VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com A CALCULATION METHOD FOR ESTIMATION OF THE MEAN TIME TO FIRST FAILURE OF THE TECHNICAL SSTEMS ON BASIS OF THE TOOLOGICAL CONVERSION OF THE MARKOV RELIABILIT MODEL avel Azzurovch Rahman Ufa Sae eroleum Technologcal Unversy, Kosmonavov S.,, Ufa, Russan Federaon E-Mal: avelar@yandex.ru ABSTRACT Ths scenfc aer deals wh he relably models of echncal sysems on bass of connuous-me Markov chans CTMC.An exsng oeraor mehod for calculang he mean me o frs falure MTT, based on he reducon of Markov chan and solvng he sysem of Kolmogorov-Chaman dfferenal equaons, s also dscussed. The work also hghlghs he oologcal mehod offered by he auhor for calculang he mean me o frs falure on bass of he secal converson of Markov chan. The calculaon examles of MTT for he asymmerc comung sysem wh rle modular redundancy by he exsng oeraor mehod and he offered oologcal mehods are also resened. The exermenal research of calculaon me by he oeraor and he oologcal mehods are also dscussed. Accordng o he research resuls, he oologcal mehod offered by he auhor s sgnfcanly faser han exsng oeraor mehod. Keywords: echncal sysems, relably model, connuous-me Markov chan CTMC, mean me o frs falure MTT, kolmogorov-chaman dfferenal equaons sysem, lalace ransform, oologcal mehod, lnear algebrac equaons sysem. INTRODUCTION In resen days he relably models [, ] on he bass of well-known connuous-me Markov chans [, 4] and a se of relably ndces lke saonary avalably facor, mean me o falure and mean o rear, are wdely used for he relably analyss of echncal sysems. For he calculaon of such relably ndces an effcen oologcal mehod s aled [5, 6]. However, for boh non-rearable and rearable sysems n mos cases he mean me o frs falure MTT also reresens an moran relably ndex, whch allows for redcng he mean me o frs falure sae of he sysem afer s saru. To calculae hs relably ndex he modern relably heory offers a reducon of he Markov chan and a calculaon mehod based on me-consumng soluon of a sysem of dfferenal equaons for he gven nal oerable sae of he echncal sysem. The oeraor mehod based on he Lalace ransform [7, 8] smlfes he soluon of dfferenal equaons however, he comuaonal comlexy s sll hgh. Whn he research work n he feld of relably models of dfferen echncal sysems on he bass of Markov chans, he auhor derved he formulas for calculang varous relably ndces, ncludng he mean me o frs falure, for dfferen daa ransmsson, rocessng and sorage sysems [9-]. As a resul, by summarzng he research resuls and negrang he exsng mehods for he calculaon of he relably ndces, he auhor develoed an effecve oologcal mehod for calculang he mean me o frs falure on bass of converson of Markov chan. Exsng oeraor mehod for calculaon of he mean me o frs falure In general case, he relably models Fgure- based on connuous-me Markov chans conan a fne se of saes E, ncludng a subse of oerable saes E + and a subse of falure saes E. The subse E +, n s urn, conans he subse of border oerable saes H +, whch has he drec ranson lnks o he falure saes. The subse E, n s urn, conans he subse of border falure saes H, whch has drec ranson lnks o he oerable saes. Each sae E Fgure- can have eher nbound lnks wh he gven ranson raes γ from he subse of saes { r R}, whch have he ranson o he -h sae, and oubound lnks wh he gven ranson raes γs o he subse of saes { s S}, whch have he ranson from he -h sae. For calculaon of he mean me o frs falure n he relably model, based on a connuous-me Markov chan, modern relably heory suggess reducng he base relably model and solvng he Kolmogorov-Chaman dfferenal equaons sysem for he reduced chan for he gven nal oerable sae of he echncal sysem. The nal oerable sae of he sysem should belong o he subse of oerable saes E +. The nal sae n Markov chan s ycally desgnaed as. The reducon of he base Markov chan ncludes he followng ses: removng all oubound lnks from all falure saes whn subse E o any oher saes removng all falure saes ha have no drec ransons o he oerable saes. r 89
VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com As a resul, only «dead-end» border falure saes whn subse H reman n he reduced Markov chan Fgure-, and hey have drec nbound ranson lnks from he oerable saes and no oubound ranson lnks. E + E H + H Fgure-. Base relably model of he sysem on bass of Markov chan. r γ r γ s s Fgure-. Inbound and oubound ransons lnks for -h sae. H E + H + Fgure-. Reduced Markov chan for he oeraor mehod. Furher, akng no consderaon ha he oerable saes n he reduced Markov chan have no nbound ransons from he falure saes and n he 8
VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com dfferenal equaons for he oerable saes he robably funcons of he falure saes are no used n equaons, only he equaons for he oerable saes whn subse E + are ncluded n he Kolmogorov-Chaman sysem of dfferenal equaons. As for he nal condons, he nal robably of he nal sysem sae s consdered, and for he all oher saes he nal robables are consdered : E \ {}: d γrr γs d rr ss d E \ {}: γrr γs. d rr ss Fnally, he mean me o frs falure of he sysem wh he gven nal sae can be calculaed as he mean me sen by he sysem n he oerable saes before he frs falure: T d. E I mus be menoned ha, solvng he sysem of dfferenal equaons, obanng unknown robably funcons and calculang he defne negral are que me-consumng asks. Therefore, n modern relably heory, he oeraor mehod based on he Lalace ransform s used for calculaon of he mean me o frs falure. In accordance wh he oeraor mehod, he dfferenal equaons sysem s convered by he Lalace ransform, akng no consderaon nal condon = : γrr γs rr ss E \ {}: γrr γ rr ss s. Furher, he mahemacal exressons for he ransformed robably funcons for all oerable saes are algebracally derved from he convered equaons sysem. Fnally, he mean me o frs falure s calculaed as a lm of he sum of all hese robably funcons a : T lm E. 4 EXAMLE OF CALCULATION OF THE MEAN TIME TO FIRST FAILURE B THE OERATOR METHOD FOR THE ASMMETRIC COMUTING SSTEM WITH TRILE MODULAR REDUNDANC Le us assume ha an asymmerc comung sysem wh hree funconal dencal comuer nodes and dfferen relably arameers for each node are gven. The comung nodes rovde he same calculaons, and here s a secal vong crcu, whch selecs fnal resul by usng he majory rncle. The vong crcu s consdered o be hghly relable, and he falures of he voer are no aken no consderaon. The comung sysem s consdered as oerable when a leas wo of he hree nodes are oerable and rovde correc calculaons. The comung nodes are consdered as smle rearable elemens wh exonenal dsrbuon of falure and rear mes. Moreover, he nodes are ndeenden from he vewon of falure and rear evens. The node falure raes are: / 876 hour, /876 hour and / 876 hour. The node rear raes are: / 4 hour, / 4 hour and / 4 hour. Accordngly, he base Markov chan fgure 4, whch reresens he dscussed above relably model of he asymmerc comung sysem wh rle modular redundancy, s as follows: 4 5 7 6 Fgure-4. Base Markov chan for he asymmerc comung sysem wh rle modular redundancy. Sae - all nodes are oerable. Sae - only node s faled, sae - only node s faled, sae - only node s faled. Sae 4 - nodes and are faled, sae 5 -nodes and are faled, sae 6 - nodes and are faled. Sae 7 - all hree nodes are faled. Saes,, and are consdered as oerable, because n hese saes a leas wo of hree nodes majory quany are oerable and rovde correc resuls. Sae s he nal sae of he sysem. For calculaon of he mean me o frs falure by exsng oeraor mehod for he dscussed above asymmerc comung sysem wh rle modular redundancy, we delee all oubound ranson lnks from 8
VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com 8 he falure saes 4-7 n he base Markov Chan Fgure-4. Nex, we also delee sae 7, because has no drec ranson lnks o any of oerable saes. As a resul, we oban he followng reduced Markov chan Fgure-5 for a comung sysem. The algorhm for calculaon of he mean me o frs falure ncludes he followng ses: Formng he Kolmogorov-Chaman dfferenal equaons sysem for he reduced Markov chan wh due consderaon of nal sae of he sysem. Alyng he Lalace ransform o he dfferenal equaons sysem and obanng he sysem of ransformed funconal algebrac equaons. Solvng he ransformed algebrac equaons sysem and obanng he algebrac exresson for he ransformed robably funcons for all oerable saes. Calculang he mean me o frs falure as a lm of sum of he ransformed robably funcons for all oerable saes a argumen. 4 5 6 Fgure-5. Reduced Markov chan for he asymmerc comung sysem wh rle modular redundancy for calculaon of he mean me o frs falure by he exsng oeraor mehod. For he above-dscussed reduced Markov chan of he comung sysem wh rle modular redundancy, he Kolmogorov-Chaman dfferenal equaons sysem, consderng he sae as he nal one, s as follows:. d d d d d d d d 5 In accordance wh he relably heory, he mean me o frs falure of he sysem wh he gven nal sae can be calculaed as he mean me sen by he sysem n he oerable saes - before he frs falure:. d T 6 By alyng he Lalace ransform o he dfferenal equaon sysems, we oban he followng sysem of he ransformed funconal algebrac equaons:. 7 Accordngly, he mean me o frs falure can be calculaed as he lm of he sum of ransformed robably funcons for he oerable saes - for argumen :. lm T 8 Nex, we oban algebrac exresson for he ransformed robably funcons for all oerable saes - by solvng he sysem of he ransformed funconal algebrac equaons 7:. 9 Fnally, we oban he formula for calculang he mean me o frs falure by subsung he obaned funcons no he equaon 8:. T By usng he gven values for he falure and rear raes for he comung nodes, we oban value T 6765 hours for MTT of he asymmerc comung sysem wh rle modular redundancy.
VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com OERED TOOLOGICAL METHOD FOR CALCULATION OF THE MEAN TIME TO FIRST FAILURE Now, should be noed ha f we offer some knd of addonal converson of he reduced Markov chan, whch convers he reduced chan o some anoher «rearable» chan, for whch he mean me o frs falure of he reduced chan wll be equal o he mean me o falure of he convered chan, hen we wll be able o aly oologcal aroach. The oologcal aroach s based on solvng he saonary Kolmogorov-Chaman sysem of lnear algebrac equaons, and calculaes he mean me o frs falure. The calculaed mean me o falure of he convered chan n our case wll be equal o he mean me o frs falure of he reduced chan. The auhor has obaned such addonal converson, when he mean me o frs falure of he reduced chan s equal o he mean me o falure of he convered chan. The converson s based on he unon of all border falure saes whn subse H o he sngle aggregae falure sae F, and usage of he addonal fcous ranson lnk from he aggregae falure sae F o he nal sae wh some ranson rae δ >. Fnally, by unng he descrbed above reducon of he base Markov chan and he offered by he auhor addonal converson of reduced Markov chan, we oban nex algorhm of oologcal converson of he base Markov chan, whch ncludes: removng all oubound lnks from all falure saes whn subse E o any oher saes removng all falure saes, whch have no drec ransons o he oerable saes unng all border falure saes whn subse H o one aggregae falure sae F. For each border oerable sae whn subse H + all oubound ranson lnks o he falure saes should be also uned and reroued o he aggregae falure sae F, and he arorae ranson raes should be summed addnga fcous ranson lnk from he aggregae falure sae F o he nal sae wh some ranson rae δ >. As a resul, he convered Markov chan Fgure- 6 consss of a subse of oerable saes E + of he base Markov chan and one aggregaed falure sae F. E * E {F}. The ranson lnks beween he oerable saes and her raes reman unchanged. However, for each border oerable sae whn subse H + all oubound ranson lnks o he falure saes are uned and reroued o he aggregae falure sae F, and he arorae ranson raes are summed: H : γ F γj. je Moreover, a fcous ranson lnk from aggregae falure rae F o he nal sae wh some rae δ > s added: γ F δ. Now, akng no consderaon ha he convered Markov chan s «rearable» due o he fcous ranson lnk from aggregae falure rae F o he nal sae, we can wre a saonary Kolmogorov-Chaman sysem of lnear algebrac equaons, whch allows us o oban he saonary robables for all oerable saes whn subse E + and aggregae falure sae F: E E {F}: γ rr r F r γ ss s. 4 Fnally, usng he obaned saonary robables for all oerable saes whn subse E +, we can easly calculae he mean o he falure of he convered chan by he oologcal formula, and hs value s also equal o mean me o frs falure of he base and reduced Markov chans: T E + γ F. 5 E H H + F δ Fgure-6. Convered Markov chan for he oologcal mehod. 8
VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com The mean me o frs falure s calculaed as a rao of sum of he saonary robables of all oerable saes o a weghed sum of he saonary robables of all border oerable saes. The wegh for each border oerable sae s calculaed as a sum of he raes of oubound ransons o he aggregae falure sae F. Equaly beween he mean me o frs falure and he mean me o falure n he convered Markov chan s guaraneed by he usage of fcous ranson lnk from aggregae falure rae F o he nal sae. Due o he fcous ranson lnk n relably model, he sysem afer he frs falure always asses o he nal sae, and mean me o nex falures wll be same as he mean me o frs falure. Thus, he secal converson of base Markov chan offered by he auhor allows for he alcaon of a faser oologcal mehod for calculaon of mean me o frs falure, based on solvng he saonary Kolmogorov-Chaman sysem of lnear algebrac equaons. EXAMLE OF CALCULATION OF THE MEAN TIME TO FIRST FAILURE B THE TOOLOGICAL METHOD FOR THE ASMMETRIC COMUTING SSTEM WITH TRILE MODULAR REDUNDANC For calculaon of he mean me o frs falure by oologcal mehod for he dscussed above asymmerc comung sysem wh rle modular redundancy, we remove all ranson lnks from falure saes 4-7 and delee he falure sae 7, whch has no drec ransons lnks o he oerable saes, n he base Markov chan Fgure-4. Nex, we une all falure saes o he aggregae falure sae F and for each oerable sae - we une all oubound ransons o he falure saes 4-6 and reroue hem o he aggregae falure sae F wh summarzaon of he arorae ranson raes. Fnally, we add a fcous ranson lnk from he aggregae falure sae F o he nal sae wh some rae δ >.As a resul, we oban he followng convered Markov chan Fgure-7 for he asymmerc comung sysem wh rle modular redundancy: + + + Fgure-7. Convered Markov chan for he asymmerc comung sysem wh rle modular redundancy for calculaon of he mean me o frs falure by he offered oologcal mehod. F δ Accordngly, he saonary Kolmogorov- Chaman equaons sysem for he convered Markov chan s as follows: F δf δf. 6 The mean me o frs falure, whch s equal o he mean me o falure, for he convered chan can be calculaed as a rao of sum of he saonary robables of oerable saes - o he weghed sum of he saonary robables of all border oerable saes -, where he wegh for each border oerable sae s calculaed as a sum of he raes of oubound ransons o he sae F: T. 7 Nex, by solvng he saonary equaons sysem 6, we oban saonary robables for all oerable saes - of he convered Markov Chan: δ δ δ δ δ. 8 Fnally, we oban formula for calculaon of he mean me o frs falure by subsung he obaned exresson no he formula 7: T. 9 I s easy o see, ha he equaon 9, obaned by he offered oologcal mehod, s smlar o equaon, obaned by he exsng oeraor mehod. By usng he above-gven values for he falure and rear raes and equaon 9, we oban value T 6765 hours, whch s equal o he calculaed above value. EXERIMENTAL RESEARCH OF MTT CALCULATION TIMES For comarson of he exsng oeraor mehod and he oologcal mehod offered by he auhor from he vewon of calculaon me, boh mehods were 84
VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. www.arnjournals.com rogrammed n Malesof Male 5, and an exermenal research of he calculaon me was carred ou on he desko comuer based on rocessor Inel enum IV. GHz. As an examle of he relably model for calculaon of he mean me o frs falure, a well-known generalzed Markov brh-deah chan Fgure-8 was used wh a subse of oerable saes s and one falure sae s wh nal oerable sae.for such Markov chan s que easy o rogram he equaons sysem and carry ou he research of calculaon mes by boh mehods for dfferen number of saes s +. s s s- s Fgure-8. Generalzed Markov Brh-Deah Chan. To conrol he correcness of he calculaed by he boh mehods MTT values he auhor also obaned and used nex analycal formula for esmaon of he MTT of he Markov Brh-Deah Chan: s s k q k j T. k k q j k j Whn he exermenal research, a seres of calculaons of mean me o frs falure for he relably model based on Markov brh-deah chan was carred ou usng he oeraor and oologcal mehods. For each s = 5,, 5,, 5, he seres of calculaons by boh mehods was carred ou for he dfferen ranson raes and wh he measuremen of calculaon mes. j Furher, he measured mes were averaged whn each seres for he oeraor and oologcal mehods. The resuls of he me measuremens for boh mehods and dfferen s are shown n Table-. Table-. Average mes of MTT calculaons by he oeraor and oologcal mehods. s Oeraor mehod sec s Toologcal mehod sec 5.95..597.6 5.9.94.59. 5.46.76.597.9 I s easy o see, ha he oologcal mehod offered by he auhor s faser han he exsng oeraor mehod. s CONCLUSIONS Ths arcle covers such ssues as he relably models of echncal sysems on he bass of Markov chans, he exsng oeraor mehod for calculang he mean me o frs falure, based on he reducon of Markov chan and soluon of he dfferenal equaon sysems. The oologcal mehod offered by he auhor for calculang he mean me o frs falure, based ona secal converson of Markov chan, s also hghlghed. The calculaon examles of MTT for he asymmerc comung sysem wh rle modular redundancy by he exsng oeraor mehod and he offered oologcal mehods are also resened. The exermenal research of calculaon me by he oeraor and he oologcal mehods are also dscussed. Accordng o he research resuls, he oologcal mehod offered by he auhor s faser han he exsng oeraor mehod. The oologcal mehod was aled for calculang MTT for varous echncal sysems. In all cases gave values smlar o he values obaned by he exsng oeraor mehod. ACKNOWLEDGEMENTS The auhor s graeful o rofessor I. I. Ladygn from Moscow ower Engneerng Insue for scenfc suor and heorecal base n he feld of relably models of echncal sysems. REFERENCES [] Shooman M.L.. Relably of comuer sysems and neworks. John Wley & Sons. [] Koren I. and Krshna C.M. 7. Faul-Toleran Sysems. Morgan Kaufmann ublshers. [] Anderson W.J. 99. Connuous-Tme Markov Chans. Srnger-Verlag. [4] Wang Z., Wang T. and ang X. 99. Brh and deah rocesses and Markov chans. Srnger-Verlag. [5] Cherkesov G.N. 5. Relably of Hardware and Sofware Sysems. er, San-eersburg. [6] olovko A.M. and Gurov S.V. 6. Bass of Relably Theory. BHV-eersburg, San- eersburg. [7] Graf Urs. 4. Aled Lalace Transforms and z- Transforms for Scenss and Engneers. Srnger Basel AG. [8] Dyke hl. 4. An Inroducon o Lalace Transforms and Fourer seres. Srnger-Verlag. 85
VOL., NO. 5, MARCH 8 ISSN 89-668 ARN Journal of Engneerng and Aled Scences 6-8 Asan Research ublshng Nework ARN. All rghs reserved. [9] Rahman.A. and Bobkova E. u. 7. The relably model of he faul-oleran border roung wh wo Inerne servces rovders n he enerrse comuer nework. J. hys.: Conf. Ser. 8: 4. [] Rahman.A. and Bobkova E.u. 7. The relably model of he faul-oleran comung sysem wh rle-modular redundancy based on he ndeenden nodes. J. hys.: Conf. Ser. 8: 5. [] Rahman.A. 7.Usng a secalzed Markov chan n he relably model of dsk arrays RAID- wh daa mrrorng and srng. IO Conf. Ser.: Maer. Sc. Eng. 77: 87. [] Rahman.A. 7. Analyss of he mean me o daa loss of nesed dsk arrays RAID- on bass of a secalzed mahemacal model. IO Conf. Ser.: Maer. Sc. Eng. 77: 88. www.arnjournals.com 86