com-ponent and system degradation. Future research will continue to combine Markov evel l p ( l ) . . . . . . . . . . . . . . . . . . . . . . . . . Table 7.
Probabilities, number of generators and times to computemultigraded Betti numbers for the data in Example 5.4and other models of movement between states with the ideal theory describing thedetailed structure of failure.Finally, we should declare that the importance of energy storage, and energynetworks, is likely to lead to more work in that area. We hope also to facilitate theapplication to genomics, with suitable collaborations.
Acknowledgments
The authors are partially funded by grant MTM2017-88804-P of Ministerio deEconom´ıa, Industria y Competitividad (Spain).
Appendix A. A very short introduction to the algebraic method inreliability
In order to illustrate the algebraic method for system reliability analysis, we willuse a simple example in which we will use all the concepts involved. A generaldetailed description and plenty of more elaborate examples can be found in [42, 43,44, 45, 46, 34] where the interested reader can find full proofs of the relevant resultsfor this approach.Our simple example is a multi-state parallel system S depicted in Figure 2 (i.e. itis a 1-out-of-2 multi-state system). Let { c , c } be the components of S and for eachcomponent let S = { , , } and S = { , , , } be the performance levels of c and c respectively. The structure function of S is given by φ ( s ) = max { s , s } for s = ( s , s ) ∈ S × S . Since we have a two-component system, we can algebraicallymodel its states in a polynomial ring with two variables, R = k [ x , x ] with k a c c Figure 2.
Multi-state parallel systemsuitable field, we can consider k = R . First of all, we observe the correspondencebetween states of the system S and monomials in R . states c states c (0,0) (1,0) (2,0)(0,1)(0,2)(0,3) (1,3) (2,3)(2,2)(2,1)(1,2)(1,1) (a) State space of the system S x x x x x x x x x x x x x x x x x x (b) Equivalence between state space andmonomials
Figure 3.
Relation between state space of the system and monomialsFigure 3a shows the state space of system S i.e. { ( s , s ) : s ∈ S and s ∈ S } .Now, we make each state ( s , s ) ∈ S × S correspond with the monomial x s x s in R . These monomials are represented in 3b so that the correspondence becomesclear.Let us consider now the j -working states of S for each j , i.e. F S,j consists of thetuples s = ( s , s ) such that φ ( s ) ≥ j, j ∈ S . We have F S, = { (0 , , (0 , , (0 , , (1 , , (1 , , (1 , , (1 , , (2 , , (2 , , (2 , , (2 , } , F S, = { (0 , , (0 , , (1 , , (1 , , (2 , , (2 , , (2 , , (2 , } , F S, = { (0 , , (1 , , (2 , } . The minimal j -working states, denoted F S,j are the tuples in which if any com-ponent decreases its performance level, the performance of all the system decreasesto j (cid:48) < j . Then, we obtain S, = { (0 , , (1 , } , F S, = { (0 , , (2 , } , F S, = { (0 , } . Having the relation between tuples of components’ states and monomials intoaccount and the coherence property of the system, we have that the j -workingstates correspond to the monomials in an ideal of R which we will denote I S,j it iseasy to see that the unique minimal monomial generating set of I S,j , the j -reliabilityideal of S is the one corresponding to the minimal j -working states of the system.In our example we have that I S, = (cid:104) x , x (cid:105) ,I S, = (cid:104) x , x (cid:105) ,I S, = (cid:104) x (cid:105) . That ideals are represented in Figures 4a, 4b and 4c respectively. x x (a) S x x (b) S x x (c) S Figure 4. j -reliability ideals for system S Observe that while the set of possible states of the system is finite, we have an in-finite number of monomials in our ideal. We will deal with this issue when assigningthe probability distribution to the system’s components and describe its reflectionat the ideal level. A powerful tool in commutative algebra to describe the structureof a monomial ideal is the Hilbert series, which is a short way to enumerate the setof monomials in a monomial ideal. It is based on the inclusion-exclusion principleand consists in adding up all the multiples of the minimal generators of the ideal,substract the multiples of the pairwise least common multiple of minimal generators,add again the multiples of the threefold least common multiples of minimal gener-ators, and so on. There are compact ways to obtain the Hilbert series, which are eyond the scope of this paper. For full details we refer the reader to the referencesat the beginning of this Appendix.Finally, to use the Hilbert function in order to obtain the j -reliability of thesystem we assign probabilities to monomials. Let’s say that p i,j is the probabilitythat component i is in state at least j , we then assign to the j ’th power of variable i the probability p i,j and the probability of a monomial is given by the product of theprobabilities assigned to its individual powers. Observe that if a variable is raised toa power that does not correspond to any state of the corresponding component, thenits assigned probability is 0 and this removes all except a finite set of monomialsfrom the final result, except exactly those corresponding to possible states of thesystem.As for this example, let us assign p , = 0 . , p , = 0 . , p , = 0 . , p , = 0 . , p , =0 . H I S, = x + x − x x . Graphi-cally, this can be seen as: • The ideal (cid:104) x (cid:105) contains the monomials in the shaded area in Figure 5a • The ideal (cid:104) x (cid:105) contains the monomials in the shaded area in Figure 5b • The ideal (cid:104) x x (cid:105) (i.e. generated by the pairwise least common multiples ofthe generators of the ideal -just one such pair in this case-) contains themonomials in the shaded area in Figure 5cAssigning the corresponding probabilities to the monomials in H I S, we obtainthat the 1-reliability for S is 0 . H I S, = x + x − x x and H I S, = x , respectively and the2-reliability of S is 0 .
38 and the 3-reliability of S is 0 . x x (a) Monomials in (cid:104) x (cid:105) x x (b) Monomials in (cid:104) x (cid:105) x x (c) Monomials in (cid:104) x x (cid:105) Figure 5.
Using H I S, to obtain the monomials in (cid:104) x , x (cid:105) Appendix B. Mayer-Vietoris trees
Let I ⊆ S = k [ x , . . . , x n ] be a monomial ideal and G = { g , . . . , g r } a monomialgenerating set (unless otherwise stated we will always consider that G is the uniqueminimal monomial generating set of I ). Fix any numbering of the elements in G nd let I i = (cid:104) g . . . , g i (cid:105) be the subideal generated by the first i generators of I . Foreach i we have the following exact sequence(B.1) 0 −→ I i − ∩ (cid:104) g i (cid:105) j −→ I i − ⊕ (cid:104) g i (cid:105) l −→ I i −→ . Assume that free resolutions F (cid:48) i and (cid:101) F i are known for I (cid:48) i = I i − and (cid:101) I i = I i − ∩ (cid:104) g i (cid:105) respectively. Then, a (not necessarily minimal) resolution F i of I i is obtained as themapping cone of the chain complex morphism ψ : (cid:101) F i −→ F (cid:48) i that lifts the inclusion j , cf. [8, 20].Using recursively sequence (B.1) on i we can compute a free resolution F = F r of I that is called an iterated mapping cone resolution . Observe that this processpreserves (multi) degrees. The ideals involved in this process can be displayedas a binary tree. The root of this tree is I and every node J = (cid:104) f . . . , f j (cid:105) has J (cid:48) = (cid:104) f , . . . , f j − (cid:105) as right child and (cid:101) J = J (cid:48) ∩ (cid:104) f j (cid:105) as left child. This is called a Mayer-Vietoris tree of I , cf. [41].Each node in a Mayer-Vietoris tree is assigned a position and a dimension. Theroot has position 1 and dimension 0 and the right and left children of a node withposition p and dimension d are given positions 2 p + 1 and 2 p respectively and di-mensions d and d + 1 respectively. We say that a node is relevant if it is either theroot or if its position is even. The multidegrees of the minimal generators of therelevant nodes of dimension d in a Mayer-Vietoris tree are then the multidegreesof the generators of the d -th module of the iterated mapping cone resolution F of I described by the tree. Let MVT( I ) d,µ be the set of the positions of the relevantnodes of dimension d of a given Mayer-Vietoris tree of I having x µ as a minimalgenerator. If a monomial x µ appears only once as generator of a relevant node in thetree then if d is the dimension of that node and p its position let MVT( I ) (cid:48) d,µ = { p } otherwise MVT( I ) (cid:48) d,µ = ∅ for all d . Note that if MVT( I ) (cid:48) d,µ is not empty, thenMVT( I ) (cid:48) d,µ = MVT( I ) d,µ . Since the minimal free resolution of I is a subresolutionof F we have that for any Mayer-Vietoris tree the following result holds [41]. Proposition B.1.
For any Mayer-Vietoris tree of I I ) (cid:48) d,µ ≤ β d,µ ( I ) ≤ I ) d,µ . The generators of the relevant nodes of MVT( I ) provide upper and lower boundsfor the Betti numbers of the ideal without actually computing the resolution. Thesebounds can be improved using several criteria and are sharp in several families ofideals, see [41] for details. A simple useful criterion is the following: Proposition B.2.
Let µ be a multidegree such that there are generators of multi-degree µ in relevant nodes of MVT( I ) of dimensions d . . . d k such that no two ofthem are consecutive, then β d i ,µ ( I ) = I ) d i ,µ . We say that two generators e ( i ) σ and e ( i − τ of F with the same multidegree form a reduction pair if the coefficient of e ( i − τ in ϕ ( e ( i ) σ ) is a non-zero scalar, i.e. if we canreduce F by deleting e ( i ) σ and e ( i − τ and adjusting ϕ i . Reduction pairs appear only in ompatible nodes. Let J and J (cid:48) two nodes of MVT( I ) whose first common ancestoris K and such that J is a descendant of (cid:101) K and J (cid:48) is a descendant of K (cid:48) we say J and J (cid:48) are compatible if dim( J ) − dim( (cid:101) K ) = dim( J (cid:48) ) − dim( K (cid:48) ). Compatibilityof J and J (cid:48) can be read from the binary expression of their positions. We cantherefore ensure that β d,µ ( I ) is bigger than or equal to the number of generators ofmultidegree µ in relevant nodes of dimension d in MVT( I ) such that they have nocompatible generator. Hence, if there are no compatible generators, we obtain theBetti numbers of I directly from MVT( I ). Example B.3.
Let us consider Mayer-Vietoris trees of ideals of consecutive linear k -out-of- n :G systems. Theses systems work if at least k consecutive componentsof the n components of the system work. The corresponding ideal is of the form I k,n = (cid:104) x · · · x k , . . . , x n − k +1 · · · x n (cid:105) . The Mayer-Vietoris tree of the ideal of theconsecutive linear 2-out-of-5 system, taking as pivot always the last generator, is(1 , x x , x x , x x , x x (2 , x x x x , x x x (4 , x x x x x (5 , x x x x (3 , x x , x x , x x (6 , x x x (7 , x x , x x (14 , x x x (15 , x x From this tree we obtain that β , ( I , ) = 4, β , ( I , ) = 3, β , ( I , ) = 1 and β , ( I , ) = 1. Moreover, the numerator of the Hilbert series of this ideal is HN I , = ( x x + x x + x x + x x ) − ( x x x x + x x x + x x x + x x x )+ x x x x x As one can see, the node at position 3 of
M V T ( I k,n ) is just I k,n − so the contributionof this branch of the tree is just a smaller case of the same kind. The analysis ofthe other branch of the tree is also straightforward and we can easily come up witha recursive formula for the Betti numbers of I k,n as was shown in [42]. Using thiskind of reasoning on Mayer-Vietoris trees we come out with recursive formulas like(4.1). References [1] Abramson M.
Restricted combinations and compositions , Fibonacci Quarterly 14:5 (1976), pp.439–452[2] Al-Seedy R., Habib A. and Radwan, T.,
Reliability evaluation of multi-state consecutive k-out-of-r-from-n: G system.
Applied Mathematical Modelling, Vol 31 (2007), pp. 2412–2423.[3] Al-Seedy R., Elsherbeny A., Habib, A. and Radwan, T.,
Bounds for increasing multi-stateconsecutive k-out-of-r-from-n: F system with equal components probabilities.
Applied Mathe-matical Modelling, Vol 35 (2011), pp. 2366-2373.
4] Amari S. V., Zuo M. J., Dill G.
A fast and robust reliability evaluation algorithm for generalizedmulti-state k -out-of- n systems. IEEE Transactions on Reliability, Vol 58, No 1 (2009), pp. 88–97.[5] Amari, S. V., Dugan, J. B., Xing, L. and Mo, Y.
Efficient analysis of multi-state k-out-of-nsystems.
Reliability Engineering & System Safety, Vol 133, (2015), pp. 95-105.[6] Boedigheimer R. A., Kapur K. C.
Customer-driven reliability models for multi-state coherentsystems.
IEEE Transactions on reliability, Vol 43, No 1 (1994), pp. 96–50.[7] Chaturvedi S. K., Besha S. H., Amari S. V., Zuo M. J.
Reliability analysis of generalizedmulti-state k -out-of- n systems. J Risk Reliability, Vol 226, No 3 (2012), pp.327–336.[8] Charalambous H. and Evans E.G.,
Resolutions obtained as iterated mapping cones , Journal ofAlgebra 176 (1995), pp. 750–754[9] Ding, Y., Lisnianski, A., Li, W. and Zuo, M. J.
A framework for reliability approximationof multi-state weighted k-out-of-n systems.
IEEE Transactions on Reliability, Vol 59, No. 2(2010), pp. 297-308.[10] Ding, Y., Li, W., Tian, Z. and Zuo, M. J.
The hierarchichal weighted multi-state k-out-of-n system model and its application for infrastructure management.
IEEE Transactions onReliability, Vol 59, No. 3, (2010), pp. 593-603.[11] Dolan, D., Zupanic, A., Nelson, G., Hall, P., Miwa, S., Kirkwood, T. and Shanley, D.P.
Integrated stochastic model of DNA damage repair by non-homologous end joining andp53/p21-mediated early senescence signalling , PLoS computational biology, Vol. 11, No. 5(2015), e1004246[12] Eger, S.
Review of recent advances in reliability of consecutive k -out-of- n and related systems ,Journal of Risk and Reliability Vol 224, No. 3 (2010), pp. 225-237.[13] Eger, S. Restricted weighted integer compositions and extended binomial coefficients , Journalof Integer Sequences 16 (2013), article 13.1.3[14] Eger, S.
Reliability analysis of multi-state system with three-state components and its applica-tion to wind energy , Reliability Engineering and System Safety Vol 172 (2018), pp. 58-63.[15] Eisenbud, D.
Commutative algebra with a view towards algebraic geometry , Springer, 1995[16] El-Neweihi E., Proschan F., Sethuraman J.
Multi-state coherent system.
J Applied Probability,Vol 15 No 4 (1978), pp. 675–688[17] Fenton, N. and Bieman, J.
Software Metrics: A Rigorous and Practical Approach, ThirdEdition , CRC Press, 2014[18] Flajolet, P. and Sedgewick, R.
Analytic combinatorics , Cambridge University Press, Cam-bridge, 2009[19] Gasemir, J. and Natvig, B.,
Improved availability bounds for binary and multi-state systemswith independent component processes , Journal of Applied Probability 54(3), (2017), pp. 750–762.[20] Herzog J., and Takayama Y.,
Resolutions by mapping cones , Homology, Homotopy and Ap-plications, 4 (2002), pp. 277–294.[21] Huang, J. and Zuo, M. J. and Fang, Z.,
Multi-state consecutive k-out-of-n systems.
IIE Trans-actions, Vol 35 (2003), pp. 527–534[22] Huang J., Zuo M. J., Wu Y.
Reliability evaluation of combined k -out-of- n :f, consecutive- k -out-of- n :f and linear connected- ( r, s ) -out-of- ( m, n ) :f system structures. IEEE Transactions onreliability, Vol 49, No 1 (2000), pp. 99–104.[23] Huang J., Zuo M. J., Wu Y.
Generalized Multi-state k -out-of- n :G systems. IEEE Transactionson reliability, Vol 49, No 1 (2000), pp. 105–111.[24] G. Jaklic, V. Vitrih and E. Zagar,
Closed form formula for the number of restricted composi-tions , Bull. Aus. Math. Soc. 81 (2010), pp. 289–297[25] Kumar, A. and Singh, SB.
Computations of the signature reliability of the coherent system ,Vol. 34, No. 6 (2017), pp. 785-797[26] Kuo W., Zuo M. J.
Optimal reliability modeling: principles and applications.
New York: JohnWiley & Sons; 2003.
27] Lisnianski, A. and Ding, Y.
Redundancy analysis for repairable multi-state system by usingcombined stochastic processes methods and universal generating function technique , ReliabilityEngineering & System Safety, Vol. 94, No. 11 (2009), pp. 1788–1795[28] Lisnianski, A., Frenkel, I. and Karagrigoriou, A.,
Recent advances in multi-state systems reli-ability: Theory and applications , Springer; 2017.[29] Lisnianski, A. and Levitin, G.
Multi-state system reliability: Assesment, Optimization andApplications.
World Scientific Publishing; 2003.[30] Liu, Y., Pedrielli, G., Li, H., Lee, L. H., Chen, C-H. and Shortle, J. F.
Optimal ComputingBudget Allocation for Stochastic N– k Problem in the Power Grid System.
IEEE Transactionson Reliability, Vol. 68, No.3 (2019) pp. 778–789.[31] Grayson, D. R., and Stillman, M. E.,
Macaulay2, a software system for research in algebraicgeometry , Available at [32] Mo Y., Liudong X., Amari A. V., Bechta J.
Efficient analysis of multi-state k -out-of- n systems. Reliability Engineering and System Safety, Vol 133 (2015), pp. 95–105.[33] Mohammadi, L.,
The joint reliability signature of order statistics , Communications inStatistics-Theory and Methods (2019), pp.1–22[34] Mohammadi, F., Pascual-Ortigosa, P., S´aenz-de-Cabez´on, E. and Wynn, H.P.
Polarizationand depolarization of monomial ideals with application to multi-state system reliability , Journalof Algebraic Combinatorics (2019), DOI: 10.1007/s10801-019-00887-6[35] Cui, L., Mo, Y., Si, S. and Xing, L.
MDD-based performability analysis of multi-state linearconsecutive-k-out-of-n: F systems.
Reliability Engineering & System Safety, Vol 166 (2017),pp. 124-131.[36] Natvig B. multi-state systems reliability theory with applications.
John Wiley & Sons; 2011.[37] Ram, M. and Dohi, T.
Systems Engineering: Reliability Analysis Using K-out-of-n Structures.
CRC Press; 2019[38] Rizk, G., Lavenier, D.and Chikhi, R.
DSK: k-mer counting with very low memory usage ,Bioinformatics, Vol. 29, No. 5 (2013), pp. 652–653[39] Rushdi, A. M. A.,
Utilization of symmetric switching functions in the symbolic reliabilityanalysis of multi-state k-out-of-n systems , International Journal of Mathematical, Engineeringand Management Sciences (IJMEMS), Vol4, No2 (2019), pp. 306–326.[40] S´aenz-de-Cabez´on, E.,
Combinatorial Koszul Homology: Computations and Applications , PhDThesis, Universidad de La Rioja, 2008.[41] S´aenz-de-Cabez´on, E.,
Multigraded Betti numbers without computing minimal free resolutions ,Applicable Algebra Eng. Commun. Comput., vol. 20 (2009), pp. 481–495.[42] S´aenz-de-Cabez´on, E. and Wynn, H. P.,
Betti numbers and minimal free resolutions for multi-state system reliability bounds , Journal of Symbolic Computation 44 (2009), pp. 1311–1325.[43] S´aenz-de-Cabez´on, E. and Wynn, H. P.,
Mincut ideals of two-terminal networks , ApplicableAlgebra Eng. Commun. Comput., vol. 21 (2010), pp. 443–457.[44] S´aenz-de-Cabez´on, E. and Wynn, H. P.,
Computational algebraic algorithms for the reliabilityof generalized k -out-of- n and related systems , Math. Comput. Simulation, vol. 82, no. 1 (2011),pp. 68–78.[45] S´aenz-de-Cabez´on, E. and Wynn, H. P., Algebraic reliability based on monomial ideals: Areview , in Harmony of Gr¨obner Basis and The Modern Industrial Society, Wiley and sons(2012), pp. 314–335.[46] S´aenz-de-Cabez´on, E. and Wynn, H. P.,
Hilbert functions for design in reliability , IEEE Trans-actions on Reliability, vol. 64, no. 1 (2015), pp. 83–93.[47] Singh, Ch., Jirutitijaroen, P. and Mitra, J.
Introduction to Power System Reliability.
Wiley-IEEE Press, 2019[48] Sturmfels, B., Trung, N. V. and Vogel, W.,
Bounds on degrees of projective schemes , Math.Ann., 302(3) (1995), pp. 417–432[49] Sullivant, S.
Algebraic Statistics , American Mathematical Soc., 2018
50] Tian Z., Zuo M.J. and Yam R.,
Multi-state k-out-of-n systems and their performance evalua-tion , IIE Transactions 41 (2008), pp. 32–44[51] Yeh, W-Ch.
The k-out-of-n acyclic multistate-node networks reliability evaluation using theuniversal generating function method , Reliability Engineering & System Safety, Vol. 91, No. 7(2006), pp. 800–808[52] Yingkui, G. and Jing, L.
Multi-State System Reliability: A New and Systematic Review ,Procedia Engineering 29 (2012), pp. 531–536.[53] Zhao X., Cui L. R.
Reliability evaluation of generalized multi-state k -out-of- n systems basedon FMCI approach. Int J Syst Sci , Vol. 41(2010), pp. 1437–1443.[54] Zuo M J, Tian Z.
Performance evaluation of generalized multi-state k -out-of- n systems. IEEETransactions on Reliability, Vol 55, No 2 (2006), pp. 319–327.
Departamento de Matem´aticas y Computaci´on, Universidad de La Rioja, Spain
E-mail address : [email protected] Departamento de Matem´aticas y Computaci´on, Universidad de La Rioja, Spain
E-mail address : [email protected] Department of Statistics, London School of Economics, UK
E-mail address : [email protected]@lse.ac.uk