Central Server Network 

Ten requests circulate in a central server network with three nodes. Balance the load in the network by adjusting the service rate at node 1, which is the central server.

In[1]:=

X centralserver = QueueingNetworkProcess[\!\(\* TagBox[ RowBox[{"(", "", TagBox[GridBox[{ {"0"}, {"0"}, {"0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], Column], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"0", "0.3", "0.7"}, {"1", "0", "0"}, {"1", "0", "0"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", TagBox[GridBox[{ {"\[Mu]"}, {"0.5"}, {"1.25"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], Column], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", TagBox[GridBox[{ {"1"}, {"1"}, {"1"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, "ItemsIndexed" -> {}}], Column], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), 10];

If , the load is concentrated at the central server.

In[2]:=

X data = RandomFunction[centralserver /. {\[Mu] -> 1}, {0, 50}]; ListLinePlot[data, PlotLegends -> Range[3], InterpolationOrder -> 0, Filling -> Axis, PlotRange -> {0, 10}]

Out[2]=

By increasing to 2, the balance of the load shifts to the remaining servers.

In[3]:=

X data = RandomFunction[centralserver /. {\[Mu] -> 2}, {0, 50}]; ListLinePlot[data, PlotLegends -> Range[3], InterpolationOrder -> 0, Filling -> Axis, PlotRange -> {0, 10}]

Out[3]=