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{"sheet":"Make & Check Formulas","index":0,"n_rows":42,"n_cols":15,"rows_kept":42,"cut":false,"hidden":false,"merged":[[20,1,20,10],[18,1,18,13],[17,1,17,13],[16,1,16,13],[13,1,13,12],[5,1,5,13],[22,10,22,12],[22,2,22,3],[22,4,22,5],[0,0,0,14],[2,1,2,2],[3,1,3,13],[7,1,7,10],[8,1,8,10],[4,1,4,13],[10,1,10,12],[11,1,11,12],[9,1,9,12],[12,1,12,12],[21,1,21,9],[19,1,19,9]]}
["REPRODUCE selected RMA examples in \"An Intro to Reliability & Maintainability Engineering, 1st Ed,\" C. E. Ebeling"]
[]
[null,"Objective:"]
[null,"1.  Make and validate formulas in Excel to calculate Reliability for a new item and three types of Inherent Availability for one or more items (elements) in a series configuration, and"]
[null,"     where each item's distribution of times to failure (end of each uptime state) and distribution of times to complete a repair (return to an uptime state) are modeled with the "]
[null,"     exponential probability distribution.  This type of probability distribution treats the failure rate and repair rate as a constant (fixed) value over time."]
[]
[null,"Nomenclature, Assumptions, and Observations:"]
[null,"Input Data (yellow-shaded headers or cell): N, λ, μ, t, and r where …"]
[null,"1.  N = number of identical elements in a series configuration."]
[null,"2.  λ = 1/MTBF and μ = 1/MTTR where MTBF = mean time between failure and MTTR = mean time to repair.  Mean is the same as average."]
[null,"3.  λ and μ are constant over time since the exponential distribution is used to model the failure distribution and the repair distribution."]
[null,"4.  λ, μ, and t must have the same units of time (e.g., hours).  Mission time = t.  "]
[null,"5.  For design purposes, view MTBF as a lower-bound parameter and MTTR as an upper-bound parameter. "]
[null,"6.  r is the number repairs (or spares needed) during the mission (from 0 to t).  When r = 0, then P = R.  When r > 0, then P > R since spares increase mission success."]
[null,"Output Data: R, A, and P where …"]
[null,"7.  R = Reliability, the probability an item will perform its intended function with no failure during the time interval from 0 to t.  R is 0.3679 or less when Nλt is 1.0 or more."]
[null,"8.  A = Inherent Availability, the probability of mission readiness at point t, the average from 0 to t, and as t becomes large (limiting or steady-state)."]
[null,"9.  As mission time (t) increases, first Point Availability and then Average Availability approximate Limiting Availability."]
[null,"10. P = the probability of r or less repairs (spares needed) from 0 to t via the cumulative Poisson distribution.",null,null,null,null,null,null,null,null," Let r = ",2]
[null,"11. The Poisson distribution does not account for repair time since it assumes failures are immediately repaired or replaced."]
[]
[null,null,"Life (Uptime) Parameter",null,"Repair (Downtime) Parameter",null,null,null,"Probability Measure",null,"Probability Measures for Inherent Availability",null,null,null,"Simulation Generated (2nd check)"]
["GIVEN: Textbook Examples","Number of Identical Serial Elements (N)"," Element Failure Rate (λi)","Element MTBF (MTBFi)","Element Repair Rate (μi)","Element MTTR (MTTRi)","Mission Time (t)","Nλt","Reliability From 0 to t (R)","(λ+μ)t","Point Availability, at t (AP)","Average Availability, from 0 to t (AAvg)","Limiting Availability, as t→∞ (AL)","Probability of r or Less Repair Events from 0 to t (P)","Mean Availability, Raptor v 7, 5000 iterations"]
["Ebeling, p. 258, ex 11.2",1,{"$f":"1/200","v":5.0000000000000001E-3},{"$f":"1/C25","v":200},{"$f":"1/10","v":0.1},{"$f":"1/E25","v":10},10,{"$f":"B25*C25*G25","v":0.05},{"$f":"(EXP(-C25*G25))^B25","v":0.95122942450071402},{"$f":"(C25+E25)*G25","v":1.05},{"$f":"((E25/(E25+C25))+(C25/(E25+C25))*EXP(-(C25+E25)*G25))^B25","v":0.96904465471957879},{"$f":"((E25/(E25+C25))+(C25/(((E25+C25)^2)*G25))*(1-EXP(-(C25+E25)*G25)))^B25","v":0.98186223360040104},{"$f":"(E25/(E25+C25))^B25","v":0.95238095238095233},{"$f":"POISSON($L$20,H25,TRUE)","v":0.99997993250637562},0.98042499999999999]
["Same as above except t increases",1,{"$f":"1/200","v":5.0000000000000001E-3},{"$f":"1/C26","v":200},{"$f":"1/10","v":0.1},{"$f":"1/E26","v":10},110,{"$f":"B26*C26*G26","v":0.55000000000000004},{"$f":"(EXP(-C26*G26))^B26","v":0.57694981038048665},{"$f":"(C26+E26)*G26","v":11.55},{"$f":"((E26/(E26+C26))+(C26/(E26+C26))*EXP(-(C26+E26)*G26))^B26","v":0.95238141124014775},{"$f":"((E26/(E26+C26))+(C26/(((E26+C26)^2)*G26))*(1-EXP(-(C26+E26)*G26)))^B26","v":0.95650377391860186},{"$f":"(E26/(E26+C26))^B26","v":0.95238095238095233},{"$f":"POISSON($L$20,H26,TRUE)","v":0.98153586490980294},0.95609699999999997]
["Ebeling, p. 259, ex 11.3",2,0.1,{"$f":"1/C27","v":10},0.2,{"$f":"1/E27","v":5},10,{"$f":"B27*C27*G27","v":2},0.1353352832366127,{"$f":"(C27+E27)*G27","v":3.0000000000000004},{"$f":"((E27/(E27+C27))+(C27/(E27+C27))*EXP(-(C27+E27)*G27))^B27","v":0.46684744729423577},{"$f":"((E27/(E27+C27))+(C27/(((E27+C27)^2)*G27))*(1-EXP(-(C27+E27)*G27)))^B27","v":0.5963637011731675},{"$f":"(E27/(E27+C27))^B27","v":0.44444444444444442},{"$f":"POISSON($L$20,H27,TRUE)","v":0.6766764161830634},0.60206199999999999]
["Ebeling, p. 267, ex 11.9",1,{"$f":"1/2100","v":4.7619047619047619E-4},{"$f":"1/C28","v":2100},{"$f":"1/5.25","v":0.19047619047619047},{"$f":"1/E28","v":5.25},10,{"$f":"B28*C28*G28","v":4.7619047619047623E-3},0.99524941513136345,{"$f":"(C28+E28)*G28","v":1.9095238095238094},{"$f":"((E28/(E28+C28))+(C28/(E28+C28))*EXP(-(C28+E28)*G28))^B28","v":0.99787568807445659},{"$f":"((E28/(E28+C28))+(C28/(((E28+C28)^2)*G28))*(1-EXP(-(C28+E28)*G28)))^B28","v":0.99861871696848914},{"$f":"(E28/(E28+C28))^B28","v":0.99750623441396513}]
[null,1,{"$f":"1/3200","v":3.1250000000000001E-4},{"$f":"1/C29","v":3200},{"$f":"1/8","v":0.125},{"$f":"1/E29","v":8},10,{"$f":"B29*C29*G29","v":3.1250000000000002E-3},{"$f":"(EXP(-C29*G29))^B29","v":0.99687987773020814},{"$f":"(C29+E29)*G29","v":1.2531249999999998},0.99821848096474597,0.99892789548947958,0.99750623441396513]
[null,1,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/12.5","v":0.08},12.5,10,2E-3,0.99800199866733308,0.80200000000000005,0.99862451671882313,0.99922130084934757,0.99750623441396502]
[null,1,{"$f":"1/1700","v":5.8823529411764701E-4},1700.0000000000002,{"$f":"1/4.25","v":0.23529411764705882},4.25,10,5.8823529411764705E-3,0.99413491422311084,2.3588235294117648,0.9977419734942149,0.99846350250868698,0.99750623441396513]
["series system calculation",null,null,null,null,null,null,null,null,null,{"$f":"PRODUCT(K28:K31)","v":0.99248171563679755},{"$f":"PRODUCT(L28:L31)","v":0.99523976687750304},0.99006218886173925,null,0.99512199999999995]
["Same as above except t increases",1,{"$f":"1/2100","v":4.7619047619047619E-4},{"$f":"1/C33","v":2100},{"$f":"1/5.25","v":0.19047619047619047},{"$f":"1/E33","v":5.25},50,{"$f":"B33*C33*G33","v":2.3809523809523808E-2},{"$f":"(EXP(-C33*G33))^B33","v":0.97647168665224326},{"$f":"(C33+E33)*G33","v":9.5476190476190457},{"$f":"((E33/(E33+C33))+(C33/(E33+C33))*EXP(-(C33+E33)*G33))^B33","v":0.99750641239648852},{"$f":"((E33/(E33+C33))+(C33/(((E33+C33)^2)*G33))*(1-EXP(-(C33+E33)*G33)))^B33","v":0.99776740817792398},{"$f":"(E33/(E33+C33))^B33","v":0.99750623441396513}]
[null,1,{"$f":"1/3200","v":3.1250000000000001E-4},{"$f":"1/C34","v":3200},{"$f":"1/8","v":0.125},{"$f":"1/E34","v":8},50,{"$f":"B34*C34*G34","v":1.5625E-2},0.98449643700540845,{"$f":"(C34+E34)*G34","v":6.265625},0.99751097387835141,0.99790348546579932,0.99750623441396513]
[null,1,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/12.5","v":0.08},12.5,50,0.01,0.99004983374916811,4.0100000000000007,0.99755145485096453,0.99811684417681679,0.99750623441396502]
[null,1,{"$f":"1/1700","v":5.8823529411764701E-4},1700.0000000000002,{"$f":"1/4.25","v":0.23529411764705882},4.25,50,2.9411764705882349E-2,0.97101655179243718,11.794117647058822,0.99750625323898379,0.99771767428896407,0.99750623441396513]
["series system calculation",null,null,null,null,null,null,null,null,null,{"$f":"PRODUCT(K33:K36)","v":0.99011197149145891},{"$f":"PRODUCT(L33:L36)","v":0.99153238547476685},{"$f":"PRODUCT(M33:M36)","v":0.99006218886173925},null,0.99021800000000004]
["Same as above except t increases",1,{"$f":"1/2100","v":4.7619047619047619E-4},{"$f":"1/C38","v":2100},{"$f":"1/5.25","v":0.19047619047619047},{"$f":"1/E38","v":5.25},100,{"$f":"B38*C38*G38","v":4.7619047619047616E-2},{"$f":"(EXP(-C38*G38))^B38","v":0.95349695483347674},{"$f":"(C38+E38)*G38","v":19.095238095238091},{"$f":"((E38/(E38+C38))+(C38/(E38+C38))*EXP(-(C38+E38)*G38))^B38","v":0.99750623442666797},{"$f":"((E38/(E38+C38))+(C38/(((E38+C38)^2)*G38))*(1-EXP(-(C38+E38)*G38)))^B38","v":0.99763683061605979},{"$f":"(E38/(E38+C38))^B38","v":0.99750623441396513}]
[null,1,{"$f":"1/3200","v":3.1250000000000001E-4},{"$f":"1/C39","v":3200},{"$f":"1/8","v":0.125},{"$f":"1/E39","v":8},100,{"$f":"B39*C39*G39","v":3.125E-2},0.96923323447634413,{"$f":"(C39+E39)*G39","v":12.53125},0.99750624342143668,0.99770523743270334,0.99750623441396513]
[null,1,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/12.5","v":0.08},12.5,100,0.02,0.98019867330675525,8.0200000000000014,0.99750705441402188,0.99781707550947352,0.99750623441396502]
[null,1,{"$f":"1/1700","v":5.8823529411764701E-4},1700.0000000000002,{"$f":"1/4.25","v":0.23529411764705882},4.25,100,5.8823529411764698E-2,0.94287314385487497,23.588235294117645,0.99750623441410724,0.99761195514952672,0.99750623441396513]
["series system calculation",null,null,null,null,null,null,null,null,null,{"$f":"PRODUCT(K38:K41)","v":0.99006301169542632},{"$f":"PRODUCT(L38:L41)","v":0.99080297676500373},{"$f":"PRODUCT(M38:M41)","v":0.99006218886173925},null,0.99036500000000005]

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[null,"EXPLORE the relationships of R, M, and A (based on constant failure and repair rates, λ and μ)"]
[]
[null,"IF",null,null,null,null,null,null,null,null,null,null,null,null,"Interim Calculations"]
[null,null,"Number of Identical Serial Elements (N)"," Element Failure Rate (λi)","Element MTBF (MTBFi)","Element Repair Rate (μi)","Element MTTR (MTTRi)","Mission Time (t)--enter time only in cell H5","Nλt","Reliability From 0 to t (R)","(λ+μ)t","Point Availability, at t (AP)","Average Availability, from 0 to t (AAvg)","Limiting Availability, as t→∞ (AL)","λi * MTTRi","λi/μi"]
[null,null,1,{"$f":"1/2100","v":4.7619047619047619E-4},{"$f":"1/D5","v":2100},{"$f":"1/5.283061","v":0.18928420474418145},{"$f":"1/F5","v":5.283061},10,{"$f":"C5*D5*H5","v":4.7619047619047623E-3},{"$f":"(EXP(-D5*H5))^C5","v":0.99524941513136345},{"$f":"(D5+F5)*H5","v":1.8976039522037191},{"$f":"((F5/(F5+D5))+(D5/(F5+D5))*EXP(-(D5+F5)*H5))^C5","v":0.99786680215846868},{"$f":"((F5/(F5+D5))+(D5/(((F5+D5)^2)*H5))*(1-EXP(-(D5+F5)*H5)))^C5","v":0.99861472309998056},{"$f":"(F5/(F5+D5))^C5","v":0.99749056974909089},{"$f":"D5*G5","v":2.5157433333333331E-3},{"$f":"D5/F5","v":2.5157433333333331E-3}]
[null,null,1,{"$f":"1/3200","v":3.1250000000000001E-4},{"$f":"1/D6","v":3200},{"$f":"1/8.050378","v":0.12421776965007109},{"$f":"1/F6","v":8.0503780000000003},{"$f":"H$5","v":10},{"$f":"C6*D6*H6","v":3.1250000000000002E-3},0.99687987773020814,{"$f":"(D6+F6)*H6","v":1.2453026965007108},0.99821291883645624,0.99892562760827897,0.99749056995637997,{"$f":"D6*G6","v":2.5157431250000003E-3},{"$f":"D6/F6","v":2.5157431250000003E-3}]
[null,null,1,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/12.578716","v":7.9499370205989225E-2},12.578716,{"$f":"H$5","v":10},2E-3,0.99800199866733308,0.79699370205989228,0.99862152439797525,0.99922016397824853,0.99749056988175577,{"$f":"D7*G7","v":2.5157432E-3},2.5157432E-3]
[null,null,1,{"$f":"1/1700","v":5.8823529411764701E-4},1700.0000000000002,{"$f":"1/4.276763","v":0.23382170113237513},4.2767629999999999,10,5.8823529411764705E-3,0.99413491422311084,2.3440993642649275,0.99773130869050897,0.99845840083113602,0.997490570139282,{"$f":"D8*G8","v":2.5157429411764702E-3},2.5157429411764702E-3]
[null,null,null,{"$f":"SUM(D5:D8)","v":1.5769257703081233E-3},{"$f":"1/D9","v":634.14525834314009},{"$f":"SUM(F5:F8)","v":0.62682304573261693},{"$f":"AVERAGE(G5:G8)","v":7.5472294999999994},null,null,{"$f":"PRODUCT(J5:J8)","v":0.98435442605404755},null,{"$f":"PRODUCT(L5:L8)","v":0.99245376537675234},{"$f":"PRODUCT(M5:M8)","v":0.99522730952169847},0.98999999999184596]
[null,null,null,"Sys λ","Sys MTBF","Σ(1/MTTRi)","Avg. MTTRi",null,null,"Sys R",null,"Sys Inherent A"]
[null,null,null,null,null,"Not used with Sys A"]
[]
[]
[null,"THEN"]
[null,"System MTBF is the reciprocal of the sum of element failure rates (λ); Ebeling (5.5) p. 85",null,{"$f":"SUM(D5:D8)","v":1.5769257703081233E-3},{"$f":"1/D15","v":634.14525834314009}]
[]
[null,"System MTTR via limiting availability and system MTBF; Ebeling (11.29) p. 266",null,null,null,null,{"$f":"((1-N9)/N9)*E15","v":6.4055076653075549},"¬ Exact Sys MTTR"]
[]
[null,"System MTTR via Σ(λi*MTTRi)/Σλi, a weighted average; Ebeling, (9.23) p. 202",null,null,null,null,{"$f":"(SUM(O5:O8))/D15","v":6.3813863588160835},"¬ Approx. Sys MTTR"]
[]
[null,"System μ is not the sum of element repair rates (μi).  System μ is the reciprocal of System MTTR.",null,null,null,{"$f":"1/G17","v":0.156115651131921}]
[]
[null,"Approx. system limiting availability via 1-Σ(λi/μi); Lewis (10.92) p. 315 ",null,null,null,null,null,null,null,null,null,null,null,{"$f":"1-SUM(P5:P8)","v":0.98993702740049017}]
[]
[null,"Same as above except finds approx. limiting availability via 1-Σ(Sys MTTR/Sys MTBF)",null,null,null,null,null,null,null,null,null,null,null,{"$f":"1-(G19/E15)","v":0.98993702740049017}]
[]
[null,"Limiting availability via exact system MTBF and exact system MTTR; Ebeling (11.5) p. 255",null,null,null,null,null,null,null,null,null,null,null,{"$f":"E15/(E15+G17)","v":0.98999999999184585}]
[]
[null,"Limiting availability via exact system λ and exact system μ; Lewis (10.88) p. 315",null,null,null,null,null,null,null,null,null,null,null,{"$f":"F21/(D15+F21)","v":0.98999999999184596}]
[]
[null,"System Reliability for serial elements uses sys MTBF = Σ(1/λi); Ebeling (5.5) p. 85",null,null,null,null,null,null,null,{"$f":"EXP(-D15*H5)","v":0.98435442605404744}]

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[null,"EXPLORE the four combinations of \"R\" and \"M\" (in the form of MTBF and MTTR respectively)"]
[]
[null,"IF",null,null,null,null,null,null,"THEN"]
[null,"Combination","Number of Identical Serial Elements (N)"," Element Failure Rate (λi)","Element MTBF (MTBFi)","Element Repair Rate (μi)","Element MTTR (MTTRi)","Mission Time (t)","Nλt","Reliability From 0 to t (R)","(λ+μ)t","Point Availability, at t (AP)","Average Availability, from 0 to t (AAvg)","Limiting Availability, as t→∞ (AL)","PERFORMANCE (effectiveness metric) in terms of uptime or mission readiness"]
[null,"High MTBF, Low MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D5","v":10000},{"$f":"1/50","v":0.02},{"$f":"1/F5","v":50},10,{"$f":"C5*D5*H5","v":0.05},{"$f":"(EXP(-D5*H5))^C5","v":0.95122942450071402},{"$f":"(D5+F5)*H5","v":0.20100000000000001},{"$f":"((F5/(F5+D5))+(D5/(F5+D5))*EXP(-(D5+F5)*H5))^C5","v":0.95569548769998847},{"$f":"((F5/(F5+D5))+(D5/(((F5+D5)^2)*H5))*(1-EXP(-(D5+F5)*H5)))^C5","v":0.97686054469678874},{"$f":"(F5/(F5+D5))^C5","v":0.77928606825164748},"Best"]
[null,"High MTBF, High MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D6","v":10000},{"$f":"1/100","v":0.01},{"$f":"1/F6","v":100},10,{"$f":"C6*D6*H6","v":0.05},0.95122942450071402,{"$f":"(D6+F6)*H6","v":0.10099999999999999},0.95353367586406623,0.97610510242807402,0.60803882468895165,"Good"]
[null,"Low MTBF, Low MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/50","v":0.02},50,10,0.1,0.90483741803595963,0.20199999999999999,0.91335636429604672,0.95426046655986729,0.60803882468895165,"Same as \"Good\" but unsafe"]
[null,"Low MTBF, High MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/100","v":0.01},100,10,0.1,0.90483741803595963,0.10200000000000001,0.90922784377410726,0.95278513208643445,0.37152788212696103,"Poor"]
[]
[null,"IF",null,null,null,null,null,null,"THEN"]
[null,"Combination","Number of Identical Serial Elements (N)"," Element Failure Rate (λi)","Element MTBF (MTBFi)","Element Repair Rate (μi)","Element MTTR (MTTRi)","Mission Time (t)","Nλt","Reliability From 0 to t (R)","(λ+μ)t","Point Availability, at t (AP)","Average Availability, from 0 to t (AAvg)","Limiting Availability, as t→∞ (AL)","PERFORMANCE (effectiveness metric) in terms of uptime or mission readiness"]
[null,"High MTBF, Low MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D12","v":10000},{"$f":"1/50","v":0.02},{"$f":"1/F12","v":50},100,{"$f":"C12*D12*H12","v":0.5},{"$f":"(EXP(-D12*H12))^C12","v":0.6065306597126342},{"$f":"(D12+F12)*H12","v":2.0099999999999998},{"$f":"((F12/(F12+D12))+(D12/(F12+D12))*EXP(-(D12+F12)*H12))^C12","v":0.80582303311169623},{"$f":"((F12/(F12+D12))+(D12/(((F12+D12)^2)*H12))*(1-EXP(-(D12+F12)*H12)))^C12","v":0.86781208615031424},{"$f":"(F12/(F12+D12))^C12","v":0.77928606825164748},"Best"]
[null,"High MTBF, High MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D13","v":10000},{"$f":"1/100","v":0.01},{"$f":"1/F13","v":100},100,{"$f":"C13*D13*H13","v":0.5},0.6065306597126342,{"$f":"(D13+F13)*H13","v":1.01},0.72925057010330363,0.83213552769179411,0.60803882468895165,"Good"]
[null,"Low MTBF, Low MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/50","v":0.02},50,100,1,0.36787944117144111,2.02,0.64970766401314228,0.75330363336317796,0.60803882468895165,"Same as \"Good\" but unsafe"]
[null,"Low MTBF, High MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/100","v":0.01},100,100,1,0.36787944117144111,1.02,0.53214967050457529,0.69269828395112099,0.37152788212696103,"Poor"]
[]
[null,"IF",null,null,null,null,null,null,"THEN"]
[null,"Combination","Number of Identical Serial Elements (N)"," Element Failure Rate (λi)","Element MTBF (MTBFi)","Element Repair Rate (μi)","Element MTTR (MTTRi)","Mission Time (t)","Nλt","Reliability From 0 to t (R)","(λ+μ)t","Point Availability, at t (AP)","Average Availability, from 0 to t (AAvg)","Limiting Availability, as t→∞ (AL)","PERFORMANCE (effectiveness metric) in terms of uptime or mission readiness"]
[null,"High MTBF, Low MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D19","v":10000},{"$f":"1/50","v":0.02},{"$f":"1/F19","v":50},1000,{"$f":"C19*D19*H19","v":5},{"$f":"(EXP(-D19*H19))^C19","v":6.7379469990854444E-3},{"$f":"(D19+F19)*H19","v":20.100000000000001},{"$f":"((F19/(F19+D19))+(D19/(F19+D19))*EXP(-(D19+F19)*H19))^C19","v":0.77928606861499261},{"$f":"((F19/(F19+D19))+(D19/(((F19+D19)^2)*H19))*(1-EXP(-(D19+F19)*H19)))^C19","v":0.78903798871720687},{"$f":"(F19/(F19+D19))^C19","v":0.77928606825164748},"Best"]
[null,"High MTBF, High MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D20","v":10000},{"$f":"1/100","v":0.01},{"$f":"1/F20","v":100},1000,{"$f":"C20*D20*H20","v":5},6.7379469990854444E-3,{"$f":"(D20+F20)*H20","v":10.1},0.60805131379688637,0.63888033262482169,0.60803882468895165,"Good"]
[null,"Low MTBF, Low MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/50","v":0.02},50,1000,10,4.5399929762484671E-5,20.2,0.6080388252019937,0.62327328773307433,0.60803882468895165,"Same as \"Good\" but unsafe"]
[null,"Low MTBF, High MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/100","v":0.01},100,1000,10,4.5399929762484671E-5,10.200000000000001,0.37154169218826616,0.40975667234065533,0.37152788212696103,"Poor"]
[]
[null,"IF",null,null,null,null,null,null,"THEN"]
[null,"Combination","Number of Identical Serial Elements (N)"," Element Failure Rate (λi)","Element MTBF (MTBFi)","Element Repair Rate (μi)","Element MTTR (MTTRi)","Mission Time (t)","Nλt","Reliability From 0 to t (R)","(λ+μ)t","Point Availability, at t (AP)","Average Availability, from 0 to t (AAvg)","Limiting Availability, as t→∞ (AL)","PERFORMANCE (effectiveness metric) in terms of uptime or mission readiness"]
[null,"High MTBF, Low MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D26","v":10000},{"$f":"1/50","v":0.02},{"$f":"1/F26","v":50},10000,{"$f":"C26*D26*H26","v":50},{"$f":"(EXP(-D26*H26))^C26","v":1.9287498479639202E-22},{"$f":"(D26+F26)*H26","v":201},{"$f":"((F26/(F26+D26))+(D26/(F26+D26))*EXP(-(D26+F26)*H26))^C26","v":0.77928606825164748},{"$f":"((F26/(F26+D26))+(D26/(((F26+D26)^2)*H26))*(1-EXP(-(D26+F26)*H26)))^C26","v":0.78025592048468639},{"$f":"(F26/(F26+D26))^C26","v":0.77928606825164748},"Best"]
[null,"High MTBF, High MTTR",50,{"$f":"1/10000","v":1E-4},{"$f":"1/D27","v":10000},{"$f":"1/100","v":0.01},{"$f":"1/F27","v":100},10000,{"$f":"C27*D27*H27","v":50},1.9287498479639202E-22,{"$f":"(D27+F27)*H27","v":101},0.60803882468895165,0.61105623117222563,0.60803882468895165,"Good"]
[null,"Low MTBF, Low MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/50","v":0.02},50,10000,100,3.7200759760208441E-44,202,0.60803882468895165,0.60954569815924386,0.60803882468895165,"Same as \"Good\" but unsafe"]
[null,"Low MTBF, High MTTR",50,{"$f":"1/5000","v":2.0000000000000001E-4},5000,{"$f":"1/100","v":0.01},100,10000,100,3.7200759760208441E-44,102.00000000000001,0.37152788212696103,0.37518786531520287,0.37152788212696103,"Poor"]

{"sheet":"Tool, Calculate R & A","index":3,"n_rows":35,"n_cols":15,"rows_kept":35,"cut":false,"hidden":false,"merged":[[20,1,20,9],[8,1,8,12],[9,1,9,12],[10,1,10,12],[11,1,11,12],[12,1,12,12],[15,1,15,13],[16,1,16,13],[17,1,17,13],[19,1,19,10],[18,1,18,9],[7,1,7,10],[0,0,0,13],[2,1,2,2],[3,1,3,13],[4,1,4,13],[6,1,6,10]]}
["CALCULATE and COMPARE Reliability and Inherent Availability (based on constant failure and repair rates)"]
[]
[null,"Objective:"]
[null,"1.  Use element failure rate, element repair rate, and mission time to calculate and compare the reliability for a new item and the three types of inherent availability for"]
[null,"     one or two or more identical items (elements) in a series configuration.  Use column B to specify the number of serial items."]
[]
[null,"Nomenclature, Assumptions, and Observations:"]
[null,"Input Data (yellow-shaded headers or cell): N, λ, μ, t, and r where …"]
[null,"1.  N = number of identical elements in a series configuration."]
[null,"2.  λ = 1/MTBF and μ = 1/MTTR where MTBF = mean time between failure and MTTR = mean time to repair.  Mean is the same as average."]
[null,"3.  λ and μ are constant over time since the exponential distribution is used to model the failure distribution and the repair distribution."]
[null,"4.  λ, μ, and t must have the same units of time (e.g., hours).  Mission time = t.  "]
[null,"5.  For design purposes, view MTBF as a lower-bound parameter and MTTR as an upper-bound parameter. "]
[null,"6.  r is the number repairs (or spares needed) during the mission (from 0 to t).  When r = 0, then P = R.  When r > 0, then P > R since spares increase mission success."]
[null,"Output Data: R, A, and P where …"]
[null,"7.  R = Reliability, the probability an item will perform its intended function with no failure during the time interval from 0 to t.  R is 0.3679 or less when Nλt is 1.0 or more."]
[null,"8.  A = Inherent Availability, the probability of mission readiness at point t, the average from 0 to t, and as t becomes large (limiting or steady-state)."]
[null,"9.  As mission time (t) increases, first Point Availability and then Average Availability approximate Limiting Availability."]
[null,"10. P = the probability of r or less repairs (spares needed) from 0 to t via the cumulative Poisson distribution.",null,null,null,null,null,null,null,null," Let r = ",2]
[null,"11. The Poisson distribution does not account for repair time since it assumes failures are immediately repaired or replaced."]
[]
[]
["                                                Scenario","Number of Identical Serial Elements (N)"," Element Failure Rate (λi)","Element MTBF (MTBFi)","Element Repair Rate (μi)","Element MTTR (MTTRi)","Mission Time (t)","Nλt","Reliability From 0 to t (R)","(λ+μ)t","Point Availability, at t (AP)","Average Availability, from 0 to t (AAvg)","Limiting Availability, as t→∞ (AL)","Probability of r or Less Repair Events from 0 to t (P)"]
[]
["Check: Ebeling (ex 11.2) p. 258",1,{"$f":"1/200","v":5.0000000000000001E-3},{"$f":"1/C25","v":200},{"$f":"1/10","v":0.1},{"$f":"1/E25","v":10},10,{"$f":"B25*C25*G25","v":0.05},{"$f":"(EXP(-C25*G25))^B25","v":0.95122942450071402},{"$f":"(C25+E25)*G25","v":1.05},{"$f":"((E25/(E25+C25))+(C25/(E25+C25))*EXP(-(C25+E25)*G25))^B25","v":0.96904465471957879},{"$f":"((E25/(E25+C25))+(C25/(((E25+C25)^2)*G25))*(1-EXP(-(C25+E25)*G25)))^B25","v":0.98186223360040104},{"$f":"(E25/(E25+C25))^B25","v":0.95238095238095233},{"$f":"POISSON($L$19,H25,TRUE)","v":0.99997993250637562},"← Baseline Calculation Check (ref: 1st worksheet, row 25)"]
["Scenario 1",1,{"$f":"1/2000","v":5.0000000000000001E-4},{"$f":"1/C26","v":2000},{"$f":"1/10","v":0.1},{"$f":"1/E26","v":10},85,{"$f":"B26*C26*G26","v":4.2500000000000003E-2},{"$f":"(EXP(-C26*G26))^B26","v":0.95839046552094698},{"$f":"(C26+E26)*G26","v":8.5425000000000004},{"$f":"((E26/(E26+C26))+(C26/(E26+C26))*EXP(-(C26+E26)*G26))^B26","v":0.99502584578181541},{"$f":"((E26/(E26+C26))+(C26/(((E26+C26)^2)*G26))*(1-EXP(-(C26+E26)*G26)))^B26","v":0.9956071588198051},{"$f":"(E26/(E26+C26))^B26","v":0.99502487562189057},{"$f":"POISSON($L$19,H26,TRUE)","v":0.99998760669476083}]
["Scenario 2",1,{"$f":"1/2000","v":5.0000000000000001E-4},2000,0.1,10,86,4.3000000000000003E-2,0.95791139006703063,8.6430000000000007,0.99502575302008323,0.9956003988175306,0.99502487562189057,0.99998716892002992]
["Scenario 3",1,5.0000000000000001E-4,2000,0.1,10,87,4.3500000000000004E-2,0.95743255409096695,8.7435000000000009,0.9950256691277537,0.99559379320320773,0.99502487562189057,0.99998672106916331]
["Scenario 4",1,5.0000000000000001E-4,2000,0.1,10,88,4.3999999999999997E-2,0.95695395747304668,8.8440000000000012,0.99502559325677975,0.99558733680950029,0.99502487562189057,0.99998626303269467]
["Scenario 5",1,5.0000000000000001E-4,2000,0.1,10,89,4.4499999999999998E-2,0.95647560009362076,8.9444999999999997,0.99502552464020055,0.99558102469224663,0.99502487562189057,0.99998579470132953]
["Scenario 6",1,5.0000000000000001E-4,2000,0.1,10,90,4.4999999999999998E-2,0.95599748183309996,9.0449999999999999,0.99502546258438773,0.99557485211891794,0.99502487562189057,0.99998531596594542]
["Scenario 7",1,5.0000000000000001E-4,2000,0.1,10,91,4.5499999999999999E-2,0.95551960257195445,9.1455000000000002,0.99502540646203463,0.99556881455775692,0.99502487562189057,0.9999848267175907]
["Scenario 8",1,5.0000000000000001E-4,2000,0.1,10,92,4.5999999999999999E-2,0.95504196219071469,9.2460000000000004,0.99502535570581385,0.99556290766755207,0.99502487562189057,0.99998432684748528]
["Scenario 9",1,5.0000000000000001E-4,2000,0.1,10,93,4.65E-2,0.9545645605699703,9.3465000000000007,0.99502530980264359,0.99555712728800694,0.99502487562189057,0.99998381624702015]
["Scenario 10",1,5.0000000000000001E-4,2000,0.1,10,94,4.7E-2,0.95408739759037109,9.447000000000001,0.99502526828849969,0.99555146943066586,0.99502487562189057,0.99998329480775716]

{"sheet":"TOOL, Graph R & A","index":4,"n_rows":105,"n_cols":14,"rows_kept":105,"cut":false,"hidden":false,"merged":[[0,0,0,21],[2,1,2,11]]}
["GRAPH Reliability and the three forms of Inherent Availability over time for a single item (element or system)"]
[]
[1,"RMA Input & Output Data"]
["­             Scale Factor for Time Axis"," Failure Rate (λ)","Uptime (MTBF)","Repair Rate (μ)","Downtime (MTTR)","Mission Time (t)","Nλt","Reliability","(λ+μ)t","Point Availability","Average Availability","Limiting Availability"]
[0,{"$f":"1/C5","v":5.0000000000000001E-4},2000,{"$f":"1/E5","v":0.1},10,{"$f":"$A$3*A5","v":0},{"$f":"#REF!*B5*F5","v":{"$error":"#REF!"}},{"$f":"EXP(-B$5*F5)","v":1},{"$f":"(B5+D5)*F5","v":0},{"$f":"(D$5/(D$5+B$5))+(B$5/(D$5+B$5))*EXP(-(B$5+D$5)*F5)","v":1},null,{"$f":"D$5/(D$5+B$5)","v":0.99502487562189057}]
[1,null,null,null,null,{"$f":"$A$3*A6","v":1},{"$f":"#REF!*B6*F6","v":{"$error":"#REF!"}},{"$f":"EXP(-B$5*F6)","v":0.99950012497916929},{"$f":"(B6+D6)*F6","v":0},{"$f":"(D$5/(D$5+B$5))+(B$5/(D$5+B$5))*EXP(-(B$5+D$5)*F6)","v":0.99952430404185455},{"$f":"(D$5/(D$5+B$5))+(B$5/(((D$5+B$5)^2)*F6))*(1-EXP(-(B$5+D$5)*F6))","v":0.99975816873776535},{"$f":"D$5/(D$5+B$5)","v":0.99502487562189057}]
[2,null,null,null,null,2,{"$f":"#REF!*B7*F7","v":{"$error":"#REF!"}},0.99900049983337502,0,0.99909409169927299,0.99953188209316957,0.99502487562189057]
[3,null,null,null,null,3,{"$f":"#REF!*B8*F8","v":{"$error":"#REF!"}},0.99850112443771089,0,0.99870501406146595,0.99932001969663065,0.99502487562189057]
[4,null,null,null,null,4,{"$f":"#REF!*B9*F9","v":{"$error":"#REF!"}},0.99800199866733308,0,0.99835313803826042,0.99912154716850643,0.99502487562189057]
[5,null,null,null,null,5,{"$f":"#REF!*B10*F10","v":{"$error":"#REF!"}},0.99750312239746008,0,0.99803490660145844,0.99893550925082908,0.99502487562189057]
[6,null,null,null,null,6,{"$f":"#REF!*B11*F11","v":{"$error":"#REF!"}},0.997004495503373,0,0.99774710282771295,0.99876102350296359,0.99502487562189057]
[7,null,null,null,null,7,{"$f":"#REF!*B12*F12","v":{"$error":"#REF!"}},0.99650611786041488,0,0.99748681737944134,0.99859727451394265,0.99502487562189057]
[8,null,null,null,null,8,{"$f":"#REF!*B13*F13","v":{"$error":"#REF!"}},0.99600798934399148,0,0.99725141909504689,0.99844350858825026,0.99502487562189057]
[9,null,null,null,null,9,{"$f":"#REF!*B14*F14","v":{"$error":"#REF!"}},0.99551010982957056,0,0.99703852839115292,0.99829902886550259,0.99502487562189057]
[10,null,null,null,null,10,{"$f":"#REF!*B15*F15","v":{"$error":"#REF!"}},0.99501247919268232,0,0.99684599320798017,0.99816319083783067,0.99502487562189057]
[11,null,null,null,null,11,{"$f":"#REF!*B16*F16","v":{"$error":"#REF!"}},0.99451509730891907,0,0.99667186725470314,0.99803539823183807,0.99502487562189057]
[12,null,null,null,null,12,{"$f":"#REF!*B17*F17","v":{"$error":"#REF!"}},0.99401796405393528,0,0.99651439033487466,0.99791509922481381,0.99502487562189057]
[13,null,null,null,null,13,{"$f":"#REF!*B18*F18","v":{"$error":"#REF!"}},0.99352107930344769,0,0.99637197055303317,0.99780178296744504,0.99502487562189057]
[14,null,null,null,null,14,{"$f":"#REF!*B19*F19","v":{"$error":"#REF!"}},0.99302444293323511,0,0.9962431682226226,0.99769497638761728,0.99502487562189057]
[15,null,null,null,null,15,{"$f":"#REF!*B20*F20","v":{"$error":"#REF!"}},0.99252805481913842,0,0.99612668131255522,0.99759424125203633,0.99502487562189057]
[16,null,null,null,null,16,{"$f":"#REF!*B21*F21","v":{"$error":"#REF!"}},0.99203191483706066,0,0.99602133228529943,0.99749917146436606,0.99502487562189057]
[17,null,null,null,null,17,{"$f":"#REF!*B22*F22","v":{"$error":"#REF!"}},0.99153602286296671,0,0.99592605619344199,0.99740939058036759,0.99502487562189057]
[18,null,null,null,null,18,{"$f":"#REF!*B23*F23","v":{"$error":"#REF!"}},0.99104037877288365,0,0.99583988991439731,0.99732454952216842,0.99502487562189057]
[19,null,null,null,null,19,{"$f":"#REF!*B24*F24","v":{"$error":"#REF!"}},0.99054498244290046,0,0.99576196241443726,0.99724432447528821,0.99502487562189057]
[20,null,null,null,null,20,{"$f":"#REF!*B25*F25","v":{"$error":"#REF!"}},0.99004983374916811,0,0.99569148594362589,0.99716841495342001,0.99502487562189057]
[21,null,null,null,null,21,{"$f":"#REF!*B26*F26","v":{"$error":"#REF!"}},0.98955493256789928,0,0.99562774807264875,0.99709654201722409,0.99502487562189057]
[22,null,null,null,null,22,{"$f":"#REF!*B27*F27","v":{"$error":"#REF!"}},0.98906027877536873,0,0.99557010449104,0.99702844663453638,0.99502487562189057]
[23,null,null,null,null,23,{"$f":"#REF!*B28*F28","v":{"$error":"#REF!"}},0.98856587224791304,0,0.99551797249400698,0.99696388817044912,0.99502487562189057]
[24,null,null,null,null,24,{"$f":"#REF!*B29*F29","v":{"$error":"#REF!"}},0.98807171286193052,0,0.99547082509201001,0.99690264299667919,0.99502487562189057]
[25,null,null,null,null,25,{"$f":"#REF!*B30*F30","v":{"$error":"#REF!"}},0.98757780049388144,0,0.99542818568355496,0.99684450321052542,0.99502487562189057]
[26,null,null,null,null,26,{"$f":"#REF!*B31*F31","v":{"$error":"#REF!"}},0.98708413502028758,0,0.99538962323734603,0.99678927545451745,0.99502487562189057]
[27,null,null,null,null,27,{"$f":"#REF!*B32*F32","v":{"$error":"#REF!"}},0.98659071631773265,0,0.9953547479350966,0.99673677982859898,0.99502487562189057]
[28,null,null,null,null,28,{"$f":"#REF!*B33*F33","v":{"$error":"#REF!"}},0.98609754426286189,0,0.99532320723095125,0.99668684888736636,0.99502487562189057]
[29,null,null,null,null,29,{"$f":"#REF!*B34*F34","v":{"$error":"#REF!"}},0.98560461873238236,0,0.99529468228768725,0.99663932671549593,0.99502487562189057]
[30,null,null,null,null,30,{"$f":"#REF!*B35*F35","v":{"$error":"#REF!"}},0.98511193960306265,0,0.99526888475366682,0.99659406807506912,0.99502487562189057]
[31,null,null,null,null,31,{"$f":"#REF!*B36*F36","v":{"$error":"#REF!"}},0.98461950675173293,0,0.99524555384796176,0.99655093761901403,0.99502487562189057]
[32,null,null,null,null,32,{"$f":"#REF!*B37*F37","v":{"$error":"#REF!"}},0.98412732005528514,0,0.995224453724183,0.99650980916536602,0.99502487562189057]
[33,null,null,null,null,33,{"$f":"#REF!*B38*F38","v":{"$error":"#REF!"}},0.98363537939067236,0,0.99520537108636742,0.99647056502747855,0.99502487562189057]
[34,null,null,null,null,34,{"$f":"#REF!*B39*F39","v":{"$error":"#REF!"}},0.98314368463490964,0,0.99518811303282151,0.996433095395721,0.99502487562189057]
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{"sheet":"TOOL, Allocate Availability","index":5,"n_rows":28,"n_cols":12,"rows_kept":28,"cut":false,"hidden":false,"merged":[[15,0,15,1],[0,0,0,7],[2,0,2,2],[2,3,2,5],[2,6,2,7],[5,7,6,7],[14,0,14,1]]}
["TOOL:  Allocate Availability and Add Margin as a Function of Confidence"]
[null,null,"See Note 3"]
["IF (from management)",null,null,"AND (analyst inputs and interim calculations)",null,null,"THEN"]
["Number of Serial Elements (n)","Availability Requirement        (A Req)","Mgt Confidence in obtaining                 \"A req\"                       (50% ≤ p ≤ 100%)","z score and error for Perfect A","Standard Deviation for Adj Avail Req","Margin (M)","Availability Requirement with Margin","Element Availability with Margin"]
[37,0.96,0.8,5.2,{"$f":"(1-D6-B5)/(D5-C6)","v":6.9876138587679146E-3},{"$f":"G5-B5","v":3.6644079344069214E-3},{"$f":"IF(C5<0.55, (1+B5)/2, 1-(D5*E5))","v":0.96366440793440689},{"$f":"G5^(1/A5)","v":0.99900017125594442}]
[null,{"$f":"B5^(1/A5)","v":0.99889731125933123},{"$f":"_xlfn.NORM.S.INV(C5-0.5)","v":-0.52440051270804078},{"$f":"1-_xlfn.NORM.S.DIST(D5,TRUE)","v":9.9644263173992442E-8},null,null,"See \"A Req\"","See \"A Req allocated no Margin\""]
[null,"Allocated, no Margin","z score for p","Right Tail Error, ε "]
["Minimum Mission Time (t)","Lower Bound System Reliability (RL)",null,"Lower Bound Element Reliability (RLi)",null,null,"Lower Bound System Life (MTBFL)","Lower Bound Element Life (MTBFLi)"]
[24,0.97629999999999995,null,{"$f":"B9^(1/A5)","v":0.99935195702330015},null,"See Note 1",{"$f":"-A9/LN(B9)","v":1000.610257582549},{"$f":"-A9/LN(D9)","v":37022.579530555362}]
[]
[]
[null,null,null,null,null,null,"Upper Bound System Repair Time (MTTRU)","Upper Bound Element Repair Time (MTTRUi)"]
[null,null,null,null,null,"See Note 1",{"$f":"G9*(1-G5)/G5","v":37.728659310038864},{"$f":"H9*(1-H5)/H5","v":37.053286134271573}]
[]
["Input data uses entries from above",null,null,null,null,"See Note 2",{"$f":"G9/(G9+G13)","v":0.963664407934407},{"$f":"(H9/(H9+H13))^A5","v":0.963664407934409}]
["IF ",null,"THEN"]
["Avail Req","Mgt Conf, p ","A Req w. M"]
[{"$f":"B5","v":0.96},0.5,{"$f":"IF(B18<0.55, (1+$B$5)/2, 1-($D$5*(1-$B$5)/($D$5-_xlfn.NORM.S.INV(B18-0.5))))","v":0.98}]
[null,0.55000000000000004,{"$f":"IF(B19<0.55, (1+$B$5)/2, 1-($D$5*(1-$B$5)/($D$5-_xlfn.NORM.S.INV(B19-0.5))))","v":0.96961220628867728}]
[null,0.6,0.96790891842846527,null,null,null,null,null,null,null,null," "]
[null,0.65,0.96664760336406264]
[null,0.7,0.96557215489707349]
[null,0.75,0.96459266951771305]
[null,0.8,0.9636643174183489]
[null,0.85,0.96275952270760501]
[null,0.9,0.9618582870178215]
[null,0.95,0.96094381778089799]
[null,1,0.96]