Availability and Maintainability. Piero Baraldi

Availability and Maintainability 1

Introduction: reliability and availability System tyes Non maintained systems: they cannot be reaired after a failure (a telecommunication satellite, a F1 engine, a vessel of a nuclear ower lan. Performance arameters Reliability (R( M )) It quantifies the system caability of satisfying a secified mission within an assigned eriod of time ( M ): R M = P > M Maintained systems: they can be reaired after the failure (um of an energy roduction lant, a comonent of the reactor emergency cooling systems) Availability (A() It quantifies the system ability to fulfill the assigned mission at any secific moment of the lifetime 2

Definition of Availability X( = indicator variable such that: X( = 1, system is oerating at time t X( =, system is failed at time t X( 1 F F F R R R t F = Failed R = under Reair Instantaneous availability ( and unavailability q( ( P X ( 1 E X ( q( P X ( 1 ( 3

Contributions to Unavailability Reair A comonent is unavailable because under reair 4

Contributions to Unavailability Reair A comonent is unavailable because under reair esting / reventive maintenance A comonent is removed from the system because: a. it must undergo reventive maintenance b. it has to be tested (safety system, standby comonents) Unrevealed failure A stand-by comonent fails unnoticed. he system goes on without noticing the comonent failure until a test on the comonent is made or the comonent is demanded to function 6

Average availability descritors How to comare different maintenance strategies? 7

Average availability descritors How to comare different maintenance strategies? We need to define quantities for an average descrition of the system robabilistic behavior: If after some initial transient effects, the instantaneous availability assumes a time indeendent value Limiting or steady state availability: lim t ( 8

Average Availability ( 1 1 ( dt ( dt ( dt ( 1 ( dt 2 t ( 1 t ( dt UPIME DOWNIME ( dt t 1

Maintenability 11 How fast a system can be reaired after failure? Reair time R deends from many factors such: time necessary for the diagnosis time required to extract the comonent from the system time necessary for installing the new comonent or reair the comonent Reair time R varies statistically from one failure to another, deending on the conditions associated to the articular maintenance events Let R denotes the downtime random variable, distributed according to the df g R ( Maintenability: P t t g R R d 11

Failure classification 12 Failure can be: revealed unrevealed (stanby comonents, safety systems) 12

REVEALED FAILURE 13

Availability of a continuously monitored comonent (1) Objective: Comutation of the availability ( of a comonent continuosly monitored Hyotheses: Restoration starts immediately after the comonent failure Probability density function of the random time duration R of the reair rocess = g R ( Constant failure rate λ CONCEPUAL EXPERIMEN N = number of identical comonents at time t = (assumtion) Balance equation between time t and time t+t

Availability of a continuously monitored comonent (3) working at t+δt = working at t - Failure between t and t+ Δt N ( t N ( N ( t N ( ) g( t ) t + t Reair between t and t+ Δt (1) (2) (3) (4) 1) Number of items UP at time t+t 2) Number of items UP at time t 3) Number of items failing during the interval t 4) Number of items that had failed in (, +) and whose restoration terminates in (t, t+;

Availability of a continuously monitored comonent (3) he integral-differential form of the balance d( dt t ( ( ) g( t ) d ( ) 1 he solution can be obtained introducing the Lalace transforms ime domain t d t dt t t g t = න τ g t τ dτ L t L Lalace ransform d t dt = න L t g t + e st t dt = (s) = s s = (s) g (s)

Availability of a continuously monitored comonent (4) Alying the Lalace transform we obtain: s which yields: ~ 1 ( s) s 1 g~ ( s) Inverse Lalace transform ( Limiting availability: ~ ( s) 1 ~ ( s) ~ ( s) g~ ( s ) lim t ( lim s ~ ( s) lim s s s 1 s s g~ ( )

Availability of a continuously monitored comonent (5) As s, the first order aroximation of g(s) is: g~ ( s) MR= e s g( ) d E R g R (1 s...) g( ) d 1 s g( ) d 1 s MR R s lim s s s R 1 1 R 1 1 R MF MF MR average time the comonent isup average time of a failure/reair "cycle" General result!

Exercise 1 Find instantaneous and limiting availability for an exonential comonent whose restoration robability density is g( e t

Exercise 1: Solution Find instantaneous and limiting availability for an exonential comonent whose restoration robability density is he Lalace transform of the restoration density is t e t g ) ( s t g L s g ) ( ) ~ ( ) ( 1 ) ~ ( s s s s s s s t = L 1 (s) = L 1 s + μ s(s + λ + μ)

Exercise 1: Solution Find instantaneous and limiting availability for an exonential comonent whose restoration robability density is g( e t L L 1 = 1 s 1 s + a = e at L 1 s + μ s(s + λ + μ) s = A s + B A s + μ + λ + Bs = (s + μ + λ) s(s + λ + μ) s(a + B) + Aμ + Aλ s(s + λ + μ) = s + μ s(s + λ + μ) = 1 = A + B ቊ μ = Aλ + Aμ A = μ μ + λ B = μ μ + λ t = A + Be μ+λ t = μ μ + λ + μ μ + λ e μ+λ t

Exercise 1: Solution ( 1.99.98.97.96 constant failure rate =1/5 [day -1 ] constant reair rate = 1/5 [day -1 ].95.94.93.92.91.9 2 4 6 8 1 12 14 16 18 2 t [day]

UNREVEALED FAILURE

Safety Systems: Examles Risks: Ignition of the gases in the oil tank, lightning strike or generation of sarks due to electrostatic charges. Fire extinguisher Fire rotection

Exercise 2: Instantaneous Availability of an Unattended Comonent Find the instantaneous unavailability of an unattended comonent (no reair is allowed) whose cumulative failure time distribution is F(

Exercise 2: Solution Find the instantaneous availability of an unattended comonent (no reair is allowed) whose cumulative failure time distribution is F( SOLUION he istantaneous unavailability, i.e. the robability q( that at time t the comonent is not functioning is equal to the robability that it fails before t q( F(

Availability of a safety system under eriodic maintenance (1) Safety systems are generally in standby until accident and their comonents must be eriodically tested (interval between two consecutive test = ) he instantaneous unavailability is a eriodic function of time ( 1 R( 2 3 t 27

Availability of a safety system under eriodic maintenance (1) Safety systems are generally in standby until accident and their comonents must be eriodically tested he instantaneous unavailability is a eriodic function of time he erformance indicator used is the average unavailability over a eriod of time q q( dt DOWNtime Average time the system is not working comlete maintenance cycle 28

EX. 3: Availability of a comonent under eriodic maintenance Suose the unavailability is due to unrevealed random failures, e.g. with constant rate Assume also instantaneous and erfect testing and maintenance rocedures erformed every hours Draw the qualitative time evolution of the instantaneous availability Find the average unavailability of the comonent 29

Ex 3. Solution (1) Suose the unavailability is due to unrevealed random failures, e.g. with constant rate Assume also instantaneous and erfect testing and maintenance rocedures he instantaneous availability within a eriod coincides with the reliability ( 1 R( t 2 3 3

Ex. 3: Solution (2) he average unavailability and availability are then: q q( dt F ( dt ( dt R( dt 1 q For different systems, we can comute and q by first comuting their failure robability distribution F ( and reliability R( and then alying the above exressions 31

32 32 he average unavailability and availability are then: Ex. 3: Solution (3) q 2 1 1 1 t dt t dt e dt t F dt t q q 2 1 2 ) (1 ) ( ) ( 2

Ex.4: A more realistic case 33 he test is erformed after a time τ of oeration Assume a finite test time R : Draw the qualitative time evolution of the instantaneous availability Estimate the average unavailability and availability over the comlete maintenance cycle eriod = + R 33

Ex.4: Solution (1) 34 Assume a finite test time R : Draw the qualitative time evolution of the instantaneous availability ( 1 R R R t 34

Ex. 4: Solution (2) Assume a finite test time R he average unavailability and availability over the comlete maintenance cycle eriod = + R are: q F ( dt F ( dt R R R R R( dt R( dt R R 35

Comonent under eriodic maintenance: a more realistic case Objective: comutation of the average unavailability over the lifetime [, M ] Hyoteses: he comonent is initially working: q() = ; () = 1 Failure causes: q M [, ] DOWNtime random failure at any time F ( on-line switching failure on demand Q maintenance disables the comonent (human error during insection, testing or reair) M 36

Comonent under eriodic maintenance: a more realistic case maintenance maintenance A B C F R R R t (1) (2) (3) (4) 1) he robability of finding the comonent DOWN at the generic time t is due either to the fact that it was demanded to start but failed or to the fact that it failed unrevealed randomly before t. he average DOWNtime is: DOWNtime[ A 1 q t Q Q F t A, ] q ( dt Q (1 Q ) F ( dt Q (1 Q ) F ( dt D ( A ) A 37

Comonent under eriodic maintenance: a more realistic case 2) During the maintenance eriod the comonent remains disconnected and the average DOWNtime is the whole maintenance time: DOWNtime D( AB [ A, B) ] 3) he comonent can be found failed because, by error, it remained disabled from the revious maintenance or because it failed on demand or randomly before t. he average DOWNtime is: R q ( (1 ) Q (1 Q ) F ( BC DOWNtime [ B, C] D( BC ) qbc ( dt (1 ) Q (1 Q ) F ( dt 38

Comonent under eriodic maintenance: a more realistic case 4) he normal maintenance cycle is reeated throughout the comonent lifetime M. he number of reetitions, i.e. the number of AB-BC maintenance cycles, is: k = M τ τ + τ r he total exected DOWNtime is: M DOWNtime D ( D M ) Q (1 Q ) F ( dt R (1 ) Q (1 Q) F ( dt R DOWNtime D( M ) Q 1 Q 1 q M q F ( ) (1 ) (1 ) ( ) M M t dt R Q Q F t dt M M M M R 39

Comonent under eriodic maintenance: a more realistic case DOWNtime q F ( dt (1 ) Q (1 Q ) F ( dt D( M ) Q 1 Q 1 q M M R M M M M R q M τ M τ F t dt τ τ R τ DOWNtime D( M ) Q 1 Q 1 q F ( ) (1 ) (1 ) ( ) M t dt R Q Q F t dt M M M M R q M 4

Comonent under eriodic maintenance: a more realistic case Q and F ( are generally small, and since tyically R and M, the average unavailability can be simlified to: q M q Q F t dt R 1 Q q (1 ) ( ) M M M DOWNtime Consider an exonential comonent with small, constant failure rate t F ( 1 e t Since tyically 1, Q 1, the average unavailability reads: Maintenance q M DOWNtime q M 1 Q 2 R M Error after test Switching failure on demand Random, unrevealed failures between tests 41

Where to study? 42 Slides Lecture 5: Reliability of simle systems Chater 5 (Red Book): heory Chater 5 (Green Book): Exercises Lecture 6: Availability and Maintenability Chater 6 (Red Book): heory Chater 6 (Green Book): Exercises 42