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will need experience in markov chains. please try to solve as soon as possible. 4 Bobs. in a. Straw Imagine that Wan Fung is drinking

will need experience in markov chains. please try to solve as soon as possible.

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4 Bobs. in a. Straw Imagine that Wan Fung is drinking milk tea and he has a very short straw: it has enough room to t two boba (see Figure 1). Figure 1: A straw with one boba currently inside. The straw only has enough room to t two boba. Here is a formal description of the drinking process: We model the straw as having two \"com- ponents\" (the top component and the bottom component). At any given time, a component can contain nothing, or one boba. As Wan Fung drinks from the straw, the following happens every second: 1. The contents of the top component enter Wan Fung's mouth. 2. The contents of the bottom component move to the top component. 3. With probability p, a new boba enters the bottom component; otherwise the bottom compo- nent is now empty. Help Wan Fung evaluate the consequences of his incessant drinking! (a) At the very start, the sh'aw starts off completely empty. What is the expected number of seconds that elapse before the straw is completely lled with boba for the rst time? [Write down the equations; you do not have to solve them.] (b) Consider a slight variant of the previous part: now the straw is narrower at the bottom than at the top. This a'ects the drinking speed: if either (i) a new boba is about to enter the bottom component or (ii) a boba from the bottom component is about to move to the top component, then the action takes two seconds. If both (i) and (ii) are about to happen, then the action takes three seconds. Otherwise, the action takes one second. Under these conditions, answer the previous part again. [Write down the equations; you do not have to solve them.] (c) Wan Fung was annoyed by the straw so he bought a fresh new straw (the straw is no longer narrow at the bottom). What is the long-run average rate of Wan Fung's calorie consumption? (Each boba is roughly 10 calories.) ((1) What is the long-run average number of boba which can be found inside the straw? [Maybe you should rst think about the long-run distribution of the number of boba.] CS 70, Spring 2017,HW 13 2

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