Research on the Algorithm of Evaluating and Analyzing Stationary Operational Availability Based on Mission Requirement

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1 Reseach on he Algoihm of Evaluaing and Analyzing Saionay Opeaional Availabiliy Based on ission Requiemen Wang Naichao, Jia Zhiyu, Wang Yan, ao Yilan, Depamen of Sysem Engineeing of Engineeing Technology, BeiHang Univesiy, Beijing, China Absac The popula mahemaical and simulaion mehods boh have some shoage in evaluaing he saionay opeaional availabiliy. The fome assumes ha he iem demand is independen of he opeaing sysems, which can inoduce a seious undeesimaion of he opeaional availabiliy. The lae in ode o ge accepable confidence level equies a lage of ials unde he pe-assigned scenaio. This pape addesses he issue of deemining he saionay opeaional availabiliy basing on mahemaical models. No only he sysem passivaion and opeaion mission, bu also ohe influencing facos, such as he sysem design paamees, he numbe of woing sysems, un ound ime and mainenance ime, ae all consideed in he model. In ode o achieve he esul, a poweful appoximaion mehod o he opeaional availabiliy is given. A las, an example is used on eseaching he elaionship among afoemenioned facos. Expeimens show ha he model wos well wih one Calo simulaion esul and so, he feasibiliy and aionaliy of he mahemaical mehod ae validaed. Keywods- Opeaional Availabiliy; Saionay Sae; Reliabiliy; ainainabiliy; Spae pas I. INTRODUCTION Opeaional availabiliy (Ao is an impoan evaluaion paamee of sysem efficiency, which is he funcion of sysem design chaaceisics, opeaion mission and mainenance scheme. The mehod used fo calculaing Ao usually can be classified ino wo ypes. One is simulaion mehod, which is esablished on he base of sysem funcion model, mission scenaio model and mainenance and suppo model. Some sofwae, such as Simlox2. [], SS [2], ec., has such funcions. Though simulaion mehod has many meis, bu is efficiency is low and which maes i is moe feasible o evaluae sysem dynamic availabiliy ohe han sysem saionay availabiliy. Anohe is he model mehod, which s pinciple is ecuing o consuc he elaionship of Ao wih sysem opeaion mission and mainenance scheme. Some sofwae, such as ETRIC [3], VARI-ETRIC [4], SPAREL [5] and OPRAL [6] has such funcions. Though above menioned models can be used o calculae saionay availabiliy, bu passivaion is no consideed in hese models. When consideing passivaion, sysem has no spaes demand because of mainenance, so he availabiliy is lowe han which conside passivaion. Lau ec [7] eseach dynamic availabiliy on he condiion of passivaion. In his pape, he auhos conside he elaionship of uilizaion vesus availabiliy ohe han mission scenaio vesus availabiliy. Lau ec [8] also eseached he sysem availabiliy consideing he sysem damage in one of his ohe pape. This pape esolves he mehod of how o calculaion sysem saionay availabiliy consideing mission scenaio. The auho expaiae he echnique of ansfoming mission scenaio o sysem equiemens. Then, such echniques ae binging ino an example on calculaion he sysem availabiliy basing on feedbac heoy. Afe validaion, i can be seen ha he mehod wos well wih diffeen paamees. II. BASIC PRINCIPLE The value of Ao is decided by he sysem design chaaceisic, opeaion and mainenance policy, ec. Fom he ime axis, i can be seen ha Ao is dynamic changes along wih ime and is final value apes off o saionay value as. When no consideing managemen delays, Ao is he funcion of such facos: ( ( λ (, (, ( A f b ( O n n n n Whee, λ( n is he failue ae a ime n ; ( n is he epai ae a n ; b( n is he iem bacodes a n ; n is he ime ineval and which has inege value. Fom fomula (, i is obvious ha Ao( n is deemined by he value of sysem failue ae, epai ae and he expeced bacodes a n. When he failue ae and epai ae ae consan values, Ao( n is a funcion only of b( n. So b( n is deemined by Ao( n- a ime n-. The fomula is: O( n ( λ,, O( n A g A (2 In equaion (2, denoes he iniial ime when n. I can be seen fom (2 ha, along wih he incease of n, he diffeence beween Ao( n- and Ao( n is gadually educing and in he end, he sabilizaion sae is obained. So Ao( n- and Ao( n consuc he minus feed bac egulae sysem. I is shown in Figue /9/$ IEEE 59

2 I Zi S ( { i( i /( i } (4 i A EBO s N Z Figue. Feed bac sech map of Ao( In Figue, - denoes he compaison opeao. The inpu signal afe egulae epesens he diffeence beween Ao( n- and Ao( n. When he sysem failue ae is lage han he sysem epai ae, Ao( is deceasing wih ime. Conay, if he sysem failue ae is smalle han he sysem epai ae, Ao( is inceasing wih ime. Sysems coninuously execuing mission consiues sequen egulaion acions of deceasing he diffeence. Clealy, he diecion of adjusmen is owads deceasing he diffeence beween hese wo values. So he pocess menioned above consiues he negaive feed bac pocess of Ao(. The ieaive pocess is displayed in Figue 2. Ao Ieaive numbe Anicipae Ao Acual Ao Figue 2. The ieaive pocesses of Ao( Based on he pinciple, he elaionship of Ao( wih mission equiemens as well as spaes invenoy levels can be esablished. Basing on hese facos, he saionay Ao( when can be calculaed. III. CALCULATION ODEL A. Calculae A O ( Ao( can be seen as he aio beween he numbe of available sysems and he numbe of nominal sysems a ime. When spae bacodes ae place, some numbe of sysems will be unavailabiliy because of he shoage of spaes. Fuhemoe, spae bacodes ae place andomly wihin sysems unde he condiion of no inefeence. So Ao( can be calculaed by [9] : A o ( + A A S ( ( Whee, A S ( is he supply availabiliy and A ( is he mainenance availabiliy(o achieved available. The expession of A S ( and A ( ae given by: (3 A ( TB TB + CT + PT Whee, TB is he mean ime beween mainenance; CT is he mean ime beween coecive mainenance; PT is he mean ime beween pevenive mainenance; EBO i ( s i is he expeced bacodes of iem i a ime ; s i is he soc level of iem i; N is he numbe of sysems; Z is he numbe of iems pe sysem; i is he index numbe. In (4, EBO( can be wien as: ( ( ( ( (5 ( d T e d TAT (6 s+! EBO s Whee, d( is he demand ae of spaes; T AT is he odeing ime fo epaiable iems o he un ound ime fo discadable iems and i is eplaced by anspo ime in his pape. B. Calculae A S ( Lemma [] : Suppose he non-homogeneous Poisson inpu inensiy funcion has he foms λ ( ohe and non-saionay sevice disibuion G. Then he numbe of aivals undegoing sevice ime a has a Poisson disibuion wih mean ( ( G( v λ ( v Λ, Whee, he sevice ime Y is andom vaiable and a ime has he pobabiliy disibuion [ ] (, dv (7 P Y y G + y (8 Supposing he anspoaion ime T has consan value, he epai ime T is andom vaiable and has exponen disibuion wih he mean value of /μ. Then he numbe of iems in anspoaion and epai pocess can be calculaed fom Lemma. Leing he anspoaion ime and he epai ime as a whole one, hen he disibuion of YT +T is: (, P{ + } G v T T v { } ( v T P T v T e μ + (9 + v T ( ( μ( Λ d v e dv ( 6

3 The expeced numbe of bacodes of iems i a ime is: S ( ( Λ ( EBO EBO s i i i Λi ( e Λi ( s s+! ( ( I Zi i (2 i ( { i( /( } A EBO N Z The value of A S ( is depended on d( and EBO(. oeove, he EBO( has elaion wih d(, so when, he saionay sae can be expessed by: ( lim ( μ( v T lim Λ d v e dv ( lim ( μ( lim d v dv+ d v e dv T T + v T Le T a T +T, so i ( T d T + ( ( EBO s s I + ( ( (3 ( di Ta e di Ta (4 s+! ( { i( /( i i } Z AS EBO s N Zi (5 C. Calculae A ( If sysem has wo possible saes: availabiliy and nonavailabiliy, hen he sae ansiion ae diagam beween hese wo saes can be achieved by use of λ and μ. I is shown in Figue 3. If P O (, hen P O ( P ( + (7 D ( λ μ + μ λe PO ( + μ + λ λ + μ μ μ + PO ( e μ + λ λ + μ ( λ μ + The value of A ( in unsable Poisson pocess is: ( A 2 μ + λ( 2 μ λ( 2 + μ ( 2 + A ( e λ( 2 + μ μ (8 (9 D. The Tansfom of Sysem ission Because he saionay availabiliy can be obained on he basis of giving sysem mission scenaio, so he calculaion model should involve mission equiemens. This can be down by ansfom he mission scenaio o sysem uilizaion ae. If he sysems numbe N (N and he numbe of sysems equied o pefom mission ( N ae all given, leing sysems as a goup, so he numbe of goups and is coesponding pobabiliy can be obained. Because can be exacly divided by N o no a specified mission scenaios, some analyze echnique mus be used. Given and N, hee mus exis an inege ( saisfying he following expession: ( N < + (2 Obviously, in (2, he numbe of sysem goups is lage han bu is less han +, ha is Figue 3. The ansiion ae diagam of sysem saes diagam Accoding o Figue 3, P O ( can be calculaed by [] : ( λ+ μ μ e PO( + λpo P D μ + λ λ + μ ( ( (6 N < + N < ( (2 Se q N Q - ( q<, by Benoulli Lemma, he pobabiliy disibuion of sysem goups is Whee, P O ( is sysem availabiliy a ime ; P D ( is he sysem un-availabiliy a ime. 6

4 q P( g q g g + else Whee, g is he numbe of sysem goups. (22 Equaion (22 shows ha he pobabiliy of having sysem goups is -q and he pobabiliy of having + sysem goups is q. The saionay Ao( unde diffeen siuaion is given in (8, and he aveage saionay Ao( can be gained by muliplying saionay Ao( wih coesponding pobabiliy. IV. APPLICATION OF THE APPROACH A. Inpu Daa Suppose hee ae 8 aiplanes used o pefom a mission and he mission pofile equies 3 aiplanes calling ou 4 pe day. The sysem and iem daa design daa ae shown in Table : ID λ /h μ /h TABLE I. T AT/d ITE DATA Iem Numbes pe mission LRU LRU LRU LRU LRU LRU LRU LRU Invenoy Level B. Resuls Though (3 ~ (22, he values of saionay Ao( changing along wih he numbe of sysems and he value of T AT can be go. Basing on he daa in Table, he esul of saionay lim A O (.394, in ohe wods, hee ae abou 3.5 aiplanes available. Using Simlox2., he value dynamic Ao ( can be go and i is shown in Figue 4. Figue 4. The Ao changes along wih ime Though compaing, i is can be seen ha he elaive diffeence beween hese wo esuls is abou 2%. C. Sensiive Analysis Basing on he heoy menioned above, he elaionship of saionay Ao( wih T AT, sysem numbes, epai ime ec has been esablished. So he effec by changing one o wo seleced vaiables on he saionay Ao( will be full analyzed. In his pape, he T AT and sysem numbe ae seleced as equied paamees and he saionay Ao( changing ange along wih hese facos is shown in Table 2. TABLE II. Changing scale faco of TAT Ao THE CALCULATION RESULTS OF STATIONARY AO(T Sysem Numbes Figue 5. The Ao( changes along wih ime In Table 2, i is can be seen ha he saionay Ao( inceases along wih he inceasing of sysem numbes and so on wih he deceasing of T AT. If he scale faco of T AT is seing o.6, hen accoding o Table 2, he saionay lim A O (.879, which is shown in Figue 5. Afe compaing, i is can be seen he elaive diffeence beween hese wo esuls is abou 5.7%. Though a sequence of changing scale facos, a numbe of saionay Ao( can be go by using simulaion. Compaing analyical esuls wih hese simulaion esuls, i is can be seen ha, fis, he diffeences is lage when he sysem numbe is fewe and he diffeences is geing smalle when he sysem numbe is inceasing. Second, he ime equied fo eaching 62

5 saionay Ao( is lage when he T AT is lage and o else he opposie esul. Because he epai ime of faul iems is always shoe in his pape, so i has lile effec on he saionay Ao(. On he conay, if he epai ime is becoming lage, i s effec on he saionay Ao( can no be neglec. V. CONCLUSIONS Sysem design paamees, opeaion equiemens and mainenance concep ae all ciical facos affecing he saionay Ao(. In his pape, he auhos calculae he saionay Ao( including passivaion and mission equiemens. Fis, he shoage of sepaae analyical and simulaion esuls can be avoid. Second, he model pesened in his pape can consuc he saionay Ao( as funcion of sysem design paamees, opeaion equiemens, mainenance concep. So i is vey useful fo designes and opeaos eseach saionay Ao( consideing hese facos. A las, fo fuhe eseach, he model can also include ohe facos and heefoe, he eseach can gealy exend model s applicaion scope. ACKNOWLEDGENT The auhos deeply han Pofesso Kang Rui fo eviewing he manuscip caefully. REFERENCES [] Simlox2 Use s Refeences vesion 2, Sysecon AB, Jan 24. [2] SS Use s Refeence, Depamen of Sysem Engineeing of Engineeing Technology of Beihang Univesiy, 26. [3] Caig C. Shebooe, eic: A uli-echelon Technique fo Recoveable Iem Conol. Opeaional Reseach, Vol. 6, No.. (Jan. Feb., 968, pp [4] Caig C. Shebooe, Vai-eic: Impoved Appoximaions fo uli- Indenue, uli-echelon Availabiliy odels. Opeaions Reseach, Vol. 34, No.2. (a. Ap., 986, pp [5] Nodin, A., aie, F.F., SPAREL: A odel fo Reliabiliy and Spaing in he Wold of Redundancies. 989 Reliabiliy and ainainabiliy Symposium Poceedings, Page(s: [6] Alfedsson, P., Opimizaion of muli-echelon epaiable iem invenoy sysems wih simulaneous locaion of epai faciliies. Euopean Jounal of Opeaional Reseach 99 ( [7] Lau, H.C., Song, H.W., See, C.T., Cheng, S.Y. Evaluaion of imevaying availabiliy in muli-echelon spae pas sysems wih passivaion. Euopean Jounal of Opeaional Reseach, Vol.7, No., 26: 9-5. [8] Lau, H.C., Song, H., Evaluaion of Time-Vaying Availabiliy in uli- Echelon Invenoy Sysem wih Comba Damage, Poc, IEEE Confeence on Auomaion Science and Engineeing (CASE, , Edmonon, Canada, Augus 25. [9] Shebooe, C.C., OPTIAL INVENTORY ODELING OF SYSTES: uli-echelon Techniques.Second Ediion. Ameican: John Wiley & Sons, 24:36-4. [] Caillo,.J., Exensions of Palm s Theoem: a Review. anagemen Science, 37( [] R. Billinon, R. Allan, Reliabiliy Evaluaion of Engineeing Sysems, New Yo, Plenum Pess,

STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION

Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE