START Selected Topics in Assurance

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1 START Selected Toics in Assurance Related Technologies Table of Contents Introduction Statistical Models for Simle Systems (U/Down) and Interretation Markov Models for Simle Systems (U/Down) and Interretation A Markov Model for a Simle Redundant System Model Extensions and Comarisons For Further Study About the Author Other START Sheets Available Introduction There is a fundamental difference between the aroaches used to erform statistical and reliability analyses of nonmaintained (some times called one-shot ) and those used for maintained systems. Non-maintained systems either fulfill their missions (by surviving beyond mission time) or fail it (by erishing before the mission is comleted). In contrast, maintained systems can be reaired (maintained) and ut back into oeration. During maintenance, however, the system is down and unavailable for its intended use. This situation changes the analysis aroach because maintenance introduces the related and new concet of availability (that the system will be u and available for use, when needed, in lieu of undergoing maintenance). The main objective of this START sheet is to hel engineers better understand the meaning and imlications of the statistical methods used to develo erformance measures (PM) for assessing system availability (A). We start by reviewing some relevant definitions. RAC s Reliability Toolkit defines availability as a measure of the degree to which an item is in an oerable state at any time. It defines maintainability as a measure of the ability of an item to be retained in, or restored to, a secified condition, when maintenance is erformed using rescribed rocedures and technician skill levels. From these definitions, we can deduce that system availability is a robabilistic concet. It is based on the system life (X), a random variable (RV). Since the system can fail at any random time, availability is also based on a second RV: the Availability maintenance time (Y). We calculate their long run averages (i.e., results obtained as time goes to infinity, t ) or exected values and denote them resectively E(X) and E(Y). E(X) is the exected system life or the u time, often measured as Mean Time Between Failures (MTBF). E(Y) is the exected maintenance time (which includes the activities of fault isolation, reair or removal and function check), often measured as Mean Time To Reair (MTTR). The random nature of system life (X) and maintenance times (Y) demands the use of statistics for obtaining system PM. Therefore, we need to redefine Availability in statistical terms. Hoyland et al (Reference 1), for examle, define availability at time t, A(t), as the robability that the system is functioning at time t. If we call X(t) the state of a system at time t (which can be either u and running [X(t) 1], or down and failed [X(t) 0]), then this definition of A(t) can be written as: A(t) P{X(T) 1}; t > 0 Volume, Number 6 The availability concet becomes even more comlex when we realize that it is divided into several classes. For examle, Blanchard (Reference ) states that availability may be exressed differently, deending on the system and its mission and defines three tyes: 1. Inherent availability (A i ) is the robability that a system, when used under stated conditions will oerate satisfactorily at any oint in time, as required. A i excludes reventive maintenance, logistics, and administrative delays, etc. MTBF A i MTBF + MTTR. Achieved availability (A a ) is the robability that a system when used under stated conditions will oerate satisfactorily at any oint in time. A a includes reventive maintenance but excludes logistics and administrative delays, etc. 3. Oerational availability (A o ) is the robability that a system when used under stated conditions will oerate satisfactorily when called uon. A o includes all the fac- START 004-6, AVAILSTAT A ublication of the DoD Reliability Analysis Center

2 tors that contribute to system downtime for all reasons (i.e., maintenance actions and delays, access, diagnostics, active reair, suly delays, etc.). We call it Mean Down Time (MDT). MTBM A o MTBM + MDT Notice that, as oosed to A(t), these formulae do not include any reference to a random time t. The reason is that they are based on the long run averages of the system life (X) and maintenance times (Y) and, hence, about the long run average (or exected) Availability. To better understand this statistical concet, consider the whole cycle of u-time lus maintenance (i.e., X + Y). The cycle reeats itself again and again, throughout the entire system life, and constitutes the totality of ossibilities for the RV system state X(t) within a single cycle. Now consider system u-time (X(t) 1) as our event of interest. Since the robability of any event is defined as the ratio of its favorable to its total ossibilities, we can state that the long run availability A is: Favorable Cases A P(System U) Total Cases P(X(t) 1); as t U Time Cycle Time Since we are not interested in the u-time during a secific cycle but in the long run (t ), we substitute U (X) or Down (Y) times by their resective long run averages, E(X) and E(Y). For examle, if a system has MTBF 500 hours and MTTR 30 hours we obtain: E(X) A A( ) P{X( ) 1} E(X) + E(Y) MTBF MTBF + MTTR Finally, we must select which Availability definition we want to discuss. The three classes of availability differ only in their resective definition scoe. For examle, in A i, Inherent Availability, the average u-time (MTBF) includes only design and manufacturing failures, and the average maintenance (MTTR) only includes active reair time. In Oerational availability, average u-time (MTBM) includes all downing events, whatever the cause (e.g., design or manufacturing failure, induced failure, reventive maintenance event, etc.), and average maintenance (MDT) includes all ossible downtime (Reference 3). Without diminishing the ractical imortance and imact of the differences, the concetual treatment of availability, from a strictly statistical oint of view, is similar in all these cases. Their arameters, exected values (and erhas even the distributions of the variables involved) may change, and with them, also the interretation and form of their results. However, such differences will not modify the basic statistical hilosohy used for obtaining them, nor the concets on which their aroaches rest. Since the main objective of this START sheet is heling the engineer better understand such statistical hilosohy and aroaches, we will consider all three cases as a single one and adot the nomenclature given reviously for A i when referring and dealing with availability. In the remainder of this START sheet, we first overview and give numerical examles of the statistical treatment of the availability of a simle, reairable system in discrete times. Then, we will consider the case of a simle, reairable, arallel redundant system, comaring two different statistical modeling and analysis aroaches. We will illustrate these aroaches by develoing more numerical examles, erforming systems analyses, and comaring their results. Finally, we will mention several ways to imrove system availability and rovide additional bibliograhy for further study of these toics. Statistical Models for Simle Systems (U/Down) and Interretation In the Introduction, the long run average Availability was obtained as the ratio of the long run averages of U-Time to Cycle Time. However, Availability is a (cycle) RV itself. Hence, like any other RV, it has its own distribution, density function, etc. In this section, we overview a statistical model that describes Availability (A) as the RV resulting of the algebraic combination of the RV time between failures (X) and the RV time to reair (Y), at every cycle. In the following section, we describe Availability as a two-state, discrete time Markov Chain. The objective for resenting two contrasting models is to enhance the understanding of different statistical aroaches, so engineers can better use them and get meaningful results from their imlementation. We will briefly illustrate their mathematical derivations using a ractical examle. In For Further Study, we reference documents that rovide more in-deth information on the subject. We start by defining random variable (cycle) Availability as the following ratio: X A i i ; Xi, Yi > 0, i 1,..., n Xi + Yi The roblem of obtaining the density function or statistical descrition of A is resolved using a variable transformation of the joint distribution of the Availability (A) and of some other convenient function (denoted B), of time between failures X and time to reair Y, such as B(X, Y) X + Y. Assume that system times between failures and to reair (X and Y) are indeendent of each other and Exonentially distributed,

3 with mean µ MTBF MTTR 1 hour. Hence, their individual density functions are f 1 (X) Ex (-X) and f (Y) Ex (-Y). Their joint density function, denoted f(x, Y), is just the roduct of the two individual Exonential densities, since both (failure and reair) times X and Y are indeendent of each other. Therefore: f(x, Y) f1 (X) x f (Y) Ex{-X} x Ex{-Y} Ex{-(X + Y)} Define the (cycle) Availability function A(X, Y) X/(X + Y). Define auxiliary function B(X, Y) X + Y. Then, their inverse functions are X W (A, B) AB and Y Z (A, B) B (1 - A). Finally, obtain the matrix of the artial derivatives of the inverses W (A, B) and Z (A, B) and denote it J (W, Z). To derive the joint distribution g(a, B) of A and B, just substitute the values of X and Y in the original joint distribution function f(x,y), with their inverses [X W(A, B) AB and Y Z(A, B) B(1 - A)] and multily this by the absolute value of the matrix of the artial derivatives J(W,Z) B. That is: g(a, B) f(ab; B(1- A)) x J(W, Z) Ex{-(AB + B(1- A))}x B BEx{-B} This variable transformation yields the density of Availability. For, density function g 1 (A) of the resulting system Availability (A X/(X + Y)) is just the marginal distribution of the above derived, bivariate density function g (A, B). Hence, the desired marginal (Availability density) is obtained by integrating function g (A, B) out on B: X g1(a) g1 g(a, B)dB BEx{-B}db 1; X + Y 0 for 0 < A < 1 Hence, g 1 (A) 1 is the theoretical density of Availability and corresonds to the density of the Uniform (0, 1). As a result, all erformance measures (PM) of interest, such as the exected value, variance, ercentiles, robabilities, etc., are now obtained from the theoretical Uniform (0, 1) distribution arameters. For the simle examle given, where both times between failure (X) and to reair (Y) are distributed Exonentially with mean µ MTBF MTTR 1, we obtain: 1. Exected Availability E{Uniform(0,1)} ½ Variance of Availability Var {Uniform(0,1)} 1/ L 10 Percentile of 10% of Availabilities P{A < 0.1} First and Third Quartiles of Availability 0.5 and The Uniform distribution is a secial case of the Beta, which is the general distribution of Availability when the failure and reair times are exonentially distributed. Such theoretical results allow us to obtain emirically the Availability distribution via Monte Carlo (MC) simulation. To verify this for the results just given, we generate n 5,000 Exonentially distributed random failure and reair times, X i and Y i, i 1,, n, with µ MTBF MTTR 1. We then obtain the corresonding Availabilities A i X i /( X i + Y i ), sort them, and calculate the n 5,000 results numerically, via MC. The Exected Value is obtained from the samle average (0.5067); the Variance, from the samle variance (0.086). Percentile L 10 (Availability achieved 90% of the times) and all other robabilities are obtained by maniulating the sorted ranks of the total number of data oints n, of the MC generated values. For examle, L 10 corresonds to the 500 th sorted rank (10% of the n 5,000 MC data oints) and yields a value The quartiles are and Emirical and theoretical results agree closely. For Exonential means different than unit (µ MTBF MTTR 1) the mathematical treatment is more difficult and we use the Beta distribution, directly. For a more realistic examle, we reuse the examle of times between failure (X) with µ MTBF 500 hours, and to reair (Y) with µ MTTR 30 hours. We generate n 5,000 random Beta values with arameters corresonding to the said failure and reair times and obtain the (cycle) MC Availabilities, A i. Results are given in Figure 1 and Table 1. Frequency Figure 1. Histogram of Realistic Examle Table 1: Realistic Availability Results MC Results for Beta (500,30) Examle: Average Availability Variance of Availability 9.9x10-5 Life L Quartiles and P{A > 0.95} For examle, the robability that the Availability is greater than value 0.95, P{A > 0.95}, is obtained by looking at the sorted rank corresonding to a MC Availability closer to 0.95: (A 0.94 Beta

4 Rk 3,653). Then, we divide this Rank by n 5,000 and subtract it from unit: 3,653 P{A > 0.95} 1- P{A 0.95} ,000 Markov Models for Simle Systems (U/Down) and Interretation Now, consider the revious roblem, aroached as a two-state Markov Chain (References 4, 5, 6, and 7). Here we monitor the status of the system at time T, denoted X(T), instead of its availability A(T). Denote State 0 (Down) and State 1 (U), and assess the status X(T) of your system S every hour (T 0,1, ). Hence, X(T) 0 means that system S was Down at time T and X(T) 1, that system S was U at time T. We are interested in studying how the System S develos (transitions) over time. That is, we want to know what is the robability q (or ) that system S is U (or Down) at time T, given that it was Down (or U) an hour earlier (at time T - 1). We reresent this roblem using the Markov Chain state diagram shown in Figure. P P Figure. State Diagram for System S The transition robabilities are: P 01 P P{X(T) 1X(T -1) 0} q 10 P{X(T) 0 X(T -1) 1} q For examle, let system S be Down at time T - 1. Then, either the system is U at time T, with robability q, or it will be Down with robability 1 - q. If the system was U at T - 1 then it is either Down at T with robability, or it is U at T, with robability 1 -. This occurs because there are only two ossibilities (U or Down) for S at any given time T. The two state robabilities have to add u to Unit. Let s analyze this situation further. Each time unit (hour) T, transitioned by system S, can be considered as an indeendent trial, and the robability ij of moving from i into the other state j, as the robability of success. For examle, let system S be in state U. Then, moving to state Down by one ste with robability yields a Geometric distribution with Mean µ 1/ 500 hours. If, instead, system S is in state Down then, moving to state U by one-ste (hour), with robability 01 q 0.033, yields a Geometric, with Mean µ 1/q 30 hours. These are the same arameters of the realistic examle of the revious section. The Geometric distribution is the discrete counterart of the continuous Exonential and, as the units of time T become smaller (hours to minutes, seconds, etc.), the two distributions converge. Therefore, this numerical examle is equivalent to the one given in the revious section, which used similar time arameters, and will serve as a vehicle for comarison and contrast. One imortant roerty of Markov Chains is their lack of Memory. This means that only the system status at the immediately revious time has any bearing on the status at the current time, and every other ast history goes into oblivion. In addition, these Markov Chains are time homogeneous (the transition robabilities ij do not change over time). Hence, it is enough to know the one-ste state transition robabilities or the robabilities ij of going from any state i to any other state j in one ste, to resolve the roblems. Markov Chains can be reresented by a Transition Probability Matrix P, where rows reresent every system state we can be in at time T, and columns reresent every other state we can go to, in one ste (i.e., where we will be, at T + 1). Entries of Matrix P( ij ) corresond to the Markov Chain s one-ste transition robabilities and must add u to unit, on every matrix row. For our numerical examle, the Transition Probability matrix P is: States 0 1 States (1 - q q) 0 ( ) 1 ( 1 - ) 1 ( ) If we need the robabilities of moving from one state to any other, in two stes, we raise matrix P to the second ower. For examle, moving from states U to Down in two stes, entails either moving from U to Down in first ste, and remaining in Down state another ste. Or it may entail first remaining U for one ste, before moving from states U to Down in the second ste. In matrix language, this is exressed in the following way: 1- q (1- q) + q (1- q) + (1- ) q 1- q 1- q 1- q 1- x 1- q(1- q) + q(1- ) 00 q + (1- ) 10 q 1-01 In our examle, the 10 result rovides the robability that system S is Down, if it was initially U, after oerating for two hours (T ): 10 (1 - q) + (1 - ) The result (robability that S is U after hours, given that it started U) can be obtained as one minus the robability that S is down, after hours: We can then interret A(T) as the system Availability, after T hours of oeration, if it started in state U (at T 0). To obtain the robability of moving from one state to another in n stes, we raise the matrix P to the nth ower (P n ). For examle, the robability that S is Down after T 10 hours 4

5 (stes) if it was initially U, is 10 (10) (this includes that S could have gone Down or U, then restored, and this may have occurred more than once during the T 10). where both units are oerating and the system is working at full caacity. The state diagram for this model is shown in Figure 3. For a sufficiently large n matrix P n yields quasi identical rows. Results are interreted as long run averages or limiting robabilities i of S being in the state corresonding to column i. These results are similar to the ones obtained using the Exected Availability and Unavailability and the variable transformations aroaches. To obtain these limiting robabilities (i.e., to calculate P n ) we need a ractical result. For any two-state (e.g., U, Down) system, as the one described above, this ractical method is as follows. P 00 P 0 P 10 P 1 P 01 P 1 1 P n 1- q q n q n q (1- - q) q + + q q - -q Figure 3. Markov Chain for Redundant System The state equations are: Then, for a sufficiently large n, the second term goes to zero and the matrix P n reduces to: Limit P n Limit n n 1 + q /( + q) /( + q) n q (1- - q) q + + q q/( + q) q/( + q) q - -q Verify, for our given numerical examle, that the robability of being in state U at any arbitrary time T is q/( + q) 0.943, and the robability of being in state Down, is /( + q) These two state occuancy rates, E(X) and E(Y), can also be interreted as the ercent of the time that the system S will send in states U and Down. A Markov Model for a Simle Redundant System In Reference 8, we develoed a statistical model for a non-maintained, simle redundant system, comosed of two identical devices in arallel. The aroach was based on the two RV, corresonding to the two device lives. In this section we also analyze a simle redundant system comosed of two identical devices in arallel. The differences now are that we use a Markov Chain aroach, and that system S is maintained and can function at a degraded level with only one unit. The advantages of Markov modeling of system Availability, as will become aarent from the numerical examle that follows, increase as the system becomes more comlex (as also do the mathematics behind the analyses involved). Let, as before, X(T) be the state of the system at time T ( 0,1,, hours). Let State 0 be the Down state, where both devices have failed and one of them is being reaired. Let State 1 be the Degraded state, where one device has failed and is being reaired and the second is working (and the system is oerable but with lesser caabilities). Finally, let State be the U state, 01 P{X(T) 1X(T -1) 0} q 10 P{X(T) 0 X(T -1) 1} 1 P{X(T) X(T -1) 1} q 1 P{X(T) 1X(T -1) } ii P{X(T) i X(T -1) i} 1- ij j 1 As before, we can consider every ste (hour) T as an indeendent trial, having robability of success ij corresonding to the feasible transitions from our current state i into state j 0,1,. Hence, we can again think of the distribution of every change of state (roduced by the occurrence of a failure or a reair) as being geometric, the discrete counterart of the Exonential. It will have robability of success ij (corresonding to the change into that state) and a mean time to accomlishing such change of µ 1/ ij. The transition robability matrix P for this model is given by: States 0 1 States q q q q Rows must add to one (robability is unity because the system is always in one of its three states). And, if we want to know the robability ij (n) of being in some state j after n stes, given that we started in some state i of the system, we raise matrix P to the ower n as we did before, and look at entry ij of the resulting matrix P n. With the advent of modern comuters and math software, these oerations are no longer tedious or difficult. Modify the numerical examle of revious section, now using two units instead of one. The robability of either unit failing 5

6 in the next hour is The robability q of the reair crew comleting a maintenance job in the next hour is Only one failure is allowed in each unit time eriod, and only one reair can be undertaken at a time. With these new conditions, the robability that a degraded system (State 1) remains degraded after two hours is the sum of the robabilities corresonding to three events. First, that system status has never changed. Second, that one unit is first reaired and then another unit fails during the second hour. Third, that remaining unit fails in the first hour (the entire system goes down) then, a reair is comleted in the second hour (system goes u, at degraded level): P [P x P] Average times until System S goes down yield V 1 4,65 hours (starting in state Degraded) and V 4,875 (starting U). For comarison, the non maintained system version referred to initially, would work an Exected 3/λ 3/ hours in U state, before going Down (Reference 8). The fact that main- q + (1- - q) + q 0.00 x ( ) + x 0.00 x We are also interested in the mean time that the system sends in any given state. For examle, System S can change to U or Down, from state Degraded, in one ste, with robabilities and q. Hence, S will remain in the state Degraded with robability q. Then, on average, S will send a sejour of length 1/{1 - (1 - - q)} 1/ consecutive hours in the Degraded state, before moving out to either U or Down states. Let s now analyze Availability at time T A(T) P{S is Available at T}. But this just means that system S is not Down at time T (it can be U or Degraded). In addition, S could have initially been U, Down or Degraded. Hence, A(T) deends on the initial state of S (States 0,1,), actual system availability level (States 1,) and time (T). Assume we are interested in S being Degraded Available at T, given it was Degraded at T 0: (T). Since for matrix P T every row has to add to unit, we can obtain such Availability via: (T) (T) (T) (T) (T) (T) (T) (T) (T) A(T) P{X(T) 1 X(0) 1} We may instead be interested in long run averages or state occuancies. These are the asymtotic robabilities of system S being in each one of its ossible states at any time T, or the ercent time sent in these states, irresective of the state they were in, initially. These results are obtained by considering the Vector (denoted Π) of long run robabilities: Limit T [Prob{X(T) 0}, Prob{X(T) 1}, Prob{X(T) }] ( 1; ; 3) Vector Π fulfills two imortant roerties that allow the calculation of such values: Limit (1) : x P ; i 1; with : i T Prob{X(T) i} In lain English, Π P Π (Vector Π times the matrix P equals Π) defines a system of linear equations, that are normalized by the second roerty (that robabilities in the comonents of Vector Π add to Unit). For our examle, we have the following x P ( 0, 1, ) x ( 0, 1, ) ; with : i i The solution of this linear system of equations yields the long run or asymtotic occuancy rates: ( 0, 1, ) (0.0065, , ) A Π indicates that the system S is oerating at full caacity 88% of the time. A Π means that S is oerating at a Degraded caacity 10% of the time. Only Π 0, the robability corresonding to State 0 (Down state), is associated with the system being Unavailable. The long run system Availability is then: 1 - Π Finally, we are also interested in the exected times for System S to go Down if initially S was in State U (denoted V 1 ) or Degraded (V ), or in the average time S sent in each of these states before going Down. We obtain them by assuming Down is an absorbing state (one that, once entered, can never be left) and solving the linear system of equations leading to all such ossible situations. That is, one ste is taken at minimum (when the system goes Down, directly). If S is not absorbed in one ste, then it will necessarily move on to any of other, non-absorbing (U or Degraded) states, with the corresonding robability, and the rocess restarts. V 1 1+ V1 + 1V V V V 1+ 1V1 + V V V 6

7 tenance is now ossible, while S continues oerating in a Degraded state (with a single unit), results in an increase of µ/λ 0.033/ x ,15 hours in its Exected Time to go Down (from U). Verify that the new Exected Time is due to the sum of Exected times to failures, lus maintenance: V 3/λ + µ/λ ,15 4,875. Model Extensions and Comarisons We have seen how a stochastic rocess is just a R.V. X(T), indexed in some arameter T called time. The rocesses overviewed here are collectively known as discrete time arameter Markov Chains, because transitions only occur at regular intervals (in our examles, every hour). Hourly time intervals can be shortened (to minutes, seconds, etc.) and X(T) aroximates a continuous time arameter Markov Chain (also known as a Markov Processes) just like a Riemann sum aroximates an Integral. We have not dealt with continuous time arameter Markov Chains in this START sheet, because their mathematical treatment requires using differential equations, Lalace Transforms and other tools of advanced calculus and mathematics. The objective of this START sheet is not to discuss mathematics, but to convey imortant statistical rinciles to the engineer who uses software and tools that imlement them. The reader interested in learning more about these advanced methods is referred to the sources in the Bibliograhy. Reference (6), is a START sheet that discusses the mathematical derivation of a simle continuous time arameter Markov Chain and some uses. It is available on our web site at: <htt://rac. alionscience.com/df/markov.df>. Reference 7, also discusses these models in more detail. Reference 5, Chater 10, is an older but classic reliability book that treats this roblem at introductory level. Reference 1, Chater 6, is a recent textbook that treats the subject extensively and in a more mathematically advanced way. Reference 4, is a mathematics book about stochastic modeling with a clear aroach to the toic. Finally, Reference 8 develos the system examle used for comarison here. We have discussed extensively, however, the understanding and use of several imortant statistical models. Among them, Availability via RV transformation and via defining a Markov Chain that reresents the system as it moves through time. In doing so, we have shown how different but comlementary statistical modeling aroaches rovide different answers to different tyes of roblems and questions. For examle, if the roblem is one of characterizing the RV Availability (A) via finding a confidence interval, a ercentile (Life L 10 ) or the secific robability of some events (say, A > 0.9) then we may want to derive the distribution of A, directly. Obtaining such theoretical distributions may not always be easy. But then, one can resort to MC methods, which will rovide working aroximations to the exact but unavailable solutions. If the system is more comlex, involving redundancy, degradation, etc. and one is more interested in asymtotic or steady state results, we may want to imlement a Markov model. PM such as long run Availability (state occuancies), Exected time to failure, etc. can also be obtained as the system X(T) moves through time. Markov Model assumtions (e.g., that distributions of times to failure, to reair, etc. should be Exonential) are some times unrealistic. But here too, one can resort to Monte Carlo methods. Some software ackages (e.g., BlockSim) imlement some of these models and methods. Knowledge of the mathematics involved in model develoment is no longer necessary for the engineer. But a better understanding of the nature and imlications of the methods they imlement rovides a safer use of such software ackages. Finally, it becomes clear that there are two ways of imroving Availability: either extending the system life or imroving its Maintainability. Logistics deals directly with the latter issue and its imlications. Due to its comlexity, Logistics will be the toic of a forthcoming START sheets. For Further Study 1. System Reliability Theory: Models and Statistical Methods, Hoyland, A. and M. Rausand, Wiley, NY, Logistics Engineering and Management, Blanchard, B.S., Prentice Hall, NJ, Criscimagna, N., RAC, Personal communication. 4. An Introduction to Stochastic Modeling, Taylor, H. and S. Karlin, Academic Press, NY, Methods for Statistical Analysis of Reliability and Life Data, Mann, N., R. Schafer, and N. Singurwalla, John Wiley, NY, Alicability of Markov Analysis Methods to Availability, Maintainability and Safety, Fuqua, N., RAC START Sheet, Volume 10, No., <htt://rac.alionscience. com/df/markov.df>. 7. Aendix C of the Oerational Availability Handbook (OPAH), Manary, J. RAC. 8. Understanding Series and Parallel Systems Reliability, Romeu, J.L., RAC START Sheet, Volume, No. 5 <htt://rac.alionscience.com/df/s&psysrel.df>. About the Author Dr. Jorge Luis Romeu has over thirty years of statistical and oerations research exerience in consulting, research, and teaching. He was a consultant for the etrochemical, construction, and agricultural industries. Dr. Romeu has also worked in statistical and simulation modeling and in data analysis of software and hardware reliability, software engineering, and ecological roblems. 7

8 Dr. Romeu has taught undergraduate and graduate statistics, oerations research, and comuter science in several American and foreign universities. He teaches short, intensive rofessional training courses. He is currently an Adjunct Professor of Statistics and Oerations Research for Syracuse University and a Practicing Faculty of that school s Institute for Manufacturing Enterrises. For his work in education and research and for his ublications and resentations, Dr. Romeu has been elected Chartered Statistician Fellow of the Royal Statistical Society, Full Member of the Oerations Research Society of America, and Fellow of the Institute of Statisticians. Romeu has received several international grants and awards, including a Fulbright Senior Lectureshi and a Seaker Secialist Grant from the Deartment of State, in Mexico. He has extensive exerience in international assignments in Sain and Latin America and is fluent in Sanish, English, and French. Romeu is a senior technical advisor for reliability and advanced information technology research with Alion Science and Technology reviously IIT Research Institute (IITRI). Since rejoining Alion in 1998, Romeu has rovided consulting for several statistical and oerations research rojects. He has written a State of the Art Reort on Statistical Analysis of Materials Data, designed and taught a three-day intensive statistics course for racticing engineers, and written a series of articles on statistics and data analysis for the AMPTIAC Newsletter and RAC Journal. Other START Sheets Available Many Selected Toics in Assurance Related Technologies (START) sheets have been ublished on subjects of interest in reliability, maintainability, quality, and suortability. START sheets are available on-line in their entirety at <htt://rac. alionscience.com/rac/js/start/startsheet.js>. For further information on RAC START Sheets contact the: Reliability Analysis Center 01 Mill Street Rome, NY Toll Free: (888) RAC-USER Fax: (315) or visit our web site at: <htt://rac.alionscience.com> About the Reliability Analysis Center The Reliability Analysis Center is a Deartment of Defense Information Analysis Center (IAC). RAC serves as a government and industry focal oint for efforts to imrove the reliability, maintainability, suortability and quality of manufactured comonents and systems. To this end, RAC collects, analyzes, archives in comuterized databases, and ublishes data concerning the quality and reliability of equiments and systems, as well as the microcircuit, discrete semiconductor, and electromechanical and mechanical comonents that comrise them. RAC also evaluates and ublishes information on engineering techniques and methods. Information is distributed through data comilations, alication guides, data roducts and rograms on comuter media, ublic and rivate training courses, and consulting services. Located in Rome, NY, the Reliability Analysis Center is sonsored by the Defense Technical Information Center (DTIC). Alion, and its redecessor comany IIT Research Institute, have oerated the RAC continuously since its creation in Technical management of the RAC is rovided by the U.S. Air Force's Research Laboratory Information Directorate (formerly Rome Laboratory). 8