MYUNG HWAN NA, MOON JU KIM, LIN MA

BAYESIAN APPROACH TO MEAN TIME BETWEEN FAILURE USING THE MODULATED POWER LAW PROCESS MYUNG HWAN NA, MOON JU KIM, LIN MA Abstract. The Reewal process ad the No-homogeeous Posso process (NHPP) process are probably the most popular models for descrbg the falure patter of reparable systems. But both these models are based o too restrctve assumptos o the effect of the repar acto. For these reasos, several authors have recetly proposed pot process models whch corporate both reewal type behavor ad tme tred. Oe of these models s the Modulated Power Law Process (MPLP). The Modulated Power Law Process s a sutable model for descrbg the falure patter of reparable systems whe both reewal-type behavor ad tme tred are preset. I ths paper we propose Bayes estmato of the ext falure tme after the system has expereced some falures, that s, Mea Tme Betwee Falure for the MPLP model. Numercal examples llustrate the estmato procedure. 1. Prelmares The the reewal process ad Nohomogeeous Posso Process (NHPP) are models that are ofte used to descrbe the radom occurrece of evets tme. For a reewal process, the tmes betwee falure are depedet ad detcally dstrbuted(d). Sce the tmes betwee falure are d, a repared ut s always brought to a lke-ew codto. For ths reaso, the reewal process caot be used to model a system experecg deterorato or relablty mprovemet. The reewal process has also bee called the good-as-ew model. For a NHPP, the probablty of a evet a small terval of tme, (t, t + t), depeds oly o t ad ot o the prevous patter of evets. The lmt, P (a evet (t, t + t)) λ(t) = lm, t >, t t s defed the testy fucto of a NHPP. If the Posso process s used to model the falure tmes of a reparable system, the the fact that λ depeds o t ad ot o the prevous patter of falures meas that a faled ut s exactly the same codto after a repar as t was just before the falure. For ths reaso, the NHPP s ofte called the same-as-old model whe t s appled to a reparable system. The homogeeous 2 Mathematcs Subject Classfcato: 65D5,65D7,65D17. Key words ad phrases: NHPP; Modulated Power Law Process; Mea Tme Betwee Falure. c The Korea Socety for Idustral ad Appled Mathematcs, 26

Posso Process (HPP), for whch the testy fucto s a costat ad the tmes betwee falures are depedet ad detcally expoetally dstrbuted, s a specal case of both. If the testy fucto has the form λ(t) = β ( t )β1, t >, the the process s called the power law process. Ths process has bee called the Webull process. Typcally, the parameters β ad are ukow ad must be estmated from data from oe or more systems. Crow (1974) developed may of the ferece propertes of the power law process. Rgdo ad Basu (1989) preseted a revew of propertes of the power law process. Ufortuately, reewal process ad NHPP are based o too restrctve assumptos o the effect of the repar acto. For these reasos, several authors have recetly proposed pot process models whch corporate both reewal type behavor ad tme tred. The Modulated Power Law Process (MPLP) s compromse betwee the Reewal process ad the NHPP wth power testy fucto, sce ts falure probablty at a gve tme t depeds both o the age t of a system ad o the dstace of t from the last falure tme. Thus the MPLP s a sutable model for descrbg the falure patter of reparable systems whe both reewal-type behavor ad tme tred are preset. Lakey ad Rgdo (1992) have troduced the MPLP whch s specal case of the homogeeous gamma process troduced by Berma (1981). Lakey ad Rgdo (1992) propose Maxmum lkelhood pot estmators of the three parameters of MPLP. Black ad Rgdo (1996) descrbe a algorthm for obtag the MLEs of the three parameters ad derve the asymptotc varace of these pot estmates. I relablty aalyss, t s qute mportat to be able to determe the ext falure tme after the system has expereced some falures durg the developmet ad test process. I order words, the mea tme to the ext falure at the -th observed falure tme t, or the Mea Tme Betwee Falures at t, MT BF (t ), s of sgfcat terest. I ths paper we derve Mea Tme Betwee Falure (MTBF) for the MPLP model ad propose Bayes estmato of Mea Tme Betwee Falure (MTBF) at the -th falure tme. Numercal examples llustrate the estmato procedure. 2. Mea Tme Betwee Falure for Modulated Power Law Process Suppose that a system falure occurs ot at every shock but at every k-th shock, where k s a postve teger ad suppose that shocks occur accordg to the NHPP wth testy fucto: λ(t) = lm t P (a shock (t, t + t)), t >. t If for example k = 3, the the system falure occur at every thrd shock. Thus, a faled ad repared system would be better codto tha t was before the falure

occurrece, sce 3 other shocks must occur order to observe the ext falure. Eve f the explaato gve prevously terms of shocks does ot carry over whe k s ot a teger, the MPLP ca be stll defed for ay postve value of k. Let T 1 < T 2 < < T deote the frst falure tmes of a falure trucated MPLP. The codtoal relablty fucto of T gve T 1, s; R(t t 1 ) = Pr(T > t T 1 = t 1 ) = Pr(o falure the terval t 1, t]) = Pr(N(t) N(t 1 ) k 1) k1 (U(t) U(t 1 )) j ] = exp U(t) + U(t 1 ), j! j= where U(t) = t λ(u)du s mea value fucto of NHPP. Thus the codtoal pdf of T gve T 1 ca be easly computed from the above codtoal relablty fucto as follows; f(t t 1 ) = 1 ( ) β β1 (t ) β ( ) ] β k1 ( ) β ( ) ] β t t1 t t1 (1.1) exp +. Γ(k) The the jot pdf of falure tmes T 1 < T 2 < < T, s; 1 β ( ) ] β (1.2) f(t 1.t 2,, t β,, k) = (Γ(k)) exp βk (t β tβ 1 )k1. Whe k = 1 the MPLP reduces to the PLP, whe β = 1 the process becomes a Gamma reewal process, whe k = 1 ad β = 1 the MPLP reduces to the HPP. Thus β s a measure of the system mprovemet or deterorato over the system lfe, whereas k s a measure of the mprovemet of worseg troduced by the repar actos. I relablty aalyss, oe of mportat characterstcs s the ext falure tme after the system has expereced some falures. I order words, the mea tme to the ext falure at the -th observed falure tme t. Now we derve Mea Tme Betwee Falure (MTBF) for the MPLP model. Let T 1 < T 2 < < T < be the successve system falure tmes. The the MTBF at the -th falure tme t s defed as; MT BF (t ) = ET +1 T T = t ]. I order to express the MT BF (t ) more explctly, we eed the codtoal dstrbuto of T +1 gve T = t. From (1), the codtoal pdf of T +1, gve T = t, s f +1 (t t 1,, t ) = 1 β Γ(k) ( t ) β1 ( ) t β The MTBF at T = t for the MPLP s gve by MT BF (t ) = ET +1 T T = t ] ( ) ] β k1 exp ( ) t β + ( ) ] β.

= = t t tf +1 (t t 1,, t )dt t 1 t Γ(k) t β ( t Usg tegral by parts, we get β ( ) t β ( t MT BF (t ) = Γ(k) Let y = (t/) β (t /) β, the (1.3) MT BF (t ) = Γ(k) ) β1 ( ) t β ) β y k1 y + ( ) ] β k1 exp ( ) ] β k1 exp ( ) t β + ( ) ] 1 β β exp(y)dy t. ( ) t β + ( ) ] β dt t. ( ) ] β dt t. 3. Bayes Estmato Let t 1 < t 2 < < t deote the frst falure tmes of a falure trucated MPLP sample. From (2), the lkelhood fucto, gve falure tmes t 1 < t 2 < < t, s; 1 β ( ) ] β L(β,, k t 1, t 2,, t ) = (Γ(k)) exp βk (t β tβ 1 )k1. Usg argumets Calabra ad Pulc (1997), we have Jeffrey s (1961) prors as follows; π(β) = 1 β π() = 1 π(k) = ψ (k), where ψ (k) = d 2 l Γ(k)/dk 2 s the tr-gamma fucto. Usg the a pror depedece assumpto, the o-formatve jot pror s ψ π(β,, k) = (k) β ad the correspodg jot posteror pdf results ; π(β,, k t 1,, t ) = 1 ψ (k) β 1 ( C (Γ(k)) exp βk+1 where the deomator C = g(, )dkdβ ) β ] (t β tβ 1 )k1

ad g(a, b) = k a β2+b t βk The margal posteror pdf of β s ad the posteror mea s (1.4) ψ (k)γ(k) (Γ(k)) π(β t 1,, t ) = 1 C E(β t 1,, t ) = 1 C The margal posteror pdf of s π( t 1,, t ) = 1 C 1 ad the posteror mea s E( t(1.5) 1,, t ) = 1 C β 1 β 2 exp Fally, the margal posteror pdf of k s ad the posteror mea s (1.6) (1.7) π(k t 1,, t ) = 1 C E(k t 1,, t ) = 1 C (t β tβ 1 )k1. g(, )dk, g(, 1)dkdβ. ( ) ] β ψ (k) 1 (Γ(k)) βk (t β tβ 1 )k1 dkdβ ψ (k)γ(k 1/β) (tβ tβ 1 )k1 (Γ(k)) dkdβ. g(, )dβ, g(1, )dkdβ. We ow propose a Bayes estmator of MT BF (t ) from (3): MT BF B (t ) = ˆ Γ(ˆk) yˆk1 y + ( ) tṋ ˆβ] 1ˆβ exp(y)dt t. t βk1 where ˆ, ˆβ ad ˆk are the Bayes estmator of, β ad k, obtaed from (4), (5) ad (6), uder square error lose, respectvely. 4. Examples Two umercal examples llustrate the proposed estmato procedure. The frst s the falure tmes of a arcraft geerator take from Duae (1964). These falure tmes have bee read from a plot Duae s paper by Black ad Rgdo (1996). For ths system there were = 14 falures, ad these falure tmes are show Table 4.1.

Table 4.1. Falure Tmes for Arcraft Geerator Duae (1964) 1 55 166 25 341 488 567 731 138 25 2453 3115 417 4596 The pot estmates for the parameters are ˆ = 1.3, ˆβ =.419 ad ˆk = 4.2. Thus, from (7), the estmate of MTBF s MT BF B (4596) = 653 ad the estmated mea tme to the ext falure at t = 4596 s 5249. The secod example cossts of the falure tmes from the secod arcraft arcodtog ut from Proscha (1963). The falure tmes are show Table 4.2. Table 4.2. Falure Tmes of Arcraft Arcodtog Equpmet gve by Proscha (1963) 413 427 485 522 622 687 696 865 1496 1532 1733 1851 1885 1916 1934 1952 219 276 2145 2167 221 The pot estmates for the parameters are ˆ = 34.4, ˆβ = 2.15 ad ˆk = 1.4. Thus MT BF B (221) = 67, that s, the mea tme to the ext falure at t = 221 s 2268. Ackowledgemet Ths work was supported by the Korea Research Foudato Grat fuded by the Korea Govermet (KRF-24-3-C43) REFERENCES 1]: Berma, M. (1981), Ihomogeeous ad modulated gamma processes. Bometrka, 68, pp. 143 152. 2]: Black, S. E. ad Rgdo, S. E. (1996), Statstcal ferece for a modulated power law process, Joural of Qualty Techology, 28, 1, 81 9. 3]: Calabra, R. ad Pulc, G. (1997), Bayes Iferece for the Modulated Power Law Process, Commucato Statstcs, Theory ad Methods, 26, 1, 2421 2438. 4]: Crow, L. (1974), Relablty Aalyss of Complex, Reparable Systems, I Relablty ad Bometry,(Eds., F. Proscha ad R. J. Serflg), 379 41, SIAM, Phladelpa. 5]: Duae, J. T. (1964), Learg Curve Approach to Relablty, IEEE Trasactos o Aerospace, V2, 563 566. 6]: Jeffreys H. (1961), Theory of Probablty. Claredo Press. 7]: Lakey, M. J. ad Rgdo, S. E. (1992), The modulated power law process, Proc. 45th Aual Qualty Cogress, Mlwaukee 1992, 559 563.

8]: Proscha, F. (1963), Theortcal Explaato of Observed Decreasg Falure Rate, Techometrcs, 5, pp. 375 383. 9]: Rgdo, S. E. ad Basu, A. P. (1989) The power law process: A model for the relablty of reparable systems, Joural of Qualty Techology, 2, pp. 251-26. Myug Hwa Na, Moo Ju KIm: Departmet of Statstcs, Choam Natoal Uversty, Yogbog-dog Buk-gu Gwagju L Ma: School of Egeerg Systems, Queeslad Uversty of Techology, Australa