Copyright 2010 IEEE. Reprinted from 2010 Reliability and Maintainability Symposium, San Jose, CA, USA, January 25-28, 2010.

Transcription

1 opyrght 00 IEEE. Reprnted from 00 Relablty and antanablty Symposum, San Jose, A, USA, January 5-8, 00. hs materal s posted here wth permsson of the IEEE. Such permsson of the IEEE does not n any way mply IEEE endorsement of any of RelaSoft orporaton's products or servces. Internal or personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve wors for resale or redstrbuton must be obtaned from the IEEE by wrtng to pubs-permssons@eee.org. By choosng to vew ths document, you agree to all provsons of the copyrght laws protectng t.

2 Pecewse HPP odels wth axmum Lelhood Estmaton for Reparable Systems Huaru Guo, PhD, RelaSoft orporaton Adamantos ettas, RelaSoft orporaton Georgos Saraas, RelaSoft orporaton Pengyng u, RelaSoft orporaton Key Words: pecewse HPP model, maxmum lelhood estmaton, reparable system SUARY & OLUSIOS on-homogeneous Posson process (HPP) models are wdely used for reparable system analyss. Dfferent HPP models have been developed for dfferent applcatons. It has been notced that almost all the exstng models apply only a sngle model for the entre system development or operaton perod. However, n some crcumstances, such as when the system desgn or the system operaton envronment experences major changes, a sngle model wll not be approprate to descrbe the falure behavor for the entre tmelne. In ths paper, we proposed a pecewse HPP model for reparable systems wth multple stages. he maxmum lelhood estmaton (LE) for the model parameters s also provded. IRODUIO onsder a reparable system, such as a vehcle or an ar condtoner. Its falure process usually can be modeled as a non-homogeneous Posson process. Ascher and Fengold s boo [], one of the most nfluental boos on reparable system analyss, dscussed many research results publshed before the 990s. In the past several decades, dfferent HPP models have been developed for modelng dfferent applcatons. For example, the row-asaa used by row [] and the log-lnear model proposed by ox and Lews [3] are two of the most popular ones. hese two models are used for the cases when the falure ntensty monotoncally decreases, ncreases or eeps constant wth tme. he bounded falure ntensty models [4, 5] are desgned for the stuatons when the falure ntensty has an upper bound when tme goes to nfnty. S-shaped models such as the Yamada model [6] and the generalzed GO-model [7] are used to analyze software relablty where the falure ntensty vs. tme plot usually s S-shaped. All of the aforementoned models apply one sngle model for the entre system development or operaton perod. In some stuatons, however, such as when the system under development experences major changes or the system operates under dfferent envronments, a sngle model may not be accurate enough. o overcome ths drawbac, pecewse HPP (homogeneous Posson process) models were ntroduced for an HPP process and the nonparametrc method was used for the parameter estmaton [8-0]. he pecewse HPP models assume that the falure ntensty s constant wthn a gven tme perod and would be dfferent for dfferent perods. However, ths assumpton s not always true. For a gven tme nterval, especally for systems deployed n the feld, the falure ntensty can change wth tme. In ths paper, we propose a pecewse HPP model for reparable systems wth multple development or operaton stages. If the separaton or change ponts between stages cannot be dentfed based on engneerng nowledge and have to be found by analyzng the data, the teraton method proposed n ths paper can be used to estmate them. In ths paper, the row-asaa model s appled as the base model for each stage. Unle the pecewse HPP model, whch gnores the damages from the prevous stages, the damages accumulated from the prevous stages are consdered n the proposed pecewse HPP model. hs s dfferent from the method that analyzes each stage separately. wo well-nown methods have been used for model parameter estmaton n reparable system modelng. One s the least squares estmaton (LSE) and another s the maxmum lelhood estmaton (LE). For a gven HPP model, for nstance for the row-asaa model, t s found that the confdence ntervals for the model parameters estmated from LSE are usually tghter than the confdence ntervals estmated from LE. One of the reasons s that n the applcaton of the least squares method, each observaton (cumulatve number of falures) needs to be ndependent and wth constant varance. However, on page 85 of Ascher and Fengold s boo [], t s stated that for a HPP the ndependency assumpton usually does not hold true from a statstcal vewpont. he dependency between the observed values results n the underestmaton of the varablty of the model parameters when the LSE s used. hs also results n narrow confdence ntervals for the parameters and the functons of these parameters such as the number of falures. We use the LE method n ths paper. For the LE method, the condtonal probablty wll be used n constructng the /0/$ IEEE

3 lelhood functon. By usng the condtonal probablty, the dependency between the observatons s consdered n the model. However, there are some barrers n the use of LE. he pecewse HPP model s a dscrete functon. It conssts of dfferent HPP models for dstnct tme regons. hs causes the maxmum lelhood functon for the overall model to be a dscrete functon too. he tradtonal method that uses the frst order dervatves to get the soluton for the model parameters cannot be appled drectly when the change ponts are not predefned. Lmted research has been done on how to obtan the LE soluton for a pecewse functon. For example, shler and Zang [] employed a contnuous functon to approxmate the pecewse lnear functon, then obtaned analytcal dervatves of the lelhood functon from the approxmated functon to get the LE solutons. However, gettng a suffcently accurate approxmated functon s not an easy tas. In ths paper, a practcal method s proposed for obtanng the change ponts frst. Once the change ponts are obtaned, a contnuous lelhood functon can be formed. Solutons for other model parameters such as the scale and shape parameters n the row-asaa model can then be drectly estmated usng the dervatves. hs paper s organzed as follows. In secton, an example s provded to llustrate why the pecewse HPP model s needed. Secton 3 dscusses how to buld the lelhood functon. In Secton 4, parameter estmaton and confdence ntervals are dscussed. Secton 5 provdes the solutons for an llustraton example. Fnally, we provde conclusons n Secton 6.. Example PROBLE SAEE onsder a reparable system wth the followng falure data. um. um. of Falures Falure me um. um. of Falures Falure me able Falure Data for the Illustraton Example For the data n able, the cumulatve number of falures vs. tme plot s shown n Fgure. In Fgure, the ponts are the observed number of falures and the curve represents the predcted values usng the row- um. of Falures um. of Falures vs. me Obs. um. of Falures me Fgure Falure umber vs. me Plot ASAA model, whch wll be dscussed n secton.. learly, from Fgure, one can see that the sngle model cannot ft the data well. he ramér-von ses (V) test [, ] shows that at a sgnfcance level of 0., the model fals the goodness-of-ft test. he calculated V s 0.534, whch s larger than the crtcal value of 0.7. By examnng the curve, we can see that there are two segments wth the change pont at around 0. It wll be possble to ft an ndvdual HPP model for each segment. However, these two ndvdual models are not ndependent. hey are connected by the change pont. In the followng secton, we wll dscuss the row- ASAA model and the pecewse HPP model.. Pecewse Power Law HPP odel Pred. um. of Falures he falure ntensty functon of a HPP can be descrbed by a power functon: λ() t λβt β () where λ s the scale parameter and β s the shape parameter. hs model s also nown as the row-asaa model []. Under ths model, the cumulatve number of falures s calculated by: t () λt β () By tang the logarthmc transformaton of both sdes, equaton () can be lnearzed as: ln[ t ( )] ln( λ) + βln( t) (3) Equaton (3) s a standard smple lnear regresson model. herefore, the least squares method can be appled f the cumulatve numbers of falures are assumed to be ndependent and wth constant varance. hs s the basc assumpton of usng least squares estmaton. For the data gven n able, the ndvdual row- ASAA model wll be appled to each of the segments. Assume the change pont s. he pecewse HPP model wll be: β () t λt t < (4) β () t λt t > he two curves ntercept at tme. herefore, ) ( ) (

4 and the model parameters have the followng relatonshp: β λ λ β (5) From equaton (5), t can be seen that f s nown, there are only three ndependent parameters. For example, we can use λ, β and β. If s unnown and needs to be estmated, wll be an addtonal parameter. hs s smlar to the strategy used by the tradtonal pecewse lnear regresson [3]. Once we have the pecewse HPP model, the next step s to estmate the model parameters. Secton 3 wll gve the LE soluton. 3 PARAEER ESIAIO AD SAISIAL IFEREES In order to use LE, the lelhood functon should be establshed frst. he example n secton s for only one system. In some cases, falure data from multple systems wll be avalable. For example, several prototypes mght be bult n the development stage, or several systems mght be deployed n the feld at the same tme. In the followng sectons, we wll gve the lelhood functon for the case of multple systems and then provde the closed form solutons for the case of a sngle system. 3. Lelhood Functon o mae the dscusson smple, let s assume there s only one change pont and the change pont s nown. For a system, t s used to denote the th falure tme. he probablty of the th falure s condtonal on the (-)th falure tme. he condtonal probablty of the th falure n segment s: R ( t ) β β F( t t ) exp λt λt R( t ) (6) where F () and R () are the probablty of falure and the relablty at segment. From equaton (6), the condtonal probablty densty functon can be obtaned as: β β β f ( t ) f( t t ) λβ t exp λt λt (7) R( t ) For the falures at segment, we only need to use the model for segment to replace the model n segment n equaton (6) and (7). Assumng that the system starts at tme S and ends at tme, the lelhood functon wll be: LKV f ( t S) f ( t t ) R ( t ) f ( t ) f ( t t ) R ( t ) + + where: f () s the probablty densty functon for segment. s the total number of falures. s the total number of falures n segment. Equaton (8) can be expanded as: (8) f( t) f( t) R( ) LKV R( S) R( t) R( t ) (9) f( t + ) f( t + ) R ( ) R( ) R( t + ) R( t) At tme, the system relablty can only have one value. So R ( ) R ( ), thus they cancel-out n equaton (9). If the system end tme s the same as the last falure tme, the last term n equaton (9) wll equal to. herefore, equaton (9) s for both tme and falure termnated stuatons. If we tae the logarthmc transform of equaton (9), and by cancelng some terms, we get the log lelhood functon as: [ ] [ ] β ln( LKV ) ln( λ ) + ln( β ) + ( β ) ln( t ) + λ S + ln( λ ) + ln( β ) + ( β ) ln( t ) λ + β (0) Generalzng equaton (0) to multple systems, the log lelhood functon s: K q, β Λ q, [ ln( λ ) + ln( β )] + ( β ) ln( tq, ) + λ Sq q () K q β + q, [ ln( λ) ln( β) ] ( β ) ln( tq, ) λ + + q q q, + where: K s the total number of systems. q... K. q s the total number of falures for the qth system. q, s the number of falures n segment for the qth system. q, s the number of falures n segment for the qth system. S q s the start tme of the qth system. q s the end tme of the qth system. Recall that equaton (5) shows that there are only three ndependent parameters, f the change pont s gven. herefore, we replace λ usng the other parameters. Equaton () then becomes: K q, β Λ q, [ ln( λ ) + ln( β )] + ( β ) ln( tq, ) + λ Sq q K () + q, [ ln( λ ) + ln( β )] + q,( β β) ln( ) q K q β β q + ( β ) ln( tq, ) λ q q, + When s unnown, t can be seen from equaton () that t s not possble to get the lelhood functon. hs s because the number of falures for each segment, whch s requred n the lelhood functon, s not avalable wthout nowng the change pont. herefore, t s not possble to construct the lelhood functon wthout havng as gven. In secton 3.3, a heurstc method wll be proposed for teratvely solvng for.

5 Equaton () s a contnuous functon for λ, β and β. So we can get the frst order dervatve for each parameter. By settng the dervatves equal to 0, the followng nonlnear equatons are obtaned: λ q q q q q q β β β β S (3) q, q (4) β q, β β ln( ) q, + λ Sq ln( tq, ) λsq ln( Sq) q q q q β q, q q β β β q q, + λ q q, q q q q, + ln( ) ln( ) ln( t ) (5) By solvng the nonlnear equatons (3-5), we can get the L estmators for the model parameters. When all the systems start from 0 and end at the same tme,, closed form solutons exst. hese are: λ q q β β β q K q, β q, ln( ) q, ln( tq, ) q q q q, β q ln( ) + ln( ) ln( t ) q, q q, q q q q, + (6) (7) (8) So far the dscusson s based on the case of one change pont. If there are multple change ponts, the same methodology s stll vald. Once the model parameters are estmated, the next step s to obtan the varance and covarance matrx for them. o get the varance and covarance matrx, we frst need to calculate the Fsher nformaton matrx [4], whch s: Λ Λ Λ λ λ β λ β Λ Λ F (9) β β β Λ β F s a symmetrc matrx. he varance and covarance matrx s obtaned by: Σ F (0) If there s only one system and t starts from tme 0, the calculaton wll be even smpler. In secton 3., we wll provde the LE solutons for a sngle system and also the confdence bounds of the model parameters. 3. LE Solutons for a Sngle System For a system that starts from tme 0 and has one change pont, the log lelhood functon n equaton () can be smplfed as: L ln( λ) + ln ( β) + ln ( β) + ( β β) ln( ) () β β β + β ln( t ) + ( β ) ln( t ) λ c ( ) + he closed form solutons for the model parameters are: λ () β β β β (3) ln( ) ln( t ) + β ln( ) (ln t ) ln( ) (4) he Fsher nformaton matrx s: β β β β ln( ) ln( ) λ F + ln ( ) ln( ) ln( ) (5) β + ln ( ) β By tang the nverse of matrx F, we can get the varance and covarance matrx for λ, β and β. he -sded ( α)00 % confdence bounds on model parameters can be obtaned by: B ˆ θexp ± z Var( ˆ θ) / ˆ θ (6) θ ( α/ ) Where: θ s a gven parameter. Var (θˆ ) s the varance of that parameter. θˆ s the L estmator. z α / s the ( α / ) percentle of a standard normal dstrbuton. he confdence bounds for a functon of the model parameters can also be calculated. For example for λ whch s a functon of λ, β and β, ts varance s: dλ dλ dλ Var( λ) Var( λ) + Var( β) + Var( β) dλ dβ dβ dλ dλ dλ dλ (7) + ov( λ, β) + ov( λ, β) dλ dβ dλ dβ dλ dλ + ov( β, β) dβ dβ Usng the varance from equaton (7) n equaton (6), we

6 can get the bounds for λ. he method n equatons (6) and (7) s also vald for other functons of the model parameters such as the cumulatve number of falures, BF and falure ntensty. her expected values can be calculated usng the L estmators of λ, β and β and ther confdence bounds can be calculated usng the method n equatons (6) and (7). In the above dscusson, we assume that the change pont s nown. Sometme, t s dffcult to now the exact value of. In secton 3.3, we wll dscuss how to estmate the value. 3.3 Dscusson on hange Pont If the change pont cannot be determned upfront, the followng procedure can be used to estmate t. Step : Plot the falure number vs. tme plot n both lnear and logarthmc scale. Step : From the plots, dentfy the range for, denoted as [_n, _ax]. Step 3: Set _ n +. alculate the LE soluton for ˆλ, ˆβ and ˆβ usng. Step 4: alculate the log lelhood value usng ˆλ, ˆβ and ˆβ. Step 5: Set + and repeat steps 3 and 4 untl reaches _ax. he soluton of ˆλ, ˆβ and ˆβ and the value of that provde the largest lelhood value wll be the L soluton. It has been found that several local maxmal ponts may exst n the search range. If t s necessary, further refned search wth smaller should be conducted at the vcnty of the local maxmal ponts. Searchng from _n to _ax wll ensure that the optmum soluton n ths range wll be found. In secton 4, we wll use an example to llustrate how the value affects the lelhood values. In the dscusson of equaton (), we ponted that t s not possble to get the lelhood functon because the values for number of falures q, and q, at each stage cannot be determned f s unnown. ow, let s assume that we now that the range for s from _n to _ax and there s no falure between _n and _ax. hs means we now q, and q,. If ths s the case, can we get the L estmator for by tang ts dervatve from the lelhood functon? he answer s no. Recall that the log lelhood functon for one system s equaton (). he frst order dervatve for from ths equaton s: β β β Λ ( β β) λ ( β β) (8) From equaton (), we now: β β β λ So equaton (8) s smplfed as: Λ ( β β) ( β β) ( β β) (9) By settng equaton (9) equal to 0, we wll have β β and there s no soluton for. hs proves that usng the dervatve to get the soluton for change pont s mpossble. he heurstc method gven n ths secton should be used to estmate. 4 SOLUIO FOR HE EXAPLE In ths secton, we wll provde the results for the example gven n secton. 4. Estmate the Range for hange Pont From Fgure, we can estmate that s wthn 0 and 40. Usng the method descrbed n secton 3.3, the global optmum of s found to be o chec how the lelhood value changes wth, we calculated the lelhood values for values from 50 to 45. he results are gven n Fgure. Ln(LKV) Ln(LKV) vs hange Pont Fgure Ln(LKV) vs. hange Pont From Fgure, we can see that there are several local maxmal ponts. 4. L Estmators and Statstcal Inferences Based on the estmated value, usng the equatons gven n secton 3., the LE results for the parameters and ther standard devatons are gven n able. LE LSE Parameter ean Std. Devaton ean Std. Devaton λ β β able LE and LSE Solutons For comparson, able also provdes the LSE solutons. As noted n the ntroducton secton, the LSE method usually provdes smaller standard devatons for model parameters n the HPP model. hs can be seen n able. Usng the nformaton n able, the confdence bounds for each of the model parameters can be calculated. able

7 also can be used to get statstcal nferences for the functons such as the number of falures, BF and the falure ntensty, based on the model parameters. For example, we can calculate the cumulatve number of falures and ts confdence ntervals usng the procedure gven n secton 3. he results are gven n Fgure 3. um. of Falure Fgure 3 LE Soluton: umulatve um. of Falures In Fgure 3, the dashed lnes are the upper and lower bounds for the cumulatve number of falures. For comparson, the predcted number of falures and ther bounds from the LSE are gven n Fgure 4. Usng LSE, the estmated value for the change pont s um. of Falures Fgure 4 LSE Soluton: umulatve um. of Falures 4.3 Goodness-of-Ft est um. of Falures me Obs. um. of Falures Upper Bounds um. of Falures Pred. um. of Falures Lower Bounds me Pred. um. of Falures Lower Bounds Upper Bounds Obs. um. of Falures Snce the ndvdual falure tmes are nown, the ramérvon ses statstc s used to test the null hypothess that the pecewse HPP model wth power law falure ntensty functons can descrbe the falure behavor well. It s nown that for a sngle HPP model, the V goodness-of-ft s gven by []: β t (30) where f the test s tme termnated; f the test s falure termnated. s the total number of falures. For the pecewse HPP model of ths example, we modfed the V statstc and calculated t as: β t (3) β β t β β + where. f the test s tme termnated; f the test s falure termnated. s the total number of falures at the th segment. For ths example, 7 and 3. Usng equaton (3), the s calculated to be he crtcal value at a sgnfcant level of 0. s 0.7. Snce s less than the crtcal value, we cannot reject the null hypothess. he pecewse HPP model passes the goodness-of-ft test. In fact, by examnng Fgure 3, we can see that the proposed model can ft the data very well whch s a bg mprovement that than the model n Fgure. 5 OLUSIOS In ths paper, we examned the case of reparable systems wth multple stages for whch a sngle model s not adequate to descrbe the falure behavor for the entre tmelne. We proposed a pecewse HPP model to address ths. Such cases often occur when there s a desgn change, natural operator learnng, or when there s a change n the system s operatonal envronment. We provded the maxmum lelhood functon for the general case of multple reparable systems. losed form solutons for the model parameters for sngle system were also gven. We proposed an teraton method to dentfy the change pont n the cases that the separaton between stages cannot be dentfed based on engneerng nowledge. Parameter bounds and an ntegrated test statstc for the goodness of ft of the proposed model were also dscussed. he proposed methodology allows the data analyst to apply a pecewse HPP model for reparable systems wth multple stages and uses a robust statstcal methodology to assure that the model provdes a good ft even f the change ponts are unnown. REFEREES. H. Ascher and H. Fengold, Reparable Systems Relablty, arcel Deer, Inc L. H. row, Relablty Analyss for omplex Reparable Systems n Relablty and Bometry, Socety for Industral and Appled athematcs, ed. by F. Proschan & R.J. Serflng, Phladelpha, 974, pp D. R. ox and P. A. Lews, he Statstcal Analyss of Seres of Events, ethouen, London, 966.

8 4. L. Attard and G. Pulcn, A ew odel for Reparable Systems wth Bounded Falure Intensty, IEEE rans. Relablty, vol. 54, no. 4, 005, pp G. Pulcn, A Bounded Intensty Process for the Relablty of Reparable Equpment, J. Qual. echnol. vol. 33. no. 4, 00, pp S. Yamada. Ohba, and S. Osa, S-Shaped Relablty Growth odelng for Software Error Detecton, IEEE rans. Relablty, vol. R-3, no., 983, pp L. Goel and K. Oumoto, A me Dependent Error Detecton Rate odel for Software Relablty and Other Performance easures, IEEE rans. on Relablty, vol. R-8. no. 3, 979, pp L.. Leems, onparametrc Estmaton and Varate Generaton for a onhomogeneous Posson Process from Event ount Data, IIE rans. vol. 36, 004, pp L.. Leems, onparametrc Estmaton of the Intensty Functon for a onhomogeneous Posson Process, anagement Scence, vol. 37, no. 7, 99, pp L.. ang and. Xe, A Smple Graphcal Approach for omparng Relablty rends of Dfferent Unts n a Fleet, Proc. Ann. Relablty & antanablty Symp., 00, pp A. shler and I. Zang, A axmum Lelhood ethod for Pecewse Regresson odels wth a ontnuous Dependent Varable, Appl. Statst. vol. 30, no., 98, pp D. Kececoglu, Relablty & Lfe estng Handboo, vol., DEStech Publcatons, Inc, S. E, Ryan and L. S. Porth, A utoral on the Pecewse Regresson Approach Appled to Bedload ransport Data, Gen. ech. Rep. RRS-GP-89. Fort ollns, O. U. S. Department of Agrculture, Forest Servce, Rocy ountan Research Staton, W. Q. eeer, and L. A. Escobar, Statstcal ethod for Relablty Data, John Wley & Sons, Inc., 998. Huaru Guo, Ph.D. RelaSoft orporaton 450 S. Eastsde Loop ucson, Arzona 8570, USA BIOGRAPHIES e-mal: Harry.Guo@RelaSoft.com Dr. Huaru Guo s the Drector of heoretcal Development at RelaSoft orporaton. He receved hs PhD n Systems and Industral Engneerng from the Unversty of Arzona. Hs dssertaton focuses on process modelng, dagnoss and control for complex manufacturng processes. He has publshed numerous papers n the areas of qualty engneerng ncludng SP, AOVA and DOE and relablty engneerng. Hs current research nterests nclude reparable system modelng, accelerated lfe/fegradaton testng, warranty data analyss and robust optmzaton. Dr. Guo s a member of SRE, IIE and IEEE. Adamantos ettas RelaSoft orporaton 450 S. Eastsde Loop ucson, Arzona 8570, USA e-mal: Adamantos.ettas@RelaSoft.com r. ettas s the Vce Presdent of product development at RelaSoft orporaton. He flls a crtcal role n the advancement of RelaSoft's theoretcal research efforts and formulatons n the subjects of lfe data analyss, accelerated lfe testng, and system relablty and mantanablty. He has played a ey role n the development of RelaSoft's software ncludng, Webull++, ALA and BlocSm, and has publshed numerous papers on varous relablty methods. r. ettas holds a B.S. degree n echancal Engneerng and an.s. degree n Relablty Engneerng from the Unversty of Arzona. Georgos Saraas RelaSoft orporaton 450 S. Eastsde Loop ucson, Arzona 8570, USA e-mal: Georgos.Saraas@Relasoft.com Georgos Saraas s a research scentst at RelaSoft orporaton, where he s nvolved wth the development of RelaSoft's software products, publcatons and consultng projects. Hs research nterests nclude relablty and qualty management and the development of relablty program plans. r. Saraas receved hs.s. n Relablty Engneerng from the Unversty of Arzona. He s a ertfed Relablty Engneer, ertfed Project anagement Professonal and ertfed Sx Sgma Green Belt. Pror to jonng RelaSoft, he wored as a relablty engneer at Hewlett-Pacard. Pengyng u RelaSoft orporaton 450 S. Eastsde Loop ucson, Arzona 8570, USA e-mal: Pengyng.u@Relasoft.com Pengyng u s a research scentst at RelaSoft orporaton. She played a ey role n the development of Lambda Predct. Before jonng RelaSoft, she wored at exas Instruments where she was nvolved n I desgn and testng. She receved her asters degree from the atonal Unversty of Sngapore and. E n Electrcal and omputer Engneerng from the Unversty of Arzona. She has done extensve wor on A/D and D/A converters. Her current research nterests nclude relablty predcton and physcs of falure for electronc components such as OSFE, IGB and electronc systems.