Recommendation T/N (Edinburgh 1988)

Transcription

1 o T/N -0 E B Page Reommedatio T/N -0 (Ediburgh ) TESTNG THE COMPLANCE OF AN EQUPMENT WTH TS RELABLTY, MANTANABLTY AND AVALABLTY SPECFCATONS Reommedatio proposed by Workig Group T/WG Network Aspets (NA) Text of the Reommedatio adopted by the <Teleommuiatios Commissio: The Europea Coferee of Postal ad Teleommuiatios Admiistratios, osiderig P that for various equipmets, besides the ormal futioal aspets, availability, reliability ad maitaiability aspets are speified, that whe aeptig these equipmets from the maufaturer or deliverer, the Admiistratios must be sure that these availability, reliability ad maitaiability speifiatios are met, that it is therefore advisable to use testig methods, most of whih are based o statistial elaboratio of various iformatio i this field; whih methods must be agreed with the maufaturer or deliverer of the equipmet oered, that this subjet remais the total resposibility of the Admiistratio oered, i.e. this Reommedatio will ot ditate what the Admiistratio should or should ot do i this matter, that it is, however, reommedable to have some guidae o the matter of testig the ompliae of the speified reliability, maitaiability ad availability aspets, reommeds the followig guidae, whih desribes a method for testig the ompliae of a equipmet with its reliability, maitaiability y ad availability speifiatios. m Editioof Jue 0.

2 T/N -0 E Page. NTRODUCTON This Reommedatio is made of three parts: the first part idiates the appliatio area of the Reommedatio by listig reliability, maitaiability, availability harateristis to whih the Reommedatio applies; the seod part itrodues three testig priiples whih a be used for testig the ompliae of a equipmet to its reliability, maitaiability, or availability speifiatios ad idiates for eah priiple to whih harateristi it is best fitted; the third part is devoted to the desriptio of methods ad mathematial tools refereed i the seod part CHARACTERSTCS TO BE VERFED The followig list gives the mai reliability, maitaiability ad availability harateristis for whih a method for ompliae testig is proposed i this Reommedatio. Reliability Global failure harateristis rate This parameter is used to evaluate the umber of repairs whih will have to be doe, for a give period, o the osidered equipmet. Futioal failure rate This parameter is used to evaluate the umber of times whe oe equipmet is ot able to work as speified durig a give period. For a swithig system, several futioal failure rates a be spedified aordig to the osequee of the osidered failure: failure affetig a give group of subsribers; failure affetig a give group of iruits; failure whih do ot affet speifially a give group of subsribers or iruits but whih lower the tratlability performae of the system. Maitaiability harateristis Probability of failed state detetio t is the probability that the existee of a failure iside the equipmet is deteted, whether the required futios of the equipmet are fulfilled or ot. E#iiey of loalizatio of a failed item Whe a failure is deteted, a loalizatio proedure takes plae whih will idetify a give set of presumably failed items as a failure loalizatio. The eflliey of the loalizatio proedure is defied by two probabilities orrespodig to the followig defiitio: For preise loalizatio, where o more tha k items should be idetified per failure (k= i most appliatios), a probability P is speified. However, as loalizatio pr~edures aot be perfet, a larger set (betwee k + ad, say, m items) a be idetified, but with low probability P (osequetly, the probability that the umber of idetified items per failure is more tha m is equal to (Pl + P)). summary, the etliiey of loalizatio is defied by the two probabilities P ad P suh that: P = P (loalizatio amog or or... k items) P = P (loalizatio amog k+, or k+,... or m items) with m <, beig the total umber of items buildig the equipmet. Duratios related to the itrisi maitaiability of the equipmet These duratios (failed state detetio time, failure orretio time...) reat o the value of the availability of the equipmet: they a be speified diretly of idiretly through the speifiatio of the itrisi availability of the equipmet. Availability harateristis The uavailability of a equipmet a be omplete or partial. Complete uavailability is related to the omplete failure of the equipmet. For a swithig system, several partial uavailabilities a be speified, depedig o the osequee of the failures, as for the futioal failure rate. Editioof Jue 0,

3 T/N -0 E Page. METHODS FOR TESTNG.. Testig from field data The method osist i estimatig the harateristi to be verified from field data olleted o oe (or several) equipmet(s) ad i usig statistial tools to proess the olleted data so as to deide (with previously aepted risks) whether the equipmet omplies or ot with its speifiatios. This method applies partiularly to the test of the global failure rate of equipmets. t a be hose oly if the quatity of data whih a be olleted durig field operatio is of a magitude ompatible with the use of the statistial tests proposed i paragraph... Testig from preditios Whe testig from field data aot be osidered, it is possible to test the ompliae of a equipmet to its reliability or availability speifiatios by omparig the speified value of eah harateristi to the orrespodig value obtaied from a preditio. This method must be hose whe the quatity of field data whih ould be olleted durig field testig is so low that sigifiat olusios ould ot be draw from them. t applies partiularly to the test of the global failure rate (whe field testig appears impossible) ad to the test of futioal failure rates ad of availability harateristis. t is reommeded to preset the previsios as idiated i paragraph.. Remark. As the futioal failure rate ad the availability harateristis deped o the mea values of duratios whih are itrisi to the equipmet (failed state detetio time, failure orretio time...), it is eessary to test that the values adopted for these mea duratios i the preditio are osistet with the atual possibilities of the equipmet: the orrespodig test a be made by failure simulatio (as idiated i paragraph..) to ollet data o the osidered duratios, followed by statistial tests o the mea values (as desribed i paragraph..)... Test by failure simulatio The method osists i simulatig failure loated i the differet parts of the equipmet i quatities whih reflet the mea umber of failures whih are likely to our i field operatio i that part of the equipmet. This method is partiularly adapted to the test of maitaiability parameters. As a matter of fat, these parameters are difllult to predit ad their field testig may be log ad hasardous. So, testig by failure simulatio applies partiularly to the test of failed state detetio probability, of loalizatio effiiey ad to the test of mea duratios related to the maitaiability of the equipmet. The method to selet the failures to be simulated is desribed i paragraph.. The statistial test for the failed state detetio probability is desribed i paragraph.. The statistial test for the effiiey of loalizatio of a failed item is desribed i paragraph.. The statistial test for the mea duratios take ito aout i the omputatios of futioal failure rates or i the omputatios of uavailability harateristis is desribed i paragraph.. SUMMARY TABLE Charateristi to be tested Reommeded test method Remarks Probability of failed state detetio Effiiey of failed item loalizatio Failure simulatio Failure simulatio Uavailability Preditio () () Uder the oditio that the quatity of failures likely to our is osistet with the quatities required for statistial testig (see paragraph.). () t a be eessary, aordig to the situatio, to test that the mea values of the atual repair times are osistet with the values adopted i the previsio (for the duratios whih are itrisi to the systems). Editioof Jue 0,

4 T/N -0 E Page. METHODS AND MATHEMATCAL TOOLS.. Test of a failure rate from field data... Data olletio t is reommeded that reliability data be olleted aordig to a systemati method of olletio ad that this method be well kow from operatig people. For this, see CCTT Hadbook o the quality of servie ad etwork maiteae, hapter, setio ad revised EC Publiatio (i preparatio).... Failures of the equipmet or system Eah test item failure shall be lassified as a relevat or a o-relevat failure. All test item failures that aot be learly lassified as o-relevat failures aordig to sub-lauses...,...,... below or to ay additioal rule give i the detailed reliability test speifiatio shall be osidered relevat test item failures. f two or more idepedet failure auses are preset, eah of these shall be osidered as oe test item failure. A test item failure may be regarded as a o-relevat failure oly if the irumstaes at the ourree show lear evidee to lassify it ito oe of the lasses defied i sub-lauses...,..., or... below. The evidee shall be doumeted ad iluded i the test report. Additioal lasses of o-relevat failures appliable i a partiular ase may be defied i the detailed reliability test speifiatios.... SeodPry failures A seodary failure is defied as a failure of a item aused either diretly or idiretly by the failure of aother item. Seodary failures are osidered o-relevat. The orrespodig primary failure is always a relevat failure if it is loated i the test item. Observe that a seodary failure may our after a time delay from the ourree of the primary failure. The duratio of the time delay shall be approved by the ustomer or test agey. However, seodary failures a be useful for the lassifiatio of failures i terms of safety aspets, osts of failure, et.... Misuse failures A misuse failure is defied as a failure attributable to the appliatio of stresses beyod the stated apabilities of the item. Misuse failures durig field testig may be due to uitetioal operatig oditios, e.g. operatig oditios exeedig those speified for the equipmet (lightig), rough hadlig by operatig or repair persoel, et. Misuse failures are osidered o-relevat.... Failure elimiated by desig orretio A type of failure observed early i the test may result i a desig hage or other remedy implemeted o all equipmets i the populatio. f suh a orretive atio is prove to be effetive, the failures of this type may be relassified as o-relevat failures upo agreemet.... Test plas the followig test plas, the failure rate is supposed to be ostat. These plas are based o a parametri hypothesis test whih osists i opposig the followig hypothesis: the true failure rate k is equal to the speified value LO, the true failure rate h is equal to the maximum aeptable value k,. Suh a statistial test ivolves the followig false deisio risks: a: Suppliers risk: it is the probability y of rejetio of a equipmet or system (or of a bath) whose true failure rate his equal to the speified value ~. (the probability of rejetio whe k<& is less tha a). B: Admiistratio s risk: it is the probability of aeptae of a equipmet or system (or of a bath) whose true failure rate k is equal to the maximum aeptable value k, (the aeptae probability whe > Lis less tha ~). The ratio D = ll/ko is alled the disrimiatio fator.... Stadard test plas Whe the values of a, ~, 0 ad D are give, oe a derive the operatig test time (T) whih has to be aumulated by the equipmets or systems ad the maximum umber (C) of failures ourrig durig the aumulated test time T ompatible with the deisio that the equipmet omplies with its failure rate speifiatio. Correspodig test plas are desribed i EC 0- Publiatio. Editioof Jue 0,

5 T/N -0 E Page f Other test plas t a be oveiet to hoose beforehad the value of the aumulated test time T (whih allows to deide beforehad of the quatity of equipmets or systems to be moitored ad of the duratio of the test). this ase, a, ~, LOad T are give ad the value of the disrimiatio fator (D) as well as the maximum umber (C) of failures whih a our durig the aumulated test time T are derived. The method for derivig D ad C from the values of a, ~, A ad T is give i the supplemet to the EC 0- Publiatio: Proedure for the desig of time termiated test plas, to be published. Presetatio of reliability, maitaiability ad availability preditios Related doumets The presetatio of reliability, maitaiability ad availability preditios is overed by the EC Publiatio. However, some eessary adaptatios have bee made i the followig paragraphs. Objet The objet of this doumet is to provide the writer of a preditio report with a omplete listig of all items to be osidered i makig a proper ad full presetatio of preditio iformatio. this Reommedatio, the way of presetatio is iteded to failitate ompliae testig of reliability, maitaiability ad availability y harateristis by omparig the speified values of the required harateristis to the orrespodig predited values. Appliatio area This Reommedatio is geerally appliable to all reliability, maitaiability ad availability preditios of teleommuiatio equipmet or systems, iludig hardware, software ad huma elemets. Cotets of the presetatio Aordig to EC Publiatio. Detail requiremets of the presetatio For the detail requiremets of the presetatio, refer to EC Publiatio, exept for the followig: Charateristis The system or equipmet reliability, maitaiability ad availability harateristis whih ostitute the fial objetive of the preditio shall be stated by referee to relevat system or equipmet doumets, suh as speifiatios of reliability, maitaiability ad availability requiremets. Assumptios, defiitios ad oditios All the assumptios, defiitios ad oditios eessary for the preditio shall be stated: System/equipmet futios. A system or equipmet maybe iteded to futio i may modes or to arry out sequees of futios. Ay suh futio or sequee of futios, overed by the preditio, shall be stated. Ay futio or equipmet exluded from the preditio shall be idetified ad the reaso give. Failure defiitios. The failures of the system/equipmet to be osidered i the preditio are those stated i the reliabilityy/availability speifiatio of the equipmet/system. Ay deviatio from these defiitios shall be learly idiated. Quality/reliability programme, The quality ad maturity of the system or equipmet shall be stated, for istae, i terms of: a) system or equipmet bur-i; b) referee to quality/reliability programme of system or equipmet ad ompoets; ) ompoet sreeig. Ay assumptio regardig reliability or maitaiability growth shall be stated. Evirometal oditios. The evirometal oditios for whih the preditio is performed shall be those speified for the equipmet/system operatio. Operatioal oditios. The operatioal oditios for whih the preditio is performed shall be those stated for the equipmet/system i its relevat speifiatio. Defiitio of maiteae atios. The equipmet/system speifiatio defies as maiteae requiremets o whih equipmet/system omplexity level orretive maiteae is to be performed, suh as failure loalizatio of replaeable uits or failure loalizatio dow to ompoet level. Aordigly, the expeted mea values of the duratios of the orrespodig maiteae atios, whe used for the preditio, shall be stated (see paragraph Maitaiability data ). Editioof Jue 0,

6 T/N -0 E Page Prevetive maiteae oditios. The prevetive maiteae oditios for whih the preditio is performed shall be stated i the form of: a) ategories ad stadards of prevetive maiteae resoures; b) ategories ofprevetive maiteae atios; ) riteria goverig the shedulig of prevetive maiteae, for example fixed itervals betwee atios or degree of wear-out; d) effets o system operatioal readiess. Corretive maiteae resores. Categories ad stadards of orretive maiteae resoures shall be defied. These may ilude: a) replaemet uits; b) spare ompoets; ) software media; d) test equipmet; e) tools; f,) test programs; g) doumetatio; h) persoel. Maiteae support oditios. The maiteae support oditios for whih the preditio is performed shall be i aordae with those stated by the Admiistratio i the equipmet/system speifiatio. Aalysis A aalysis has to be made to determie: a) the struture of the system/equipmet; b) the stresses applied to the system/equipmet ad its parts; ) the maitaiability properties of the system/equipmet; d) the properties of the maiteae support. Based o this aalysis models are built for: the reliability struture, the maitaiability struture, the availability struture. The mathematial model used for eah harateristi ad the derivatio of applied formulas shall be stated or refereed. f the preditio is performed by a proedure whih preeeds stepwise through several futioal levels of the system/equipmet, the mathematial models uses shall be preseted separately for eah harateristi. Data soures Reliability data. The soures of reliability data shall be agreed by the Admiistratio. Reliability data used, suh as failure rates or mea times betwee failures at uit level, shall be stated. Maitaiability data. Maitaiability data used, suh as mea ative repair times at differet levels, failure detetio probability, failure loalizatio etliiey, shall be stated. Maiteae support data. Maiteae support data used, suh as umbers of repair me ad spare parts, shall be stated either diretly or i probabilisti terms. They shall be osistet with the maiteae support oditios. Preditio results The umerial results shall be learly preseted for eah speified harateristi, i orrespodee with the orrespodig required value. Failure simulatio Failure simulatio is merely used for testig maitaiability related parameters. Quatity offailures to be simulated The total umber of failures to be simulated shall be determied from the adequate test pla (paragraphs..,.. or..). Distributio offailures For eah group of ompoets belogig to a give family (trasistor, itegrated iruits) ad belogig to a give part of the equipmet, the umber of simulated failures shall be proportioal to the mea umber of failures whih are likely to our i field operatio amog the ompoets of this group. Whe the failure rates of the ompoets are ostat, the umber of failures to simulate i eah ompoet is proportioal to its failure rate: the followig paragraphs give the details of the method aordig to this hypothesis. The failure rates to be used i this respet are to be agreed upo by the supplier ad by the Admiistratio. Editio of Jue0,

7 T/N -0 E Page... Appliatio a) Classify the p parts of the equipmet of their ompoets. b) Classify theqfamilies of ompoets of their ompoets i eah family. ) Give to eah group family oft ype q i dereasig order aordig to the sum P of the failure rates i dereasig order aordig to the sum A~ of the failure rates belogig to part p a weight ~ where: LP%is the sum of the failure rates of the ompoets belogig to the group pq. A the sum of the failure rates of all the ompoets of the equipmet. A table as the followig oe summarises the tasks a), b) ad ) above. Compoet family Equipmet parts ~ k,, per family P P, k,, k,,?$, xpl x?l L?L pz q A L k L zq q L Pq d) Determie for eah group (ase pq of the table) the umber pq of failures to be simulated. f N is the total umber of failures to be simulated, pq is give by: A pq=nx~ A with A = Al + Az +... Az ad A~ = A~ + Az~ + kp~ The omputatios geerally lead to o-iteger values of pq. These figures will be systematially rouded to the earest lower iteger, thus leadig for ertai ases to zero. Thus, over the N failures to be simulated, some remai beig ot assiged: the pq groups i whih a remaiig failure will be simulated will be hose at radom amog those for whih pq <. e) Selet at radom amog the ompoets of eah group the pq oes for whih a failure will be simulated. Remark. The hoie of the failure modes for eah ompoet shall be guided by the distributio of the failure modes of the family to whih the ompoet belogs, whe this distributio is kow... Test of suess (failure) ratios... Priiple This test is based o the properties of the biomial law ad is iteded, i this Reommedatio, to the test of the failed state detetio probability. t osists i reordig the results of N failures simulatios ad i omparig the observed umber r of times where the failure is ot deteted to a deisio riteria rre. Oe oludes the the equipmet omplies with its speifiatio if r < rre ad that it does ot omply with its speifiatio if r > rre. The failures to be simulated shall be distributed iside the equipmet aordig to the method of paragraph.. This test ivolves the followig two false deisio risks: to risk a of the supplier is the probability that r > rre eve whe the true (but ukow) peretage of suess p haraterizig the equipmet is equal to the speified value PO; the risk ~ of the Admiistratio is the probability that r.< rre eve whe the true peretage of suess p haraterizig the equipmet is equal to the miimum aeptable value P defied as P = D ~ PO) where D is alied the disrimiatio fator. the test plas desribed below, the risks a ad B are equal.... Stadard test plas Whe the values of a = ~, PO ad D are give, oe derives the umber as well as the deisio riteria rre. The orrespodig test plas are desribed i EC 0- Publiatio.... Other test plas N of simulatios P t a be oveiet to deide beforehad of the total quatity of simulatios to be performed. this ase, a= ~, PO ad N are give ad oe derives the values of D ad rre. to be performed The method for derivig the values of D ad rre from the values of a = (,PO ad N is give i Appedix. Editio of Jue0,

8 T/N -0 E Page Test of the efl iiey of loalizatio of a failed item The followig statistial test is iteded to test the ompliae of a equipmet to its speifiatio of etlliey of loalizatio of failed uit expressed as two probabilities F ad P, for whih the values pl ad p are required: P = P (loalizatio amog, or or... k items) P = P (loalizatio amog k + or k+ or m items) with m <, beig the total umber of items buildig the equipmet Priiple This test is based o the properties of the multiomial distributio. t osists i testig the hypothesis HO: P] = p, P = p, p ad p beig the required values agaist the hypothesis H : P= ql, P = q, ql ad q beig miimum aeptable values Performig a) ) the test Perform the N failure simulatios, the simulated failures beig distributed aordig to paragraph.. b) Reord the quatities: Xl= umber of suessful loalizatio withi or or... k items X= umber of suessful loalizatio withi k+ or k+ or.. m items Compute the quatities A ad A from the values of P, P, by (. = Nepriam logarithm): d) Compare the quatity Al Xl + AX to a riteria C: f AX + AX > C, the HO hypothesis a be admitted ad the oe osiders that the equipmet omplies with its speifiatio of failed item loudlizatio effiiey. f AX + AX < C, oe osiders that the equipmet does ot omply with its speifiatio of failed item loalizatio eflliey. As ay statistial test, this test ivolves the followig false deisio risks: the risk a of the supplier is the probability that A + AX < C, i.e. oe oludes that the equipmet does ot omply to its speifiatio of failed item loalizatio effiiey, eve whe the HO hypothesis is true, whih meas that P = pl ad P = p (speified values); the risk ~ of the Admiistratio is the probability that AX + AX > C, i.e. oe oludes that the equipmet omplies with its speifiatio of failed item loalizatio effiiey, eve whe the H hypothesis is true whih meas that the failed item loalizatio is araterised by the miimum aeptable values ql ad q. The risks a ad ~ are geerally equal Quatity of tests to be performed (Uder study.) Deisio (Uder ritevia study.) Test of mea duratios The statistial test desribed below does ot eed ay hypothesis o the distributios of the duratios. this test, the mea values ~ of observed duratios is ompared to a deisio riteria L. Oe oludes that the true mea value of the osidered duratios is less tha or equal to the value mo proposed by the supplier, as a basis for availability omputatios, if X> L. This test ivolves two risks of false olusios: the risk a of the supplier is the probability that ~ > L, eve whe the true mea m of the osidered duratios is equal to the value mo so that oe oludes wrogly that the equipmet does ot omply with the value take ito aout i the omputatios of futioal failure rate or of availability; the risk ~ of the Admiistratio is the probability that;< L, eve whe the true mea m of the osidered duratios is equal to a value ml = D x mo so that oe oludes wrogly that the equipmet omplies with the value take ito aout i the omputatios, D beig the disrimiatio fator. Editio of Jue0,

9 T/N -0 E Page... Quatity of observatios requiyed The test is based o the etral limit theorem ad requires a miimum of 0 observatios. O the other had, the umber N of observatios is related to the risks a ad ~ by: N= U0 where (ml mo) { u is the uit ormal variable: u = U_a = u~ Oz is the (ukow) variae of the duratios ml = D mo For the test of the duratios proposed by the supplier, the value D= will be admitted. A. * f a estimatio of o, say o s available, oe omputes the quatity of observatios required from: N= U($* mo * f o iformatio is available o the variae of the duratios, oe exeutes the 0 first observatios, from whih oe omputes the observed variae (S~O)ad the quatity of observatios to be performed is obtaied from: U(S0)* N= z mo... Deisio riteria The deisio riteria L is give by: =mo+ % S~ beig the stadard deviatio of the N observed duratio. Editio of Jue0,

10 T/N -0 E Page 0 Appedix GUDANCE ON THE DESGN OF COMPLANCE TEST PLANS FOR FALURE RATO. PURPOSE This doumet is iteded to be used for testig the ompliae of a equipmet to its speifiatio of failed state detetio probability. t a be used more geerally i either or situatios, e.g. for ompliae evaluatio of a failure ratio. The speivied failure ratio is the probability that a item aot perform a required futio or that a evet will be usuessful uder stated oditios. A observed failure ratio may be defied as the ratio of the umber of failed items or usuessful evets at the ompletio of testig to the total umber of test items or evets.. APPLCATON AREA The method is appliable to ases where the followig quatities are give: aeptable failure ratio; produer s omial risk: osumer s omial risk; total umber of test items or evets; possibly, the maximum value of the disrimiatio fator. The method gives the followig output quatities: disrimiatio fator (atual value); ritial value (maximally allowed umber of failed items or usuessful evets).. RELATED DOCUMENTS EC: Equipmet reliability testig, part : ompliae test plas for suess ratio ). (Publiatio 0-,. SYMBOLS PO: aeptable suess ratio qo: aeptable failure ratio: PO P: uaeptable ratio ql: uaeptable failure ratio: P D = ql/qo disrimiatio fator a: produer s risk ~: osumer s risk : total umber of test items or evets r: observed umber of failed items or usuessful evets C: ritial value. Maximally allowed umber of failed items or usuessful evets Note. The ritial value C is related to the quatity rre used i EC 0- Publiatio by: C= rre., CALCULATON PROCEDURE O the basis of the iput quatities: po(orqo=l po), a=~, the derived parameters D ad C are determied by meas of graphs,, (D values) ad tables,, (C values). Editio of Jue0,

11 T/N -0 E Page The tables also show roughly the disrimiatio fator aordig to: D> <D< ~: <D< D.: D<: Note. f D>, the must be ireased. f D >, it is reommeded to irease. f D is larger tha the (possibly) speified maximum value, the should be ireased. P. DECSON CRTERA The alulated value C is ompared with r, the observed umber of failed items or usuessful evets. f r< C, the the speified requiremets are regarded as havig bee omplied with r> C, the the speified requiremets are regarded as havig ot bee omplied with. MATHEMATCAL BACKGROUND. The biomial distributio f the probability of a evet is q (approximately ostat), the the probability that the evet will our exatly r times i observatios is: P(r)= () ~ q (l q) -, r=o,,... () where The probability P(r) of fidig r or less evets i observatios is: ~=p(r)=i~o (~)qi( qy-i () For give, qo ad a om the C-value is alulated as the lowest iteger satisfyig p()= ~ [~) qo ( qo~--i> a om,=0 () The tables, ad show are alulated aordig to () for all... Approximatio formulas A very oveiet ad rather good approximatio for the desribed test plas is the arsi trasformatio (give by R.A. Fisher) for ofidee limits for the biomial distributio (ref. ) with slight modifiatios: For give qo, a ad, the followig formula () a be applied to fid the C value with extremely good approximatio for the rages: 0.00 <qo<00. ~<a<so~0 > C= si arsi (J@)-}& 0. [ Jl () The alulated C value shall be rouded to the earest iteger. u = U_a is the a fratile i the ormal distributio. Editio of Jue0,

12 T/N -0 E Page.. Auray The value of D for a = ~ a be alulated The graphs, ad show are alulated aordig to (). as: D=si[arsi(J@+*)l qo order to hek the auray of the result obtaied by the approximatio formulas () ad () the exat formula () may be used, as show i the followig example. () Example: Give: qo=o.0, u om=~ om= 0%, = Requested: C, D, true a ad ~ Formula () gives: C=., rouded to Formula () gives: D=. Formula () gives for qo= 0.0: exat a=.% ql =0.0: exat ~=. /o For po= 0.0, a= ~ = 0% ad D =, EC 0- Publiatio gives: = rre=, that is C =. Editio of Jue0,

13 GRAPH (T/N -0). Disrimiatio fator D = ql/qo versus observed umber of evets, for fixed po (or qo) ad for differet risk levels a om= ~ om (0/0, z. /0, So/O, 0 /0, 0/o,ZOo/O).

14 T/N -0 E Page v-l Editio of Jue0,

15 GRAPH (T/N -0). Disrimiatio fator D = ql/qo versus observed umber of evets, for fixed PO (or qo) ad for differet risk levels a om= B om (0/0,. /0, /0, 0 /0, /0, 00/0).

16 T/N -0 E Page TABLE (T/N -0). Critial value C versus observed umber of evets, for fixed po (or qo) ad for differet risk levek a om= B om (l O/.,.0/., 0/., 00/., ]so/.,.00/.). PO=0. qo=o.0 a (0/0) a (0/0),0, 0 0 a ( /0), J ll

17 T/N -0 E Page TABLE (T/N -O) (Cotiued). Critial value C versus observed umber of evets, for fixed po (or qo) ad for differet risk levels a om= ~ om (%,.T 0, Y0, 0%, ~o, 0 Ko). PO=0. q = 0.0 a ( /0) l(%) a (%).O. 0 0.O. 0 0.O : t t 0 0 S , D< D< D< Editio of Jue0,

18 T/N -0 E Page TABLE (T/N -0). Critial value C versus observed umber of evets, for fixed po (or qo) ad for differet risk levels a om= B om (lyo,.y0, Y0, 00A, 0A, 0 Yo). PO=0.0 qo=o.lo a (0/0) a (0/0) a (0/0).0, ll] -r lo ll t <D< D< D< D< (To be otiued) Editioof Jue0.

19 TN -0 E Page TABLE (T/N -0) (Cotiued). Critial value C versus observed umber of evets, for fixed po (or qo) ad for differet risk levels a om= ~ om (l%,.%, %, 0 +!o,y0, 0%). P(,=0.0 qo=o.lo Cl(%) a ( %) (%) S /--= i D< D< D< Editio of Jue0,

20 T/N -0 E Page 0 e TABLE (T/N -O). Critial value C versus observed umber of evets, for fixed po (or qo) ad for differet risk levels a om= ~ om (lo/o,. /0, /0, 0?Ao, /0, 00/0) PO=0. qo=o. a ( ~) a (0/0) a (0/0), ol S D< D< D< (To be otiued) Editio of Jue0,

21 T/N -0 E Page TABLE (T/N -0) (Cotiued). Critial value C versus observed umber of evets, for fixed PO (or qo) ad for differet risk levels a om= ~ om (?., Z.? 0, Y0, 0Yo, Y0, ZOYO). - PO=0. qo=o. a (0/0) a (%) a (O/.) O ( 0 0 ( ( : : 0 :i 0 : } ( f D< D< D< Editio of Jue0,