1. A photo detector counts photons for 10 nano seconds to decide if a distant light source is emitting light. When the light source is not emitting light, some photons are still counted by the detector due to the ambient background radiation. The number of photons counted in 10 nanoseconds is modeled as a random variable X whose parameter has value ln (9) if the light source is not emitting light, and value in (27) if the source is emitting light. The maximum-likelihood detector decides that detection of light is the true if and only if the likelihood ratio = probability of detection when light is present (x)/probability of detection when light is not present (x) exceeds 1. (a) What value(s) of X result in a decision in favour of detecting of light? (b) Compute the false alarm probability PFA and the missed detection or false dismissal probability PMD of the maximum-likelihood decision rule. (Sent for Answer) 2. 100 AIDS patients were given a new drug test. The results were as follows: Year on drug Number of deaths Number of withdrawals 1 5 2 2 8 4 3 12 2 4 18 10 5 24 12 Withdrawals occurred when patients left the test area or died from causes not related to the AIDS disease. Construct a life table to estimate the probability (reliability) that a patient will survive at least five years. 3. Determine the minimum-cost trade-off between the MTBF and the MTTR for a system requiring 0.99 availability. The MTTR must be between 10 and 30 hr, and the MTBF must be at least 1000 hr. Consider the cost functions as: a. C(x) = 0.0002x2; C(y) = 12500/y b. C(x) = 2x; C(y) = 2000-y2 4. The Breakit Company manufactures anti-clock disk brakes (ABS) for use in automobiles. An essential part of the manufacturing process is the factory assemble and integration linking (FAIL) system. Unfortunately, this system has experienced numerous failures over the years under minimal repair concept that is a non-homogeneous Poisson process having an intensity function (t)=1.1510-7t1.3 with t measured in operating hours. The system averages 2000 operating hours a year and it currently 5 years old. When the system fails, the repair time is lognormal with tmed = 6.2 hours and s = 1.2. a) When the system was new, its manufacturer offered a 500-operating-hr warranty. What is the system reliability during the warranty period? b) When the system fails, it takes a crew of two to repair it. If the labor rate is $50 per hour and based upon the MTTR, what is the maintenance cost of the FAIL system during its 6th year of operation? c) If the cost of the system is $30,000, what is its optimal (minimum cost) replacement time? d) Management believes that a preventive maintenance (PM) program may be cost effective. A PM consists primarily of replacing certain parts having a cost of $700 and thus restoring it to as-good-as-new condition. If it takes one technician 4 5. hours to do this, what is the optimum (minimum cost) PM interval? External corrosion is a potential failure mechanism of buried natural gas pipeline. Once the protective coating and/or the cathodic protection system fail the external corrosion rate depends largely on the characteristics of the surrounding soil. The corrosion rate is typically modeled using an empirical power-law relationship as: n = Where y is the corrosion depth, t is the exposure time, and A and n are site dependent constants, with n typically less than 1. Below is the data available from a specific site. Estimate the failure probability of 40 mm pipeline in 20 years. Pipeline failure may be defined as loss of 66% of the pipeline thickness. Time (years) Corrosion depth (mm) 0 0 1.5 2.5 3.5 5.0 5.0 8.5 8.5 11.0 12.0 13.5 15 14.0 17 14.5 19 15.0 6. An offshore platform consists of three large main power generators (MPGs), two operating and one spare (standby mode). The lifetime of each MPG follows the Weibull distribution with beta = 2.3 and theta = 40 years. In a given time, two MPGs are used, while third one can be used to replace a failed unit in 12 days with a cost of $ 5 million. The failure of each additional unit requires the procurement and installation of a brand new transformer with a cost of $ 280 million. Estimate failure probability and associated risk with the MPGs over the next 20 years of operation considering both scenarios (spare is available as good as new and spare is in failed state). 7. The load (S) and resistance (R) are modeled as log normally distributed random variables with the following parameters: the mean and coefficient of variation (CoV) of the load are 160 kN and 0.15, respectively, while the CoV of resistance is given as 0.28. a) If the design criterion is R = S, then what should be the mean resistance? Assume the nominal resistance (R) is the 5th percentile of the distribution; b) what is the probability of failure for design criterion R = S?; and c) what is reliability index? 8. A repairable system consists of three non-identical components A, B and C, all of which must work for system success. When one component fails, no further component failures can occur. Construct the relevant state space diagram and hence evaluate general expressions for the individual limiting state probabilities in terms of component failure and repair rates. Evaluate the unavailability of the system in hour/year if the three components have the following reliability data. Component Mean time to Failure (yr) MTTR (hours) A 1.0 180 B 2.0 12 C 1.0 52 9. The figure below shows the pipe layout in a lawn sprinkler system. Sprayers are located at every intersection of the pipe and also at the corners of the system. Thus there are 16 sprayers. Two things can happen to a sprayer: it can break off or it can get clogged. If any one of the sprayers breaks off, a great stream of water will pour forth and the system will have failed. If one of the sprayers clogs, it will fail to water the lawn in the immediate vicinity; however, adjacent sprayers can reach the affected areas. The system is judged to be failed if any two adjacent sprayers are both failed. For one sprayer, let the probability of breakage be 0.01 and the probability of clogging be 0.05. Find the reliability of the system (the probability that it does not fail in either mode). Layout of the sprinkler system 10. A gas-fired furnace is shown below. The hot combustion gases pass through heat exchanger to heat fresh air for space heating. The gas flow is controlled by an electric solenoid valve connected to a thermoset. The gas is ignited by a pilot light flame. A high temperature switch shuts off all gas in the event of high temperature in the fresh air. Develop fault tree using step-by-step approach for an event (scenario) \"No Space Heating\