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ErlangB[c,a]

computes the Erlang B loss probability for an M/M/c/c queue.

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ErlangB

ErlangB[c,a]

computes the Erlang B loss probability for an M/M/c/c queue.

Details

Examples

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Basic Examples  (2)

Compute a loss probability using ErlangB:

Wolfram Language code: λ = 12;μ = 15;c = 3;
Wolfram Language code: a = λ / μ;
Wolfram Language code: ErlangB[c, a]

Obtain the same result using Probability:

Wolfram Language code: Probability[n == c, nStationaryDistribution[QueueingProcess[λ, μ, c, c]]]

Plot for a different number of servers c:

Wolfram Language code: Plot[Evaluate@Table[ErlangB[c, a], {c, 1, 80, 15}], {a, 0.1, 80}]

Scope  (4)

Use exact values for the parameters:

Wolfram Language code: c = 4;a = 12 / 7;
Wolfram Language code: ErlangB[c, a]

Machine precision:

Wolfram Language code: c = 4;a = 12.3 / 7;
Wolfram Language code: ErlangB[c, a]

Higher precision:

Wolfram Language code: c = 4;a = 12.3`20 / 7;
Wolfram Language code: ErlangB[c, a]

Symbolic parameters:

Wolfram Language code: ErlangB[c, a]

Applications  (2)

A company has five 1 Mbps lines to carry video conferences between two company sites. Suppose that each video conference requires 1 Mbps and lasts for an average of one hour. Assume that requests for video conferences arrive according to a Poisson process with a rate of three calls per hour. Find the probability that a call request is blocked due to lack of lines:

Wolfram Language code: ErlangB[5, 3.]

A modem pool consists of four modems and the offered traffic intensity is 2 Erlangs. Find the probability that a connection fails due to blocking:

Wolfram Language code: ErlangB[4, 2.]

Blocking probability with six modems:

Wolfram Language code: ErlangB[6, 2.]

Properties & Relations  (3)

ErlangB gives the loss probability for an M/M/c/c queue:

Wolfram Language code: Probability[n == c, nStationaryDistribution[QueueingProcess[λ, μ, c, c]], Assumptions -> c∈ℤ && c > 0]
Wolfram Language code: ErlangB[c, λ / μ]
Wolfram Language code: FullSimplify[%% - %, λ / μ > 0 && c > 0 && c∈ℤ]

ErlangB satisfies a nonlinear difference equation:

Wolfram Language code: x[c] /.  RSolve[{x[c] == (x[c - 1]/(c/a) + x[c - 1]), x[0] == 1}, x, c][[1]]
Wolfram Language code: ErlangB[c, a]//Refine[#, a > 0 && c > 0 && c∈ℤ]&

ErlangB is related to ErlangC:

Wolfram Language code: (ErlangC[c, a] == (ErlangB[c, a]/1 - (a (1 - ErlangB[c, a])/c)))//FullSimplify[#, 0 < a < c && c∈ℤ]&
Wolfram Research (2012), ErlangB, Wolfram Language function, https://reference.wolfram.com/language/ref/ErlangB.html.

Text

Wolfram Research (2012), ErlangB, Wolfram Language function, https://reference.wolfram.com/language/ref/ErlangB.html.

CMS

Wolfram Language. 2012. "ErlangB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ErlangB.html.

APA

Wolfram Language. (2012). ErlangB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ErlangB.html

BibTeX

@misc{reference.wolfram_2026_erlangb, author="Wolfram Research", title="{ErlangB}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ErlangB.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_erlangb, organization={Wolfram Research}, title={ErlangB}, year={2012}, url={https://reference.wolfram.com/language/ref/ErlangB.html}, note=[Accessed: 13-June-2026]}