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| | comment4, | | comment4, |
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| − | = Providing a guarantee through limiting =
| + | comment5, |
| − | The idea of getting a guarantee is simple:
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| − | | |
| − | {{out|if any group <math>g_i</math> requires a <math>G_i</math> units of resource from <math>R</math> units available then limiting all the rest groups with <math>R - G_i</math> units provides a desired guarantee}}
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| − | | |
| − | For <math>N</math> groups in the system this implies solving a linear equation set to get limits <math>L_i</math> like this:
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| − | | |
| − | <center><math>
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| − | \begin{cases}
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| − | L_2 + L_3 + \ldots + L_N = R - G_1 \\
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| − | L_1 + L_3 + \ldots + L_N = R - G_2 \\
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| − | \ldots \\
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| − | L_1 + L_2 + \ldots + L_{N-1} = R - G_N \\
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| − | \end{cases}
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| − | </math></center>
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| − | | |
| − | In a matrix form this looks like
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| − | | |
| − | <center><math>
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| − | A L = G\;
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| − | </math></center>
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| − | | |
| − | where
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| − | | |
| − | <center><math>
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| − | A = \begin{bmatrix}
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| − | 0 & 1 & 1 & \cdots & 1 & 1 \\
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| − | 1 & 0 & 1 & \cdots & 1 & 1 \\
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| − | & & \cdots \\
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| − | 1 & 1 & 1 & \cdots & 1 & 0 \\
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| − | \end{bmatrix}
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| − | , | |
| − | L = \begin{bmatrix} L_1 \\ L_2 \\ \vdots \\ L_N \end{bmatrix}
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| − | ,
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| − | G = \begin{bmatrix} R - G_1 \\ R - G_2 \\ \vdots \\ R - G_N \end{bmatrix}
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| − | </math></center>
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| − | | |
| − | | |
| − | and thus the solution is
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| − | | |
| − | <center><math>
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| − | L = A^{-1}G\;
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| − | </math></center>
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| − | | |
| − | | |
| − | Skipping boring calculations, the reverse matrix <math>A^{-1}\;</math> is
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| − | | |
| − | <center><math>
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| − | A^{-1} = \frac {1} {N - 1} \left( A - (N - 2) E \right)
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| − | </math></center>
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| − | | |
| − | This solutions looks huge, but the <math>L</math> vector is calculated in <math>O(N)</math> time:
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| − |
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| − | <pre>
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| − | void calculate_limits(int N, int *g, int *l)
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| − | {
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| − | int sum;
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| − | int i;
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| − | | |
| − | if (N == 1) {
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| − | l[0] = R;
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| − | return;
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| − | }
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| − | | |
| − | sum = 0;
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| − | for (i = 0; i < N; i++)
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| − | sum += R - g[i];
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| − | for (i = 0; i < N; i++)
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| − | l[i] = (sum - (R - g[i]) - (N - 2) * (R - g[i]))/(N - 1);
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| − | }
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| − | </pre>
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| − | | |
| − | == Disadvantages of this approach ==
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| − | | |
| − | This approach has only one disadvantage: O(n) time needed to start a new container.
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