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Bucket Brigades is inspired by the technique of the same name used in the days before fire engines (https : //en. wikipedia. org/wiki/Bucket_brigadel). Its goal is to capture the advantages of both meth-ods, without the disadvantages. Bucket Brigades can be implemented as follows: • Order workers from the slowest to the fastest (assuming that this order is maintained across all tasks). • Forward motion: Each worker carries out tasks until they are interrupted by the worker ahead of them. If they are blocked by the worker ahead (for example because the worker ahead is using a resource they need) then they must wait, until the resource becomes available. • Backward motion: When the last worker in the line finishes a job, they walk back to the worker behind them and take the job they are working on from them and continue working where the previous guy left off. Then, the worker who has been relieved of his work walks back and takes the partially completed job from the person before them, and so on until the first worker is relieved. The first worker goes back and begins work on a new job. We call this process a reset. Of course, this technique requires that there be more workstations than workers (such as in order-picking, where each location can be treated as a workstation). It also requires that hand-offs can be done at almost any time and can be done almost instantaneously. We will model this system under the following assumption: • We normalize the total work required for each job to be equal to 1. • We assume work is infinitely divisible and hand-offs are instantaneous (so when a job is completed, the workers instantly pick-up exactly where the worker behind them was). • We assume each worker works at a constant speed vk. • We will observe the system only at discrete times, immediately after each reset. • We define xk(t) as the fraction of work completed on the job that worker k is holding immediately after a reset. 
• Workers are ordered from slowest to fastest. v1 < v2 < < vn. We will use the methods learned in this course to study stochastic systems, in order to study this fully deterministic system. Before you start:

Learn more about bucket brigades from Ihttps : //www2. isye gatech. edu/-33b/1 (Dr. Bartholdi passed away, so the website is no longer updated). The problem below asks you to write a step-by-step proof of Theorem 3 in the paper "A production line that balances itself'. J. Bartholdi and D. Eisenstein, Operations Research 44(1):21-34 (1996), available in the resource section of the website above. 

1. Assuming that each worker did all the tasks for a job, provide a mathematical expression for the total speed of the production line. Note that this is the maximum possible speed for any configuration. 

2. Explain (in your own words) the following: (a) Why the time between reset (t — 1) and reset t is given by 1 — xn(t — 1)  V, 
(b) The following recursion holds (make sure you justify each component of the right hand side). 
(t) = 0 xk(t) = xk_i(t — 1) + vk-i1 — xn(t — 1) 2  vn  (la) k = 2, ... ,n (lb) 

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