# [ORDER SOLUTION] Computing Poisson Probabilities

CHAPTER 3 Constant Failure Rate Model 75Poisson & Gamma (Section 3.5)Redundancy (Section 3.6)Exercises3.1 A component experiences chance (CFR) failures with an MTTF of 1100 hr. Findthe following:(a) The reliability for a 200-hr mission(b) The design life for a 0.90 reliability(c) The median time to failure(d) The reliability for a 200-hr mission if a second, redundant (and independent)component is added3.2 A CFR system with ? = 0.0004 has been operating for 1000 hr. What is the prob-ability that it will fail in the next 100 hr.? The next 1000 hr.?3.3 A gearbox has two independent failure modes: a constant failure rate of 0.0003 anda linearly increasing (wear-out) failure rate given by ? = t/(5 X 10 5 ). Find the reliability ofthe gearbox for 100-hr of operation.3.4 A hydraulic system is comprised of five components having the following constantfailure rates (times are in days): ?= 0.001, ? = 0.005, ? = 0.0007, ? 0.0025, and ? = 0.00001.(a) Find the system MTTF and standard deviation.(b) Find the system design life if a 0.99 reliability is desired.3.5 A complex power system has a nonlinear failure rate that has historically been greaterthan 0.001 failures per day Determine an upper bound on the system`s reliability fora 60-day operating period.3.6 A landing gear system has repetitive stresses placed on it twice a day as a result oflandings. The probability of a failure during landing is 0.0028. Determine the reliability ofthe landing gear system over a 30-day contingency operation. What is the probability of afailure occurring between days 10 and 20 of the operation?3.7 A system contains 20 identical and critical components that will be replaced onfailure (renewal process). As a result, a constant failure rate for the system will beobserved. If a design life of 10 yr. with a reliability of 0.99 is required, what should thesystem MTTF and median time to failure be? If each component has a CFR, whatwill be the component MTTF and median time to failure?3.8 Two identical and CFR computers are placed in a redundant configuration. If thesystem reliability is to be 0.95 at 3000 operating hours, determine the correspondingMTTF (design specifications) for each computer.76 Part Basic Reliability Models3.9 Specifications for a power unit consisting of three independent and serially relatedcomponents (failure modes) require a design life of 5 yr. with a 0.95 reliability(a) Let each component have a constant failure rate such that the first componentsrate is twice that of the second and the third components rate is three times thatof the second. What should be the MTTF of each component and the system?(b) If two identical power units are placed in parallel, what is the system reliabilityat 5 yr., and what is the system MTTF?3.10 The time to failure of fluorescent lights in a large office building is exponentiallydistributed with a failure rate of 0.03125 hr. How many spare tubes must the buildingcustodian maintain to have at least a 0.95 probability of replacing all failures on a givenday? Assume continuous 24-hour use of the lights.3.11 A flashlight contains two batteries each having an MTTF of 5 operating hours (assumeCFR).(a) What is the probability of battery failure occurring within the first 2 hr. ofoperation?(b) If failed batteries are immediately replaced, what is the probability of more thanone failure occurring during the first 5 hr. of operation?(c) Would you expect batteries to have a constant failure rate?3.12 Consider two redundant components having constant but different failure rates.(a) Derive the reliability function and the MTTF(b) Find the reliability at 1000 hr. and the MTTF of a two-component redundantsystem where ? 1 = 0.000356 per hour and ? 2 = 0.00156 per hour.3.13 An electronic circuit board with ?(t) = 0.00021 per hour is replaced on failure. Whatis the probability that the third failure will occur by 10000 hours?3.14 ln reliability testing it is of interest to know how long the test must run in order togenerate a specified number of failures. A new condenser fan motor is believed tohave a constant failure rate of 3.4 failures per 100 operating hours. A single test stand is tobe used in which a motor is operated until failure and then replaced with a newmotor from production. What is the expected test time if 10 failures are desired?3.15 Consider two identical and redundant CFR components having a guaranteed life of2 months and a failure rate of 0. 15 failures per year. What is the system reliability for10,000 hr. of continuous operation?3.16Derive a general expression for R(t) and the MTTF for the two-component systemdescribed in Exercise 3.15.3.17 A microwave link in a communications network has a high failure rate. Althoughseveral pieces of equipment have been used in the link; the link always seems to fail atabout the same rate regardless of the age of the equipment and its prior maintenancehistory. ln general, microwave transmissions are subject to fading. Selective fadingoccurs when atmospheric conditions bend a transmission to the extent that signals reachthe receiver in slightly different paths. The merging paths can cause interference and createdata errors. Other channels in the microwave transmission are not affected by selectivefading. Selective fading occurs when there is an electrical storm.Flat fading occurs during fog and when the surrounding ground is very moist. It ismore serious since it may last several hours and affect surrounding channels. If during thecurrent season, electrical storms occur about once every week and fog alerts -are issued at the rate of one every two months, what is the reliability of the link overa 24-hour period? What assumptions, if any are necessary?3.18 A 60-watt outdoor lightbulb is advertised as having an average life (i.e., MTTF) of1000 (operating) hours. However, experience has shown that it will also fail on de-mand an average of once every 120 cycles. A particular bulb is turned on once eachevening for an average of 10 hr. If it is desired to have a reliability of 90 percent, whatis its design life in days?3.19 A pump used in a water filtration system operates continuously It has experiencedfrequent failures as a result of abnormally heavy loads occurring at random times. ltsCDF has been found to beF(twhere t is measured in days. If it has been operating without failure for the last90 days, what is the probability it will fail within the next 90 days?3.20 Repetitive loading. A packaging machine (cartoner) in a food processing facilitywill jam with a constant probability of 0.005 per application (per carton). Twelvecans of coffee are combined into a single ca.se for shipment to buyers. The productionate is 30 cans of coffee every minute. What is the probability (reliability) of no jamsduring a 1-hr production run?3.21 For the reliability function R(l) = cio/20m)], use Eqs. (3.13) and (3.14) and comparethe upper and lower bounds with the actual reliabilities at 100, 200, 500, 800, 1000,2000, 5000, and 10,000 hr. This failure distribution has an IFR with an MTTF of1772.46.3.22 Sources at NASA have suggested that an asteroid of sufficient size to cause destructionequivalent to the power of thousands of hydrogen bombs could smash into theEarth within the next 300,000 years. If 300,000 years is the mean time between suchcatastrophic natural failure events (known as the return period) and assuming anexponential distribution, determine the probability that someone living to the age S0will not encounter such an event.3.23 Airbags used in automobiles for safety deploy as a result of vehicle accidents. Astateside distributor of these airbags receives an average (mean) of four demandsper month for replacement airbags from automotive service centers. The distributormust order replenishment bags from overseas, which takes 2 months. How many air-bags should the distributor have on hand when placing a replenishment order so theprobability of a stock-out is no more than S percent? Assume that the time betweensuch accidents is exponential.78 Part Basic Reliability Models.24 A brake component has been in use for one year. During that time, 125 incidentshave been recorded over an estimated 4,120,000 miles. It is assumed that the time tofailure distribution is exponential.(a) What is the estimated reliability of the component over a 12,000-mile warranty?(b) Determine the design life if a 0.90 reliability is required.(c) Find the MTTF and the median time to failure.(d) If one of these components is installed on each wheel brake subsystem, what isthe reliability of the automobile (system) over a 12,000-mile warranty? Assumea failure of any component results in a system failure (i.e., there are 4 failuremodes).(e) The Rely Able trucking company operates two trucks in which this componenthas been installed on each wheel subsystem. Over the next year, they expect toaccumulate 12,360 miles on each truck. They have two spare components to useas a replacement in the event of a failure. What is the 1-year reliability of thetrucks with respect to this critical component if a system failure occurs when thethird failure occurs?3.25 A critical part used on a manufacturing machine has an exponential failuredistribution with a mean of 1,000 (operating) days, When the part fails it is immediately re-placed with a spare. All spares must be purchased now since the manufacturer of thispart will be terminating its production. The life of the machine is 3,650 (operating)days. How many spares must be purchased in order to have a 99 percent reliability ofno stock out resulting in machine failure?3.26An electronic device consists of the following components all having CFR:ailure RateComponent type Count (in 10âÂ° operating hours)connectors 9 0.214transformers 1 0.121rectifiers 1 0.057inverters 2 0.084diodes 12 0.013resistors 1S 0.008capacitors 24 0.051coils 2 0.002relays 3 0.066Determine the device`s MTTE median time to failure, standard deviation, and reliabilityover 1 year of continuous operation. Which component type is the most likelyto fail?3.27 A system has a hazard rate of .015 failures per day If2 more identical redundantsystems are added, resulting in 3 parallel systems, what is the reliability function? What isthe reliability over 100 days? What is the MTTF? What is the reliability over 100days without the redundancy?3.28 A parking lot has 10 floodlights consisting of 100-watt metal halide bulbs sittingtop 20-foot poles. The failure distribution of the bulbs is best modeled with thetwo-parameter exponential distribution having a guaranteed lifetime of 1000 hoursand a failure rate of .002 failures per day. The floodlights operate for 10 hours perday Because of the cost to send a crew to replace bulbs, a scheduled replacementinterval must be established.a) Determine a replacement interval in days so that the average number of failedbulbs equal 2 (20 percent).(b) The parking lot is considered unsafe if half or more of the floodlights are inoperable.Given the replacement interval in (a), what is the probability of more than4 bulbs having failed before replacement?3.29 A mean of`0.074 demands per day is observed for components that can be repaired.Repair takes 30 days. How many spare components are needed to meet demandsgenerated during a repair cycle with a probability of O.98?SUPPLEMENTARY EXERCISES3.30 A more general exponential reliability model may be defined byR(t) = um where u > 1 /2 > 0and a and b are parameters to be determined. Find the hazard rate function, andshow how this model is equivalent to R(t) = il'.3.31 Show for the exponential distribution that the residual mean life is 1/}. regardless ofthe length of time the system has been operating..32 Derive the CDF for the Erlang distribution (Eq. 3.22) from its PDF, f(YQ),where kis assumed to be an integer:Consult a table of integrals.3.33 Consider the general case in problem 3.27 where there are 3 identical and redundantCFR components. Derive the reliability function, hazard rate function, and theMTTR3.34 For the 3-component redundant system in 3.33, plot the probability density functionin the interval (0 S IS 10) for}. = 0.3. Compare the mode and median values to the mean.3.35 Computing Poisson probabilities. Numerical issues can arise when computingPoisson probabilities particularly when both the mean (lu) and the number of failures(n) are large. Derive a recursive relationship for computing R, = & , n = O,nl3.36 For the 2-component redundant system (Eq. 3.24), show that the residual MTTF(Eq. 2.18) approaches 1/I as Tu 6 Â¤>Â¤.3.37 Show for the 2-component redundant system (Eq. 3.24) that g : g and theâ1Â¤(0.5) 2*mode = T