Balanced Sequences and Optimal Routing

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In summary, the speaker is reading a paper on Balanced Sequences and Optimal Routing and has encountered some proofs they do not understand. They are hoping for someone to help clarify the proofs for them. They also mention they are looking for applications of the results presented in the paper and would appreciate any resources or links provided.
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rhj23
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I've been reading the paper on Balanced Sequences and Optimal Routing (Altman, Gaujal, Hordijk; 2000). However, there are a couple of proofs given that I don't quite follow. There are statements made that are assumed to trivially follow, but I can't see how and am hoping someone will be able to help me.

The first is in the proof of Proposition 2.16. The fact that l_i >= (n-1)l_1 + n is easily shown, but 'on the other hand' l_i <= n(l_1) - 3 does not seem to follow from any similar method.

The second is in the proof of Theorem 2.21, Step (2). Where does the fact that |s_1| >= max{4, 2(n+m)+1} come from. (Obviously the 4 is trivial, but I do not understand the 2(n+m)+1)

The paper is attached; I hope that someone with a better understanding than me will be able to follow the proofs and let me know where I'm missing something obvious!

Thanks
 

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A question

Sorry not to provide any answer to your questions? But if you don't mind, I hav a question about that paper on Balanced sequences and optimal routing. I hav been looking for some particle applications of the results presented in that paper, but unfornately I havn't been abled to find a good one, so far. May be it is due to my lack of knowledge about queuing networks.
Therefore, I will be more than please, if you can provide me with an application, or any link or article where I will be able to find some.
Thanks.

ml7
 

FAQ: Balanced Sequences and Optimal Routing

What are balanced sequences and how are they related to optimal routing?

Balanced sequences refer to a set of instructions or steps that are organized in a way that minimizes the overall cost or time required to complete a task. In the context of optimal routing, balanced sequences are essential in determining the most efficient route to reach a destination.

How are balanced sequences and optimal routing used in real-world applications?

Balanced sequences and optimal routing are used in various real-world applications such as transportation planning, logistics, supply chain management, and network routing. These techniques help in minimizing costs, reducing travel time, and improving overall efficiency.

What factors are considered when creating a balanced sequence for optimal routing?

The factors that are considered when creating a balanced sequence for optimal routing include the distance between locations, traffic conditions, mode of transportation, and any constraints such as road closures or weight limits. These factors are taken into account to determine the most cost-effective and time-efficient route.

What are some challenges in creating balanced sequences for optimal routing?

One of the main challenges in creating balanced sequences for optimal routing is obtaining accurate and up-to-date data. This includes information on traffic patterns, road conditions, and any potential obstacles that may affect the route. Additionally, balancing the trade-offs between cost and time can also be a challenge in creating an optimal route.

How can advancements in technology improve the use of balanced sequences and optimal routing?

Advancements in technology, such as the use of real-time data and advanced algorithms, can greatly improve the use of balanced sequences and optimal routing. With access to more accurate and timely information, routes can be constantly adjusted to account for changing conditions, resulting in more efficient and cost-effective routing.

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