ESE503 - Simulation Modeling & Analysis (Homework #2) -University of Pennsylvania

ESE503 - Simulation Modeling & Analysis (Homework #2) Spring Semester, 2018 M. Carchidi –––––––––––––––––––––––––––––––––––– Instructions: 1.) Do the following six (6) Problems and Simulation Problem #1. 2.) Please indicate in the table below whether the problem is submitted electronically (via Canvas) or in hard copy (or both). 3.) Indicate any optional comments that might assist us in interpreting your submission. 4.) For electronic submissions please create a separate file for each problem. Also, use the following naming format for all electronic submissions: Example : Your Name HW2 P. 5.) Submit this cover sheet in order with you assignment. 6.) Homework is due on 02/15/18. No late submissions allowed. –––––––––––––––––––––––––––––––––––– Your Name _____________________ Problem Electronic Paper Number Submission Submission Both Optional Comments 1 2 3 4 5 6 SP#1 X Must be Done –––––––––––––––––––––––––––––––––––– –––––––––––––––––––––––––––––––––––– Problem #1 (20 points) - Computing Probabilities a.) (13 points) Suppose that the coordinates of a point (X, Y ) are such that X ∼ U[0, 1) and Y ∼ U[0, 1). Compute the probability that the distance from this point to the origin is less than or equal to L. You should consider the two cases when 0 ≤ L ≤ 1 and when 1 ≤ L ≤ √ 2. and you should check your results using a Monte-Carlo simulation of 3000 trials for various values of L. b.) (7 points) Compute the average distance a point will be from the origin and you should check your results using a Monte-Carlo simulation of 3000 trials. –––––––––––––––––––––––––––––––––––– Problem #2 (15 points) - Computing Probabilities Consider the quadratic equation Ax2 + Bx + C = 0 where A, B and C are independent standard uniform random variables so that A ∼ U[0, 1) , B ∼ U[0, 1) and C ∼ U[0, 1). Compute the probability that the solution to the above equation yields real results for x and you should check your results using a Monte-Carlo simulation of 3000 trials. –––––––––––––––––––––––––––––––––––– 2 –––––––––––––––––––––––––––––––––––– Problem #3 (20 points) - A Reliability Problem A certain component of a machine has a lifetime that is exponentially distributed with a mean of μ = 10 hours. When this component fails, the machine is stopped and the component must be replaced. Suppose that the replacement time is uniformly distributed between a = 25 and b = 35 minutes. In addition, company policy is to replace the component after T = 15 hours of operation even if it has not failed. a.) (5 points) Compute (in terms of μ and T) the average time (in hours) that a component is used. b.) (5 points) Compute (in terms of μ, a, b, N and T) the time for N components to be replaced. c.) (5 points) Compute (in terms of μ, a, b and T) the average percent of the time the machine is idle. d.) (5 points) Compute (in terms of a, b, μ, n and T) the number of components that must be replaced during the first n days of operation. You should check your analytical results in this problem using the simulation you constructed for Problem #3 of Homework #1. –––––––––––––––––––––––––––––––––––– 3 –––––––––––––––––––––––––––––––––––– Problem #4 (15 points) - Checking Light Bulbs a.) (5 points) Suppose that X is a discrete random variable with range space RX and pmf p(x), and suppose that Y is a continuous random variable with cdf F(y). Show that Pr(Y ≤ X) = X x∈RX F(x)p(x) where the sum is over all points in RX. b.) (5 points) Suppose the lifetime of a light bulb is exponentially distributed with a mean lifetime of 1000 hours. The light bulb is turned on and it will be checked once and only once to see if it is still on. The light bulb will be checked sometime at or after one hour in one-hour increments, and suppose this “checking time” (in hours) is geometrically distributed with a mean checking time of 1500 hours. Use the result from part (a) to compute the probability that the light bulb will not be on when it is checked. You may use the fact that X∞ x=1 rx = r 1 − r for |r| 1. c.) (5 points) Run a Monte-Carlo Simulation using 5000 trials to check your result. You may use the fact that an exponential distribution with mean μ can be sampled from R ∼ U[0, 1) using X = −μ ln(R) and the geometric distribution with success probability p can be sampled from R ∼ U[0, 1) using Y = » ln(1 − R) ln(1 − p) ¼ .