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Math Help - Emergency - A quantitative criterion for judging the degree of spectral fragmentation

  1. #1
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    Emergency - A quantitative criterion for judging the degree of spectral fragmentation

    Greeting,

    My name is Pooria, an Optical Telecommunications student. I have been working on Flexible Bandwidth Optical Networks since 2009. However, in my current project, I got a question that I thought maybe you can help me to solve. Well, I need to introduce a quantitative criterion for judging the degree of spectral fragmentation in the network links. I will explain the problem in a simple way, hence, it is highly appreciated if you could guide me.


    Assuming, we have a row of twelve chairs. Five people will randomly sit on them. We take two possible cases:

    • They sit on the five first chairs respectively. We named this case “The Compact case”.


    • In the second case, which we named “The Fragmented case”, people sit on chairs in the following order:


    • Man 1 – Chair 1, Man 2 – Chair 4, Man 3 – Chair 6, Man 4 – Chair 8, Man 5 – Chair 10.


    Now, I would like to introduce a factor that shows How far is the fragmented case form the compact case. More difficult, what will be this factor if people need random number of chairs as well. For example:

    • Man 1 needs 2 chairs, Man 2 needs 1 chair, Man 3 needs 1 chair, Man 4 needs 1 chair, Man 5 needs 3 chairs.



    I will be grateful if you could guide me through this problem. Thank you very much.

    Looking forward to hear you,
    Cheers,
    Pooria
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  2. #2
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    Re: Emergency - A quantitative criterion for judging the degree of spectral fragmenta

    For the "The Fragmented Case" you could perhaps quantify it by taking the squared deviations from their expect placements (so that the sum is always positive). Or you could take the signed difference in case direction matters.

    Not sure how to approach the random number of chairs other than combining it with the above.
    Thanks from Pooria
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  3. #3
    mfb
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    Re: Emergency - A quantitative criterion for judging the degree of spectral fragmenta

    Now, I would like to introduce a factor that shows How far is the fragmented case form the compact case.
    You want a single number for the "distance" between the cases? Fine, take "1".
    Do you look for a general way to define a "distance" for arbitrary seat combinations? Even that is easy: Take 0 for the same combinations, else 1. It is possible to show that this is a well-defined metric on the set of seat options.
    Do you look for a general way to define a "distance" for arbitrary seat combinations in a meaningful way? In this case, please define "meaningful way".

    Is the seat setup you gave as "fragmented case" the only fragmented case, or is it an example of a fragmented case? In the latter case, I agree with ANDS! that the squared deviations from the compact case might be an interesting value to look at.
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  4. #4
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    Re: Emergency - A quantitative criterion for judging the degree of spectral fragmenta

    Not sure how to approach the random number of chairs other than combining it with the above.

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