REVISTA INVESTIGACION OPERACIONAL Vol., No. 3, 000 A NOTE ON THE PREDICTION AND TESTING OF SYSTEM RELIABILITY UNDER SHOCK MODELS C. Bouza, Departaento de Mateátca Aplcada, Unversdad de La Habana ABSTRACT The realblty of fnte systes s studed. The functonng of a syste s characterzed by a vector of bnary varables. The a pror dstrbuton of the rando vector perts to establsh a lnear probablty odel. It s used for dervng a predctor of the realblty and ts error. Under a set of ld condtons T- Student based nferences can be ade when a suffcently large saple sze s uded. The ndependent case and a shock dependent odel are studed. Key words and phrases: realblty, Bayesan procedures, predcton. AMS: 6N05. INTRODUCTION RESUMEN La fabldad de ssteas fntos es estudada. Su funconaento es caracterzado por vectores de varables bnaras. La dstrbucón a pror de los vectores aleatoros perte establecer un odelo probablístco lneal. Este es utlzado para dervar un predctor de la fabldad y de su error. Bajo un conjunto de condcones suaves se pueden realzar nferencas basadas en la T-Student cuando el taaño de la uestra es sufcenteente grande. Palabras clave y frases: fabldad, procedento Bayesano, predccón. A systec analyss, of dfferent probles, provdes a theoretc frae that allows to use coon technques for predctng the probablty that t works. We wll assue that we cope wth a fnte set of enttes whch are clustered n coponents. A certan functon, evaluated n the observed workng condton of the enttes, evaluated the functonng of the syste. For each evaluaton of the functon a set of weghts can be coputed. The realblty can be evaluated when the nvolved probabltes are known, whch s the coon case. The a pror nforaton perts to use Bayes Theore and soe predctors are proposed. Secton s devoted to the descrpton of the syste and of the realblty of ts functonng. The ndependence of the coponents s assued. Under a lnear probablty odel a predctor s proposed. It s odel ubased. If a suffcently large nuber or realzatons of the syste are observed the related test statstc s a T-test. Secton 3 ntroduces the noton of shock. The ndependence s not longer vald. The realblty can be predcted when an adequate rando experent s perfored. Agan T-Student based nferences can be pleented. Secton 4 dscusses soe exaples, where ths approach can be used. They odel envronental, econoc and techncal probles.. MODELING THE FUNCTIONING Take a fnte set ζ = {,...,n} and a partton ℵ = {A} defned on t, [ A ℵ A = ζ A A = Φ f A, A ℵ],. Each A functons correctly (FC) wth a probablty p, P = {p : A}. A set A ℵ functons correctly (A - FC) f every A FC. Hence π A = Π A p s the probablty that A - FC. The perforance of s characterzed by a bnary rando varable X = 0 f FC otherwse eal:bouza@atco.uh.cu 50
It s a Bernoull rando varable.then A - FC whenever X A = Π A x = To analyze the functonng of a syste Ψ = {ζ, φ} s the objectve of ths paper. The functon φ(x) = 0 f ζ FC otherwse evaluates the status of the syste where X = {X A : X A = (X,...,X na ) T &A ℵ} n A = nuber of coponents of A. An adequate set of weghts W = {δ(a): A ℵ} can be deterned by solvng the equaton φ(x) = Σ A ℵ δ(a)π A X The functon h(p) = Σ A ℵ δ(a)π A p easures the realblty of the syste f the coponents of A are ndependent. Note that ths odels, n techncal probles, the fact that ζ represents a devce wth equvalent parallel coponents. Then t functons f at least one A works. Hence δ(x A ) = for at least one A ℵ ζ - FC and the probablty that t functons s gven by E[φ(X)] = h(p). Dfferent econocal, techncal and socal systes use weghts, for ther coponents, that are not necessarly equal to one. Engeland-Huseby (99) denoted by h(p) the realblty of a network syste. P s naed 'realblty vector'. It ay be unknown. Generally the syste s constructed but ts perforance ust be studed. Hence, the decson aker (DM) observes a rando X and evaluates φ(x), whch s an unbased estate of h(p). Then ts error s the varance V[φ(X)] = Σ A ℵ δ (A)Π A p ( - p ) f the rando varables X are utually ndependent. For obtanng the saple the D desgns an experent and w,...,w rando and ndependent events, fro the correspondng probablty space (Ω, σ, μ), are observed. We copute the saple estates φ[x(w )],...,φ[x(w )]. An estator of the ultlnear P-functon h(p) s. ĥ(p) = wth error t= φ[x(w t )] 5
A.. V [ ĥ(p) ] δ (A) Π p ( p ) = A ℵ Note that ths estate s ore accurate than the prevous one. Experts can cobne ther opnons about the bahavor of the coponents of ζ. Gasyr-Natug (996) analyzed ths proble by usng a Standard Bayes Theore approach. Scott (977) proposed a Bayesan ethod for fnte populaton nferences. We wll use both deas for odelng a Bayesan approach to ths proble. Suppose that a vector of real nubers Z = (Z,...,Z N ), the data (D), wch s related wth X s copletely or partally known. Usng a saplng desgn d(ο: Z), whch depends on the Z - vector, we observe, the saple s. Gasyr-Natug (995) assued that P s unknown but that the DM s able to odel the uncertanty n ters of a pror dstrbuton Q. The use of ths pror yelds Q ( = d(s : D Z:X(w) ) Ω d(s : Z)Q(X(w) : Z) Z)Q(X(w) : 0 Z) μ(x(w) : Z) f X(w) Ω otherwse perts to use Bayes Theore. w denotes a rando event that dentfes the actual populaton, μ s the Lebesgue easure and D denote the data set. If Z s avalable at the data analyss stage Q D (X(w): Z) s proportonal to Q(X(w): Z). Note that E[h(p)] = P[Φ(X) = : Z]Q D (X(w : Z)) The saplng desgn s unportant for the nferences. Hence the DM should only observe experental ponts generated by soe devse and to use the nforaton provded by the data through the guessed pror. Another ethod s to use the functonal relaton between X(w) and Z. The DM ay postulate that for the pror Q[X(w)] the observatons are d rando vectors dstrbuted wth E(X(w) : Z) = βz t + ε and σ Cov [ X (w), X (w):z] = t t t 0 f t = t otherwse In ters of our proble we can wrte the lnear probablty odel by fxng A as an ndex: () X A = βz A + ε A where Z s a known varable, or at least t can be easured, β s an unknown paraeter and ε s an unobservable rando varable wth odel expectaton E μ (ε A ) = 0 and covarance Cov μ [ ε, ε ] A σ = A A 0 = Π A p ( p ) = Π σ A f A = A otherwse under the hypothess of ndependence of the odel resduals. The easureent of the Z A 's perts to predct 5
P(X A = ) = βz A Note that we are dealng wth s a Generalzed Lnear Model wth bnary rando coponent and dentty lnk functon, see Agrest (996). The use of φ(x) as a predctor of h(p) was suggested prevously. As the saplng desgn s non nforatve we can rely on the results of a reasonable experental desgn for obtanng the requred nforaton. Proposton. Suppose that a certan experental desgn generates rando results of a syste ζ wth a odel structure ζ = Σ A ℵ A. Take the functon φ 0 [X(w t )] and the odel gven by (). The hypothess that ζ s relable [h(p) γ] can be tested by usng a ℵ ĥ(p) γ δ (A) Π A pˆ ( pˆ ) = t(p) where: t= ĥ(p) = pˆ = = φ[x(w )] X t Proof: φ(w t ) s a Bernoull rando varable wth weght w =. If P = (p,...,p n ) s unchanged durng the experent then h(p) = E μ [ĥ(p)]. Take the weghts θ = w w n = w n where w = w then s also the 'equvalent saple sze'. Then t(p), for a suffcently large follows a = T-Student dstrbuton wth approxatelly - degrees of freedo, see Bouza (995) an the proposed test can be perfored 3. SHOCK MODELS A syste can be affected n ts perforance by external causes that we wll denonate as 'shocks'. Forally t s gven by: Defnton. A shock s an applcaton S j A a v(a) that represents an exogeneous cause that destroys all A. Note that the nfluence of nternal causes s not odeled. For exaple ths, eans that the products are out of the 'nfant ortalty perod' or that the coalton does not reach to a consensus because of dscrepances aong the coaltoned agents. Followng the deas of Gaseyr-Natvg (995) we have: ℵ A = { S j A : j =,...,J }, set of shocks for A E = {B: s shocked}, set of shocks for. E A = {B : B A Φ} 53
The status of A s evaluated by the Bernoull rando varable: Y = A 0 f S A E A otherwse wth E(Y A ) = θ A. Currently the shocks can be consdered as utually ndependent. Real lfe probles behavor are odeled by the observaton of a sequence a of sgnals. The structure of X leads to consderng t as a functon of the Y A 's. They are consstent wth X. The dscrete product easure pror p = P(X : w) = Π A E Q(Y A ) = Π A E q A can be used for Bayesan nferences. q A s an unspecfed argnal for Y. Ths s a Bayesan approach that has been used by several authors n Saplng theory, see Chadhur-Voos (988). The sequence of sgnals are characterzed by Y {0,} a. If the DM assgns a probablty q [0,] a we can odel the functonng of t by X = Π A E Y A The syste can be redefned by usng the functon ς : {0,} a {0,} n such that ς(y) = X, ς : {0,} a [0,] n and ς (Y) = q. n j= Take A = {j}. An structure ϕ : {0,} a {0,}. Then ϕ(y) = φ ο ς(y) perts to dentfy the coponents of the syste. We can express ϕ(y) = φ(ς(y)) = Σ A ℵ Π A (Π B E Y ) B = ΣA ℵδ(A)Π B A Y B. We avod the use of natural nubers for characterzng the ntal syste by usng Θ = {ℵ *, ϕ}. The coponents of Θ are ndependent because ϕ s Y - evaluated. The realblty of Θ s g(θ) = Σ A ℵ δ(a)π B EA θ B Hence the procedure used prevously perts to predct n ths fraework under the hypothess of d because E μ [ϕ(y)] = g(θ) Note that the knowledge of the sgned donaton functon δ(a) for (ζ,φ) perts to predct the realblty of the dependent case. Take the set of shocks for the subsets a of ζ and a faly ℵ of the. Then: ĝ( θ ) = ϕ(y) s odel unbased and ts error Vμ [ĝ( θ)] = Σ A ℵ δ (A)Π B EA θ A ( - θ A ) s estated by 54
Vμ [ĝ( θ)] = Σ A ℵ δ (A)Π B EA θ ˆ ( ˆ A θ A ) where θˆ A = Y = A If a rando experent s perfored and the results t α α YA (w t ), t =,..., are observed then ĝ( θ) t (, ) Vˆ μ[ĝ( θ),ĝ( θ) + t (, ) Vˆ μ[ĝ( θ)] s a confdence nterval at the confdence level α. 4. EXAMPLES The descrbed systes can be used for odelng dfferent practcal probles. We descrbe soe of the. Exaple 3. Evaluaton of a eal network. Suppose that we have k dfferent boxes. The network conssts of n devses for sendng a essage. We can dentty ℵ = A wth the network. A j s a path, p s the probablty k U j= that a coponent works. δ(a j ) descrbes the effcency of the path. Once the essage s successfully sent φ(x) =. Though p hould be gven by the producer of the devses the real proble ay be such that the operatonal condtons are far fro beng accoplshed. The ndependence of the ' boxes' n an eal network can be accepted. The DM evaluates the techncal condton of the devses and establshes the values of X,...,X n. The h(p) can be predcted and the councaton network s evaluated. The DM uses the evaluaton for establshng the realblty of sendng successfully a essage. The exstence of shocks s present when daages n the lnes or the coputers are expected or changes n the electrc power are usual. Exaple 4. Desgn and operaton of a ontorng network. The DM studes the envronent of a regon. A network of ontorng statons s desgned. Each staton obtans nforaton fro a set of contanatng sources C () () () () = { c,..., ch } and provdes nforaton on a set of levels of a pollutant Λ ={ L,...,LI }. The () () condtonal probabltes Prob ( c j : Lt ) characterzes the relaton between the observed level of the pollutant and the sources of t. For a certan essage X the DM coputes j φ(x) = Σ A ℵ δ(a)π A X where φ(x) = establshes that the envronent s beng serously affected. A s related wth a ontorng staton and X = 0 f the observed L h s 'large'. otherwse ACKNOWLEDGEMENTS Ths paper was partally developed dutrng a vst to the Centre Prospectve of Econoe Matheàtque Aplquées supported by the Project Théory des Jeux et Mateàtque Econoque. CNRS - MINVEL 597. REFERENCES BOUZA, C. (995): "Lnear rank tests derved fro a superpopulaton odel", Boetrcal, J. 37, 497-506. CHADHURI, A. and J.W. VOOS (988): "Unfed Theory and Strateges of Survey Saplng", N. Holland, Asterda. GASEMYR, J. and B. NATUIG (995): "Soe aspects of relablty analyss n shock odels", Scandnavan J. of Stat. Theory and App., 385-393. 55
MI, J. (994): "Maxzaton of the survval probablty and ts applcatons", J. App. Probablty, 3, 06-033. SCOTT, A.J. (977): "On the proble of randozaton n survey saplng", Sankhya, 39, -9. 56