for Shipboard EquipIllent

Transcription

1 R. J. Hunt Quantitative evaluatin f the usefulness and effectiveness f fleet shipbard guided-missile def ense systems is essential fr assessment f present and f uture fleet defense capability. The measure discussed, called equipment availability, is the average prbability that an equipment will perfrm in a specified manner at sme randm pint in time. Such variable parameters as equipment failure, repair time, practical perating cnditins, and lgistics are taken int accunt. AVALABLTY MODEL fr Shipbard Equipllent Guided-missile weapn systems cnstitute a significant fighting frce fr defense f the fleet. Quantitative evaluatin f the usefulness and effectiveness f these systems is imprtant in lngrange planning f future systems, in assessment f the status f current systems, and in assessment f imprvements. n particular, a prject engineer has a need t quantify his area f cncern in terms f the ultimate intended use f a system s that varius engineering changes and imprvements can be cmpared in terms f cst/effectiveness. n this article, we develp a mathematical mdel that gives an effectiveness measure f shipbard equipments, taking int accunt practical perating cnditins, equipment failure, and repair phenmena. During the develpment f the mdel, we shw the effect f changes in parameters that determine effectiveness. The measure develped in this article is called equipment availability. Availability is defined here as the average prbability that an equipment will perfrm in a specified manner at sme randm pint in time. The majr interest is in what we shall term tactical availability; that is, the chance that an equipment will perfrm satisfactrily when invlved in a tactical envirnment. Many authrs have discussed at sme time the term availability under the guise f reliability, readiness, dependability, r sme ther name. Many f the mdels develped under these names are useful fr particular equipments that characteristically perate in peculiar ways. The availability mdel develped here is riented tward tactical shipbard equipment peratin. As we define availability, it des nt take int accunt what is ften referred t as "missin reliability" r the prbability that an equipment r system will perfrm thrughut an engagement perid. Results f ther availability mdels are liberally drawn upn insfar as they pertain t ur mdel. l 2 1 R. E. Berlw and L. C. Hunter, "Mathematical Mdels fr System Reliability, "Sylvania T echnlgist, J a n P. M. Mrse, Queues,nvent1'ies and Jlaintenance, Jhn Wiley and Sns, nc., New Yrk, May - Ju.ne

2 Measures f Availability Our discussin begins by cnsidering sme practically desirable and simple measures f availability, pinting ut the shrtcmings in these measures. deally, if a particular tactical use f a weapn system were defined in terms f pal ticular threat characteristics, ne f the best measures f the success f the system against this particular threat culd then be btained by cnducting repeated tests incrprating this threat against the weapn system. f we then ran the test n times, and the weapn system successfully achieved its gal s times, we culd say that si n, the fractin f times that the system was successful, is a measure f the usefulness f the system. There are, f curse, many pssible types f threat that a surface-t-air guided missile weapn system may encunter; therefre, t measure adequately the usefulness f the system, many tests must be cnducted. Unfrtunately, rganizing tests n this scale is difficult. n particular, the cst f such tests can ften becme as high as that f the system itself. Tests f this nature are really cncerned with exercising the entire weapn system. Experience with cmplex systems has shwn that it is in fact necessary t test the system as a whle rather than t try t cmbine the results f tests f its cmpnents. On many ccasins, equipments have been reprted as capable f a satisfactry level f perfrmance, but, when actually put t a system test, have been fund t he unavailable. This difficulty is usually referred t as an interface prblem. n this cntext, the interface prblem means that even thugh an equipment is apparently peratinal by itself, its functins are impaired when used with the ther required equipments in the system. We are then faced with the fllwing dichtmy. First, the cst f testing the cmplete system is prhibitive, and secnd, the lack f a cmplete system test fails t check equipment interfaces. Cmprmises can be made, hwever. The system can be brken dwn int subsystems, each cnsisting f perhaps several equipments that can be tested tgether, thus taking int accunt the interfaces between them. The subsystem test can ften be dne at reasnable cst. Our cmprmise cnsists, then, f a test that is within reasnable cst cnstraints and that gives a high cnfidence in the perability f the system at the end f the test, but which at the same time is nt cmpletely definitive. The mdel develped here takes this cmprmise int accunt, which leads t an incmplete test. nputs t the mdel depend n a systematized failure and repair reprting sys- tem, as well as engineering analysis f the way in which equipments are mnitred during nrmal peratin. Sme authrs use as a measure f equipment effectiveness the percentage f time an equipment is reprted as peratinal. That is, effectiveness r availability is Up T ime (1) Up Time + Dwn Time Fr sme equipments this is indeed a practical measure f effectiveness. Fr example, sme lngrange search radars used in cntinental air defense can be mnitred in such a way that they are either perating r dwn fr repair; n ther cnditins are pssible. Simply recrding these times, then, is sufficient t btain a measure f effectiveness. Unfrtunately, mst shipbard equipments cannt be cnsidered in this manner. Smetimes Eq. () is written as MTBF (2) MTBF + MTTR' where MTBF is mean time between equipment failures and MTTR is the mean time t repair. t shuld be understd here and in what fllws that by failures we mean thse phenmena that prevent the equipment frm perfrming in sme specified manner. Basic Mathellatical Develpllent f A vailabili ty We will nw develp an expressin fr availability, which has been used by many authrs,!. 2 and shw that the limiting value f this expressin is simply that cntained in Eq. (2). The methd f derivatin is by the use f difference equatins. We define A as the equipment failure rate (the reciprcal f the mean time between failures) and J... as the repair rate (the reciprcal f mean time t repair). Further, p et) is defined as the prbability that an equipment will perfrm at time t. f A and J... are independent f time, then in a small time interval At, the prbabilities f failure and repair are respectively AAt and J...At. The expressin fr the prbability that an equipment is up at time t + At can then be written pet + At) = P(t)[ 1 - AAtJ + [1 - P(t)JJ...At. Rearranging terms and taking the limit as At 0 gives P'Ct) + P(t)(J... + A) = J..., where P(t) is the derivative f p et) with respect t time. The slutin f this equatin is pet) = _J..._ + c exp [ - (J... + A)tJ. (3) J...+A 10 APL Technical Digest

3 - :::::i 1.0 _-r..., kl = :!+---=-...t--=--+--l =--"""""'+--l Z w it ,--+--l ::::: w kl = FRACTON O F FALUR ES DETECTED _ --+ _ --'1 REPAR RATE,j.t = 0.5 FALURE RATE A = A= 0.05 O L- L TM E (hurs) Fig. -Equipment availability as a functin f tilne, fr selected values f equipment failure rate and repair ra te T determine c, the cnstant f integratin, it is assumed that the equipment is perating at the start f the interval. That is, P(O) = 1. With this assumptin, Thus, c A J..+A The curves labelled kl = 1.0 n Fig. (where kl is the fractin f failures detectable during nrmal peratin) shw Pct) as a functin f time fr indicated values f J.. and A. Ntice that if we take the limit f p et) as t 00, then the availability becmes simply J../ ( J.. + A), which is equal t Eq. (2). Figure 2 shws availability as a functin f MTTR fr several values f MTBF fr the steady-state case. This expressin is particularly useful fr equipments which have perfect mnitring; that is, equipments in which failures are discvered as they ccur. The resulting expressin is simple, and the m athematics used in the derivatin is quite straightfrward. The assumptins f time independence fr A have been prved by experience t be valid. Hwever, experience indicates that the distributins f repair rates are lg nrmal. The effect f this difference has nt been evaluated. A v ail a bili ty Prblells The prblem that is mst bvius t the analyst r engineer cncerned with shipbard equipments is that such equipments can be in a nnperating state withut the fact being apparent t perating persnnel. This is, in fact, ne f the majr stumbling blcks in measuring availability. We have intrduced this phenmenn int the availability mdel in the fllwing way. Since sme equipment failures may be seen in the nrmal curse f peratin and thers nt until peridic tests, we will cnsider the equipment as cnsisting pet) = _1_ {J.. + A exp [- (J.. + A)tJ}. (4) J..+A We will define availability as A = 1T PCt) dn(t). n this expressin, T is the length f time under cnsideratin and N (t) is the distributin functin fr the time f need f the equipment. We will assume that the equipment is equally likely t be needed any time in the interval 0 t T; that is, This gives A Finally, 1 it - P(t)dt. T 0 (fr t 0) (fr 0 t T) (fr T t). A = J.. A + T(J.. A)2 {1 - exp [- (J.. + A)rJ}. 1.0 ;;;;;;;;;;;;;;::::::::--r-:-:::-r---'' it 0.4 ::::: L- L 4 10 MEAN TM E TO REPAR (hurs) Fig. 2-Equipment availability a s a functin f mean time t repair, based n selected values f mean time befre failure. May - J Uli e

4 f tw sectins. n the first, failure f a part can be bserved immediately. n the secnd, a part that fails is nt bserved until a system test is cnducted. We will define the failure rate fr the first sectin as kla and the failure rate fr the secnd as (1 - k l ) A. Cnsider first nly bserved failures. Let pet) be the prbability that the first sectin f the equipment is up (in perating cnditin) at time t; then, using the same derivatin as abve, we find that fj. + ki A [( k exp - fj. + ki A)t]. fj. + la Further, let PL(t) be the prbability that the secnd sectin f the equipment is up at time t. Then, PL(t) = exp [ - (1 - kl)at]. Finally, the prbability that the equipment is up at time t is This gives pet) = fj. k exp [ - (1 - kl)at] fj. + la ki A [ l A + + k exp -(fj. + A)t]. fj. Figure 1 shws p et) as a functin f t fr sme selected values f the parameters kl, A, and fj.. Ntice in this figure that the steady-state phenmenn that ccurs as a result f taking the limit f Eq. (4) des nt exist. This is because f the parameter k i. Obviusly, if there are undetected failures in the system, availability will apprach zer. Hwever, if a definitive test f the equipment is perfrmed, r a trial f the real use f the equipment is perfrmed, then the accrued failures during nrmal peratin will be discvered and the equipment can be restred t a perfect cnditin. This indicates the necessity f intrducing scheduled test prcedures. We will intrduce test prcedures by means f the parameter T. That is, if we schedule a definitive test every day, fr example, then the equipment is restred n a daily basis t an availability f ne. This satisfies the initial cnditins assumed in the riginal derivatin f p et). n rder t simplify the mathematics, we will assume the duratin f the test t be f zer length. During sme f the (5) (6) test perids there will be n failures in the equipment, and during thers there will, f curse, be failures that require repair. n rder t take accunt f dwn time fr repairs during tests, we will cmpute an expected dwn time Ts during each test, and define availability as where T8 1 it - A = T 0 pet) dt, (7) (1 - kl)xt fj. + (1 - k 1 )X' The result f the integratin in Eq. (7) gives A = fj. (fj. + kl A) (1 - k 1)AT. {1 - exp [- (1 - kl)a(t - Ts)]} + ki A T(fJ. + kl X)(fJ. + X). {1 - exp [-(fj. + A)(T - Ts)]}. (8) Figures 3 and 4 shw availability cmputed frm Eq. (8) as a functin f MTTR. Figure 3 shws the effect f the time between system maintenance tests, as well as the effect f changing the f ::; c :s 0.6 ""'-=c :( < f- as kl = FRACTON O F DETECTABLE FALUR ES T = NTERVAL BETWEEN TESTS MEAN TM E BETW EEN FALUR ES = 25 HOURS 1.0 r'--r.---' -- T FOR kl = kl FOR T = 24 HR O 0.4 rtf====f======t===j.-j T =:: 96 w T = MEAN TM E TO REPA R (hurs) Fig. 3-EquiplDent availability as a functin f ldean tilde t repair, shwing the effect f tilde between systeld ldaintenance tests and the effect f changing the fractin f detectable failures. 12 APL Technical Digest

5 .O,-.--.-,---r---, 0.8 p""-==---=f-===+t--+--j - ::J < =...r.=-+t-=-+-..;;:. -' :( Z t-1rt---j :J UJ FRACTON OF DETECTABLE FALURES = t- NTERVAL BETWEEN TESTS = 24 HR O MEAN TME TO REPAR (hurs) Fig.4-Equipment availability as a functin f mean time t repair, shwing the effect f variatin in the mean time between breakdwns. fractin f detectable failures. Figure 4 shws the effect f a variatin in the mean time between breakdwns. With the parameter values chsen, ntice frm Fig. 3 that with weekly r biweekly tests the equipment can be available less than 25 % f the time, and that availability is insensitive t MTTR. This is because during such a time perid the expected number f unbserved failures is high and there is a gd chance that the equipment will be dwn and unperceived by the perating persnnel. As T decreases, the availability is mre sensitive t MTTR. The value f kl will vary cnsiderably frm ne equipment t anther and will depend particularly n the custmary r nrmal perating mde. Figure 3 shws the effect f this parameter. f the equipment MTTR is 8 hr, then increasing kl frm 0.4 t 1.0 is equivalent t increasing simultaneusly the mean time t failure frm 25 t 45 hr and decreasing the repair time t 2 hr. Figure 4 shws the effect f failure rate. A significant thing t ntice here is that availability is mre sensitive t a decrease in MTBF than it is t the same percentage f increase in MTBF. Availability-Test Prblellls The next difficulty encuntered in perating the system we have described is that the test prcedures may nt be cmplete. That is t say, because f either high cst r physical space limitatins r a cmbinatin f these tw factrs, definitive ship- bard equipment tests can be difficult t achieve. T accunt fr this fact in ur mdel, we have intrduced the parameter k 2 -the percentage f failures detectable during a test. This means that when a test is cnducted, instead f restring the equipment t an availability f, the availability after tests cannt exceed exp[ - (1 - k 2 );\t]. Lacking a definitive test, we still have the prblem that equipment availability can apprach zeralthugh at a much slwer average rate than withut any test. There is, therefre, a need t perfrm smehw a definitive test that will in fact restre the equipment t an perating cnditin. Thus, we intrduce the parameter n, the number f nrmal test cycles befre a definitive test is cnducted. A definitive test, as used here, is ne that checks thse parts f the equipment nt tested in the nrmal test prcedure. n the case f a missile system, a definitive test may be a missile firing. Figure 5A indicates the phenmenn just described. Here we see availability as a functin f time. At time zer, availability is 1 and decays accrding t Eq. (6) until time T - T s. At this time the nrmal test is perfrmed. After an expected dwn time T s, the equipment is restred t an perating cnditin with the prbability exp[ - ( 1 - k 2 );\(T - T s) ]. n this figure, we have assumed that three test cycles precede a definitive test, that is n = 3. Thus, at time 3 T the equipment is restred t full perating cnditin. Availability is defined as befre in Eq. (7). Thus, where and lnt A = nt 0 p(t) = P(t) fr 0 ::s; (T - p(t) dt, Ts) p(t) = 0 fr (T - T s) < t < T p(t) = exp[ - (1 - k 2 );\( T - T s)jp(t) fr T ::s; t ::s; (2 T - T s) p(t) = 0 fr2t- Ts < t < 2T p(t) = exp[ - 2(1 - k2);\( T - T s)jp(t) fr 2 T ::s; t ::s; (3 T - T s) p(t) = exp[ - (n - 1) (1 - k 2);\( T - T s)jp(t) fr (n - 1) T ::s; t ::s; (n T - Tl$) p(t) = 0 fr n T - T il < t < n T. (9) May - June

6 p(t) p(t) exp [- ( - k2p. (T - Tsll!, exp [- ( - k2) 2A (T - T.ll " A -- exp [-( - k2)aft] -0 ex'p [ - (-k2) A fr] :'"... t l=ft B --J ,J -.A..._ ""'--J Ts T, T..J T 2T 3T TM E (t) - f = EQU PMENT DUTY CYC LE k2 = FRACTON OF FA LURES DETECTA BLE AT TEST A = FA LURE RATE T = TME BETWEE N TESTS T. = DOWN TM E FOR REPAR AT TESTS '" ".J "'-"""" t = T + 2fT "'-.J--.. t = 2T + 3fT Ts T. T, -""-- T 2T 3T TM E (t)-- "' "- Fig. 5-Equipment availahility as a functin f time fr incmplete tests (A) when the equipment perates cntinuusly, and (B) when the equipment is turned ff fr perids f time hetween tests. ntegratin f Eq. (9) gives A = (_1 ) 1 - exp [-( - k2)xn(t - Ts)] nt 1 - exp [-(1 - k 2 )'A(T - Ts)] ( JL{1 - exp [-( - k1)x(t - Ts)]} (10) (1 - k1)'a(p, + kl 'A) + kf 1 - exp [-(JL + X)(T - Ts)]}). (p, + kl 'A) (JL + 'A) Figure 6 shws the effect f k 2 Ntice that it must be greater than k 1 ; that is, ur test prcedure must be at least as gd as mnitring during nrmal peratin. Shipbard Operatinal Prblells The next phenmenn t be intrduced ccurs because many equipments in the fleet d nt perate cntinuusly; that is, the stress n the electrnic cmpnents ccurs nly at certain times during the day. Fr example, the diagram fllwing indicates a typical equipment's peratin fr ne day. SYSTEM TEST DOWN DOWN AND REPAR OFF ON FOR REPA R ON FOR REPA R OFF ON OFF - - (TME) -2 4 hr f we assume that stress n the equipment ccurs nly during the indicated ON perids, and that during all ther perids the equipment will nt fail, it is cnvenient fr analysis t regrup the perids abve as shwn belw. SYSTEM TEST DOWN DOWN AND REPAR ON FOR REPAR ON FOR REPAR ON OFF (TM E) 24 hr Of curse, the pattern and cycle fr each day will be different. That is, the number f ON and OFF cycles, as well as repair perids, during bth test and nrmal peratin, will vary each day. These cycles, averaged ver many days f peratin, are represented in terms f prbability by Eq. (6). Hwever, in this equatin we have nt taken OFF perids int accunt. An OFF perid means that the equipment des nt fail. The reasn fr regruping the perids, with the OFF perids 14 APL Technical Dige t

7 tgether at the end f the day, is t simplify the mathematical treatment f nnstress perids. n rder t deal with OFF perids, we intrduce the parameter j, the duty cycle f the equipment r the fractin f the time between test perids during which the equipment is subject t failures. Thus, assuming the equipment des nt fail during nnperative perids, we can recnstruct Fig. 5A in the frm f Fig. 5B, in which we have again assumed that tests ccur T cycles apart. The time frm zer t jt represents the stressed perid f the equipment. At time jt we will assume that the equipment is OFF. Hwever, we will further assume that bserved failures (which ccur at the rate f k1a) will be fixed befre turn-ff. This accunts fr the discntinuity and increase in availability at time jt. At time T, the same phenmenn as in Fig. 5A repeats again and cntinues thrugh n cycles f tests. Using the definitin f availability given in Eq. (9), and taking accunt f OFF perids, we find A = (_1) 1 - exp [-(1 - k2)anjt] nt 1 - exp [- (1 - k2)xjt]. (JL{1 - exp r - (1 - k1)ajt]} (1 - k1)x(jl + kl X) + kl A{ 1 - exp [-(JL + A)jT]} (JL + kl X) (JL + X) (11) + [T(i - j) - T,] exp [- (i - k1)xjt]), w here, in this case, Figure 7 shws the effect f the tw parameters j and n. t is particularly relevant t ntice that the duty cycle, the nrmal test cycle, and the definitive test cycle are three items that can be cntrlled by ship's perating dctrine; that is, j, k, and T are independent f failure rates, repair rates, and the mnitring prcedures f equipments. Equatin (11) is a mathematical expressin that takes int accunt mst f the phenmena that ccur in shipbard equipments. We have nt included such imprtant items as turn-n stress r test-equipment reliability. The frmer has nt been included because f the difficulty in btaining useful data. t is interesting that testequipment reliability can be cnsidered as a part f k 2 Thus, k 2 ' is defined as the percentage f failures detectable during a nrmal test. f reliability f the test gear is r, then we culd say that k2' = rk2 and simply replace k2 in Eq. (11) by k 2'. n a similar way, if p is the prbability that failure will ccur at turn-n, then by knwing the average number f equipment turn-ns per day, m, we culd say that A' = (mp/ T) + A, and simply replace A by A' in Eq. (11). Anther usual shipbard ccurrence is preventive maintenance events. Sme preventive maintenance can ccur withut dismantling the equipment. Thse scheduled preventive maintenance actins that require ding s can be included in the mdel if the average dwn time between test cycles is knwn. Let us call this time td ; then, Ts = td + (1 - k1)ajt. JL Equatin (11) is valid fr the fllwing range f parameter values: 0 < A; 0 :::; JL; 0 < kl < ; k 1 :::; k2 < 1; 1:::; n;tt:::; T; and tt/ T :::;j:::; 1, where tt is the minimum time required t run tests. t is assumed in the mdel that the equipment is stressed during the time tt, and minimum values fr T and j are determined by the test time. Expressins fr availability fr limiting values f the parameters can be btained easily by cnsideratin f the basic derivatin given at the utset. f we have a perfect test, that is k2 = 1 and n = 1, then Eq. (11) reduces t Eq. (8). f, in additin, we have perfect mnitring f failures, that is kl = 1, then Eq. (11) reduces t the simple defini- f- f- z ill 0.4 :::J ill r--- r--- k2 1.0 r ,\ = FALURE RATE = 0.05 k, = FRACT ON OF FALURES DETECTED DURNG NORMAL OPERATON = 0.6 k2 = FRACTON OF FALURES DETECTED AT TEST n = NUMBER O F NORMAL TEST CYCLES BETW EEN DEFNTVE TEST S = 7 r--- - T = T ME BETWEEN NORMAL TESTS = 24 HR f = EQUPMENT DUTY CYCLE = 1/ MEAN TME TO REPAR (hurs) Fig. 6-Equipment availability as a functin f mean time t repair, shwing the effect f test cmpleteness. - May - June

8 1.0 i: 0.8 ::::i 0.6 f- z w = 8" r--- r r 1/ DUTY C YCLE = 1/24-1/12 r- r---.- = 1 n =1 n = 4 n = 7 n = 14 A = FALURE RATE = 0.05 k, = FRACTON OF FALURES DETECTED DURNG NORMAL OPERATON = 0.6 k2 = FRACTON OF FALURES DETECTED AT TEST = 0.85 n = NUMBER OF NORMAL TEST CYCLES BETWEEN DEFNTVE TESTS T = TME BETWEEN NORMAL TESTS = 24 HR f = EQU PMENT DUTY C YCLE -- f FOR n=7 -- n FOR f= 1/3 4 M EAN TME TO REPAR (hurs) 10 Fig.7-Equipment availability as a functin f mean time t repair, shwing the effect f selected numbers f nrmal test cycles befre definitive tests and f the duty cycles f the equipment. tin f availability given at the utset in Eq. (2). Sme care shuld be exercised in using Eq. (11) in ptimizatin f parameters, particularly with respect t the duty cycle. t wuld seem frm the mdel that the greatest availability is btained by nt perating the equipment at all. This, f curse, is due t the assumptin f n failures during OFF perids. n any investigatin f ptima, the duty cycle is best treated as a cnstant. The mdel as develped s far assumes that there are n lgistic prblems; that is, the availability is cmputed with the assumptin that spare parts are in ample supply. f a given equipment has a lgistic plicy, the prbability that the equipment is awaiting spares can be easily cmputed. Thus, frm a mdel fr lgistics it is pssible t cmpute the lgistic availability, which we may call AL. The verall availability f an equipment is then the prduct f AL times the quantity A as cmputed in Eq. (11). ' Engineering estimates f kl and k2 require extensive effrt. One methd f btaining input values fr these parameters is t establish frm a reprting system a list f failures fr an equipment. This failure list is then analyzed by system engineers and perating persnnel t determine (1) whether the failure wuld be bserved in nrmal peratin, and (2) whether it wuld be bserved during system tests. When an equipment is in the design stage, the mnitring f particular equipment subassemblies shuld be cnsidered in terms f theretical failure rate and the cst f cntinuus mnitring. This mdel fr availability has been used t determine quantitatively the current level f effectiveness f guided-missile systems in the fleet, as well as t predict effectiveness levels n the basis f planned r prpsed changes. Anther useful utput f the mdel is its capability fr determining the directin that affrds the best payff in availability. Fr example, we have previusly indicated the effect f changing the rutine mnitring aspects f the equipment as cmpared t changing simultaneusly the failure rate and the repair rate. Additinally, the mdel has prved useful t manufacturing cntractrs as a guideline tward new equipment design, which will give ptimum equipment fr the lwest cst. 16 APL T echnical Digest