Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co Abstract The perforance of a relablty syste can be proved by dfferent ethods e.g. the relablty of one or ore coponents can be proved hot or cold redundant coponents can be added to the syste. Soetes these easures can be equvalent as they wll have the sae effect on soe perforance easure of the syste. Ths paper dscusses the relablty equvalences of a seres-parallel syste. The syste consdered here conssts of subsystes connected n parallel wth subsyste consstng of n ndependent and dentcal coponents n seres for. The falure rates of the syste coponents are functons of te wth a lfe dstrbuton of Webull dstrbuton. Three dfferent ethods are used to prove the gven syste relablty. The relablty equvalence factor s obtaned usng the relablty functon. The fractles of the orgnal and proved systes are also obtaned. Nuercal exaple s presented to nterpret how to utlze the obtaned results. Keywords: Relablty MTTF equvalence factors seres- parallel syste Webull dstrbuton. Introducton In relablty theory one way to prove the perforance of a syste s to use the redundancy ethod. There are two an such ethods:. ot duplcaton ethod: n ths case t s assued that soe of the syste coponents are duplcated n parallel.. old duplcaton ethod: n ths case t s assued that soe of the syste coponents are duplcated n parallel va a perfect swtch. Unfortunately for any dfferent reasons such as space ltaton hgh cost etc t s not always possble to prove a syste by duplcatng soe or all of ts
94 M. A. El-Dacese coponents. For exaple satelltes and space arcrafts have lted space whch doesn't allow coponent duplcaton. Also soe crochps are so expansve that anufacturers cannot afford to duplcate the. In such cases where duplcaton s not possble the engneer turns to another well-known ethod n relablty theory the so-called reducton ethod. In ths ethod t s assued that the falure rates of soe of the syste coponents are reduced by a factor ρ < ρ <. Now once the reducton ethod s adopted the an proble facng the engneer s to decde to what degree the falure rate should be decreased n order to prove the syste. To solve ths proble one can ake equvalence between the reducton ethod and the duplcaton ethod based on soe relablty easures. In other words the desgn of the syste proved by the reducton ethod should be equvalent to the desgn of the syste proved by one of the duplcaton ethods. The coparson of the desgns produces the so-called relablty equvalence factors []. The concept of the relablty equvalence factors was ntroduced n the report [] and appled to varous relablty systes n [] and [4]. Rade [5 6] appled ths concept for the two-coponent parallel and seres systes wth ndependent and dentcal coponents whose lfetes follow the exponental dstrbuton. Sarhan [7- ] derved the relablty equvalence factors of other ore general systes. The systes studed by Sarhan are the seres syste [7] a basc seres-parallel syste [8] a brdge network syste [9] the parallel syste [] a parallel-seres syste [] and a general seres-parallel syste []. All these systes have ndependent and dentcal exponental coponents. In ths paper we hope to dscuss ore lfe dstrbutons Webull dstrbuton for exaple. Dfferent fro the constant falure rate of exponental dstrbuton Webull dstrbuton has a te varyng falure rate. So the relablty equvalence factors should be generalzed accordngly. In the current study we consder a general seres-parallel syste and assue that all coponents are ndependent and dentcally Webull dstrbuted. Frst we coputed the relablty functon and the ean te to falure MTTF of the syste. Second we coputed the sae relablty easures when the syste s proved usng the reducton ethod. Thrd we coputed the sae easures when the syste s proved usng the hot and cold duplcaton ethods. Fnally we equate the relablty functon of the syste proved by duplcaton wth the relablty functon of the syste proved by reducton to get the survval relablty equvalence factors. These factors can be used by the engneer to decde to what degree the falure rate of soe of the syste coponents should be decreased n order to prove the perforance of the syste wthout duplcatng any coponent.. Seres parallel syste The syste consdered here conssts of subsystes connected n parallel wth subsyste consstng of n coponents n seres for. Fgure shows the dagra of a seres parallel syste. lock n lock n lock n Fgure a Seres-parallel syste.
Relablty equvalence factors 94 Let R t be the relablty of subsyste and r j t be the relablty of coponent j j n n subsyste =. Then n = R t r t j= The syste relablty says Rt s gven by R t = j R t Usng the syste relablty takes the followng for: n R t = r t j= j Assung that the syste coponents are ndependent and dentcal. The lfete of each coponent s Webull dstrbuted wth falure rate z t = λ t ; λ >. That s r j t = exp λt for j= n and. Thus the syste relablty becoes R t = exp n λ t Usng equaton the syste ean te to falure MTTF can be derved n the followng for: MTTF = R t dt 4. The proved systes The relablty of the syste can be proved accordng to one of the followng two dfferent ethods: - Reducton ethod. - Standby redundancy ethod: a ot standby redundancy called hot duplcaton ethod b old standby redundancy called cold duplcaton ethod. In the followng sectons we wll derve the relablty functons and the ean te to falures of the systes proved accordng to the ethods entoned above... The reducton ethod It s assued n ths ethod that the syste can be proved by reducng the falure rates of the set A of syste coponents by a factor s < s <. ere we consder that reducng the falure rate by reducng only the scale paraeter λ of the set A of syste coponents by a factor ρ. Assung that the set A conssts of k coponents; k n where n denotes the total nuber of the syste coponents. Assung that the coponents belongng to A can be dstrbuted nto the subsystes of the syste such that k coponents of the subsyste belong to the set A where k n. We denote such a A A K A k k K k set by ether A A or A k. Let R Aρ t denote the relablty functon of the syste proved by reducng the scale paraeter λ of the set A of ts coponents by a factor ρ. Let MTTF Aρ t be the ean te to falure of the syste that has the relablty functon R Aρ t. The relablty functon of coponent j n the subsyste j= n ; after reducng ts scale paraeter λ by a factor ρ s t = exp ρ λt. 5 r j ρ
944 M. A. El-Dacese Thus R Aρ t can be derved as follows: That s R k n k A ρ t = rj ρ t rj t j= j= A ρ R t = exp [ n + k ρ ] λt 6 Usng equaton 6 MTTF Aρ t can be derved n the followng for: A t = RA ρ t MTTF ρ dt 7.. The hot duplcaton ethod Assung that n the hot duplcaton ethod each coponent of the set s proved by assung a hot duplcaton of another dentcal one. Suppose that the set conssts of h coponents h n. Thus the set can be wrtten as a unon of dsjont subsets such that the subset contans those coponents belongng to the subsyste ;. That s h n and h = K We denote such a set by ether or. Let r j t denote the relablty functon of the coponent j n the subsyste j= n and. when t s proved accordng to hot duplcaton ethod. Thus t = exp λt exp λt. 8 r j h h h Let R t and MTTF t denotes respectvely the relablty functon and ean te to falure of the desgn obtaned by provng the coponents belongng to the set accordng to the hot duplcaton ethod. Thus the functon R t can be derved as follows: R t = = h n h rj t rj j= j= t K h h [ exp λ t ] exp n λt 9 Usng 9 MTTF t can be derved n the followng for:.. The cold duplcaton ethod MTTF t = R t dt In the cold duplcaton ethod t s assued that each coponent of the set s connected wth an dentcal coponent va a perfect swtch. Assue that conssts of c coponents c n. Thus the set can be wrtten as a unon of dsjont subsets such that the subset contans those coponents belongng to the subsyste ;. That s = c = = c c n. We denote such a set by ether K c c c or K c Let r t j denote the relablty functon of the coponent j n the subsyste ; j= n and when t s proved accordng to cold duplcaton ethod. Thus.
Relablty equvalence factors 945 r j t = + λt exp λt Let R t and MTTF t denotes respectvely the relablty functon and ean te to falure of the desgn obtaned by provng the coponents belongng to the set accordng to the cold duplcaton ethod. Thus the functon R t can be derved as follows: R t = = c n c rj t j= j= r t j c + λ t exp n λt Usng MTTF t can be derved n the followng for: MTTF t = R t dt 4. Relablty equvalence factors Now we gve the defnton of relablty equvalence factor: A relablty equvalence factor s a factor by whch a characterstc of coponents of a syste desgn has to be ultpled n order to reach equalty of a characterstc of ths desgn and a dfferent desgn regarded as a standard [8]. As enton above the relablty equvalence factor s defned as the factor by whch the falure rates of soe of the syste s coponents should be reduced n order to reach equalty of the relablty of another better syste. Dfferent fro the constant falure rate of exponental dstrbuton the falure rate of Webull dstrbuton s te varyng accordngly the ethod used to obtan the relablty equvalence factors n the case of usng Webull dstrbuton s dfferent than the ethod used n the exponental case. For convenence of calculaton whle te varyng falure rate s reduced by factor s we consder that the scale paraeter of Webull dstrbuton reduced fro λ to ρλ only. Fro the falure rate of Webull dstrbuton z t = λ t we know s z t = ρλ t 4 Obvously s wll ncrease as ρ ncreases and they fall n nterval also. In what follows we wll present how to calculate ρ only and we obtan s by takng ρ n equaton 4. Next we present soe of relablty equvalence factors of the proved seres-parallel syste studed here. 4.. ot relablty equvalence factor The hot relablty equvalence factor say s A α s defned as that factor by whch the falure rate of the set A coponents should be reduced so that one could obtan a desgn of the syste coponents wth a relablty functon that equals the relablty functon of a desgn obtaned fro the orgnal syste by assung hot duplcatons of the set coponents. As entoned before the falure rate reduced by s s equal to the scale A α
946 M. A. El-Dacese paraeter reduced fro λ to ρ A α λ. That s ρ A α s the soluton of the followng syste of two equatons R A ρ t = α R t = α 5 Therefore fro equatons 6 9 and 5 ρ A α s the soluton of the followng non-lnear syste of equatons wth respect to x = exp λ t and ρ for a gven α n + k ρ α = x 6 h n α = x x 7 As t sees the above syste of non-lnear equatons has no closed-for soluton. So a nuercal technque ethod s requred to get the soluton of such a syste. So we have ρ = ρa α. ence the hot relablty equvalence factor s A α s obtaned fro equaton 4. 4.. old relablty equvalence factor The cold relablty equvalence factor say s A α s defned as that factor by whch the falure rate of the set A coponents should be reduced so that one could obtan a desgn of the syste coponents wth a relablty functon that equals the relablty functon of a desgn obtaned fro the orgnal syste by assung cold duplcatons of the set coponents. As entoned before the falure rate reduced by sa α s equal to the scale paraeter reduced fro λ to ρ A α λ. That s ρ A α s the soluton of the followng syste of two equatons R A ρ t = α R t = α 8 Therefore fro equatons 6 and 8 ρ A α s the soluton of the followng non-lnear syste of equatons wth respect to x = exp λ t and ρ for a gven α n + k ρ α = x 9 c n α = + ln/ x x As t sees the above syste of non-lnear equatons has no closed-for soluton. So a nuercal technque ethod s requred to get the soluton of such a syste. So we have ρ = ρa α. ence the cold relablty equvalence factor s A α s obtaned fro equaton 4. 5. α-fractles In ths secton we deduce the α-fractles of the orgnal desgn and the proved desgns whch are a popular easure of relablty n echancal ndustry. D Let L α be the α-fractle of the orgnal syste. Let L α be the α- fractle of the desgn obtaned by provng the set coponents accordng to hot D = or cold D = duplcaton ethod. The α-fractle of a syste havng relablty functon Rt say L α s defned
Relablty equvalence factors 947 as the soluton of the followng equaton wth respect to L: R L α/ λ = α Usng equatons and one can obtan L of the orgnal syste by solvng the followng equaton wth respect to L: α = exp n L Also the α-fractle of the proved syste that has the relablty functon R D t D say L α can be obtaned by solvng the followng equaton wth respect to L: R D L α/ λ = α D = Thus accordng to 9 and one can fnd L α by solvng the followng equaton wth respect to L: h α = exp L exp n L 4 Fnally usng and one can copute L α by solvng the followng equaton wth respect to L: c = + L α exp n L 5 Equatons 4 and 5 have no closed-for solutons n L so a nuercal technque ethod s needed to get the values of α-fractles. 6. A nuercal results Soe nuercal results are gven n ths secton to llustrate how to nterpret the theoretcal results prevously obtaned. In the followng exaple we assue a seres-parallel syste wth n =5 = n = n = and =. The coponents are ndependent and dentcal. For such syste ρ D A A α ; D= for dfferent sets A A = A A = when α =. are coputed. Tables and gve ρ A α and ρ A α respectvely. Notce that negatve value of ρ D A α eans that t s not possble to reduce the falure rate of the set A coponents n order to prove the desgn of syste to be equvalent wth that desgn of the syste whch can be obtaned by provng the set coponents accordng to the redundancy ethods. Table gves the α-fractles of the orgnal syste for α =.. Table 4 gves the α-fractles of the systes proved accordng to hot and cold duplcaton ethods for α =..
948 M. A. El-Dacese Table α ρ A α 4 4 5 A..86.75 7 89 A..85.7 56 56.876.779.897.75.88.68.684.79.48.84.85.67.74.654.6 A..874.85.744 64 5.865 5.796.6.854.744 77.7.459.8.65.85.6.7.6.776.47.694.665 A..886.87 78 4 5.877.787.854.77.87.775.6 8.85.69.89.649.867.76.67.89.77 99.86.744 7.85.7 8 A..788.758.699 5.88.8.844.76 7.76.679 8.66.4.74 67.758 44.66..67.7 88 6 A. 6.866 8 5 55.894.8 7.87.88.875.794.67.84.7 5.88.689.87.78.645.895.789.67.866.76.69.857.77 6 A..8 6.765.744.68.44.758.67.746.495.65.7.669.7.67 8 A..87.854.85 5 4.896.84.77.856.85.756.788.7.69.87.769.67.854.76.66.784.77 8.8.76 7.777.687 46.767.66.498 A 4..89 7.85.84.88.7.856.777.85.77.86.697.84.689.788.66.764.64 A 4..875 9.8.89.78.685.87.747.89.7.768.658.777.649.747.6.78 68 A 5. 47 84 6 9.89 7 7.889.885.877
Relablty equvalence factors 949 Table α ρ A α 4 4 5 A. A..84 48.47.64 A..8.694.64.476.75.647 A..4..86.4 7.49.85.757.86.85.747 7.6 54.89.76.654.848.79 7.87.44.695 7.78.665.769.67.45 A..76.67.47 66.4.6 44 A..4.45.55. 8.854.794.97.84.784. 88.4.78.69.85.746.68.97.4 65.4.79 59.78.677.77.646.474 A..79.7.758.74.655 9 65.48 9.6.49 A..46.78..8.5.85.84.79.79.54.769.79.66. 5.58.79.6.76.67 44.55..58.6.498.689 99.469.678 7.48 A 4..6 4.54 5.48..874.89 8.856.8.446.86.64.87.74.77.66 9.469.69.76.64.789.7 8.777.67 A 4. 9 7.7 9.447.78.848.86.495.87.797.4.49 98.499.769.77.79.69.496.4 74.499.687 68.7.649 6.76.6.474 A 5..87.775.86.747.794.78 Table The α-fractles α L α. α L.8.798 8
95 M. A. El-Dacese D Table 4 The α-fractles α L α 4 4 5 L L...4.859 9.87.88 99.99.87 68..86 7.5.88.69..6. 45.696..7.758..866.6.6.895.66.6 57.74.5.48.778.67.659.7 55.68.54 5.685.4 9.74. 74.74.4.69.8.88 49.7.79.9.78.4 99.78.6..874 Fgure the relablty functons of the orgnal and soe odfed systes. Fgure shows the relablty functons of the orgnal syste and of the systes odfed by provng the sets and 5 of coponents accordng to hot and cold duplcaton ethods. Fro fgure we fnd that: a. ot duplcaton of the set coponents gves an proved desgn wth the lowest relablty functon aong all of those proved desgns whch can be obtaned by provng any other set of coponents accordng to ether the hot or cold duplcaton ethods. b. old duplcaton of the set 5 coponents gves an proved desgn wth a hghest relablty functon aong all of those proved desgns whch can be obtaned by provng any other set of coponents accordng to ether the hot or cold duplcaton ethods. c. Iprovng coponents of the set accordng to the hot cold duplcaton ethod ncreases the te at whch the syste relablty s.4 fro. te easure to.9.. d. Iprovng coponents of the set accordng to the hot cold duplcaton 5 ethod ncreases the te at whch the syste relablty s.4 fro. te easure to.87.4.
Relablty equvalence factors 95 7. oncluson In ths paper we dscussed the relablty equvalence of a seres-parallel syste wth dentcal and ndependent coponents. It s assued that the coponents of the syste had a te varyng falure rates. Three ways naely the reducton hold duplcaton and cold duplcaton ethods are used to prove the syste relablty. A relablty equvalence factor was derved. A nuercal exaple s used to llustrate how the results obtaned can be appled. In the future we hope that we wll be able to study the relablty equvalence of ore coplcated systes wth ndependent and dentcal or non dentcal coponents. Also we hope that we can deterne the optal nuber of coponents to duplcate n the duplcaton ethods and the optal nuber of coponents whose falure rate s to be reduced n the reducton ethod. References [] Sarhan A.M. Tadj L. Al-khedhar A. and A. Mustafa Equvalence Factors of a Parallel-Seres Syste Appled Scences 8 9-. [] Rade L. Relablty Equvalence Studes n Statstcal Qualty ontrol and Relablty 989- Matheatcal Statstcs halers Unversty of Technology. [] Rade L. Relablty Systes of -state oponents Studes n Statstcal Qualty ontrol and Relablty 99- Matheatcal Statstcs halers Unversty of Technology. [4] Rade L. Perforance Measures for Relablty Systes wth a old Standby wth a Rando Swtch Studes n Statstcal Qualty ontrol and Relablty 99 halers Unversty of Technology. [5] Rade L. Relablty Equvalence Mcroelectroncs & Relablty No. 99-5. [6] Rade L. Relablty Survval Equvalence" Mcroelectroncs & Relablty No. 6 99 88-894. [7] Sarhan A.M. Relablty Equvalence of Independent and Non-dentcal oponents Seres Systes Relablty Engneerng & Syste Safety 67 No. 9-. [8] Sarhan A.M. Relablty Equvalence wth a asc Seres-Parallel Syste Appled Matheatcs & oputaton No. 5-. [9] Sarhan A.M. Relablty Equvalence Factors of a rdge Network Syste Internatonal Journal of Relablty & Applcatons No. 4 8-. [] Sarhan A.M. Relablty Equvalence Factors of a Parallel Syste Relablty Engneerng & Syste Safety 87 No. 5 45-4. [] Sarhan A.M. Relablty equvalence factors of a general seres-parallel syste Relablty Engneerng & Syste Safety 94 No. 9 9-6. Receved: October 8