Load Balancing for Future Internet: An Approach Based on Game Theory

Algorithm 2

INPUT:   Load Managers’ load vector/arriving rate: 𝑅¹,𝑅²,𝑅³,…,𝑅^𝑛;

   Processors’ processed rate: 𝑃¹₁⋯𝑃^𝑛₁𝑃⋮⋱⋮_1𝑚⋯𝑃^𝑛_𝑚;

OUTPUT:   Processors’ load vector/arriving rate: 𝑆¹₁⋯𝑆^𝑛₁𝑆⋮⋱⋮_1𝑚⋯𝑆^𝑛_𝑚

Initialization:

 𝛽←0; 𝑞←1;

   For 𝑗=1,2,3,…,𝑛 do

 If 𝑅^𝑗>0

For 𝑘=1,2,3,…,𝑚 do // Sort the processors in increasing order of the equation below

such that 𝑡^𝑗₁<𝑡^𝑗₂<𝑡^𝑗₃<⋯<𝑡^𝑗_𝑚

       𝑡^𝑗_𝑘←1𝑃^𝑗_𝑘;

RIGHT ←0; // Find the minimum 𝑓 to satisfy the inequality

    𝑅^𝑗≤∑^𝑓_𝑘=1(𝑃^𝑗_𝑘−1𝑡^𝑗_𝑓)

While (𝑓≤𝑚) do

For 𝑘=1,2,…,𝑓 do

         RIGHT←RIGHT+𝑃^𝑗_𝑘−1𝑡^𝑗_𝑓;

If (RIGHT ≥ 𝑅^𝑗)

     BREAK;

Else

       𝑓←𝑓+1

      𝑎←0; 𝑏←1; SUM←0; // Calculate 𝛼 using a binary search; sort 𝜎^𝑖 in

      increase order

While (1) do

      𝛽←𝑎+𝑏2;

     For 𝑘=1,2,3,…,𝑞 do

        SUM←SUM+𝑃^𝑗_𝑘−1𝛽;

     If SUM−𝑅^𝑗<𝜀 BREAK

If (SUM>𝑅^𝑗)  𝑏←𝛽;

     Else    𝑎←𝛽;

    For 𝑖=1,2,3,…,𝑚 do

If (𝑖≤𝑞)

     𝑆^𝑗_𝑘=𝑃^𝑗_𝑘−1𝛽;