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Fundamentals of Queueing Theory

Verify Little's law relating the system size and waiting time for a queue:
Wolfram Language code: 𝒬 = QueueingProcess[λ, μ];
Wolfram Language code: QueueProperties[𝒬 , "QueueDiagram"]
Wolfram Language code: L = QueueProperties[𝒬, "MeanSystemSize"]
Wolfram Language code: W = QueueProperties[𝒬, "MeanSystemTime"]
Wolfram Language code: L == λ W
Verify Burke's theorem for a feedforward network with three queues in series:
Wolfram Language code: 𝒩 = QueueingNetworkProcess[(⁠| | | -- | | 3. | | 0 | | 0 |⁠), (⁠| | | | | - | - | - | | 0 | 1 | 0 | | 0 | 0 | 1 | | 0 | 0 | 0 |⁠), (⁠| | | --- | | 4 | | 7 | | 8.2 |⁠), (⁠| | | - | | 1 | | 1 | | 1 |⁠)];
Compute a probability for the steady state of the network:
Wolfram Language code: Probability[x == 2 && y == 1 && z == 3, {x, y, z}𝒩[∞]]
Verify the result using Burke's theorem:
Wolfram Language code: Probability[x == 2 && y == 1 && z == 3, {x, y, z}ProductDistribution[QueueingProcess[3, 4][∞], QueueingProcess[3, 7][∞], QueueingProcess[3, 8.2][∞]]]
Verify the definition of the Erlang B loss probability for an M/M/c/c queue:
Wolfram Language code: Probability[n == c, Distributed[n, StationaryDistribution[QueueingProcess[λ, μ, c, c]]], Assumptions -> Element[c, Integers] && c > 0]
Use the built-in ErlangB function to compute the same result:
Wolfram Language code: ErlangB[c, λ / μ]
Confirm that the two answers are indeed the same:
Wolfram Language code: FullSimplify[% - %%]