IEOR E4106: Stochastic Models (Spring 2024)
Practice Final Exam
Problem 1 (40 points) A researcher becomes interested in a new topic at time 0. Since then, papers on this topic are produced according to an nonhomogeneous Poisson process with intensity function λ(t) = 12/(t + 2)2, t ≥ 0. Denote by N(t) the number of papers produced before or at time t.
1. (20 points) What is the cumulative distribution function of the time at which the first paper is produced?
2. (20 points) Let n be a positive integer. What is the conditional distribution of N(1) given that N(2) = n?
Problem 2 (30 points) A computer has 2 parts and runs continuously as long as both of them are working. Parts 1 and 2 have independent lifetimes with distributions Exp(λ1) and Exp(λ2), respectively. When part i fails, it takes a random time Ti and a deterministic cost $Ci to fix that part and get the computer running again. During that time, the other working part is not subject to failure. Suppose that T1 ∼ Exp(1/µ) and T2 is uniformly distributed between µ and 3µ. Find the long-run average cost per unit of time.
Problem 3 (30 points) A repairman maintains two printers. Each printer works for Exp(1) units of time and then fails. The repairman works on each failed printer until it is repaired, so that a second failed printer must wait for the repairman to become free before it can receive attention. The repair times of printers 1 and 2 have distributions Exp(1) and Exp(2), respectively. All failures and repairs are independent.
1. (15 points) Find the long-run average number of working printers.
2. (15 points) Let T be the random time between two successive repairs that leave both printers working. Compute ET.