STATISTICS DEPARTMENT. Technical Report

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1 STATISTICS DEPARTMENT Technicl Report RT-MAE Miniml repir ge replcement in heterogeneous popultion: n optiml stopping problem by Vnderlei d Cost Bueno Institute of Mthemtics nd Sttistics University of São Pulo São Pulo - Brzil

2 This pper is of exclusive responsibility of the uthors. Reproduction of this text nd the dt it contins is llowed s long s the source is cited. Reproductions for commercil purposes re prohibited. 2

3 Miniml repir ge replcement in heterogeneous popultion: n optiml stopping problem Vnderlei d Cost Bueno Abstrct In this note we nlyse miniml repir ge replcement policy s solution of n optiml stopping problem under point process mrtingle theory. Keywords: Semi-mrtingle, optiml stopping rule, miniml repir, ge replcement policy. 1 Introduction The most common, populr nd importnt replcement policy might be the ge dependent preventive mintennce policy. Under this policy, unit is lwys replced t its ge T or its filure, whichever occurs first nd T is constnt. Lter, s the concepts of miniml repirs becme more nd more stblish, lrge vriety of extensions nd modifictions of the ge-replcement policy were proposed. An overview on the mintennce theory from theoreticl point of view cn be found in Ncgw (25). More sophisticted models hve lso been developed in Mi, J.(1994), Ch, J.H. (21), Ch, J.H. (23), Ebrhimi, N. (1997), Aven nd Jensen (2) nd Bdi, F.G. et l. ( 21). Most of the previous reserch on mintennce models hs been focused on those for items from homogeneous popultion, however popultion heterogeneity hs been widely recognized in relibility re nd lot of studies on the stochstic properties of heterogeneous popultions hve been recently performed in relibility context, s in Bdi, F.G. et l. ( 23), Ch nd Finkelstein (211) nd Finkelstein nd Ch (213). In this pper we consider mixed popultion which is composed of ordered subpopultions through its intensities. An item is selected of n unknown subpopultion nd during its opertion is minimlly repired on ech filure nd it is replced t fixed ge. As the subpopultion re stochsticlly ordered, the ge t which the corresponding item is replced should be different depending on the subpopultion from which the item in opertion is selected. When the opertionl history of the item, which my contin some importnt informtion bout the corresponding subpopultion, cn be obtined during the field opertion, it is resonble to employ it in determining the replcement policy of the item. To be more specific, the items from stronger subpopultions nd those from weker subpopultions would exhibit different filure ptterns during field opertions. Thus, news replcement policies cn be nlysed, the ge t which the corresponding item will be replced is not determined before the opertion of the item, but it will be determined bsed on the filure/repir history of the observed item during its initil opertion, which is n stopping time problem. This problem ws studied in Ch, J. H. (215) considering 1

4 week nd strong sub-popultions in stochstic process frmework. However, the nturl pproch to consider effectively the informtion of pssed events in time is the point process mrtingle theory which consider the incresing informtion in time through incresing fmilies of σ-lgebrs. In this context, in this pper, we resumes to solve nd generlizes the bsic miniml repir ge replcement optiml stopping problem using n infinitesiml-look-hed stopping rule. In Section 2 we describe the miniml repir process in n heterogeneous popultions nd in Section 3 we nlyse miniml repir ge replcement optiml stopping problem in heterogenious popultions. 2 Miniml repir processes in heterogeneous popultions In this section we describe the heterogeneous popultion nd the corresponding miniml repir process s in Ch nd Finkelstein (211). Let filures of repirble items be repired instntneously. Then the process of repirs cn be described by stochstic point process or, equivlently, by its stochstic intensity λ t, t, (see Aven nd Jensen (1991) nd Bremud (1981)). The stochstic intensity of point process (N t ) t is defined s P (N t,t+ t = 1 I t ) λ t = lim t t = lim t E[N t,t+ t I t ]. t where N t,t+ t = N t+ t N t nd I t = σ{s i, 1 i N t }. A clssicl exmple is the deterministic intensity λ t = λ(t) which defines the nonhomogeneous Poisson process (NHPP) of repirs with intensity λ(t). It is well known tht the NHPP cn be interpreted s the miniml repir process. To describe n heterogeneous popultion we let S be n item lifetime com cumultive distribution function F (t) nd ssume tht it is indexed by rndom vrible Z, i.e. F (t, z) = P (S t Z = z) = P (S t z) nd tht the probbility density function f(t, z) exists. Then the corresponding filure rte λ(t, z) is f(t,z). F (t,z) Let Z be non negtive rndom vrible with support in [, b], < b, nd probbility density function π(z). The setting leds nturlly to considering mixtures of distribution functions which re useful to describing heterogeneity: F m (t) = F (t, z)π(z)dz. In ccordnce with this definition, the mixture filure rte is λ m (t) = f(t, z)π(z)dz = F (t, z)π(z)dz 2 λ(t, z)π(z t)dz,

5 where F (t, z) π(z t) = π(z T > t) = π(z). F (t, z)π(z)dz is the conditionl probbility function on {S > t}. Let return to the stochstic intensity nd modify it with respect to the heterogeneous cse, when the orderly point process is indexed (conditioned) by the rndom vrible Z: P (N t,t+ t = 1 I t, σ(z)) λ t = lim t t = lim E[P (N t,t+ t = 1 I t, σ(z)) ] = E[λ t,z ]. t t where the expecttion is with respect to the conditionl distribution (Z I t ). We, now, consider the miniml repir process of n item selected, t time t =, from our heterogeneous popultion. At ech filure, perform miniml repir. In this cse if Z = z t time t =, the corresponding intensity reliztion is deterministic, λ t,z = λ(t, Z), t, in view tht t t =, the informtion is the degenerted σ- field, σ{ω, } nd the distribution π(z). So, if t 1 is the reliztion of the first filure T 1 in the intervl [, t 1 ), λ(t, Z) = λ m (t) = λ 1 m(t) is just the mixture of the filure rte. The corresponding stochstic intensity λ t is the expecttion of λ(t, Z) with respect to the distribution (Z I t ). This opertions mens tht lthough the vlue of Z is chosen t t = nd is fixed, its distribution is updted with time s informtion bout the filures emerges. In the next intervl [t 1, t 2 ), where t 2 is the reliztion of T 2, given the dditionl informtion tht n item hs filed t t = t 1, the probbility density function of Z is ctulized s π 2 (z) = λ(t 1, z) exp{ t 1 λ(s, z)ds}π(z) λ(t 1, z) exp{ t 1. λ(s, z)ds}π(z)dz The conditionl distribution of (Z I t ) is λ(t 1, z) exp{ t λ(s, z)ds}π(z) λ(t 1, z) exp{ t λ(s, z)ds}π(z)dz nd the corresponding stochstic intensity is λ 2 m(t) = λ(t, z) λ(t 1, z) exp{ t λ(s, z)ds}π(z) λ(t 1, z) exp{ t λ(s, z)ds}π(z)dz in [t 1, t 2 ). More generlly, for t [t n 1, t n ), the conditionl distribution (Z I t ) is defined by π n (z t 1,..., t n 1 ) = λ(t 1, z)...λ(t n 1, z) exp{ t λ(s, z)ds}π(z) λ(t 1, z)...λ(t n 1, z) exp{. t λ(s, z)ds}π(z)dz 3

6 Therefore in [t n 1, t n ) nd λ n m(t) = λ(t, z)π n (z t 1,..., t n 1 )dz λ t = λ n m(t)1 {Tn 1 t<t n}, T =. n=1 3 Miniml repir ge replcement optiml stopping problem in heterogeneous popultion. An item is rndomly selected of unknown subpopultion nd during its opertion is minimlly repired in ech filure nd replced t fixed ge T. We suppose cost c m for ech miniml repir nd cost c r for replcement. If N t is the number of miniml repirs in (, t], the totl cost up to time t, for the item, is c m N t + c r. The totl cost for unit time is C(t) = cmnt+cr. The gol is to find the replcement ge tht minimizes the long run t verge cost per unit time C(T ) = c me[n T ] + c r. E[T ] The replcement policies cn be strongly connected to the following stopping problem: Minimize C τ = E[Z τ] E[X τ ], in suitble clss of stopping time relted to filtrtion (I t ) t which represents our observtions, in the probbility spce (Ω, I, P ), ssumed to fulfill the usul conditions of right continuity nd completeness. The stochstic processes Z t nd X t re observble in (I t ) t, tht is, Z t nd X t re I t -mesurble. As before, T represents the ge replcement nd S the item lifetime. We let (Z t ) t, with Z t = c m N t + c r nd (X t ) t, with X t = t be rel right continuous stochstic processes dpted to I t nd such tht E[Z S ] > nd E[ X S ] <. We intend to minimize the rte C τ = E[Z τ] E[X τ ] over the clss of I t -stopping time C It S = {τ : τ is n I t stopping time, τ S, E[Z τ ] >, E[ X τ ] < }, tht is, to find stopping time σ C It S, with C = C σ = inf{c τ : τ C It S }. 4

7 It is well known tht smooth semi-mrtingle (see ppendix 2.1) representtions for the processes (Z t ) t nd (X t ) t, is n excellent tool to crry out the stopping problem. Under Section 2, the semi-mrtingle representtion for Z t is : = c m n=1 t Z t = c m N t + c r = c m { t λ n m(s)ds + M t } + c r n=1 1 {tn 1 t<t n}λ(t, z)π n (z t 1,..., t n 1 )dzds + M t } + cr, where M t is n uniformly integrl zero men I t -mrtingle. Also X t = t = t ds = t 1 {tn 1 t<t n}π n (z t 1,..., t n 1 )dzds. n=1 To solve the bove stopping problem is equivlent to solve the following mximiztion problem. Observe tht the inequlity C τ = E[Zτ ] E[X τ ] C is equivlent to C E[X τ ] E[Z τ ] for ll τ C It S, where the equlity holds for n optiml stopping time. We hve the mximiztion problem: Find σ C It S, with E[Y σ ] = sup{e[y τ ] : τ C It S } =, where Y t = C X t Z t nd C = inf{c τ : τ C It S }. A smooth semi-mrtingle representtion for Y t for the miniml repir ge replcement policy is where λ s = Therefore n=1 1 {tn 1 t<t n} Y t = c r + t (C c m λ s )ds + M t λ(t, z)π n (z t 1,..., t n 1 )dz. E[Y t ] = E[C X t Z t ] = c + E[ t (C c m λ s )ds] To find n explicit solution of the stopping problem we dopt condition clled the monotone cse: Definition 3.1 (MON) Let Y = (f, M) be n SSM. Then the following condition {f t } {f t+h }, t, h R +, {f t } = Ω t R + 5

8 is sid to be the monotone cse nd the stopping time σ = inf{t R + : f t } is clled the ILA-stopping rule ( infinitesiml-look-hed). Obviously in the monotone cse the process f driving the smooth semi-mrtingle Y t remins non-positive if once crosses zero from bove nd the ILA-stopping rule σ is cndidte to solve the mximiztion problem. Aven nd Jensen (1991), proves tht Theorem 3.2 Let Y = (f, M) be n SSM nd σ the ILA-stopping rule. Then, in the monotone cse, E[Y σ ] = sup{e[y τ ] : τ C It }, where C It = {τ : τ is n I t stopping time, τ <, E[Y τ ] > }. Clerly, the monotone cse holds when λ s is incresing,.s., λ < C c m nd lim s λ s > C c m. However it is seem too restrictive to demnd tht λ s is incresing.s.. We would like the monotone cse to cover cses s the bth-tub-shped functions, which decrese first up to sme vlue, nd increse fter tht vlue. The definition of (, b)-incresing function llows such cses, see ppendix A.2.2. The min ide to solve the stopping problem, for the bsic replcement policy, using the monotonicity condition is, insted to considering ll stopping time in C It T we my restrict the serch for n optiml stopping time to the clss of index stopping times ρ x = inf{s R + : x c m λ s } T, inf =, x R. The optiml stopping level cn be determined from E[Y τ ] = nd coincides with C s in the following Theorem from Aven nd Jensen (1991). Theorem 3.3 If the process (Y t ) t in its SSM representtion hs n intensity with (, b) incresing pths on (, T ], then σ = ρ x, with x = inf{x R : xe[ρ x ] E[Z ρx ] } is n optiml stopping time with x = C. 6

9 nd, In our context λ s = t n=1 λ(t, z)π n (z t 1,..., t n 1 )dz. t = c r + E{ n=1 E[Y t ] = c r + E[ t (C c m λ s )ds] 1 {tn 1 s<t n}π n (z t 1,..., t n 1 )[C c m λ(s, z)]dzds. ρ x = inf{s R + : x c m λ(s, z) } T, inf =, x R considering vlues of x in [ cr + qλ E[T ], cme[ T λ(s,z)ds]+cr ] obtined from Lemm A.2.3. E[T ] This mens: Replce the items the first time the sum of the intensities reches level x. This level hs to be determined s x = inf{x R : xe[ρ x ] E[c m N(ρ x ) + c r ] }. The cse of deterministic intensities is of specil interest nd is stted s corollry under the ssumption of the lst theorem. Lemm 3.4 If λ(t, z) is deterministic with inverse λ 1 = inf{t R : λ(t, z) x}, x R nd X =, then σ = t T is optiml with t = λ 1 (C ) R + { } nd C = inf{x R : where S is the item lifetime. λ 1 (x) [x c m λ(s, z)]p (S > s)ds c r }, We observe tht the bove result is lso pplied for doubly stochstic Poisson process, lso clled Cox process, when the intensity is rndom vrible which is known t the time origin, tht is, λ is I mesurble. We cn tke I = σ{λ s,z, (s, z) R 2 +} nd I t = I σ{n s, s t}. Consider subpopultions of items with Weibull distribution F (t, ) = 1 exp[ ( t η ) β ], η < t <, < < 1, β >. The index is rndom with probbility density π() = (β 1) β, < < 1, An item of n unknown subpopultion is selected nd during its opertion is minimlly repired 7

10 following ordered Weibull process, forming Cox Process. In prcticl we consider the ordered lifetimes with conditionl survivl function given by F (t i t 1, t 2,..., t i 1 ) = exp[ ( t i η i ) β + ( t i η i 1 ) β ] for η i t i 1 < t i where t i re the ordered observtions. The density function is f(t 1, t 2,..., t n ) = π n i=1f(t i t 1, t 2,..., t i 1 ) = Follows tht ( β )(t 1 η 1 ) β 1 exp[( t 1 η 1 exp[ ( t i η i ) β ]πi=2( n β )(t i η i ) β 1 ) β + ( t i η i 1 ) β ]. λ(t i, t 1, t 2,..., t i 1 ) = f(t 1, t 2,..., t n ) F (t i t 1, t 2,..., t i 1 ) = (β )(t η i ) β 1, t i 1 t < t i, t = is deterministic nd λ 1 (t) = inf{s R + : c m λ(s, z) x c r } T, inf =, x R We ssumes β > 1, which mke λ(t, ) incresing in t, nd λ 1 (t) = ( (t c r) β 1 ) 1 β 1 + ηi. c m β As, in this cse, the intensity is deterministic, we cn pply Corollry 3.4 nd get the ge replcement T, indexed by. 4 Appendix. 4.1 A.1 An extended nd positive rndom vrible τ is n I t -stopping time if, nd only if, {τ t} I t, for ll t ; n I t -stopping time τ is clled predictble if n incresing sequence (τ n ) n of I t -stopping time, τ n < τ, exists such tht lim n τ n = τ; n I t -stopping time τ is totlly inccessible if P (τ = σ < ) = for ll predictble I t -stopping time σ. For bsis of stochstic processes see the book of Bremud [2]. 8

11 4.2 A.2 The stopping problem. Definition A.2.1 A stochstic process Z = (Z t ) t is clled smooth semimrtingle representtion (SSM) if it hs decomposition of the form Z t = Z + t f s ds + M t, where (f t ) t, is progressively mesurble process with E[ t f s ds < for ll t R, E[ Z ] < nd (M t ) t n zero men uniformly integrble I t mrtingle. We denote SSM by Z = (f, M). Definition A.2.2 Let, b R {, }, b. Then function f : R + R is clled (, b)-incresing if for ll t, h R +, f(t), implies f(t + h) min{f(t), b}. Roughly spoken, n (, b)-incresing function f(t) psses with incresing t the levels, b from bellow nd never flls bck bellow such level. The first step to detect the prmeters nd b is to estblish bounds for C : Lemm A.2.3 Let X = (g, L) nd Z = (f, M) be smooth semimrtingles under the bove ssumptions nd Then q = inf{ f t(w) g t (w) holds true, where the bounds re given by : t < S(w), w Ω} >. b l C b u b u = E[Z S] E[X S ], b l = E[Z qx E[X S ] + q, if E[Z qx ] > ;E[Z ] E[X ],otherwise. 9

12 5 Conclusions News replcement policies cn be nlysed, the ge t which the corresponding item will be replced is not determined before the opertion of the item, but it will be determined bsed on the filure/repir history of the observed item during its initil opertion, view s n stopping time problem. We conjecture the vlidtion of such model for more sophisticted policies. 6 Acknowledgements This work ws prtilly supported by FAPESP, Proc No. 215/ nd by the University of São Pulo. 7 References [1] Aven, T. nd Jensen, U. Stochstic Models in Relibility. Springer Verlg, New York, [2] Aven, T. nd Jensen, U. A generl miniml repir model, J. Appl. Probbility, 37,, , 2. [3] Bdi, F. G., Berrde, M.D. nd Cmpos, C.A. Optimiztion of inspection intervls bsed on cost, J. Appl. Probbility, 38, , 21. [4] Bdi, F. G., Berrde, M.D. nd Cmpos, C.A. Why does the filure rte decrese?, in Proc. VIIth Zrgoz-Pu Conf. Appl. Mth. Sttistics, 97-14, 23. [5] Brlow nd Proschn,F. Sttisticl Theory of Relibility nd Life Testing: Probbility models. Hold, Reinhrt nd Wiston, Inc. Silver Spring, MD [5] Bremud, P. Point Processes nd Queues: Mrtingles Dynmics. Springer Verlg, New York, [6] Ch, J. H. nd Finkelstein, M. Stochstic intensity for miniml repirs in heterogeneous popultions. J. Al. Prob., 48, , 211. [7] Ch, J. H. Burn-in procedures for generlized model, J. Appl. Probbility, 38, , 21. [8] Ch, J. H. A further extension of generlized burn-in model, J. Appl. Probbility, 4, , 23. 1

13 [9] Ch, J. H. Optiml replcement of heterogeneous items with minimlrepirs, IEEE Trnsctions on Relibility, 1-11, 215. [1] Ebrhimi, N. Multivrite ge replecement, J. Appl. Probbility, 34, , [11]Finkelstein, M. nd Ch, J.H. Stochstic modelling for relibility (Shocks, Burn-in nd Heterogeneous popultions) Springer, LOndon, Uk, 213. [12] Mi, J. Burn-in nd mintennce policies. Adv. Appl. Probbility, 26, , [13] Nkgw, T. Mintennce theory of relibility. Springer, London, Uk,