ESE 503 - University of Pennsylvania _ ESE503 - Simulation Modeling & Analysis (Final Exam) Spring Semester, 2021

ESE503 - Simulation Modeling & Analysis (Final Exam) Spring Semester, 2021 M. Carchidi Problem #1 (25 points) - Chapter 8 Suppose that X is a mixed random variable having pdf/pmf fx  Ae3x, when −  x  0 A, ...

ESE503 - Simulation Modeling & Analysis (Final Exam) Spring Semester, 2021 M. Carchidi Problem #1 (25 points) - Chapter 8 Suppose that X is a mixed random variable having pdf/pmf fx  Ae3x, when −  x  0 A, when x  0 Ae−6x, when 0  x   . a.) (5 points) Determine the value of A. b.) (15 points) Determine how a single sample of X can be constructed from a single random number R  U0,1. b.) (5 points) Determine the numerical values of EX and VX. Hint: You may use the fact that  − 0 xneaxdx  −1nn! an1 and 0 xne−axdx  ann!1 for n  0,1,2,... and a  0. Problem #2 (15 points) - A Non-Stationary Poisson Process Consider a non-stationary Poisson process having time-dependent rate function t  12t, when 0 hours  t  2 hours 24, when 2 hours  t  5 hours 64 − 8t, when 5 hours  t  8 hours in units of customers per hour, over an 8-hour day from 9:00 AM - 5:00 PM. Compute the probability that more that 111 customers arrive into the store between the hours of 10:00 AM and 3:00 PM. Problem #3 (20 points) - Chapter 8 a.) (12 points) Consider a continuous random variable X having pdf fx  2 Γ1/4 e− 14 x4 for −  x  . Develop an acceptance-rejection algorithm for sampling X using the symmetric two-sided exponential distribution with pdf gy  1 2 e−|y| for −  y  , as a basis. b.) (8 points) Using the fact that Γ1/4 ≃ 3.6256, compute the probability that your algorithm in part (a) will require fewer than three (3) iterations to produce a sample of X. Problem #4 (20 points) - Chapter 6 Suppose that customers enter a store so that the interarrival time between customers is exponential with a mean of 0.8 minutes per customer. In the store, there are c identical servers all working in parallel and all with a common service-time distribution that is geometric, ps  1 − ps−1p for s  1,2,3,..., in minutes, with a mean service time of 3 minutes per customer. a.) (5 points) Determine the smallest value of c so that steady-state conditions will exist for this queueing system. b.) (15 points) Using your value of c determined in part (a), compute the steady-state values of: , LQ, wQ, w and L, each worth 3 points. ——————————————————————————————————————— Problem #5 (20 points) - Chapter 9 Consider a random variable X with pdf fx,  1 2 3x−1lnx2 for 0  x  1 and 1  . If an n-point sample of X Sn  X1,X2,X3,… ,Xn, is taken, a.) (10 points) determine an expression for the maximum-likelihood estimator of  (ML) in terms of 〈lnX  1 n ∑ k1 n lnXk. b.) (5 points) Given also that  xnlnx2dx  xn1 n  12lnx n2 − 21n3  1lnx  2 for n  0, and lim x→0 xn lnx  0 and lim x→0 xnlnx2  0 for n  0, determine an expression for the sample-mean estimator of  (SM) in terms of 〈X  1 n ∑ k1 n Xk. c.) (5 points) Given the 5-point data sample S5  0.2,0.8,0.3,0.4,0.1, compute the numerical values of ML and SM.