Compute Loss Probabilities for a Queue 

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X Plot[Evaluate@Table[ErlangB[c, a], {c, {1, 15, 30, 45}}], {a, 0.1, 80}, Filling -> Axis, ImageSize -> 300, AxesLabel -> {"a", "prob"}, PlotLegends -> LineLegend[ Table[StringJoin["c = ", ToString[c]], {c, {1, 15, 30, 45}}]]]

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X DynamicModule[{NumberOfCallLines, bht = 21, pr = 0.02}, Grid[{{"", Style["Erlang B Calculator", Bold], ""}, {"Traffic", "Blocking", "Lines"}, {InputField[Dynamic[bht], Number, FieldSize -> 7], InputField[Dynamic[pr], Number, FieldSize -> 7], Style[Dynamic@NumberOfCallLines[bht, pr], {Bold, Blue}]}, {Button[ "Calculate"], SpanFromLeft}}, Alignment -> {Left, Center}, BaseStyle -> {FontFamily -> "Helevetica"}, Spacings -> {3.5, 0.5}, Frame -> True, FrameStyle -> Thick, Background -> Lighter[Yellow, 0.85]], Initialization :> (NumberOfCallLines[a_, p_] /; Positive[a] && Positive[p] && 0 < p < 1 := Block[{n, n1, res}, res = Catch[Do[n1 = n; res = Quiet[ErlangB[n1, a]]; If[res <= p, Throw[n1]], {n, 10^6}]; Throw[$Failed]]; res /; FreeQ[res, $Failed]]; NumberOfCallLines[__] := "Failed"), SaveDefinitions -> True]

The Erlang B formula computes the probability that a customer is lost from an M/M/c/c queue because all the servers are busy. For example, if there are 30 call lines and the traffic is 25 Erlangs, then the blocking or loss probability is approximately 0.05.

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X ErlangB[30, 25.]

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Study the variation of the loss probability for different numbers of lines.

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Calculate the number of lines required, given the traffic and blocking probability.

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