ON DETERMINATION OF SOME CHARACTERISTICS OF SEMI-MARKOV PROCESS FOR DIFFERENT DISTRIBUTIONS OF TRANSIENT PROBABILITIES ABSTRACT

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1 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June ON DETERMINATION O SOME CHARACTERISTICS O SEMI-MARKOV PROCESS OR DIERENT DISTRIBUTIONS O Maeusz Zajac Wrocław Universiy of Technology, Wroclaw, Poland maeusz.zajac@wr.wroc.l Tymoeusz Budny Gdynia Mariime Universiy, Gdynia, Poland asymo@w.l ABSTRACT There is a model of ransor sysem resened in he aer. The ossible semi - Markov rocess definiions are included. The sysem is defined by semi Markov rocesses, while funcions disribuions are assumed. There are aems o assess facors for oher han exonenial funcions disribuions. The aer consis discussion on Weibull and Gamma disribuion in semi Markov calculaions. I aears ha some forms of disribuion funcions makes comuaions exremely difficul. INTRODUCTION The reliabiliy model of inermodal ransor was resened during ESREL 6 conference (Zajac 26b. The model is described by semi Markov rocesses. During he resenaion assumed, ha, robabiliies of ransiion beween saes were exonenial. Comlex echnical sysems are usually assumed, ha robabiliies of ransiion beween saes or sojourn imes robabiliies are exonenial. Lack of informaion, oo lile number of samles or inaccurae assessmen of daa may cause ha such assumion is abused. In some cases, when exonenial disribuion is assumed, here is also ossibiliy o assess facors according o differen disribuions (Weibull, Gamma, ec.. Probabiliies of ransiion beween saes are one of he fundamenal reliabiliy characerisic. The aer includes examle of deerminaion of above menion characerisic for one of he hases of combined ransoraion sysems reliabiliy model. 2 TRANSHIPMENT PHASE CHARACTERISTIC There are hree mehods o define semi Markov rocesses (Grabski 22, Grabski&Jazwinski 23: - by air (, (, when: vecor of iniial disribuion, ( marix of disribuion funcions of ransiion imes beween saes; - by hrees (, P, (, where: vecor of iniial disribuion, P marix of ransiion robabiliies, ( marix of disribuion funcions of sojourn imes in sae i-h, when j-h sae is nex; - by hrees (, e(, G(, 74

2 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June where: vecor of iniial disribuion, e( marix of robabiliies of ransiion beween i-h and j-h saes, when sojourn ime in sae i-h is x, G( marix of sojourn imes disribuion funcions. or ransshimen hase semi Markov rocess is defined by (, P, (. Phase of ransshimen includes following saes:. sandby, 2. dislocaion works, 3. ransshimen, 4. revenive mainenance, 5. reair (afer failure. Aciviies which are involved ino each of above saes are described in aers (Zajac 26a, Zajac 27. The grah of sae is resened in igure. igure. Grah of saes in ransshimen hase 3 CONDITIONS DETERMINATION OR TRANSSHIPMENT PHASE RELIABILITY Transshimen hase elemens can say in reliabiliy saes from he se S (,, where: unserviceabiliy sae, serviceabiliy sae. Oeraion saes akes values from he se T (,2,3,4,5. Caresian roduc of boh saes creaes following airs: (,, (,, (,2, (,2, (,3, (,3, (,4, (,4, (,5, (,5. The model allows for exisence of following airs, only: S - (,, S 2 - (,2, S 3 - (,3, S 4 - (,4, S 5 - (,5. Means of ransor are in firs oeraion sae (sandby during ime described by random variable ζ. The disribuion funcion of random variable is { } ζ = P ζ,. The ime of he second sae (dislocaion is described by ζ 2. The disribuion funcion of random variable akes form { } ζ 2 = P ζ 2, The ime of hird sae (ransshimen is described by ζ 3, where disribuion funcion of random variable is given by formula: 75

3 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June { } ζ 3 = P ζ 3, If he ime of realizaion of revenive mainenance is known (and lass γ, han he disribuion funcion of sojourn ime in he fourh sae (revenive mainenance is { } ( = P γ,. γ Some of aciviies can be inerrued by failures. I was assumed, ha ime of work wihou failure in saes 2-nd and 3-rd is described by η i, i = 2,3. The disribuion funcion is given by formula: { } ( = P η,, i = 2,3. η i i If here is known ime, when he sysem is broken down, and ha ime is given by χ, hen he disribuion funcion of sae 5-h (reair is { } ( = P χ,. χ Saes 4-h and 5-h are saes of unserviceabiliy, however only sae 5-h requires reair afer failure. We assume ha random variables ζ i, η i and χ are indeenden. 3. Kernel deerminaion and he definiion of semi Markov rocess in ransshimen hase The hase of ransshimen can be described by semi Markov rocess {X(: } wih he finie se of saes S = {, 2, 3, 4, 5}. The kernel of he rocess is described by marix (= ( Transshimens from -s sae o 2-nd, 3-rd and 4-h can be described by Transshimens from 2-nd sae o -s and 5-h: = 2ζ (, 2 = 3ζ (, 3 4 ( 4 ζ = (. = 2ζ 2 (, 2 76

4 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June Transshimens from 3-rd sae o -s and 5-h: Transshimen from 4-h sae o -s: Transshimen from 5-h sae o -s: = 25ζ 2 (. 25 = 3ζ 3 (, 3 = 35ζ 3 (. 35 = P( γ < = (. 4 γ = P( χ < = (. 5 χ The vecor = [, 2, 3, 4, 5 ] is iniial disribuion of he rocess. In his case vecor akes values = [,,,, ]. The marix of ransien robabiliies is given by P = (2 3.2 The ransien robabiliies Transien robabiliies are one of he mos imoran characerisics of semi Markov rocesses. They are defined as condiional robabiliies P ij { X = j X ( = i}, i j S = P, (3 Above robabiliies obey eller s equaions (Grabski 22, Grabski&Jazwinski 23 P = δ [ G ] P ( x d ( x (4 ij ij i k S kj ik Soluion of ha se of equaions can be found by alying he Lalace Sieljes ransformaion. Afer ha ransformaion he se akes form In marix noaion his se of equaion has form ~ ( s = δ [ g~ ( s] q~ ( s ~ ( s, i, j S. (5 ij ij i k S 77 ik kj

5 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June ~ ( s = [ I ~ g( s] q ~ ( s ~ ( s, (6 hence ( s = [ I q ~ ( s] ~ [ I ~ g( s]. (7 Deerminaion of ransien robabiliies requires finding of he reverse Lalace Sieljes ransformaion of he elemens of marix (s. 4 DATA AND ASSUMPTIONS OR CALCULATIONS Daa were colleced in 26 in one of he Polish conainers erminals. The daa includes informaion abou numbers of ransien beween saes during 5 succeeded days. Seleced daa are resened in Table. Table. Seleced daa abou ime of saes [h] sae 2 sae dissand by locaion sae 3 ransshimen sae 4 sae 5 revenive reair mainenance average variance min. value max. value disersion Colleced daa didn allow for verifying robabiliies disribuion. The informaion gave ossibiliy o esimae necessary arameers o assess facors for exonenial, Weibull and Gamma disribuion funcions. acors are resened in Table 2. Table 2. Disribuion arameers for differen disribuion funcion sae sae 2 sae 3 sae 4 sae 5 Parameer of exonenial disribuion λ Parameers of gamma disribuion λ Parameers of Weibull disribuion λ A firs calculaion has been done wih assumion, ha ransien robabiliies are exonenial. The disribuion funcion of sojourn imes and heir Lalace Sieljes ransformaion resecively, ake form,28 w = e,,28 f w * =, s,28 78

6 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June,27 w2 = e,,27 f w 2 * =, s,27,26 w3 = e,,26 f w 3 * =, s,26,86 w4 = e,,86 f w 4 * =, s,86,86 w5 = e,,86 f w 5 * =. s,86 Then, kernel of he rocess is given by marix,98( e ( =,99( e ( e ( e,27,26,86,86,8( e,28,6( e,28,4( e,28,2( e,( e,27,26 (8 Marices q (s and g (s have been deermined according o equaions (5 (7. In considered examle we obain and,28,28,28,8,6,4 s,28 s,28 s,28,27,27,98,2 s,27 s,27,26,26 q ~ ( s =,99, (9 s,26 s,26,86 s,86,86 s,86 79

7 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June ~ g(s = s,28,28,27 s,27,26 s,26,86 s,86,86 s,86 ( According o (7, marix (s is a resul of mulilying of wo marices. Elemens from firs column of obained marix (s are shown on igure 2. igure 2. irs column of marix (s Deerminaion of ransien robabiliies requires finding of reverse Lalace Sieljes ransformaion of each elemen of he (s marix. or elemens of he firs column of marix (s reverse ransformaions are as follow: P 86,856,262,539,873 =,4966,87 e,2 e,533 e, e ( P2 4,856,262,539,8734 =,4966,29 e,6 e,522 e, e (2 P3 4,856,262,539,873 =,4966,8 e,3 e,4769 e, e (3 P4 894,856,262,539,873 =,4966,62 e,2 e,787 e, e (4 8

8 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June P5 73,856,262,539,873 =,4966,838 e,3 e,3486 e, e (5 All oher ransien robabiliies (for columns 2 5 have been calculaed similar way. On he basis of above resuls, characerisics of ransien robabiliies from sae -s (sandby o oher, boh serviceabiliy and unserviceabiliy, saes were calculaed. The values of hose robabiliies sabilize afer few days of work of he sysem. The ransien robabiliies funcions o serviceabiliy saes are shown on igure 3, o unserviceabiliy saes on igure 4.,6,5,4,3,2, ime [days] igure 3. Grah of ransien robabiliies o serviceabiliy saes or assumed condiions of hase of he sysem and disribuion arameers, ransien robabiliy o serviceabiliy saes is: 2 3 =.498. Transien robabiliies o unserviceabiliy saes achieve sable value for = 4 days and don change unil = 3. The calculaion hasn been done for greaer values of. igure 4. Grah of ransien robabiliies o unserviceabiliy saes 8

9 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June 5 METHODS O DETERMINING O OR OTHER DISTRIBUTION UNCTIONS Gamma disribuion is aroriae for describing age hardening rocesses of echnical objec. There exiss an assumion ha sum of n indeenden random variables (wih exonenial disribuions, wih arameer λ, has wo arameers gamma disribuion (where is shae arameer, and λ is scale arameer (Jazwinski&iok 99. Weibull disribuion very ofen is used o objec s durabiliy modeling. According o Table 2, colleced daa can be described by Weibull or gamma disribuions. In he aer, for hose disribuions, only Lalace - Sieljes ransformaion are resened. Sojourn imes for Weibull disribuion funcions ake form b( x,269,2 = e, b2 ( x,26, = e, b3 ( x 2,66,4 = e, b4 ( x,538,985 = e, b5 ( x,833,2 = e. Derivaive of Weibull disribuion funcion (i.e. densiy funcion is resened by (6 = λ e Lalace Sieljes ransformaions of Weibull disribuion funcion can be obained by using formula f = s * e '( d = e ( e ' d = s e ( e d ( e λ (7 Hence f * = s e ( e d = s e d s e e d = s e e d (8 Using Maclaurin series for elemen ex(-λ we obain Lalace Sieljes ransformaion of he Weibull disribuion funcion (9 f 2 3 Γ( λ 2 Γ(2 λ 3 Γ(3 = λ... = 2 s 2! s 3! s * 3 n λ n= n! n Γ( n n S 82

10 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June or considered examle, Weibull disribuion Lalace Sieljes ransformaions ake form, resecively,273,774,2,59 =,2 2,4 3,6 4, s s s s f * b 8,26,689,83,49 =, 2,2 3,3 4, s s s s f * b 2 4 2,676 7,269 9,845 54,489 =,4 2,28 3,42 4, s s s s f * b 3 56,5284 2,33 3,439 5,39 =,985,97 2,96 3, s s s s f * b 4 94,845,726,626,5468 =,2 2,42 3,63 4, s s s s f * b 5 84 According o equaion (7, afer deermining of reverse Lalace - Sieljes ransformaion of elemens of he marix (s, ransien robabiliies can be calculaed. Gamma disribuion is given by formula Densiy funcion of gamma disribuion has form Γλ ( =. (2 Γ( λ Γ( '( = e. (2 Lalace Sieljes ransformaion can be obain by using formula f = s λ λ ( s λ * e '( d = e e d = e d Γ( Γ( (22 Taking ino accoun equaion a Γ( a e d =, (23 a s Lalace Sieljes ransformaion akes form (24 λ Γ( λ f * = =. Γ( ( s λ ( s λ 83

11 Zajac, Budny ON DETERMINATION O SOME CHARACTERISTICS O SEMI MARKOV PROCESS OR DIERENT DISTRIBUTIONS O R&RATA # 2(3 ar 2 (Vol. 2 29, June In his case sojourn imes disribuion funcions and resecive Lalace Sieljes ransformaions are as follows 9,26 Γ33,2 (9,26 33,2 =, f * =, 9, 26 Γ(9,26 ( s 33,2 52,6 Γ53,46 (52,6 53,46 =, f * =, 52, 6 Γ(52,6 ( s 53,46 8,99 Γ665,3 (8,99 665,3 =, f * =, 8, 99 Γ(8,99 ( s 665,3 3,97 Γ9,3 (3,97 9,3 =, f * =, 3, 97 Γ(3,97 ( s 9,3 4,2 Γ4,88 (4,2 4,88 =, f * =. 4, 2 Γ(4,2 ( s 4,88 Using of equaion (7 and calculaing reverse Lalace - Sieljes ransformaions ransien robabiliies can be obained. In he case of gamma disribuion, here are numerical roblems wih calculaing of incomlee gamma funcions values. Moreover, even values of gamma funcion for argumens larger han 5 canno be easy obained. Used sofware ools (SciLab 4.. and Derive 6. don allow for calculaing such grea values. 6 CONCLUSIONS. Semi - Markov rocesses allow for esimae basic reliabiliy characerisics like availabiliy or ransien robabiliies for sysems, where disribuions funcions are discreional. 2. Usages of disribuion funcions oher han exonenial in case of semi Markov rocesses causes ha furher calculaions are very comlicaed. 3. There is no easy available sofware which allow for calculaions conneced wih semi Markov rocesses. Because of ha, rofis from usage of semi Markov rocesses are limied. 4. Lack of informaion abou ye of disribuion and rouine assessmen of exonenial disribuion can bring no accurae assumions and consequenly false resuls. REERENCES. Grabski,. 22. Semi Markov models of reliabiliy and mainenance (in olish. Warszawa: Polish Academy of Sciences, Sysem Research Insiue. 2. Grabski,. & Jazwinski, J. 23. Some roblems of ransoraion sysem modeling (in olish, Warszawa. 3. Jazwinski, J. & iok Wazynska, K. 99. Reliabiliy of echnical sysems (in olish. Warszawa. 4. Zajac, M. 26a. Modeling of combined ransoraion erminals srucure. inal reor of rojec ZPORR for Ph.D. sudens of Wroclaw Universiy of Technology (in Polish, Wroclaw. 5. Zajac, M. 26b. Alicaion of five-hases model of reliabiliy of combined ransoraion sysem, Proc. Euroean Safey and Reliabiliy Conference, Esoril. 6. Zajac, M. 27. Reliabiliy model of inermodal ransor sysem. PhD hesis (in olish, Wroclaw Universiy of Technology. 84

Chapter 3 Common Families of Distributions

Chapter 3 Common Families of Distributions Chaer 3 Common Families of Disribuions Secion 31 - Inroducion Purose of his Chaer: Caalog many of common saisical disribuions (families of disribuions ha are indeed by one or more arameers) Wha we should