Practice Midterm2 - IEOR 3106 Solutions to Midterm Exam I...

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IEOR 3106 Solutions to Midterm Exam I 1. (40 points) A small business sells professional digital cameras one by one. On any given day, either a camera is delivered to them from the factory (causing the inventory level to go up by 1) with probability p = 0 . 20, or they sell a camera (causing the inventory level to go down by 1), with probability 1 p = 0 . 80. (a) (10 points) Suppose that there are initially 3 cameras in inventory. What is the probability that the inventory level drops to 0 before reaching 5? SOLUTION: Gambler’s ruin problem with N = 5, i = 3, p = 0 . 20; q/p = 4. We want 1 P 3 , where P 3 = 1 ( q/p ) 3 1 ( q/p ) 5 = 63 / 1023 = 0 . 062 (b) (10 points) Suppose that there are initially 3 cameras in inventory. What is the probability that the inventory level reaches 4 before going down to 1? SOLUTION: Gambler’s ruin problem ( a = 1 , b = 2) with N = 3, i = 2, p = 0 . 20. We want P 2 = 1 ( q/p ) 2 1 ( q/p ) 3 = 15 / 63 = 0 . 24 For the next three questions, suppose further that if the inventory level ever reaches the high value of 5 , then they sell 4 of them at once (same day) to a wholesaler at a reduced rate. If the inventory level ever drops to 0 , then 2 cameras are immediately delivered (same day). (c) (10 points) Find the long-run proportion of days that the inventory level hits 0. SOLUTION: Model as a MC on S = { 1 , 2 , 3 , 4 } : X n = inventory level at the end of the n th day. We then will solve for π = πP yielding our answer as (1 p ) π 1 , since the level hits 0 if and only if it does so by reaching level 1 and then selling a camera that day. For our chain however, P 1 , 2 = 1 since when X n = 1 either X n +1 = 2 due to 1 camera arriving (prob.= p ), or X n +1 = 2
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