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Solving a system of Inequalities

  1. Nov 17, 2012 #1
    Hello, I'm having some trouble with a Queuing Networks question, not the networks but solving a system of inqualities based on the network.

    1. The problem statement, all variables and given/known data
    I have to find the value of α that gives maximum γ, and then use the value. The system is defined by
    [tex]5\gamma< 1[/tex]
    [tex]20\gamma \alpha<1[/tex]
    [tex](60/0.9) \gamma (1-\alpha)<1[/tex]

    Now [itex]\alpha[/itex] is a probability and lies in the region [itex] 0<\alpha<1[/itex]
    While [itex]\gamma[/itex] is a rate and is non-zero.

    2. Relevant equations

    3. The attempt at a solution
    Now I've got so far as to put the system in this form and to solve through to find that in the region
    [tex] 0< \alpha ≤ 10/13, that 0 < \gamma < -3/(200(\alpha-1)) [/tex]
    while in the region
    [tex] 10/13 < \alpha <1, that 0 < \gamma < 1/(20 \alpha) [/tex]

    Thus the maximum value of [itex]\gamma[/itex] lies in the region [itex] \gamma < 13/200[/itex] when [itex]\alpha = 10/13 [/itex].

    This much is fine, but I need to use an actual value of [itex]\gamma[/itex] in the next part of the question and I can't think how to get a [itex]\gamma = [/itex] expression. Any help would be gratefully appreciated.
     
  2. jcsd
  3. Nov 17, 2012 #2

    haruspex

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    γ = 13/200 satisfies all the constraints;
    for α < 10/13, γ < 3/(200(1-α)) < 3/(200(1-10/13)) = 13/200, right?
    and for α > 10/13 etc.
     
  4. Nov 17, 2012 #3
    Yes but isn't that only if [tex] \gamma ≤ 13/200 [/tex] whereas in this case [tex] \gamma < 13/200 [/tex] so surely [itex] \gamma [/itex] must be infinitesimally less than this value?
     
  5. Nov 17, 2012 #4

    haruspex

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    Ok, now I understand your question properly.
    If you take the constraints as strict, there is no maximum value that satisfies them. No matter what value you pick, you can always get a slightly higher one.
     
  6. Nov 17, 2012 #5

    Ray Vickson

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    In general, optimization problems subject to strict inequality constraints do not have solutions: you may have an "inf" but not a minimum, or a "sup" but not a maximum. For example, what is the solution of the simple problem
    minimize x, subject to x > 0?
    Answer: the problem is ill-posed, and does not have a solution!

    Often, when students are first introduced to such problems they are sloppy and write ">" when they should write "≥" or they write "<" when they should write "≤".

    RGV
     
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