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Markov Chains and Queues
Fundamentals of Queueing Theory
Verify Little's law
relating the system size and waiting time for a queue.
In[1]:=
X
\[ScriptCapitalQ] = QueueingProcess[\[Lambda], \[Mu]];
In[2]:=
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QueueProperties[\[ScriptCapitalQ] , "QueueDiagram"]
Out[2]=
In[3]:=
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L = QueueProperties[\[ScriptCapitalQ], "MeanSystemSize"]
Out[3]=
In[4]:=
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W = QueueProperties[\[ScriptCapitalQ], "MeanSystemTime"]
Out[4]=
In[5]:=
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L == \[Lambda] W
Out[5]=
Verify Burke's theorem for a feedforward network with three queues in series.
In[6]:=
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\[ScriptCapitalN] = QueueingNetworkProcess[\!\(\* TagBox[ RowBox[{"(", "", TagBox[GridBox[{ {"3."}, {"0"}, {"0"} }, GridBoxAlignment>{ "Columns" > {{Center}}, "ColumnsIndexed" > {}, "Rows" > {{Baseline}}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}, GridBoxSpacings>{"Columns" > { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" > {}, "Rows" > { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}], Column], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"0", "1", "0"}, {"0", "0", "1"}, {"0", "0", "0"} }, GridBoxAlignment>{ "Columns" > {{Center}}, "ColumnsIndexed" > {}, "Rows" > {{Baseline}}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}, GridBoxSpacings>{"Columns" > { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" > {}, "Rows" > { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", TagBox[GridBox[{ {"4"}, {"7"}, {"8.2"} }, GridBoxAlignment>{ "Columns" > {{Center}}, "ColumnsIndexed" > {}, "Rows" > {{Baseline}}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}, GridBoxSpacings>{"Columns" > { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" > {}, "Rows" > { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}], Column], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", TagBox[GridBox[{ {"1"}, {"1"}, {"1"} }, GridBoxAlignment>{ "Columns" > {{Center}}, "ColumnsIndexed" > {}, "Rows" > {{Baseline}}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}, GridBoxSpacings>{"Columns" > { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" > {}, "Rows" > { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" > {}, "Items" > {}, "ItemsIndexed" > {}}], Column], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\)];
Compute a probability for the steady state of the network.
In[7]:=
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Probability[ x == 2 && y == 1 && z == 3, {x, y, z} \[Distributed] \[ScriptCapitalN][\[Infinity]]]
Out[7]=
Verify the result using Burke's theorem.
In[8]:=
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Probability[ x == 2 && y == 1 && z == 3, {x, y, z} \[Distributed] ProductDistribution[QueueingProcess[3, 4][\[Infinity]], QueueingProcess[3, 7][\[Infinity]], QueueingProcess[3, 8.2][\[Infinity]]]]
Out[8]=
Verify the definition of the Erlang B loss probability for an M/M/c/c queue.
In[9]:=
X
Probability[n == c, Distributed[n, StationaryDistribution[QueueingProcess[\[Lambda], \[Mu], c, c]]], Assumptions > Element[c, Integers] && c > 0]
Out[9]=
Use the builtin
ErlangB
function to compute the same result.
In[10]:=
X
ErlangB[c, \[Lambda]/\[Mu]]
Out[10]=
Confirm that the two answers are indeed the same.
In[11]:=
X
Probability[n == c, Distributed[n, StationaryDistribution[QueueingProcess[\[Lambda], \[Mu], c, c]]], Assumptions > Element[c, Integers] && c > 0]; ErlangB[c, \[Lambda]/\[Mu]]; FullSimplify[%  %%]
Out[11]=