BASIC RELIABILITY “bottom-up” MODELS 1.Series model The series model is used to go from individual components to the entire system, assuming the system fails when the first component fails and all components fail or survive independently of one another. R (t ) R (t ) n sys i i 1 h (t ) h (t ) n sys i 1 i Example 1 What is the reliability of the series system shown below? A = .001 fph, B = .002 fph , mission time t = 50 hours For the system to work, both devices must work. If one device fails, the system fails. RA = reliability of device A = probability that device A will work for at least 50 hours RA = = e -(.001)(50) = .9512 RB = reliability of device B = probability that device B will work for at least 50 hours RB = = e -(.002)(50) = .9048 RS = reliability of system = probability that the system will work for at least 50 hours RS = RA X RB RS = .9512 X .9048 = .8606 2.Parallel or redundant mode The parallel model assumes all n components that make up a system operate independently and the system works as long as at least one component still works. If there are n components, any (n-1) of them may be considered redundant to the remaining one (even if the components are all different). When the system is turned on, all the components operate until they fail. The system reaches failure at the time of the last component failure. For a parallel model, the CDF Fs(t) for the system is just the product of the CDF's Fi(t) for the components or F (t ) F (t ) n sys i 1 i So, Rs(t) and hs(t) can be evaluated using basic definitions, once we have Fs(t). The schematic below represents a parallel system with 5 components and the (reliability) equivalent 1 component system with a CDF Fs equal to the product of the 5 component CDF's. Example 2 What is the reliability of the parallel system shown below? A = .001 fph, B = .002 fph ,mission time t = 50 hrs For the system to work, one or both devices must work. The system will fail when both devices fail. From example 1, RA = .9512 and RB = .9048 FA = 1 - RA = probability that device A fails and FB = 1 - RB = probability that device B fails. Fsys = FAFB = (1-RA)(1-RB) ,so: Rsys = reliability of system = RA + RB - RARB = .9512 +.9048 – (.9512)(.9048) = .9954 For n identical components: 1 1 1 1 MTTF (1 ) 2 3 n 3. r out of n model An r out of n model is a system that survives when at least r of its components are working (any r) An "r out of n" system contains both the series system model and the parallel system model as special cases. The system has n components that operate or fail independently of one another and as long as at least r of these components (any r) survive, the system survives. System failure occurs when the (n-r+1)th component failure occurs. When r = n, the r out of n model reduces to the series model. When r = 1, the r out of n model becomes the parallel model. We treat here the simple case where all the components are identical. Formulas and assumptions for r out of n model (identical components): 1. All components have the identical reliability function R(t). 2. All components operate independently of one another (as far as failure is concerned). 3. The system can survive any (n-r) of the components failing. The system fails at the instant of the (n-r+1)th component failure. System reliability is given by adding the probability of exactly r components surviving to time t to the probability of exactly (r+1) components surviving, and so on up to the probability of all components surviving to time t. These are binomial probabilities (with p = R(t)), so the system reliability is given by: n i ni Rsys(t ) R(t ) 1 R(t ) i n i r Note: If we relax the assumption that all the components are identical, then Rs(t) would be the sum of probabilities evaluated for all possible terms that could be formed by picking at least r survivors and the corresponding failures. The probability for each term is evaluated as a product of R(t)'s and F(t)'s. For example, for n = 4 and r = 2, the system reliability would be (abbreviating the notation for R(t) and F(t) by using only R and F) Rsys = R1R2F3F4 + R1R3F2F4 + R1R4F2F3 + R2R3F1F4 + R2R4F1F3 + R3R4F1F2 + R1R2R3F4 + R1R3R4F2 + R1R2R4F3 + R2R3R4F1 + R1R2R3R4 Example 3 2 4 What is the reliability of the out of system if: all λi = 0.001 fph and the mission time t = 50 hours ? Solution All Ri 0.9512 ,so: 4 i 4i Rsys(t ) 0.9512 0.0488 2 4 i 2 R 0.9996 sys 4. Standby model The Standby Model evaluates improved reliability when backup replacements are switched on when failures occur. A Standby Model refers to the case in which a key component (or assembly) has an identical backup component in an "off" state until needed. When the original component fails, a switch turns on the "standby" backup component and the system continues to operate. In the simple case, assume the non-standby part of the system has CDF F(t) and there are (n-1) identical backup units that will operate in sequence until the last one fails. At that point, the system finally fails. The total system lifetime is the sum of n identically distributed random lifetimes, each having CDF F(t). In other words, Tn = t1 + t2+ ... + tn, where each ti has CDF F(t) and Tn has a CDF we denote by Fn(t). This can be evaluated using convolution formulas: In general, convolutions are solved numerically. However, for the special case when F(t) is the exponential model, the above integrations can be solved in closed form. Special Case: The Exponential (or Gamma) Standby Model If F(t) has the exponential CDF (i.e., F(t) = 1 - e-λt ), then R e sys (t ) 2 (t ) 3 (t ) [1 t ] 2! 3! (n 1)! n 1 t MTTF n and the PDF fn(t) is the well-known gamma distribution. For n = 2: t Rparallel 2e Rstandby (1 t ) e For t 1 ; e 2 t t Fparallel 2 Fstandby Example 4: An unmanned space probe sent out to explore the solar system has an onboard computer with reliability characterized by the exponential distribution with a Mean Time To Failure (MTTF) of 1/ = 30 months (a constant failure rate of 1/30 = .033 fails per month). The probability of surviving a two year mission is only e-24/30 = .45. If, however, a second computer is included in the probe in a standby mode, the reliability at 24 months (using the above formula for F2) becomes .8 × .449 + .449 = .81. The failure rate at 24 months (f2/[1-F2]) reduces to [(24/900) ×.449]/.81 = .015 fails per month. At 12 months the failure rate is only .0095 fails per month, which is less than 1/3 of the failure rate calculated for the nonstandby case. Standby units (as the example shows) are an effective way of increasing reliability and reducing failure rates, especially during the early stages of product life. Their improvement effect is similar to, but greater than, that of parallel redundancy. The drawback, from a practical standpoint, is the expense of extra components that are not needed for functionality. 5.Complex systems Often the reliability of complex systems can be evaluated by successive applications of Series and/or Parallel model formulas. Many complex systems can be diagrammed as combinations of Series components, Parallel components, R out of N components and Standby components. By using the formulas for these models, subsystems or sections of the original system can be replaced by an "equivalent" single component with a known CDF or Reliability function. Proceeding like this, it may be possible to eventually reduce the entire system to one component with a known CDF. Below is an example of a complex system composed of both components in parallel and components in series is reduced first to a series system and finally to a one-component system. Note: The reduction methods described above will work for many, but not all, systems. Some systems with a complicated operational logic structure will need a more formal structural analysis methodology. For example let’s solve the system bellow by the help of Baye’s theorem. About gamma distribution There are two ways of writing (parameterizing) the gamma distribution that are common in the literature. In addition, different authors use different symbols for the shape and scale parameters. Below we show three ways of writing the gamma, with a = = , the "shape" parameter, and b =1/ , the scale parameter. EXCEL uses . Note: When α = 1, the gamma reduces to an exponential distribution with β = 1/λ. For the standby model α = n and β =1/λ. EXCEL has built-in functions to evaluate the gamma pdf and cdf. The syntax is: =GAMMADIST(t,α,β,FALSE) for the PDF =GAMMADIST(t,α,β,TRUE) for the CDF