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Markov Chains and Queues
Analyze the Performance of a Queueing Network
Define an open queueing network.
In[1]:=
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\[Gamma] = {3, 6}; \[Mu] = {7, 17}; r = {{1/6, 1/9}, {1/4, 1/8}}; c = {1, 1};
In[2]:=
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\[ScriptCapitalN] = QueueingNetworkProcess[\[Gamma], r, \[Mu], c];
Simulate the network.
In[3]:=
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data = RandomFunction[\[ScriptCapitalN], {0, 10}];
Plot the simulated values at the nodes in the network.
In[4]:=
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ListLinePlot[data, Filling > Axis]
Out[4]=
Performance measures at the nodes in the network.
In[5]:=
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Table[QueueProperties[{\[ScriptCapitalN], i}], {i, 2}] // N
Out[5]=
Stationary distribution for the network.
In[6]:=
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\[ScriptCapitalD] = StationaryDistribution[\[ScriptCapitalN]];
Probability density function for the steady state of the network.
In[7]:=
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PDF[\[ScriptCapitalD], {m, n}]
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In[8]:=
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DiscretePlot3D[ PDF[\[ScriptCapitalD], {m, n}] // Evaluate, {m, 0, 4}, {n, 0, 4}, ExtentSize > 0.5, PlotRange > All]
Out[8]=
Cumulative distribution function for the steady state of the network.
In[9]:=
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CDF[\[ScriptCapitalD], {m, n}]
Out[9]=
In[10]:=
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DiscretePlot3D[ CDF[\[ScriptCapitalD], {m, n}] // Evaluate, {m, 0, 35}, {n, 0, 35}, ExtentSize > Right, PlotRange > {0, 1.05}]
Out[10]=